CN1934512A - Numerical control method and device - Google Patents

Abstract

A design method for industrial products using a clothoid curve, wherein (A). the route of the motion of a mechanical element is designed by using a three-dimensional curve (called the three-dimensional clothoid curve) in which each of a pitch angle and a yaw angle in tangential direction is given by the quadratic of a curve length or curve length variables and (B). the route of a tool or the profile of a work is expressed by using a three-dimensional curve (called the three-dimensional clothoid curve) in which each of a pitch angle and a yaw angle in tangential direction is given by the quadratic of a curve length or curve length variables, and the motion of the tool is controlled by the three-dimensional curve.

Description

The numerical control method and the device of the method for designing of the industrial product of employing clothoid and the industrial product that designs with this method for designing, employing clothoid

Technical field

In the following description book, relevant " adopting the method for designing of industrial product of clothoid and the industrial product that designs with this method for designing " (below, the method for designing that only is called the industrial product that adopts clothoid) remarks A in the explanation, remarks B in relevant the numerical control method and the device of clothoid " adopt " (below's, only be called the numerical control method that adopts clothoid) explanation.

A. adopt the method for designing of the industrial product of clothoid

The present invention relates to adopt the method for designing of shape of the industrial product of clothoid, relate in particular in the machinery that comprises the mechanism that makes the mechanical organ motion with quality, design makes the method for tracks of the motion smoothness of this mechanical organ.

B. adopt the numerical control method of clothoid

In addition, the present invention relates to adopt clothoid, the numerical control method and the device of the motion of the instrument (comprising handle part or various tools such as handle) in the Work machines (being called robot etc.) such as control robot, lathe, make-up machinery, inspection machinery.

Background technology

A. adopt the method for designing of the industrial product of clothoid

Along with the miniaturization and the high precision int of machinery, the mechanism of high-speed motion mechanical organ becomes important.Rational smooth movement locus on the strong request design mechanics reduces and vibrates or kinematic error, suppresses timeliness variation or damage, realizes high speed, high-precision motion.

About the method for designing of free movement track, adopt to connect the method for the curve that straight line or circular arc etc. resolve in the past, or SPL interpolation (method of the point range that provides with the clothoid interpolation) (with reference to non-patent literature 1).

B. adopt the numerical control method of clothoid

Weld, in the robot of the numerical control of application, bonding agent coating etc., general as discrete some column data input tablet pattern.Therefore, generate continuous figure, need to adopt any method interpolation point range.

Method between the point range that provides arbitrarily as interpolation, known method or B spline interpolation, the cubic expression tertiary spline interpolation etc. that the bight of fillet processing broken line is arranged, but as can be strict interpolation by the each point that provides, known have a cubic expression tertiary spline interpolation (with reference to non-patent literature 1).

But the cubic expression tertiary spline interpolation is not because performance has the independent variable of the geometric meaning as parameter, so have unstable this big defective of relation of geometric all amounts of independent variable and curve.This cubic expression tertiary spline interpolation from the complexity that concerns of the displacement of initial point and curvature, is not suitable in making the fixing control of linear velocity.

Non-patent literature 1: fringe slope Wei assistant Tian Dengzhi work, " integration CAD/CAM system " Ohm publishing house, 1997

Non-patent literature 2: enemy Time rain, herd wild ocean, Shall field spring, the respectful man in Hengshan Mountain, " the free curve interpolation that utilization is circled round " (No. 6, association of Japanese robot will 8 volumes, pp40-47)

Non-patent literature 3:Li Guiqing, Li Xianmin, Li Hua, " 3D Discrete ClothoidSplines ", (CGI ' 01, pp321-324)

Summary of the invention

A. adopt the method for designing of the industrial product of clothoid

In the method for the curve that connects parsings such as straight line or circular arc, difficult tie point at straight line and circular arc is connected curvature continuously.If the method according to the SPL interpolation can connect curvature continuously, but because from the complexity that concerns of the displacement of initial point and curvature, therefore be difficult to the reasonably distribution of curvature on the orbit Design mechanics, can not get good movement locus.

Therefore, the objective of the invention is to, in the machinery that comprises the mechanism that makes mechanical organ motion, provide a kind of method for designing of tracks of the motion smoothness that makes this mechanical organ with quality.This method is the new and brand-new method that is proposed by present inventors.

Herein, so-called smoothness, the meaning are that the variation of tangent line, contact plane (normal) or curvature etc. of track is continuous along track, thereby the power that acts on the mechanical organ that moves on the track changes continuously.

, the form of the recurrence road warp of the ball-screw that robot, lathe, make-up machinery, inspection machinery etc. are used more connects with straight line or circular arc, and the tangent line or the curvature of curve are discontinuous, and the degree of freedom of orbit Design is also not enough in addition.

Another object of the present invention is to, in the design of the ball circulation paths of ball-screw, be the motion loss of energy on the circulating path that alleviates ball-screw, prevent from addition parts to be caused damage along circulating path, establish the tangent line of circulating path or curvature continuous and the curved transition method for designing of circulating path stably.The method for designing of the circulating path of ball-screw is the application examples of method of track of the motion of the design motion smoothness that makes mechanical organ.

B. adopt the numerical control method of clothoid

As the interpolating method by the each point that provides in two dimension, the interpolation that circles round that the known inventors of having propose, interpolation glibly (with reference to non-patent literature 2).Therefore think,, compare, can realize easily that retention wire speed fixes, or change the control of linear velocity according to line length with feature as the clothoid of the function representation of length of curve if three-dimensional expansion clothoid is used for the interpolation of free point range.In addition since with curve length as parameter, so different with other method, do not need to obtain from behind the advantage of line length with ining addition, wish that three-dimensional to expand clothoid be useful in fields such as Numerical Control.In the past, about three-dimensional expansion clothoid, known " 3D Discrete ClothoidSplines " (with reference to the non-patent literature 3) that people such as Li is arranged etc., but also do not find the three-dimensional expansion of form clothoid with formula.With the expansion of the mode of formula, has advantage on this point calculating easily respectively to be worth.

Therefore, the objective of the invention is to, for the motion of Numerical Control instrument, provide a kind of definition of three-dimensional clothoid newly, it can take over the characteristic of simple two-dimentional clothoid as far as possible with respect to the curved transition figure of independent variable.In addition, the objective of the invention is to, by this three-dimensional clothoid interpolation point range.

A. adopt the method for designing of the industrial product of clothoid

Below, the invention of method for designing of the industrial product of the described clothoid of claim 1～10 is described.

The 1st invention, for solving the above problems, a kind of method for designing of industrial product is provided, it is characterized in that: adopt the inclination angle (pitchangle) and deflection angle (yaw angle) three-dimensional curve (being called three-dimensional clothoid) separately that provide tangential direction by the quadratic expression of curve length or the long variable of curve, the shape of design industrial product.

The 2nd invention, the method for designing as the described industrial product of the 1st invention is characterized in that: described industrial product is the machinery that comprises the mechanism that makes the mechanical organ motion with quality; Adopt the tracks of the described mechanical organ of described three-dimensional curve (being called three-dimensional clothoid) design.

The 3rd invention, the method for designing as the described industrial product of the 2nd invention is characterized in that: described machinery is the screw device that comprises the mechanism that makes the ball motion as described mechanical organ; Described screw device possesses, have the lead screw shaft of spiral helicine rolling body raceway groove and have and the opposed load rolling body of described rolling body raceway groove raceway groove at outer peripheral face at inner peripheral surface, have simultaneously the end that connects described load rolling body raceway groove and the other end return path nut and be arranged in the described rolling body raceway groove of described lead screw shaft and the described load rolling body raceway groove of described nut between and a plurality of rolling bodys on the return path; Adopt described three-dimensional curve (being called three-dimensional clothoid), design the described return path of described screw device.

The 4th invention, as the method for designing of inventing any one described industrial product in 1～3 are by define described three-dimensional clothoid with following formula.

[numerical expression 1]

$P={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+{\&Integral;}_0^s$ $\mathrm{uds}=P_0+h{\&Integral;}_0^S\mathrm{udS},$ 0≤s≤h， $0\&le;S=\frac{s}{h}\&le;1$

(1)

$u=E^{\mathrm{k\&beta;}}{E}^{\mathrm{j\&alpha;}}\left(i\right)=$ $\left[\begin{array}{ccc}\cos \mathrm{\&beta;}& \sin \mathrm{\&beta;}& 0\\ \sin \mathrm{\&beta;}& \cos \mathrm{\&beta;}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos \mathrm{\&alpha;}& 0& \sin \mathrm{\&alpha;}\\ 0& 1& 0\\ -\sin \mathrm{\&alpha;}& 0& \cos \mathrm{\&alpha;}\end{array}\right]\left\{\begin{array}{c}1\\ 0\\ 0\end{array}\right\}=\begin{array}{c}\left\{\begin{array}{c}\cos \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ \sin \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ -\sin \mathrm{\&alpha;}\end{array}\right\}\end{array}$

(2)

α＝a ₀+a ₁S+a ₂S ² (3)

β＝b ₀+b ₁S+b ₂S ² (4)

Herein,

[numerical expression 2]

$P=\left\{\begin{array}{c}x\\ y\\ z\end{array}\right\},$ $P_0=\left\{\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right\}---\left(5\right)$

Position vector and the initial value thereof of representing the point on the three-dimensional clothoid respectively.

To be made as s from the length of a curve of initial point, its length overall (length from the initial point to the terminal point) will be made as h.The value of representing to remove s with S with h.S is the value of no guiding principle amount, is referred to as the long variable of curve.

I, j, k are respectively x axle, y axle and the axial unit vector of z.

U is the unit vector of the tangential direction of the curve on the expression point P, is provided by formula (2).E ^{K β}And E ^{J α}Be rotation matrix, represent the rotation of the angle beta that the k axle is and the rotation of the angle [alpha] that the j axle is respectively.The former is called deflection (yaw) rotation, the latter is called inclination (pitch) rotation.Formula (2), expression is only rotated α by at first making the axial unit vector of i in j axle system, and then, system only rotates β at the k axle, obtains tangent line vector u.

a ₀, a ₁, a ₂, b ₀, b ₁, b ₂It is constant.

The 5th invention, the method for designing as the described industrial product of the 4th invention is characterized in that: specify a plurality of spatial point in three-dimensional coordinate, by adopting described three-dimensional these spatial point of clothoid interpolation, design the shape of described industrial product.

The 6th invention, method for designing as the described industrial product of the 5th invention, it is characterized in that: with in described a plurality of spatial point, with three-dimensional line segment (constituting the unit curve of the group of curves that generates by interpolation) and the next three-dimensional line segment (constituting unit curve) that circles round that circles round by the group of curves of interpolation generation, the mode that connects both position, tangential method, normal direction and curvature is calculated circle round 7 parameter a of line segment of described three-dimensional ₀, a ₁, a ₂, b ₀, b ₁, b ₂, h.

The 7th invention, the method for designing as the described industrial product of the 6th invention is characterized in that: specify the initial point in described a plurality of spatial point and tangential direction, normal direction and the curvature of terminal point; By between preassigned described spatial point, inserting the interpolation object again, make the conditional of the tangential direction, normal direction and the curvature that add described initial point of computing and described terminal point and circle round with a three-dimensional on described a plurality of spatial point line segment and the next three-dimensional line segment that circles round be connected the conditional number of conditional of both position, tangential direction, normal direction and curvature, with circle round 7 parameter a of line segment of described three-dimensional ₀, a ₁, a ₂, b ₀, b ₁, b ₂, h the unknown number unanimity; By making the several consistent of conditional and unknown number, calculate circle round 7 parameter a of line segment of described three-dimensional ₀, a ₁, a ₂, b ₀, b ₁, b ₂, h.

The 8th invention is a kind of industrial product, with the method for designing design as any one described industrial product in the invention 1～7.

The 9th invention is a kind of program, be used in order to design the shape of industrial product, make computing machine as the three-dimensional curve (being called three-dimensional clothoid) that the inclination angle of adopting tangential direction and deflection angle quadratic expression long by curve respectively or the long variable of curve provide, design industrial product shape means and play a role.

The 10th invention is a kind of recording medium of embodied on computer readable, logging program thereon, this program is used in order to design the shape of industrial product, make computing machine as the three-dimensional curve (being called three-dimensional clothoid) that the inclination angle of adopting tangential direction and deflection angle quadratic expression long by curve respectively or the long variable of curve provide, design industrial product shape means and play a role.

B. adopt the numerical control method of clothoid

Below, the invention of the numerical control method that adopts the described clothoid of claim 11～27 is described.

The 11st invention; for solving the above problems; a kind of numerical control method is provided; wherein; adopt the inclination angle of tangential direction and the three-dimensional curve (being called three-dimensional clothoid) that deflection angle quadratic expression long by curve respectively or the long variable of curve provides; the contour shape of performance tool path or workpiece is by the motion of this three-dimensional curve control tool.

The 12nd invention is as the described numerical control method of the 11st invention, wherein by circling round with following formula definition three-dimensional.

[numerical expression 3]

$P={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+{\&Integral;}_0^s$ $\mathrm{uds}=P_0+h{\&Integral;}_0^S\mathrm{udS},$ 0≤s≤h， $0\&le;S=\frac{s}{h}\&le;1$

(1)

$u=E^{\mathrm{k\&beta;}}{E}^{\mathrm{j\&alpha;}}\left(i\right)=\left[\begin{array}{ccc}\cos \mathrm{\&beta;}& -\sin \mathrm{\&beta;}& 0\\ \sin \mathrm{\&beta;}& \cos \mathrm{\&beta;}& 0\begin{array}{}\end{array}\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos \mathrm{\&alpha;}& 0& \sin \mathrm{\&alpha;}\\ 0& 1& 0\\ -\sin \mathrm{\&alpha;}& 0& \cos \mathrm{\&alpha;}\end{array}\right]\left\{\begin{array}{c}1\\ 0\\ 0\end{array}\right\}=\left\{\begin{array}{c}\cos \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ \sin \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ -\sin \mathrm{\&alpha;}\end{array}\right\}$

(2)

α＝a ₀+a ₁S+a ₂S ² (3)

β＝b ₀+b ₁S+b ₂S ² (4)

Herein,

[numerical expression 4]

$P=\left\{\begin{array}{c}x\\ y\\ z\end{array}\right\},$ ${\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0$ $=\left\{\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right\}---\left(5\right)$

Position vector and the initial value thereof of representing the point on the three-dimensional clothoid respectively.

To be made as s from the length of a curve of initial point, its length overall (length from the initial point to the terminal point) will be made as h.The value of representing to remove s with S with h.S is the value of no guiding principle amount, is referred to as the long variable of curve.

I, j, k are respectively x axle, y axle and the axial unit vector of z.

U is the unit vector of the tangential direction of the curve on the expression point P, is provided by formula (2).E ^{K β}And E ^{J α}Be rotation matrix, represent the rotation of the angle beta that the k axle is and the rotation of the angle [alpha] that the j axle is respectively.The former is called deflection (yaw) rotation, the latter is called inclination (pitch) rotation.Formula (2), expression is only rotated α by at first making the axial unit vector of i in j axle system, and then, system only rotates β at the k axle, obtains tangent line vector u.

a ₀, a ₁, a ₂, b ₀, b ₁, b ₂It is constant.

The 13rd invention is a kind of numerical control device; wherein; adopt the inclination angle of tangential direction and the three-dimensional curve (being called three-dimensional clothoid) that deflection angle quadratic expression long by curve respectively or the long variable of curve provides; the contour shape of performance tool path or workpiece is by the motion of this three-dimensional curve control tool.

The 14th invention is a kind of program; be used for motion for the Numerical Control instrument; make computing machine as the three-dimensional curve (being called three-dimensional clothoid) that the inclination angle of adopting tangential direction and deflection angle quadratic expression long by curve respectively or the long variable of curve provide, show tool path or workpiece contour shape means and play a role.

The 15th invention is a kind of recording medium of embodied on computer readable; logging program or the result of calculation that draws by this program thereon; this program is used for the motion for the Numerical Control instrument; make computing machine as the three-dimensional curve (being called three-dimensional clothoid) that the inclination angle of adopting tangential direction and deflection angle quadratic expression long by curve respectively or the long variable of curve provide, show tool path or workpiece contour shape means and play a role.

The 16th invention is for addressing the above problem, providing is a kind of numerical control method, wherein, adopt the inclination angle of tangential direction and the three-dimensional curve that deflection angle quadratic expression long by curve respectively or the long variable of curve provides (three-dimensional circle round line segment), between the point range that interpolation provides arbitrarily in three-dimensional coordinate, by the circle round motion of line segment control tool of this three-dimensional.

The 17th invention is a kind of numerical control method, wherein, many of the three-dimensional curve that the inclination angle of tangential direction and deflection angle quadratic expression long by curve respectively or the long variable of curve are provided (three-dimensional circle round line segment) connections are by the circle round motion of line segment control tool of these many three-dimensionals.

The 18th invention, as invent 16 or 17 described numerical control methods, wherein by define three-dimensional clothoid with following formula.

[numerical expression 5]

$P={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0$ ${\&Integral;}_0^s$ $\mathrm{uds}={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+h{\&Integral;}_0^S\mathrm{udS},$ 0≤s≤h， $0\&le;S=\frac{s}{h}\&le;1$

(1)

$u=E^{\mathrm{k\&beta;}}{E}^{\mathrm{j\&alpha;}}\left(i\right)=\left[\begin{array}{ccc}\cos \mathrm{\&beta;}& -\sin \mathrm{\&beta;}& 0\\ \sin \mathrm{\&beta;}& \cos \mathrm{\&beta;}& 0\begin{array}{}\end{array}\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos \mathrm{\&alpha;}& 0& \sin \mathrm{\&alpha;}\\ 0& 1& 0\\ -\sin \mathrm{\&alpha;}& 0& \cos \mathrm{\&alpha;}\end{array}\right]\left\{\begin{array}{c}1\\ 0\\ 0\end{array}\right\}=\left\{\begin{array}{c}\cos \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ \sin \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ -\sin \mathrm{\&alpha;}\end{array}\right\}$

(2)

α＝a ₀+a ₁S+a ₂S ² (3)

β＝b ₀+b ₁S+b ₂S ² (4)

Herein,

[numerical expression 6]

$P=\left\{\begin{array}{c}x\\ y\\ z\end{array}\right\},$ ${\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0=\begin{array}{c}\left\{\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right\}\end{array}---\left(5\right)$

Position vector and the initial value thereof of representing the point on the three-dimensional clothoid respectively.

To be made as s from the length of a curve of initial point, its length overall (length from the initial point to the terminal point) will be made as h.The value of representing to remove s with S with h.S is the value of no guiding principle amount, is referred to as the long variable of curve.

I, j, k are respectively x axle, y axle and the axial unit vector of z.

U is the unit vector of the tangential direction of the curve on the expression point P, is provided by formula (2).E ^{K β}And E ^{J α}Be rotation matrix, represent the rotation of the angle beta that the k axle is and the rotation of the angle [alpha] that the j axle is respectively.The former is called deflection (yaw) rotation, the latter is called inclination (pitch) rotation.Formula (2),

Expression is only rotated α by at first making the axial unit vector of i in j axle system, and then, system only rotates β at the k axle, obtains tangent line vector u.

a ₀, a ₁, a ₂, b ₀, b ₁, b ₂It is constant.

The 19th invention, as the described numerical control method of the 18th invention, it is characterized in that: circle round on the joint of line segment in circle round line segment and next three-dimensional of a three-dimensional,, calculate circle round 7 parameter a of line segment of described three-dimensional in both continuous modes in position, tangential direction (and according to circumstances curvature) ₀, a ₁, a ₂, b ₀, b ₁, b ₂, h.

The 20th invention is a kind of numerical control device, wherein, adopt the inclination angle of tangential direction and three-dimensional that deflection angle quadratic expression long by curve respectively or the long variable of curve the provides line segment that circles round, between the point range that interpolation provides arbitrarily, by the circle round motion of line segment control tool of this three-dimensional in three-dimensional coordinate.

The 21st invention is a kind of program, be used for motion for the Numerical Control instrument, the three-dimensional that computing machine is provided as the inclination angle of adopting tangential direction and deflection angle quadratic expression long by curve respectively or the long variable of curve the line segment that circles round, means between the point range that interpolation provides arbitrarily in three-dimensional coordinate and playing a role.

The 22nd invention is a kind of recording medium of embodied on computer readable, logging program or the result of calculation that draws by this program thereon, this program is used for the motion for the Numerical Control instrument, the three-dimensional that computing machine is provided as the inclination angle of adopting tangential direction and deflection angle quadratic expression long by curve respectively or the long variable of curve the line segment that circles round, means between the point range that interpolation provides arbitrarily in three-dimensional coordinate and playing a role.

The 23rd invention is a kind of numerical control method, wherein: adopt the inclination angle of tangential direction and the three-dimensional curve (being called three-dimensional clothoid) that deflection angle quadratic expression long by curve respectively or the long variable of curve provides, the contour shape of performance tool path or workpiece; Moving of the instrument that appointment is moved along described three-dimensional curve; According to the motion of appointment, calculate the shift position of instrument by time per unit.Herein, so-called motion refers to the positional information as the function of time.

The 24th invention as the described numerical control method of the 23rd invention, wherein defines three-dimensional clothoid as follows.

[numerical expression 7]

$P={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+{\&Integral;}_0^s$ $\mathrm{uds}={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+h{\&Integral;}_0^S\mathrm{udS},$ 0≤s≤h， $0\&le;S=\frac{s}{h}\&le;1$

(1)

$u=E^{\mathrm{k\&beta;}}{E}^{\mathrm{j\&alpha;}}\left(i\right)=\left[\begin{array}{ccc}\cos \mathrm{\&beta;}& -\sin \mathrm{\&beta;}& 0\\ \sin \mathrm{\&beta;}& \cos \mathrm{\&beta;}& 0\begin{array}{}\end{array}\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos \mathrm{\&alpha;}& 0& \sin \mathrm{\&alpha;}\\ 0& 1& 0\\ -\sin \mathrm{\&alpha;}& 0& \cos \mathrm{\&alpha;}\end{array}\right]\left\{\begin{array}{c}1\\ 0\\ 0\end{array}\right\}=\left\{\begin{array}{c}\cos \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ \sin \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ -\sin \mathrm{\&alpha;}\end{array}\right\}$

(2)

α＝a ₀+a ₁S+a ₂S ² (3)

β＝b ₀+b ₁S+b ₂S ² (4)

Herein,

[numerical expression 8]

$P=\left\{\begin{array}{c}x\\ y\\ z\end{array}\right\},$ ${\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0=\begin{array}{c}\left\{\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right\}\end{array}---\left(5\right)$

Position vector and the initial value thereof of representing the point on the three-dimensional clothoid respectively.

To be made as s from the length of a curve of initial point, its length overall (length from the initial point to the terminal point) will be made as h.The value of representing to remove s with S with h.S is the value of no guiding principle amount, is referred to as the long variable of curve.

I, j, k are respectively x axle, y axle and the axial unit vector of z.

U is the unit vector of the tangential direction of the curve on the expression point P, is provided by formula (2).E ^{K β}And E ^{J α}Be rotation matrix, represent the rotation of the angle beta that the k axle is and the rotation of the angle [alpha] that the j axle is respectively.The former is called deflection (yaw) rotation, the latter is called inclination (pitch) rotation.Formula (2), expression is only rotated α by at first making the axial unit vector of i in j axle system, and then, system only rotates β at the k axle, obtains tangent line vector u.a ₀, a ₁, a ₂, b ₀, b ₁, b ₂It is constant.

The 25th invention is a kind of numerical control device, wherein: adopt the inclination angle of tangential direction and the three-dimensional curve (being called three-dimensional clothoid) that deflection angle quadratic expression long by curve respectively or the long variable of curve provides, the contour shape of performance tool path or workpiece; Moving of the instrument that appointment is moved along described three-dimensional curve; According to the motion of appointment, calculate the shift position of instrument by time per unit.Herein, so-called motion refers to the positional information that changes as the function of time.

The 26th invention is a kind of program, be used for motion for the Numerical Control instrument, and computing machine is played a role as following means: the three-dimensional that the inclination angle of employing tangential direction and deflection angle quadratic expression long by curve respectively or the long variable of curve provides is returned curve (being called three-dimensional clothoid), the means of the contour shape of performance tool path or workpiece; The means of moving of the instrument that appointment is moved along described three-dimensional curve; According to the motion of appointment, calculate the means of the shift position of instrument by time per unit.Herein, so-called motion refers to the positional information that changes as the function of time.

The 27th invention is a kind of recording medium of embodied on computer readable, logging program or the result of calculation that draws by this program thereon, this program is used for the motion for the Numerical Control instrument, and computing machine is played a role as following means: the three-dimensional that the inclination angle of employing tangential direction and deflection angle quadratic expression long by curve respectively or the long variable of curve provides is returned curve (being called three-dimensional clothoid), the means of the contour shape of performance tool path or workpiece; The means of moving of the instrument that appointment is moved along described three-dimensional curve; According to the motion of appointment, calculate the means of the shift position of instrument by time per unit.Herein, so-called motion refers to the positional information that changes as the function of time.

A. adopt the method for designing of the industrial product of clothoid

According to the described invention of the 1st～10 invention, the track of the motion by adopting three-dimensional clothoid, can design the motion smoothness that makes mechanical organ.If can so design track, can realize reasonably motion on the mechanics, can make the machinery that function descends or the track damage is little that kinematic error causes.

Especially for screw device, the method for generation of the wide usefulness of essential space curve in the time of can being provided at the circulating path of design rolling body.At the space curve of rolling body, follow under the situation of acceleration-deceleration motion, but design constraint power changes smooth design along circulating path.According to this feature, because rolling body carries out steady, smooth motion, therefore can improve the power transmission efficiency of screw device, suppress excessive friction or be used to the generation of power.Thereby can prevent the damage of parts, realize the high screw device of reliability.

In addition, application can be controlled the feature of curved transition figure, can be increased in the application of industrial field.For example, in the pattern form design that requires aesthetic artistic conception, can use the curve design method of this wide usefulness effectively.

B. adopt the numerical control method of clothoid

According to the described invention of the 11st～27 invention, because the master variable of curve is curve length or the long variable of curve, and quadratic expression long by curve respectively or the long variable of curve provides the inclination angle and the deflection angle of its tangential direction, therefore can guarantee the long or long variable of curve about curve, its normal direction that obtains of a subdifferential, to reach its curvature that obtains of second differential be continuous.In other words, in a clothoid, normal direction and curvature connect.Therefore, can obtain smooth well-behaved curve, realize that the mathematical control mode of rational velocity variations on the mechanics becomes possibility.

Description of drawings

Fig. 1 is the diagram of the two-dimentional clothoid on the expression xy coordinate.

Fig. 2 is the diagram of the shape of the typical two-dimentional clothoid of expression.

Fig. 3 is the diagram of the definition of expression inclination alpha of three-dimensional clothoid and deflection angle β.

Fig. 4 is the diagram of the shape of the typical three-dimensional clothoid of expression.

Fig. 5 is the diagram of the variable quantity of representation unit normal vector.

Fig. 6 is an expression size, shape is identical but towards 2 the opposite two dimensions or the diagram of three-dimensional clothoid.

Fig. 7 is the diagram of cutting apart of the three-dimensional clothoid of expression.

Fig. 8 is expression G ²The diagram of the condition of continuous interpolation.

Fig. 9 is the diagram of the notion of expression contact plane.

Figure 10 is the circle round diagram of concise and to the point flow process of method of interpolation of expression.

Figure 11 is that G is satisfied in expression ²The diagram of the concise and to the point flow process of the method for the interpolation of circling round of condition for continuous.

Figure 12 is expression point P ₁, P ₂, P ₃The circle round diagram of interpolation of three-dimensional.

Figure 13 is the diagram of the 3D Discrete Clothoid Splines of expression r=4.

Figure 14 is the diagram of explanation 3D Discrete Clothoid Splines.

Figure 15 is the skeleton view by the three-dimensional clothoid of interpolation generation.

Figure 16 gets displacement from initial point at transverse axis, gets the curved transition curve map of curvature at the longitudinal axis.

Figure 17 is the circle round diagram of concise and to the point flow process of interpolation of the three-dimensional that is illustrated in each value of two-end-point control.

Figure 18 is the circle round sketch of interpolation of the three-dimensional that is illustrated in each value of two-end-point control.

Figure 19 is the actual diagram of carrying out the result of interpolation of expression.

Figure 20 is expression from the curve map of the relation of the displacement of the initial point of each curve and curvature.

Figure 21 is the diagram of the control of the value on the expression intermediate point.

Figure 22 is the diagram that the concise and to the point flow process of the interpolation that circles round in the three-dimensional of initial point and each value of end points control is adopted in expression.

Figure 23 is the diagram of the 3D Discrete Clothoid Splines of expression r=4.

Figure 24 is polygonal diagram that expression generates.

Figure 25 is expression point P ₁, P ₂, P ₃The circle round diagram of interpolation of three-dimensional.

Figure 26 is curve and the polygonal diagram that expression generates.

Figure 27 is the diagram of insertion point.

Figure 28 is the diagram of the three-dimensional clothoid cut apart of expression.

Figure 29 is the diagram of the curve of expression generation.

Figure 30 is expression from the curve map of the relation of the displacement s of the initial point of each curve and curvature κ.

Figure 31 is that the expression reverser is the diagram of the ball-screw of nut and other reverser mode.

Figure 32 be the expression reverser for the diagram of the nut of the ball-screw of nut one.

Figure 33 A is the stereographic map of nut of seeing the state of ball circulating groove.

Figure 33 B is the stereographic map of nut of seeing the state of load ball road groove.

Figure 34 is the diagram that is illustrated in the state of assembling nut on the lead screw shaft.

Figure 35 is the stretch-out view of the circulating path of ball-screw in the past.

Figure 36 is the curve map of curvature of representing the circulating path of ball-screw in the past.

Figure 37 is the diagram of the track at expression ball center.

Figure 38 is the diagram of denotation coordination system.

Figure 39 is the diagram of expression from the coordinate system of z axle.

Figure 40 is the diagram of curve of the track at the expression center of drawing the ball that moves along the leading screw groove.

Figure 41 is the curve C of expression from the y axle ₀And C ₁Diagram.

Figure 42 is the some P of expression from the z axle _sNear curve C ₀And C ₁Diagram.

Figure 43 is insertion point P ₂Diagram.

Figure 44 is return path and the curve C that expression generates ₀Diagram.

Figure 45 is that expression is from a P _eDisplacement and the diagram of the relation of curvature.

