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Application of Improved BQGA in Robot Kinematics Inverse Solution

Application of Improved BQGA in Robot Kinematics Inverse Solution Hindawi Journal of Robotics Volume 2019, Article ID 1659180, 7 pages https://doi.org/10.1155/2019/1659180 Research Article Application of Improved BQGA in Robot Kinematics Inverse Solution 1 2 Xiaoqing Lv and Ming Zhao School of Electronics and Information Engineering, University of Science and Technology Liaoning, Anshan , China School of Applied Technology, University of Science and Technology Liaoning, Anshan , China Correspondence should be addressed to Ming Zhao; as zmyli@163.com Received 26 October 2018; Accepted 9 January 2019; Published 3 February 2019 Academic Editor: Raffaele Di Gregorio Copyright © 2019 Xiaoqing Lv and Ming Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In view of the problem that Bloch Quantum Genetic Algorithm (BQGA) is easy to fall into local optimum, an improved BQGA is proposed. eTh algorithm can control the step size and the mutation probability in real time in the iterative process, avoiding over the optimal solution and guaranteeing search efficiency. In addition, the improved algorithm further completes the anti-degradation mechanism, which maintains the diversity of the population while preserving the dominant gene to the maximum extent, so that the algorithm is not easy to fall into the local extremum and finally approaches the global optimal solution. eTh application in the inverse solution of robot kinematics shows that the improved BQGA eeff ctively avoids the premature problem and accelerates the convergence of understanding and the search result is close to the complete solution, which provides a new idea for solving complex nonlinear and multivariate functional equations. 1. Introduction structure, space, expressions, etc., so it is not convenient and efficient to solve inverse kinematics problem. With the rapid The kinematics problem is the basis for the intelligent devel- development of computer programming technology, various opment of today’s robotic arms. Research in related fields intelligent algorithms, such as neural network algorithm [9], has made tremendous contributions to the development of genetic algorithm [10, 11], particle swarm optimization [12], science and technology. In document [1] the solutions of quantum optimization algorithms, and their various hybrid both the forward and inverse kinematics problems for Adept algorithms, have emerged, which highlight the ability to solve Six 300 6R robot are presented. A matrix method was used complex problems. For instance, in order to simplify the in computations. Algorithm based on the derived formulas controller design of the manipulator, document [13] proposes and logical conditions has been implemented in Adept Smart to train the NN algorithm with entropy clustering method, Controller Cx. Literatures [2] are based on kinematics for which effectively solves the multi-value problem; applying more in-depth path planning studies. This paper innovatively an improved genetic algorithm to the inverse kinematics can applies BQGA to the inverse kinematics program, which not overcome the structural limitations of the manipulator [14]; only expands the application eld fi of intelligent algorithms, in the inverse solution of the EOD manipulator, the simulated but also provides a new perspective for solving kinematic annealing algorithm was used, and the expected effect was problems. obtained [15]. With the development of quantum theory, In robotics [3], the positive solution problem is relatively people are increasingly interested in quantum optimization simple and will get a unique solution, while the inverse algorithms. Quantum Genetic Algorithm (QGA) is popular among scholars because of its convenience and ecffi iency. solution problem [4] is relatively complex, and there may be multiple sets of solutions. The traditional methods of Therefore, many improved algorithms such as real-coded solving them are geometry and algebra methods [5, 6], QGA [16], adaptive double-chain QGA [17], and Bloch numerical analysis method [7, 8], etc., but they are limited by Quantum Genetic Algorithm (BQGA) [18] have emerged. 2 Journal of Robotics 𝑡 𝑡 ∇𝑓 (𝑋 )−∇𝑓 min Among them, BQGA is a probability optimization algorithm 󵄨 󵄨 󵄨 ̂ 󵄨 󵄨 󵄨 Δ𝜃 =𝜃 × exp (− ) (6) 󵄨 󵄨 𝑡 𝑡 󵄨 󵄨 that uses the qubits to represent population elements [19]. ∇𝑓 max −∇𝑓 min Its three-chain coding method is more efficient than the 𝑡 𝑡 traditional double-chain structure, which makes the GA where ∇𝑓 and ∇𝑓 represent the maximum and mini- max min algorithm take a new step. However, in practical applications, mum tness fi gradient values of the generation, respectively. It BQGA also hasproblemssuch asbeing easy to passthe should be pointed out that the discrete optimization problem optimal solution and population degradation, and sometimes [20] and the continuous optimization problem [21] are cal- the optimization result is not ideal. Therefore, this paper culated differently when acquiring the relevant gradient. For proposes an improvement plan for these problems in BQGA. the discrete optimization problem, since 𝑓(𝑋) does not have a gradient, we can replace the gradient with the difference of the tness fi of two adjacent generations [22]; the expression is 2. Improvement Scheme of BQGA 󵄨 󵄨 𝑡 𝑡 𝑡−1 󵄨 󵄨 󵄨 󵄨 ∇𝑓 (𝑋 )= 𝑓(𝑋 )−𝑓(𝑋 ) (7) 󵄨 󵄨 .. Adjustment of Adaptive Corner Value. In the process of 𝑖 b 󵄨 󵄨 updating BQGA’s quantum chromosome, the value of the 󵄨 󵄨 𝑡 𝑡 𝑡−1 󵄨 󵄨 󵄨 󵄨 ∇𝑓 max = max {𝑓(𝑋 )−𝑓(𝑋 ) ,⋅⋅⋅ , corner is more complicated. If the planning is unreasonable, 󵄨 󵄨 1𝑏 1𝑏 󵄨 󵄨 (8) it will seriously affect the optimization effect of the algorithm. 