Trajectory Optimization of High-Speed Robotic Positioning with Suppressed Motion Jerk via Improved Chicken Swarm Algorithm
Abstract
:1. Introduction
2. Improved Chicken Swarm Algorithm
2.1. Improved CSO
2.1.1. Parallel Policy-Based X-Best Bootstrap and Levy Flight Rooster Update Mechanism
2.1.2. Dynamic Constrained Hen-Following Goal Mechanism
- (1)
- The total number of roosters is constant, and the number of roosters followed by hens as they transition from following all roosters to following high-quality roosters drops;
- (2)
- In the early iterations, the number of followed roosters should be quickly reduced to determine the approximate range of the optimal solution;
- (3)
- In the late iterations, the variation is kept constant or reduced to achieve a precise search and improve the accuracy of the solution;
- (4)
- There is always a worst rooster among good roosters, which improves the diversity of solutions and helps to avoid falling into local optima.
2.2. Implementation Steps of the Algorithm
Algorithm 1: Pseudocode for the PDCSO algorithm. |
Initialize a population of N chickens and define the related parameters; Evaluate the chickens’ fitness values, While If Rank the chickens’ fitness values and establish a hierarchal order in the swarm Divide the swarm into different groups and determine the relationship between the chicks and mother hens in a group End For If Select rooster update method using Equation (13) Update its solution/location using Equation (7) or Equation (9) End If Update its solution/location using Equation (15) End If Update its solution/location using Equation (6) End Evaluate the new solution If the new solution is better than the previous one, update it End End |
2.3. Improved Algorithm Performance Testing
3. Construction of the Optimization Objective Function
3.1. Description of Optimal Time-Shock Planning Problem
3.2. B-Spline Interpolation Trajectory Construction
4. Simulation and Experimentation
4.1. ADAMS Simulation
4.2. Experiment
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Function | Range | Dimension | Theoretically Optimal Value |
---|---|---|---|
[−100, 100] | D | 0 | |
[−10, 10] | D | 0 | |
[−100, 100] | D | 0 | |
[−100, 100] | D | 0 | |
[−100, 100] | D | 0 | |
[−1.28, 1.28] | D | 0 | |
[−5.12, 5.12] | D | 0 | |
[−32, 32] | D | 0 | |
[−600, 600] | D | 0 | |
[−50, 50] | D | 0 | |
[−4, 5] | D | 0 | |
[−5, 10] | D | 0 | |
[−5, 5] | 4 | 0.0003075 | |
[(−5, 0), (10, 15)] | 2 | 0.398 | |
[−2, 2] | 2 | 3 | |
[−10, 10] | 2 | 0 | |
[0, ] | 10 | −9.66015 | |
[−512, 512] | 2 | −959.6407 |
Algorithm | Parameter Settings |
---|---|
CSO | |
PDCSO | |
ICSO | |
ASCSO-S |
Function | CSO | PDCSO | ICSO | ASCSO-S | |||||
---|---|---|---|---|---|---|---|---|---|
F1 | Ave | 0 | 0 | 2 × 10−131 | 1.04 × 10−31 | 2.1 × 10−142 | 1.86 × 10−45 | 0 | 0 |
Std | 0 | 0 | 9.5 × 10−131 | 4.53 × 10−31 | 8.6 × 10−142 | 7.68 × 10−45 | 0 | 0 | |
Best | 0 | 0 | 6.2 × 10−136 | 3.79 × 10−50 | 7.5 × 10−151 | 7.47 × 10−50 | 0 | 0 | |
Worst | 0 | 0 | 5.2 × 10−130 | 2.47 × 10−30 | 4.6 × 10−141 | 4.22 × 10−44 | 0 | 0 | |
F2 | Ave | 0 | 0 | 7.57 × 10−78 | 4.88 × 10−35 | 2.41 × 10−91 | 4.89 × 10−38 | 0 | 0 |
Std | 0 | 0 | 3.23 × 10−77 | 1.72 × 10−34 | 1.27 × 10−90 | 8.78 × 10−38 | 0 | 0 | |
Best | 0 | 0 | 2.58 × 10−80 | 6.16 × 10−40 | 4.03 × 10−96 | 1.96 × 10−40 | 0 | 0 | |
