CN111913441B - Corner smooth transition method based on track mode - Google Patents

Corner smooth transition method based on track mode Download PDF

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CN111913441B
CN111913441B CN202010792762.5A CN202010792762A CN111913441B CN 111913441 B CN111913441 B CN 111913441B CN 202010792762 A CN202010792762 A CN 202010792762A CN 111913441 B CN111913441 B CN 111913441B
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corner
axis
transition curve
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track
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CN111913441A (en
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李�浩
王保升
闫注文
潘龙
吕东升
陆玄鸣
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Huai'an Aote Technology Co.,Ltd.
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Nanjing Institute of Technology
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B19/00—Programme-control systems
    • G05B19/02—Programme-control systems electric
    • G05B19/18—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form
    • G05B19/416—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by control of velocity, acceleration or deceleration
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B19/00—Programme-control systems
    • G05B19/02—Programme-control systems electric
    • G05B19/18—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form
    • G05B19/19—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by positioning or contouring control systems, e.g. to control position from one programmed point to another or to control movement along a programmed continuous path
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B2219/00—Program-control systems
    • G05B2219/30—Nc systems
    • G05B2219/36—Nc in input of data, input key till input tape
    • G05B2219/36171—Edit velocity, motion profile, graphic plot of speed as function of time, position

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Abstract

The invention discloses a corner smooth transition method based on a track mode, which belongs to the technical field of numerical control machining and comprises the following steps: constructing a track mode containing only three low harmonic components by utilizing a trigonometric function; initializing parameters; respectively establishing an equality constraint condition about boundary limitation, an inequality constraint condition about motion contour limitation and an objective function about contour error limitation on a coordinate axis, calculating a track parameter by utilizing linear programming and substituting the track parameter into a track mode to obtain an axis motion track; and adjusting the motion tracks of the shafts, and synthesizing the motion tracks of the shafts to obtain a corner transition curve. The invention utilizes the track mode to simultaneously complete the construction of the corner transition curve and the planning of the shaft speed, effectively improves the smoothing efficiency and avoids the resonance of a mechanical structure; the shaft motion track is smooth and only comprises three low-frequency components, so that the high-frequency components contained in the driving force/moment can be remarkably reduced, the excitation of the vibration mode of the system is avoided, and the high-speed and high-quality machining is realized.

Description

Corner smooth transition method based on track mode
Technical Field
The invention belongs to the technical field of numerical control machining, and particularly relates to a corner smooth transition method based on a track mode.
Background
In the early stage, in order to solve the problem of machine tool vibration caused by frequent start and stop of a driving shaft, a learner calculates the allowable maximum speed at the joint of continuous small line segments according to the constraint conditions of the acceleration and deceleration performance of the machine tool, load power, contour error and the like, and inserts a straight line segment for transition, so that the continuous processing of a numerical control program is realized, and the processing efficiency is effectively improved. But sudden changes in the speed direction at the connection can still cause the machine to vibrate. Therefore, some scholars continuously improve the smoothness of the connection part of the transition curve and the small line segment by constructing various curves (such as circular arcs, polynomials, Bezier curves, B splines and the like) at the corners and generate processing tracks of G1 (tangent continuity), G2 (curvature continuity) and even G3 (curvature change rate continuity), and the method is called a local track smoothing method. The method has the advantages of good locality and simple error control, and is widely used in actual processing; however, when a continuous processing track above G3 is constructed, the algorithm complexity of the method is increased remarkably, and the improvement of the processing efficiency and the processing quality is not obvious any more.
With the rise of spline interpolation methods, researchers have proposed a global trajectory smoothing method for obtaining smoother processing trajectories, which mainly uses an approximation or interpolation mode to fit a broken line processing path composed of a large number of discrete data points into one or more smooth spline curves under the condition of meeting a contour error limit value, thereby improving the smoothness of the processing path and improving the processing efficiency and the processing quality. For example, the scholars first find a continuous region which can be fitted from a large number of discrete data points, and then fit a broken line processing path in the continuous region by using a polynomial, a B-spline or NURBS curve and the like to obtain continuous processing tracks of G1, G2 and even G3. Therefore, the method can achieve a good effect in improving the overall processing path smoothness and compression ratio. However, as a large number of discrete data points need to be processed, the method becomes more complex with the improvement of the smoothness of the machining track, particularly for the machining track with the continuity of G3 or more, the iteration times and the processing time of the method are obviously increased, and it is difficult to ensure that one interpolation operation is completed within the interpolation period (2ms) of the numerical control system, so that the machine tool is out of step.