󵄨 󵄨 𝑡 𝑡−1 󵄨 󵄨 󵄨 󵄨 𝑓(𝑋 )−𝑓(𝑋 ) } The look-up table method in literature [18] provides an 󵄨 󵄨 𝑚𝑏 𝑚𝑏 󵄨 󵄨 effective convergence strategy. However, the look-up table 󵄨 󵄨 𝑡 𝑡 𝑡−1 󵄨 󵄨 󵄨 󵄨 ∇𝑓 min = min{𝑓(𝑋 )−𝑓(𝑋 ) ,⋅⋅⋅ , 󵄨 󵄨 method does not take into account the differences in individ- 1𝑏 1𝑏 󵄨 󵄨 (9) ual chromosomes and the trend of the evaluation function, 󵄨 󵄨 𝑡 𝑡−1 󵄨 󵄨 󵄨 󵄨 𝑓(𝑋 )−𝑓(𝑋 ) } 󵄨 󵄨 making the algorithm easy to cross the optimal solution and 𝑚𝑏 𝑚𝑏 󵄨 󵄨 less adaptive. Based on the traditional table lookup method, The gradient in the continuous optimization problem is this paper further proposes an adaptive adjustment strategy for corner values. The expression is as follows: 󵄨 𝑡 󵄨 󵄨 󵄨 (𝑋 ) 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∇𝑓 (𝑋 ) = (10) 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑡 󵄨 󵄨 󵄨 󵄨 󵄨 ̂ 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 Δ𝜙 󵄨 ∇𝑓 (𝑋 )󵄨 ≥𝑎 { 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 Δ𝜑 = (1) 󵄨 󵄨 { 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑡 𝑡 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ̃ 󵄨 󵄨 󵄨 󵄨 (𝑋 ) (𝑋 ) Δ𝜙 ∇𝑓 (𝑋 ) <𝑎 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑡 󵄨 󵄨 1𝑏 𝑚𝑏 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 { 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∇𝑓 max = max { ,⋅⋅⋅ , } (11) 󵄨 󵄨 󵄨 󵄨 𝑡 𝑡 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 1𝑏 𝑚𝑏 󵄨 󵄨 󵄨 𝑡 󵄨 󵄨 󵄨 ̂ 󵄨 󵄨 󵄨 󵄨 Δ𝜃 󵄨 ∇𝑓 (𝑋 )󵄨 ≥𝑎 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑡 󵄨 󵄨 𝑡 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 Δ𝜃 = (2) (𝑋 ) (𝑋 ) 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 { 󵄨 󵄨 𝑡 1𝑏 𝑚𝑏 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ̃ 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∇𝑓 min = min{ ,⋅⋅⋅ , } (12) Δ𝜃 ∇𝑓 (𝑋 ) <𝑎 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑡 𝑡 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 { 󵄨 󵄨 󵄨 󵄨 󵄨 1𝑏 󵄨 󵄨 𝑚𝑏 󵄨 where 𝑡 represents evolutionary algebra (same number of In the above adaptive strategy, the gradient size (measured iterations), |Δ𝜑 | and |Δ𝜃 | are the corner sizes of 𝜑 and 𝜃 by 𝑎) determines the calculation formula of the corner. In the formula related to the gradient, the larger the gradient of the 𝑗 th qubit on the 𝑖 rd chromosome, 𝑋 represents the optimal solution among the three chains of each chromosome is, the smaller the step size becomes and, in the formula related to the number of iterations, the step size decreases in the𝑡 nd generation,|∇(𝑓 𝑋 )| represents the tness fi gradient monotonically with the increase of the number of iterations. value of the optimal solution for each chromosome in the 𝑡 nd generation, 𝑎 is an evaluation value whose size is u Th s, the size of the corner varies with the gradient and is again limited by the number of iterations. The purpose is to determined by the maximum and minimum gradient values ̃ control the step size in a reasonable range in real time. in each generation, and |Δ𝜑̃ | and |Δ𝜃 | are the corner values associated with the number of iterations, and the expression is .. Adaptive Mutation Probability. During the quantum chromosomal mutation of BQGA, the qubits rotate greatly 𝑡 −𝑡 󵄨 󵄨 max 󵄨 󵄨 along the Bloch sphereand arenot constrained by other 󵄨 Δ𝜑 󵄨 =𝜑 × (3) 𝑖 0 󵄨 󵄨 max chromosomes, which helps to increase population diversity and avoid premature convergence. However, if the probability 󵄨 󵄨 𝑡 −𝑡 max 󵄨 󵄨 󵄨 󵄨 Δ𝜃 =𝜃 × (4) of mutation is constant throughout the evolution process, it 󵄨 𝑖 󵄨 0 󵄨 󵄨 max is not conducive to the late convergence of the algorithm. In order to enable the algorithm to expand the global search where 𝜑 and 𝜃 are initial iteration values, which range 0 0 ability in the early iteration and focus on local search in from 0.005𝜋 to 0.05𝜋 ,and 𝑡 is the maximum number of max the late iteration to speed up the convergence, the adaptive iterations. |Δ𝜑̂ | and |Δ𝜃 | are the angle values associated mutation probability [23] is adjusted to with the gradient. Referring to literature [18], the expressions constructed are as follows: 𝑡 −𝑡 max 𝑝 =𝑝 × (13) 𝑚 𝑚0 𝑡 𝑡 max 󵄨 󵄨 ∇𝑓 (𝑋 )− ∇𝑓 min 󵄨 󵄨 󵄨 󵄨 Δ𝜑̂ =𝜑 × exp (− ) (5) 󵄨 󵄨 0 󵄨 󵄨 𝑡 𝑡 where 𝑝 ∈ (0.01,0.5). ∇𝑓 max −∇𝑓 min 𝑚0 𝑖𝑗 𝑖𝑏 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑏 𝑖𝑏 𝑖𝑗 𝑖𝑗 𝜕𝑋 𝜕𝑋 𝑖𝑏 𝑖𝑗 𝑖𝑗 𝜕𝑓 𝜕𝑓 𝑖𝑏 𝑖𝑗 𝜕𝑋 𝜕𝑋 𝑖𝑏 𝑖𝑗 𝜕𝑓 𝜕𝑓 𝑖𝑗 𝑖𝑏 𝑖𝑗 𝑖𝑏 𝜕𝑋 𝑖𝑏 𝑖𝑏 𝜕𝑓 𝑖𝑏 𝑖𝑏 𝑖𝑗 𝑖𝑏 Journal of Robotics 3 .. Anti-Degradation Mechanism. In the process of popula- set of inverse solutions, we have established (16) [4, 29] which tion evolution, the anti-degradation mechanism can increase transforms the inverse solution problem into a multivariate the chances of dominant individuals participating in inheri- function to find the extremum problem. tance and variation and reduce the possibility of population 󵄩 󵄩 󵄩 󵄩 𝐹= A − T (16) 󵄩 󵄩 degradation, thus ensuring that the algorithm converges with 󵄩 󵄩 maximum probability within a limited number