CN110962130A - Heuristic RRT mechanical arm motion planning method based on target deviation optimization - Google Patents

Heuristic RRT mechanical arm motion planning method based on target deviation optimization Download PDF

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CN110962130A
CN110962130A CN201911346837.0A CN201911346837A CN110962130A CN 110962130 A CN110962130 A CN 110962130A CN 201911346837 A CN201911346837 A CN 201911346837A CN 110962130 A CN110962130 A CN 110962130A
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CN110962130B (en
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袁成人
刘桂峰
张文群
郭文勇
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Naval University of Engineering PLA
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
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    • B25J9/1656—Programme controls characterised by programming, planning systems for manipulators
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Abstract

The invention belongs to the technical field of robot path planning and generation in a three-dimensional space, and particularly relates to a heuristic RRT mechanical arm motion planning method based on target deviation optimization. The method comprises the following steps: determining a starting point qstartQ is prepared bystartStore to qnodesPerforming the following steps; full map random sampling point q in spacerandLooking for qnodesReach qrandOne of the nearest points is taken as qnear(ii) a At qnearTo qrandIn the direction of (d) by a step delta, using a target bias factor
Figure DDA0002333607850000011
Controlling the random point generation direction and distance to advance to the node qnewCollision detection is performed during the forward process, and if a collision is detected, a2 is returned; if no collision is detected, q isnewStore to qnodesNeutralization and conversion to a 4; according to the updated qnodesRepeating A2-A3 until obtainingUp-to-date node qgoalSatisfy | qnew‑qgoal|<Error, as immediate to target point, treat qnewIs stored in qnodesPerforming the following steps; at qnodesAnd reversely searching to find the planned path according to the parent-child relationship of each node. The invention can reduce the length of the search path and avoid the phenomenon of falling into a local optimal value or oscillating near an obstacle.

Description

Heuristic RRT mechanical arm motion planning method based on target deviation optimization
Technical Field
The invention belongs to the technical field of robot path planning and generation in a three-dimensional space, and particularly relates to a heuristic RRT mechanical arm motion planning method based on target deviation optimization.
Background
Motion planning can be divided into path planning and trajectory planning. The path planning focuses on the generated path, the trajectory planning gives path time information, for the mechanical arm, an end effector is a technical object, the path point tracing problem of the mechanical arm is mainly technically solved in the path planning, and the practical application of inverse kinematics solution (pose inverse solution, speed inverse solution and acceleration inverse solution) is realized in the trajectory planning according to the requirements of operation tasks, so the motion planning of one system has important significance for solving the practical problem.
The RRT algorithm is a rapid path planning algorithm based on random sampling, can effectively search a high-dimensional space, and can effectively avoid the difficulty of three-dimensional modeling, so that the RRT algorithm is widely adopted. Because the RRT algorithm search has randomness and blindness, scholars at home and abroad propose a plurality of improved algorithms based on RRT aiming at different technical objects and performance requirements. The LaValle introduces probability-based sampling, so that the search efficiency of the basic RRT algorithm is improved, but the LaValle is easy to fall into local minimum. Karaman proposes an RRT algorithm for path optimization, and the planning path has high stability and is close to the optimal path, but consumes a great deal of time. The above-mentioned technique can not completely avoid the phenomenon of local optimum or oscillation near the obstacle, and the algorithm reliability is not strong for the mechanical arm. Meanwhile, the motion planning is separated, most algorithms stay in a two-dimensional space, and the feasibility of practical application in a three-dimensional space cannot be guaranteed.
Disclosure of Invention
The invention aims to provide a heuristic RRT mechanical arm motion planning method (PBG-RRT) based on target deviation optimization, which combines heuristic probability to plan a path by using a target deviation factor, avoids the phenomenon of falling into a local optimal value or oscillating near an obstacle, improves the quality and efficiency of trajectory planning while ensuring the original path characteristics.
In order to achieve the purpose, the invention adopts the following technical scheme.