The speed planning method mainly ensures the stable operation of the numerical control machine tool by flexibly controlling the speed, the acceleration or the jerk, thereby improving the processing efficiency and the processing quality. For example, in the linear acceleration/deceleration control method, the machining speed is changed in a linear manner in an acceleration/deceleration stage, and the acceleration is kept constant; the method is simple to control and small in calculation amount, but sudden changes exist in the acceleration at the beginning and the end of the acceleration/deceleration stage, and machine tool vibration can be caused. Some scholars propose S-type and cubic polynomial acceleration and deceleration control methods to construct smooth speed curves and continuous acceleration curves, but the acceleration curves still have step situations. In order to further realize flexible control, researchers provide a quartic polynomial type, a quintic polynomial type and a trigonometric function type speed planning method to obtain smooth speed, acceleration and jerk curves; however, the method is complex in flow, needs to involve a plurality of parameters and equations, and takes a large amount of time to perform numerical calculation, so that the methods mostly adopt an off-line mode to perform speed planning.
From the above analysis, in order to reduce machine tool vibration and achieve high-speed, high-precision and high-quality machining, researchers use various polynomial base curves, such as arcs, polynomials, a splines, B splines, C splines, and the like, and by improving the fitting method, the smoothness of the machining path is continuously improved; meanwhile, the flexible control capability of the driving shaft is continuously improved by constructing smooth speed and acceleration curves and even acceleration curves. However, currently, either smoothness of the machining path or the flexible control capability of the drive shaft has reached a limit and is difficult to continue to lift.
Disclosure of Invention
The purpose of the invention is as follows: aiming at the problems in the prior art, the invention discloses a corner smooth transition method based on a track mode, which utilizes the track mode to complete the construction of a corner transition curve and the planning of the axial speed in one step according to a given base frequency, effectively improves the efficiency of smooth processing, further improves the processing efficiency and precision of a complex curve curved surface and avoids the resonance of a mechanical structure.
The technical scheme is as follows: the invention adopts the following technical scheme: a corner smooth transition method based on a track mode is characterized by comprising the following steps:
s1, setting a track mode: constructing a track mode only containing three low harmonic components by utilizing a trigonometric function, wherein the track mode comprises a speed function, an acceleration function and a displacement function;
s2, setting parameter initial values: the parameters comprise the fundamental frequency of the track mode, the contour error limit of the corner transition curve, the motion time on the corner transition curve, and the corner speed, the corner acceleration and the corner jerk at the starting point and the ending point of the corner transition curve;
s3, planning axis movement: establishing a plane direct coordinate system, respectively establishing an equality constraint condition about boundary limitation, an inequality constraint condition about motion contour limitation and an objective function about contour error limitation on two coordinate axes according to the trajectory mode in the step S1, solving a trajectory parameter in the trajectory mode by utilizing linear programming, and respectively obtaining an axis motion trajectory and a corner velocity for performing motion planning on the two coordinate axes;
s4, shaft movement adjustment: when the corner speeds of the motion planning on the two coordinate axes are equal, synthesizing the axis motion tracks on the two coordinate axes to obtain a corner transition curve and an axis speed planning when the corner transition curve is processed; when the corner speeds of the motion planning on the two coordinate axes are not equal, the axis motion track on one coordinate axis is re-planned until the two corner speeds are equal, and the axis motion tracks on the two coordinate axes are synthesized to obtain a corner transition curve and the axis speed planning when the corner transition curve is processed.