Worst | 0 | 0 | 1.78 × 10−76 | 9.37 × 10−34 | 6.94 × 10−90 | 4.05 × 10−37 | 0 | 0 | |
F3 | Ave | 0 | 0 | 1.59 × 10−46 | 149.2781 | 4.75 × 10−70 | 19.57705 | 0 | 0 |
Std | 0 | 0 | 8.66 × 10−46 | 457.8035 | 1.38 × 10−69 | 98.51109 | 0 | 0 | |
Best | 0 | 0 | 1.27 × 10−71 | 1.29 × 10−6 | 7.13 × 10−81 | 1.93 × 10−8 | 0 | 0 | |
Worst | 0 | 0 | 4.74 × 10−45 | 2038.464 | 6.39 × 10−69 | 539.5373 | 0 | 0 | |
F4 | Ave | 0 | 0 | 4.14 × 10−47 | 18.97023 | 1.21 × 10−59 | 17.23904 | 0 | 0 |
Std | 0 | 0 | 2.21 × 10−46 | 6.744785 | 2.52 × 10−59 | 6.530568 | 0 | 0 | |
Best | 0 | 0 | 2.03 × 10−52 | 1.201056 | 2.89 × 10−68 | 0.125546 | 0 | 0 | |
Worst | 0 | 0 | 1.21 × 10−45 | 28.29731 | 8.94 × 10−59 | 25.15158 | 0 | 0 | |
F5 | Ave | 1.13 × 10−26 | 0.073115 | 0.043912 | 4.416301 | 0.019847 | 5.616788 | 0.966116 | 10.02114 |
Std | 2.58 × 10−26 | 0.09572 | 0.031135 | 0.376249 | 0.048784 | 0.338539 | 0.152327 | 0.330825 | |
Best | 9.21 × 10−31 | 0.00795 | 0.001415 | 3.844698 | 0.000216 | 5.041785 | 0.655284 | 8.975014 | |
Worst | 1.18 × 10−25 | 0.282306 | 0.124232 | 5.418371 | 0.270026 | 6.263295 | 1.259867 | 10.49705 | |
F6 | Ave | 5.64 × 10−5 | 6.44 × 10−5 | 0.000227 | 0.001258 | 0.000344 | 0.001001 | 7.63 × 10−5 | 0.000118 |
Std | 5.37 × 10−5 | 6.66 × 10−5 | 0.000133 | 0.000405 | 0.000222 | 0.000536 | 7.68 × 10−5 | 0.000112 | |
Best | 9.17 × 10−7 | 2.49 × 10−6 | 3.51 × 10−5 | 0.000596 | 1.9 × 10−5 | 0.000415 | 2.75 × 10−6 | 4.85 × 10−6 | |
Worst | 0.000262 | 0.000263 | 0.000475 | 0.002215 | 0.000832 | 0.003155 | 0.000332 | 0.00047 | |
F7 | Ave | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Std | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
Best | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
Worst | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
F8 | Ave | 8.88 × 10−16 | 8.88 × 10−16 | 1.6 × 10−15 | 1.14 × 10−14 | 8.88 × 10−16 | 4.20 × 10−15 | 8.88 × 10−16 | 8.88 × 10−16 |
Std | 0 | 0 | 1.45 × 10−15 | 2.90 × 10−14 | 0 | 9.01 × 10−16 | 0 | 0 | |
Best | 8.88 × 10−16 | 8.88 × 10−16 | 8.88 × 10−16 | 4.44 × 10−15 | 8.88 × 10−16 | 8.88 × 10−16 | 8.88 × 10−16 | 8.88 × 10−16 | |
Worst | 8.88 × 10−16 | 8.88 × 10−16 | 4.44 × 10−15 | 1.64 × 10−13 | 8.88 × 10−16 | 4.44 × 10−15 | 8.88 × 10−16 | 8.88 × 10−16 | |
F9 | Ave | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Std | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
Best | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
Worst | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
F10 | Ave | 1.84 × 10−27 | 0.000681 | 0.002161 | 0.216774 | 0.001866 | 0.329059 | 0.317632 | 0.855516 |
Std | 5.35 × 10−27 | 0.001182 | 0.001818 | 0.04494 | 0.002993 | 0.035752 | 0.096942 | 0.040541 | |
Best | 4.74 × 10−32 | 5.48 × 10−5 | 1.29 × 10−5 | 0.15463 | 7.2 × 10−7 | 0.233194 | 0.107997 | 0.743935 | |
Worst | 2.69 × 10−26 | 0.004427 | 0.007248 | 0.344528 | 0.012607 | 0.389021 | 0.49548 | 0.924134 | |
F11 | Ave | 0 | 0 | 1.07 × 10−6 | 6.97 × 10−7 | 3.05 × 10−9 | 1.82 × 10−30 | 0 | 0 |
Std | 0 | 0 | 5.85 × 10−6 | 1.67 × 10−6 | 1.66 × 10−8 | 7.41 × 10−30 | 0 | 0 | |
Best | 0 | 0 | 3.5 × 10−42 | 9.37 × 10−36 | 3.18 × 10−71 | 1.54 × 10−48 | 0 | 0 | |
Worst | 0 | 0 | 3.2 × 10−5 | 5.83 × 10−6 | 9.12 × 10−8 | 3.99 × 10−29 | 0 | 0 | |
F12 | Ave | 0 | 0 | 1.02 × 10−57 | 3.239553 | 4.47 × 10−65 | 3.550345 | 0 | 0 |