Figure 46 is the diagram of the two-dimentional clothoid on expression x, the y coordinate.

Figure 47 is the diagram of the two-dimentional clothoid of expression.

Figure 48 is the diagram of definition of α, the β of the three-dimensional clothoid of expression.

Figure 49 is the diagram of the figure of the typical three-dimensional clothoid of expression.

Figure 50 is expression G ²The diagram of the condition of continuous interpolation.

Figure 51 is the diagram of the notion of expression contact plane.

Figure 52 is the circle round diagram of concise and to the point flow process of interpolating method of expression.

Figure 53 is that G is satisfied in expression ²The diagram of the concise and to the point flow process of the method for the interpolation of circling round of condition for continuous.

Figure 54 is expression point P ₁, P ₂, P ₃The circle round diagram of interpolation of three-dimensional.

Figure 55 is the diagram of the 3D Discrete Clothoid Splines of expression r=4.

Figure 56 is the diagram of explanation 3D Discrete Clothoid Splines.

Figure 57 is the skeleton view by the three-dimensional clothoid of interpolation generation.

Figure 58 is a curved transition curve map of obtaining curvature at transverse axis from the displacement of initial point, at the longitudinal axis.

Figure 59 is the circle round diagram of concise and to the point flow process of interpolation of the three-dimensional that is illustrated in each value of two-end-point control.

Figure 60 is the circle round sketch of interpolation of the three-dimensional that is illustrated in each value of two-end-point control.

Figure 61 is the actual diagram of carrying out the result of interpolation of expression.

Figure 62 is expression from the curve map of the relation of the displacement of the initial point of each curve and curvature.

Figure 63 is the diagram of the control of the value on the expression intermediate point.

Figure 64 is the diagram that the concise and to the point flow process of the interpolation that circles round in the three-dimensional of initial point and each value of end points control is adopted in expression.

Figure 65 is the diagram of the 3D Discrete Clothoid Splines of expression r=4.

Figure 66 is polygonal diagram that expression generates.

Figure 67 is expression point P ₁, P ₂, P ₃The circle round diagram of interpolation of three-dimensional.

Figure 68 is curve and the polygonal diagram that expression generates.

Figure 69 is the diagram of insertion point.

Figure 70 is the diagram of the three-dimensional clothoid cut apart of expression.

Figure 71 is the diagram of the curve of expression generation.

Figure 72 is expression from the curve map of the relation of the displacement s of the initial point of each curve and curvature κ.

Figure 73 is the process chart of expression numerical control method.

Figure 74 is a comparison diagram of representing SPL in the past.

Embodiment

A. adopt the method for designing of the industrial product of clothoid

Below, working of an invention mode about the method for designing of the industrial product that adopts clothoid, divide the definition of 1. three-dimensional clothoids and feature, 2. adopt the interpolation of three-dimensional clothoid, 3. adopt the three-dimensional interpolation of circling round, design is as the method for the return path of the ball-screw of screw device, 4. adopt the numerical control method of the three-dimensional interpolation of circling round, explanation successively.

1. the definition of three-dimensional clothoid and feature

(1-1) the three-dimensional basic mode of circling round

Clothoid (Clothoid curve), another name also are called the spiral (Cornu ' s spiral) of Ke's knob, are the curves that changes curvature with length of a curve with being directly proportional.

The clothoid of the two dimension that the inventor has proposed is a kind of of plane curve (two-dimensional curve), on xy coordinate shown in Figure 1, represents with following formula.

[numerical expression 9]

$P={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+{\&Integral;}_0^s$ $e^{\mathrm{j\&phi;}}\mathrm{ds}={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+h{\&Integral;}_0^S{e}^{\mathrm{j\&phi;}}\mathrm{dS},$ 0≤s≤h， $0\&le;S=\frac{s}{h}\&le;1$

(1-1)

φ＝c ₀+c ₁s+c ₂s ²＝φ ₀+φ _vS+φ _uS ² (1-2)

Herein,

[numerical expression 10]

P＝x+jy， $j=\sqrt{-1}$ $---(1-3)$

Be the position vector of the point on the expression curve,

[numerical expression 11]

P ₀＝x ₀+jy ₀ (1-4)

Be its initial value (position vector of initial point).

[numerical expression 12]

e ^{J φ}=cos ^φ+ jsin ^φBe the position vector (length is 1 vector) of the tangential direction of expression curve (1-5), this direction Φ measures counterclockwise from former line (x direction of principal axis).If in this unit vector, multiply by tiny length ds integration, can obtain the some P on the curve.

To be made as s along the length from initial point of the curve of curve determination, its length overall (length from the initial point to the terminal point) will be made as h.The value of representing to remove s with S with h.S is the value of no guiding principle amount, is referred to as the long variable of curve.

The feature of clothoid shown in (1-2), is to represent tangent directional angle Φ with the quadratic expression of long s of curve or curve variable S.c ₀, c ₁, c ₂Or Φ _o, Φ _v, Φ _uBe quadratic coefficient, length overall h these are several and curve is called the parameter of circling round.Fig. 2 represents the shape of general clothoid.

The above relation of three-dimensional expansion, the formula of making three-dimensional clothoid.Do not know to provide the formula of three-dimensional clothoid, so initial its formula that derives of inventors in the past.

Define three-dimensional clothoid by following formula.

[numerical expression 13]

$P={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+{\&Integral;}_0^s$ $\mathrm{uds}={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+h{\&Integral;}_0^S\mathrm{udS},$ 0≤s≤h， $0\&le;S=\frac{s}{h}\&le;1$

(1-6)

$u=E^{\mathrm{k\&beta;}}{E}^{\mathrm{j\&alpha;}}\left(i\right)=\left[\begin{array}{ccc}\cos \mathrm{\&beta;}& -\sin \mathrm{\&beta;}& 0\\ \sin \mathrm{\&beta;}& \cos \mathrm{\&beta;}& 0\begin{array}{}\end{array}\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos \mathrm{\&alpha;}& 0& \sin \mathrm{\&alpha;}\\ 0& 1& 0\\ -\sin \mathrm{\&alpha;}& 0& \cos \mathrm{\&alpha;}\end{array}\right]\left\{\begin{array}{c}1\\ 0\\ 0\end{array}\right\}=\left\{\begin{array}{c}\cos \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ \sin \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ -\sin \mathrm{\&alpha;}\end{array}\right\}$

(1-7)

α＝a ₀+a ₁S+a ₂S ² (1-8)

β＝b ₀+b ₁S+b ₂S ² (1-9)，

Herein,

[numerical expression 14]

$P=\left\{\begin{array}{c}x\\ y\\ z\end{array}\right\},$ ${\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0=\begin{array}{c}\left\{\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right\}\end{array}---(1-10)$

Position vector and the initial value thereof of representing the point on the three-dimensional clothoid respectively.I, j, k are respectively x axle, y axle and the axial unit vector of z.

U is the unit vector of the tangential direction of the curve on the expression point P, and (1-7) provides by formula.In formula (1-7), E ^{K β}And E ^{J α}Be rotation matrix, as shown in Figure 3, represent the rotation of the angle beta that k axle (z axle) is and the rotation of the angle [alpha] that j axle (y axle) is respectively.The former is called deflection (yaw) rotation, the latter is called inclination (pitch) rotation.Formula (1-7), the expression by at first make i axle (x axle) to unit vector be only to rotate α at j axle (y axle), be only to rotate β then at k axle (z axle), obtain tangent line vector u.

Just, when two dimension, obtain representing the unit vector e of the tangential direction of curve by angle of inclination Φ from the x axle ^{J Φ}When three-dimensional, can be by the tangent line vector u that obtains curve from inclination alpha and deflection angle β.If inclination alpha is 0, can obtain the two-dimentional clothoid rolled with x y plane, if deflection angle β is 0, can obtain the two-dimentional clothoid of rolling with the x z-plane.If in tangential direction vector u, multiply by small long ds ground integration, can obtain three-dimensional clothoid.

In three-dimensional clothoid, the inclination alpha of tangent line vector and deflection angle β suc as formula shown in (1-8) and the formula (1-9), can be provided by the quadratic expression of the long variable S of curve respectively.So, can freely select the variation of tangential direction, and can also in it changes, make it have continuity.

Shown in above formula, three-dimensional clothoid is defined as " being to represent the inclination alpha of tangential direction and the curve of deflection angle β with the quadratic expression of the long variable of curve respectively ".

From P ₀A three-dimensional clothoid of beginning, by

[numerical expression 15]

a ₀，a ₁，a ₂，b ₀，b ₁，b ₂，h (1-11)

These 7 parameters are determined.a ₀～b ₂6 units that variable has angle, the expression clothoid shape.In contrast, h has the unit of length, the size of expression clothoid.As the typical example of three-dimensional clothoid, spiral helicine curve shown in Figure 4 is arranged.

(1-2) Fu Leinie frame and the curvature on the three-dimensional clothoid

When having arbitrarily the ternary curve, regulation is represented with R (t) as parameter with t.Especially, during as parameter, to represent with R (s) from the displacement s of initial point.

If the absolute value of 2 on the curve with ds degree difference relative position vector dR (s) is regarded as linear element ds, between ds and dt, has the relation of following formula (2-1).For simplification utilizes the differential of the R of parametric t, additional round dot is represented on letter.

[numerical expression 16]

$\mathrm{ds}=\left|\mathrm{dR}\left(t\right)\right|=$ $\left|\frac{\mathrm{dR}\left(t\right)}{\mathrm{dt}}\right|\mathrm{dt}$ $=\left|\stackrel{\&CenterDot;}{R}\right|=\sqrt{\stackrel{\&CenterDot;}{R}\&CenterDot;\stackrel{\&CenterDot;}{R}\mathrm{dt}}---(2-1)$

Because unit tangent vector u (t) makes linear element vector dR (t) standardization of curve, so if with reference to formula (2-1), available formula (2-2) expression.

[numerical expression 17]

$u\left(t\right)=\frac{\mathrm{dR}\left(t\right)}{\left|\mathrm{dR}\left(t\right)\right|}=\frac{\mathrm{dR}\left(t\right)}{\mathrm{ds}}=\frac{\stackrel{\&CenterDot;}{R}}{\left|\stackrel{\&CenterDot;}{R}\left(t\right)\right|}---(2-2)$

Then, consider the variable quantity du of unit tangent vector.The variable quantity of Fig. 5 representation unit normal vector.Because tangential direction do not change when being straight line, therefore be du (t)=0,0,0}, but when curve not like this, the variable quantity du of the locational unit tangent vector of separating distance ds and tangent line vector u quadrature.If this also can just can obtain understanding fully among the orthogonality relation udu=0 from the relation of differential uu=1.Making the variable quantity du of this unit tangent vector standardized, is the principal normal vector n of unit (t).Just, with formula (2-3) representation unit principal normal vector n (t)

[numerical expression 18]

$n\left(t\right)$ $=\frac{\stackrel{\&CenterDot;}{u}\left(t\right)}{\left|\stackrel{\&CenterDot;}{u}\left(t\right)\right|}$ $---(2-3)$

Normal direction with the people left during towards tangential direction to for just.More properly say, in the plane that makes by vector du and unit tangent vector u (t), will revolve the positive dirction of the principal normal vector n of the direction unit of being defined as (t) that turn 90 degrees to counter clockwise direction from unit tangent vector u (t).

In addition,, be and the vector of both sides' quadrature of unit tangent vector u (t) and the principal normal vector n of unit (t), define by formula (2-4) from normal vector b (t).

[numerical expression 19]

b(t)＝u(t)×n(t) (2-4)

Unit tangent vector u (t), the principal normal vector n of unit (t), the normal vector b (t) of definition are defined as 3 set of vectors { u (t), n (t), b (t) }, are called as the Fu Leinie frame (Frenet Frame) on the position R (t) of curve.

Then, narration unit tangent vector is curvature κ along the ratio of the linear element bending of curve.Curvature on the three-dimensional defines with formula (2-5).

[numerical expression 20]

$\mathrm{\&kappa;}\left(t\right)=\frac{\left|\right|\stackrel{\&CenterDot;}{R}\left(t\right)\&times;\stackrel{\&CenterDot;\&CenterDot;}{R}\left(t\right)\left|\right|}{{\left|\right|\stackrel{\&CenterDot;}{R}\left(t\right)\left|\right|}^3}$ $---(2-5)$

About the basic amount on the three-dimensional curve of above definition, be used in the three-dimensional clothoid and adopt the performance of the long variable S of curve to record and narrate as parameter.

When considering three-dimensional arbitrarily clothoid P (S), unit tangent vector u (S) can be by formula (2-2), with formula (2-6) expression.

[numerical expression 21]

$u\left(S\right)=$ $\frac{P^{\&prime;}\left(S\right)}{|P^{\&prime;}\left(S\right)|}$ $---(2-6)$

In addition, if unit tangent vector u (S) considers definition (1-7), (1-8), (1-9) of three-dimensional clothoid, also can enough following formulas (2-7) expression.In this manual, mainly adopt these performances.

[numerical expression 22]

$u\left(S\right)=\left\{\begin{array}{c}\cos \mathrm{\&beta;}\left(S\right)\cos \mathrm{\&alpha;}\left(S\right)\\ \sin \mathrm{\&beta;}\left(S\right)\cos \mathrm{\&alpha;}\left(S\right)\\ -\sin \mathrm{\&alpha;}\left(S\right)\end{array}\right\}---(2-7)$

By the long variable S1 of the curve rank differential of the unit tangent vector u (S) of three-dimensional clothoid, (2-9) represents its size with formula with formula (2-8) expression.

[numerical expression 23]

$u^{\&prime;}\left(S\right)=\left\{\begin{array}{c}-{\mathrm{\&alpha;}}^{\&prime;}\left(S\right)\cos \mathrm{\&beta;}\left(S\right)\sin \mathrm{\&alpha;}\left(S\right)-{\mathrm{\&beta;}}^{\&prime;}\left(S\right)\sin \mathrm{\&beta;}\left(S\right)\cos \mathrm{\&alpha;}\left(S\right)\\ -{\mathrm{\&alpha;}}^{\&prime;}\left(S\right)\sin \mathrm{\&beta;}\left(S\right)\sin \mathrm{\&alpha;}\left(S\right)-{\mathrm{\&beta;}}^{\&prime;}\left(S\right)\cos \mathrm{\&beta;}\left(S\right)\cos \mathrm{\&alpha;}\left(S\right)\\ -{\mathrm{\&alpha;}}^{\&prime;}\left(S\right)\cos \mathrm{\&alpha;}\left(S\right)\end{array}\right\}$

(2-8)

$\left|\right|u^{\&prime;}\left(S\right)\left|\right|=\sqrt{{\mathrm{\&alpha;}}^{\&prime;}{\left(S\right)}^2+{\mathrm{\&beta;}}^{\&prime;}{\left(S\right)}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&alpha;}\left(S\right)}---(2-9)$

Then, consider the principal normal vector n of unit (S).Since with the normal vector of formula (2-3) expression three-dimensional curve, thus the normal vector of three-dimensional clothoid, with formula (2-10) expression.

[numerical expression 24]

$n\left(S\right)=\frac{u^{\&prime;}\left(S\right)}{\left|\right|u^{\&prime;}\left(S\right)\left|\right|}$

$=\frac{1}{\sqrt{{\mathrm{\&alpha;}}^{\&prime;}{\left(S\right)}^2+{\mathrm{\&beta;}}^{\&prime;}{\left(S\right)}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&alpha;}\left(S\right)}}\left\{\begin{array}{c}-{\mathrm{\&alpha;}}^{\&prime;}\left(S\right)\cos \mathrm{\&beta;}\left(S\right)\sin \mathrm{\&alpha;}\left(S\right)-{\mathrm{\&beta;}}^{\&prime;}\left(S\right)\sin \mathrm{\&beta;}\left(S\right)\cos \mathrm{\&alpha;}\left(S\right)\\ -{\mathrm{\&alpha;}}^{\&prime;}\left(S\right)\sin \mathrm{\&beta;}\left(S\right)\sin \mathrm{\&alpha;}\left(S\right)+{\mathrm{\&beta;}}^{\&prime;}\left(S\right)\cos \mathrm{\&beta;}\left(S\right)\cos \mathrm{\&alpha;}\left(S\right)\\ -{\mathrm{\&alpha;}}^{\&prime;}\left(S\right)\cos \mathrm{\&alpha;}\left(S\right)\end{array}\right\}$

(2-10)

About from normal vector b (S), stipulate to obtain from the unit tangent vector u (S) of formula (2-7) and the principal normal vector n of unit (S) of formula (2-10) by formula (2-4).

[numerical expression 25]

b(S)＝u(S)×n(S) (2-11)

Be about curvature at last, if deformation type (2-5), with formula (2-12) expression.

[numerical expression 26]

$\mathrm{\&kappa;}\left(S\right)=\frac{\left|\right|P^{\&prime;}\left(S\right)\&times;P^{\&prime;\&prime;}\left(S\right)\left|\right|}{{\left|\right|P^{\&prime;}\left(S\right)\left|\right|}^3}=\frac{\left|\right|u^{\&prime;}\left(S\right)\left|\right|}{h}=\frac{\sqrt{{\mathrm{\&alpha;}}^{\&prime;}{\left(S\right)}^2+{\mathrm{\&beta;}}^{\&prime;}{\left(S\right)}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&alpha;}\left(S\right)}}{h}$

(2-12)

By more than, can obtain Fu Leinie frame and curvature κ on the each point on the three-dimensional clothoid from the long variable S of curve.

(1-3) towards the generation of opposite three-dimensional clothoid

It is identical with certain three-dimensional clothoid and towards opposite three-dimensional clothoid consider to generate size, shape shown in Figure 6.

Suppose to have initial point P _sWith terminal point P _e, the parameter of circling round of three-dimensional clothoid has by h, a ₀, a ₁, a ₂, b ₀, b ₁, b ₂Deng 7 three-dimensional clothoid C that value is definite ₁At this moment, tangent line rotation angle α ₁, β ₁, (2-14) represent with following formula (2-13).

[numerical expression 27]

α ₁＝a ₀+a ₁S+a ₂S ² (2-13)

β ₁＝b ₀+b ₁S+b ₂S ² (2-14)

Generate size, shape is identical with this three-dimensional clothoid and towards opposite three-dimensional clothoid C ₂In, if initial point is set at P ' _s, with P ' _eAs terminal point, be respectively P ' _s=P _e, P ' _e=P _sAt first consider the long h of curve, if but consider it is that size is identical, curve length is in curve C ₁, C ₂In equate.Then, three-dimensional clothoid C ₂On tangent line t, if consider towards with the three-dimensional clothoid C of identical usually coordinate ₁On tangent line t opposite, then in curve C ₁Tangent line rotation angle α ₁, β ₁And curve C ₂Tangential direction rotation angle α ₂, β ₂Between, have down note relation.

[numerical expression 28]

α ₂(S)＝α ₁(1-S)+π (2-15)

β ₂(S)＝β ₁(1-S) (2-16)

If put these formulas in order, note formula (2-7) (2-18) is represented under using.

[numerical expression 29]

α ₂(S)＝(a ₀+a ₁+a ₂+π)-(a ₁+2a ₂)S+a ₂S ² (2-17)

β ₂(S)＝(b ₀+b ₁+b ₂)-(b ₁+2b ₂)S+b ₂S ² (2-18)

Owing to determine remaining parameter thus, so curve C ₂Circle round parameter h ', a ' ₀, a ' ₁, a ' ₂, b ' ₀, b ' ₁, b ' ₂, adopt curve C ₁Parameter, available formula (2-19) expression.

[numerical expression 30]

$\left\{\begin{array}{c}{p}_s^{\&prime;}=p_e\\ a_0^{\&prime;}=a_0+a_1+a_2+\mathrm{\&pi;}\\ a_1^{\&prime;}=-(a_1+2a_2)\\ a_2^{\&prime;}=a_2\\ {\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">b</figure-callout>}}_0^{\&prime;}={\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">b</figure-callout>}}_0+{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}}_1+b_2\\ {\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^{\&prime;}=-({\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}}_1+2b_2)\\ {\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}}_2^{\&prime;}=b_2\\ h^{\&prime;}=h\end{array}\right.----(2-19)$

If adopt this relational expression, can generate size, shape is identical and towards opposite three-dimensional clothoid.

(1-4) three-dimensional clothoid cuts apart

Suppose to have initial point P ₁With terminal point P ₂, the parameter of circling round of three-dimensional clothoid has by h, a ₀, a ₁, a ₂, b ₀, b ₁, b ₂Deng 7 three-dimensional clothoid C that value is definite ₀This moment, the long variable of the curve in the purposes was S=S as shown in Figure 7 _dSome P _mCut apart point of contact P ₁, P ₂Three-dimensional clothoid C ₀, consider to be divided into curve C below ₁And C ₂Method.

In the curve of considering to cut apart with a P ₁Curve C for initial point ₁If consider the long h of curve, learn curve C by the definition of three-dimensional clothoid ₁The long h of curve ₁Equal curve C ₀The long h of curve ₀S _dDoubly.In addition, if will represent curve C ₁On some the time curve C ₀The long variable of curve be made as S ₀, with curve C ₁The long variable of curve be made as S ₁, note relation under between them, setting up.

[numerical expression 31]

S ₁＝S _dS ₀ (2-20)

Just, learn, in curve C ₀Tangent line rotation angle α ₀, β ₀And curve C ₁Tangent line rotation angle α ₁, β ₁Between, have down note relation.

[numerical expression 32]

α ₁(S ₁)＝α ₀(S _dS ₀)

β ₁(S ₁)＝β ₀(S _dS ₀) (2-21)

If put these formula in order, with following formula (2-22) expression.

[numerical expression 33]

α ₁(S)＝a ₀+a ₁S _dS+a ₂S _d ²S ²

β ₁(S)＝b ₀+b ₁S _dS+b ₂S _d ²S ² (2-22)

Owing to determine tangential direction thus, so curve C ₁Circle round parameter h ', a ' ₀, a ' ₁, a ' ₂, b ' ₀, b ' ₁, b ' ₂, adopt curve C ₀Parameter, available formula (2-23) expression.

[numerical expression 34]

$\left\{\begin{array}{c}{a}_0^{\&prime;}=a_0\\ a_1^{\&prime;}=a_1{S}_d\\ a_2^{\&prime;}=a_2{S_d}^2\\ {\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">b</figure-callout>}}_0^{\&prime;}=b_0\\ {\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^{\&prime;}=b_1{\mathrm{<figure-callout\; id="2"\; label="S\; d\; b"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">S</figure-callout>}}_d& \\ b_{}^{}{}_2^{\&prime;}=b_2{S_d}^2\\ h^{\&prime;}=h{S}_d\end{array}\right.----(2-23)$

Below, consider with cut-point P _mCurve C as initial point ₂About curve C ₂, can be created on the size described in the 1-3 by combination, shape is identical and towards the method for opposite curve with in curve C ₁Generation in the method that adopts generate.

At first, with size, shape and curve C ₀Identical and be curve C ' towards opposite curve setting ₀On this curve, use P _m=C ' ₀(1-S _d) expression cut-point P _mHerein, if consider with some P _mCut apart curve C ' ₀, with the P that does in this curve of cutting apart ₂Curve C for initial point ' ₂, become and curve C ₂Size, shape is identical and towards opposite curve.Owing to utilize in method described in the 1-3 and curve C ₁Used method can formation curve C ' ₂So,, herein, if in addition to curve C ' ₂Employing can formation curve C in the method described in the 1-3 ₂

This curve C ₂The parameter h that circles round ", a " ₀, a " ₁, a " ₂, b " ₀, b " ₁, b " ₂, adopt curve C ₀Parameter, with following formula (2-24) expression.

[numerical expression 35]

$\left\{\begin{array}{c}{a}_0^{\&prime;\&prime;}=a_0+a_1{S}_d+a_2{S_d}^2\\ a_1^{\&prime;\&prime;}=(1-S_d)\{a_1+2a_2{S}_d\}\\ a_2^{\&prime;\&prime;}=a_2{(1-S_d)}^2\\ {\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">b</figure-callout>}}_0^{\&prime;\&prime;}={\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">b</figure-callout>}}_0+b_1{S}_d+b_2{S_d}^2\\ {\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^{\&prime;\&prime;}=(1-S_d)\{{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}}_1+2b_2{S}_d\}\\ {\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}}_2^{\&prime;\&prime;}=b_2{(1-S_d)}^2\\ h^{\&prime;\&prime;}=h(1-S_d)\end{array}\right.----(2-24)$

By more than, can enough three-dimensional clothoid C ₀On the curve variable be S=S _dSome P _m, curve segmentation is become curve C ₁And C ₂

(1-5) feature of three-dimensional clothoid

(a) continuity of curve

In a clothoid (with the clothoid of same parametric representation), because quadratic expression long by curve respectively or the long variable S of curve provides the inclination angle and the deflection angle of its tangential direction, so about the long variable S of curve, it is continuous can guaranteeing its normal direction that obtains of 1 subdifferential, reach its curvature that obtains of 2 subdifferentials.In other words, in a clothoid, normal direction and curvature are continuous.Therefore, can obtain smoothness, well-behaved curve.Even under the situation that links two clothoids,,, can make the curve of a smooth connection by selecting parameter for tangent line, normal, curvature on its joint reach continuously.Be referred to as the rondo line-group.

(b) applicability

Because the tangential direction that enough two angles of energy (inclination angle and deflection angle) are shared curve, so can make the three-dimensional curve that meets various conditions arbitrarily, can be used in various uses, the method for generation of the wide usefulness of the essential space curve of the design of industrial product can be provided.Follow along space curve under the situation that acceleration-deceleration moves at object, can carry out constraining force and change design stably.In addition, because can be, thereby can be used for a plurality of industrial fields of the aesthetic artistic conception curve design of requirement etc. effectively with respect to the variation of the long suitably design of curve curvature.

(c) with the conformability of geometrical curve

Geometrical curves such as Straight Line and Arc spiral curve can place 0 by the several of the parameter of will circling round, or set specific funtcional relationship and make between Several Parameters.These curves are a kind of of clothoid, can adopt the form performance of circling round.

In addition, owing to, can make two dimension and circle round, so the two dimension resource of circling round and having obtained just before can using by any among α or the β placed 0 usually.

Just,, comprise that the two dimension of having known circles round, can also show other curve such as circular arc or straight line by suitable setting α or β.Owing to for such other curve, can adopt the three-dimensional clothoid formula of same form, therefore can simplify the calculating formality.

(d) formedness of Tui Ceing

In interpolations in the past such as spline interpolation, when making the free curve formulation, many difficult separately its whole forms or local form, but in three-dimensional is circled round, by imagining inclination angle and deflection angle separately, can be than being easier to hold overall image.

In addition, at the end midway as clothoid performance, the value of line length tangential direction curvature etc. is known, need recomputate unlike in the past interpolation.Just, corresponding with the parameter S of curve, by formula (1-7), (2-10) and (2-12), directly obtain the tangent line of curve or normal, curvature.

(e) easiness of motion control

The master variable of curve is length s or standardized length S, and the curve's equation formula uses the natural equation with respect to this length to provide.Therefore,, kinetic characteristics such as acceleration-deceleration can be provided arbitrarily,, the high speed of operation can be sought to process by adopting the used good curve movement of characteristic such as cam in the past by determining length s as the function of time t.Because the value that can be used as in the in esse cartesian space provides length s, obtains speed, acceleration with respect to tangential direction,, synthesize by each value that provides so do not need picture interpolation in the past.In addition,, thereby also obtain the centrifugal acceleration when moving easily, can meet the control of movement locus because the calculating of curvature is easy.

2. adopt the interpolation of three-dimensional clothoid

(2-1) mathematic condition of Liu Chang connection

In 1 three-dimensional clothoid, the performance of the shape of curve has boundary.Herein, be fundamental purpose with the motion control of the instrument that utilizes Numerical Control, many connect three-dimensional clothoid (three-dimensional circle round line segment), by the shape of these many three-dimensional clothoids design industrial products.Below will adopt the interpolation of three-dimensional clothoid to be called the three-dimensional interpolation of circling round.Below, will be called three-dimensional clothoid by the group of curves integral body that interpolation generates, the unit curve that constitutes it is called the three-dimensional line segment that circles round.

Connect 2 three-dimensionals line segment that circles round glibly at its end points, being defined as is continuous connection end point position, tangent line and curvature.Adopt above-mentioned definition, by this condition of following narration.The initial locative continuity of 3 formulas, next 2 formulas are represented the continuity of tangent line, and next 1 formula is represented the unanimity of normal, and last formula is represented the continuity of curvature.

[numerical expression 36]

Px _i(1)＝Px _i+1(0)

Py _i(1)＝Py _i+1(0)

Pz _i(1)＝Pz _i+1(0)

α _i(1)＝α _i+1(0)

β _i(1)＝β _i+1(0) (3-1)

tanγ _i(1)＝tanγ _i+1(0)

κ _i(1)＝κ _i+1(0)

This be satisfy tangent line vector and normal vector continuously, curvature and α, β be condition for continuous at tie point, condition is too strict sometimes.Therefore, also can satisfy condition by change condition shown below singlely.

[numerical expression 37]

Px _i(1)＝Px _i+1(0)

Py _i(1)＝Py _i+1(0)

Pz _i(1)＝Pz _i+1(0)

cos[α _i(1)-α _i+1(0)]＝1

cos[β _i(1)-β _i+1(0)]＝1 (3-2)

tanγ _i(1)＝tanγ _i+1(0)

κ _i(1)＝κ _i+1(0)

Herein, in addition,

[numerical expression 38]

cos[α _i(1)-α _i+1(0)]＝1

Also take into account if will go up relation of plane,

[numerical expression 39]

tanγ _i(1)＝tanγ _i+1(0)

Used the conditional substitutions of note down.

[numerical expression 40]

tanγ _i(1)＝tanγ _i+1(0)

$\frac{{{\mathrm{\&alpha;}}^{\&prime;}}_i\left(1\right)}{{{\mathrm{\&beta;}}^{\&prime;}}_i\left(1\right)\cos {\mathrm{\&alpha;}}_i\left(1\right)}=\frac{{{\mathrm{\&alpha;}}^{\&prime;}}_{i+1}\left(0\right)}{{{\mathrm{\&beta;}}^{\&prime;}}_{i+1}\left(0\right)\cos {\mathrm{\&alpha;}}_{i+1}\left(0\right)}$

∵α′ _i(1)β′ _i+1(0)＝α′ _i+1(0)β′ _i(1)

The result learns, if satisfy the condition of note down, can achieve the goal.

