Fixed-Time Control with an Improved Sparrow Search Algorithm for Robotic Arm Performance Optimization
Abstract
:1. Introduction
- (1)
- The proposed fixed-time control strategy achieves rapid convergence and enhanced vibration suppression for a n-degree-of-freedom robotic arm. This strategy addresses input saturation, output constraints, and system uncertainties to broaden the arm’s application. A novel adaptive law using logarithmic barrier Lyapunov functions (BLFs) was developed within this framework to manage parameter uncertainties and ensure that outputs stay within set limits.
- (2)
- The improved sparrow search algorithm enhances the traditional method by more effectively searching the solution space and identifying optimal control parameters, thus improving the robotic arm’s motion control performance. This algorithm serves as an optimization mechanism in this paper, boosting the effectiveness of the fixed-time control method in order to optimize the arm’s performance.
2. Theoretical Background and Problem Statement
2.1. Input Saturation Nonlinearity
2.2. Barrier Lyapunov Function
2.3. Radial Basis Function Neural Network
2.4. Improved Sparrow Search Algorithm (ISSA)
- (1)
- The good point set (GPS) is a strategy for generating initial solution sets characterized by high uniformity and low correlation. It is suitable for use in global optimization problems. This strategy involves selecting the smallest prime number greater than , coupled with the use of the cosine function, to create a point set. The specific formula for generating points is as follows:
- (2)
- By dynamically adjusting the proportion of discoverers and joiners, this improvement aims to balance exploration and exploitation capabilities. The proportional coefficient changes dynamically with the number of iterations, allowing for more explorers to investigate the solution space early in the search and increasing the number of joiners later on in order to refine potential solution areas already identified [27].
- (3)
- This enhancement uses an exponential decay factor to adjust the position updates of explorers in order to optimize their efficiency when searching the solution space [25].
- (4)
- This improvement increases population diversity by combining a tent map and Cauchy mutation, thereby enhancing the algorithm’s ability to avoid local optima and explore new areas [29,30,31]. Cauchy mutation is used when an individual’s fitness value is less than the average fitness value of the population. Tent mapping is used when an individual’s fitness value is greater than or equal to the average fitness value of the population.
2.5. Preliminaries
3. Fixed-Time Control Design and Lyapunov Function Proof Process
3.1. Fixed-Time Control Design
3.2. Lyapunov Function Proof
4. Simulation
4.1. Simulation on ISSA Performance
4.2. Two-DOF Robotic Arm Simulation
4.3. Summary
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Appendix B
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Function Number | Test Functions | Domain | Theoretical Value |
---|---|---|---|
F1 | [−100, 100] | 0 | |
F2 | [−10, 10] | 0 | |
F3 | [−10, 10] | 0 | |
F4 | [−100, 100] | 0 | |
F5 | [−5, 10] | 0 | |
F6 | [−32, 32] | 0 | |
F7 | [−50, 50] | 0 | |
F8 | [−5, 5] | 0.000309 |
Test Functions | Algorithm | Best Value | Worst Value | Average Value |
---|---|---|---|---|
F1 | SSA | 0.0000 × 100 | 4.0130 × 10−267 | 4.0130 × 10−268 |
ISSA | 0.0000 × 100 | 0.0000 × 100 | 0.0000 × 100 | |
PSO | 6.3177 × 10−1 | 2.5751 × 100 | 1.6997 × 100 | |
Chimp | 7.2228 × 10−10 | 5.0203 × 10−7 | 8.2238 × 10−8 | |
GWO | 4.4064 × 10−32 | 2.8157 × 10−30 | 6.5511 × 10−31 | |
WOA | 1.4104 × 10−89 | 6.9762 × 10−79 | 6.9780 × 10−80 | |
F2 | SSA | 0.0000 × 100 | 1.0519 × 10−103 | 1.0519 × 10−104 |
ISSA | 0.0000 × 100 | 0.0000 × 100 | 0.0000 × 100 | |
PSO | 2.4067 × 100 | 4.9044 × 100 | 3.8305 × 100 | |
Chimp | 2.9009 × 10−7 | 1.9247 × 10−5 | 5.7236 × 10−6 | |
GWO | 1.5962 × 10−19 | 2.7536 × 10−18 | 1.3757 × 10−18 | |