The invention relates to a heuristic RRT mechanical arm motion planning method (PBG-RRT) based on target deviation optimization, which comprises the following steps:
step one, determining RRT path nodes based on target deviation optimization; the method comprises the following steps:
a1, determining a starting point qstartQ is prepared bystartStore to qnodesPerforming the following steps;
a2, sampling point q at random on full map in spacerandLooking for qnodesReach qrandOne of the nearest points is taken as qnear；
A3 at qnearTo qrandIn the direction of (d) by a step delta, using a target bias factor
Figure BDA0002333607830000021
Controlling the random point generation direction and distance to advance to the node qnewCollision detection is performed during the forward process, and if a collision is detected, a2 is returned; if no collision is detected, q isnewStore to qnodesNeutralization and conversion to a 4; wherein:
Figure BDA0002333607830000022
Figure BDA0002333607830000023
Figure BDA0002333607830000024
a4, according to the updated qnodesRepeating A2-A3 until the latest node q is obtainednewSatisfy | qnew-qgoal|<Error, as tracing to target point, let qnewIs stored in qnodesPerforming the following steps;
a5 at qnodesAnd reversely searching to find the planned path according to the parent-child relationship of each node.
Further improvement of the heuristic RRT mechanical arm motion planning method (PBG-RRT) based on target deviation optimization further comprises:
step two, performing thinning processing on the initial path node based on the Douglas-Peucker algorithm; the method comprises the following steps:
b1, connecting a line segment between the head and the tail AG points of the path according to the planning path point obtained by A5;
b2, determining a point C with the maximum distance from other path points to the line segment AG, and acquiring the distance d from the point C to the line segment AG;
b3, comparing the given gamma with d, if d is small, deleting the point C, if d is large, keeping the point C as a key point, and dividing the original path into two segments of AC and CG by using the point C; wherein, gamma belongs to (0, delta);
b4, repeating B1-B3 for the segments formed after division; deleting or keeping as a key point until all points on the initial path have been traversed;
b5, derived keypoint qiI.e. the path point after the rarefaction.
The further improvement of the heuristic RRT mechanical arm motion planning method (PBG-RRT) based on the target deviation optimization also comprises
Step three, a track optimization step based on non-uniform B spline fitting comprises the following steps:
c1, optimizing the path after rarefaction obtained in the B5 by adopting cubic non-uniform B spline interpolation;
b-spline curve equation:
Figure BDA0002333607830000031
wherein d isi(i-1, 2, …, N) is a control point, Ni,kNormalizing the B-spline basis function for k times; and is
Figure BDA0002333607830000032
Wherein li＝|di-di-1|(i＝1,2,...,n)；
Figure BDA0002333607830000033
Wherein k is an even number of times:
Figure BDA0002333607830000034
where k is an odd number of times:
Figure BDA0002333607830000035
path point q obtained based on B5iObtaining a control point d by inverse solution of the non-uniform B-splinei(i∈qcurve) And then generating a smooth track through the control points, namely realizing the track optimization on the basis of the original path planning.
Further improvement of the heuristic RRT mechanical arm motion planning method (PBG-RRT) based on target deviation optimization further comprises,
in step a3, when there is no obstacle effect, control is performed when β ═ pi
Figure BDA0002333607830000036
And local optimal solution caused by too strong directivity of a target point is avoided.
Wherein β is
Figure BDA0002333607830000037
And
Figure BDA0002333607830000038
angle between delta e C (arbitrary constant)
The further improvement of the heuristic RRT mechanical arm motion planning method (PBG-RRT) based on the target deviation optimization further comprises that in the part C1 of the step three, the inverse solution formula of cubic non-uniform B splines is as follows:
Figure BDA0002333607830000041
wherein, the element is the value of the basic function in the coefficient matrix, only related to the node vector u, and is simplified as follows:
Figure BDA0002333607830000042
wherein
Figure BDA0002333607830000043
The beneficial effects are that:
1. the invention provides a heuristic RRT mechanical arm motion planning method (PBG-RRT) based on target deviation optimization aiming at a mechanical arm. The algorithm combines heuristic probability, carries out path planning by using a target deflection factor, obviously reduces the length of a search path, and avoids the phenomenon of falling into a local optimal value or oscillating near an obstacle;
2. and thinning is performed after the path points are obtained, and a smooth track is generated by adopting non-uniform B-spline interpolation, so that the quality and efficiency of track planning are improved while the original path characteristics are ensured, and the oscillation phenomenon of nearby obstacles in the path planning process is avoided. At the same time, the number of inflection points due to non-uniform B-spline fitting can be reduced.
