CN111814269A - RES-SCA algorithm-based inverse kinematics real-time analytical solution optimization method - Google Patents

RES-SCA algorithm-based inverse kinematics real-time analytical solution optimization method Download PDF

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CN111814269A
CN111814269A CN202010553649.1A CN202010553649A CN111814269A CN 111814269 A CN111814269 A CN 111814269A CN 202010553649 A CN202010553649 A CN 202010553649A CN 111814269 A CN111814269 A CN 111814269A
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李春泉
罗族
江逸冰
何永华
熊维进
王子航
陶艺程
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Abstract

The invention provides an inverse kinematics real-time analytical solution optimization method based on an RES-SCA algorithm. Aiming at closed analytic solution real-time optimization of inverse kinematics, an improved sine and cosine algorithm is provided: rebalance-enhanced search SCA (RES-SCA) algorithm. The algorithm realizes balanced exploration and development by combining a Rebalance Strategy (RS) and an enhanced search strategy (ES), enhances the group search capability, improves the convergence rate and ensures the rapidness and the accuracy of closed analytic solution optimization.

Description

RES-SCA algorithm-based inverse kinematics real-time analytical solution optimization method
Technical Field
The invention relates to RES-SCA algorithm-based inverse kinematics real-time analytical solution optimization. The method is used for solving the inverse kinematics solution of the tail end position and the posture of the robot in real time in a man-machine teleoperation system, and the robot can be ensured to stably and accurately complete teleoperation instructions.
Background
In teleoperation scenarios, in order to make a robot flexibly grasp or manipulate a target object like a human, a kinematic redundancy of the robot arm is usually required. Thus, many robot manufacturers have manufactured robotic arms with seven degrees of freedom to mimic human arms. However, for each given end target, redundancy will result in an infinite solution. The collection of these infinite solutions is called a closed-form solution. The closed solution can be used for avoiding joint limitation, avoiding obstacles, singular configuration, optimizing mechanical arm dynamics and the like.
However, it is difficult to obtain an inverse kinematics solution for the redundant robotic arm. A common method of solving the inverse kinematics is a numerical iterative method using a jacobian matrix. However, this method requires the inverse of the jacobian matrix to be calculated, which is not a simple task, especially for redundant robotic arms, where the jacobian matrix will no longer be a square matrix. There are other iterative methods that do not rely on jacobian matrices, but still require multiple iterations to determine the inverse kinematics solution, and the given solution may not be optimal or even ideal. Also, numerical iteration can only obtain one solution. Therefore, how to obtain an analytic solution in a closed form becomes a hot point of research.
Unlike numerical iteration methods, analytical methods provide a closed solution by establishing redundant parameters according to the geometry of the mechanical arm links and joints. Lee et al propose a joint parameterization technique to derive closed solutions (see: Reductant armkinematical control based on parameter-part [ C ]// proceedings.1991EEInternational Conference on Robotics and analysis. IEEE 1991: 458-465.). Tolani and Badler derive a closed solution by specifying the elbow position and the fixed shoulder or wrist joint (see: Real-time inverting mechanics of the human arm [ J ]. Presence: Tele-operators & Virtual environment, 1996,5(4): 393. 401.). Dahm and Joublin consider that for any given pose (position and orientation), the elbow of the robot is free to move along a circle called a redundant circle (see: Closed form solution for the inverse kinematics of a redundant robot arm [ M ]. Ruhr-Univ., Inst. fur Neuroformation ik, 1997.). Moradi et al proposed a closed form solution based on closed form inverse kinematics based on Dahm and Joublin work and achieved minimization of the elbow motion of the robot (see: Joint limited analysis and elastic movement for reduced manipulators used method [ C ]// International Conference on Intelligent Comm-rendering. Springer, Berlin, Heidelberg,2005: 423-432.). For the seven-degree-of-freedom Inverse Kinematics solution problem with Joint offset, Sinha and Chakraborty compute Multiple Inverse Kinematics solutions using two-parameter Search by using redundant Manipulator geometry (see: geometry Search-Based Inverse Kinematics of 7-Dofridgendant Manipulator with Multiple joints Offsets [ C ]//2019International conference on Robotics and analysis (ICRA), IEEE,2019: 5592-.
However, how to find the optimal solution in the closed solution in real time is the key to realize the smooth and real-time slave-end control of the teleoperation system. The most common method is the traditional exhaustive search algorithm, the grid search method, by traversing the redundant parameters (here redundant circular redundant corners)
Figure BDA0002543512860000026
) To find the most suitable redundancy parameter. The calculation cost of the method is too large and rough, if the set parameter traversal interval is smaller, the iteration times are more and more time-consuming, and if the set traversal interval is too large, the potential optimal solution between the intervals can be ignored.