[numerical expression 41]

Px _i(1)＝Px _i+1(0)

Py _i(1)＝Py _i+1(0)

Pz _i(1)＝Pz _i+1(0)

cos[α _i(1)-α _i+1(0)]＝1

cos[β _i(1)-β _i+1(0)]＝1 (3-3)

α _i′(1)β _i+1＇(0)＝α _i+1′(0)β _i′(1)

κ _i(1)＝κ _i+1(0)

In formula (3-3), the initial locative continuity of 3 formulas, next 2 formulas are represented the continuity of tangent line, and next 1 formula is represented the unanimity of normal, and last formula is represented the continuity of curvature.Carry out G ²Continuous interpolation needs 2 three-dimensional clothoids to satisfy 7 conditionals of formula (3-3) at its end points.

About G ²(G is the prefix of Geometry) replenishes continuously.Fig. 8 represents G ²The condition of continuous interpolation.

So-called G ⁰Refer to 2 three-dimensional clothoids continuously at its endpoint location unanimity, so-called G ¹Refer to the tangential direction unanimity continuously, so-called G ²Refer to contact plane (normal) and curvature unanimity continuously.The used C of contrast SPL in following table 1 ⁰～C ²The used G of clothoid continuous and of the present invention ⁰～G ²Continuously.

Table 1

C 0: the position	G 0: the position
		C 1: a differential coefficient	G 1: tangential direction
C 2: the second differential coefficient	G 2: contact plane (normal), curvature

When considering the continuity of 2 three-dimensional clothoids, along with reaching C ⁰→ C ¹→ C ², G ⁰→ G ¹→ G ², it is tight that the interpolation condition becomes.At C ¹Need the size and the direction of tangent line all consistent continuously, but at G ¹Can have only the tangential direction unanimity continuously.When connecting tangent line glibly, preferably use G with 2 three-dimensional clothoids ¹Make conditional continuously.As SPL, if use C ¹Make conditional continuously, make in the condition of the same size of unallied tangent line geometrically, so condition is tight excessively owing to increase.If use G ¹Make conditional continuously, have the advantage that freely to set the size of a differential coefficient.

At G ²Make contact plane (normal) unanimity continuously.So-called contact plane as shown in Figure 9, refers to planar S 1, S2 that curve C is contained in the part.It is continuous that Fig. 9 is illustrated in a P tangential direction, but the discontinuous example of contact plane S1, S2.When considering the continuity of three-dimensional curve, what must consider after the unanimity of tangential direction is the unanimity of contact plane.When words curvature, do not mean that contact plane is inconsistent, need after making the contact plane unanimity, make the curvature unanimity.Make coordinate, tangential direction, contact plane (normal) and curvature unanimity with 2 three-dimensional curves, can reach and satisfy G ²Condition for continuous.

(2-2) concrete computation sequence

Have following 2 kinds of computation sequences.

(a) provide parameter h, α, the β of curve, 1 three-dimensional clothoid takes place,,, determine the parameter of next three-dimensional clothoid to satisfy the mode of formula (3-3) at its end points.So, the smooth one by one three-dimensional clothoid that connects can take place.According to this computation sequence, calculate parameter of curve easily, be referred to as along separating.According to this mode, the curve of multiple shape can take place easily, but the tie point that can not clear and definite assignment curve passes through.

(b) can become the mode of the tie point of curve with preassigned point group, connect three-dimensional clothoid.Make short clothoid (section of circling round) in each interval of each point range that provides arbitrarily discretely herein.In such cases, determine that in the mode that satisfies formula (3-3) computation sequence of parameter of curve is more complicated than (a), for repeating to bring together calculating.Owing to determine parameter of curve on the contrary from condition of contact, this computation sequence is called contrary separating.

Separate about the contrary of above-mentioned (b), at length narrate computing method.The computational problem that solves is by following quilt formulism.

Unknown parameter: parameter of curve

Constraint condition: formula (3-3) or one portion

Problem as requested, the quantity of variation constraint condition can be used as the parameter of curve that unknown parameter is set the quantity that conforms to it.For example, can freely make a part of parameter of curve work under the successional situation of curvature not requiring.Perhaps, in curvature continuously and specify under the situation of tangential direction, need increase corresponding unknown parameter of curve by cutting apart the quantity that increases the used three-dimensional clothoid of interpolation.

Above-mentionedly to repeat to bring together calculation stability and bring together in order to make, need on calculating, work hard.For fear of dispersing of calculating, accelerate to bring together, about unknown parameter, effective method is to set better initial value.Therefore, effective method is, interpolation curve constraint conditions such as tie point, more single that provides takes place to satisfy, and line transect curve etc. is for example calculated the parameter of curve of three-dimensional clothoid from its curve shape, as the initial value that repeats to bring together calculating.

Perhaps, the gas that differs satisfies the constraint condition that should satisfy, and increases the mode of conditional successively, also is effective as the stable method that obtains separating.For example, the order that curve is taken place is divided into three following STEP, carries out successively.As the 1st STEP after the mode interpolation consistent with tangential direction with positional information, as the 2nd STEP so that the mode of normal direction unanimity is carried out interpolation, in the mode interpolation of the 3rd STEP with the curvature unanimity.Figure 10 represents the concise and to the point flow process of this method.The necessary three-dimensional clothoid formula and the definition of tangent line, normal or curvature thereof have been shown.

(2-3) embodiment of the interpolation of the three-dimensional clothoid of employing

(a) flow process of interpolation

Describe in detail to adopt a three-dimensional clothoid embodiment of the method between the point range that provides of interpolation glibly.

As the circle round basic flow process of interpolation of three-dimensional, circle round each parameter of line segment as unknown number with the three-dimensional between the point that links the interpolation object, the point by the interpolation object closely, and obtain the satisfied G that becomes with Newton-Raphson method ²Separating of condition for continuous, formation curve.Figure 11 is a diagram of concluding the summary of this flow process.So-called G ²Continuously, refer to 2 three-dimensional clothoids at its end points, position, tangential direction, normal direction and curvature unanimity.

(b) G ²The condition of continuous interpolation

In three-dimensional is circled round interpolation,, and become G about the point by the interpolation object closely ²Condition for continuous is considered concrete condition.

Now, have 3 some P simply ₁={ Px ₁, Py ₁, Pz ₁, P ₂={ Px ₂, Py ₂, Pz ₂And P ₃={ Px ₃, Py ₃, Pz ₃, consider with three-dimensional this point of line segment interpolation that circles round.Figure 12 represents a P ₁, P ₂And P ₃The three-dimensional interpolation of circling round.If with point of contact P ₁, P ₂Between curve setting be curve C ₁, with point of contact P ₂, P ₃Between curve setting be curve C ₂, in such cases, unknown number is a curve C ₁Parameter a0 ₁, a1 ₁, a2 ₁, b0 ₁, b1 ₁, b2 ₁, h ₁, curve C ₂Parameter a0 ₂, a1 ₂, a2 ₂, b0 ₂, b1 ₂, b2 ₂, h ₂14.In addition, the subscript of the literal that occurs in explanation later is corresponding with the subscript of each curve.

Consider closely point below, and reach G by the interpolation object ₂Condition for continuous.At first, the condition of the point by the interpolation object closely at initial point, if consider from the definition of three-dimensional clothoid, owing to when providing initial point, must reach, so there is not the interpolation condition.Then at tie point P ₁, aspect the position, set up 3, aspect the tangent line vector, set up 2, aspect the size and Orientation of curvature condition for continuous formula, set up 2, add up to and set up 7.In addition about terminal point, at a P ₂Be 3 aspect the position.Add up to 10 by the above conditional that draws.But, after this manner with respect to 14 of unknown numbers, because only there are 10 in conditional, so can not obtain separating of unknown number.Therefore, in this research, provide the tangent line vector of two-end-point,, increase each two condition, the number of conditional and unknown number is equated for two-end-point.In addition, if determine tangential direction, owing to can obtain a0 from its definition at initial point ₁, b0 ₁So, can not handle as unknown number.Below, consider each condition.

At first, if consider the condition of position, set up 3 formulas (4-1), (4-2), (4-3) of note down.

(below, regulation natural number i＜3.)

[numerical expression 42]

$P{x}_i+h_i{\&Integral;}_0^1\cos ({a0}_i+a{l}_{i}S+a{2}_j{S}^2)\cos (b{0}_i+b{l}_{i}S+b{2}_i{S}^2)\mathrm{dS}-P{x}_{i+1}=0$

(4-1)

$P{y}_i+h_i{\&Integral;}_0^1\cos (a{0}_i+a{l}_{i}S+a{2}_j{S}^2)\sin (b{0}_i+b{l}_{i}S+b{2}_i{S}^2)\mathrm{dS}-P{y}_{i+1}=0$

(4-2)

$P{z}_i+h_i{\&Integral;}_0^1(-\sin ({a0}_i+a{1}_{i}S+a{2}_j{S}^2))\mathrm{dS}-P{z}_{i+1}=0$

(4-3)

Then, if consider tangential direction, then set up 2 formulas (4-4), (4-5).

[numerical expression 43]

cos(a0 _i+a1 _i+a2 _i-a0 _i+1)＝1 (4-4)

cos(b0 _i+b1 _i+b2 _i-b0 _i+1)＝1 (4-5)

About the size of curvature κ, set up next formula (4-6)

[numerical expression 44]

κ _i(1)-κ _i+1(0)＝0 (4-6)

Consider normal direction vector n at last.The normal vector n of three-dimensional clothoid is represented by formula (2-10).

Herein, with the tangent line vector u of three-dimensional clothoid determine equally, adopting circles round considers normal vector n.For initial stage tangential direction (1,0,0), regulation adopts constant γ, represents the initial stage normal direction with (0, cos γ ,-sin γ) i.If with tangent line it is circled round, normal n is suc as formula shown in (4-7).

[numerical expression 45]

$n\left(S\right)=\left[\begin{array}{ccc}\cos \mathrm{\&beta;}\left(S\right)& -\sin \mathrm{\&beta;}\left(S\right)& 0\\ \sin \mathrm{\&beta;}\left(S\right)& \cos \mathrm{\&beta;}\left(S\right)& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos \mathrm{\&alpha;}\left(S\right)& 0& \sin \mathrm{\&alpha;}\left(S\right)\\ 0& 1& 0\\ -\sin \mathrm{\&alpha;}\left(S\right)& 0& \cos \mathrm{\&alpha;}\left(S\right)\end{array}\right]\left\{\begin{array}{c}0\\ \cos \mathrm{\&gamma;}\\ -\sin \mathrm{\&gamma;}\end{array}\right\}$

$=\left\{\begin{array}{c}-\sin \mathrm{\&gamma;}\cos \mathrm{\&beta;}\left(S\right)\sin \mathrm{\&alpha;}\left(S\right)-\cos \mathrm{\&gamma;}\sin \mathrm{\&beta;}\left(S\right)\\ -\sin \mathrm{\&gamma;}\sin \mathrm{\&beta;}\left(S\right)\sin \mathrm{\&alpha;}\left(S\right)+\cos \mathrm{\&gamma;}\cos \mathrm{\&beta;}\left(S\right)\\ -\sin \mathrm{\&gamma;}\cos \mathrm{\&alpha;}\left(S\right)\end{array}\right\}$

(4-7)

If comparison expression (2-10), (4-7), sin γ, cos γ are corresponding with formula (4-8) as can be known.

[numerical expression 46]

$\sin \mathrm{\&gamma;}=\frac{{\mathrm{\&alpha;}}^{\&prime;}\left(S\right)}{\sqrt{{\mathrm{\&alpha;}}^{\&prime;}{\left(S\right)}^2+{\mathrm{\&beta;}}^{\&prime;}{\left(S\right)}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&alpha;}\left(S\right)}}$

$\cos \mathrm{\&gamma;}=\frac{{\mathrm{\&beta;}}^{\&prime;}\left(S\right)\cos \mathrm{\&alpha;}\left(S\right)}{\sqrt{{\mathrm{\&alpha;}}^{\&prime;}{\left(S\right)}^2+{\mathrm{\&beta;}}^{\&prime;}{\left(S\right)}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&alpha;}\left(S\right)}}---(4-8)$

That is, (4-8) learns by formula, and the normal that reach the tie point in three-dimensional is circled round interpolation is continuous, as long as tan γ is continuous passable.

[numerical expression 47]

$\mathrm{tan\&gamma;}=\frac{{\mathrm{\&alpha;}}^{\&prime;}\left(S\right)}{{\mathrm{\&beta;}}^{\&prime;}\left(S\right)\cos \mathrm{\&alpha;}\left(S\right)}---(4-9)$

That is, learning the normal condition for continuous, is formula (4-10).

[numerical expression 48]

tanγ _i(1)＝tanγ _i+1(0) (4-10)

Herein, in addition, as

[numerical expression 49]

cos[α _i(1)-α _i+1(0)]＝1 (4-11)

Also in considering, conditional (4-10), the conditional displacement of available note down.That is, the normal condition for continuous is formula (4-12).

[numerical expression 50]

α′ _i(1)β′ _i+1(0)＝α′ _i+1(0)β′ _i(1) (4-12)

In sum, learn, the point by the interpolation object closely, and become G ₂Condition for continuous, at tangent point suc as formula (4-13).In addition, even at the initial point terminal point, also can select these several conditions wherein.

[numerical expression 51]

Px _i(1)＝Px _i+1(0)

Py _i(1)＝Py _i+1(0)

Pz _i(1)＝Pz _i+1(0)

cos[α _i(1)-α _i+1(0)]＝1

cos[β _i(1)-β _i+1(0)]＝1

α _i′(1)β _i+1′(0)＝α _i+1′(0)β _i′(1)

κ _i(1)＝κ _i+1(0) (4-13)

Learn by above, for unknown number a1 ₁, a2 ₁, b1 ₁, b2 ₁, h ₁, a0 ₂, a1 ₂, a2 ₂, b0 ₂, b1 ₂, b2 ₂And h ₂Deng 12, conditional is set up 12 that remember down.(some P ₃The tangential direction rotation angle be set at α ₃, β ₃)

[numerical expression 52]

Px ₁(1)＝Px ₂(0)

Py ₁(1)＝Py ₂(0)

Pz ₁(1)＝Pz ₂(0)

cos[α ₁(1)-α ₂(0)]＝1

cos[β ₁(1)-β ₂(0)]＝1

α′ ₁(1)β′ ₂(0)＝α′ ₂(0)β′ ₁(1)

κ ₁(1)＝κ ₂(0)

Px ₂(1)＝Px ₃(0)

Py ₂(1)＝Py ₃(0)

Pz ₂(1)＝Pz ₃(0)

cos[α ₂(1)-α ₃]＝1

cos[β ₂(1)-β ₃]＝1 (4-14)

So, owing to set up 12 formulas for 12 unknown numbers, so can find the solution.This available Newton-Raphson method is explained, obtained and separate.

In addition, generally when considering n point range of interpolation, conditional is just passable as long as above-mentioned natural number i is expanded as i＜n.Be the problem of the quantity of unknown number and conditional then.

For example, when having n-1 point range, regard as and set up N unknown number and N relational expression.Herein, if hypothesis increases by 1 point again, unknown number just increases the three-dimensional line segment P that circles round _N-1, P _nThe parameter a0 that circles round _n, a1 _n, a2 _n, b0 _n, b1 _n, b2 _nAnd h _nDeng 7.On the one hand, conditional is because tie point increases by 1, so at a P _N-1, aspect the position, increase by 3, aspect the tangent line vector, increase by 2, aspect the size and Orientation of curvature condition for continuous formula, increase by 2, adding up to increases by 7.

Owing to learn that when n=3 unknown number, relational expression all are 12,, also be 7 (n-2)+5 to the formula of this establishment so in n 〉=3 o'clock, unknown number is 7 (n-2)+5.So, because the number of unknown number and relevant with it condition equates, so when the individual free point range of n, also can same method find the solution with 3 the time.As solving method, adopt and utilize between unknown number and conditional, to become the newton-pressgang Shen method of the relation of vertical (4-15), (4-16) to find the solution.(condition is made as F, unknown number is made as u, the error Jacobi matrix is made as j.)

[numerical expression 53]

ΔF＝[J]Δu (4-15)

Δu＝[J] ^-1ΔF (4-16)

Learn by above, also can carry out closely point for n point range, and reach G by the interpolation object ²The continuous three-dimensional interpolation of circling round.

(C) initial value determines

In newton-pressgang Shen method, when the exploration that begins to separate, need provide suitable initial value.Initial value how to provide can, but an example of only narrating this initial value herein provides mode.

In interpolation, at first, need determine the initial value of each unknown number from point range, but in this research, be created on and have 4 summits between the interpolation object point range of single shape of polygon Q of 3D Discrete Clothoid Splines of Li etc., calculate its initial value, determine from this polygon Q.3DDiscrete Clothoid Splines closely by the interpolation object-point, has curvature with respect to the character from the displacement smooth change of initial point.In this manual, be used for the circle round initial value of interpolation of three-dimensional, by making as the polygon Q of the 3D Discrete Clothoid Splines of the r=4 of Figure 13, definite by calculating from here.

Below, supplementary notes 3D Discrete Clothoid Splines.At first as shown in figure 14, make the polygon P that classifies the summit with the point of interpolation object as, between each summit of P, insert each identical several r new summits, be made as the polygon Q of P  Q.,, under the situation that polygon Q closes, have rn summit herein, under the situation that polygon Q opens, have r (n-1)+1 summit if be n with the vertex of P.Regulation as the consecutive number from initial point, is represented each summit with qi with subscript later on.In addition, on each summit, determine to determine to have the vector k of curvature κ as size from normal vector b as direction.

At this moment, reach equidistant formula (4-17) each other with satisfying the summit of note down, during condition that curvature displacement the most approaching and from initial point is directly proportional (when making the function minimization of formula (4-18)) polygon Q, be called 3D Discrete Clothoid Splines.

[numerical expression 54]

|q _i-1q _i|＝|q _i+1q _i|，(q _iP) (4-17)

$\underset{i=1}{\overset{r-1}{\mathrm{\&Sigma;}}}{\left|\right|{\mathrm{\&Delta;}}^2{k}_{\mathrm{ir}+1}\left|\right|}^2,$ i＝{0...n-1}，Δ ²k _i＝k _i-1-2k _i+k _i+1

(4-18)

In 3D Discrete Clothoid Splines, obtained the Fu Leinie frame on each summit.Therefore, obtain parameter a from its unit tangent direction vector t ₀, b ₀T is known when obtaining polygon Q for this tangential direction vector, and the formula of the tangent line by this t and three-dimensional clothoid is obtained tangential direction rotation angle α, the β on the summit of polygon Q.Obtain a of each curve thus ₀, b ₀Initial value.In addition, on the three-dimensional that begins from initial point is circled round line segment, provide this value.

[numerical expression 55]

$u=\left\{\begin{array}{c}\cos \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ \sin \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ -\sin \mathrm{\&alpha;}\end{array}\right\}---(4-19)$

Herein, about 3D Discrete Clothoid Splines, if consider equidistant arrangement summit, at the some q of Figure 13 _4i+1, can be similar to the long variable S of curve is 1/4.Equally at a q _{4 (i+1)-1}, can be similar to the long variable S of curve is 3/4.If lump together with the formula of the α of three-dimensional clothoid and to consider these, set up following formula (4-20).

[numerical expression 56]

$\left\{\begin{array}{c}{a0}_{4i}+\frac{1}{4}a{1}_{4i}+{\left(\frac{1}{4}\right)}^{2}a{2}_{4i}=a{0}_{4i+1}\\ a{0}_{4i}+\frac{3}{4}a{1}_{4i}+{\left(\frac{3}{4}\right)}^2{a2}_{4i}=a{0}_{4(i+1)-1}\end{array}\right.---(4-20)$

It is a1 that this formula becomes unknown number _4iAnd a2 _4iTwo-dimentional simultaneous equations, it is found the solution, as parameter a ₁, a ₂Initial value.Equally also can determine parameter b ₁, b ₂Initial value.

Remaining unknown number is the long h of curve, but can be calculated by the formula of three-dimensional rondo curvature of a curve about its initial value.Three-dimensional rondo curvature of a curve, available formula (4-21) expression.

[numerical expression 57]

$\mathrm{\&kappa;}=\frac{\sqrt{{\mathrm{\&alpha;}}^{\&prime;2}+{\mathrm{\&beta;}}^{\&prime;2}{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&alpha;}}}{h}---(4-21)$

If change this formula, become formula (4-22), can determine the initial value of h.

[numerical expression 58]

${\mathrm{<figure-callout\; id="4i"\; label="h"\; filenames="A20058000605301142.png"\; state="\{\{state\}\}">h</figure-callout>}}_{4i}=\frac{\sqrt{{({a1}_{4i}+2a{2}_{4i})}^2+{(\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}{1}_{4i}+2\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}{2}_{4i})}^2{\cos }^2(a{0}_{4i}+a{1}_{4i}+a{2}_{4i})}}{k_{4(i+1)}}$

(4-22)

With above method, can determine initial value for 7 three-dimensionals parameter of circling round.Adopt this initial value of determining, that narrates in (b) reaches G ²Under the condition for continuous, obtain the approximate value of the parameter of each curve with newton-pressgang Shen method.Generate the three-dimensional line segment that circles round from the parameter that obtains thus, with between three-dimensional clothoid interpolation point range.

(d) interpolation example

As the example of reality with above-described method interpolation point range, circle round interpolation (0.0,0.0,0.0), (2.0,2.0,2.0), (4.0,0.0,1.0) and (5.0,0.0,2.0) this example of 4 of three-dimensional for example.The skeleton view of the three-dimensional clothoid that generates by interpolation shown in Figure 15.Solid line among Figure 15 is three-dimensional clothoid, and dotted line, a dot-and-dash line, two dashdotted straight lines are that the size of getting on the each point on the curve is log (radius-of-curvature+natural logarithm e), and getting direction is the radius-of-curvature changing pattern of normal direction.

In addition, the parameter of each curve shown in the table 2, this appearance 3 are illustrated in coordinate, tangent line, the normal of each tangent point, the deflection of curvature.Find out from these tables, become G at each tangent point ²The continuous three-dimensional clothoid.In addition, Figure 16 gets displacement from initial point at transverse axis, gets the curved transition curve map of curvature at the longitudinal axis.

Table 2

The circle round pattern of line segment of each three-dimensional

Curve 1 (radius-of-curvature changing pattern dotted line)
	α＝-0.657549-1.05303S+1.84584S 2 β＝1.03297+1.29172S-2.55118S 2 h＝3.82679 P 0＝(0.0、0.0、0.0)
Curve 2 (radius-of-curvature changing pattern one dot-and-dash line)
	α＝0.135559+2.18537S-2.69871S 2 β＝-0.226655-3.15603S+3.03298S 2 h＝3.16932 P 0＝(2.0、2.0、2.0)
Curve 3 (radius-of-curvature changing pattern two dot-and-dash lines)
	α＝-0.377569-1.45922S+0.984945S 2 β＝-0.349942+1.32198S-0.873267S 2

h＝1.43987 P 0＝(4.0、0.0、1.0)

Table 3

Coordinate, tangent line, normal, the skew curve 1 of curvature and the tie point of curve 2 at each tangent point
	Coord： (1.16×10 -5，2.00×10 -6，3.82×10 -6) Tvector： (7.59×10 -5，1.50×10 -5，2.95×10 -4) Nvector： (2.93×10 -4，9.19×10 -5，-7.57×10 -6) Curvature：3.06×10 -7
The tie point of curve 2 and curve 3
	Coord： (-4.33×10 -6，-1.64×10 -6，1.11×10 -5) Tvector： (2.06×10 -6，2.33×10 -4，1.97×10 -4) Nvector： (3.30×10 -4，1.19×10 -5，-3.23×10 -5) Curvature：5.96×10 -6

(2-4) the respectively G of the control of value that considers at two ends ²The continuous three-dimensional interpolation of circling round

(a) interpolation condition and unknown number

As described in (2-3), under the situation that curve is opened, when the point of interpolation object has n, with n-1 the curve three-dimensional interpolation point range of circling round.If strictly by each point, about each three-dimensional line segment that circles round, because unknown number has a ₀, a ₁, a ₂, b ₀, b ₁, b ₂, 7 of h etc., so unknown number integral body is that 7 (n-1) are individual.On the other hand,, all there are 3 of coordinate on coordinate, tangent line, normal, curvature each 7 and the terminal point, so all be 7 (n-2)+3 owing to have the tie point of n-2 about conditional.In the method for (2-3), by it being provided the tangent line vector on the initial point terminal point, increase by 4 conditions, make the number of conditional and unknown number relative.

Herein, if the tangent line normal curvature on the control initial point terminal point, and to reach G ²Compare when continuous mode interpolation, the tangent line at condition and control two ends,, respectively increasing by 2 aspect the normal curvature, add up to increase by 4 in addition at the initial point terminal point.So conditional all reaches 7n-3.In such cases, because the number of unknown number lacks than condition, so can not find the solution with newton-pressgang Shen method.Therefore, need how to increase unknown number.

Therefore,, the number of unknown number and conditional is equated herein by inserting the interpolation object-point again.For example,, just insert 2 new points, as 2 in the coordinate of unknown number processing each point if a side of 4 unknown numbers is many.

In such cases, because tie point increases by 2, so increase each 7 14 of coordinate, tangent line, normal, curvature for each tie point condition.On the other hand, the line segment because 2 three-dimensionals of unknown number increase are circled round is so increase a ₀, a ₁, a ₂, b ₀, b ₁, b ₂, h each 14 of total of 7.Because the number of the contained point of point range this moment is n+2, so if wholely consider that it is individual that unknown number reaches 7 (n+1), conditional reaches 7 (n+1)+4.In addition, suppose that unknown number just increases by 4 as 2 in the coordinate of the new point that inserts of unknown number processing herein.So unknown number, conditional all reach 7 (n+2)-3, can obtain separating of unknown number.So, by inserting new point, can carry out the each point by providing closely, G ²The interpolation of the continuous and tangent line normal curvature of having controlled two-end-point.

In addition, consider general situation.When interpolation n point range, consider the number of the point that when m project of two-end-point control, inserts and at this number as the coordinate of unknown number processing.The front was was also recorded and narrated, but when curve is opened, with n-1 curve interpolating point range.If closely by each point, owing to the line segment unknown number that circles round for each three-dimensional has a ₀, a ₁, a ₂, b ₀, b ₁, b ₂, 7 of h, so unknown number integral body has 7 (n-1) individual.On the one hand, about conditional, all have 3 on coordinate on coordinate, tangent line, normal, curvature each 7 and the terminal point owing to have the tie point of n-2, so all be 7 (n-2)+3, conditional is few, is 4.Just, in project that two-end-point will be controlled more than 4.Below, the natural number, the k that are described in m in the explanation and are more than 4 are the natural number more than 2, the method that the number of conditional and unknown number is equated.

(i) during m=2k

When lumping together m=2k project of control at two ends, unknown number integral body has 7 (n-1) individual, and conditional integral body is 7 (n-1)-4+2k.At this moment, Guo Sheng conditional is 2k-4.Now, if consider to insert again k-2 point, line segment increases the k-2 root because three-dimensional is circled round, and tie point increases k-2, so unknown number integral body has 7 (n+k-3) individual, conditional integral body is 7 (n+k-3)-4+2k., suppose that in addition unknown number integral body is that 7 (n+k-3)+2 (k-2) are individual as 2 (for example x, y) in the seat target value of the new each point that inserts of unknown number processing herein, conditional integral body is that 7 (n+k-3)+2 (k-2) are individual, and the number of unknown number and conditional equates.

(ii) during m=2k+1

When lumping together m=2k+1 project of control at two ends, unknown number integral body is that 7 (n-1) are individual, and conditional integral body is 7 (n-1)+2k-3.At this moment, Guo Sheng conditional is 2k-3.Now, if consider to insert again k-1 point, line segment increases the k-1 root because three-dimensional is circled round, and tie point increases k-1, so unknown number integral body is that 7 (n+k-2) are individual, conditional integral body is 7 (n+k-2)-3+2k., suppose that in addition unknown number integral body has 7 (n+k-2)+2 (k-2) individual as 2 (for example x, y) in the seat target value of the new each point that inserts of unknown number processing herein, conditional integral body is 7 (n+k-2)+2k-3, many 1 of the number of conditional.Therefore, on 1 point in the point that when m=2k+1, insert, only handle 1 that sits in the target value as unknown number.So, unknown number integral body is that 7 (n+k-2)+2 (k-2) are individual, and conditional integral body is that 7 (n+k-2)+2 (k-2) are individual, and the number of unknown number and conditional equates.

Method as previously discussed, even contrasting by number with the condition of appending, the number that becomes unknown number in the coordinate of the point that adjustment is inserted, under all situations during for example tangent line swing angle α beyond control tangent line, normal, the curvature etc., also can make the number of unknown number and conditional relative, can control each value of two-end-point in theory.In addition, about the number of control project and unknown number, conditional, table 4 is listed the number of conclusion

Table 4

In the interpolation that n is ordered at the number of the control project at two ends and unknown number, conditional

Want the item number controlled	The conditional that increases	The number of the point that inserts	Be added in 1 number of going up the coordinate of handling as unknown number	The unknown number that increases
					4	0	0	0	0
5	1	1	1	1
					6	2	1	2	2
7	3	2	1 point: 21 points: 1	3
					
2k	2k-4	k-2	K-2 point: 2	2k-4
					2k+1	2k-3	k-1	K-2 point: 21 some t:1	2k-3

^*The natural number that k:2 is above

(b) method

The interpolation that employing is circled round in the three-dimensional of each value of initial point terminal point control as Figure 17 and shown in Figure 180, is undertaken by following flow process.

Step1) only adopt in the condition that will control 4, carry out closely by the interpolation object-point, and G ²Continuous interpolation, formation curve.

Step2) on the curve that generates, insert new point, the number of regularization condition formula and unknown number.

Step3) with the parameter of curve of Step1 as initial value, obtain the approximate value of the parameter of each curve that satisfies the purpose condition with newton-pressgang Shen method.

Below, each Step is remarked additionally.At first in Step1,, just can adopt the method formation curve of (2-3) as long as control tangential direction.In addition, even under the situation of not controlling tangential direction, the initial value during as the parameter of obtaining this curve also adopts the initial value identical with the method for (2-3).

Then, in Step2, insert new point, carry out the adjustment of the number of conditional and unknown number.At this moment, the new point that inserts, between each interpolation object-point as much as possible below 1.In addition, as the point that inserts, insert with linking the circle round point of centre of line segment of the mutual three-dimensional of interpolation object in the Step1 generation.In addition, the point of insertion will insert successively from two ends.Just, the initial point that the inserts point that is initial point and its adjacency between and the point of terminal point and its adjacency between.