of iterations. where A is the actual position and posture matrix of the However, the principle of replacement of individual elimina- robot arm established by the D-H parameter method, andT is tion will cause the population diversity to decrease rapidly the set target position and posture matrix, ‖Α −T‖ is the two with the increase of the number of iterations, so that it is norm of A and T [30] which represents the distance between easy to fall into local optimum, and it is difficult to jump the two matrices. out only by relying on mutation. In the improved BQGA, Available from (16), when the actual position and posture the replacement principle of the chain change is adopted. of the robot arm are equal to or infinitely close to the target The description is that the chain is changed first, and the position and posture, the minimum value of the evaluation entire chromosome is replaced when the chain number is not function 𝐹 is zero or approximately zero. the same. That is, it is the rfi st to find the optimal solution (corresponding to the optimal chain) and the worst solution .. Coding Method. According to the basis of quantum com- (corresponding to the worst chain) in one generation. If the puting and the three-strand gene coding scheme of quantum chain numbers are the same, the worst chain is replaced by the chromosomes [18, 31], the initial population expression is as optimal chain, and the other two chains remain unchanged; follows: if thetwochainnumbers aredieff rent, replacetheentire chromosome. The replacement expression is 𝑡 BAX ←󳨀 BEX = 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 cos𝜑 sin 𝜃 cos𝜑 sin 𝜃 ⋅⋅⋅ cos 𝜑 sin 𝜃 𝑖1 𝑖1 󵄨 𝑖2 𝑖2 󵄨 󵄨 󵄨 󵄨 󵄨 (14) (17) 󵄨 󵄨 󵄨 [ ] 󵄨 󵄨 󵄨 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 󵄨 󵄨 󵄨 [ ] 󵄨 󵄨 󵄨 BAC ←󳨀 BEC =𝑔̸ 𝑠𝑏𝑒𝑠𝑡 = sin 𝜑 sin 𝜃 sin𝜑 sin 𝜃 ⋅⋅⋅ sin 𝜑 sin 𝜃 󵄨 󵄨 󵄨 [ 𝑖1 𝑖1 𝑖2 𝑖2 ] 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑡 󵄨 𝑡 󵄨 󵄨 𝑡 󵄨 󵄨 󵄨 ⋅⋅⋅ cos 𝜃 cos 𝜃 cos 𝜃 󵄨 󵄨 󵄨 𝑖1 𝑖2 where BEX is the most suitable solution called the con- [ ] temporary optimal solution, and its chromosome is called where 𝜑 ∈ (𝜋/2, 3𝜋/2), 𝜃 ∈ (0,)𝜋 , 𝑖 = 1,2,⋅⋅⋅ ,𝑚 , 𝑗= the contemporary optimal chromosome [24, 25] recorded 1,2,⋅⋅⋅ ,𝑛 ; 𝑚 is the population size, which is the number of as BEC. The same reason BAX is the worst solution and chromosomes; 𝑛 is the number of qubits (representing 𝑛 to- BAC is the worst chromosome, gbads and gbests repre- be-turned corners); 𝑡 is an evolutionary algebra. sent the corresponding chain number. The anti-degradation mechanism of the improved BQGA ensures that the worst .. Transformation of Unit Space Solution to Joint Space Solu- chromosomes of each generation have an optimal gene chain tion. According to the theory of Bloch spherical coordinates, while maintaining population diversity. a population can represent the m-group solutions of the search space and map them into the actual solution space 3. Applying Improved BQGA for Ω=[−2,𝜋 2]𝜋 to obtain m sets of feasible solutions (𝑋 , Inverse Kinematics 𝑋 ,and 𝑋 ). The mapping formula [18] isasfollows: .. Establishment of Evaluation Function. Using the 𝑗 𝑗 𝑗 𝑋 = [𝑉 (1 + 𝑝 )+𝑉 (1 − 𝑝 )] (18) max min improved BQGA to solve the inverse kinematics problem of the robot, an evaluation function needs to be established. 𝑗 𝑗 𝑗 According to the kinematics principle of the n-degree-of- 𝑋 = [𝑉 (1 + 𝑝 )+ 𝑉 (1 − 𝑝 )] (19) max min freedom manipulator [26], the kinematics positive solution equation established by the D-H parameter method [27, 28] 𝑗 𝑗 𝑗 (20) 𝑋 = [𝑉 (1 + 𝑝 )+𝑉 (1 − 𝑝 )] is max min 𝑛 0 1 2 3 A = A (𝜔 )⋅ A (𝜔 )⋅ A (𝜔 )⋅ A (𝜔 )⋅ ... 1 2 3 4 1 2 3 4 where 𝑗 = 1,2,⋅⋅⋅ ,𝑛 ; 𝑉 and 𝑉 correspond to the max min maximum and minimum values in the range of values for 𝑛 𝑜 𝑎 𝑝 𝑗 𝑗 𝑥 𝑥 𝑥 𝑥 each angle (check Table 1 to get its value); 𝑝 , 𝑝 ,and [ ] (15) [ 𝑛 𝑜 𝑎 𝑝 ] 𝑗 𝑦 𝑦 𝑦 𝑦 n−1 𝑝 represent the 𝑋 solution, the 𝑌 solution, and the 𝑍 [ ] 𝑖 z ⋅ A (𝜔 )= 𝑛 [ ] 𝑛 𝑜 𝑎 𝑝 solution of the 𝑗 th gene position of the 𝑖 th chromosome, [ ] 𝑧 𝑧 𝑧 𝑧 respectively. According to the decoding process, each of the 00 0 1 [ ] chains of each chromosome decodes n corresponding robot arm corner values and their values directly fall within the Thatis, given a setof angles from 𝜔 to 𝜔 ,matrix A 1 𝑛 respective range of values. of position and posture of the end eeff ctor is uniquely determined. However, in practice it is the opposite; that is, the end position and posture are known to solve the value of the .. Update and Mutation of Quantum Chromosomes. The angle of each joint (inverse solution). In order to evaluate each quantum chromosome is updated by the quantum revolving 𝑖𝑦 𝑖𝑥 𝑖𝑧 𝑖𝑧 𝑖𝑧 𝑖𝑦 𝑖𝑦 𝑖𝑦 𝑖𝑥 𝑖𝑥 𝑖𝑥 𝑖𝑧 𝑖𝑦 𝑖𝑥 𝑖𝑗 𝑖𝑗 𝑖𝑛 𝑖𝑛 𝑖𝑛 𝑔𝑏𝑎𝑑𝑠 𝑖𝑛 𝑖𝑛 𝑡𝑠𝑔𝑏𝑒𝑠 𝑔𝑏𝑎𝑑𝑠 4 Journal of Robotics Table 1: Connecting rod parameters of PUMA560. 𝑖𝛼 (rad) 𝑎 (m) 𝑑 (m) 𝜔 (rad) Joint angle range(rad) i-1 i-1 i i 10 0 0 𝜔 -8𝜋 /9 ∼8𝜋 /9 2-𝜋 0 d 𝜔 -5𝜋 /4 ∼𝜋 /4 2 2 30 a 0 𝜔 -𝜋 /4 ∼5𝜋 /4 2 3 4-𝜋 a d 𝜔 -11𝜋 /18 ∼17𝜋 /18 3 4 4 5 𝜋 00 𝜔 -5𝜋 /9 ∼5𝜋 /9 6-𝜋 00 𝜔 -133𝜋 /90 ∼133𝜋 /90 where𝑎 =0.4318,𝑎 =0.02032,𝑑 =0.14909,𝑑 =0.43307. 