3. The three-dimensional space obstacle avoidance simulation experiment shows that: compared with RRT with heuristic probability, the PBG-RRT has the advantages that the PBG-RRT search efficiency is improved by 217%, and the search time efficiency is improved by 168%. And after the thinning, applying the tail end track of the mechanical arm generated by three times of non-uniform B splines, and obtaining the stable change of each joint of the mechanical arm through inverse kinematics solution. And finally, performing a physical operation experiment on the ROS, wherein the result shows that the actual operation is consistent with the simulation experiment.
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FIG. 1 is a schematic diagram of the growth process of RRT;
FIG. 2 is a schematic diagram of the RRT searching process in three-dimensional space;
FIG. 3 is a schematic diagram of a random sampling that may occur during a search;
FIG. 4 is a schematic diagram of a planned trajectory obtained based on the first, second, and third steps of the present invention;
FIG. 5 is a search schematic diagram in a route planning process based on the RRT method;
FIG. 6 is a search schematic diagram in a path planning process based on the P-RRT method;
FIG. 7 is a schematic diagram of search in a path planning process based on the BG-RRT method;
FIG. 8 is a schematic diagram of the search in the path planning process based on the PBG-RRT method of the present invention;
fig. 9 is a schematic diagram of path planning based on the PBG-RRT method of the present invention (δ ═ 5);
fig. 10 is a schematic diagram of path planning based on the PBG-RRT method of the present invention (δ ═ 10);
fig. 11 is a schematic diagram of path planning based on the PBG-RRT method of the present invention (δ -15);
fig. 12 is a schematic diagram of path planning based on the PBG-RRT method (δ ═ 20) of the present invention.
Detailed Description
The invention is described in detail below with reference to specific embodiments.
The first part
The heuristic RRT mechanical arm motion planning method (PBG-RRT) based on target deviation optimization is characterized in that the RRT algorithm is a rapid path planning algorithm based on random sampling, can effectively search a high-dimensional space, can effectively avoid the difficulty of three-dimensional modeling and is widely adopted, the RRT is an algorithm which is convenient to search in the high-dimensional space by traversing a whole graph with probability, and FIG. 1 is a growth process graph of the RRT.
The basic principle is as follows: given a starting point qstartQ is prepared bystartStore to qnodesIn space, full map random sampling point qrandLooking for qnodesReach qrandOne nearest point in is qnearAt q isnearTo qrandIn the direction of (a) advances to q with a certain step deltanewDuring which collision detection is performed, if no collision is detectedThen q isnewStore to qnodesIf a collision is detected, resampling is performed and the above process is repeated. When | qnew-qgoal|<Error is regarded as finding the target point, and q is regarded asnewIs stored in qnodesIn (1). Finally at qnodesAnd reversely searching to find the planned path according to the parent-child relationship of each node. The RRT has strong searching capability, but has obvious deficiency in searching efficiency due to randomness and blindness caused by random sampling. Fig. 2 shows the search process of the RRT in the three-dimensional space. As the tracking space becomes larger, although traversing the entire space, the algorithm can track to a path, but consumes a significant amount of computational resources and time.
Therefore, the invention provides an improved heuristic RRT mechanical arm motion planning method (PBG-RRT) based on target deviation optimization, which specifically comprises the following steps:
step one, determining RRT path nodes based on target deviation optimization; the method comprises the following steps:
a1, determining a starting point qstartQ is prepared bystartStore to qnodesPerforming the following steps;
a2, sampling point q at random on full map in spacerandLooking for qnodesReach qrandOne of the nearest points is taken as qnear；
A3 at qnearTo qrandIn the direction of (d) by a step delta, using a target bias factor
Figure BDA0002333607830000061
Controlling the random point generation direction and distance to advance to the node qnewCollision detection is performed during the forward process, and if a collision is detected, a2 is returned; if no collision is detected, q isnewStore to qnodesNeutralization and conversion to a 4; wherein:
Figure BDA0002333607830000062
target bias factor
Figure BDA0002333607830000063
The strategy is used for controlling the generation direction and distance of random points by combining the idea of artificial potential field. When the target point is far away, the shock near an obstacle can be avoided by using the search capability of the RRT and the heuristic probability as guidance, and when the target point is close, the target point can be quickly pointed and converged.