Be about Step3 at last, but need redefine the initial value that is used for the newton-pressgang Shen method of carrying out at Step3.Therefore, cut apart curve, from determining respectively being worth of curve that generates for inserting the curve of newly putting, adopting by (1-4) described method of cutting apart three-dimensional clothoid.For the curve of insertion point not, directly adopt the value of the curve that generates at Step1.More than, determined the initial value of each parameter of the curve in Step3.Adopt this initial value, from generating three-dimensional clothoid, with satisfying between the three-dimensional clothoid interpolation point range of purpose condition with the parameter that newton-pressgang Shen method obtains.

(C) interpolation example

Illustrate in the mode of reality, carry out the circle round example of interpolation of three-dimensional with the tangent line at the condition of table 5 control two ends, normal, curvature.Point to the interpolation object that should pass through is closely shared consecutive number, forms P ₁, P ₂And P ₃

The condition of table 5 interpolation object each point and initial point terminal point

	Coordinate	The unit tangent vector	The principal normal vector	Curvature
					P 1	(0，0，0)	(Cos(θ)，Sin(θ)，0)	(-Sin(θ)，Cos(θ)、0)	0.2
P 2	(4，-4，-4)	-	-	-
					P 3	(8，-4，-5)	(1，0，0)	(0，-1，0)	0.2

^*θ＝-(π/6)

Figure 19 represents the actual with this understanding result who carries out interpolation.The three-dimensional clothoid of the curve representation of solid line, dotted line one dot-and-dash line two dot-and-dash lines three dot-and-dash lines represent that the radius-of-curvature of each curve changes.In addition, Figure 20 is expression from the curve map of the relation of the displacement of the initial point of each curve corresponding with the line kind of the curve of Figure 19 and curvature.By finding out among the figure, the curve of generation satisfies the given condition of table 6.

The tangent line of the initial point terminal point of the value that table 6 provides and the curve of generation, normal, curvature poor

		Coordinate	The unit tangent vector	The principal normal vector	Curvature
						P 1	The value of providing	{0.0，0.0，0.0}	{0.8660，-0.5，0.0}	{0.5，0.8660，0.0}	0.20
Value according to formation curve	{0.0，0.0，0.0}	{0.8660，-0.5，0.0}	{0.5000，0.8660，0.0}	0.20
					Difference		{0.0，0.0，0.0}	{0.0，0.0，0.0}	{0.0，0.0，0.0}	0
P 3	The value of providing	{8.0，-4.0，-5.0}	{1.0，0.0，0.0}	{0.0，-1.0，0.0}							0.20
					Value according to formation curve	{8.0，-4.0，-5.0}	{1.0，0.0，0.0}	{0.0，-1.0，0.0}	0.20
	Difference	{0.0，0.0，0.0}	{0.0，0.0，0.0}	{0.0，0.0，0.0}						0

(d) in the control of the value of intermediate point

Utilize the method for (b), continue each value on the control two-end-point, carry out G ²Continuous interpolation.Herein, consider not at two-end-point and in the intermediate point controlling value.

For example under the situation of the point range of interpolation such as Figure 21, consider at intermediate point P _cControl tangent line, normal.But, can not control the value on the intermediate point in the described in front method.Therefore, by this point range is divided into 2, be controlled at the value of intermediate point herein.

Just, for point range, not to carry out interpolation at one stroke, but clip intermediate point P _cBe divided into curve C ₁And curve C ₂Ground carries out interpolation.In such cases, owing to put P _cBe equivalent to end points, so as long as the method for employing (b) just can controlling value.

So separately distinguish on the point that the value that will control is arranged, control the value on its two ends, carry out the result of interpolation, as long as connect the curve that generates, can carry out in theory can be in the interpolation of circling round of the three-dimensional of each point control tangent method line curvature.

(2-5) three-dimensional of the tangent line of control on the two-end-point, normal, the curvature interpolation of circling round

(a) flow process of method

The interpolation that employing is circled round in the three-dimensional of each value of initial point terminal point control can be undertaken by following flow process shown in Figure 22.Below, along this process description.

(b-1) provide the point of interpolation object

In this example, provide three-dimensional 3 points 0.0,0.0,0.0}, 5.0,5.0,10.0}, 10.0,10.0,5.0}.Table 7 is listed in the condition of tangent line that other each point provides, normal, curvature etc. inductively.

The condition of table 7 interpolation object each point and initial point terminal point

	Coordinate	The unit tangent vector	The principal normal vector	Curvature
					P 1	(0.0，0.0，0.0)	{0.0，1.0，0.0}	{1.0，0.0，0.0}	0.1
P 2	(5.0，5.0，10.0)	-	-	-
					P 3	(10.0，10.0，5.0)	{1.0，0.0，0.0}	{0.0，-1.0，0.0}	0.1

(b-2) generation of the 3DDCS of r=4

In newton-pressgang Shen method, when the exploration that begins to separate, need provide suitable initial value.Draw the preparation of this initial value herein.Research in advance is 3D Discrete Clothoid Splines, has closely by the interpolation object-point character that changes reposefully with respect to the displacement curvature from initial point.Therefore, in this research, make polygon Q, be identified for the circle round initial value of interpolation of three-dimensional by calculating from here as the 3D Discrete Clothoid Splines of the r=4 of Figure 23.In addition, Figure 24 illustrates actual polygon by this column-generation, and the coordinate on summit is listed table 8 in.

Polygonal apex coordinate that table 8 generates

	Apex coordinate
		P 1	{0.0，0.0，0.0}
	{0.4677，0.4677，3.1228}
			{0.9354，0.9354，6.2456}
	{2.3029、2.3029、9.4966}
		P 2	{5.0，5.0，10.0}
	{6.7095，6.7095，9.9244}
			{8.0655，8.0655，8.4732}
	{9.0327，9.0327，6.7366}
		P 3	{10.0，10.0，5.0}

(b-3) initial value determines

To find the solution with newton-pressgang Shen method, need to determine the initial value of each unknown number.In the method, use the polygon Q that generates at (b-2), obtain the approximate value of each unknown number, determine this value.In 3D Discrete Clothoid Splines, obtained the Fu Leinie frame on each summit.Therefore the unit tangent direction vector t by the polygon Q that generates at (b-2) obtains parameter a ₀, b ₀T is known when obtaining polygon Q for this tangential direction vector, and the formula of the tangent line by this t and three-dimensional clothoid is obtained tangential direction swing angle α, the β on the summit of polygon Q.Obtain a of each curve thus ₀, b ₀Initial value.In addition, on the three-dimensional that begins from initial point is circled round line segment, provide this value.

[numerical expression 59]

$u=\left\{\begin{array}{c}\cos \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ \sin \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ -\sin \mathrm{\&alpha;}\end{array}\right\}$

Herein, about 3D Discrete Clothoid Splines, if consider that the summit is with equidistant arrangement, at the some q of Figure 23 _4i+1On, can be similar to the long variable S of curve is 1/4.Equally at point _{Q4 (i+1)-1}On, can be similar to the long variable S of curve is 3/4.If lump together with the formula of the α of three-dimensional clothoid and to consider these, set up following formula.

[numerical expression 60]

$\left\{\begin{array}{c}a{0}_{4i}+\frac{1}{4}a{1}_{4i}+{\left(\frac{1}{4}\right)}^{2}a{2}_{4i}=a{0}_{4i+1}\\ a{0}_{4i}+\frac{3}{4}a{1}_{4i}+{\left(\frac{3}{4}\right)}^{2}a{2}_{4i}=a{0}_{4(i+1)-1}\end{array}\right.$

It is a1 that this formula becomes unknown number _4iAnd a2 _4iTwo-dimentional simultaneous equations, it is found the solution, as parameter a ₁, a ₂Initial value.Equally also can determine parameter b ₁, b ₂Initial value.

Remaining unknown number is the long h of curve, but can be calculated by the formula of three-dimensional rondo curvature of a curve about its initial value.Three-dimensional rondo curvature of a curve, available note expression down.

[numerical expression 61]

$\mathrm{\&kappa;}=\frac{\sqrt{{\mathrm{\&alpha;}}^{\&prime;2}+{\mathrm{\&beta;}}^{\&prime;2}{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&alpha;}}}{h}$

If change this formula, become following formula, can determine the initial value of h.

[numerical expression 62]

${\mathrm{<figure-callout\; id="4i"\; label="h"\; filenames="A20058000605301142.png"\; state="\{\{state\}\}">h</figure-callout>}}_{4i}=\frac{\sqrt{{(a{1}_{4i}+2a{2}_{4i})}^2+{(\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}{1}_{4i}+2\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}{2}_{4i})}^2{\cos }^2(a{0}_{4i}+a{1}_{4i}+a{2}_{4i})}}{k_{4(i+1)}}$

With above method, can determine initial value for 7 three-dimensionals parameter of circling round.

Table 9 illustrates the initial value that this method of actual usefulness is obtained.

Table 9 initial value

Point of contact P 1And P 2Curve	a 0(0.0 known)
		a 1 -0.2684
	a 2 1.0739
		b 0Pi/2 (known)
	b 1 0.0
		b 2 0.0
	h 12.7684
		Point of contact P 2And P 3Curve	a 0 -0.1648
a 1 3.2061
	a 2 -2.6327
b 0 0.7853
	b 1 0.0
b 2 0.0
	h 9.6752

(b-4) closely by each point, G ²The continuous three-dimensional interpolation of circling round

Adopt by (b-3) definite initial value, reaching G ²Under the condition for continuous, obtain the approximate value of the parameter of each curve with newton-pressgang Shen method.Generate the three-dimensional line segment that circles round from the parameter that obtains thus, with between three-dimensional clothoid interpolation point range.

In 3 three-dimensional is circled round interpolation, by the interpolation object-point, and reach G herein, about closely ²Condition for continuous is considered concrete condition.Figure 25 represents a P ₁, P ₂, P ₃The three-dimensional interpolation of circling round.If with point of contact P ₁, P ₂Between curve as curve C ₁, with point of contact P ₂, P ₃Between curve as curve C ₂, because a0 ₁And b0 ₁Be known, so unknown number is a curve C ₁Parameter a1 ₁, a2 ₁, b1 ₁, b2 ₁, h ₁, curve C ₂Parameter a0 ₂, a1 ₂, a2 ₂, b0 ₂, b1 ₂, b2 ₂, h ₂12.The subscript of the literal that occurs in explanation later is corresponding with the subscript of each curve, as the function of the long variable S of curve, as Px _i, Py _i, Pz _i, α _i, β _i, n _i, κ _i, represent coordinate, tangent line swing angle α, β, normal, curvature on each curve.

At first, at a P ₁On the condition by the interpolation object-point closely, if consider, when providing initial point, must reach from the definition of three-dimensional clothoid.In addition, about tangential direction, owing to provide as known value, so do not specify at a P ₁On condition.

Then, consider some P ₂Point P ₂Be the mutual tie point of curve, reach G ²Continuous to-be position, tangent line, normal, curvature are continuous.Promptly at a P ₂On the condition that should set up as follows.

[numerical expression 63]

Px ₁(1)＝Px ₂(0)

Py ₁(1)＝Py ₂(0)

Pz ₁(1)＝Pz ₂(0)

cos[α ₁(1)-α ₂(0)]＝1

cos[β ₁(1)-β ₂(0)]＝1

n ₁(1)·n ₂(0)＝1

κ ₁(1)＝κ ₂(0)

Consider some P at last ₃Point P ₃Be terminal point, because the condition that should satisfy is position, tangent line, so set up 5 following conditions.Regard α as herein, ₃, β ₃Be tangential direction swing angle α, the β that determines the tangent line vector on the terminal point that provides.

[numerical expression 64]

Px ₂(1)＝Px ₃

Py ₂(1)＝Py ₃

Pz ₂(1)＝Pz ₃

cos[α ₂(1)-α ₃]＝1

cos[β ₂(1)-β ₃]＝1

Learn by above, for unknown number a1 ₁, a2 ₁, b1 ₁, b2 ₁, h ₁, a0 ₂, a1 ₂, a2 ₂, b0 ₂, b1 ₂, b2 ₂, h ₂12, conditional is set up 12 of note down.It is as follows to conclude the conditional of setting up.

[numerical expression 65]

Px ₁(1)＝Px ₂(0)

Py ₁(1)＝Py ₂(0)

Pz ₁(1)＝Pz ₂(0)

cos[α ₁(1)-α ₂(0)]＝1

cos[β ₁(1)-β ₂(0)]＝1

n ₁·n ₂＝1

κ ₁(1)＝κ ₂(0)

Px ₂(1)＝Px ₃

Py ₂(1)＝Py ₃

Pz ₂(1)＝Pz ₃

cos[α ₂(1)-α ₃]＝1

cos[β ₂(1)-β ₃]＝1

Owing to set up 12 formulas for 12 unknown numbers, so can find the solution.Find the solution this formula, obtain and separate with newton-pressgang Shen method.Table 10 is listed initial value and is conciliate.

Table 10 initial value is conciliate

		Initial value	Separate
				Point of contact P 1And P 2Curve C 1	a 0	(0.0 known)	-
a 1	-0.2684	-5.4455
			a 2		1.0739	5.4122
b 0	Pi/2 (known)	-
			b 1		0.0	-3.8590
b 2	0.0	3.1003
			h		12.7684	13.5862
Point of contact P 2And P 3Curve C 2	a 0	-0.1648					-0.033258
			a 1	3.2061	3.6770
	a 2	-2.6327				-3.6437
			b 0	0.7853	0.8120
	b 1	0.0				1.6006
			b 2	0.0	-2.4126
	h	9.6752				9.2873

(b-5) generation of curve

Figure 26 represents that simultaneously with the parameter of obtaining at (b-4) be the curve of basis generation and the polygon that generates at (b-2).The curve of solid line is a curve C ₁, the curve of dotted line is a curve C ₂In this stage, be formed on the G of initial point terminal point control tangential direction ²The continuous three-dimensional clothoid.

(b-6) conditional and unknown number

Herein, consider also with initial point P in addition ₁With terminal point P ₃On normal and curvature be defined as the value that table 7 provides.To control normal and curvature again at the initial point terminal point, need 2 conditions that increase on the initial point terminal point respectively.But, under the state of 4 of condition increases, from considering to obtain to satisfy separating of this condition with the relation of unknown number.Therefore, for the number that makes unknown number and conditional is relative, as shown in figure 27, in curve C ₁The position insertion point DP again of the long variable S=0.5 of curve ₁In addition, for curve C ₂, also insertion point DP again in the position of the long variable S=0.5 of curve ₂

At this moment, with point of contact P ₁With a DP ₁Curve as curve C ' ₁, with point of contact DP ₁With a P ₂Curve as curve C ' ₂, with point of contact P ₂With a DP ₂Curve as curve C ' ₃, with point of contact DP ₂With a P ₃Curve as curve C ' ₄The subscript of the literal that occurs in explanation later on is corresponding with each curve name, for example as the function of the long variable S of curve, as Px _c, Py _c, Pz _c, α _c, β _c, n _c, κ _c, coordinate, tangent line swing angle α, β, normal, curvature on the expression curve C.In addition, on the initial point terminal point, at initial point such as Px _s, Py _s, Pz _s, α _s, β _s, n _s, κ _s, at terminal point such as Px _e, Py _e, Pz _e, α _e, β _e, n _e, κ _e, denotation coordination, tangent line swing angle α, β, normal, curvature.

The condition of setting up on each point below is described.

[numerical expression 66]

Point P ₁: tangent line, normal, curvature: 4

cos[α _C′1(0)-α _s]＝1

cos[β _C′1(0)-β _s]＝1

n _C′1(0)·n _s＝1

κ _C′1(0)＝κ _s

Point DP ₁: position, tangent line, normal, curvature: 7

Px _C′1(1)＝Px _C′2(0)

Py _C′1(1)＝Py _C′2(0)

Pz _C′1(1)＝Pz _C′2(0)

cos[α _C′1(1)-α _C′2(0)]＝1

cos[β _C′1(1)-β _C′2(0)]＝1

n _C′1(1)·n _C′2(0)＝1

κ _C′1(1)＝κ _C′2(0)

Point P ₂: position, tangent line, normal, curvature: 7

Px _C′2(1)＝Px _C′3(0)

Py _C′2(1)＝Py _C′3(0)

Pz _C′2(1)＝Pz _C′3(0)

cos[α _C′2(1)-α _C′3(0)]＝1

cos[β _C′2(1)-β _C′3(0)]＝1

n _C′2(1)·n _C′3(0)＝1

κ _C′2(1)＝κ _C′3(0)

Point DP ₂: position, tangent line, normal, curvature: 7

Px _C′3(1)＝Px _C′4(0)

Py _C′3(1)＝Py _C′4(0)

Pz _C′3(1)＝Pz _C′4(0)

cos[α _C′3(1)-α _C′4(0)]＝1

cos[β _C′3(1)-β _C′4(0)]＝1

n _C′3(1)·n _C′4(0)＝1

κ _C′3(1)＝κ _C′4(0)

Point P ₃: position, tangent line, normal, curvature: 7

Px _C′4(1)＝Px _e

Py _C′4(1)＝Py _e

Pz _C′4(1)＝Pz _e

cos[α _C′4(1)-α _e]＝1

cos[β _C′4(1)-β _e]＝1

n _C′4(1)·n _e＝1

κ _C′4(1)＝κ _e

More than, all the conditional that should set up is 32.Herein, the parameter of circling round that each curve has is a ₀, a ₁, a ₂, b ₀, b ₁, b ₂, each 7 of h, and, because curve is 4, so unknown number is 28.But, like this one, because the number of unknown number and conditional is unequal, separate so can not obtain.Therefore handle 2 some DP that insert again as unknown number ₁, DP ₂Y, z coordinate, increase by 4 unknown numbers.By such processing, unknown number, conditional all are 32, can obtain and separate.

(b-7) initial value determines

To satisfy in (b-6) separating of the conditional set up in order obtaining, to adopt newton-pressgang Shen method, but determine the initial value of unknown number in order to improve its rate of bringing together.As method, by being segmented in the three-dimensional clothoid that generates in (b-5) in the front and back of the new point that inserts as shown in figure 28, make 4 three-dimensional clothoids, provide its parameter of circling round.

About the split plot design of curve, if explanation is with curve C ₁Be divided into curve C ' ₁And curve C ' ₂Method, curve C ' ₁Circle round parameter h ', a ' ₀, a ' ₁, a ' ₂, b ' ₀, b ' ₁, b ' ₂, adopt curve C ₁Parameter, represent with following formula.S herein _dBe the long variable of curve on the cut-point, be 0.5 herein.

[numerical expression 67]

$\left\{\begin{array}{c}{a}_0^{\&prime;}=a_0\\ a_1^{\&prime;}=a_1{S}_d\\ a_2^{\&prime;}=a_2{S_d}^2\\ {\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">b</figure-callout>}}_0^{\&prime;}=b_0\\ {\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^{\&prime;}=b_1{\mathrm{<figure-callout\; id="2"\; label="S\; d\; b"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">S</figure-callout>}}_d& \\ b_{}^{}{}_2^{\&prime;}=b_2{S_d}^2\\ h^{\&prime;}=h{S}_d\end{array}\right.$

Then consider with cut-point DP ₁Curve C as initial point ' ₂At first, if with size, shape and curve C ₁Identical and towards opposite curve as curve C " ₁, the parameter h that circles round of this curve ", a " ₀, a " ₁, a " ₂, b " ₀, b " ₁, b " ₂, adopt curve C ₁The parameter of curve, represent with following formula.

[numerical expression 68]

$\left\{\begin{array}{c}{p}_S^{\&prime;\&prime;}=P\left(1\right)\\ a_0^{\&prime;\&prime;}=a_0+a_1+a_2+\mathrm{\&pi;}\\ a_1^{\&prime;\&prime;}=-(a_1+2a_2)\\ a_2^{\&prime;\&prime;}=a_2\\ {\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">b</figure-callout>}}_0^{\&prime;\&prime;}={\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">b</figure-callout>}}_0+{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}}_1+b_2\\ {\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^{\&prime;\&prime;}=-({\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}}_1+2b_2)\\ {\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}}_2^{\&prime;\&prime;}=b_2\\ h^{\&prime;\&prime;}=h\end{array}\right.$

On this curve, cut-point DP ₁Use DP ₁=C " ₁(1-S _d) expression.Herein, if consider at a DP ₁Cut apart curve C " ₁, with the some P in this curve of cutting apart ₂Curve C as initial point " ₂, become size, shape and curve C " ₂Identical and towards opposite curve.Can utilize formation curve C ' ₁Method formation curve C " ₂Herein, in addition as long as with respect to curve C " ₂Generate size, shape is identical and towards opposite curve, just can formation curve C ₂

With above method, can be at three-dimensional clothoid C ₁On the some DP of the long variable S=0.5 of curve ₁, with curve C ₁Be divided into C ' ₁And C ' ₂Use the same method, also can be in curve C ₂On the some DP of the long variable S=0.5 of curve ₂, with curve C ₂Be divided into C ' ₃And C ' ₄

Table 11 is listed the parameter of 4 curves cutting apart with this method.The parameter of this curve is used at the initial value of obtaining newton-pressgang Shen method used when satisfying the separating of the conditional set up at b-4.

Table 11 is cut apart the parameter of the curve of generation

Curve C ' 1	a 0	(0.0 known)	Curve C ' 2	a 0	4.9134
						a 1	-2.7227	a 1	-0.016629
	a 2	1.3530		a 2	1.3530
						b 0	Pi/2 (known)	b 0	0.41633
	b 1	-1.9295		b 1	-0.37938
						b 2	0.7750	b 2	0.77507
	h	6.7931		h	6.7931
						Initial point	{0.0，0.0，0.0}	Initial point	{1.8431，3.0860，4.9597}
	Curve C ' 3	a 0		-0.033258	Curve C ' 4					a 0	7.1774
a 1			1.8385			a 1	0.016629
		a 2		-0.91093				a 2	-0.91093
b 0			0.81202			b 0	1.0091
		b 1		0.80031				b 1	-4.40601
b 2			-0.60316			b 2	-0.60316
		h		4.6436				h	4.6436
Initial point			{5，0，5.0，10.0}			Initial point	{7.0029，8.1298，7.5337}

(b-8) obtain the parameter of circling round that satisfies condition

Based on the initial value of in (b-7), determining, obtain separating of the conditional that satisfies establishment in (b-6) with newton-pressgang Shen method.Table 12 is parameters of each curve of calculating.The tangent line of the initial point terminal point of the value that provides shown in the table 13 in addition, and the curve of generation, normal, curvature poor.

The parameter of the curve that table 12 generates

Curve C ' 1	a 0	(0.0 known)	Curve C ' 2	a 0	5.3846
						a 1	0.0000	a 1	-3.4602
	a 2	-0.89854		a 2	4.341
						b 0	Pi/2 (known)	b 0	0.47690
	b 1	-0.51836		b 1	-3.2143
						b 2	-0.57552	b 2	3.4613
	h	5.1836		h	9.9808
						Initial point	{0.0，0.0，0.0}	Initial point	{1.8431，4.1726，1.4653}
	Curve C ' 3	a 0		-0.017740	Curve C ' 4					a 0	6.8553
a 1			3.4572			a 1	-1.1443
		a 2		-2.8673				a 2	0.57219
b 0			0.72385			b 0	0.76315
		b 1		2.4551				b 1	-1.1942
b 2			-2.4158			b 2	0.43108
		h		6.60818				h	3.3206
Initial point			{5.0，5.0，10.0}			Initial point	{7.0029，9.0734，5.6186}

The tangent line of the initial point terminal point of the value that table 13 provides and the curve of generation, normal, curvature poor

		Coordinate	The unit tangent vector	The principal normal vector	Curvature
						P 1	The value of providing	{0.0，0.0，0.0}	{0.0，1.0，0.0}	{1.0，0.0，0.0}	0.10
Value according to formation curve	{0.0，0.0，0.0}	{0.0，1.0，0.0}	{1.0，0.0，0.0}	0.10
					Difference		{0.0，0.0，0.0}	{0.0，0.0，0.0}	(0.0，0.0，0.0)	0
P 3	The value of providing	{10.0，10.0，5.0}	{1.0，0.0，0.0}	{0.0，-1.0，0.0}							0.10
					Value according to formation curve	{10.0，10.0，5.0}	{1.0，0.0，0.0}	{0.0，-1.0，0.0}	0.10
	Difference	{0.0，0.0，0.0}	{0.0，0.0，0.0}	{0.0，0.0，0.0}						0

(b-9) generation of curve

Figure 29 represents the curve by the parameter generation of obtaining in (b-8).Solid line is represented three-dimensional clothoid, and dotted line one dot-and-dash line two dot-and-dash lines three dot-and-dash lines represent that each direction of curve in the principal normal direction, is of a size of radius, satisfy natural logarithm, the radius-of-curvature changing pattern of taking the logarithm.In addition, Figure 30 is expression from the curve map of the relation of the displacement s of the initial point of each curve corresponding with the line kind of Figure 29 and curvature κ.By finding out among the figure, the curve of generation satisfies the given condition of table 12.

More than, illustrated and adopted at the circle round example of interpolation formation curve of the three-dimensional of two ends control tangent line, normal, curvature.

3. adopt the method for designing of return path of the ball-screw of the three-dimensional interpolation of circling round

As Application in Machine Design example, carry out the design of return path of the ball-screw of reverser mode at three-dimensional clothoid.

(3-1) explanation of the ball-screw of reverser mode

Figure 31～Figure 35 represents the ball-screw of reverser mode.Reverser constitutes along the return path of the ball of leading screw groove rotation.Reverser has with nut and separates the mode that is fixed on the mode on the nut and forms with nut after the formation.Figure 31 represents to separate the mode that forms reverser with nut.

Below, the ball screw of the mode of reverser and nut one is described.Figure 32 be the expression reverser for the nut 1 of the ball-screw of the mode of nut one.At the inner peripheral surface of nut 1, as a week less than spiral helicine load rolling body raceway groove, form load ball road groove 2.Load ball road groove 2 has the helical pitch consistent with the ball road groove of lead screw shaft described later.As an end and the other end of the ball road groove 3 connected load raceway grooves of return path, has the opposite helical pitch of direction and load ball road groove 2.Constitute one one volume groove 4 with these load ball road grooves 2 and ball road groove 3.Figure 33 A is the stereographic map of nut 1 of seeing the state of ball circulating groove 3, and Figure 33 B is the stereographic map of nut 1 of seeing the state of load ball road groove 2.

Figure 34 is illustrated in the state of this nut 1 of assembling on the lead screw shaft.

At the outer peripheral face of lead screw shaft 5,, form ball road groove 6 as spiral helicine rolling body raceway groove with regulation helical pitch.The load ball road groove 2 of nut 1 is opposed with the ball road groove 6 of lead screw shaft 5.Between the ball road groove 6 of the load ball road groove 2 of nut 1 and ball circulating groove 3 and lead screw shaft 5,, arrange a plurality of balls as a plurality of rolling bodys of rotatable motion.Along with the relative rotation of nut 1 with respect to turning axle 5, a plurality of balls rotatablely move while accept load between the ball road groove 6 of the load ball road groove 2 of nut 1 and lead screw shaft 5.

The ball circulating groove 3 of nut 1 shown in Figure 32 is the part corresponding with reverser shown in Figure 31.Ball circulating groove 3, the ball that rolls with load ball road groove 2 along turning axle 5 change around the lead screw shaft 5 one touring, turn back to the mode of original load ball road groove, cross the ridge 7 of lead screw shaft 5 along ball.

When the circulating path of pattern in the past, stretch-out view by coiling Figure 35 on lead screw shaft, separate from lead screw shaft 5 centers with the degree of not running into ridge and ball and to make, but find out that from the curved transition of Figure 36 this path is that curvature is discontinuous along the path.Therefore, adopt the three-dimensional interpolation of circling round, design cycle path again on the continuous path of curvature.

Figure 37 represents the track at ball center.Make the circulating path of ball reach G as a whole ²Continuously, need on the point that moves on the return path, reach G at ball ²Continuously.Therefore in the design of return path, thinking has the necessity of controlling tangent line, normal, curvature at the two-end-point of return path.

(3-2), illustrate and adopt three-dimensional clothoid, the example of the return path of the ball-screw of design reverser mode,

(a-1) lead screw shaft and ball

Table 14 is listed the lead screw shaft used among the design and the size of ball.

The size of table 14 lead screw shaft and ball

Lead screw shaft external diameter (mm)	28.0
		Footpath, ball center (mm)	28.0
Minor diameter of thread (mm)	24.825
		Spacing (mm)	5.6
Ball footpath (mm)	3.175

(a-2) symmetry and coordinate

The return path of the ball-screw of reverser mode considers it need is axisymmetric from its use.Therefore the coordinate system that the design is used is described.

At first, as shown in figure 38, getting the z axle is the lead screw shaft direction.The solid line of Figure 38 is the track that draw at the center of ball when the leading screw groove moves ball.In addition, will enter the point of return path as some P _s, will turn back to the point of leading screw groove as some P from return path _e, will put P _sWith a P _eMid point as a P _mAs shown in figure 39, as use to the perspective view on xy plane and see P _sWith a P _e, by initial point 0, P _sWith a P _ePlot two equilateral triangles, but get the ∠ P of this two equilateral triangle _sOP _eThe direction of vertical halving line be the y direction of principal axis.Consider from symmetry that in addition regulation y axle is by some P _mAbout the direction of each, shown in Figure 38,39.So design return path to adopt coordinate system to reach axisymmetric mode.

When actual design, with the coordinate of θ=15 ° definite each point.Table 15 is listed coordinate, tangent line, normal, the curvature of determining thus.

The coordinate of table 15 each point, tangent line, normal, curvature

	Coordinate	Tangent line	Normal	Curvature
					Point P s	{-3.6088，-13.5249，2.5563}	{0.96397，-025829，0.063533}	{0.25881，0.96592，0.0}	0.071428
Point P e	{3.6088，-13.5249，-2.5563}	{0.96397，025829，0.063533}	{-0.25881，0.96592，0.0}	0.071428
					Point P m	{0.0，-13.5249，0.0}	-	-	-

(a-3) constraint condition

Constraint condition in the design of the return path of the ball-screw of research reverser mode.At first, with at a P _sWith a P _eThe curve that the track at the center of the ball that moves along the leading screw groove is described must be G ₂Continuously.