2 3 2 4 Table 2: Optimization results of BQGA and improved BQGA. Optimization results 𝜔 (rad) 𝜔 (rad) 𝜔 (rad) 𝜔 (rad) 𝜔 (rad) 𝜔 (rad) 1 2 3 4 5 6 BQGA -1.2877 -3.5668 2.8165 1.9572 -1.2525 3.0360 Improve BQGA -1.1290 -3.4995 2.8803 2.0680 -1.2586 3.1212 Table 3: Comparison of convergence between basic BQGA and improved BQGA. Number of iterations 50 100 200 400 800 1000 The F value of BQGA 0.0716 0.0643 0.0479 0.0373 0.0373 0.0373 The F value of improve BQGA 0.0273 0.0210 0.0148 0.0085 0.0081 0.0071 gate [24, 32], which causes the current chromosome to According to the flowchart of the inverse kinematics evolve toward the direction of the contemporary optimal solution in Figure 1, the initial parameters are 𝑚 = 100; chromosome, producing the next generation. The symbol of 𝑛=6 ;𝑝 =0.1;𝜑 =0.05;𝜋 𝜃 = 0.05𝜋 ; 𝑡 = 1000. 𝑚0 0 0 max the two corners of the quantum revolving door is selected in The program is written in Matlab. Under the same initial [18], and the size is determined by using the adaptive corner conditions, the optimization results of BQGA and improved value adjustment scheme of Section 2.1. It should be pointed BQGA are shown in Table 2. The convergence is shown in out that since each corner of the robot arm has a specific range Table 3 and Figure 2. of values, the F formed by it cannot be continuously taken. The values of the two sets of optimal solution in Table 2 Therefore, the inverse solution is a discrete optimization are substituted into the kinematics positive equation (15), and problem, and it is necessary to replace the gradient with two the actual position and posture matrix of the BQGA and the generations of difference. improved BQGA are A and A , respectively. 1 2 The mutation of quantum chromosomes is realized by the non-gate [24, 33], and its role can be seen as an unconstrained 0.4988 −0.1207 −0.8583 0.1199 [ ] large rotation. The adaptive mutation probability scheme in [ −0.5406 −0.8174 −0.1992 0.1215 ] Section 2.2 is used to control the usage of non-gates. [ ] A = (22) [ ] −0.6775 0.5633 −0.4730 −0.4811 [ ] .. Steps to Find Inverse Kinematics. Based on the above- 0 0 0 1.0000 [ ] mentioned theory of improving BQGA, the flow chart for 0.5336 −0.1143 −0.8379 0.0764 finding the inverse kinematics is shown in Figure 1. [ ] −0.4743 −0.8608 −0.1846 0.1872 [ ] [ ] A = (23) [ ] 4. Application Example of Improved BQGA −0.7002 0.4959 −0.5136 −0.4921 [ ] In this paper, the general PUMA560 is taken as an example, 0 0 0 1.0000 [ ] and the inverse solution is obtained by using BQGA and improved BQGA, respectively. It is known that the connect- It can be seen that the deviation of the corresponding element ing rod parameters [34, 35] of the PUMA 560 are as shown values in A and T is smaller. That is, the optimal solution in Table 1. In the operable space, it is assumed that the target of the improved algorithm is closer to the actual solution of position and posture matrix T are randomly generated as the target equation (this can also be seen from the F value corresponding to the same number of iterations in Table 3), 0.5322 −0.1137 −0.8390 0.0833 indicating that the actual position and posture of the end [ ] eeff ctor of the mechanical arm with improved BQGA can be [ −0.4697 −0.8641 −0.1809 0.1890 ] [ ] T = (21) closer to the set target pose. [ ] −0.7044 0.4903 −0.5133 −0.4925 [ ] It can be seen from Figure 2 and Table 3 that aer ft the BQGA is iterated 250 times, the F value does not change, 0 0 0 1.0000 [ ] Journal of Robotics 5 Start InitializationQ(t) Solution space transformation Calculating fitness values F(BEX)<=F(GX)? Generate Q(t+1) BEX replaces GX; GX replaces BAX BEC replaces GC GC replaces BAC At the same time, Update and variation Q(t) the replacement principle is enabled. Ｎ>=Ｎ ? Ｇ；Ｒ Save the global optimal solution End Figure 1: Flow chart for finding inverse kinematics. where, as for calculating the tnes fi s value with the evaluation function F, the smaller F is, the greater tnes fi s is; GX and GC represent the global optimal solution and the global optimal chromosome, respectively. Comparison of evaluation function curve indicating that it has fallen into the local optimal solution. 0.5 Not only does the improved BQGA find faster speed, but also 0.45 the curve gradually approaches the zero axis with the increase of the number of iterations, which highlights the superiority 0.4 of the improved algorithm not easily falling into the local 0.35 extremum. 0.3 0.25 5. Conclusions 0.2 (i) The improved BQGA solves the problems of the BQGA 0.15 better and has the advantages of high search efficiency, 0.1 difficulty in overcoming the optimal solution, and difficulty in falling into local extremum. 0.05 (ii) The improved BQGA not only solves the problem of 0 100 200 300 400 500 600 700 800 900 1000 inverse kinematics of robots, but also effectively improves the speed and accuracy of robotic control. It provides a method Number of iterations for solving similar complex multivariate functions. Improved BQGA (iii) Although this paper solves the typical discrete opti- BQGA mization problem, it is instructive and versatile for some Figure 2: Comparison of evaluation function curve. continuous optimization and other fields of application. Evaluation function value F 6 Journal of Robotics Data Availability algorithm,” Journal of Northeastern University (Natural Science), vol.39,no.6,pp.787–791, 2018. 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Application of Improved BQGA in Robot Kinematics Inverse Solution

Lv, Xiaoqing; Zhao, Ming
Journal of Robotics , Volume 2019: 7 – Feb 3, 2019