WOA | 2.3018 × 10−59 | 6.1251 × 10−53 | 6.9522 × 10−54 | |
F3 | SSA | 0.0000 × 100 | 0.0000 × 100 | 0.0000 × 100 |
ISSA | 0.0000 × 100 | 0.0000 × 100 | 0.0000 × 100 | |
PSO | 0.0000 × 100 | 1.3654 × 10−10 | 1.4841 × 10−11 | |
Chimp | 3.2463 × 10−10 | −1.1122 × 10−7 | 2.3298 × 10−8 | |
GWO | 8.4082 × 10−10 | 1.9413 × 10−7 | 3.1146 × 10−8 | |
WOA | 0.0000 × 100 | 1.2829 × 10−12 | 1.4072 × 10−13 | |
F4 | SSA | 0.0000 × 100 | 4.2410 × 10−252 | 4.2410 × 10−253 |
ISSA | 0.0000 × 100 | 0.0000 × 100 | 0.0000 × 100 | |
PSO | 1.4490 × 100 | 2.0563 × 100 | 1.6774 × 100 | |
Chimp | 2.2011 × 10−2 | 6.0196 × 10−1 | 2.0611 × 10−1 | |
GWO | 1.7687 × 10−8 | 2.1929 × 10−7 | 5.8221 × 10−8 | |
WOA | 9.8640 × 10−2 | 8.5463 × 101 | 4.5762 × 101 | |
F5 | SSA | 0.0000 × 100 | 3.0119 × 10−124 | 3.0119 × 10−125 |
ISSA | 0.0000 × 100 | 0.0000 × 100 | 0.0000 × 100 | |
PSO | 6.2717 × 101 | 1.7519 × 102 | 1.2573 × 102 | |
Chimp | 1.7803 × 100 | 2.2152 × 101 | 9.1301 × 100 | |
GWO | 1.4177 × 10−11 | 4.5254 × 10−9 | 1.0949 × 10−9 | |
WOA | 5.0206 × 102 | 2.9749 × 102 | 4.3256 × 102 | |
F6 | SSA | 8.8818 × 10−16 | 8.8818 × 10−16 | 8.8818 × 10−16 |
ISSA | 8.8818 × 10−16 | 8.8818 × 10−16 | 8.8818 × 10−16 | |
PSO | 1.0527 × 100 | 3.0465 × 100 | 2.2359 × 100 | |
Chimp | 1.9961 × 101 | 1.9964 × 101 | 1.9963 × 101 | |
GWO | 5.0626 × 10−14 | 7.5495 × 10−14 | 6.1995 × 10−14 | |
WOA | 8.8818 × 10−16 | 7.9936 × 10−15 | 4.0856 × 10−15 | |
F7 | SSA | 1.4742 × 10−6 | 1.4673 × 10−2 | 5.0658 × 10−3 |
ISSA | 5.7401 × 10−5 | 2.4232 × 10−2 | 3.3781 × 10−3 | |
PSO | 2.1128 × 10−1 | 9.0880 × 10−1 | 4.6949 × 10−1 | |
Chimp | 2.6782 × 100 | 2.9905 × 100 | 2.8295 × 100 | |
GWO | 3.0732 × 10−1 | 7.2904 × 10−1 | 5.0930 × 10−1 | |
WOA | 1.2416 × 10−1 | 9.4357 × 10−1 | 3.1349 × 10−1 | |
F8 | SSA | 3.1419 × 10−4 | 1.2254 × 10−3 | 5.2091 × 10−4 |
ISSA | 3.0750 × 10−4 | 3.6394 × 10−4 | 3.1646 × 10−4 | |
PSO | 7.9768 × 10−4 | 9.9879 × 10−4 | 9.1620 × 10−4 | |
Chimp | 1.2354 × 10−3 | 1.3394 × 10−3 | 1.2804 × 10−3 | |
GWO | 3.0755 × 10−4 | 2.0363 × 10−2 | 4.4465 × 10−3 | |
WOA | 3.1812 × 10−4 | 6.9223 × 10−4 | 4.9395 × 10−4 |
Parameters | Meaning | Value | Unit |
---|---|---|---|
Mass of link 1 | 3.64 | kg | |
Mass of link 2 | 0.42 | kg | |
Length of link 1 | 0.301 | m | |
Length of link 2 | 0.094 | m | |
Inertia of link 1 | ; | ||
Inertia of link 2 | |||
Time (s) | Error of First Link with Fixed-Time Control (rad) | Error of Second Link with Fixed-Time Control (rad) | Error of First Link with Fixed-Time Control + ISSA (rad) | Error of Second Link with Fixed-Time Control + ISSA (rad) |
---|---|---|---|---|
3 | ||||
7 |
Time (s) | Error of First Link with Fixed-Time Control (rad) | Error of Second Link with Fixed-Time Control (rad) | Error of First Link with Fixed-Time Control + ISSA (rad) | Error of Second Link with Fixed-Time Control + ISSA (rad) |
---|---|---|---|---|
From 0 to 5 s |
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Zhang, R.; Choi, H.-S.; Jung, D.; Cho, H.; Anh, P.H.N.; Vu, M.T. Fixed-Time Control with an Improved Sparrow Search Algorithm for Robotic Arm Performance Optimization. Appl. Sci. 2024, 14, 10096. https://doi.org/10.3390/app142210096
Zhang R, Choi H-S, Jung D, Cho H, Anh PHN, Vu MT. Fixed-Time Control with an Improved Sparrow Search Algorithm for Robotic Arm Performance Optimization. Applied Sciences. 2024; 14(22):10096. https://doi.org/10.3390/app142210096
Chicago/Turabian StyleZhang, Ruochen, Hyeung-Sik Choi, Dongwook Jung, Hyunjoon Cho, Phan Huy Nam Anh, and Mai The Vu. 2024. "Fixed-Time Control with an Improved Sparrow Search Algorithm for Robotic Arm Performance Optimization" Applied Sciences 14, no. 22: 10096. https://doi.org/10.3390/app142210096
APA StyleZhang, R., Choi, H.-S., Jung, D., Cho, H., Anh, P. H. N., & Vu, M. T. (2024). Fixed-Time Control with an Improved Sparrow Search Algorithm for Robotic Arm Performance Optimization. Applied Sciences, 14(22), 10096. https://doi.org/10.3390/app142210096