Then, as considering the height of ball for example, consider that owing to needing only return path is the y rotational symmetry, the center of ball is just by certain point on the y axle, so this is put as putting P _h(with reference to Figure 38,39).At this moment, ball will be crossed ridge, needs some P _hThe absolute value of y coordinate satisfied at least:

(some P _hThe absolute value of y coordinate) 〉=(lead screw shaft external diameter+ball footpath)/2 therefore, in the design, be:

(some P _hThe absolute value of y coordinate) 〉=(lead screw shaft external diameter+ball footpath * 1.2)/2 in addition, the normal direction when consideration is the y rotational symmetry need be 0,1,0), tangential direction only has the degree of freedom of rotation around it.

Satisfy above condition, generate the axisymmetric return path of y with three-dimensional clothoid.In fact, in addition, also must consider interference, but, the initial value that can pass through to change interpolation when interfering is being arranged, or increasing the interpolation object-point, or solve in the redesign path about interfering the return path that to check design to lead screw shaft.

(a-4) for fear of interference

Occur in easily on the limit that enters return path with the interference of lead screw shaft,, therefore cause interference easily owing to make the path by free interpolation.Require return path to leave lead screw shaft, cross ridge and get back to original position, but will avoid interfering, be preferably in leave lead screw shaft to a certain degree after, cross ridge and get back to original position.As the method that generates this return path, the interpolation of increasing object-point is arranged, avoid the method for interfering and enter the 1st curve of this return path, forcibly from the method for lead screw shaft separation with manually generating.Wherein in the design, adopt with manually generating and enter the 1st curve of return path, forcibly the method for separating from lead screw shaft.

Herein, illustrate and enter from a P _sThe 1st curve C of the return path of beginning ₁As the variable of the long variable S of curve, as Px ₁, (S), Py ₁, (S), Pz ₁(S), α ₁(S), β ₁(S), n ₁(S), κ ₁(s), expression curve C ₁On coordinate, tangent line swing angle α, β, normal, curvature.In addition, at a P _sPoint P _hOn, at a P _sAs Px _s, Py _s, Pz _s, α _s, β _s, n _s, κ _s, at a P _hAs Px _h, Py _h, P _{Z h}, α _h, β _h, n _h, κ _h, denotation coordination, tangent line swing angle α, β, normal, curvature.The curve of describing with the track at the center of the ball that moves along the leading screw groove is G ²Condition for continuous is at a P ^sLast establishment following formula.

[numerical expression 69]

Point P _s: tangent line, normal, curvature: 4

cos[α ₁(0)-α _s]＝1

cos[β ₁(0)-β _s]＝1

n ₁(0)·n _s＝1

κ ₁(0)＝κ _s

In addition, the curve that the track at the center of the ball that moves along the leading screw groove is described adopts three-dimensional clothoid to represent, but from point shown in Figure 40, the three-dimensional clothoid C of the length of a circle degree ₀Formula represent with following formula., the pedometer of screw thread is decided to be pit herein, the lead screw shaft profile is defined as R, the angle of pitch of screw thread is defined as α ₀

[numerical expression 70]

α ₀(S)＝-α ₀

β ₀(S)＝β _e+2πS

${\mathrm{<figure-callout\; id="0"\; label="h"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">h</figure-callout>}}_0=\sqrt{{\mathrm{<figure-callout\; id="2"\; label="pit"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">pit</figure-callout>}}^2+{\left(2\mathrm{\&pi;R}\right)}^2}$

$P_0\left(S\right)=P_e+{\mathrm{<figure-callout\; id="0"\; label="h"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">h</figure-callout>}}_0{\&Integral;}_0^{1}u\left(S\right)\mathrm{dS}$

In curve C ₀Formula in, the some P _sBe expressed as P _s=P ₀(11/12).Now, if as from a P _sBeginning is at a P _sWith curve C ₀Reach G ²Continuous curve C ₁, generate curve with parameter of remembering down, can it be left from lead screw shaft.

[numerical expression 71]

$\left\{\begin{array}{c}{\mathrm{\&alpha;}}_1\left(S\right)=-{\mathrm{\&alpha;}}_0\\ {\mathrm{\&beta;}}_1\left(S\right)={\mathrm{\&beta;}}_0\left(\frac{\mathrm{\&pi;}}{12}\right)+\frac{1}{60}\\ P_1\left(S\right)=P_s+\frac{{\mathrm{<figure-callout\; id="0"\; label="h"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">h</figure-callout>}}_0}{60}{\&Integral;}_0^1{u}_1\left(S\right)\mathrm{dS}\end{array}(\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}{1}_0+\frac{11}{6}{2}_0)S-\frac{1}{15}(\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}{1}_0+\frac{11}{6}\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}{2}_0){\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">S</figure-callout>}}^2\right.$

For example, as the curve C that satisfies this condition ₁, generation has the three-dimensional clothoid of the parameter of table 16.

Table 16 curve C ₁Parameter

Curve C 1	a 0	6.2196
			a 1	0.0
	a 2	0.0
			b 0	6.0213
	b 1	0.10472
			b 2	-0.41887
	h	1.4631
			Initial point	{-3.6088，-13.5249，2.5563}

At this moment, if comparison point P _sOn curve C ₀And curve C ₁Tangent line, normal, the value of curvature, form table 17, judge to reach G ²Continuously.

Table 17 is at a P _sTangent line, normal, the skew of curvature

	Coordinate	The unit tangent vector	The principal normal vector	Curvature
					Curve C 0	{-3.6088，-13.5249，2.5563}	{0.96397，-025829，0.063533}	{0.25881，0.96592，0.0}	0.071428
Curve C 1	{-3.6088，-13.5249，2.5563}	{0.96397，-025829，0.063533}	{0.25881，0.96592，0.0}	0.071428
					Difference	{0.000，0.000，0.000}	{0.000，0.000，0.000}	{0.000，0.000，0.000}	0

In addition, this curve as judging from Figure 41,42, only forms the shape of separating from lead screw shaft.Therefore, about entering from a P _sThe 1st curve C of the return path of beginning ₁, adopt the curve of this parameter.

(a-5) the three-dimensional condition and the unknown number of interpolation that circle round

Add in the condition described in (a-3), reaching G ²Under the condition for continuous, adopt newton-pressgang Shen method to obtain the approximate value of the parameter of each curve.Herein, owing to generate from a P _sThe curve C of beginning ₁So,, in explanation, narrate curve C later on ₁Terminal point P ₁With a P _hBetween the design of pathway.The subscript of the literal that occurs in explanation is corresponding with the subscript of each curve, as the function of the long variable S of curve, as Px _i, (S) Py _i, (S) Pz _i(S), α _i(S), β _i(S), n _i(S), κ _i(S), represent coordinate, tangent line swing angle α, β, normal, curvature on each curve.In addition, at a P _hOn, coordinate, tangent line swing angle α, β, normal, curvature are expressed as Px _h, Py _h, Pz _h, α _h, β _h, n _h, h _h

In the design in path, because the point that should pass through closely is a P ₁With a P _hThis 2 point is so be the interpolation of circling round of the three-dimensional of 2 of this interpolations.Herein, if consideration is in the interpolation condition of two-end-point, because the number of conditional is Duoed 2 than unknown number, so in order to carry out G ²The continuous three-dimensional interpolation of circling round as shown in figure 43, is determined at a P ₁With a P _hBetween insertion point P ₂In addition, with point of contact P ₁With a P _hCurve as curve C ₂, with point of contact P ₂With a P _eCurve as curve C ₃

Interpolation condition on the each point below is described.

[numerical expression 72]

Point P ₁: tangent line, normal, curvature: 4

cos[α ₂(0)-α ₁(1)]＝1

cos[β ₂(0)-β ₁(1)]＝1

n ₂(0)·n ₁(1)＝1

κ ₂(0)＝κ ₁(1)

Point P ₂: position, tangent line, normal, curvature: 7

Px ₃(1)＝Px ₂(0)

Py ₃(1)＝Py ₂(0)

Pz ₃(1)＝Pz ₂(0)

cos[α ₃(1)-α ₂(0)]＝1

cos[β ₃(1)-β ₂(0)]＝1

n ₃(1)·n ₂(0)＝1

κ ₃(1)＝κ ₂(0)

Point P _h: position, β, normal: 5

Px ₃(1)＝Px _h

Py ₃(1)＝Py _h

Pz ₃(1)＝Pz _h

cos[β ₃(1)]＝1

n ₃(1)·{0，1，0}＝1

Show that more than all the conditional that should set up is 16.Herein, the parameter of circling round that each curve has is a ₀, a ₁, a ₂, b ₀, b ₁, b ₂, 7 of h etc., and, because curve is 2, so unknown number is 14.But, like this one, because the number of unknown number and conditional is unequal, separate so can not obtain.Therefore handle 2 some P that insert again as unknown number ₂Y, z coordinate, increase by 2 unknown numbers.By such processing, unknown number, conditional all are 16, can obtain and separate.In addition, though do not carry out in the design's example, the number of this unknown number and conditional as long as provide the point that should pass through closely on the way, is reached G in the front and back of this point ²Continuously, set up usually, so even at a P ₁With a P _hBetween increase the interpolation object-point, also can obtain and separate.

(a-6) obtain the parameter of circling round that satisfies condition

Obtain with newton-pressgang Shen method and to satisfy in (a-5) separating of the conditional set up.The generation method of interpolating method, initial value is followed the circle round method of interpolation of three-dimensional.Table 18 is listed the parameter of each curve of calculating, and table 19 is listed in the skew of coordinate on the tie point of retouching out, tangent line, normal, curvature.

The parameter of the curve that table 18 generates

Curve C 2	a 0	-0.063576
			a 1	0.0000
	a 2	0.62696
			b 0	-0.57595
	b 1	-0.98004
			b 2	0.77916
	h	1.9561
			Initial point P 1	{-2.2429，-14.021，2.6492}
	Curve C 3	a 0			6.8465
a 1			1.729
		a 2		-0.86450
b 0			-0.77684
		b 1		0.79736
b 2			-0.020523
		h		2.69723
Initial point P 2			{-0.93007，-15.389，2.3720}

Table 19 is in the skew of the coordinate of each tangent point, tangent line, normal, curvature

		Coordinate	The unit tangent vector	The principal normal vector	Curvature
						P 1	Curve C 1	{-2.2429，-14.021，2.6492}	{0.83697-0.54353，0.063533}	{-0.54463，-0.83867，0.0}	0.50
Curve C 2	{-2.2429，-14.021，2.6492}	{0.83697-0.54353，0.063533}	{-0.54463，-0.83867，0.0}	0.50
					Difference		{0.0，0.0，0.0}	{0.0，0.0，0.0}	{0.0，0.0，0.0}	0
P 2	Curve C 2	{-0.93007，-15.389，2.3720}	{0.60291，-0.59268，-0.53405}	{-0.10017，0.60786，-0.78769}							0.68803
					Curve C 3	{-0.93007，-15.389，2.3720}	{0.60291，-0.59268，-0.53405}	{-0.10017，0.60786，-0.78769}	0.68803
	Difference	{0.0，0.0，0.0}	{0.0，0.0，0.0}	{0.0，0.0，0.0}						0
					P h	Curve C 3	{0.000-15.905，0.0}	{0.14241，0.0000，-0.98980}	{0.000，1.0，0.000}		0.039934
The value of providing	{0.0，-15.905，0.0}	-	{0.0，1.0，0.0}	-
						Difference	{0.0，0.0，0.0}	-	{0.0，0.0，0.0}	-

(a-7) generation in path

Parameter by being obtained by (a-5), (a-6) can design from a P _sTo a P _hThe path.In addition, because from a P _sTo a P _eThe path, because of the path is that y is axisymmetric, and refetch coordinate system, will put P _eRegard a P as _sThe path that generates is identical, so these also can be generated by same curve.

Figure 44 represents the path with above method generation.Solid line is that the central orbit of the ball on the lead screw shaft is a curve C ₀, to a P _s～P _nDotted line, a dot-and-dash line, these 3 curves of two dot-and-dash lines be respectively curve C ₁, C ₂, C ₃In addition, to a P _n～P _eTwo dot-and-dash lines, a dot-and-dash line, these 3 curves of dotted line be respectively curve C ₃, C ₂, C ₁With with the axisymmetric curve of y.

Figure 45 is that expression is seen from a P from the positive dirction of z axle _eThe counterclockwise curve map of the relation of displacement that moves along circulating path and curvature κ.The line kind of curve map is corresponding with the line kind of the curve of Figure 44.

With above method, adopt three-dimensional clothoid, the circulating path of the ball-screw of design reverser formula.In addition, adopt three-dimensional clothoid design cycle route method, certainly be not limited to the ball-screw of reverser formula, also be applicable to the ball-screw that constitutes the what is called backflow tubular type of return path with pipe, perhaps with the end gap that is located on the nut end face, pick up ball from the ball road groove of lead screw shaft, in nut, turn back to the ball-screw of so-called end gap formula of the ball road groove of lead screw shaft from the end gap of opposition side.

, when active computer was carried out the program that realizes method for designing of the present invention, stored programme in the auxilary units such as hard disk unit of computing machine was encased in the primary memory and carries out.In addition, program so can be stored in the movable recording medium such as CD-ROM and sell, or be stored in the pen recorder of the computing machine that connects via network, also can send other computing machine by network to.

B. adopt the numerical control method of clothoid

Below, divide the definition of 1. three-dimensional clothoids and feature, 2. utilize the interpolation of three-dimensional clothoid, 3. adopt the numerical control method of the three-dimensional interpolation of circling round, the working of an invention mode of the numerical control method of clothoid is adopted in explanation successively.

1. the definition of three-dimensional clothoid and feature

(1) basic form that circles round of three-dimensional

Clothoid (Clothoid curve), another name also are called the spiral (Cornu ' s spiral) of Ke's knob, are the curves that changes curvature with length of a curve with being directly proportional.Known in the past two-dimentional clothoid is a kind of of plane curve (two-dimensional curve), on x y coordinate shown in Figure 46, represents with following formula.

[numerical expression 73]

$P={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+{\&Integral;}_0^s{e}^{\mathrm{j\&phi;}}\mathrm{ds}={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+h{\&Integral;}_0^S{e}^{\mathrm{j\&phi;}}\mathrm{dS},$ 0≤s≤h， $0\&le;S=\frac{s}{h}\&le;1$

(1)

φ＝c ₀+c ₁s+c ₂s ²＝φ ₀+φ _vS+φ _uS ² (2)

Herein,

[numerical expression 74]

P＝x+jy，

$j=\sqrt{-1}---\left(3\right)$

Be the position vector of the point on the expression curve,

[numerical expression 75]

P ₀＝x ₀+jy ₀ (4)

Be its initial value (position vector of initial point).

[numerical expression 76]

e ^{J φ}=cos φ+jsin φ (5) is the position vector (length is 1 vector) of the tangential direction of expression curve, and this direction Φ measures counterclockwise from former line (x direction of principal axis).If in this unit vector, multiply by tiny length ds integration, can obtain the some P on the curve.

To be made as s along the length from initial point of the curve of curve determination, its length overall (length from the initial point to the terminal point) will be made as h.The value of representing to remove s with S with h.S is the value of no guiding principle amount, is referred to as the long variable of curve.

The feature of clothoid as the formula (2), is to represent tangent directional angle Φ with the quadratic expression of long s of curve or curve variable S.c ₀, c ₁, c ₂Or Φ _o, Φ _v, Φ _uBe quadratic coefficient, length overall h these are several and curve is called the parameter of circling round.Figure 47 represents the shape of general clothoid.

The above relation of three-dimensional expansion, the formula of making three-dimensional clothoid.Do not know to provide the formula of three-dimensional clothoid, so initial its formula that derives of inventors in the past.

Define three-dimensional clothoid by following formula.

[numerical expression 77]

$P={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0$ $+{\&Integral;}_0^s\mathrm{uds}={\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0+h{\&Integral;}_0^S\mathrm{udS},$ 0≤s≤h， $0\&le;S=\frac{s}{h}\&le;1$

(6)

$u=E^{\mathrm{k\&beta;}}{E}^{\mathrm{j\&alpha;}}\left(i\right)=\left[\begin{array}{ccc}\cos \mathrm{\&beta;}& -\sin \mathrm{\&beta;}& 0\\ \sin \mathrm{\&beta;}& \cos \mathrm{\&beta;}& 0\begin{array}{}\end{array}\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos \mathrm{\&alpha;}& 0& \sin \mathrm{\&alpha;}\\ 0& 1& 0\\ -\sin \mathrm{\&alpha;}& 0& \cos \mathrm{\&alpha;}\end{array}\right]\left\{\begin{array}{c}1\\ 0\\ 0\end{array}\right\}=\left\{\begin{array}{c}\cos \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ \sin \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ -\sin \mathrm{\&alpha;}\end{array}\right\}$

(7)

α＝a ₀+a ₁S+a ₂S ² (8)

β＝b ₀+b ₁S+b ₂S ² (9)

Herein,

[numerical expression 78]

$P=\left\{\begin{array}{c}x\\ y\\ z\end{array}\right\},$ ${\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">P</figure-callout>}}_0=\begin{array}{c}\left\{\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right\}\end{array}---\left(10\right)$

Position vector and the initial value thereof of representing the point on the three-dimensional clothoid respectively.I, j, k are respectively x axle, y axle and the axial unit vector of z.

U is the unit vector of the tangential direction of the curve on the expression point P, is provided by formula (7).In formula (7), E ^{K β}And E ^{J α}Be rotation matrix, as shown in figure 48, represent the rotation of the angle beta that k axle (z axle) is and the rotation of the angle [alpha] that j axle (y axle) is respectively.The former is called deflection (yaw) rotation, the latter is called inclination (pitch) rotation.Formula (7), the expression by at first make i axle (x axle) to unit vector be only to rotate α at j axle (y axle), be only to rotate β then at k axle (z axle), obtain tangent line vector u.

Just, when two dimension, obtain representing the unit vector e of the tangential direction of curve by angle of inclination Φ from the x axle ^{J Φ}When three-dimensional, by the tangent line vector u that obtains curve from inclination alpha and deflection angle β.If inclination alpha is 0, can obtain the two-dimentional clothoid rolled with the xy plane, if deflection angle β is 0, can obtain the two-dimentional clothoid of rolling with the xz plane.If in tangential direction vector u, multiply by small long ds integration, can obtain three-dimensional clothoid.

In three-dimensional clothoid, the inclination alpha of tangent line vector and deflection angle β suc as formula shown in (8) and the formula (9), can be provided by the quadratic expression of the long variable S of curve respectively.So, can freely select the variation of tangential direction, and can also in it changes, make it have continuity.

Shown in above formula, three-dimensional clothoid is defined as " being to represent the inclination angle of tangential direction and the curve of deflection angle with the quadratic expression of the long variable of curve respectively ".

From P ₀A three-dimensional clothoid of beginning, by

[numerical expression 79]

a ₀，a ₁，a ₂，b ₀，b ₁，b ₂，h (11)

These 7 parameters are determined.a ₀～b ₂6 units that variable has angle, the shape of expression rondo line segment.In contrast, h has the unit of length, the size of expression rondo line segment.

As the typical example of three-dimensional clothoid, spiral helicine curve shown in Figure 49 is arranged.

(2) moving frame

In formula (7), if replace basic tangential direction vector i, substitution basic coordinates [i, j, k] obtains next moving frame (moving frame) E.

[numerical expression 80]

$E=\left[\begin{array}{ccc}u& v& w\end{array}\right]=E^{\mathrm{k\&beta;}}{E}^{\mathrm{j\&alpha;}}\left[\mathrm{i\; j\; k}\right]=E^{\mathrm{k\&beta;}}{E}^{\mathrm{j\&alpha;}}\left[\begin{array}{c}I\end{array}\right]=E^{\mathrm{k\&beta;}}{E}^{\mathrm{j\&alpha;}}$

$=\left[\begin{array}{ccc}\cos \mathrm{\&beta;}\cos \mathrm{\&alpha;}& -\sin \mathrm{\&beta;}& \cos \mathrm{\&beta;}\sin \mathrm{\&alpha;}\\ \sin \mathrm{\&beta;}\cos \mathrm{\&alpha;}& \cos \mathrm{\&beta;}& \sin \mathrm{\&beta;}\sin \mathrm{\&alpha;}\\ -\sin \mathrm{\&alpha;}& 0& \cos \mathrm{\&alpha;}\end{array}\right]---\left(12\right)$

$u=\left\{\begin{array}{c}\cos \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ \sin \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ -\sin \mathrm{\&alpha;}\end{array}\right\},$ $v=\left\{\begin{array}{c}-\sin \mathrm{\&beta;}\\ \cos \mathrm{\&beta;}\\ 0\end{array}\right\},$ $w=\left\{\begin{array}{c}\cos \mathrm{\&beta;}\sin \mathrm{\&alpha;}\\ \sin \mathrm{\&beta;}\sin \mathrm{\&alpha;}\\ \cos \mathrm{\&alpha;}\end{array}\right\}---\left(13\right)$

Herein, v and w are the face contained unit vector vertical with the tangent line of curve, and be mutually orthogonal, simultaneously with tangential direction unit vector u quadrature.The group of these 3 unit vectors (3 fluorescence groups) is and the frame (coordinate system, frame) of moving some P one same-action, is referred to as moving frame.

Because available following formula is obtained moving frame, so can carry out the calculating of principal normal, binormal easily, carries out the shape analysis of curve easily.

In addition, can adopt E to obtain the posture of the tool point of robot, can obtain the posture of the object of holding by the robot handle.

If with the initial value of E and end value respectively as E ₀, E ₁, for:

[numerical expression 81]

${\mathrm{<figure-callout\; id="0"\; label="E"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">E</figure-callout>}}_0=E^{k{b}_0}{E}^{j{a}_0}---\left(14\right)$

${\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">E</figure-callout>}}_1=E^{k({\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">b</figure-callout>}}_0+{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}}_1+b_2)}{E}^{j(a_0+a_1+a_2)}---\left(15\right)$

(3) roll

By considering moving frame, can handle the 3rd rotation " (roll) rolls ".Rolling is the rotation around the tangential direction.The existence of rolling does not influence circle round shape of three-dimensional itself, but the three-dimensional moving frame that circles round and induce of influence.By the thin abacus bead of the metal of complications, can freely around wiry, rotate, but be not to change the thin shape of metal by it.

When considering to roll rotation, moving frame is a following formula.

[numerical expression 82]

E＝E ^kβE ^jαE ^iγI＝E ^kβE ^jαE ^iγ (16)

About roll angle γ, can be as the function performance of S.

[numerical expression 83]

γ＝c ₀+c ₁S+c ₂S ² (17)

(4) the geometric character of three-dimensional clothoid

(a) normal of three-dimensional clothoid

Known, the normal vector of three-dimensional curve adopts tangential direction u, represents with following formula.

[numerical expression 84]

$n=\frac{u^{\&prime;}}{\left|\right|u^{\&prime;}\left|\right|}---\left(18\right)$

Herein, as follows by 1 an amount of subdifferential of the tangent line of the three-dimensional clothoid of formula (7).

[numerical expression 85]

$u^{\&prime;}\left(S\right)=\left\{\begin{array}{c}-{\mathrm{\&alpha;}}^{\&prime;}\left(S\right)\cos \mathrm{\&beta;}\left(S\right)\sin \mathrm{\&alpha;}\left(S\right)+{\mathrm{\&beta;}}^{\&prime;}\left(S\right)\sin \mathrm{\&beta;}\left(S\right)\cos \mathrm{\&alpha;}\left(S\right)\\ -{\mathrm{\&alpha;}}^{\&prime;}\left(S\right)\sin \mathrm{\&beta;}\left(S\right)\sin \mathrm{\&alpha;}\left(S\right)+{\mathrm{\&beta;}}^{\&prime;}\left(S\right)\cos \mathrm{\&beta;}\left(S\right)\cos \mathrm{\&alpha;}\left(S\right)\\ -{\mathrm{\&alpha;}}^{\&prime;}\left(S\right)\cos \mathrm{\&alpha;}\left(S\right)\end{array}\right\}$

$\left|\right|u^{\&prime;}\left(S\right)\left|\right|=\sqrt{{\mathrm{\&alpha;}}^{\&prime;}{\left(S\right)}^2+{\mathrm{\&beta;}}^{\&prime;}{\left(S\right)}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&alpha;}\left(S\right)}$

(19)

Just, the normal of three-dimensional clothoid is an amount of, adopts S, represents with following form.

[numerical expression 86]

$n\left(S\right)=\frac{u^{\&prime;}\left(S\right)}{\left|\right|u^{\&prime;}\left(S\right)\left|\right|}$

$=\frac{1}{\sqrt{{\mathrm{\&alpha;}}^{\&prime;}{\left(S\right)}^2+{\mathrm{\&beta;}}^{\&prime;}{\left(S\right)}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&alpha;}\left(S\right)}}\left\{\begin{array}{c}-{\mathrm{\&alpha;}}^{\&prime;}\left(S\right)\cos \mathrm{\&beta;}\left(S\right)\sin \mathrm{\&alpha;}\left(S\right)-{\mathrm{\&beta;}}^{\&prime;}\left(S\right)\sin \mathrm{\&beta;}\left(S\right)\mathrm{os\&alpha;}\left(S\right)\\ -{\mathrm{\&alpha;}}^{\&prime;}\left(S\right)\sin \mathrm{\&beta;}\left(S\right)\sin \mathrm{\&alpha;}\left(S\right)+{\mathrm{\&beta;}}^{\&prime;}\left(S\right)\cos \mathrm{\&beta;}\left(S\right)\cos \mathrm{\&alpha;}\left(S\right)\\ -{\mathrm{\&alpha;}}^{\&prime;}\left(S\right)\cos \mathrm{\&alpha;}\left(S\right)\end{array}\right\}$

(20)

(b) normal of the three-dimensional clothoid of employing rotation

Herein, with the tangent line u of (7) determine equally, also consider normal n.For initial stage tangential direction (1,0,0), suppose and adopt constant γ, with (0, cos γ ,-sin γ) expression initial stage normal direction.If make its rotation in the same manner with tangent line, normal n is expressed as follows.

[numerical expression 87]

$n\left(S\right)=\left[\begin{array}{ccc}\cos \mathrm{\&beta;}\left(S\right)& -\sin \mathrm{\&beta;}\left(S\right)& 0\\ \sin \mathrm{\&beta;}\left(S\right)& \cos \mathrm{\&beta;}\left(S\right)& 0\begin{array}{}\end{array}\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos \mathrm{\&alpha;}\left(S\right)& 0& \sin \mathrm{\&alpha;}\left(S\right)\\ 0& 1& 0\\ -\sin \mathrm{\&alpha;}\left(S\right)& 0& \cos \mathrm{\&alpha;}\left(S\right)\end{array}\right]\left\{\begin{array}{c}0\\ \cos \mathrm{\&gamma;}\\ -\sin \mathrm{\&gamma;}\end{array}\right\}$

$=\left\{\begin{array}{c}-\sin \mathrm{\&gamma;}\cos \mathrm{\&beta;}\left(S\right)\sin \mathrm{\&alpha;}\left(S\right)-\cos \mathrm{\&gamma;}\sin \mathrm{\&beta;}\left(S\right)\\ -\sin \mathrm{\&gamma;}\sin \mathrm{\&beta;}\left(S\right)\sin \mathrm{\&alpha;}\left(S\right)+\cos \mathrm{\&gamma;}\cos \mathrm{\&beta;}\left(S\right)\\ -\sin \mathrm{\&gamma;}\cos \mathrm{\&alpha;}\left(S\right)\end{array}\right\}$

(21)

Relatively the formula of (20), (21) is learnt, sin γ, cos γ are with note is corresponding down.

[numerical expression 88]

$\sin \mathrm{\&gamma;}=\frac{{\mathrm{\&alpha;}}^{\&prime;}\left(S\right)}{\sqrt{{\mathrm{\&alpha;}}^{\&prime;}{\left(S\right)}^2+{\mathrm{\&beta;}}^{\&prime;}{\left(S\right)}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&alpha;}\left(S\right)}}$

$\cos \mathrm{\&gamma;}=\frac{{\mathrm{\&beta;}}^{\&prime;}\left(S\right)\cos \mathrm{\&alpha;}\left(S\right)}{\sqrt{{\mathrm{\&alpha;}}^{\&prime;}{\left(S\right)}^2+{\mathrm{\&beta;}}^{\&prime;}{\left(S\right)}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&alpha;}\left(S\right)}}$

(22)

(c) normal at tie point that circles round in the interpolation of three-dimensional is continuous

It is continuous to reach the normal at tie point that three-dimensional circles round in the interpolation, draw by formula (22), as long as

[numerical expression 89]

$\tan \mathrm{\&gamma;}=\frac{{\mathrm{\&alpha;}}^{\&prime;}\left(S\right)}{{\mathrm{\&beta;}}^{\&prime;}\left(S\right)\cos \mathrm{\&alpha;}\left(S\right)}---\left(23\right)$

Be continuous just passable.

(d) three-dimensional rondo curvature of a curve

Three-dimensional rondo curvature of a curve is represented with following formula.

[numerical expression 90]

$\mathrm{\&kappa;}\left(S\right)=\frac{\left|\right|P^{\&prime;}\left(S\right)\&times;P^{\&prime;\&prime;}\left(S\right)\left|\right|}{\left|\right|P^{\&prime;}\left(S\right)\left|\right|}=\frac{\left|\right|u\left(S\right)\&times;u^{\&prime;}\left(S\right)\left|\right|}{h}=\frac{\left|\right|u^{\&prime;}\left(S\right)\left|\right|}{h}---\left(24\right)$

Drawn by formula (19), curvature is expressed as:

[numerical expression 91]

$\mathrm{\&kappa;}\left(S\right)=\frac{\sqrt{{\mathrm{\&alpha;}}^{\&prime;2}+{\mathrm{\&beta;}}^{\&prime;2}{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&alpha;}}}{h}$ $---\left(25\right)$

(5) feature of three-dimensional clothoid

(a) continuity of curve

In a rondo line segment (with the clothoid of same parametric representation), because quadratic expression long by curve respectively or the long variable S of curve provides the inclination angle and the deflection angle of its tangential direction, so about the long variable S of curve, it is continuous can guaranteeing its normal direction that obtains of 1 subdifferential, reach its curvature that obtains of 2 subdifferentials.In other words, in a rondo line segment, normal direction and curvature are continuous.Therefore, can obtain smoothness, well-behaved curve.Even under the situation that links two clothoids,,, can make the curve of a smooth connection by selecting parameter for tangent line, normal, curvature on its joint reach continuously.Be referred to as the batten that circles round.