In the random sampling process, there are three cases, as shown in FIG. 3, when there is no barrier effect, when β ∈ (0, π/2), q isnewRatio qrandCloser to qgoalThe moving step length is more efficient, and when β is equal to the element (pi/2, pi), q is equal to the elementnewRatio qrandCloser to qgoalBut the moving step efficiency is reduced, and the search for invalid interval is reduced, when β is pi, control is needed
Figure BDA0002333607830000064
And local optimal solution caused by too strong directivity of a target point is avoided.
A4, according to the updated qnodesRepeating A2-A3 until the latest node q is obtainednewSatisfy | qnew-qgoal|<Error, consider to find the target point, let qnewIs stored in qnodesPerforming the following steps;
a5 at qnodesAnd reversely searching to find the planned path according to the parent-child relationship of each node.
The search tree nodes are increased along with the expansion of the space, the path points of the track planning are correspondingly increased, the three-dimensional space is particularly obvious, the overall concave-convex performance of the curve of the B-spline fitting along with the increase of the control points is increased, the corresponding distance, time and mechanical arm energy consumption for the track planning are increased, and the method has important significance for reducing the control points to the track planning as much as possible while the original curve characteristics of the path are kept.
Therefore, the heuristic RRT mechanical arm motion planning method (PBG-RRT) based on target deviation optimization further comprises the following steps:
step two, performing thinning processing on the initial path node based on the Douglas-Peucker algorithm; the method comprises the following steps:
b1, connecting a line segment between the head and the tail AG points of the path according to the planning path point obtained by A5;
b2, determining a point C with the maximum distance from other path points to the line segment AG, and acquiring the distance d from the point C to the line segment AG;
b3, comparing the given gamma with d, if d is small, deleting the point C, if d is large, keeping the point C as a key point, and dividing the original path into two segments of AC and CG by using the point C; wherein, gamma belongs to (0, delta);
b4, repeating B1-B3 for the segments formed after division; deleting or keeping as a key point until all points on the initial path have been traversed;
b5, derived keypoint qiI.e. the path point after the rarefaction.
Due to the randomness and blindness of the RRT, a continuous line segment formed by planned path points is generally in a sawtooth shape, the vibration of the mechanical arm is unstable in the motion process due to the path, and the acceleration is influenced and damaged by sudden change of the acceleration.
Therefore, the invention provides a scheme for optimizing the track by utilizing cubic non-uniform B-spline interpolation, so that the obtained final path can be applied to a robot in a high-dimensional space, the problem of joint driving smoothness is solved, and a processing method for laying a foundation for a manipulator to execute a complex operation task is provided. Specifically, the method comprises the following steps:
step three, a track optimization step based on non-uniform B spline fitting comprises the following steps:
c1, optimizing the path after rarefaction obtained in the B5 by adopting cubic non-uniform B spline interpolation;
b-spline curve equation:
Figure BDA0002333607830000071
wherein d isi(i-1, 2, …, N) is a control point, Ni,kNormalize the B-spline basis function for k-times, and:
Figure BDA0002333607830000072
wherein li＝|di-di-1|(i＝1,2,...,n)；
Figure BDA0002333607830000073
Wherein k is an even number of times:
Figure BDA0002333607830000074
where k is an odd number of times:
Figure BDA0002333607830000075
path point q obtained based on B5iObtaining a control point d by inverse solving the non-uniform B-spline by the following formulai(i∈qcurve)
In part C1 of step three, the inverse solution formula of the cubic non-uniform B-spline is:
Figure BDA0002333607830000081
wherein, the element is the value of the basic function in the coefficient matrix, only related to the node vector u, and is simplified as follows:
Figure BDA0002333607830000082
wherein
Figure BDA0002333607830000083
Obtaining the path point in the path planning process, and obtaining the control point p of the planned path through the rarefying algorithm optimizationi(i∈qcurve) And (3) optimizing the trajectory on the basis of the original path planning, and fig. 4 is a schematic diagram of the planned trajectory obtained on the basis of the first step, the second step and the third step.
The second part
In order to further verify the practical effect of the scheme, the method is experimentally verified on the basis of Matlab simulation experiments, specifically:
two 500X 500 maps map-1 and map-2 are set. The former is used to verify the validity of the algorithm and the latter is biased towards verifying the reliability of the algorithm. Simulation environment hardware information: interi5-8265U, dominant frequency 1.8GHz and memory 8G. The initial point is (40, 40, 40), the target point is (460, 460, 460), and the heuristic probability p is 0.1.
