CN113791581A - Spherical shrub spherical interpolation algorithm based on equal-chord segmentation sampling - Google Patents

Spherical shrub spherical interpolation algorithm based on equal-chord segmentation sampling Download PDF

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CN113791581A
CN113791581A CN202110883515.0A CN202110883515A CN113791581A CN 113791581 A CN113791581 A CN 113791581A CN 202110883515 A CN202110883515 A CN 202110883515A CN 113791581 A CN113791581 A CN 113791581A
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interpolation
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point
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robot
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CN113791581B (en
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李传军
陈玺
王立萍
杨保鹏
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Tianjin University of Science and Technology
Tianjin Sino German University of Applied Sciences
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Tianjin Sino German University of Applied Sciences
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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
    • G05B19/41—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 interpolation, e.g. the computation of intermediate points between programmed end points to define the path to be followed and the rate of travel along that 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
    • G05B2219/34—Director, elements to supervisory
    • G05B2219/34083—Interpolation general

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Abstract

The invention discloses a spherical shrub spherical interpolation algorithm based on equal chord segmentation sampling, which gives a spherical center coordinate P0The radius of the sphere center R. The method comprises the following steps: firstly, a sphere interpolation auxiliary point P is obtained1‑P6The spherical surface can be regarded as a plurality of radial semicircles, and the P is obtained by using the algorithm of the invention1‑P4The coordinate of the interpolation point m +1 of the spherical surface is used for calculating the radial semicircular posture of the robot at each section; setting the starting point and the ending point of each radial semicircle to be Pstart，PendLet Pstart＝P5,Pend＝P6，PjIs P1‑P4The j of the jth interpolation point of the great circle of the sphere is 0; solving the arc P using the algorithm of the present inventionstarPjAnd the arc PjPendN +1 interpolation point coordinates in the formed radial semicircle; solving robot in arc PstarPjAnd the arc PjPendAttitude during interpolation. The invention reduces the operation amount and improves the operation speed。