(b) applicability

Because the tangential direction that enough two angles of energy (inclination angle and deflection angle) are shared curve so can make the three-dimensional curve that meets various conditions arbitrarily, can be used in various uses.

(c) with the conformability of geometrical curve

Geometrical curves such as straight line, circular arc, spiral curve can place 0 by the several of the parameter of will circling round, or set specific funtcional relationship and make between Several Parameters.These curves are a kind of of clothoid, can adopt the form performance of circling round.Therefore, do not need the NC as in the past, handle, can adopt identical form to calculate or control by described forms of variation such as straight line, circular arc, free curves.

In addition, owing to, can make two dimension and circle round, so the two dimension resource of circling round and having obtained just before can using by usually any among α or the β being placed 0.

Just,, comprise that the two dimension of having known circles round, can also show other curve such as circular arc or straight line by suitable setting α or β.Owing to for such other curve, can adopt the three-dimensional clothoid formula of same form, therefore can simplify the calculating formality.

(d) formedness of Tui Ceing

In interpolations in the past such as spline interpolation, when making the free curve formulation, many difficult separately its whole forms or local form, but in three-dimensional is circled round, by imagining inclination angle and deflection angle separately, can be than being easier to hold overall image.

In addition, at the end midway as the clothoid performance, the value of line length, tangential direction, curvature etc. is known, does not need to recomputate as interpolation in the past.Just, with the parameter S of curve accordingly, shown in formula (7), (20) and (26), directly obtain the tangent line of curve or normal, curvature.This is very effective feature for mathematical control mode described later.So, can shorten computing time significantly, save resources such as storer, in addition, can carry out real-time interpolation operation.

In NC processing, the minimum profile curvature radius of tool path is an important problem, in spline interpolation etc., to find the solution it, need complicated calculation, but in circling round, because generally at each line segment, the value of minimum profile curvature radius is known, so in the selected grade in cutter footpath, be favourable.

(e) easiness of motion control

The master variable of curve is length s or standardized length S, and the curve's equation formula uses the natural equation with respect to this length to provide.Therefore,, kinetic characteristics such as acceleration-deceleration can be provided arbitrarily,, the high speed of operation can be sought to process by adopting the used good curve movement of characteristic such as cam in the past by determining length s as the function of time t.Because the value that can be used as in the in esse cartesian space provides length s, obtains speed, acceleration with respect to tangential direction, so do not need as interpolation in the past synthetic by each value that provides.In addition, because the calculating of curvature is easy, the centrifugal acceleration when therefore also obtaining motion easily can meet the control of movement locus.

(6) character of the generation of curve and each parameter

According to definition, each parameter of three-dimensional clothoid is as follows to the influence of curve.By providing each parameter, can generate three-dimensional clothoid as shown in figure 49.

Table 20 has gathered the character of each parameter of three-dimensional clothoid.

Table 20

Parameter	The meaning
		P 0	Parallel moving three dimension clothoid
h	Determine the size of three-dimensional clothoid
		a 0，b 0	Rotate three-dimensional clothoid
a 1，a 2，b 1，b 2	Determine the shape of three-dimensional clothoid

2. adopt the interpolation of three-dimensional clothoid

(1) mathematic condition of Liu Chang connection

In 1 three-dimensional clothoid, the performance of the shape of curve has boundary.Herein, be fundamental purpose with the motion control of the instrument that utilizes Numerical Control, many connect three-dimensional clothoid (three-dimensional circle round line segment), by the circle round motion of line segment control tool of these many three-dimensionals.

Connect 2 three-dimensional clothoids glibly at its end points, being defined as is continuous connection end point position, tangent line and curvature.Adopt above-mentioned definition, by this condition of following narration.The initial locative continuity of 3 formulas, next 2 formulas are represented the continuity of tangent line, and next 1 formula is represented the unanimity of normal, and last formula is represented the continuity of curvature.

[numerical expression 92]

Px _i(1)＝Px _i+1(0)

Py _i(1)＝Py _i+1(0)

Pz _i(1)＝Pz _i+1(0)

α _i(1)＝α _i+1(0)

β _i(1)＝β _i+1(0) (26)

tanγ _i(1)＝tanγ _i+1(0)

κ _i(1)＝κ _i+1(0)

This be satisfy tangent line vector and normal vector continuously, curvature and α, β be in the tie point condition for continuous, condition is tight excessively sometimes.Therefore, also can satisfy condition by change condition shown below singlely.

[numerical expression 93]

Px _i(1)＝Px _i+1(0)

Py _i(1)＝Py _i+1(0)

Pz _i(1)＝Pz _i+1(0)

cos[α _i(1)-α _i+1(0)]＝1

cos[β _i(1)-β _i+1(0)]＝1 (27)

tanγ _i(1)＝tanγ _i+1(0)

κ _i(1)＝κ _i+1(0)

Herein, in addition,

[numerical expression 94]

cos[α _i(1)-α _i+1(0)]＝1

Also take into account if will go up relation of plane,

[numerical expression 95]

tanγ _i(1)＝tanγ _i+1(0)

Used the conditional substitutions of note down.

[numerical expression 96]

tanγ _i(1)＝tanγ _i+1(0)

$\frac{{{\mathrm{\&alpha;}}^{\&prime;}}_i\left(1\right)}{{{\mathrm{\&beta;}}^{\&prime;}}_i\left(1\right)\cos {\mathrm{\&alpha;}}_i\left(1\right)}=\frac{{{\mathrm{\&alpha;}}^{\&prime;}}_{i+1}\left(0\right)}{{{\mathrm{\&beta;}}^{\&prime;}}_{i+1}\left(0\right)\cos {\mathrm{\&alpha;}}_{i+1}\left(0\right)}$

∵α′ _i(1)β′ _i+1(0)＝α′ _i+1(0)β′ _i(1)

The result draws, if satisfy the condition of note down, can achieve the goal.

[numerical expression 97]

Px _i(1)＝Px _i+1(0)

Py _i(1)＝Py _i+1(0)

Pz _i(1)＝Pz _i+1(0)

cos[α _i(1)-α _i+1(0)]＝1

cos[β _i(1)-β _i+1(0)]＝1 (28)

α _i′(1)β _i+1′(0)＝α _i+1′(0)β _i′(1)

κ _i(1)＝κ _i+1(0)

In formula (28), the initial locative continuity of 3 formulas, next 2 formulas are represented the continuity of tangent line, and next 1 formula is represented the unanimity of normal, and last formula is represented the continuity of curvature.Carry out G ²Continuous interpolation needs 2 three-dimensional clothoids to satisfy 7 conditionals of formula (28) at its end points.

About G ²(G is the prefix of Geometry) replenishes continuously.Figure 50 represents G ²The condition of continuous interpolation.

So-called G ⁰Refer to 2 three-dimensional clothoids continuously at its endpoint location unanimity, so-called G ¹Refer to the tangential direction unanimity continuously, so-called G ²Refer to contact plane (normal) and curvature unanimity continuously.The used C of contrast SPL in following table 21 ⁰～C ²The used G of clothoid continuous and of the present invention ⁰～G ²Continuously.

Table 21

C 0: the position	G 0: the position
		C 1: a differential coefficient	G 1: tangential direction
C 2: the second differential coefficient	G 2: contact plane (normal), curvature

When considering the continuity of 2 three-dimensional clothoids, along with reaching C ⁰→ C ¹→ C ², G ⁰→ G ¹→ G ², it is tight that the interpolation condition becomes.At C ¹Need the size and the direction of tangent line all consistent continuously, but at G ¹Can have only the tangential direction unanimity continuously.When connecting tangent line reposefully, preferably use G with 2 three-dimensional clothoids ¹Make conditional continuously.As SPL, if use C ¹Make conditional continuously, make in the condition of the same size of unallied tangent line geometrically, so condition is tight excessively owing to increase.If use G ¹Make conditional continuously, have the advantage of the size of freely setting a differential coefficient.

At G ²Make contact plane (normal) unanimity continuously.So-called contact plane shown in Figure 51, refers to planar S 1, S2 that curve C is contained in the part.It is continuous that Figure 51 is illustrated in a P tangential direction, but the discontinuous example of contact plane S1, S2.When considering the continuity of three-dimensional curve, what must consider after the unanimity of tangential direction is the unanimity of contact plane.When words curvature, do not mean that contact plane is inconsistent, need after making the contact plane unanimity, make the curvature unanimity.Make coordinate, tangential direction, contact plane (normal) and consistent the reaching of curvature satisfy G with 2 three-dimensional curves ²Condition for continuous.

(2) concrete computation sequence

Have following 2 kinds of computation sequences.

(a) provide parameter h, α, the β of curve, 1 three-dimensional clothoid takes place,,, determine the parameter of next three-dimensional clothoid to satisfy the mode of formula (28) at its end points.So, the smooth one by one three-dimensional clothoid that connects can take place.According to this computation sequence, calculate parameter of curve easily, be referred to as along separating.According to this mode, the curve of multiple shape can take place easily, but the tie point that can not clear and definite assignment curve passes through.

(b) can become the mode of the tie point of curve with preassigned point group, connect three-dimensional clothoid.Make short clothoid (line segment circles round) in each interval of each point range that provides arbitrarily discretely herein.In such cases, determine that in the mode that satisfies formula (28) computation sequence of parameter of curve is more complicated than (a), for repeating to bring together calculating.Owing to determine parameter of curve on the contrary from condition of contact, this computation sequence is called contrary separating.

Separate about the contrary of above-mentioned (b), at length narrate computing method.The computational problem that solves is by following quilt formulism.

Unknown parameter: parameter of curve

Constraint condition: formula (28) or one portion

Problem as requested, the quantity of variation constraint condition can be used as the parameter of curve that unknown parameter is set the quantity that conforms to it.For example, can freely make a part of parameter of curve work under the successional situation of curvature not requiring.Perhaps, in curvature continuously and specify under the situation of tangential direction, need increase corresponding unknown parameter of curve by cutting apart the quantity that increases the used three-dimensional clothoid of interpolation.

Above-mentionedly to repeat to bring together calculation stability and bring together in order to make, need on calculating, work hard.For fear of dispersing of calculating, accelerate to bring together, for unknown parameter, effective method is to set better initial value.Therefore, effective method is, interpolation curve constraint conditions such as tie point, more single that provides takes place to satisfy, and line transect curve etc. is for example calculated the parameter of curve of three-dimensional clothoid from its curve shape, as the initial value that repeats to bring together calculating.

Perhaps, the gas that differs satisfies the constraint condition that should satisfy, but increases the mode of conditional successively, also is effective as the stable method that obtains separating.For example, the order that curve is taken place is divided into three following STEP, carries out successively.As the 1st STEP after the mode interpolation consistent with tangential direction with positional information, as the 2nd STEP so that the mode of normal direction unanimity is carried out interpolation, in the mode interpolation of the 3rd STEP with the curvature unanimity.Figure 52 represents the concise and to the point flow process of this method.The necessary three-dimensional clothoid formula and the definition of tangent line, normal or curvature thereof have been shown.

(3) embodiment of the interpolation of the three-dimensional clothoid of employing

(a) flow process of interpolation

Describe in detail to adopt a three-dimensional clothoid embodiment of the method between the point range that provides of interpolation glibly.Below, the interpolation that adopts three-dimensional clothoid is called the three-dimensional interpolation of circling round.To all be called three-dimensional clothoid by the group of curves that interpolation generates, the unit curve that constitutes it will be called the three-dimensional line segment that circles round.

As the circle round basic flow process of interpolation of three-dimensional, circle round each parameter of line segment as unknown number with the three-dimensional between the point that links the interpolation object, the point by the interpolation object closely, and obtain the satisfied G that reaches with Newton-Raphson method ²Separating of condition for continuous, formation curve.Figure 53 is a diagram of concluding the summary of this flow process.So-called G ²Continuously, refer to 2 three-dimensional clothoids at its end points, position, tangential direction, normal direction and curvature unanimity.

(b) G ²The condition of continuous interpolation

In three-dimensional is circled round interpolation,, and become G about the point by the interpolation object closely ²Condition for continuous is considered concrete condition.

Now, have 3 some P simply ₁={ Px ₁, Py ₁, Pz ₁, P ₂={ Px ₂, Py ₂, Pz ₂And P ₃={ Px ₃, Py ₃, Pz ₃, consider with three-dimensional this point of line segment interpolation that circles round.Figure 54 represents a P ₁, P ₂And P ₃The three-dimensional interpolation of circling round.If with point of contact P ₁, P ₂Between curve setting be curve C ₁, with point of contact P ₂, P ₃Between curve setting be curve C ₂, in such cases, unknown number is a curve C ₁Parameter a0 ₁, a1 ₁, a2 ₁, b0 ₁, b1 ₁, b2 ₁, h ₁, curve C ₂Parameter a0 ₂, a1 ₂, a2 ₂, b0 ₂, b1 ₂, b2 ₂, h ₂Deng 14.In addition, the subscript of the literal that occurs in explanation later is corresponding with the subscript of each curve.

Consider closely point below, and reach G by the interpolation object ₂Condition for continuous.At first, the condition of the point by the interpolation object closely at initial point, if consider from the definition of three-dimensional clothoid, owing to when providing initial point, must reach, so there is not the interpolation condition.Then at tie point P ₁, aspect the position, set up 3, aspect the tangent line vector, set up 2, aspect the size and Orientation of curvature condition for continuous formula, set up 2, add up to and set up 7.In addition about terminal point, at a P ₂Be 3 aspect the position.Add up to 10 by the above conditional that draws.But, after this manner for 14 of unknown numbers, because only there are 10 in conditional, so can not obtain separating of unknown number.Therefore, in this research, provide the tangent line vector of two-end-point, respectively increase by two conditions, the number of conditional and unknown number is equated at two-end-point.In addition, if determine tangential direction, owing to can obtain a0 from its definition at initial point ₁, b0 ₁So, can not handle as unknown number.Below, consider each condition.

At first, if consider the condition of position, set up 3 formulas of note down by formula (1-1), (1-2), (1-3).(below, regulation natural number i＜3.)

[numerical expression 98]

$P{x}_i+h_i{\&Integral;}_0^1\cos (a{0}_i+a{1}_{i}S+a{2}_i{S}^2)\cos (b{0}_i+b{1}_{i}S+b{2}_i{S}^2)\mathrm{dS}-P{x}_{i+1}=0$

(1-1)

$P{y}_i+h_i{\&Integral;}_0^1\cos (a{0}_i+a{1}_{i}S+a{2}_i{S}^2)\sin (b{0}_i+b{1}_{i}S+b{2}_i{S}^2)\mathrm{dS}-P{y}_{i+1}=0$

(1-2)

$P{z}_i+h_i{\&Integral;}_0^1(-\sin (a{0}_i+a{1}_{i}S+a{2}_i{S}^2))\mathrm{dS}-P{z}_{i+1}=0$

(1-3)

Then, if consider tangential direction, set up (1-4), (1-5) 2 formulas.

[numerical expression 99]

cos(a0 _i+a1 _i+a2 _i-a0 _i+1)＝1 (1-4)

cos(b0 _i+b1 _i+b2 _i-b0 _i+1)＝1 (1-5)

About the size of curvature κ, set up following formula (1-6).

[numerical expression 100]

κ _i(1)-κ _i+1(0)＝0 (1-6)

Consider normal direction vector n at last.The normal vector n of three-dimensional clothoid is with formula (21) expression.

Herein, with the tangent line vector u of three-dimensional clothoid determine equally, also adopt rotation, consider normal vector n.For initial stage tangential direction (1,0,0), suppose and adopt constant γ, with (0, cos γ ,-sin γ) expression initial stage normal direction.If make its rotation in the same manner with tangent line, normal n is represented suc as formula (1-7).

[numerical expression 101]

$n\left(S\right)=\left[\begin{array}{ccc}\cos \mathrm{\&beta;}\left(S\right)& -\sin \mathrm{\&beta;}\left(S\right)& 0\\ \sin \mathrm{\&beta;}\left(S\right)& \cos \mathrm{\&beta;}\left(S\right)& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos \mathrm{\&alpha;}\left(S\right)& 0& \sin \mathrm{\&alpha;}\left(S\right)\\ 0& 1& 0\\ -\sin \mathrm{\&alpha;}\left(S\right)& 0& \cos \mathrm{\&alpha;}\left(S\right)\end{array}\right]\left\{\begin{array}{c}0\\ \cos \mathrm{\&gamma;}\\ -\sin \mathrm{\&gamma;}\end{array}\right\}$

$=\left\{\begin{array}{c}-\sin \mathrm{\&gamma;}\cos \mathrm{\&beta;}\left(S\right)\sin \mathrm{\&alpha;}\left(S\right)-\cos \mathrm{\&gamma;}\sin \mathrm{\&beta;}\left(S\right)\\ -\sin \mathrm{\&gamma;}\sin \mathrm{\&beta;}\left(S\right)\sin \mathrm{\&alpha;}\left(S\right)+\cos \mathrm{\&gamma;}\cos \mathrm{\&beta;}\left(S\right)\\ -\sin \mathrm{\&gamma;}\cos \mathrm{\&alpha;}\left(S\right)\end{array}\right\}$

(1-7)

Comparison expression (21), (1-7) learn, sin γ, cos γ are corresponding with formula (1-8).

[numerical expression 102]

$\sin \mathrm{\&gamma;}=\frac{{\mathrm{\&alpha;}}^{\&prime;}\left(S\right)}{\sqrt{{\mathrm{\&alpha;}}^{\&prime;}{\left(S\right)}^2+{\mathrm{\&beta;}}^{\&prime;}{\left(S\right)}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&alpha;}\left(S\right)}}$

$\cos \mathrm{\&gamma;}=\frac{{\mathrm{\&beta;}}^{\&prime;}\left(S\right)\cos \left(S\right)}{\sqrt{{\mathrm{\&alpha;}}^{\&prime;}{\left(S\right)}^2+{\mathrm{\&beta;}}^{\&prime;}{\left(S\right)}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&alpha;}\left(S\right)}}$

(1-8)

That is, (1-8) learns by formula, and it is continuous to reach the normal at tie point that three-dimensional circles round in the interpolation, as long as tan γ is just passable continuously.

[numerical expression 103]

$\tan \mathrm{\&gamma;}=\frac{{\mathrm{\&alpha;}}^{\&prime;}\left(S\right)}{{\mathrm{\&beta;}}^{\&prime;}\left(S\right)\cos \mathrm{\&alpha;}\left(S\right)}---(1-9)$

That is, learn that the normal condition for continuous is formula (1-10).

[numerical expression 104]

tanγ _i(1)＝tanγ _i+1(0) (1-10)

Herein, in addition, if will

[numerical expression 105]

cos[α _i(1)-α _i+1(0)]＝1 (1-11)

Take conditional (1-10), conditional (1-12) displacement of available note down into account.That is, the normal condition for continuous is formula (1-12).

[numerical expression 106]

α′ _i(1)β′ _i+1(0)＝α′ _i+1(0)β′ _i(1) (1-12)

In sum, learn,, and reach G closely by the interpolation object-point ²Condition for continuous is formula (1-13) at tie point.In addition, even at the initial point terminal point, also can select several conditions wherein.

[numerical expression 107]

Px _i(1)＝Px _i+1(0)

Py _i(1)＝Py _i+1(0)

Pz _i(1)＝Pz _i+1(0)

cos[α _i(1)-α _i+1(0)]＝1

cos[β _i(1)-β _i+1(0)]＝1 (1-13)

α _i′(1)β _i+1′(0)＝α _i+1′(0)β _i′(1)

κ _i(1)＝κ _i+1(0)

Learn by above, for unknown number a1 ₁, a2 ₁, b1 ₁, b2 ₁, h ₁, a0 ₂, a1 ₂, a2 ₂, b0 ₂, b1 ₂, b2 ₂, h ₂Deng 12, conditional is set up 12 that remember down.(will put P ₃On the tangential direction rotation angle be defined as α ₃, β ₃)

[numerical expression 108]

Px ₁(1)＝Px ₂(0)

Py ₁(1)＝Py ₂(0)

Pz ₁(1)＝Pz ₂(0)

cos[α ₁(1)-α ₂(0)]＝1

cos[β ₁(1)-β ₂(0)]＝1

α′ ₁(1)β′ ₂(0)＝α′ ₂(0)β′ ₁(1)

κ ₁(1)＝κ ₂(0) (1-14)

Px ₂(1)＝Px ₃(0)

Py ₂(1)＝Py ₃(0)

Pz ₂(1)＝Pz ₃(0)

cos[α ₂(1)-α ₃]＝1

cos[β ₂(1)-β ₃]＝1

So, owing to set up 12 formulas for 12 unknown numbers, so can find the solution.This available Newton-Raphson method is explained, obtained and separate.

In addition, generally when considering n point range of interpolation, conditional is just passable as long as above-mentioned natural number i is expanded as i＜n.Be the problem of the quantity of unknown number and conditional then.

For example, when having n-1 point range, regard as and set up N unknown number and N relational expression.Herein, if hypothesis increases by 1 point again, unknown number just increases the three-dimensional line segment P that circles round _N-1, P _n7 parameter a0 that circle round _n, a1 _n, a2 _n, b0 _n, b1 _n, b2 _nAnd h _nOn the one hand, conditional is because tie point increases by 1, so at a P _N-1, aspect the position, increase by 3, aspect the tangent line vector, increase by 2, aspect the size and Orientation of curvature condition for continuous formula, increase by 2, adding up to increases by 7.

Owing to learn that when n=3 unknown number, relational expression all are 12,, also be 7 (n-2)+5 to the formula of this processing so in n 〉=3 o'clock, unknown number is 7 (n-2)+5.So, because the number of unknown number and relevant with it condition equates, so when the individual free point range of n, also can same method find the solution with 3 the time.As solving method, adopt and utilize between unknown number and conditional, to become the newton-pressgang Shen method of the relation of vertical (1-15), (1-16) to find the solution.(condition is made as F, will know that number is made as u, the error Jacobi matrix is made as j.)

[numerical expression 109]

ΔF＝[J]Δu (1-15)

Δu＝[J] ^-1ΔF (1-16)

Learn by above, also can carry out closely point for n point range, and can carry out G by the interpolation object ²The continuous three-dimensional interpolation of circling round.

(c) initial value determines

In newton-pressgang Shen method, when the exploration that begins to separate, need provide suitable initial value.Initial value how to provide can, but an example of only narrating this initial value herein provides mode.

The research of carrying out is 3D Discrete Clothoid Splines in advance, has closely by the interpolation object-point, and curvature is with respect to the character from the displacement smooth change of initial point.Therefore, in this research, be used for the circle round initial value of interpolation of three-dimensional, by making as the polygon Q of the 3D DiscreteClothoid Splines of the r=4 of Figure 55, definite by calculating from here.

Below, supplementary notes 3D Discrete Clothoid Splines.At first shown in Figure 56, make the polygon P that classifies the summit with the point of interpolation object as, between each summit of P, insert each identical several r new summits, be made as the polygon Q of P  Q.,, under the situation that polygon Q closes, have a r n summit herein, under the situation that polygon Q opens, have r (n-1)+1 summit if be n with the vertex of P.Regulation as the consecutive number from initial point, is represented each summit with qi with subscript later on.In addition, on each summit, determine to determine to have the vector k of curvature κ as size from normal vector b as direction.

At this moment, reach equidistant formula (1-17) each other with satisfying the summit of note down, during condition that curvature displacement the most approaching and from initial point is directly proportional (when making the function minimization of formula (1-18)) polygon Q, be called 3D Discrete Clothoid Splines.

[numerical expression 110]

|q _i-1q _i|＝|q _i+1q _i|，(q _iP) (1-17)

$\underset{i=1}{\overset{r-1}{\mathrm{\&Sigma;}}}{\left|\right|{\mathrm{\&Delta;}}^2{k}_{\mathrm{ir}+1}\left|\right|}^2,$ i＝{0...n-1}，Δ ²k _i＝k _i-1-2k _i+k _i+1

(1-18)

In 3D Discrete Clothoid Splines, obtained the Fu Leinie frame on each summit.Therefore, obtain parameter a from its unit tangent direction vector t ₀, b ₀This tangential direction vector t known when obtaining polygon Q, and the formula of the tangent line by this t and three-dimensional clothoid is obtained tangential direction rotation angle α, the β on the summit of polygon Q.Obtain a of each curve thus ₀, b ₀Initial value.In addition, on the three-dimensional that begins from initial point is circled round line segment, provide its value.

[numerical expression 111]

$u=\left\{\begin{array}{c}\cos \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ \sin \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ -\sin \mathrm{\&alpha;}\end{array}\right\}---(1-19)$

Herein, about 3D Discrete Clothoid Splines, if consider equidistant arrangement summit, at the some q of Figure 55 _4i+1, can be similar to the long variable S of curve is 1/4.Equally at a q _{4 (i+1)-1}, can be similar to the long variable S of curve is 3/4.If lump together with the formula of the α of three-dimensional clothoid and to consider these, set up following formula (1-20).

[numerical expression 112]

$\left\{\begin{array}{c}a{0}_{4i}+\frac{1}{4}a{1}_{4i}+{\left(\frac{1}{4}\right)}^{2}a{2}_{4i}=a{0}_{4i+1}\\ {a0}_{4i}+\frac{3}{4}a{1}_{4i}+{\left(\frac{3}{4}\right)}^{2}a{2}_{4i}=a{0}_{4(i+1)-1}\end{array}\right.---(1-20)$

It is a1 that this formula becomes unknown number _4iAnd a2 _4iTwo-dimentional simultaneous equations, it is found the solution, as parameter a ₁, a ₂Initial value.Equally also can determine parameter b ₁, b ₂Initial value.

Remaining unknown number is the long h of curve, but can be calculated by the formula of three-dimensional rondo curvature of a curve about this initial value.Three-dimensional rondo curvature of a curve can be used formula (1-21) expression.

[numerical expression 113]

$\mathrm{\&kappa;}=\frac{\sqrt{{\mathrm{\&alpha;}}^{\&prime;2}+{\mathrm{\&beta;}}^{\&prime;2}{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&alpha;}}}{h}---$ $(1-21)$

If change this formula, become formula (1-22), can determine the initial value of h.

[numerical expression 114]

${\mathrm{<figure-callout\; id="4i"\; label="h"\; filenames="A20058000605301142.png"\; state="\{\{state\}\}">h</figure-callout>}}_{4i}=\frac{\sqrt{{(\mathrm{<figure-callout\; id="1"\; label="a"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">a</figure-callout>}{1}_{4i}+2\mathrm{<figure-callout\; id="2"\; label="a"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">a</figure-callout>}{2}_{4i})}^2+{(\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}{1}_{4i}+2\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}{2}_{4i})}^2{\cos }^2(a{0}_{4i}+\mathrm{<figure-callout\; id="1"\; label="a"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">a</figure-callout>}{1}_{4i}+\mathrm{<figure-callout\; id="2"\; label="a"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">a</figure-callout>}{2}_{4i})}}{{\mathrm{\&kappa;}}_{4(i+1)}}$

(1-22)

With above method, can determine initial value to 7 three-dimensionals parameter of circling round.Adopt this initial value of determining, that narrates in (b) reaches G ²Under the condition for continuous, obtain the approximate value of the parameter of each curve with newton-pressgang Shen method.Generate the three-dimensional line segment that circles round from the parameter that obtains thus, with between three-dimensional clothoid interpolation point range.

(b) interpolation example

As the example of reality with above-described method interpolation point range, circle round interpolation (0.0,0.0,0.0), (2.0,2.0,2.0), (4.0,0.0,1.0) and (5.0,0.0,2.0) this example of 4 of three-dimensional for example.The skeleton view of the three-dimensional clothoid that generates by interpolation shown in Figure 57.Solid line among Figure 57 is three-dimensional clothoid, and dotted line, a dot-and-dash line, two dashdotted straight lines are that the size of getting on the each point on the curve is log (radius-of-curvature+natural logarithm e), and getting direction is the radius-of-curvature changing pattern of normal direction.

In addition, the parameter of each curve shown in the table 22, this appearance 23 are illustrated in coordinate, tangent line, the normal of each tangent point, the deflection of curvature.Find out from these tables, become G at each tangent point ²The continuous three-dimensional clothoid.In addition, Figure 58 gets displacement from initial point at transverse axis, gets the curved transition curve map of curvature at the longitudinal axis.

Table 22

The circle round parameter of line segment of each three-dimensional

Curve 1 (radius-of-curvature changing pattern dotted line)
	α＝-0.657549-1.05303S+1.84584S 2 β＝1.03297+1.29172S-2.55118S 2 h＝3.82679 P 0＝(0.0、0.0、0.0)
Curve 2 (radius-of-curvature changing pattern one dot-and-dash line)
	α＝0.135559+2.18537S-2.69871S 2 β＝-0.226655-3.15603S+3.03298S 2 h＝3.16932 P 0＝(2.0、2.0、2.0)
Curve 3 (radius-of-curvature changing pattern two dot-and-dash lines)
	α＝-0.377569-1.45922S+0.984945S 2 β＝-0.349942+1.32198S-0.873267S 2 h＝1.43987 P 0＝(4.0、0.0、1.0)

Table 23

Coordinate, tangent line, normal, the skew curve 1 of curvature and the tie point of curve 2 at each tangent point
	Coord：(1.16×10 -5，2.00×10 -6，3.82×10 -6) Tvector：(7.59×10 -5，1.50×10 -5，2.95×10 -4) Nvector：(2.93×10 -4，9.19×10 -5，-7.57×10 -6) Curvature：3.06×10 -7
The tie point of curve 2 and curve 3
	Coord：(-4.33×10 -6，-1.64×10 -6，1.11×10 -5) Tvector：(2.06×10 -6，2.33×10 -4，1.97×10 -4) Nvector：(3.30×10 -4，1.19×10 -5，-3.23×10 -5) Curvature：5.96×10 -6

(4) consideration is at the G of the control of each value at two ends ²The continuous three-dimensional interpolation of circling round

(a) interpolation condition and unknown number

As described in (3), under the situation that curve is opened, when the point of interpolation object has n, with n-1 the curve three-dimensional interpolation point range of circling round.If strictly by each point, about each three-dimensional line segment that circles round, because unknown number has a ₀, a ₁, a ₂, b ₀, b ₁, b ₂, 7 of h etc., so unknown number integral body is that 7 (n-1) are individual.On the other hand,, all there are 3 of coordinate on coordinate, tangent line, normal, curvature each 7 and the terminal point, so all be 7 (n-2)+3 owing to have the tie point of n-2 about conditional.In the method for (3), by it being provided the tangent line vector on the initial point terminal point, increase by 4 conditions, make the number of conditional and unknown number relative.