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Hindawi Publishing Corporation
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Copyright © 2019 Xiaoqing Lv and Ming Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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10.1155/2019/1659180
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Hindawi Journal of Robotics Volume 2019, Article ID 1659180, 7 pages https://doi.org/10.1155/2019/1659180 Research Article Application of Improved BQGA in Robot Kinematics Inverse Solution 1 2 Xiaoqing Lv and Ming Zhao School of Electronics and Information Engineering, University of Science and Technology Liaoning, Anshan , China School of Applied Technology, University of Science and Technology Liaoning, Anshan , China Correspondence should be addressed to Ming Zhao; as zmyli@163.com Received 26 October 2018; Accepted 9 January 2019; Published 3 February 2019 Academic Editor: Raffaele Di Gregorio Copyright © 2019 Xiaoqing Lv and Ming Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In view of the problem that Bloch Quantum Genetic Algorithm (BQGA) is easy to fall into local optimum, an improved BQGA is proposed. eTh algorithm can control the step size and the mutation probability in real time in the iterative process, avoiding over the optimal solution and guaranteeing search efficiency. In addition, the improved algorithm further completes the anti-degradation mechanism, which maintains the diversity of the population while preserving the dominant gene to the maximum extent, so that the algorithm is not easy to fall into the local extremum and finally approaches the global optimal solution. eTh application in the inverse solution of robot kinematics shows that the improved BQGA eeff ctively avoids the premature problem and accelerates the convergence of understanding and the search result is close to the complete solution, which provides a new idea for solving complex nonlinear and multivariate functional equations. 1. Introduction structure, space, expressions, etc., so it is not convenient and efficient to solve inverse kinematics problem. With the rapid The kinematics problem is the basis for the intelligent devel- development of computer programming technology, various opment of today’s robotic arms. Research in related fields intelligent algorithms, such as neural network algorithm [9], has made tremendous contributions to the development of genetic algorithm [10, 11], particle swarm optimization [12], science and technology. In document [1] the solutions of quantum optimization algorithms, and their various hybrid both the forward and inverse kinematics problems for Adept algorithms, have emerged, which highlight the ability to solve Six 300 6R robot are presented. A matrix method was used complex problems. For instance, in order to simplify the in computations. Algorithm based on the derived formulas controller design of the manipulator, document [13] proposes and logical conditions has been implemented in Adept Smart to train the NN algorithm with entropy clustering method, Controller Cx. Literatures [2] are based on kinematics for which effectively solves the multi-value problem; applying more in-depth path planning studies. This paper innovatively an improved genetic algorithm to the inverse kinematics can applies BQGA to the inverse kinematics program, which not overcome the structural limitations of the manipulator [14]; only expands the application eld fi of intelligent algorithms, in the inverse solution of the EOD manipulator, the simulated but also provides a new perspective for solving kinematic annealing algorithm was used, and the expected effect was problems. obtained [15]. With the development of quantum theory, In robotics [3], the positive solution problem is relatively people are increasingly interested in quantum optimization simple and will get a unique solution, while the inverse algorithms. Quantum Genetic Algorithm (QGA) is popular among scholars because of its convenience and ecffi iency. solution problem [4] is relatively complex, and there may be multiple sets of solutions. The traditional methods of Therefore, many improved algorithms such as real-coded solving them are geometry and algebra methods [5, 6], QGA [16], adaptive double-chain QGA [17], and Bloch numerical analysis method [7, 8], etc., but they are limited by Quantum Genetic Algorithm (BQGA) [18] have emerged. 2 Journal of Robotics 𝑡 𝑡 ∇𝑓 (𝑋 )−∇𝑓 min Among them, BQGA is a probability optimization algorithm 󵄨 󵄨 󵄨 ̂ 󵄨 󵄨 󵄨 Δ𝜃 =𝜃 × exp (− ) (6) 󵄨 󵄨 𝑡 𝑡 󵄨 󵄨 that uses the qubits to represent population elements [19]. ∇𝑓 max −∇𝑓 min Its three-chain coding method is more efficient than the 𝑡 𝑡 traditional double-chain structure, which makes the GA where ∇𝑓 and ∇𝑓 represent the maximum and mini- max min algorithm take a new step. However, in practical applications, mum tness fi gradient values of the generation, respectively. It BQGA also hasproblemssuch asbeing easy to passthe should be pointed out that the discrete optimization problem optimal solution and population degradation, and sometimes [20] and the continuous optimization problem [21] are cal- the optimization result is not ideal. Therefore, this paper culated differently when acquiring the relevant gradient. For proposes an improvement plan for these problems in BQGA. the discrete optimization problem, since 𝑓(𝑋) does not have a gradient, we can replace the gradient with the difference of the tness fi of two adjacent generations [22]; the expression is 2. Improvement Scheme of BQGA 󵄨 󵄨 𝑡 𝑡 𝑡−1 󵄨 󵄨 󵄨 󵄨 ∇𝑓 (𝑋 )= 𝑓(𝑋 )−𝑓(𝑋 ) (7) 󵄨 󵄨 .. Adjustment of Adaptive Corner Value. In the process of 𝑖 b 󵄨 󵄨 updating BQGA’s quantum chromosome, the value of the 󵄨 󵄨 𝑡 𝑡 𝑡−1 󵄨 󵄨 󵄨 󵄨 ∇𝑓 max = max {𝑓(𝑋 )−𝑓(𝑋 ) ,⋅⋅⋅ , corner is more complicated. If the planning is unreasonable, 󵄨 󵄨 1𝑏 1𝑏 󵄨 󵄨 (8) it will seriously affect the optimization effect of the algorithm. 