Preferably, taking the x-axis as an example, step S3 includes the following steps:
s31, boundary limitation: establishing an equality constraint condition related to the boundary limit through the displacement, the speed, the acceleration and the jerk at the starting point and the speed, the acceleration and the jerk at the end point of the corner transition curve;
s32, motion contour limitation: on the corner transition curve, according to an expected smooth shaft motion profile, namely, a shaft acceleration function is always larger than or equal to zero when the x shaft is accelerated, or the shaft acceleration function is always smaller than or equal to zero when the x shaft is decelerated, an inequality constraint condition about motion profile limitation is established;
s33, contour error limitation: establishing a target function related to the contour error limit according to the minimum distance from the corner point to the corner transition curve, namely the contour error and the set contour error limit;
s34, calculating track parameters: calculating a trajectory parameter by utilizing linear programming according to equality constraint, inequality constraint and an objective function;
s35, track parameter verification: calculating a contour error according to the calculated track parameters, and if the contour error is less than or equal to the contour error limit, obtaining an x-axis motion track according to the track parameters; otherwise, the corner speed is changed to 1/2 for the existing corner speed, and steps S31 through S35 are repeated until the contour error is less than or equal to the contour error limit.
Preferably, in step S4, the corner velocity v set on the x-axis for motion planningi，c，xThe corner velocity for the motion planning on the y-axis is vi，c，yIf v isi，c，x＝vi，c，ySynthesizing the x-axis motion trail and the y-axis motion trail to obtain a corner transition curve and an axis speed plan when the corner transition curve is processed;
if v isi，c，x＜vi，c，yLet v stand fori，c＝vi，c，yExecuting steps S31 to S35, and performing the motion planning on the y axis again until vi，c，x＝vi，c，ySynthesizing the x-axis motion trail and the y-axis motion trail to obtain a corner transition curve and an axis speed plan when the corner transition curve is processed;
if v isi，c，x＞vi，c，yLet v stand fori，c＝vi，c，xExecuting steps S31 to S35, and repeating the motion planning on the x axis until vi，c，x＝vi，c，yAnd synthesizing the x-axis motion trail and the y-axis motion trail to obtain a corner transition curve and an axis speed plan when the corner transition curve is processed.
Preferably, in step S1, the speed function is:
v(t)＝a0+a1 cos(2πft)+a3 cos(6πft)+a5 cos(10πft)
with respect to time t, the first derivative is taken of the velocity function, resulting in an acceleration function of:
a(t)＝-2πfa1 sin(2πft)-6πfa3 sin(6πft)-10πfa5 sin(10πft)
with respect to time t, the second derivative is calculated for the velocity function, resulting in a jerk function of:
j(t)＝-(2πf)2a1 cos(2πft)-(6πf)2a3 cos(6πft)-(10πf)2a5 cos(10πft)
with respect to time t, the velocity function is integrated indefinitely to obtain a displacement function as:
Figure BDA0002622411440000041
wherein, a0、a1、a3、a5And acRespectively are track parameters; f is a fundamental frequency; t is a time variable, T belongs to [0, T ∈]And T is the movement time on the corner transition curve.
Preferably, in step S2, the initialized parameters are: corner velocity vi，c＝min(li，li+1) (ii)/2, corner acceleration ai，c0, corner jerk j i，c0, the motion time T on the corner transition curve is 1/2f, where f is the fundamental frequency and liAnd li+1The lengths of the small line segments on the two sides of the corner where the corner transition curve is located are respectively shown.
Preferably, in step S3, taking the x-axis as an example, the constraint conditions of the equation regarding the boundary limit are:
Figure BDA0002622411440000042
wherein s isx(0)、vx(0)、ax(0) And jx(0) Respectively the axis displacement, the axis speed, the axis acceleration and the axis jerk of the x axis at the starting point of the corner transition curve; v. ofx(T)、ax(T) and jx(f) Respectively the axis speed, the axis acceleration and the axis jerk of the x axis at the end point of the corner transition curve; v. ofi，cAs the starting and ending points of the corner transition curveThe corner speed of (d); a isi，cThe corner acceleration at the starting point and the ending point of the corner transition curve; j is a function ofi，cThe corner acceleration at the starting point and the ending point of the corner transition curve is obtained; theta1Is the angle between the small line segment where the starting point of the corner transition curve is located and the x-axis, theta2Is the included angle between the small line segment where the end point of the corner transition curve is located and the small line segment where the starting point is located.