The RRT can be explored in a complex environment, and the time consumed increases as the environment is complex. In order to verify the effectiveness of a heuristic RRT mechanical arm motion planning method (PBG-RRT) based on target deviation optimization, the search capability of RRT, heuristic probability P-RRT and BG-RRT at map-2 is compared. Assuming that the step δ is 20 simulations for a total of 1000 times, fig. 5 to 8 are schematic diagrams of search nodes for four path plans, and simulation data are shown in table 1.
Table 1 four algorithm experimental data table
Figure BDA0002333607830000091
Fig. 5 to 8 are schematic diagrams of search nodes for four types of path planning, which can be obtained from the table, and the PBG-RRT and the RRT or RRT-Connect of the present invention are significantly improved, thereby reducing search of invalid space and reducing search paths. Compared with the P-RRT search path efficiency, the PBG-RRT provided by the invention has the advantages that the path search efficiency is improved by 217%, the search distance is greatly reduced, and the average search time efficiency is improved by 168%.
As can be seen from fig. 5 to 8, compared to the PBG-RRT of the present invention, several other algorithms exist in a complex environment, and are close to the situation of nearby oscillations of obstacles, and search for a certain directivity in the whole map range is lacking, resulting in the search for an invalid space. The PBG-RRT of the present invention performs more efficiently.
Fast search is a feature of RRT, taking different δ under map-1 to perform 1000 simulations, with the results as in table 2. Similar trajectories at different step sizes are selected as in fig. 9-12. The experimental data are shown in table 2:
TABLE 2 PBG-RRT experimental data table of four step lengths
Figure BDA0002333607830000092
As can be seen from the table, when the step length delta is longer, the PBG-RRT has stronger searching capability, fewer nodes of the search tree and fewer corresponding points optimized by the thinning algorithm, and the final track distance also has a tendency of descending. The average time reflects that a suitable step size can improve the ability of the path search, and the time consumption trend of δ ═ 5 is steep. This is by the target bias factor
Figure BDA0002333607830000093
Thereby, the effect is achieved. When the target point is far away, the target point can be as close as possible in order to prevent the searched track from falling into the local optimal solution or oscillating around the obstacle, but also having the influence of heuristic probability. When approaching the target point, is received
Figure BDA0002333607830000094
The influence can be brought quickly close to the target point.
For the path length of the motion planning, the number of path points can be reduced by 75-90% by the thinning algorithm while the original curve characteristics are ensured. The distance after the track planning is reduced by 7-16% compared with the distance of the path planning, and the path is smooth.
A Robot-Anno manipulator is taken as a research object, a visualization tool Rvizformulation is used for executing in ROS, a complete motion planning solution is simulated, a manipulator simulation environment is constructed in Rviz, and then obstacles and postures are set. Then, the parameters of the starting pose and the target pose of the manipulator end effector are as follows: (unit: m)
Figure BDA0002333607830000101
TABLE 3 Angle per joint (Unit: radian)
Joint 0 Joint 1 Joint 2 Joint 3 Joint 4 Joint 5
Initial posture -0.67 -1.12 0.60 0.00 0.52 -0.67
Target pose 0.69 -1.12 0.57 0.00 0.55 0.69
The PBG-RRT algorithm of the invention is combined with the sparse third-order non-uniform B spline processing. A smooth path is obtained in the simulation, and the track of the manipulator is smooth in the obstacle avoidance process. Meanwhile, in the simulation process, each joint angle changes smoothly without sudden change or impact. Furthermore, twenty times of simulation of different algorithms in the same environment are used, as shown in table 4.
TABLE 4 mean time of different algorithms in the same environment
P-RRT TRRT RRT-Connect RRT-Star BIT-RRT PBG-RRT
Mean planning time/s 0.637 3.352 0.521 5.028 0.025 0.019
According to the heuristic RRT mechanical arm motion planning method based on target deviation optimization, the speed of trajectory planning is greatly improved in a blocked 3D space, the local minimum phenomenon is eliminated, the redundant path points can be minimized while the characteristics of an original curve are maintained, the oscillation phenomenon of nearby obstacles in the path planning process is avoided, the number of inflection points generated by non-uniform B-spline fitting can be reduced, and the smooth trajectory of the end effector is obtained.