Std | 0 | 0 | 2.49 × 10−57 | 2.054349 | 2.43 × 10−64 | 1.95467 | 0 | 0 | |
Best | 0 | 0 | 2.78 × 10−62 | 0.525498 | 3.44 × 10−78 | 1.092479 | 0 | 0 | |
Worst | 0 | 0 | 9.85 × 10−57 | 8.636726 | 1.33 × 10−63 | 9.225863 | 0 | 0 | |
F13 | Ave | 0.000338 | 0.00062 | 0.000593 | 0.001151 | ||||
Std | 0.000167 | 0.000187 | 0.000205 | 0.000363 | |||||
Best | 0.000307 | 0.000308 | 0.000307 | 0.000521 | |||||
Worst | 0.001223 | 0.001223 | 0.001223 | 0.002194 | |||||
F14 | Ave | 0.397887 | 0.397887 | 0.397887 | 0.414706 | ||||
Std | 0 | 2.52 × 10−9 | 1 × 10−10 | 0.01935 | |||||
Best | 0.397887 | 0.397887 | 0.397887 | 0.398084 | |||||
Worst | 0.397887 | 0.397887 | 0.397887 | 0.47301 | |||||
F15 | Ave | 3 | 3 | 3 | 3.229174 | ||||
Std | 1.32 × 10−15 | 1.34 × 10−15 | 8.29 × 10−8 | 0.332011 | |||||
Best | 3 | 3 | 3 | 3.001369 | |||||
Worst | 3 | 3 | 3 | 4.782669 | |||||
F16 | Ave | 0 | 0 | 0 | 0.02634 | ||||
Std | 0 | 0 | 0 | 0.025041 | |||||
Best | 0 | 0 | 0 | 0.001075 | |||||
Worst | 0 | 0 | 0 | 0.122481 | |||||
F17 | Ave | −9.3287 | −9.22455 | −9.16256 | −3.67902 | ||||
Std | 0.174558 | 0.279614 | 0.256986 | 0.42561 | |||||
Best | −9.65524 | −9.61852 | −9.60838 | −4.55678 | |||||
Worst | −9.00491 | −8.2349 | −8.58308 | −2.91759 | |||||
F18 | Ave | −959.641 | −959.641 | −959.641 | −909.584 | ||||
Std | 5.78 × 10−13 | 5.78 × 10−13 | 1.63 × 10−6 | 45.37918 | |||||
Best | −959.641 | −959.641 | −959.641 | −959.641 | |||||
Worst | −959.641 | −959.641 | −959.641 | −820.583 |
Linkage Number | |||||
---|---|---|---|---|---|
1 | 0 | 162.5 | −363–363 | ||
2 | 0 | −425 | 0 | −363–363 | |
3 | 0 | 392.2 | 0 | −363–363 | |
4 | 0 | 133.3 | −363–363 | ||
5 | 0 | 99.7 | −363–363 | ||
6 | 0 | 0 | 99.3 | −363–363 |
Nodes | ||||||
---|---|---|---|---|---|---|
Joint 1 | Joint 2 | Joint 3 | Joint 4 | Joint 5 | Joint 6 | |
1 | 0 | 0 | 0 | 0 | 0 | 0 |
2 | 0 | −10 | −5 | 10 | 20 | 0 |
3 | 0 | −20 | 10 | 20 | 20 | 0 |
4 | 0 | −50 | 20 | 30 | 10 | 0 |
5 | 0 | −30 | 60 | 60 | 30 | 0 |
6 | 0 | −10 | 40 | 50 | 20 | 0 |
7 | 0 | 0 | 10 | 30 | 30 | 0 |
8 | 0 | 10 | 0 | 10 | 10 | 0 |
9 | 0 | 20 | 10 | 0 | 5 | 0 |
10 | 0 | 0 | 0 | 0 | 0 | 0 |
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Share and Cite
Li, Y.; Lu, Y.; Li, D.; Zhou, M.; Xu, C.; Gao, X.; Liu, Y. Trajectory Optimization of High-Speed Robotic Positioning with Suppressed Motion Jerk via Improved Chicken Swarm Algorithm. Appl. Sci. 2023, 13, 4439. https://doi.org/10.3390/app13074439
Li Y, Lu Y, Li D, Zhou M, Xu C, Gao X, Liu Y. Trajectory Optimization of High-Speed Robotic Positioning with Suppressed Motion Jerk via Improved Chicken Swarm Algorithm. Applied Sciences. 2023; 13(7):4439. https://doi.org/10.3390/app13074439
Chicago/Turabian StyleLi, Yankun, Yuyang Lu, Dongya Li, Minning Zhou, Chonghai Xu, Xiaozhi Gao, and Yu Liu. 2023. "Trajectory Optimization of High-Speed Robotic Positioning with Suppressed Motion Jerk via Improved Chicken Swarm Algorithm" Applied Sciences 13, no. 7: 4439. https://doi.org/10.3390/app13074439
APA StyleLi, Y., Lu, Y., Li, D., Zhou, M., Xu, C., Gao, X., & Liu, Y. (2023). Trajectory Optimization of High-Speed Robotic Positioning with Suppressed Motion Jerk via Improved Chicken Swarm Algorithm. Applied Sciences, 13(7), 4439. https://doi.org/10.3390/app13074439