Preferably, in step S3, taking the x-axis as an example, the inequality constraint conditions regarding the motion profile limitation are:
a(ti) Not less than 0 or a (t)i)≤0，i＝1，2，3，4
Wherein the content of the first and second substances,
Figure BDA0002622411440000051
f is the fundamental frequency.
Preferably, in step S6, the objective function for contour error limitation is:
min(εi-ε)
wherein epsiloniThe contour error of the corner transition curve is shown, and epsilon is the set contour error limit;
Figure BDA0002622411440000052
Figure BDA0002622411440000053
Figure BDA0002622411440000054
Figure BDA0002622411440000055
wherein, a0、a1、a3、a5And acRespectively are track parameters; f is a fundamental frequency; theta1In the form of corner transition curvesThe angle between the small line segment where the starting point is located and the x-axis, theta2Is the included angle between the small line segment where the end point of the corner transition curve is located and the small line segment where the starting point is located.
Has the advantages that: the invention has the following beneficial effects:
1. according to the method, the corner transition curve structure and the axle speed planning are completed in one step by utilizing the track mode according to the given fundamental frequency, so that the smoothing efficiency is effectively improved, and the mechanical structure resonance is avoided;
2. the shaft kinematic profile generated by the invention is very smooth and only contains three low-frequency components, so that the high-frequency components contained in the driving force/moment can be remarkably reduced, the excitation of the vibration mode of the system is avoided, and high-speed and high-quality processing is realized;
3. the invention can realize smooth axis motion contour control under the condition of meeting contour error and axis kinematics limitation, and further improve the processing efficiency and precision of the complex curve curved surface.
Drawings
FIG. 1 is a simplified process flow diagram of the present invention;
FIG. 2 is a flow chart of a method of the present invention;
FIG. 3 is a schematic view of a corner transition structure;
FIG. 4 is a schematic illustration of a shaft displacement profile in a smooth shaft motion profile;
FIG. 5 is a schematic illustration of a shaft velocity profile in a smooth shaft motion profile;
FIG. 6 is a schematic illustration of a shaft acceleration profile in a smooth shaft motion profile;
FIG. 7 is a schematic illustration of a shaft jerk profile in a smooth shaft motion profile;
FIG. 8 is a schematic view of acceleration component function.
Detailed Description
The present invention will be further described with reference to the accompanying drawings.
The invention discloses a corner smooth transition method based on a track mode, which utilizes the track mode to complete corner transition curve construction and axle speed planning in one step according to a given fundamental frequency, effectively improves the efficiency of smoothing treatment and avoids mechanical structure resonance; meanwhile, the generated shaft kinematic profile is very smooth and only contains three low-frequency components, so that the high-frequency components contained in the driving force/moment can be remarkably reduced, the excitation of a system vibration mode is avoided, and high-speed and high-quality machining is realized.
As shown in figure 1, the invention provides a corner smooth transition method based on a track mode, which solves the problems of processing efficiency and precision of a complex curve surface, and the method consists of 7 parts, namely track mode setting, initialization, boundary limitation, motion contour limitation, contour error limitation, track parameter calculation and motion track adjustment, so that the resonance of a mechanical structure is avoided, and the processing quality and efficiency are improved. As shown in fig. 2, the method comprises the following specific steps:
s1, trajectory mode setting: using trigonometric functions, a trajectory pattern is constructed that contains only the three low harmonic components.
The present invention expresses the velocity function as
v(t)＝a0+a1 cos(2πft)+a3 cos(6πft)+a5 cos(10πft) (1)
Wherein v (t) is a function of velocity, a0、a1、a3、a5Is a track parameter; t is a time variable, T belongs to [0, T ∈]T is the exercise time; f is the user specified base frequency.
With respect to time t, the first and second derivatives are taken from equation (1) to obtain the acceleration and jerk functions:
a(t)＝-2πfa1 sin(2πft)-6πfa3 sin(6πft)-10πfa5 sin(10πft) (2)
j(t)＝-(2πf)2a1 cos(2πft)-(6πf)2a3 cos(6πft)-(10πf)2a5 cos(10πft) (3)
where a (t) is an acceleration function and j (t) is a jerk function.
With respect to time t, the displacement function can be obtained by integrating equation (1) indefinitely:
Figure BDA0002622411440000061
wherein s (t) is a displacement function, acAre trajectory parameters.