The Sine and Cosine Algorithm (SCA) is a novel meta-heuristic proposed by Mirjalli in 2016 (see: SCA: a sine cosine algorithm for solving optimization schemes [ J ]. Knowledge-Based Systems,2016,96: 120-). Inspired by mathematical techniques, the SCA randomly initializes each individual according to a mathematical model of sine and cosine functions and then updates, i.e., the individual may fluctuate outward or toward a global optimal solution to explore the search space. As a competitive meta-heuristic algorithm, SCA is widely used in many neighborhoods such as image processing, feature selection, power control systems, multi-objective optimization, neural networks, power market planning problems, and solar devices. However, in the initial SCA algorithm, all candidate solutions are learned by the global optimal candidate solution obtained in the entire population so far, which easily leads to premature convergence.
From the above, it can be seen that how to find the optimal solution from the closed-type analytic solutions is the key to achieve smooth and real-time slave-end control of the teleoperation system. In the traditional grid search method, a search grid is formed by setting intervals to find an optimal solution. However, the presence of a grid can miss most of the potential solutions in the search space resulting in insufficient accuracy, and the exhaustive approach is time consuming. The SCA algorithm is an efficient random optimization meta heuristic algorithm, however, the original SCA algorithm cannot well balance exploration and development capacity, so that the problems of low convergence rate, easy falling into local optimization and the like exist.
Disclosure of Invention
The invention discloses an inverse kinematics real-time analytical solution optimization method based on an RES-SCA algorithm. Aiming at closed analytic solution real-time optimization of inverse kinematics, an improved sine and cosine algorithm is provided: rebalance-enhanced search SCA (RES-SCA) algorithm. The algorithm realizes balanced exploration and development by combining a Rebalance Strategy (RS) and an enhanced search strategy (ES), enhances the group search capability, improves the convergence rate and ensures the rapidness and the accuracy of closed analytic solution optimization.
The invention is realized by the following technical scheme.
Step one, establishing redundant circle redundant angle
Figure BDA0002543512860000021
Corresponding to the end position P of the next mechanical armTool_nextPosture RTool_nextInverse kinematics closed analytic solution of thetanextThe relationship is as follows:
Figure BDA0002543512860000022
wherein, thetanextIs a K-dimension joint angle vector, K is the degree of freedom of the mechanical arm,
Figure BDA0002543512860000023
performing function expression of a solution for closed analytic solution of the inverse kinematics of the mechanical arm;
and step two, establishing an analytical solution optimization objective equation. Obtaining all joint angles theta of the current mechanical armcur. With the aim of finding the nearest neighbor joint angle, namely the change minimization between the closed solution set and the previous mechanical arm joint angle, establishing the following objective equation:
Figure BDA0002543512860000024
step three, initializing RES-SCA algorithm parameters: maximum iteration number T, population size N, candidate solution dimension D, sine and cosine wave factor (lambda)1、λ2、λ3And λ4) Balance factor
Figure BDA0002543512860000025
Solution boundary candidates (ub, lb) and constants ρ, b,
And step four, randomly initializing N candidate solutions according to the candidate solution boundary. For the ith candidate solution xiRepresents the ith possible redundant circular redundancy angle
Figure BDA0002543512860000031
Where i ∈ {1,2, …, N };
step five, solving N candidate solution Fitness values Fitness (x) according to the objective function formula (2) and the combination formula (1)i) And recording the historical optimal value of each candidate solution when the current iteration number t is
Figure BDA0002543512860000032
And the value of the global historical optimal candidate solution gbestt；
Step six, updating each candidate solution by using a Rebalance Strategy (RS) and an enhanced search strategy (ES), wherein the specific expression is as follows:
(a) using the RS strategy, for each iteration, if the current iteration number t is less than or equal to
Figure BDA0002543512860000033
The RES-SCA updates the solution candidates according to the following formula:
Figure BDA0002543512860000034
otherwise, the RES-SCA updates the candidate solution according to the following formula:
Figure BDA0002543512860000035
wherein
Figure BDA0002543512860000036
Is a candidate solution updated by using the RS strategy;
(b) and further uses the ES to generate the final update:
Figure BDA0002543512860000037
where η is T/T, N (0,1) and C (0,1) are gaussian and cauchy random numbers;
and step seven, using the value of the boundary constraint candidate solution. When the individual exceeds the boundary after updating, the individual is restrained according to the following formula:
xi＝(xi<lb)×lb+(xi>ub)×ub (6)
step eight, termination condition: when the iteration process meets the termination condition, namely the iteration times of the algorithm meet the condition that T is more than or equal to T, stopping the iteration and outputting the final optimal solution
Figure BDA0002543512860000038
And its corresponding analytical solution thetanext. Otherwise, repeating the fifth step to the seventh step.