󵄨 󵄨 𝑡 𝑡−1 󵄨 󵄨 󵄨 󵄨 𝑓(𝑋 )−𝑓(𝑋 ) } The look-up table method in literature [18] provides an 󵄨 󵄨 𝑚𝑏 𝑚𝑏 󵄨 󵄨 effective convergence strategy. However, the look-up table 󵄨 󵄨 𝑡 𝑡 𝑡−1 󵄨 󵄨 󵄨 󵄨 ∇𝑓 min = min{𝑓(𝑋 )−𝑓(𝑋 ) ,⋅⋅⋅ , 󵄨 󵄨 method does not take into account the differences in individ- 1𝑏 1𝑏 󵄨 󵄨 (9) ual chromosomes and the trend of the evaluation function, 󵄨 󵄨 𝑡 𝑡−1 󵄨 󵄨 󵄨 󵄨 𝑓(𝑋 )−𝑓(𝑋 ) } 󵄨 󵄨 making the algorithm easy to cross the optimal solution and 𝑚𝑏 𝑚𝑏 󵄨 󵄨 less adaptive. Based on the traditional table lookup method, The gradient in the continuous optimization problem is this paper further proposes an adaptive adjustment strategy for corner values. The expression is as follows: 󵄨 𝑡 󵄨 󵄨 󵄨 (𝑋 ) 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∇𝑓 (𝑋 ) = (10) 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑡 󵄨 󵄨 󵄨 󵄨 󵄨 ̂ 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 Δ𝜙 󵄨 ∇𝑓 (𝑋 )󵄨 ≥𝑎 { 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 Δ𝜑 = (1) 󵄨 󵄨 { 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑡 𝑡 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ̃ 󵄨 󵄨 󵄨 󵄨 (𝑋 ) (𝑋 ) Δ𝜙 ∇𝑓 (𝑋 ) <𝑎 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑡 󵄨 󵄨 1𝑏 𝑚𝑏 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 { 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∇𝑓 max = max { ,⋅⋅⋅ , } (11) 󵄨 󵄨 󵄨 󵄨 𝑡 𝑡 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 1𝑏 𝑚𝑏 󵄨 󵄨 󵄨 𝑡 󵄨 󵄨 󵄨 ̂ 󵄨 󵄨 󵄨 󵄨 Δ𝜃 󵄨 ∇𝑓 (𝑋 )󵄨 ≥𝑎 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑡 󵄨 󵄨 𝑡 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 Δ𝜃 = (2) (𝑋 ) (𝑋 ) 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 { 󵄨 󵄨 𝑡 1𝑏 𝑚𝑏 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ̃ 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∇𝑓 min = min{ ,⋅⋅⋅ , } (12) Δ𝜃 ∇𝑓 (𝑋 ) <𝑎 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑡 𝑡 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 { 󵄨 󵄨 󵄨 󵄨 󵄨 1𝑏 󵄨 󵄨 𝑚𝑏 󵄨 where 𝑡 represents evolutionary algebra (same number of In the above adaptive strategy, the gradient size (measured iterations), |Δ𝜑 | and |Δ𝜃 | are the corner sizes of 𝜑 and 𝜃 by 𝑎) determines the calculation formula of the corner. In the formula related to the gradient, the larger the gradient of the 𝑗 th qubit on the 𝑖 rd chromosome, 𝑋 represents the optimal solution among the three chains of each chromosome is, the smaller the step size becomes and, in the formula related to the number of iterations, the step size decreases in the𝑡 nd generation,|∇(𝑓 𝑋 )| represents the tness fi gradient monotonically with the increase of the number of iterations. value of the optimal solution for each chromosome in the 𝑡 nd generation, 𝑎 is an evaluation value whose size is u Th s, the size of the corner varies with the gradient and is again limited by the number of iterations. The purpose is to determined by the maximum and minimum gradient values ̃ control the step size in a reasonable range in real time. in each generation, and |Δ𝜑̃ | and |Δ𝜃 | are the corner values associated with the number of iterations, and the expression is .. Adaptive Mutation Probability. During the quantum chromosomal mutation of BQGA, the qubits rotate greatly 𝑡 −𝑡 󵄨 󵄨 max 󵄨 󵄨 along the Bloch sphereand arenot constrained by other 󵄨 Δ𝜑 󵄨 =𝜑 × (3) 𝑖 0 󵄨 󵄨 max chromosomes, which helps to increase population diversity and avoid premature convergence. However, if the probability 󵄨 󵄨 𝑡 −𝑡 max 󵄨 󵄨 󵄨 󵄨 Δ𝜃 =𝜃 × (4) of mutation is constant throughout the evolution process, it 󵄨 𝑖 󵄨 0 󵄨 󵄨 max is not conducive to the late convergence of the algorithm. In order to enable the algorithm to expand the global search where 𝜑 and 𝜃 are initial iteration values, which range 0 0 ability in the early iteration and focus on local search in from 0.005𝜋 to 0.05𝜋 ,and 𝑡 is the maximum number of max the late iteration to speed up the convergence, the adaptive iterations. |Δ𝜑̂ | and |Δ𝜃 | are the angle values associated mutation probability [23] is adjusted to with the gradient. Referring to literature [18], the expressions constructed are as follows: 𝑡 −𝑡 max 𝑝 =𝑝 × (13) 𝑚 𝑚0 𝑡 𝑡 max 󵄨 󵄨 ∇𝑓 (𝑋 )− ∇𝑓 min 󵄨 󵄨 󵄨 󵄨 Δ𝜑̂ =𝜑 × exp (− ) (5) 󵄨 󵄨 0 󵄨 󵄨 𝑡 𝑡 where 𝑝 ∈ (0.01,0.5). ∇𝑓 max −∇𝑓 min 𝑚0 𝑖𝑗 𝑖𝑏 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑏 𝑖𝑏 𝑖𝑗 𝑖𝑗 𝜕𝑋 𝜕𝑋 𝑖𝑏 𝑖𝑗 𝑖𝑗 𝜕𝑓 𝜕𝑓 𝑖𝑏 𝑖𝑗 𝜕𝑋 𝜕𝑋 𝑖𝑏 𝑖𝑗 𝜕𝑓 𝜕𝑓 𝑖𝑗 𝑖𝑏 𝑖𝑗 𝑖𝑏 𝜕𝑋 𝑖𝑏 𝑖𝑏 𝜕𝑓 𝑖𝑏 𝑖𝑏 𝑖𝑗 𝑖𝑏 Journal of Robotics 3 .. Anti-Degradation Mechanism. In the process of popula- set of inverse solutions, we have established (16) [4, 29] which tion evolution, the anti-degradation mechanism can increase transforms the inverse solution problem into a multivariate the chances of dominant individuals participating in inheri- function to find the extremum problem. tance and variation and reduce the