S2, initialization: fig. 3 depicts the geometry of the corner transition curves. In FIG. 3, P isi，sThe point is set as the origin and the x-y coordinate axes are established therefrom, wherein the corner is defined by a small line segment Pi-1PiAnd PiPi+1The length of the small line segment is liAnd li+1，θ1Is a vector Pi-1PiAngle of inclination with respect to the x-axis, theta2Is a vector Pi-1PiSum vector PiPi+1The included angle therebetween. Point Pi，sOn a small line segment Pi-1PiUpper, point Pi，eOn a small line segment PiPi+1Upper, point Pi，sAnd point Pi，eThe dotted line between them is the resulting corner transition curve. To reduce the complexity of the corner transition curve construction, let us say the corner line segment Pi，sPiAnd PiPi，eAre all Li，cAnd assume a starting point P at the corneri，sAnd an end point Pi，eHave the same velocity vi，cAcceleration ai，c0 and jerk j i，c0, so the corner transition curve is about ≈ Pi-1PiPi+1Is symmetrical with the angle bisector of the corner transition curvei，mDistance inflection point PiIs closest to, point PiAnd point Pm，iThe distance between the two points is the contour error of the corner transition curve.
Setting initial values of various variables used in the corner smooth transition algorithm: fundamental frequency f, contour error limit ε, corner velocity vi，c＝min(li，li+1) And 2, the movement time T on the corner transition curve is 1/2 f.
S3, planning axis movement: and according to the track mode set in the step S1, respectively carrying out motion planning on the x axis and the y axis, and calculating track parameters to obtain the motion track on the x axis and the motion track on the y axis.
Taking the x axis as an example, the specific steps are as follows:
s31, boundary limitation: according to formulae (1) to (4), let sx(0)＝0、vx(0)＝vi，c，xcosθ1、ax(0)＝0、jx(0)＝0、vx(T)＝vi，c，xcos(θ1+θ2)、ax(T)＝0、jxAnd (T) ═ 0, obtaining the equality constraint of the trajectory parameters, and obtaining the following formula (5):
Figure BDA0002622411440000071
wherein s isx、vx、ax、jxRespectively, the functions of the axis displacement, the axis speed, the axis acceleration and the axis jerk of the x axis, vi，cIs the corner velocity, θ1Is the angle between the first small line segment and the positive direction of the x-axis, theta2Is the included angle of two adjacent small line segments.
Order to
Figure BDA0002622411440000072
The above formula can be simplified into
Figure BDA0002622411440000073
S32, motion contour limitation: ensuring the shaft acceleration a of the x-axis according to the expected smooth shaft motion profilex(t) is not less than 0 or axAnd (t) is less than or equal to 0, and inequality constraint of the track parameters is obtained.
Taking acceleration as an example, in order to obtain a smooth kinematic profile, as shown in fig. 4 to 7, it is necessary to ensure that the acceleration a (T) is ≧ 0, T ∈ [0, T ≧ 0]So that the velocity profile v (t) can be set from the starting velocity vsTo the end velocity veIs monotonically increasing. The acceleration function can be written in the form of a composite of three sinusoidal functions:
a(t)＝g1(t)+g2(t)+g3(t) (7)
g1(t)＝-2πfa1 sin(2πft) (8)
g2(t)＝-6πfa3 sin(6πft) (9)
g3(t)＝-10πfa5 sin(10πft) (10)
suppose g1(t)、g2(t)、g3The profiles of (t) and a (t) are shown in FIG. 8. As can be seen from FIG. 8, the extreme value of a (t) occurs at g1(t)、g2(t) and g3(t) in the vicinity of the extreme point. Thus, to obtain a smooth kinematic profile, it is only necessary to ensure that all g's are present1(t)、g2(t) and g3(t) a (t) is not less than 0 at the extreme point of (t). The calculation method of the extreme point is as follows:
in g1(t) for example, since the extreme point of the sine function sin (j) occurs at the point j ═ 2k +1 π/2, k ∈ Z, we can get g1Equation of extreme points of (t)
Figure BDA0002622411440000081
According to formula (11) and the condition T ∈ [0, T ], k can be expressed as
Figure BDA0002622411440000082
Then, all k satisfying formula (12) are calculated and substituted for formula (11) to obtain g1Extreme point of (t)
Figure BDA0002622411440000083
By the above method, we can also calculate g2(t) and g3(t) extreme point. Finally, all extreme points are
Figure BDA0002622411440000084
All extreme points are substituted into the formula (7) to obtain the corresponding a (t) value
a(t1)＝a(t7)，a(t2)＝a(t6)，a(t3)＝a(t5)，a(t4) (15)
Therefore, in order to obtain a smooth kinematic profile, it is only necessary to guarantee jerk a (t)i)≥0，i＝1，2，3，4。
Similarly, taking deceleration as an example, to obtain a smooth kinematic profile, only jerk a (t) needs to be guaranteedi)≤0，i＝1，2，3，4。
S33, contour error limitation: and obtaining an objective function of the track parameter according to the geometric relation of the corner transition structure.