The invention has the advantages that: a closed analytic solution real-time optimization method for inverse kinematics is disclosed. The RES-SCA algorithm well balances the exploration and development capabilities of the algorithm and has the characteristics of high convergence rate, high precision and the like. The RES-SCA algorithm-based inverse kinematics analytic solution optimization method ensures convergence accuracy while providing high real-time performance, and realizes the accuracy and real-time of the inverse kinematics solution of the mechanical arm.
Drawings
FIG. 1 is a RES-SCA algorithm based inverse kinematics real-time analytical solution optimization principle;
FIG. 2 is a simplified geometry of a Barrett WAM seven degree-of-freedom robotic arm;
FIG. 3 is a diagram of analytical solution optimization results based on RES-SCA algorithm;
FIG. 4 is a diagram of the self-movement of the mechanical arm optimized by an analytic solution based on RES-SCA algorithm.
Detailed Description
The invention is further explained by taking a Barrett WAM seven-degree-of-freedom mechanical arm as an example platform.
Step one, establishing redundant circle redundant angle
Figure BDA0002543512860000041
Corresponding to the end position P of the next mechanical armTool_nextPosture RTool_nextInverse kinematics closed analytic solution of thetanextThe relationship is as follows:
Figure BDA0002543512860000042
wherein, thetanextIs a seven-dimensional joint angle vector and is,
Figure BDA0002543512860000043
for the functional expression of the closed analytical solution of the inverse kinematics of the mechanical arm, taking a Barrett WAM mechanical arm as an example,
Figure BDA0002543512860000044
the specific contents are as follows:
(a) simplifying the geometric description of Barrett WAM: barrett WAM manipulatorThe arm is a seven degree of freedom redundant robotic arm with link offset. Its simplified geometry consists of a shoulder joint S, an elbow joint combination with link offset E1, E, E2, and a wrist joint W. Two link offset structure E1E and EE2Is arranged to divide the elbow joint of the WAM into E1、E、E2The three structures are combined. In the case of a fixed wrist joint, the freely intersecting E-point distribution set of the upper arm and forearm with link offset still constitutes a redundant circle. The detailed structure is shown in figure 2.
(b) Solving a redundant circle: the WAM upper and forearm have varied rod lengths in geometry due to link misalignment:
Figure BDA0002543512860000045
Figure BDA0002543512860000046
angle of WAM upper arm and forearm to horizontal l:
Figure BDA0002543512860000047
Figure BDA0002543512860000048
WAM arm redundant circle dcAnd Rc：
dc＝NL1×cos(α1) (12)
Rc＝NL1×sin(α1) (13)
Thus, there are redundant circles:
Figure BDA0002543512860000049
wherein
Figure BDA00025435128600000410
For redundancy of redundant circlesAnd the value range of the angle is 0-360 degrees. On a redundant circle, each one
Figure BDA00025435128600000411
Corresponds to a set of inverse kinematics solutions. Normalizing the rotation matrix R according to the wristWZAnd rotating the redundant circle, wherein the final redundant circle is as follows:
CR＝Circle×RWZ(15)
(c) calculating the joint angle: inverse solution of the kinematics of the arm at a given position vector PTool＝[x,y,z]And a rotation matrix RTool＝[TRx,TRy,TRz]When describing the expected hand (or tool) pose of the mechanical arm, the required joint angle theta is given1,θ2,θ3,θ4,θ5,θ6,θ7]. The first four joint angles, i.e. theta1、θ2、θ3And theta4Given the wrist position, the last three θ5、θ6And theta7The joint angle positions the hand (or tool) pose. On the premise that the redundant circle is known, the first four joint angles [ theta ] of the wrist position are required to be located1,θ2,θ3,θ4]The elbow E after the offset of the connecting rod needs to be calculated first1、E2One of the nodes. With E1Node position PE1For example, the following steps are carried out:
ECR＝CR/||CR||2(16)
ENCW＝CR×PW/||CR×PW||2(17)
Figure BDA0002543512860000051
since the first two joints specify the azimuth angle (θ) of the upper arm from the shoulder joint (base)1) And elevation angle (theta)2) So that the upper arm P can be positionedE1Theta can be determined1And theta2：
Figure BDA0002543512860000052
Figure BDA0002543512860000053
The third joint is the torsional degree of freedom of the upper arm
NE1Z＝(PE1×EZ)×PE1(21)
Figure BDA0002543512860000054
The fourth joint corresponds to the elevation of the forearm (θ) from the elbow4) And, together with the first three joint angles, these four joint angles can ultimately determine the position of the wrist. All possible poses of the WAM inverse kinematics solution problem are described by redundant circles.