possibility of population 󵄩 󵄩 󵄩 󵄩 𝐹= A − T (16) 󵄩 󵄩 degradation, thus ensuring that the algorithm converges with 󵄩 󵄩 maximum probability within a limited number of iterations. where A is the actual position and posture matrix of the However, the principle of replacement of individual elimina- robot arm established by the D-H parameter method, andT is tion will cause the population diversity to decrease rapidly the set target position and posture matrix, ‖Α −T‖ is the two with the increase of the number of iterations, so that it is norm of A and T [30] which represents the distance between easy to fall into local optimum, and it is difficult to jump the two matrices. out only by relying on mutation. In the improved BQGA, Available from (16), when the actual position and posture the replacement principle of the chain change is adopted. of the robot arm are equal to or infinitely close to the target The description is that the chain is changed first, and the position and posture, the minimum value of the evaluation entire chromosome is replaced when the chain number is not function 𝐹 is zero or approximately zero. the same. That is, it is the rfi st to find the optimal solution (corresponding to the optimal chain) and the worst solution .. Coding Method. According to the basis of quantum com- (corresponding to the worst chain) in one generation. If the puting and the three-strand gene coding scheme of quantum chain numbers are the same, the worst chain is replaced by the chromosomes [18, 31], the initial population expression is as optimal chain, and the other two chains remain unchanged; follows: if thetwochainnumbers aredieff rent, replacetheentire chromosome. The replacement expression is 𝑡 BAX ←󳨀 BEX = 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 cos𝜑 sin 𝜃 cos𝜑 sin 𝜃 ⋅⋅⋅ cos 𝜑 sin 𝜃 𝑖1 𝑖1 󵄨 𝑖2 𝑖2 󵄨 󵄨 󵄨 󵄨 󵄨 (14) (17) 󵄨 󵄨 󵄨 [ ] 󵄨 󵄨 󵄨 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 󵄨 󵄨 󵄨 [ ] 󵄨 󵄨 󵄨 BAC ←󳨀 BEC =𝑔̸ 𝑠𝑏𝑒𝑠𝑡 = sin 𝜑 sin 𝜃 sin𝜑 sin 𝜃 ⋅⋅⋅ sin 𝜑 sin 𝜃 󵄨 󵄨 󵄨 [ 𝑖1 𝑖1 𝑖2 𝑖2 ] 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑡 󵄨 𝑡 󵄨 󵄨 𝑡 󵄨 󵄨 󵄨 ⋅⋅⋅ cos 𝜃 cos 𝜃 cos 𝜃 󵄨 󵄨 󵄨 𝑖1 𝑖2 where BEX is the most suitable solution called the con- [ ] temporary optimal solution, and its chromosome is called where 𝜑 ∈ (𝜋/2, 3𝜋/2), 𝜃 ∈ (0,)𝜋 , 𝑖 = 1,2,⋅⋅⋅ ,𝑚 , 𝑗= the contemporary optimal chromosome [24, 25] recorded 1,2,⋅⋅⋅ ,𝑛 ; 𝑚 is the population size, which is the number of as BEC. The same reason BAX is the worst solution and chromosomes; 𝑛 is the number of qubits (representing 𝑛 to- BAC is the worst chromosome, gbads and gbests repre- be-turned corners); 𝑡 is an evolutionary algebra. sent the corresponding chain number. The anti-degradation mechanism of the improved BQGA ensures that the worst .. Transformation of Unit Space Solution to Joint Space Solu- chromosomes of each generation have an optimal gene chain tion. According to the theory of Bloch spherical coordinates, while maintaining population diversity. a population can represent the m-group solutions of the search space and map them into the actual solution space 3. Applying Improved BQGA for Ω=[−2,𝜋 2]𝜋 to obtain m sets of feasible solutions (𝑋 , Inverse Kinematics 𝑋 ,and 𝑋 ). The mapping formula [18] isasfollows: .. Establishment of Evaluation Function. Using the 𝑗 𝑗 𝑗 𝑋 = [𝑉 (1 + 𝑝 )+𝑉 (1 − 𝑝 )] (18) max min improved BQGA to solve the inverse kinematics problem of the robot, an evaluation function needs to be established. 𝑗 𝑗 𝑗 According to the kinematics principle of the n-degree-of- 𝑋 = [𝑉 (1 + 𝑝 )+ 𝑉 (1 − 𝑝 )] (19) max min freedom manipulator [26], the kinematics positive solution equation established by the D-H parameter method [27, 28] 𝑗 𝑗 𝑗 (20) 𝑋 = [𝑉 (1 + 𝑝 )+𝑉 (1 − 𝑝 )] is max min 𝑛 0 1 2 3 A = A (𝜔 )⋅ A (𝜔 )⋅ A (𝜔 )⋅ A (𝜔 )⋅ ... 1 2 3 4 1 2 3 4 where 𝑗 = 1,2,⋅⋅⋅ ,𝑛 ; 𝑉 and 𝑉 correspond to the max min maximum and minimum values in the range of values for 𝑛 𝑜 𝑎 𝑝 𝑗 𝑗 𝑥 𝑥 𝑥 𝑥 each angle (check Table 1 to get its value); 𝑝 , 𝑝 ,and [ ] (15) [ 𝑛 𝑜 𝑎 𝑝 ] 𝑗 𝑦 𝑦 𝑦 𝑦 n−1 𝑝 represent the 𝑋 solution, the 𝑌 solution, and the 𝑍 [ ] 𝑖 z ⋅ A (𝜔 )= 𝑛 [ ] 𝑛 𝑜 𝑎 𝑝 solution of the 𝑗 th gene position of the 𝑖 th chromosome, [ ] 𝑧 𝑧 𝑧 𝑧 respectively. According to the decoding process, each of the 00 0 1 [ ] chains of each chromosome decodes n corresponding robot arm corner values and their values directly fall within the Thatis, given a setof angles from 𝜔 to 𝜔 ,matrix A 1 𝑛 respective range of values. of position and posture of the end eeff ctor is uniquely determined. However, in practice it is the opposite; that is, the end position and posture are known to solve the value of the .. Update and Mutation of Quantum Chromosomes. The angle of each joint (inverse solution). In order to evaluate each quantum chromosome is updated by the quantum revolving 𝑖𝑦 𝑖𝑥 𝑖𝑧 𝑖𝑧 𝑖𝑧 𝑖𝑦 𝑖𝑦 𝑖𝑦 𝑖𝑥 𝑖𝑥 𝑖𝑥 𝑖𝑧 𝑖𝑦 𝑖𝑥 𝑖𝑗 𝑖𝑗 𝑖𝑛 𝑖𝑛 𝑖𝑛 𝑔𝑏𝑎𝑑𝑠 𝑖𝑛 𝑖𝑛 𝑡𝑠𝑔𝑏𝑒𝑠 𝑔𝑏𝑎𝑑𝑠 4 Journal of Robotics Table 1: Connecting rod parameters of PUMA560. 𝑖𝛼 (rad) 𝑎 (m) 𝑑 (m) 𝜔 (rad) Joint angle range(rad) i-1 i-1 i i 10 0 0 𝜔 -8𝜋 /9 ∼8𝜋 /9 2-𝜋 0 d 𝜔 -5𝜋 /4 ∼𝜋 /4 2 2 30 a 0 𝜔 -𝜋 /4 ∼5𝜋 /4 2 3 4-𝜋 a d 𝜔 -11𝜋 /18 ∼17𝜋 /18 3 4 4 5 𝜋 00 𝜔 -5𝜋 /9 ∼5𝜋 /9 6-𝜋 00 𝜔 -133𝜋 /90 ∼133𝜋 /90 where𝑎 =0.4318,𝑎 =0.02032,𝑑 =0.14909,𝑑 =0.43307. 