Description

Spherical shrub spherical interpolation algorithm based on equal-chord segmentation sampling
Technical Field
The invention belongs to the technical field of robot control, and particularly relates to a spherical shrub spherical interpolation algorithm based on equal-chord segmentation sampling.
Background
The urban greening is insufficient due to the expansion of urbanization. In urban greening, the spherical shrubs are favored in greening construction due to the advantages of vigorous vitality, easy survival, beautiful appearance, evergreen four seasons and the like. At present, spherical shrubs are mainly trimmed by relying on experience of a trimming cutter held by a florist, other post personnel cannot perform the trimming, and the spherical shrubs are trimmed in a certain seasonality, so that the labor is unstable, the management cost is increased, the efficiency of trimming the spherical shrubs is low, and the maintenance cost is high. Spherical shrubs of different specifications are trimmed depending on visual dimensions, the grasping is not accurate enough, and after the trimming is not standard, the trimming is smaller, the batch and standardization cannot be achieved, and the phenomenon of irregular is often caused.
At present, a vector method, a coordinate transformation method and an integral method are mostly adopted for processing circular arcs and spherical surfaces of robots. The obtained interpolation point needs to be subjected to coordinate transformation, integral operation and differential operation, so that the operation amount is increased. A method with simple algorithm, small operand, high operation speed and high interpolation coordinate precision is needed.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: the space circular interpolation algorithm based on the isochord segmentation sampling is provided, and is expanded into spherical shrub spherical interpolation to obtain the space spherical interpolation point coordinates and the pose of the robot in the pruning process. The algorithm has no theoretical error, improves the interpolation coordinate precision, has small operand and improves the operation speed.
In order to solve the technical problems, the technical scheme provided by the invention is as follows: a space circular interpolation algorithm based on equal chord segmentation sampling is extended to spherical interpolation of spherical shrubs, and the position P of the spherical center of the spherical shrub relative to a base coordinate system is known0＝(x0,y0,z0) And the radius R of the sphere.
Step 1, fromKnowing the conditions, solving the spherical interpolation auxiliary point coordinate P1-P6. Wherein point P1，P2；P3，P4；P5，P6Are all diameter point pairs.
Let θ be atan2 (y)0,x0). Then P is1-P6The coordinates of (a) are:
Figure BDA0003193109170000011
and 2, applying an equichord segmentation sampling space circular arc algorithm to spatial spherical interpolation based on the equichord segmentation sampling spherical interpolation algorithm. Solving P by using an equichord division sampling space circular arc algorithm1-P4The coordinates of m +1 interpolation points on the spherical surface are obtained. The principle of the equichord division sampling space circular arc algorithm is as follows:
the algorithm is divided into two cases, namely the central angle 0< theta < pi and pi < theta <2 pi of the circular arc.
(1) When the central angle of the circular arc is 0< theta < pi
Let a given arc starting point be Ps＝(xs,ys,zs) The end point is Pe＝(xe,ye,ze) Center coordinate of circle is P0＝(x0,y0,z0) The radius of the circular arc is R. Then the chord PsPeThe length L is:
Figure BDA0003193109170000021
according to the starting point P of the arcsEnd point PeFinding the interpolation point PiFirst, for the chord PsPeAnd (5) equally dividing, wherein the equally dividing precision is delta L. The number of interpolation times n is
Figure BDA0003193109170000022
The ith interpolation point on the circular arc isPiThen straight line P0PiChord PsPeHas a point of intersection of Pi'。PsPiIs of length of
PsPi'＝i×△L
PiThe coordinates of
Figure BDA0003193109170000023
Straight line P0PiIs of length of
Figure BDA0003193109170000024
Interpolation point PiThe coordinates are
Figure BDA0003193109170000025
(2) When the central angle of the circular arc is pi < theta <2 pi
Let a given arc starting point be Ps＝(xs,ys,zs) The end point is Pe＝(xe,ye,ze) Center coordinate of circle is P0＝(x0,y0,z0) The radius of the circular arc is R. Can find PeAnd PsDiameter to point coordinate Pe'，Ps' coordinates. Arc PsPe'，Pe'Ps'，Ps'PeAll central angles of (1) belong to 0<θ<Pi, each segment can be based on a central angle of 0<θ<And when the pi is larger than the standard pi, interpolation is carried out.
Step 3, the spherical surface can be made into m sections of radial semi-circles, and P is solved according to the algorithm in the step 21-P4And m +1 interpolation points of the spherical surface great circle are used for calculating the radial semicircular posture of the robot at each section. The principle is as follows:
let the jth interpolation point be Pj＝(xj,yj,zj)。
Figure BDA0003193109170000031
θjThe projection angle of the unit vector of the tail end of the robot on the base mark is shown.
The tail end gesture of the robot is as follows:
Figure BDA0003193109170000032
and 4, solving the motion trail of the robot, wherein the algorithm comprises the following steps:
setting the starting point and the ending point of each radial semicircle to be Pstart，Pend。Pstart＝P5,Pend＝P6。PjIs P1-P4The jth interpolation point of the great circle of the spherical surface. Let j equal 0.