Herein, if the tangent line normal curvature on the control initial point terminal point, and to reach G ²Compare when continuous mode interpolation, the tangent line at condition and control two ends,, respectively increasing by 2 aspect the normal curvature, add up to increase by 4 at the initial point terminal point.So conditional all reaches 7n-3.In such cases, because the number of unknown number lacks than condition, so can not find the solution with newton-pressgang Shen method.Therefore, need how to increase unknown number.

Therefore,, the number of unknown number and conditional is equated herein by inserting the interpolation object-point again.For example,, insert 2 new points, as 2 in the coordinate of unknown number processing each point if a side of 4 unknown numbers is many.

In such cases, because tie point increases by 2, so increase each 7 14 of coordinate, tangent line, normal, curvature for each tie point condition.On the other hand, the line segment because 2 three-dimensionals of unknown number increase are circled round is so increase a ₀, a ₁, a ₂, b ₀, b ₁, b ₂, h each 14 of total of 7.Because the number of the contained point of point range this moment is n+2, so if wholely consider that it is individual that unknown number reaches 7 (n+1), conditional reaches 7 (n+1)+4.In addition, suppose that unknown number just increases by 4 as 2 in the coordinate of the new point that inserts of unknown number processing herein.So unknown number, conditional all are 7 (n+2)-3, can obtain separating of unknown number.So, by inserting new point, can carry out the each point by providing closely, G ²The interpolation of the continuous and tangent line normal curvature of having controlled two-end-point.

In addition, also consider general situation.When interpolation n point range, consider the number of the point that when m project of two-end-point control, inserts and at this number as the coordinate of unknown number processing.The front was was also recorded and narrated, but when curve is opened, with n-1 curve interpolating point range.If because closely by each point, for each three-dimensional line segment that circles round, unknown number just has a ₀, a ₁, a ₂, b ₀, b ₁, b ₂, 7 of h etc., so unknown number integral body has 7 (n-1) individual.On the one hand, about conditional, all have 3 of coordinate on coordinate, tangent line, normal, curvature each 7 and the terminal point owing to have the tie point of n-2, so all be 7 (n-2)+3, conditional is few, is 4.Just, in project that two-end-point will be controlled more than 4.Below, the natural number, the k that are described in m in the explanation and are more than 4 are the natural number more than 2, the method that the number of conditional and unknown number is equated.

(i) during m=2k

When lumping together m=2k project of control at two ends, unknown number integral body is that 7 (n-1) are individual, and conditional integral body is 7 (n-1)-4+2k.At this moment, Guo Sheng conditional is 2k-4.Now, if consider to insert again k-2 point, line segment increases the k-2 root because three-dimensional is circled round, and tie point increases k-2, so unknown number integral body has 7 (n+k-3) individual, conditional integral body is 7 (n+k-3)-4+2k., suppose that in addition unknown number integral body is that 7 (n+k-3)+2 (k-2) are individual as 2 (for example x, y) in the seat target value of the new each point that inserts of unknown number processing herein, conditional integral body is that 7 (n+k-3)+2 (k-2) are individual, and the number of unknown number and conditional equates.

(ii) during m=2k+1

When lumping together m=2k+1 project of control at two ends, unknown number integral body is that 7 (n-1) are individual, and conditional integral body is 7 (n-1)+2k-3.At this moment, Guo Sheng conditional is 2k-3.Now, if consider to insert again k-1 point, line segment increases the k-1 root because three-dimensional is circled round, and tie point increases k-1, so unknown number integral body is that 7 (n+k-2) are individual, conditional integral body is 7 (n+k-2)-3+2k., suppose that in addition unknown number integral body is that 7 (n+k-2)+2 (k-2) are individual as 2 (for example x, y) in the seat target value of the new each point that inserts of unknown number processing herein, conditional integral body is that 7 (n+k-2)+2k-3) are individual, many 1 of the number of conditional.Therefore, on 1 point in the point that when m=2k+1, insert, only handle 1 that sits in the target value as unknown number.Thus, unknown number integral body is that 7 (n+k-2)+2 (k-2) are individual, and conditional integral body is that 7 (n+k-2)+2 (k-2) are individual, and the number of unknown number and conditional equates.

Method as previously discussed, even adding together by number with the condition of appending, the number that becomes unknown number in the coordinate of the point that adjustment is inserted, under all situations during for example tangent line swing angle α beyond control tangent line, normal, the curvature etc., also can make the number of unknown number and conditional relative, can control each value of two-end-point in theory.In addition, about the number of control project and unknown number, conditional, table 24 is listed the number that gathers

Table 24

In the interpolation that n is ordered at the number of the control project at two ends and unknown number, conditional

Want the item number controlled	The conditional that increases	The number of the point that inserts	Be added in 1 number of going up the coordinate of handling as unknown number	The unknown number that increases
					4	0	0	0	0
5	1	1	1	1
					6	2	1	2	2
7	3	2	1 point: 21 points: 1	3
					

2k	2k-4	k-2	K-2 point: 2	2k-4
					2k+1	2k-3	k-1	K-2 point: 21 some t:1	2k-3

^*The natural number that k:2 is above

(b) method

The interpolation that employing is circled round in the three-dimensional of each value of initial point terminal point control shown in Figure 59 and Figure 60, is undertaken by following flow process.

Step1) only adopt in the condition that will control 4, carry out closely by the interpolation object-point, and G ²Continuous interpolation, formation curve.

Step2) on the curve that generates, insert new point, the number of regularization condition formula and unknown number.

Step3) with the parameter of curve of Step1 as initial value, obtain the approximate value of the parameter of each curve that satisfies the purpose condition with newton-pressgang Shen method.

Below, each Step is remarked additionally.At first in Step1, as long as the control tangential direction just can adopt the method formation curve of (3).In addition, even under the situation of not controlling tangential direction, the initial value during as the parameter of obtaining this curve also adopts the initial value identical with the method for (3).

Then, in Step2, insert new point, carry out the adjustment of the number of conditional and unknown number.At this moment, the new point that inserts, between each interpolation object-point as much as possible below 1.In addition, as the point that inserts, insert with linking the circle round point of centre of line segment of the mutual three-dimensional of interpolation object in the Step1 generation.In addition, the point of insertion will insert successively from two ends.Just, the initial point that the inserts point that is initial point and its adjacency between and the point of terminal point and its adjacency between.

Be about Step3 at last, but need redefine the initial value that is used for the newton-pressgang Shen method of carrying out at Step3.Therefore, cut apart curve, from determining respectively being worth of curve that generates for inserting the curve of newly putting, adopting by (1-4) described method of cutting apart three-dimensional clothoid.For the curve of insertion point not, directly adopt the value of the curve that generates at Step1.More than, determined the initial value of each parameter of the curve in Step3.Adopt this initial value, from generating three-dimensional clothoid, with satisfying between the three-dimensional clothoid interpolation point range of purpose condition with the parameter that newton-pressgang Shen method obtains.

(C) interpolation example

Illustrate in the mode of reality, carry out the circle round example of interpolation of three-dimensional with the tangent line at the condition of table 25 control two ends, normal, curvature.Point to the interpolation object that should pass through is closely shared consecutive number, forms P ₁, P ₂And P ₃

The condition of table 25 interpolation object each point and initial point terminal point

	Coordinate	The unit tangent vector	The principal normal vector	Curvature
					P 1	(0，0，0)	(Cos(θ)，Sin(θ)，0)	(-Sin(θ)，Cos(θ)、0)	0.2
P 2	(4，-4，-4)	-	-	-
					P 3	(8，-4，-5)	(1，0，0)	(0，-1，0)	0.2

^*θ＝-(π/6)

Figure 61 represents the actual with this understanding result who carries out interpolation.The three-dimensional clothoid of the curve representation of solid line, dotted line one dot-and-dash line two dot-and-dash lines three dot-and-dash lines represent that the radius-of-curvature of each curve changes.In addition, Figure 62 is expression from the curve map of the relation of the displacement of the initial point of each curve corresponding with the line kind of the curve of Figure 61 and curvature.By finding out among the figure, the curve of generation satisfies the given condition of table 26.

The tangent line of the initial point terminal point of the value that table 26 provides and the curve of generation, normal, curvature poor

		Coordinate	The unit tangent vector	The principal normal vector	Curvature
						P 1	The value of providing	{0.0，0.0，0.0}	{0.8660，-0.5，0.0}	{0.5，0.8660，0.0}	0.20
Value according to formation curve	{0.0，0.0，0.0}	{0.8660，-0.5，0.0}	{0.5000，0.8660，0.0}	0.20
					Difference		{0.0，0.0，0.0}	{0.0，0.0，0.0}	{0.0，0.0，0.0}	0
P 3	The value of providing	{8.0，-4.0，-5.0}	{1.0，0.0，0.0}	{0.0，-1.0，0.0}							0.20
					Value according to formation curve	{8.0，-4.0，-5.0}	{1.0，0.0，0.0}	{0.0，-1.0，0.0}	0.20
	Difference	{0.0，0.0，0.0}	{0.0，0.0，0.0}	{0.0，0.0，0.0}						0

(d) in the control of the value of intermediate point

Utilize the method for (b), continue each value on the control two-end-point, carry out G ²Continuous interpolation.Herein, consider not at two-end-point and in the intermediate point controlling value.

For example under the situation of the point range of interpolation such as Figure 63, consider at intermediate point P _cControl tangent line, normal.But, can not control the value on the intermediate point in the described in front method.Therefore, by this point range is divided into 2, be controlled at the value of intermediate point herein.

Just, for point range, not to carry out interpolation at one stroke, but clip intermediate point P _cBe divided into curve C ₁And curve C ₂Ground carries out interpolation.In such cases, owing to put P _cBe equivalent to end points, so as long as the method for employing (b) just can controlling value.

So separately distinguish on the point that the value that will control is arranged, control the value on its two ends, carry out the result of interpolation, as long as connect the curve that generates, can carry out in theory can be in each point control tangent method

The interpolation of circling round of the three-dimensional of line curvature.

(5) three-dimensional of the tangent line of control on the two-end-point, normal, the curvature interpolation of circling round

(a) flow process of method

The interpolation that employing is circled round in the three-dimensional of each value of initial point terminal point control can be undertaken by the following flow process shown in Figure 64.Below, along this process description.

(b-1) provide the point of interpolation object

In this example, provide three-dimensional 3 points 0.0,0.0,0.0}, 5.0,5.0,10.0}, 10.0,10.0,5.0}.Table 27 gathers the condition of the tangent line that is listed in other each point and provides, normal, curvature etc.

The condition of table 27 interpolation object each point and initial point terminal point

	Coordinate	The unit tangent vector	The principal normal vector	Curvature
					P 1	(0.0，0.0，0.0)	{0.0，1.0，0.0}	{1.0，0.0，0.0}	0.1
P 2	(5.0，5.0，10.0)	-	-	-
					P 3	(10.0，10.0，5.0)	{1.0，0.0，0.0}	{0.0，-1.0，0.0}	0.1

(b-2) generation of the 3DDCS of r=4

In newton-pressgang Shen method, when the exploration that begins to separate, need provide suitable initial value.Draw the preparation of this initial value herein.Research in advance is 3D Discrete Clothoid Splines, has closely by the interpolation object-point character that changes reposefully with respect to the displacement curvature from initial point.Therefore, in this research, make polygon Q, be identified for the circle round initial value of interpolation of three-dimensional by calculating from here as the 3D Discrete Clothoid Splines of the r=4 of Figure 65.In addition, Figure 66 illustrates actual polygon by this column-generation, and the coordinate on summit is listed table 28 in.

Polygonal apex coordinate that table 28 generates

	Apex coordinate
		P 1	{0.0，0.0，0.0}
	{0.4677，0.4677，3.1228}
			{0.9354，0.9354，6.2456}
	{2.3029，2.3029，9.4966}
		P 2	{5.0，5.0，10.0}
	{6.7095，6.7095，9.9244}
			{8.0655，8.0655，8.4732}
	{9.0327，9.0327，6.7366}
		P 3	{10.0，10.0，5.0}

(b-3) initial value determines

To find the solution with newton-pressgang Shen method, need to determine the initial value of each unknown number.In the method, use the polygon Q that generates at (b-2), obtain the approximate value of each unknown number, determine this value.In 3D Discrete Clothoid Splines, obtained the Fu Leinie frame on each summit.Therefore the unit tangent direction vector t by the polygon Q that generates at (b-2) obtains parameter a ₀, b ₀T is known when obtaining polygon Q for this tangential direction vector, and the formula of the tangent line by this t and three-dimensional clothoid is obtained tangential direction swing angle α, the β on the summit of polygon Q.Obtain a of each curve thus ₀, b ₀Initial value.In addition, on the three-dimensional that begins from initial point is circled round line segment, provide this value.

[numerical expression 115]

$u=\left\{\begin{array}{c}\cos \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ \sin \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ -\sin \mathrm{\&alpha;}\end{array}\right\}$

Herein, about 3D Discrete Clothoid Splines, if consider that the summit is with equidistant arrangement, at the some q of Figure 65 _4i+1On, can be similar to the long variable S of curve is 1/4.Equally at a q _{4 (i+1)-1}On, can be similar to the long variable S of curve is 3/4.If lump together with the formula of the α of three-dimensional clothoid and to consider these, set up following formula.

[numerical expression 116]

$\left\{\begin{array}{c}a{0}_{4i}+\frac{1}{4}a{1}_{4i}+{\left(\frac{1}{4}\right)}^{2}a{2}_{4i}=a{0}_{4i+1}\\ a{0}_{4i}+\frac{1}{4}a{1}_{4i}+{\left(\frac{3}{4}\right)}^{2}a{2}_{4i}=a{0}_{4(i+1)-1}\end{array}\right.$

It is a1 that this formula becomes unknown number _4iAnd a2 _4iTwo-dimentional simultaneous equations, it is found the solution, as parameter a ₁, a ₂Initial value.Equally also can determine parameter b ₁, b ₂Initial value.

Remaining unknown number is the long h of curve, but can be calculated by the formula of three-dimensional rondo curvature of a curve about its initial value.Three-dimensional rondo curvature of a curve, available note expression down.

[numerical expression 117]

$\mathrm{\&kappa;}=\frac{\sqrt{{\mathrm{\&alpha;}}^{\&prime;2}+{\mathrm{\&beta;}}^{\&prime;2}{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&alpha;}}}{h}$

If change this formula, become following formula, can determine the initial value of h.

[numerical expression 118]

${\mathrm{<figure-callout\; id="4i"\; label="h"\; filenames="A20058000605301142.png"\; state="\{\{state\}\}">h</figure-callout>}}_{4i}=\frac{\sqrt{{(a{1}_{4i}+2a{2}_{4i})}^2+{(\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}{1}_{4i}+2\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}{2}_{4i})}^2{\cos }^2(a{0}_{4i}+a{1}_{4i}+a{2}_{4i})}}{{\mathrm{\&kappa;}}_{4(i+1)}}$

With above method, can determine initial value to 7 three-dimensionals parameter of circling round.

Table 29 illustrates the initial value that this method of actual usefulness is obtained.

Table 29 initial value

Point of contact P 1And P 2Curve	a 0(0.0 known)
		a 1 -0.2684
	a 2 1.0739
		b 0Pi/2 (known)
	b 1 0.0
		b 2 0.0
	h 12.7684
		Point of contact P 2And P 3Curve	a 0 -0.1648
a 1 3.2061
	a 2 -2.6327
b 0 0.7853
	b 1 0.0
b 2 0.0
	h 9.6752

(b-4) closely by each point, G ²The continuous three-dimensional interpolation of circling round

Adopt by (b-3) definite initial value, reaching G ²Under the condition for continuous, obtain the approximate value of the parameter of each curve with newton-pressgang Shen method.Generate the three-dimensional line segment that circles round from the parameter that obtains thus, with between three-dimensional clothoid interpolation point range.

In 3 three-dimensional is circled round interpolation, by the interpolation object-point, and reach G herein, about closely ²Condition for continuous is considered concrete condition.Figure 67 represents a P ₁, P ₂, P ₃The three-dimensional interpolation of circling round.If with point of contact P ₁, P ₂Between curve as curve C ₁, with point of contact P ₂, P ₃Between curve as curve C ₂, because a0 ₁And b0 ₁Be known, so unknown number is a curve C ₁Parameter a1 ₁, a2 ₁, b1 ₁, b2 ₁, h ₁, curve C ₂Parameter a0 ₂, a1 ₂, a2 ₂, b0 ₂, b1 ₂, b2 ₂, h ₂Deng 12.The subscript of the literal that occurs in explanation later is corresponding with the subscript of each curve, as the function of the long variable S of curve, as Px _i, Py _i, Pz _i, α _i, β _i, n _i, κ _i, represent coordinate, tangent line swing angle α, β, normal, curvature on each curve.

At first, at a P ₁On the condition by the interpolation object-point closely, if consider, when providing initial point, must reach from the definition of three-dimensional clothoid.In addition, about tangential direction, owing to provide as known value, so do not specify at a P ₁On condition.

Then, consider some P ₂Point P ₂Be the mutual tie point of curve, reach G ²Continuous to-be position, tangent line, normal, curvature are continuous.Promptly at a P ₂On the condition that should set up as follows.

[numerical expression 119]

Px ₁(1)＝Px ₂(0)

Py ₁(1)＝Py ₂(0)

Pz ₁(1)＝Pz ₂(0)

cos[α ₁(1)-α ₂(0)]＝1

cos[β ₁(1)-β ₂(0)]＝1

n ₁(1)·n ₂(0)＝1

κ ₁(1)＝κ ₂(0)

Consider some P at last ₃Point P ₃Be terminal point, because the condition that should satisfy is position, tangent line, so set up 5 following conditions.Regard α as herein, ₃, β ₃Be tangential direction swing angle α, the β that determines the tangent line vector on the terminal point that provides.

[numerical expression 120]

Px ₂(1)＝Px ₃

Py ₂(1)＝Py ₃

Pz ₂(1)＝Pz ₃

cos[α ₂(1)-α ₃]＝1

cos[β ₂(1)-β ₃]＝1

Learn by above, for unknown number a1 ₁, a2 ₁, b1 ₁, b2 ₁, h ₁, a0 ₂, a1 ₂, a2 ₂, b0 ₂, b1 ₂, b2 ₂, h ₂Deng 12, conditional is set up 12 that remember down.It is as follows to conclude the conditional of setting up.

[numerical expression 121]

Px ₁(1)＝Px ₂(0)

Py ₁(1)＝Py ₂(0)

Pz ₁(1)＝Pz ₂(0)

cos[α ₁(1)-α ₂(0)]＝1

cos[β ₁(1)-β ₂(0)]＝1

n ₁·n ₂＝1

κ ₁(1)＝κ ₂(0)

Px ₂(1)＝Px ₃

Py ₂(1)＝Py ₃

Pz ₂(1)＝Pz ₃

cos[α ₂(1)-α ₃]＝1

cos[β ₂(1)-β ₃]＝1

So, owing to set up 12 formulas for 12 unknown numbers, so can find the solution.Find the solution this formula, obtain and separate with newton-pressgang Shen method.Table 30 is listed initial value and is conciliate.

Table 30 initial value is conciliate

		Initial value	Separate
				Point of contact P 1And P 2Curve C 1	a 0	(0.0 known)	-
a 1	-0.2684	-5.4455
			a 2		1.0739	5.4122
b 0	Pi/2 (known)	-
			b 1		0.0	-3.8590
b 2	0.0	3.1003
			h		12.7684	13.5862
Point of contact P 2And P 3Curve C 2	a 0	-0.1648					-0.033258
			a 1	3.2061	3.6770
	a 2	-2.6327				-3.6437
			b 0	0.7853	0.8120
	b 1	0.0				1.6006
			b 2	0.0	-2.4126
	h	9.6752				9.2873

(b-5) generation of curve

Figure 68 represents that simultaneously with the parameter of obtaining at (b-4) be the curve of basis generation and the polygon that generates at (b-2).The curve of solid line is a curve C ₁, the curve of dotted line is a curve C ₂In this stage, be formed on the G of initial point terminal point control tangential direction ²The continuous three-dimensional clothoid.

(b-6) conditional and unknown number

Herein, consider also with initial point P in addition ₁With terminal point P ₃On normal and curvature be defined as the value that table 27 provides.To control normal and curvature again at the initial point terminal point, need 2 conditions that increase on the initial point terminal point respectively.But, under the state of 4 of condition increases, from considering to obtain to satisfy separating of this condition with the relation of unknown number.Therefore, for the number that makes unknown number and conditional is relative, shown in Figure 69, in curve C ₁The position insertion point DP again of the long variable S=0.5 of curve ₁In addition, for curve C ₂, also insertion point DP again in the position of the long variable S=0.5 of curve ₂

At this moment, with point of contact P ₁With a DP ₁Curve as curve C ' ₁, with point of contact DP ₁With a P ₂Curve as curve C ' ₂, with point of contact P ₂With a DP ₂Curve as curve C ' ₃, with point of contact DP ₂With a P ₃Curve as curve C ' ₄The subscript of the literal that occurs in explanation later on is corresponding with each curve name, for example as the function of the long variable S of curve, as Px _c, Py _c, Pz _c, α _c, β _c, n _c, κ _c, coordinate, tangent line swing angle α, β, normal, curvature on the expression curve C.In addition, on the initial point terminal point, at initial point such as Px _s, Py _s, Pz _s, α _s, β _s, n _s, κ _s, at terminal point such as Px _e, Py _e, Pz _e, α _e, β _e, n _e, κ _e, the condition that denotation coordination, tangent line swing angle α, β, normal, the following explanation of curvature are set up on each point.

[numerical expression 122]

Point P ₁: tangent line, normal, curvature: 4

cos[α _C′1(0)-α _s]＝1

cos[β _C′1(0)-β _s]＝1

n _C′1(0)·n _s＝1

κ _C′1(0)＝κ _s

Point DP ₁: position, tangent line, normal, curvature: 7

Px _C′1(1)＝Px _C′2(0)

Py _C′1(1)＝Py _C′2(0)

Pz _C′1(1)＝Pz _C′2(0)

cos[α _C′1(1)-α _C′2(0)]＝1

cos[β _C′1(1)-β _C′2(0)]＝1

n _C′1(1)·n _C′2(0)＝1

κ _C′1(1)＝κ _C′2(0)

Point P ₂: position, tangent line, normal, curvature: 7

Px _C′2(1)＝Px _C′3(0)

Py _C′2(1)＝Py _C′3(0)

Pz _C′2(1)＝Pz _C′3(0)

cos[α _C′2(1)-α _C′3(0)]＝1

cos[β _C′2(1)-β _C′3(0)]＝1

n _C′2(1)·n _C′3(0)＝1

κ _C′2(1)＝κ _C′3(0)

Point DP ₂: position, tangent line, normal, curvature: 7

Px _C′3(1)＝Px _C′4(0)

Py _C′3(1)＝Py _C′4(0)

Pz _C′3(1)＝Pz _C′4(0)

cos[α _C′3(1)-α _C′4(0)]＝1

cos[β _C′3(1)-β _C′4(0)]＝1

n _C′3(1)·n _C′4(0)＝1

κ _C′3(1)＝κ _C′4(0)

Point P ₃: position, tangent line, normal, curvature: 7

Px _C′4(1)＝Px _e

Py _C′4(1)＝Py _e

Pz _C′4(1)＝Pz _e

cos[α _C′4(1)-α _e]＝1

cos[β _C′4(1)-β _e]＝1

n _C′4(1)·n _e＝1

κ _C′4(1)＝κ _e

More than, all the conditional that should set up is 32.Herein, the parameter of circling round that each curve has is a ₀, a ₁, a ₂, b ₀, b ₁, b ₂, each 7 of h, and, because curve is 4, so unknown number is 28.But, like this one, because the number of unknown number and conditional is unequal, separate so can not obtain.Therefore handle 2 some DP that insert again as unknown number ₁, DP ₂Y, z coordinate, increase by 4 unknown numbers.By such processing, unknown number, conditional all are 32, can obtain and separate.

(b-7) initial value determines 2

To satisfy in (b-6) separating of the conditional set up in order obtaining, to adopt newton-pressgang Shen method, but bring rate together and the initial value of determining unknown number in order to improve it.As method, be segmented in the three-dimensional clothoid that generates in (b-5) by front and back as the point of new insertion that Figure 70 is shown in, make 4 three-dimensional clothoids, provide its parameter of circling round.

About the split plot design of curve, if explanation is with curve C ₁Be divided into curve C ' ₁And curve C ' ₂Method, curve C ' ₁Circle round parameter h ', a ' ₀, a ' ₁, a ' ₂, b ' ₀, b ' ₁, b ' ₂, adopt curve C ₁Parameter, represent with following formula.S herein _dBe the long variable of curve on the cut-point, be 0.5 herein.

[numerical expression 123]

$\left\{\begin{array}{c}{a}_0^{\&prime;}=a_0\\ a_1^{\&prime;}=a_1{S}_d\\ a_2^{\&prime;}=a_2{S_d}^2\\ {\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">b</figure-callout>}}_0^{\&prime;}=b_0\\ {\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^{\&prime;}=b_1{\mathrm{<figure-callout\; id="2"\; label="S\; d\; b"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">S</figure-callout>}}_d& \\ b_{}^{}{}_2^{\&prime;}=b_2{S_d}^2\\ h^{\&prime;}=h{S}_d\end{array}\right.$

Then consider with cut-point DP ₁Curve C as initial point ' ₂At first, if with size, shape and curve C ₁Identical and towards opposite curve as curve C " ₁, the parameter h that circles round of this curve ", a " ₀, a " ₁, a " ₂, b " ₀, b " ₁, b " ₂, adopt curve C ₁The parameter of curve, represent with following formula.

[numerical expression 124]

$\left\{\begin{array}{c}{p}_S^{\&prime;\&prime;}=p\left(1\right)\\ a_0^{\&prime;\&prime;}=a_0+a_1+a_2+\mathrm{\&pi;}\\ a_1^{\&prime;\&prime;}=-(a_1+2a_2)\\ a_2^{\&prime;\&prime;}=a_2\\ {\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">b</figure-callout>}}_0^{\&prime;\&prime;}={\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">b</figure-callout>}}_0+{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}}_1+b_2\\ {\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^{\&prime;\&prime;}=-({\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}}_1+2b_2)\\ {\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}}_2^{\&prime;\&prime;}=b_2\\ h^{\&prime;\&prime;}=h\end{array}\right.$

On this curve, cut-point DP ₁Use DP ₁=C " ₁(1-S _d) expression.Herein, if consider at a DP ₁Cut apart curve C " ₁, with the some P in this curve of cutting apart ₂Curve C as initial point " ₂, become size, shape and curve C " ₂Identical and towards opposite curve.Can utilize formation curve C ' ₁Method formation curve C " ₂Herein, in addition as long as with respect to curve C " ₂Generate size, shape is identical and towards opposite curve, just can formation curve C ₂

This curve C ₂The parameter h that circles round ", a " ₀, a " ₁, a " ₂, b " ₀, b " ₁, b " ₂, adopt curve C ₀The parameter of curve, represent with following formula.

[numerical expression 125]

$\left\{\begin{array}{c}{a}_0^{\&prime;\&prime;}=a_0+a_1{S}_d+a_2{S_d}^2\\ a_1^{\&prime;\&prime;}=(1-S_d)\{a_1+2a_2{S}_d\}\\ a_2^{\&prime;\&prime;}=a_2{(1-S_d)}^2\\ {\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">b</figure-callout>}}_0^{\&prime;\&prime;}={\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="A20058000605301121.png,A20058000605301162.png"\; state="\{\{state\}\}">b</figure-callout>}}_0+b_1{S}_d+b_2{S_d}^2\\ {\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}}_1^{\&prime;\&prime;}=(1-S_d)\{{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="A20058000605301121.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}}_1+2b_2{S}_d\}\\ {\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A20058000605301113.png,A20058000605301122.png"\; state="\{\{state\}\}">b</figure-callout>}}_2^{\&prime;\&prime;}=b_2{(1-S_d)}^2\\ h^{\&prime;\&prime;}=h(1-S_d)\end{array}\right.$

With above method, can be at three-dimensional clothoid C ₁On the some DP of the long variable S=0.5 of curve ₁, with curve C ₁Be divided into C ' ₁And C ₂Use the same method, also can be in curve C ₂On the some DP of the long variable S=0.5 of curve ₂, with curve C ₂Be divided into C ' ₃And C ' ₄

Table 31 is listed the parameter of 4 curves cutting apart with this method.The parameter of this curve is used at the initial value of obtaining newton-pressgang Shen method used when satisfying the separating of the conditional set up at b-6.

Table 31 is cut apart the parameter of the curve of generation

Curve C ' 1	a 0	(0.0 known)	Curve C ' 2	a 0	4.9134
						a 1	-2.7227	a 1	-0.016629
	a 2	1.3530		a 2	1.3530
						b 0	Pi/2 (known)	b 0	0.41633
	b 1	-1.9295		b 1	-0.37938
						b 2	0.7750	b 2	0.77507
	h	6.7931		h	6.7931
						Initial point	{0.0，0.0，0.0}	Initial point	{1.8431，3.0860，4.9597}
	Curve C ' 3	a 0		-0.033258	Curve C ' 4					a 0	7.1774
a 1			1.8385			a 1	0.016629
		a 2		-0.91093				a 2	-0.91093
b 0			0.81202			b 0	1.0091
		b 1		0.80031				b 1	-0.40601
b 2			-0.60316			b 2	-0.60316
		h		4.6436				h	4.6436
Initial point			{5.0，5.0，10.0}			Initial point	{7.0029，8.1298，7.5337}

(b-8) obtain the parameter of circling round that satisfies condition

Based on the initial value of in (b-7), determining, obtain separating of the conditional that satisfies establishment in (b-6) with newton-pressgang Shen method.Table 32 is parameters of each curve of calculating.In addition, list tangent line, normal, curvature poor of initial point terminal point of the curve of the value that provides and generation in the table 33.