2 3 2 4 Table 2: Optimization results of BQGA and improved BQGA. Optimization results 𝜔 (rad) 𝜔 (rad) 𝜔 (rad) 𝜔 (rad) 𝜔 (rad) 𝜔 (rad) 1 2 3 4 5 6 BQGA -1.2877 -3.5668 2.8165 1.9572 -1.2525 3.0360 Improve BQGA -1.1290 -3.4995 2.8803 2.0680 -1.2586 3.1212 Table 3: Comparison of convergence between basic BQGA and improved BQGA. Number of iterations 50 100 200 400 800 1000 The F value of BQGA 0.0716 0.0643 0.0479 0.0373 0.0373 0.0373 The F value of improve BQGA 0.0273 0.0210 0.0148 0.0085 0.0081 0.0071 gate [24, 32], which causes the current chromosome to According to the flowchart of the inverse kinematics evolve toward the direction of the contemporary optimal solution in Figure 1, the initial parameters are 𝑚 = 100; chromosome, producing the next generation. The symbol of 𝑛=6 ;𝑝 =0.1;𝜑 =0.05;𝜋 𝜃 = 0.05𝜋 ; 𝑡 = 1000. 𝑚0 0 0 max the two corners of the quantum revolving door is selected in The program is written in Matlab. Under the same initial [18], and the size is determined by using the adaptive corner conditions, the optimization results of BQGA and improved value adjustment scheme of Section 2.1. It should be pointed BQGA are shown in Table 2. The convergence is shown in out that since each corner of the robot arm has a specific range Table 3 and Figure 2. of values, the F formed by it cannot be continuously taken. The values of the two sets of optimal solution in Table 2 Therefore, the inverse solution is a discrete optimization are substituted into the kinematics positive equation (15), and problem, and it is necessary to replace the gradient with two the actual position and posture matrix of the BQGA and the generations of difference. improved BQGA are A and A , respectively. 1 2 The mutation of quantum chromosomes is realized by the non-gate [24, 33], and its role can be seen as an unconstrained 0.4988 −0.1207 −0.8583 0.1199 [ ] large rotation. The adaptive mutation probability scheme in [ −0.5406 −0.8174 −0.1992 0.1215 ] Section 2.2 is used to control the usage of non-gates. [ ] A = (22) [ ] −0.6775 0.5633 −0.4730 −0.4811 [ ] .. Steps to Find Inverse Kinematics. Based on the above- 0 0 0 1.0000 [ ] mentioned theory of improving BQGA, the flow chart for 0.5336 −0.1143 −0.8379 0.0764 finding the inverse kinematics is shown in Figure 1. [ ] −0.4743 −0.8608 −0.1846 0.1872 [ ] [ ] A = (23) [ ] 4. Application Example of Improved BQGA −0.7002 0.4959 −0.5136 −0.4921 [ ] In this paper, the general PUMA560 is taken as an example, 0 0 0 1.0000 [ ] and the inverse solution is obtained by using BQGA and improved BQGA, respectively. It is known that the connect- It can be seen that the deviation of the corresponding element ing rod parameters [34, 35] of the PUMA 560 are as shown values in A and T is smaller. That is, the optimal solution in Table 1. In the operable space, it is assumed that the target of the improved algorithm is closer to the actual solution of position and posture matrix T are randomly generated as the target equation (this can also be seen from the F value corresponding to the same number of iterations in Table 3), 0.5322 −0.1137 −0.8390 0.0833 indicating that the actual position and posture of the end [ ] eeff ctor of the mechanical arm with improved BQGA can be [ −0.4697 −0.8641 −0.1809 0.1890 ] [ ] T = (21) closer to the set target pose. [ ] −0.7044 0.4903 −0.5133 −0.4925 [ ] It can be seen from Figure 2 and Table 3 that aer ft the BQGA is iterated 250 times, the F value does not change, 0 0 0 1.0000 [ ] Journal of Robotics 5 Start InitializationQ(t) Solution space transformation Calculating fitness values F(BEX)<=F(GX)? Generate Q(t+1) BEX replaces GX; GX replaces BAX BEC replaces GC GC replaces BAC At the same time, Update and variation Q(t) the replacement principle is enabled. Ｎ>=Ｎ ? Ｇ；Ｒ Save the global optimal solution End Figure 1: Flow chart for finding inverse kinematics. where, as for calculating the tnes fi s value with the evaluation function F, the smaller F is, the greater tnes fi s is; GX and GC represent the global optimal solution and the global optimal chromosome, respectively. Comparison of evaluation function curve indicating that it has fallen into the local optimal solution. 0.5 Not only does the improved BQGA find faster speed, but also 0.45 the curve gradually approaches the zero axis with the increase of the number of iterations, which highlights the superiority 0.4 of the improved algorithm not easily falling into the local 0.35 extremum. 0.3 0.25 5. Conclusions 0.2 (i) The improved BQGA solves the problems of the BQGA 0.15 better and has the advantages of high search efficiency, 0.1 difficulty in overcoming the optimal solution, and difficulty in falling into local extremum. 0.05 (ii) The improved BQGA not only solves the problem of 0 100 200 300 400 500 600 700 800 900 1000 inverse kinematics of robots, but also effectively improves the speed and accuracy of robotic control. It provides a method Number of iterations for solving similar complex multivariate functions. Improved BQGA (iii) Although this paper solves the typical discrete opti- BQGA mization problem, it is instructive and versatile for some Figure 2: Comparison of evaluation function curve. continuous optimization and other fields of application. Evaluation function value F 6 Journal of Robotics Data Availability algorithm,” Journal of Northeastern University (Natural Science), vol.39,no.6,pp.787–791, 2018. 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