However, due to the addition of the link offset, there are two possible solutions for each point on the redundant circle, labeled as the out-of-elbow and in-elbow gestures. Here, only the extra-elbow posture is considered, and there are:
Figure BDA0002543512860000055
the fifth and sixth joints provide an azimuth angle (θ) of the hand tool5) And elevation angle (theta)6). Up to now, the first four joint angles [ theta ]1,θ2,θ3,θ4]Are known. These can be used to obtain a rotation matrix
Figure BDA0002543512860000056
As follows:
Figure BDA0002543512860000057
since the first four joint angles at which the wrist is positioned are known, with wrist PWLocal hand tip coordinates were constructed for the reference:
Figure BDA0002543512860000058
therefore, there are:
Figure BDA0002543512860000059
Figure BDA00025435128600000510
seventh joint angle θ7Providing hand rotation. At the known first six joint angles, i.e. [ theta ]1,θ2,θ3,θ4,θ5,θ6]In the case of (2), the wrist rotation matrix can be directly obtained with reference to the equation (24)
Figure BDA00025435128600000511
Further, the hand rotation joint angle can be obtained:
θ7＝arccos(WRx·TRy) (28)
and step two, establishing an analytical solution optimization objective equation. Obtaining all joint angles theta of the current mechanical armcur. With the aim of finding the nearest neighbor joint angle, namely the change minimization between the closed solution set and the previous mechanical arm joint angle, establishing the following objective equation:
Figure BDA0002543512860000061
step three, initializing RES-SCA algorithm parameters: maximum iteration number T is 20, population size N is 10, candidate solution dimension D is 1, sine and cosine wave factor (λ)1＝1-0、λ2＝2π*rand、λ32 × rand and λ 41 × rand, wherein rand is [0,1]]Random number in between), the constant ρ becomes 1. Wherein λ is1Expressed as:
Figure BDA0002543512860000062
balance factor
Figure BDA0002543512860000063
The constant is 0.1. For candidate solution boundaries (ub, lb), due to the optimal solution
Figure BDA0002543512860000064
The corresponding inverse kinematic solution is the nearest solution to the previous inverse solution, and therefore the optimal solution
Figure BDA0002543512860000065
Should surround the last optimum parameter
Figure BDA0002543512860000066
Front and back 20 degrees (obtained by experiment, the adjacent solution exists in the last optimal solution
Figure BDA0002543512860000067
Front-to-back 20 °) range expansion search, there is an initialization boundary:
Figure BDA0002543512860000068
and step four, randomly initializing N candidate solutions according to the candidate solution boundary. For the ith candidate solution xi：
xi＝lb+rand*(ub-lb) (32)
Wherein i ∈ {1,2, …, N }, and rand is a random number between [0,1 ].
Step five, solving N candidate solution Fitness values Fitness (x) according to the objective function formula (29) and the combination formula (7)i) And recording the historical optimal value of each candidate solution when the current iteration number t is
Figure BDA0002543512860000069
And the value of the global historical optimal candidate solution gbestt。
Step six, updating each candidate solution by using a Rebalance Strategy (RS) and an enhanced search strategy (ES), wherein the specific expression is as follows:
(aa) using the RS strategy, for each iteration, if the current iteration isThe number t is less than or equal to
Figure BDA00025435128600000610
The RES-SCA updates the solution candidates according to the following formula:
Figure BDA00025435128600000611
otherwise, the RES-SCA updates the candidate solution according to the following formula:
Figure BDA00025435128600000612
wherein
Figure BDA00025435128600000613
Is a candidate solution updated by using the RS strategy;
(bb) and further using the ES to generate a final update:
Figure BDA00025435128600000614
where η is T/T, N (0,1) and C (0,1) are gaussian and cauchy random numbers.
And step seven, using the value of the boundary constraint candidate solution. When the individual exceeds the boundary after updating, the individual is restrained according to the following formula:
xi＝(xi<lb)×lb+(xi>ub)×ub (36)
step eight, termination condition: when the iteration process meets the termination condition, namely the iteration times of the algorithm meet the condition that T is more than or equal to T, stopping the iteration and outputting the final optimal solution
Figure BDA0002543512860000071
And its corresponding analytical solution thetanext. Otherwise, repeating the fifth step to the seventh step.