The first step is as follows: solving a first radial semicircle according to the method of step 2: arc PstarPjAnd the arc PjPendAnd (5) coordinates of the middle n +1 interpolation points.
The second step is that: solving robot in arc PstarPjAnd the arc PjPendAttitude during interpolation.
The third step: when the robot is at PendAnd when j ≠ m +1, exchange Pstart，Pend. And returning to the first step when j is j +1 to solve the next radial semicircle pose. When the robot is at PendAnd j equals m +1, the interpolation motion is ended.
Adopt the beneficial effect that above-mentioned technical scheme brought:
(1) the invention uses the parameter equation to represent the coordinates of the interpolation points, and the interpolation points all fall on the spherical surface, thereby avoiding the accumulated error.
(2) The tail end postures of each arc robot section are the same, the tail end postures of the robots do not need to be calculated and adjusted all the time, and the algorithm is simple and reduces the calculated amount.
(3) The invention has no coordinate transformation, no matrix, differential and integral operation in the operation process, thereby improving the operation efficiency.
Drawings
FIG. 1 is a flow chart of the algorithm of the present invention
FIG. 2 is a schematic diagram of an auxiliary point for spherical interpolation according to the present invention
FIG. 3 is a schematic diagram of the interpolation algorithm of the present invention showing the principle that 0< theta < pi
FIG. 4 minor arc P of the invention1P3Interpolation point simulation schematic diagram
FIG. 5 is a schematic diagram of the principle of interpolation algorithm central angle pi < theta <2 pi
FIG. 6 invention P1-P4Schematic diagram of point simulation of great circle insertion on spherical surface
FIG. 7 is a schematic diagram of the interpolation motion of the spherical surface of the robot according to the present invention
Detailed Description
The technical solution of the present invention will be described in detail below with reference to the accompanying drawings.
Step one, as shown in figure 2, a coordinate system unit is cm, and the position P of the sphere center of the spherical shrub relative to a base coordinate system is given0The radius of the sphere is 30cm (30, -100,100). Solving the auxiliary point P1-P6And (4) coordinates.
θ＝atan2(y0,x0)＝-73.3008°
Auxiliary point P1-P6The coordinates are:
Figure BDA0003193109170000041
step two, solving P by using an equichord division sampling space circular arc algorithm1-P4The coordinates of m +1 interpolation points on the spherical surface are obtained. As shown in fig. 3: minor arc P1P3Starting point PsCoordinate is P1(38.6204, -128.7347,100), end point PeCoordinate is P3(58.7347, -91.3795,100). The coordinate of the circle center is P0(30, -100,100). The radius of the arc is 30 cm. Chord length PsPeA length L of
Figure BDA0003193109170000042
The interpolation order 50 is set.
Figure BDA0003193109170000043
The ith interpolation point on the circular arc is PiThen straight line P0PiChord PsPeHas a point of intersection of Pi'。PsPiIs of length of
PsPi'＝i×△L＝i×0.8485
PiThe coordinates of
Figure BDA0003193109170000051
Straight line P0PiIs of length of
Figure BDA0003193109170000052
Interpolation point PiThe coordinates are
Figure BDA0003193109170000053
Minor arc P1P3The interpolation points above are all found. The simulation is shown in fig. 4.
Minor arc P1P3Point of insertion for iiCoordinates, as shown in table 1:
i Pix Piy Piz
0 38.6204 -128.7347 100
1 39.2049 -128.5529 100
2 39.8092 -128.3510 100
3 40.4333 -128.1273 100
… … … …
50 58.7347 -91.3795 100
as shown in fig. 5: let a given aStarting point of arc is Ps＝(xs,ys,zs) The end point is Pe＝(xe,ye,ze) Center coordinate of circle is P0＝(x0,y0,z0) The radius of the arc is R, P can be obtainedeAnd PsDiameter to point coordinate Pe'，Ps' coordinates in this way, the minor arc P can be determined1P3,P3P2，P2P4，P4P1The simulation of the formed spherical surface great circle is shown in figure 6. To this end P1-P4The interpolation points of the big circles of the spherical surfaces are obtained.
Step 3, the spherical surface can be made into m sections of radial semi-circles, and P is solved according to the algorithm in the step 21-P4And m +1 interpolation points of the spherical surface great circle are used for calculating the radial semicircular posture of the robot at each section.
Let the jth interpolation point be Pj＝(xj,yj,zj)。
Figure BDA0003193109170000061
θjThe projection angle of the unit vector of the tail end of the robot on the base mark is shown.
The tail end gesture of the robot is as follows:
Figure BDA0003193109170000062
TABLE 2 robot PjIn the arc P1P3P2And the posture of each radial semicircle during interpolation.
j axj ayj azj
0 -0.2873 -0.9578 0
1 -0.3068 -0.9518 0
2 -0.3270 -0.9450 0
3 -0.3478 -0.9376 0
101 0.2873 0.9578 0
And 4, solving the motion track of the robot.
Setting the starting point and the ending point of each radial semicircle to be Pstart，Pend. Wherein P isstart＝P5,Pend＝P6。PjIs P1-P4The jth interpolation point of the great circle of the spherical surface. Let j equal0。
The first step is as follows: solving a first radial semicircle according to the method of step 2: arc PstarPjAnd the arc PjPendAnd (5) coordinates of the middle n interpolation points.
The second step is that: solving robot in arc PstarPjAnd the arc PjPendAttitude during interpolation.
The third step: when the robot end is at PendAnd when j ≠ m +1, exchange Pstart，Pend. And returning to the first step when j is j +1 to solve the next radial semicircle pose. When the robot is at PendAnd j equals m +1, the interpolation motion is ended.
Fig. 7 shows a motion simulation process of a robot for trimming spherical shrubs.