According to the corner transition structure, the relationship between the length of the corner section and the displacement of the corner transition curve can be obtained:
Figure BDA0002622411440000091
wherein L isi，cThe corner segments are long. The above formula is simplified to obtain
Figure BDA0002622411440000092
According to the corner transition structure, the projection of the contour error in the x-axis direction can be obtained
Figure BDA0002622411440000093
Wherein epsiloniIs the contour error of the corner transition curve, epsiloni，xIs epsiloniProjection on the x-axis.
Inflection point PiX-axis coordinate x ofiCan be expressed as
xi＝Li，c cosθ1 (19)
X-axis coordinate x of midpoint of corner transition curvei，mCan be expressed as
Figure BDA0002622411440000094
According to the corner transition structure, the following relationship can be obtained
εi，x＝xi，m-xi (21)
By substituting equations (17) - (20) for equation (21), the profile error can be expressed as
Figure BDA0002622411440000095
Figure BDA0002622411440000096
Figure BDA0002622411440000097
Figure BDA0002622411440000101
S34, calculating track parameters: and calculating the trajectory parameters by utilizing linear programming according to the equality constraint, the inequality constraint and the objective function, and ensuring that the generated axial kinematics contour is very smooth and only contains three low-frequency components.
Let ε be the user-set contour error limit, let (ε)iε) is an objective function, ax(ti) 0, i-1, 2, 3, 4 is an inequality constraint of the motion profile, where t isiFor the time point of the extreme point, the trajectory parameter a can be obtained by performing linear programming on the following formulac、a0、a1、a3And a5：
min(εi-ε) (23)
Figure BDA0002622411440000102
a(ti)≥0，i＝1，2，3，4 (25)
S35, track parameter verification: substituting the obtained trajectory parameters into equation (22) to calculate the profile error εiIf epsiloniIf epsilon is less than or equal toi，cCorner velocity v for planning the movement of the x-axisi，c，xThe group of track parameters are track parameters when the motion planning is carried out on the x axis, and the track parameters are substituted into the formulas (1) to (4) to obtain the motion track of the x axis; if epsiloniIf > epsilon, the corner velocity is changed to 1/2 for the existing corner velocity, let v bei，c＝vi，c(ii)/2, repeating steps S31 to S35 until εi≤ε。
Similarly, the motion planning of the y-axis according to the steps S31 to S35 can obtain the corner velocity v when the motion planning of the y-axis is performedi，c，yAnd the trajectory parameters and the y-axis motion trajectory when the y-axis motion is planned.
S4, shaft movement adjustment: corner velocity v when planning the movement of the x-axisi，c，xAnd the corner velocity v when planning the motion of the y-axisi，c，ySatisfy v betweeni，c，x＝vi，c，yWhen, let vi，c＝vi，c，x＝vi，c，ySynthesizing the x-axis motion trail and the y-axis motion trail to obtain the corner transition curve and the axis speed planning when the corner transition curve is processed.