In this example, with the WAM shoulder as the origin of the spatial coordinates, and with the tip fixed in the Z-axis direction, the WAM robot arm is expected to draw a circle with a center of (0m, -0.7m,0.1m) and a radius of 0.25m on a spatial wall perpendicular to the horizontal plane and-0.7 m from the Y-axis direction of the robot arm.
In the example, the results obtained by the RES-SCA algorithm-based inverse kinematics real-time analytic solution optimization method are shown in FIGS. 3 and 4, and the Cartesian space circle drawing task is smoothly and stably completed. To further illustrate the advantages of the present invention, the method of the present invention is compared with the conventional grid search method (GS), the SCA algorithm and its variant CGSCA algorithm. The results in table 1 show that the real-time analytic solution optimization method based on the inverse kinematics of the RES-SCA algorithm provides the fastest solution speed.
TABLE 1 average solution and optimization of the time consuming comparison of pose of each robot arm
Figure BDA0002543512860000072
The foregoing merely represents preferred embodiments of the invention, which are described in some detail and detail, and therefore should not be construed as limiting the scope of the invention. It should be noted that, for those skilled in the art, various changes, modifications and substitutions can be made without departing from the spirit of the present invention, and these are all within the scope of the present invention. Therefore, the protection scope of the present patent shall be subject to the appended claims.

Claims (1)

1. An inverse kinematics real-time analytic solution optimization method based on RES-SCA algorithm is characterized by comprising the following steps: the method comprises the following steps:
step one, establishing redundant circle redundant angle
Figure FDA0002543512850000011
Corresponding to the end position P of the next mechanical armTool_nextPosture RTool_nextInverse kinematics closed analytic solution of thetanextThe relationship is as follows:
Figure FDA0002543512850000012
wherein, thetanextIs a K-dimension joint angle vector, K is the degree of freedom of the mechanical arm,
Figure FDA0002543512850000013
performing function expression of a solution for closed analytic solution of the inverse kinematics of the mechanical arm;
step two, establishing an analytic solution optimization objective equation, and obtaining all joint angles theta of the current mechanical armcurWith the aim of finding the nearest neighbor joint angle, namely the minimum change between the closed solution set and the previous mechanical arm joint angle, the following objective equation is established:
Figure FDA0002543512850000014
step three, initializing RES-SCA algorithm parameters: maximum iteration number T, population size N, candidate solution dimension D, sine and cosine wave factor (lambda)1、λ2、λ3And λ4) Balance factor
Figure FDA0002543512850000015
Candidate solution boundaries (ub, lb) and constants ρ,;
step four, randomly initializing N candidate solutions according to the candidate solution boundary, and regarding the ith candidate solution xiRepresents the ith possible redundant circular redundancy angle
Figure FDA0002543512850000016
Where i ∈ {1,2, …, N };
step five, solving N candidate solution Fitness values Fitness (x) according to the objective function formula (2) and the combination formula (1)i) And recording the historical optimal value of each candidate solution when the current iteration number t is
Figure FDA0002543512850000017
And the value of the global historical optimal candidate solution gbestt；
Step six, updating each candidate solution by using a Rebalance Strategy (RS) and an enhanced search strategy (ES), wherein the specific expression is as follows:
(a) using the RS strategy, for each iteration, if the current iteration number t is less than or equal to
Figure FDA00025435128500000111
The RES-SCA updates the solution candidates according to the following formula:
Figure FDA0002543512850000018
otherwise, the RES-SCA updates the candidate solution according to the following formula:
Figure FDA0002543512850000019
wherein
Figure FDA00025435128500000110
Is a candidate solution updated by using the RS strategy;
(b) and further uses the ES to generate the final update:
Figure FDA0002543512850000021
where η is T/T, N (0,1) and C (0,1) are gaussian and cauchy random numbers;
and seventhly, using the value of the boundary constraint candidate solution, and constraining the individual according to the following formula when the individual exceeds the boundary after updating:
xi＝(xi<lb)×lb+(xi>ub)×ub (6)
step eight, termination condition: when the iteration process meets the termination condition, namely the iteration times of the algorithm meet the condition T is more than or equal to T, stopping the iteration and outputting the final optimal solution phi and the corresponding analytic solution thetanextOtherwise, repeating the fifth step to the seventh step.
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