The heuristic RRT mechanical arm motion planning method based on the target deviation optimization has high reliability, and can better meet more scenes by combining dynamic obstacle avoidance in further expanded application.
Finally, it should be noted that the above embodiments are only used for illustrating the technical solutions of the present invention, and not for limiting the protection scope of the present invention, although the present invention has been described in detail with reference to the preferred embodiments, it should be understood by those skilled in the art that modifications or equivalent substitutions can be made on the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention.

Claims (5)

1. A heuristic RRT mechanical arm motion planning method based on target deviation optimization is characterized by comprising the following steps:
step one, determining RRT path nodes based on target deviation optimization; the method comprises the following steps:
a1, determining a starting point qstartQ is prepared bystartStore to qnodesPerforming the following steps;
a2, sampling point q at random on full map in spacerandLooking for qnodesReach qrandOne of the nearest points is taken as qnear；
A3 at qnearTo qrandIn the direction of (C) advances by a step size delta, delta epsilon C, using a target bias factor
Figure FDA0002333607820000014
Controlling the random point generation direction and distance to advance to the node qnewCollision detection is performed during the forward process, and if a collision is detected, a2 is returned; if no collision is detected, q isnewStore to qnodesIs neutralized and turned toA4; wherein:
Figure FDA0002333607820000011
Figure FDA0002333607820000012
Figure FDA0002333607820000013
a4, according to the updated qnodesRepeating A2-A3 until the latest node q is obtainedgoalSatisfy | qnew-qgoal|<Error, consider to find the target point, let qnewIs stored in qnodesIn (1), Error refers to the allowable Error;
a5 at qnodesAnd reversely searching to find the planned path according to the parent-child relationship of each node.
2. The heuristic RRT robotic arm motion planning method based on target bias optimization of claim 1, further comprising
Step two, performing thinning processing on the initial path node based on the Douglas-Peucker algorithm; the method comprises the following steps:
b1, connecting a line segment between the head and tail points AG of the path according to the planned path obtained by A5;
b2, determining a point C with the maximum distance from other path points to the line segment AG, and acquiring the distance d from the point C to the line segment AG;
b3, comparing the given gamma with d, if d is small, deleting the point C, if d is large, keeping the point C as a key point, and dividing the original path into two segments of AC and CG by using the point C; wherein, gamma belongs to (0, delta);
b4, repeating B1-B3 for the segments formed after division; deleting or keeping as a key point until all points on the initial path have been traversed;
b5, derived keypoint qiI.e. the path point after the rarefaction.
3. The heuristic RRT robotic arm motion planning method based on target bias optimization of claim 2, further comprising
Step three, a track optimization step based on non-uniform B spline fitting comprises the following steps:
c1, optimizing the path after rarefaction obtained in the B5 by adopting cubic non-uniform B spline interpolation;
b-spline curve equation:
Figure FDA0002333607820000021
wherein d isi(i-1, 2, …, N) is a control point, Ni,kNormalizing the B-spline basis function for k times; and is
Figure FDA0002333607820000022
Wherein li＝|di-di-1|(i＝1,2,...,n)；
Figure FDA0002333607820000023
Wherein k is an even number of times:
Figure FDA0002333607820000024
where k is an odd number of times:
Figure FDA0002333607820000025
path point q obtained based on B5iObtaining a control point d by inverse solution of the non-uniform B-splinei(i∈qcurve) And then generating a smooth track through the control points, and realizing the track optimization on the basis of the original path planning.
4. According to claim 1The heuristic method for planning the motion of the RRT mechanical arm based on the target deviation optimization is characterized in that in the step A3, when no obstacle is affected and β is pi, the control is carried out
Figure FDA0002333607820000028
Avoid the local optimal solution caused by too strong directivity to the target point, wherein β is
Figure FDA0002333607820000026
And
Figure FDA0002333607820000027
the included angle between delta epsilon C and C is an arbitrary constant.
5. The heuristic RRT mechanical arm motion planning method based on the target deviation optimization of claim 3, wherein in the part C1 of the step three, the inverse solution formula of cubic non-uniform B-splines is as follows:
Figure FDA0002333607820000031
wherein, the element is the value of the basic function in the coefficient matrix, only related to the node vector u, and is simplified as follows:
Figure FDA0002333607820000032
wherein
Figure FDA0002333607820000033
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