When v isi，c，x≠vi，c，yWhen, if vi，c，x＜vi，c，yLet the corner velocity vi，c＝vi，c，yExecuting steps S31 to S35, and performing the motion planning on the y axis again until vi，c，x＝vi，c，yLet v stand fori，c＝vi，c，x＝vi，c，yThe corner speed of the corner transition curve is obtained by synthesizing the motion trail of the x-axis and the motion trail of the y-axisPlanning a corner transition curve and an axis speed when the corner transition curve is processed;
if v isi，c，x＞vi，c，yLet the corner velocity vi，c＝vi，c，xExecuting steps S31 to S35, and repeating the motion planning on the x axis until vi，c，x＝vi，c，yLet v stand fori，c＝vi，c，x＝vi，c，yAnd synthesizing the x-axis motion trail and the y-axis motion trail to obtain the corner transition curve and the axis speed planning when the corner transition curve is processed.
The above description is only of the preferred embodiments of the present invention, and it should be noted that: it will be apparent to those skilled in the art that various modifications and adaptations can be made without departing from the principles of the invention and these are intended to be within the scope of the invention.

Claims (8)

1. A corner smooth transition method based on a track mode is characterized by comprising the following steps:
s1, setting a track mode: constructing a track mode only containing three low harmonic components by utilizing a trigonometric function, wherein the track mode comprises a speed function, an acceleration function and a displacement function;
s2, setting parameter initial values: the parameters comprise the fundamental frequency of the track mode, the contour error limit of the corner transition curve, the motion time on the corner transition curve, and the corner speed, the corner acceleration and the corner jerk at the starting point and the ending point of the corner transition curve;
s3, planning axis movement: establishing a plane direct coordinate system, respectively establishing an equality constraint condition about boundary limitation, an inequality constraint condition about motion contour limitation and an objective function about contour error limitation on two coordinate axes according to the trajectory mode in the step S1, solving a trajectory parameter in the trajectory mode by utilizing linear programming, and respectively obtaining an axis motion trajectory and a corner velocity for performing motion planning on the two coordinate axes;
the equality constraints on the boundary limits are: the values of the track mode at the starting point and the ending point of the corner transition curve are obtained;
the inequality constraints on the motion profile constraint are: the value ranges of the axial acceleration functions on the two coordinate axes during acceleration and deceleration;
the objective function for the contour error limit is: the distance from the corner point to the corner transition curve is minimum;
s4, shaft movement adjustment: when the corner speeds of the motion planning on the two coordinate axes are equal, synthesizing the axis motion tracks on the two coordinate axes to obtain a corner transition curve and an axis speed planning when the corner transition curve is processed; when the corner speeds of the motion planning on the two coordinate axes are not equal, the axis motion track on one coordinate axis is re-planned until the two corner speeds are equal, and the axis motion tracks on the two coordinate axes are synthesized to obtain a corner transition curve and the axis speed planning when the corner transition curve is processed.
2. The corner smooth transition method based on track mode as claimed in claim 1, wherein, taking x-axis as an example, step S3 includes the following steps:
s31, boundary limitation: establishing an equality constraint condition related to the boundary limit through the displacement, the speed, the acceleration and the jerk at the starting point and the speed, the acceleration and the jerk at the end point of the corner transition curve;
s32, motion contour limitation: on the corner transition curve, according to an expected smooth shaft motion profile, namely, a shaft acceleration function is always larger than or equal to zero when the x shaft is accelerated, or the shaft acceleration function is always smaller than or equal to zero when the x shaft is decelerated, an inequality constraint condition about motion profile limitation is established;
s33, contour error limitation: establishing a target function related to the contour error limit according to the minimum distance from the corner point to the corner transition curve, namely the contour error and the set contour error limit;
s34, calculating track parameters: calculating a trajectory parameter by utilizing linear programming according to equality constraint, inequality constraint and an objective function;
s35, track parameter verification: calculating a contour error according to the calculated track parameters, and if the contour error is less than or equal to the contour error limit, obtaining an x-axis motion track according to the track parameters; otherwise, the corner speed is changed to 1/2 for the existing corner speed, and steps S31 through S35 are repeated until the contour error is less than or equal to the contour error limit.