The parameter of the curve that table 32 generates

Curve C ' 1	a 0	(0.0 known)	Curve C ' 2	a 0	5.3846
						a 1	0.0000	a 1	-3.4602
	a 2	-0.89854		a 2	4.341
						b 0	Pi/2 (known)	b 0	0.47690
	b 1	-0.51836		b 1	-3.2143
						b 2	-0.57552	b 2	3.4613
	h	5.1836		h	9.9808
						Initial point	{0.0，0.0，0.0}	Initial point	{1.8431，4.1726，1.4653}
	Curve C ' 3	a 0		-0.017740	Curve C ' 4					a 0	6.8553
a 1			3.4572			a 1	-1.1443
		a 2		-2.8673				a 2	0.57219
b 0			0.72385			b 0	0.76315
		b 1		2.4551				b 1	-1.1942
b 2			-2.4158			b 2	0.43108
		h		6.60818				h	3.3206
Initial point			{5.0，5.0，10.0}			Initial point	{7.0029，9.0734，5.6186}

The tangent line of the initial point terminal point of the value that table 33 provides and the curve of generation, normal, curvature poor

		Coordinate	The unit tangent vector	The principal normal vector	Curvature
						P 1	The value of providing	{0.0，0.0，0.0}	{0.0，1.0，0.0}	{1.0，0.0，0.0}	0.10
Value according to formation curve	{0.0，0.0，0.0}	{0.0，1.0，0.0}	{1.0，0.0，0.0}	0.10
					Difference		{0.0，0.0，0.0}	{0.0，0.0，0.0}	{0.0，0.0，0.0}	0
P 3	The value of providing	{10.0，10.0，5.0}	{1.0，0.0，0.0}	{0.0，-1.0，0.0}							0.10
					Value according to formation curve	{10.0，10.0，5.0}	{1.0，0.0，0.0}	{0.0，-1.0，0.0}	0.10
	Difference	{0.0，0.0，0.0}	{0.0，0.0，0.0}	{0.0，0.0，0.0}						0

(b-9) generation of curve

Figure 71 represents the curve by the parameter generation of obtaining in (b-8).Solid line is represented three-dimensional clothoid, and dotted line one dot-and-dash line two dot-and-dash lines three dot-and-dash lines represent that each direction of curve in the principal normal direction, is of a size of radius, satisfy natural logarithm, the radius-of-curvature changing pattern of taking the logarithm.In addition, Figure 72 is expression from the curve map of the relation of the displacement s of the initial point of each curve corresponding with the line kind of Figure 71 and curvature κ.By finding out among the figure, the curve of generation satisfies the given condition of table 33.

More than, illustrated and adopted at the circle round example of interpolation formation curve of the three-dimensional of two ends control tangent line, normal, curvature.

3. adopt the mathematical control mode of the three-dimensional interpolation of circling round

The above-mentioned three-dimensional interpolation curve that circles round is used for the generation for the Numerical Control information of the motion control of the instrument of work mechanism or other motion object effectively.It is characterized in that, can be easy to speed control, and make velocity variations steady.

(1) mathematical control mode of the three-dimensional interpolation of circling round of employing

Adopt the mathematical control mode of the three-dimensional interpolation curve that circles round, constitute by the following order shown in Figure 73.

(a) design of movement of tool track (Figure 73, S1)

The described method of joint is determined the three-dimensional that the satisfies condition interpolation curve that circles round in the utilization.When the tool work of robot etc., can consider the representative point (tool point, tool center point) of its instrument, by the time along the plane or the space the continuous geometric locus (comprising straight line) drawn go up and move.The position of tool point represents that with coordinate (x, y, z) posture of working point is for example used with respect to the anglec of rotation of x, y, z axle and represented.How complicated no matter be work, the track of tool point can be not intermittent, but connect continuously.The 1st stage of motion control is to design by three-dimensional clothoid the shape of this track.

(b) the coincideing of curve movement (Figure 73, S2)

According to requirement, along the three-dimensional interpolation curve that circles round, the distribution of the translational speed of the controlling object point on the assignment curve from Numerical Control.Just, the 2nd stage of motion control is a speed acceleration of determining the tool point that acts on the track of design.Its by tool point with what kind of function of time along acting on the track, or the speed acceleration of definite tool point is determined.The speed acceleration of tool point, relative time is determined sometimes, sometimes along with the shape of track is determined.General heterogeneous determined, but for example when carrying out Machining of Curved Surface, make its high-speed mobile that so the requirement that its low speed is moved in the part of bending is along with the shape of track is determined speed the time owing to have in flat portions.

In the present embodiment, for example adopt the good curve of the used characteristic of cam mechanism.Constitute the continuous group of curves of posture, but use curve movement in the curve one by one, specify acceleration at it with cartesian space (physical presence space) definition.So-called cartesian space is the three-dimensional system of coordinate that adopts in 3 making of the mutually orthogonal x of initial point, y, z, and posture also can be represented in position that not only can representational tool point.

(c) time is cut apart (Figure 73, S3) and utilizes the calculating (Figure 73, S3) of posture of the instrument of cartesian coordinate system

Every the unit interval of evaluation control information,, calculate the shift position and the posture of tool point herein, according to the translational speed of controlling object appointment.Owing to determined track and motion, so can provide the posture of tool point as the function of time t.Thus, when providing time t, can obtain displacement with respect to each tool point constantly by the small time interval.(c) calculating is specifically undertaken by following.On present point, learn the value of positional information or tangent line, curvature etc.As long as the translational speed in appointment multiply by the unit interval, just can learn that the moving curve in the unit interval is long, can calculate the curve long parameter after moving thus.Curve long parameter after moving by this can calculate the positional information on the point after moving or the value of tangent line, curve etc.

By above operation, can calculate position and posture with respect to the tool point of the time t on the cartesian coordinate system (physical presence space).As variable, in three-dimensional (x, y, z, γ, μ, ν, θ).But (γ, μ, ν, θ) represents posture E with rotation of equal value, and wherein (γ, μ, ν) represents the axle of rotation of equal value, and θ represents rotation angle.

In addition,, obtain, only be offset the offset point of given size to normal direction along the three-dimensional interpolation curve that circles round according to requirement from Numerical Control, with it as cutter (track of tool focus).This calculating becomes easy because of obtaining normal direction.

(d) contrary mechanism separates (Figure 73, S5)

Then, obtain the rotation angle of required each of the posture that provides above-mentioned tool point.This process is commonly referred to as contrary mechanism and separates (inverse kinematics).For example if 6 robot because 6 joints are arranged, by the joint of how many degree rotation shoulders, the joint of arm, the joint of elbow, the joint of wrist etc., determines the posture of tool point.Being referred to as contrary mechanism separates.Contrary mechanism separates, and is the rotation angle θ 1～θ 6 that obtains shaft space with it on the contrary from the posture in space.It is electric rotating machine that the gear train of each also is not limited to, and also can be straight moving gear train such as linear electric machine sometimes, but at this moment, the actual displacement of bottom line need be transformed into the calculating of electromagnetic gear of the input pulse number of linear electric machine.Contrary mechanism separates, and is that every kind of model of mechanism such as robot is intrinsic, wants single preparation to separate for various robots etc.

(e) utilize the calculating (Figure 73, S6) of each spindle motor displacement of axis coordinate system

Each tool point of cutting apart with regard to the time is obtained contrary mechanism and is separated, and makes its integer as the displacement pulse of each spindle motor (comprising straight moving gear train).Not under the pulse controlled situation, adopting the minimum decomposition unit (resolution) of each displacement, obtaining as the data by integer of suitable umber of pulse.

Above-mentioned (a) reaches (b) is the order of preparing, and only carries out once.(c)～(e) carry out every the unit interval of appointment, be performed until the condition that satisfies object time or purpose.

In numerical control device, also can carry out above-mentioned whole calculating, or utilize other COMPUTER CALCULATION and setting (a) to reach (b), also can send into this parameter of curve, in numerical control device, carry out (c) and calculating (e) to numerical control device.

(2) NC device and CNC device

Below, the situation when using numerical control device (NC device) independently is described and uses the computing machine of effect and the situation of NC device during by incorporate CNC device with program.

(a) when adopting independently the NC device

In common NC machinery in the past, hardware separation is become to carry out program design make the timer of NC data and adopt these NC data to make these two devices of NC device of mechanical hook-up work.In contrast, in nearest CNC machinery, the computing machine that will carry out program design is located in the NC device, becomes by incorporate device.

At first, the former, under the situation that adopts independent device, proposed to utilize three-dimensional mathematical control mode of circling round.In such cases, regulation adopts the parameter of circling round in the handing-over of the data of circling round, the form that definition is circled round in G code.This for example, as shown below.

G ^***

A0、A1、A2、B0、B1、B2、H

Herein, G ^{* *}The number of expression G code.A0～H represents circle round 7 parameters of line segment of three-dimensional.Before carrying out this code, instrument is come P ₀The position.Adopt this parameter in the NC device, the instantaneous tool location or the difference of tool location are calculated in operation.This operation is called " along separating ".Carrying out along the reason of separating at the NC device side is in order to prevent the big quantification of data, therefore need to carry out certain computing in the NC device.Circle round by showing, in the NC device of having established of clothoid can being packed into G code.

(b) CNC mode

Narrate the computing machine of effect and the CNC device of NC apparatus integration below with program.Under this dress situation, relevant calculating of circling round is out of question with the hardware of which part.In addition, the speed of the amount of data or conveying also solves.

Generally, in this program, comprise the process of the parameter of circling round of determining suitable each condition.Be referred to as " contrary separating ".In contrary separating, for example, also comprise and provide several discrete point ranges, calculate closely curve program stably (free point range interpolation) by these points.In addition, also upward definite programs (so-called CAM) of required tool path of processing that comprise more.

(3) feature of the mathematical control mode of the three-dimensional interpolation of circling round of employing

In the mathematical control mode that adopts the three-dimensional interpolation of circling round, has following advantage.

(a) as mentioned above, owing to show curve as independent parameter, so can generate translational speed value corresponding control information with appointment with curve length from reference point.On other curves such as SPL that utilize with the long unallied independent parameter performance of curve, even calculate point after moving, the also difficult value of the independent parameter corresponding with this point of calculating also is difficult for the translational speed value corresponding control information of generation and appointment.

In order to describe this situation in detail, shown in Figure 74, consider from the some R on the track of SPL R (t) performance ₀, make the situation of movement of tool with a certain linear velocity.When calculating the impact point of instrument every Fixed Time Interval, can learn through the instrument amount of movement Δ S after the unit interval, but because independent variable t does not relate to time or the long parameter of curve, so the variation delta t of independent variable can not obtain immediately.Owing to separating R ₀During the formula of+Δ S=R (t0+ Δ t),, just can not calculate impact point, so must repeat this calculating at interval at regular intervals if do not obtain Δ t.

(b) on three-dimensional clothoid, the variation pattern of the curvature that expectation is long with respect to curve, approximate immobilizing, value corresponding control information therewith considers that from the viewpoint of motion control expectation becomes reasonably control information on the mechanics.In general spline interpolation etc., be difficult to the variation of PREDICTIVE CONTROL curvature.

(c) three-dimensional clothoid as its special circumstances, comprises straight line, circular arc, spiral curve etc., can show the Numerical Control information with respect to multiple curve meticulously under other a curyilinear situation of not packing into.

(d) three-dimensional clothoid is the natural equation that does not rely on the establishing method of coordinate axis.Representing with x, y, z axle in the NC device in the past of curve, for example when the inclination processing work,, processing easily sometimes, difficult sometimes processing according to the installation method of workpiece.In three-dimensional clothoid, owing to provide curve, so,, process equally in the time of just can be with the process water plane as long as on the inclined-plane, make track even when the processing inclined-plane according to line length.

In addition; adopting quadratic expression of the present invention, long or the long variable of curve to provide the inclination angle and the deflection angle three-dimensional curve (being called three-dimensional clothoid) separately of tangential direction with curve; when carrying out the program of the contour shape that shows tool path or workpiece by computing machine; procedure stores in the assisted memory devices such as hard disk unit of computing machine, is loaded in the primary memory and moves.In addition, program so can be stored in the movable recording medium such as CD-ROM and sell, or be stored in the pen recorder of the computing machine that connects via network, also can send other computing machine by network to.In addition, also the result of calculation (the displacement pulse of the parameter of curve of three-dimensional clothoid, each spindle motor etc.) of utilizing program so to draw can be stored in the movable recording mediums such as CD-ROM and sell.

According to three-dimensional clothoid of the present invention, can provide a kind of design of industrial product to produce the method for generation that required space curve extensively adopts.Follow along space curve under the situation that acceleration-deceleration moves at object, can carry out constraining force and change design stably.Its feature can be widely used in the method for designing of the tracks of the mechanical organ with quality.This is as the application examples of design, the method for designing of the return path of ball-screw has been described, but in addition, for example also can be used on the sinuate track up and down the method for designing of the track of the injection inertial power device of walking, linear guides etc. rapidly.In addition, because can be with respect to the variation of the long suitably design of curve curvature, so can be used for aesthetic multiple industrial fields such as artistic conception curve design effectively.

Claims (27)

1. the method for designing of an industrial product is characterized in that: adopt the inclination angle of tangential direction and the three-dimensional curve (being called three-dimensional clothoid) that deflection angle quadratic expression long by curve respectively or the long variable of curve provides, the shape of design industrial product.

2. the method for designing of industrial product as claimed in claim 1 is characterized in that:

Described industrial product is the machinery that comprises the mechanism that makes the mechanical organ motion with quality,

Adopt the track of the motion of the described mechanical organ of described three-dimensional curve (being called three-dimensional clothoid) design.

3. the method for designing of industrial product as claimed in claim 2 is characterized in that:

Described machinery is the screw device that comprises the mechanism that makes the ball motion as described mechanical organ;

Described screw device possesses: the lead screw shaft that has spiral helicine rolling body raceway groove at outer peripheral face; Have and the opposed load rolling body of described rolling body raceway groove raceway groove at inner peripheral surface, have the nut of the return path of the end that connects described load rolling body raceway groove and the other end simultaneously; And be arranged in the described rolling body raceway groove of described lead screw shaft and described nut described load rolling body raceway groove between and a plurality of rolling bodys on the return path,

Adopt described three-dimensional curve (being called three-dimensional clothoid), design the described return path of described screw device.

4. as the method for designing of any one described industrial product in the claim 1～3, wherein by defining described three-dimensional clothoid with following formula,

[numerical expression 126]

$P=P_0+{\&Integral;}_0^s\mathrm{uds}=P_0+h{\&Integral;}_0^S\mathrm{udS},0\&le;s\&le;h,0\&le;S=\frac{s}{h}\&le;1---\left(1\right)$

$u=E^{\mathrm{k\&beta;}}{E}^{\mathrm{j\&alpha;}}\left(i\right)=\left[\begin{array}{ccc}\cos \mathrm{\&beta;}& \sin \mathrm{\&beta;}& 0\\ \sin \mathrm{\&beta;}& \cos \mathrm{\&beta;}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos \mathrm{\&alpha;}& 0& \sin \mathrm{\&alpha;}\\ 0& 1& 0\\ -\sin \mathrm{\&alpha;}& 0& \cos \mathrm{\&alpha;}\end{array}\right]\left\{\begin{array}{c}1\\ 0\\ 0\end{array}\right\}=\left\{\begin{array}{c}\cos \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ \sin \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ -\sin \mathrm{\&alpha;}\end{array}\right\}---\left(2\right)$

α＝a ₀+a ₁S+a ₂S ² (3)

β＝b ₀+b ₁S+b ₂S ² (4)

Herein,

[numerical expression 127]

$P=\left\{\begin{array}{c}x\\ y\\ z\end{array}\right\},P_0=\left\{\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right\}---\left(5\right)$

Position vector and the initial value thereof of representing the point on the three-dimensional clothoid respectively,

To be made as s from the length of a curve of initial point, its length overall (length from the initial point to the terminal point) is made as h, and represent the value except that s with h with S, S is the value of no guiding principle amount, is referred to as the long variable of curve,

I, j, k are respectively x axle, y axle and the axial unit vector of z,

U is the unit vector of the tangential direction of the curve on the expression point P, is provided E by formula (2) ^{K β}And E ^{J α}It is rotation matrix, represent the rotation of the angle beta that the k axle is and the rotation of the angle [alpha] that the j axle is respectively, the former is called deflection (yaw) rotation, the latter is called inclination (pitch) rotation, formula (2), expression is only rotated α by at first making the axial unit vector of i in j axle system, then, system only rotates β at the k axle, obtains tangent line vector u

a ₀, a ₁, a ₂, b ₀, b ₁, b ₂It is constant.

5. the method for designing of industrial product as claimed in claim 4 is characterized in that: specify a plurality of spatial point in three-dimensional coordinate, by adopting described three-dimensional these spatial point of clothoid interpolation, design the shape of described industrial product.

6. the method for designing of industrial product as claimed in claim 5, it is characterized in that: with in described a plurality of spatial point, with three-dimensional line segment (constituting the unit curve of the group of curves that generates by interpolation) and the next three-dimensional line segment (constituting unit curve) that circles round that circles round by the group of curves of interpolation generation, the mode that connects both position, tangential method, normal direction and curvature is calculated circle round 7 parameter a of line segment of described three-dimensional ₀, a ₁, a ₂, b ₀, b ₁, b ₂, h.

7. the method for designing of industrial product as claimed in claim 6 is characterized in that:

Specify the initial point in described a plurality of spatial point and tangential direction, normal direction and the curvature of terminal point,

Between preassigned described spatial point, insert the interpolation object again, thereby,

Make with the conditional of tangential direction, normal direction and the curvature of described initial point and described terminal point with line segment and the next three-dimensional line segment that circles round that circles round of a three-dimensional on described a plurality of spatial point to make both position, tangential direction, normal direction and curvature condition for continuous formula add the number of the conditional of computing, with circle round 7 parameter a of line segment of described three-dimensional ₀, a ₁, a ₂, b ₀, b ₁, b ₂, h the unknown number unanimity;

By making the several consistent of conditional and unknown number, calculate circle round 7 parameter a of line segment of described three-dimensional ₀, a ₁, a ₂, b ₀, b ₁, b ₂, h.

8. an industrial product designs with the method for designing as any one described industrial product in the claim 1～7.

9. program, be used in order to design the shape of industrial product, make computing machine as the three-dimensional curve (being called three-dimensional clothoid) that the inclination angle of adopting tangential direction and deflection angle quadratic expression long by curve respectively or the long variable of curve provide, design industrial product shape means and play a role.

10. the recording medium of an embodied on computer readable, logging program thereon, this program is used in order to design the shape of industrial product, make computing machine as the three-dimensional curve (being called three-dimensional clothoid) that the inclination angle of adopting tangential direction and deflection angle quadratic expression long by curve respectively or the long variable of curve provide, design industrial product shape means and play a role.

11. numerical control method; wherein; adopt the inclination angle of tangential direction and the three-dimensional curve (being called three-dimensional clothoid) that deflection angle quadratic expression long by curve respectively or the long variable of curve provides, the contour shape of performance tool path or workpiece is by the motion of this three-dimensional curve control tool.

12. numerical control method as claimed in claim 11 wherein circles round by defining three-dimensional with following formula,

[numerical expression 128]

$P=P_0+{\&Integral;}_0^s\mathrm{uds}=P_0+h{\&Integral;}_0^S\mathrm{udS},0\&le;s\&le;h,0\&le;S=\frac{s}{h}\&le;1---\left(1\right)$

$u=E^{\mathrm{k\&beta;}}{E}^{\mathrm{j\&alpha;}}\left(i\right)=\left[\begin{array}{ccc}\cos \mathrm{\&beta;}& -\sin \mathrm{\&beta;}& 0\\ \sin \mathrm{\&beta;}& \cos \mathrm{\&beta;}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos \mathrm{\&alpha;}& 0& \sin \mathrm{\&alpha;}\\ 0& 1& 0\\ -\sin \mathrm{\&alpha;}& 0& \cos \mathrm{\&alpha;}\end{array}\right]\left\{\begin{array}{c}1\\ 0\\ 0\end{array}\right\}=\left\{\begin{array}{c}\cos \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ \sin \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ -\sin \mathrm{\&alpha;}\end{array}\right\}---\left(2\right)$

α＝a ₀+a ₁S+a ₂S ² (3)

β＝b ₀+b ₁S+b ₂S ² (4)

Herein,

[numerical expression 129]

$P=\left\{\begin{array}{c}x\\ y\\ z\end{array}\right\},P_0=\left\{\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right\}---\left(5\right)$

Position vector and the initial value thereof of representing the point on the three-dimensional clothoid respectively,

To be made as s from the length of a curve of initial point, its length overall (length from the initial point to the terminal point) is made as h, and represent the value except that s with h with S, S is the value of no guiding principle amount, is referred to as the long variable of curve,

I, j, k are respectively x axle, y axle and the axial unit vector of z,

U is the unit vector of the tangential direction of the curve on the expression point P, is provided E by formula (2) ^{K β}And E ^{J α}It is rotation matrix, represent the rotation of the angle beta that the k axle is and the rotation of the angle [alpha] that the j axle is respectively, the former is called deflection (yaw) rotation, the latter is called inclination (pitch) rotation, formula (2), expression is only rotated α by at first making the axial unit vector of i in i axle system, then, system only rotates β at the k axle, obtains tangent line vector u

a ₀, a ₁, a ₂, b ₀, b ₁, b ₂It is constant.

13. numerical control device; wherein; adopt the inclination angle of tangential direction and the three-dimensional curve (being called three-dimensional clothoid) that deflection angle quadratic expression long by curve respectively or the long variable of curve provides, the contour shape of performance tool path or workpiece is by the motion of this three-dimensional curve control tool.

14. program; be used for motion for the Numerical Control instrument; make computing machine as the three-dimensional curve (being called three-dimensional clothoid) that the inclination angle of adopting tangential direction and deflection angle quadratic expression long by curve respectively or the long variable of curve provide, show tool path or workpiece contour shape means and play a role.

15. the recording medium of an embodied on computer readable; logging program or the result of calculation that draws by this program thereon; this program is used for the motion for the Numerical Control instrument; make computing machine as the three-dimensional curve (being called three-dimensional clothoid) that the inclination angle of adopting tangential direction and deflection angle quadratic expression long by curve respectively or the long variable of curve provide, show tool path or workpiece contour shape means and play a role.

16. numerical control method, wherein, adopt the inclination angle of tangential direction and the three-dimensional curve that deflection angle quadratic expression long by curve respectively or the long variable of curve provides (three-dimensional circle round line segment), between the point range that interpolation provides arbitrarily in three-dimensional coordinate, by the circle round motion of line segment control tool of this three-dimensional.

17. a numerical control method, wherein, many of the three-dimensional curve that the inclination angle of tangential direction and deflection angle quadratic expression long by curve respectively or the long variable of curve are provided (three-dimensional circle round line segment) connections are by the circle round motion of line segment control tool of these many three-dimensionals.

18. as claim 16 or 17 described numerical control methods, wherein by defining three-dimensional clothoid with following formula,

[numerical expression 130]

$P=P_0+{\&Integral;}_0^s\mathrm{uds}=P_0+h{\&Integral;}_0^S\mathrm{udS},0\&le;s\&le;h,0\&le;S=\frac{s}{h}\&le;1---\left(1\right)$

$u=E^{\mathrm{k\&beta;}}{E}^{\mathrm{j\&alpha;}}\left(i\right)=\left[\begin{array}{ccc}\cos \mathrm{\&beta;}& -\sin \mathrm{\&beta;}& 0\\ \sin \mathrm{\&beta;}& \cos \mathrm{\&beta;}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos \mathrm{\&alpha;}& 0& \sin \mathrm{\&alpha;}\\ 0& 1& 0\\ -\sin \mathrm{\&alpha;}& 0& \cos \mathrm{\&alpha;}\end{array}\right]\left\{\begin{array}{c}1\\ 0\\ 0\end{array}\right\}=\left\{\begin{array}{c}\cos \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ \sin \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ \sin \mathrm{\&alpha;}\end{array}\right\}---\left(2\right)$

α＝a ₀+a ₁S+a ₂S ² (3)

β＝b ₀+b ₁S+b ₂S ² (4)

Herein,

[numerical expression 130]

$P=\left\{\begin{array}{c}x\\ y\\ z\end{array}\right\},P_0=\left\{\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right\}---\left(5\right)$

Position vector and the initial value thereof of representing the point on the three-dimensional clothoid respectively,

To be made as s from the length of a curve of initial point, its length overall (length from the initial point to the terminal point) is made as h, and represent the value except that s with h with S, S is the value of no guiding principle amount, is referred to as the long variable of curve,

I, j, k are respectively x axle, y axle and the axial unit vector of z,

U is the unit vector of the tangential direction of the curve on the expression point P, is provided E by formula (2) ^{K β}And E ^{J α}It is rotation matrix, represent the rotation of the angle beta that the k axle is and the rotation of the angle [alpha] that the j axle is respectively, the former is called deflection (yaw) rotation, the latter is called inclination (pitch) rotation, formula (2), expression is only rotated α by at first making the axial unit vector of i in j axle system, then, system only rotates β at the k axle, obtains tangent line vector u

a ₀, a ₁, a ₂, b ₀, b ₁, b ₂It is constant.

19. numerical control method as claimed in claim 18, it is characterized in that: circle round on the joint of line segment in circle round line segment and next three-dimensional of a three-dimensional, in both continuous modes in position, tangential direction (and according to circumstances curvature), calculate circle round 7 parameter a of line segment of described three-dimensional ₀, a ₁, a ₂, b ₀, b ₁, b ₂, h.

20. numerical control device, wherein, adopt the inclination angle of tangential direction and three-dimensional that deflection angle quadratic expression long by curve respectively or the long variable of curve the provides line segment that circles round, between the point range that interpolation provides arbitrarily, by the circle round motion of line segment control tool of this three-dimensional in three-dimensional coordinate.

21. program, be used for motion for the Numerical Control instrument, the three-dimensional that computing machine is provided as the inclination angle of adopting tangential direction and deflection angle quadratic expression long by curve respectively or the long variable of curve the line segment that circles round, means between the point range that interpolation provides arbitrarily in three-dimensional coordinate and playing a role.

22. the recording medium of an embodied on computer readable, logging program or the result of calculation that draws by this program thereon, this program is used for the motion for the Numerical Control instrument, the three-dimensional that computing machine is provided as the inclination angle of adopting tangential direction and deflection angle quadratic expression long by curve respectively or the long variable of curve the line segment that circles round, means between the point range that interpolation provides arbitrarily in three-dimensional coordinate and playing a role.

23. a numerical control method, wherein:

Adopt the inclination angle of tangential direction and the three-dimensional curve (being called three-dimensional clothoid) that deflection angle quadratic expression long by curve respectively or the long variable of curve provides, the contour shape of performance tool path or workpiece;

Moving of the instrument that appointment is moved along described three-dimensional curve;

According to the motion of appointment, calculate the shift position of instrument by time per unit,

Herein, so-called motion refers to the positional information as the function of time.

24. numerical control method as claimed in claim 23, wherein by defining three-dimensional clothoid with following formula,

[numerical expression 132]

$P=P_0+{\&Integral;}_0^s\mathrm{uds}=P_0+h{\&Integral;}_0^S\mathrm{udS},0\&le;s\&le;h,0\&le;S=\frac{s}{h}\&le;1---\left(1\right)$

$u=E^{\mathrm{k\&beta;}}{E}^{\mathrm{j\&alpha;}}\left(i\right)=\left[\begin{array}{ccc}\cos \mathrm{\&beta;}& -\sin \mathrm{\&beta;}& 0\\ \sin \mathrm{\&beta;}& \cos \mathrm{\&beta;}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos \mathrm{\&alpha;}& 0& \sin \mathrm{\&alpha;}\\ 0& 1& 0\\ -\sin \mathrm{\&alpha;}& 0& \cos \mathrm{\&alpha;}\end{array}\right]\left\{\begin{array}{c}1\\ 0\\ 0\end{array}\right\}=\left\{\begin{array}{c}\cos \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ \sin \mathrm{\&beta;}\cos \mathrm{\&alpha;}\\ -\sin \mathrm{\&alpha;}\end{array}\right\}---\left(2\right)$

α＝a ₀+a ₁S+a ₂S ² (3)

β＝b ₀+b ₁S+b ₂S ² (4)

Herein,

[numerical expression 133]

$P=\left\{\begin{array}{c}x\\ y\\ z\end{array}\right\},P_0=\left\{\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right\}---\left(5\right)$

Position vector and the initial value thereof of representing the point on the three-dimensional clothoid respectively,

To be made as s from the length of a curve of initial point, its length overall (length from the initial point to the terminal point) is made as h, and represent the value except that s with h with S, S is the value of no guiding principle amount, is referred to as the long variable of curve,

I, j, k are respectively x axle, y axle and the axial unit vector of z,

U is the unit vector of the tangential direction of the curve on the expression point P, is provided E by formula (2) ^{K β}And E ^{J α}It is rotation matrix, represent the rotation of the angle beta that the k axle is and the rotation of the angle [alpha] that the j axle is respectively, the former is called deflection (yaw) rotation, the latter is called inclination (pitch) rotation, formula (2), expression is only rotated α by at first making the axial unit vector of i in j axle system, then, system only rotates β at the k axle, obtains tangent line vector u

a ₀, a ₁, a ₂, b ₀, b ₁, b ₂It is constant.

25. a numerical control device, wherein:

Adopt the inclination angle of tangential direction and the three-dimensional curve (being called three-dimensional clothoid) that deflection angle quadratic expression long by curve respectively or the long variable of curve provides, the contour shape of performance tool path or workpiece;

Moving of the instrument that appointment is moved along described three-dimensional curve;

According to the motion of appointment, calculate the shift position of instrument by time per unit,

Herein, so-called motion refers to the positional information that changes as the function of time.

26. a program is used for the motion for the Numerical Control instrument, and computing machine is played a role as following means:

The three-dimensional that the inclination angle of employing tangential direction and deflection angle quadratic expression long by curve respectively or the long variable of curve provides is returned curve (being called three-dimensional clothoid), the means of the contour shape of performance tool path or workpiece;

The means of moving of the instrument that appointment is moved along described three-dimensional curve;

According to the motion of appointment, calculate the means of the shift position of instrument by time per unit,

Herein, so-called motion refers to the positional information that changes as the function of time.

27. the recording medium of an embodied on computer readable, logging program or the result of calculation that draws by this program thereon, this program is used for the motion for the Numerical Control instrument, and computing machine is played a role as following means:

The three-dimensional that the inclination angle of employing tangential direction and deflection angle quadratic expression long by curve respectively or the long variable of curve provides is returned curve (being called three-dimensional clothoid), the means of the contour shape of performance tool path or workpiece;

The means of moving of the instrument that appointment is moved along described three-dimensional curve;

According to the motion of appointment, calculate the means of the shift position of instrument by time per unit,

Herein, so-called motion refers to the positional information that changes as the function of time.

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