Claims (5)

1. A spherical shrub spherical interpolation algorithm based on isochord segmentation sampling is characterized by comprising the following steps:
step 1, according to the position P of the spherical center of the spherical shrub relative to a base coordinate system0＝(x0,y0,z0) And the radius R of the sphere, and solving the coordinates P of the spherical interpolation auxiliary point1-P6Center point P of1，P2；P3，P4；P5，P6Are all diameter point pairs;
step 2, the spherical shrub spherical interpolation algorithm based on the isochord segmentation sampling applies the isochord segmentation sampling space circular arc algorithm to the spatial spherical interpolation, and the isochord segmentation sampling space circular arc algorithm is used for obtaining P1-P4The coordinates of m +1 interpolation points of the great circle of the spherical surface are located;
step 3, the spherical surface can be made into m sections of radial semi-circles, and P is solved according to the algorithm in the step 21-P4M +1 interpolation points of the spherical surface are used for calculating the radial semicircular gesture of the robot at each section;
and 4, solving the motion trail of the robot, wherein the algorithm comprises the following steps:
setting the starting point and the ending point of each radial semicircle to be Pstart，Pend，Pstart＝P5,Pend＝P6；PjIs P1-P4The j of the jth interpolation point of the great circle of the sphere is 0;
the first step is as follows: solving a radial semicircle according to the method of step 2: arc PstarPjAnd the arc PjPendCoordinates of the middle n +1 interpolation points;
the second step is that: solving robot in arc PstarPjAnd the arc PjPendAttitude in the interpolation process;
the third step: when the robot end is at PendAnd when j ≠ m +1, exchange Pstart，PendAnd j is j +1, returning to the first step to solve the next radial semicircle pose, and when the robot is positioned at PendAnd j equals m +1, the interpolation motion is ended.
2. The spherical shrub spherical interpolation algorithm based on isochord segmentation sampling as claimed in claim 1, wherein in step 1, the coordinates P of the spherical interpolation auxiliary point is solved1-P6Let θ be atan2 (y)0,x0) Then P is1-P6The coordinates of (a) are:
Figure FDA0003193109160000011
3. the spherical shrub spherical interpolation algorithm based on the equichord segmentation sampling as claimed in claim 1, wherein in the step 2, the principle of the equichord segmentation sampling space circular arc algorithm is as follows: when the central angle of the circular arc is 0< theta < pi
Let a given arc starting point be Ps＝(xs,ys,zs) The end point is Pe＝(xe,ye,ze) Center coordinate of circle is P0＝(x0,y0,z0) Radius of arc R, chord PsPeThe length L is:
Figure FDA0003193109160000012
according to the starting point P of the arcsEnd point PeFinding the interpolation point PiFirst, for the chord PsPeEqually dividing, wherein the equally dividing precision is delta L;
the number of interpolation times n is
Figure FDA0003193109160000021
The ith interpolation point on the circular arc is PiThen straight line P0PiChord PsPeHas a point of intersection of Pi'，PsPiIs of length of
PsPi'＝i×△L
PiThe coordinates of
Figure FDA0003193109160000022
Straight line P0PiIs of length of
Figure FDA0003193109160000023
Interpolation point PiThe coordinates are
Figure FDA0003193109160000024
4. The spherical shrub spherical interpolation algorithm based on the equichord segmentation sampling as claimed in claim 3, wherein in the step 2, the principle of the equichord segmentation sampling space circular arc algorithm is as follows: when the central angle of the circular arc is pi < theta <2 pi
Is provided toDefining the starting point of a certain circular arc as Ps＝(xs,ys,zs) The end point is Pe＝(xe,ye,ze) Center coordinate of circle is P0＝(x0,y0,z0) The radius of the arc is R, P can be obtainedeAnd PsDiameter to point coordinate Pe'，Ps' coordinate, arc PsPe'，Pe'Ps'，Ps'PeAll central angles of (1) belong to 0<θ<Pi, each segment can be based on a central angle of 0<θ<And when the pi is larger than the standard pi, interpolation is carried out.
5. The spherical shrub spherical interpolation algorithm based on the isochord segmentation sampling as claimed in claim 1, wherein in the step 3, the principle of calculating the radial semicircular attitude of each segment is as follows:
let the jth interpolation point be Pj＝(xj,yj,zj)
Figure FDA0003193109160000031
θjThe projection included angle of the unit vector at the tail end of the robot on the base mark is shown;
the tail end gesture of the robot is as follows:
Figure FDA0003193109160000032
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