3. The method for corner smooth transition based on track mode as claimed in claim 1, wherein in step S4, the corner velocity for motion planning on the x-axis is vi,c,xThe corner velocity for the motion planning on the y-axis is vi,c,yIf v isi,c,x＝vi,c,ySynthesizing the x-axis motion trail and the y-axis motion trail to obtain a corner transition curve and an axis speed plan when the corner transition curve is processed;
if v isi,c,x<vi,c,yLet v stand fori,c＝vi,c,yExecuting steps S31 to S35, and performing the motion planning on the y axis again until vi,c,x＝vi,c,ySynthesizing the x-axis motion trail and the y-axis motion trail to obtain a corner transition curve and an axis speed plan when the corner transition curve is processed;
if v isi,c,x>vi,c,yLet v stand fori,c＝vi,c,xExecuting steps S31 to S35, and repeating the motion planning on the x axis until vi,c,x＝vi,c,yAnd synthesizing the x-axis motion trail and the y-axis motion trail to obtain a corner transition curve and an axis speed plan when the corner transition curve is processed.
4. The corner smooth transition method based on track pattern as claimed in claim 1, wherein in step S1, the speed function is:
v(t)＝a0+a1cos(2πft)+a3cos(6πft)+a5cos(10πft)
with respect to time t, the first derivative is taken of the velocity function, resulting in an acceleration function of:
a(t)＝-2πfa1sin(2πft)-6πfa3sin(6πft)-10πfa5sin(10πft)
with respect to time t, the second derivative is calculated for the velocity function, resulting in a jerk function of:
j(t)＝-(2πf)2a1cos(2πft)-(6πf)2a3cos(6πft)-(10πf)2a5cos(10πft)
with respect to time t, the velocity function is integrated indefinitely to obtain a displacement function as:
Figure FDA0003251841210000021
wherein, a0、a1、a3、a5And acRespectively are track parameters; f is a fundamental frequency; t is a time variable, T belongs to [0, T ∈]And T is the movement time on the corner transition curve.
5. The corner smooth transition method based on track pattern as claimed in claim 4, wherein in step S2, the initialized parameters are: corner velocity vi,c＝min(li，li+1) (ii)/2, corner acceleration ai,c0, corner jerk ji,c0, the motion time T on the corner transition curve is 1/2f, where f is the fundamental frequency and liAnd li+1The lengths of the small line segments on the two sides of the corner where the corner transition curve is located are respectively shown.
6. The method for corner smooth transition based on track mode as claimed in claim 4, wherein in step S3, taking x-axis as an example, the constraint conditions of equation about the boundary limit are:
Figure FDA0003251841210000031
wherein s isx(0)、vx(0)、ax(0) And jx(0) Respectively the axis displacement, the axis speed, the axis acceleration and the axis jerk of the x axis at the starting point of the corner transition curve; v. ofx(T)、ax(T) and jx(T) axis velocity, axis acceleration and axis jerk of the x-axis at the end point of the corner transition curve, respectively; v. ofi,cThe corner speeds at the starting point and the ending point of the corner transition curve; a isi,cThe corner acceleration at the starting point and the ending point of the corner transition curve; j is a function ofi,cThe corner acceleration at the starting point and the ending point of the corner transition curve is obtained; theta1Is the angle between the small line segment where the starting point of the corner transition curve is located and the x-axis, theta2Is the included angle between the small line segment where the end point of the corner transition curve is located and the small line segment where the starting point is located.
7. The method for corner smooth transition based on track pattern as claimed in claim 4, wherein in step S3, taking x axis as an example, the inequality constraint conditions for motion contour limitation are:
a(ti) Not less than 0 or a (t)i)≤0，i＝1,2,3,4
Wherein the content of the first and second substances,
Figure FDA0003251841210000032
f is the fundamental frequency.
8. The method for corner smooth transition based on track pattern as claimed in claim 4, wherein in step S6, the objective function for contour error limitation is:
min(εi-ε)
wherein epsiloniThe contour error of the corner transition curve is shown, and epsilon is the set contour error limit;
Figure FDA0003251841210000041
Figure FDA0003251841210000042
Figure FDA0003251841210000043
Figure FDA0003251841210000044
wherein, a0、a1、a3、a5And acRespectively are track parameters; f is a fundamental frequency; theta1Is the angle between the small line segment where the starting point of the corner transition curve is located and the x-axis, theta2Is the included angle between the small line segment where the end point of the corner transition curve is located and the small line segment where the starting point is located.
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