Inverse Kinematics: Identifying a Functional Model for Closed Trajectories Using a Metaheuristic Approach
Abstract
:1. Introduction
- The formulation of an optimization problem whose solution represents the position values for inverse kinematics in a 6-DOF serial robot.
- The determination of a mathematical function model for closed paths.
- The proposal and incorporation of an adaptive operator that biases populations in the DE algorithm to achieve the conditions for angular movements in a closed trajectory that coincide with a periodic function.
2. Materials and Methods
2.1. Optimization Approach
2.2. Differential Evolution Algorithm
2.3. DE Considerations for the Case Study
2.4. Bias Operators, Differential Evolution, and Singularities
2.5. Identification
2.6. Methodology Description
3. Experiments and Results
3.1. First Experiment
3.2. Second Experiment
3.3. Discussion
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Links | ||||
---|---|---|---|---|
1 | 0 | 0 | 0 | |
2 | 0 | 75 | ||
3 | 0 | 0 | 400 | |
4 | 410 | 75 | ||
5 | 0 | 0 | ||
6 | 80 | 0 |
Run | Sum of the Objective Function Fixed Weghts | Sum of the Objective Function Dynamic Weghts |
---|---|---|
1 | 28.7365 | 11.4222 |
2 | 23.6704 | 11.4222 |
3 | 28.3489 | 14.8810 |
4 | 15.0127 | 11.8869 |
5 | 127.8434 | 12.6072 |
6 | 37.3099 | 15.8484 |
7 | 51.4377 | 9.3323 |
8 | 43.1509 | 3.4832 |
9 | 34.3967 | 3.7549 |
10 | 14.0598 | 15.0776 |
Mean | 40.3967 | 11.0166 |
Standard deviation | 32.8647 | 4.3624 |
Joint | m Pairs Sine, Cosine in Fourier Representation | Error of the Objective Function |
---|---|---|
1 | 1 | 0.2134 |
1 | 2 | 0.1696 |
1 | 4 | 0.1396 |
1 | 8 | 0.1096 |
1 | 16 | 0.0706 |
1 | 24 | 0.0155 |
2 | 1 | 0.2700 |
2 | 2 | 0.2617 |
2 | 4 | 0.2214 |
2 | 8 | 0.1417 |
2 | 16 | 0.1103 |
2 | 24 | 0.0059 |
3 | 1 | 1.1012 |
3 | 2 | 1.0062 |
3 | 4 | 0.8996 |
3 | 8 | 0.6349 |
3 | 16 | 0.5365 |
3 | 24 | 0.0334 |
4 | 1 | 5.2442 |
4 | 2 | 4.7759 |
4 | 4 | 4.0156 |
4 | 8 | 2.7904 |
4 | 16 | 1.8654 |
4 | 24 | 0.1361 |
5 | 1 | 5.6003 |
5 | 2 | 4.9198 |
5 | 4 | 4.4202 |
5 | 8 | 3.1937 |
5 | 16 | 2.7098 |
5 | 24 | 0.1612 |
Parameter | Values for Joint 1 Model | Values for Joint 2 Model | Values for Joint 3 Model | Values for Joint 4 Model | Values for Joint 5 Model |
---|---|---|---|---|---|
0.0035 | −0.0143 | −0.2579 | −0.0720 | −0.2515 | |
−0.0093 | 0.0072 | 0.1173 | 0.0973 | 0.1680 | |
0.1316 | −0.0007 | 0.0107 | 0.0786 | −0.0031 | |
0.0012 | 0.0045 | −0.0186 | −0.0525 | 0.1036 | |
0.0045 | 0.0016 | −0.0040 | 0.0340 | 0.0175 | |
−0.0008 | −0.0046 | 0.0124 | 0.0202 | −0.0470 | |
0.0024 | 0.0015 | −0.0056 | 0.0501 | 0.0290 | |
0.0012 | 0.0004 | −0.0031 | −0.0490 | 0.0209 | |
0.0021 | −0.0040 | 0.0070 | 0.0503 | −0.0234 | |
−0.0001 | 0.0026 | −0.0095 | 0.0074 | 0.0487 | |
0.0008 | 0.0032 | −0.0150 | 0.0576 | 0.0789 | |
−0.0014 | −0.0014 | 0.0005 | 0.0273 | 0.0081 | |
0.0019 | 0.0010 | −0.0046 | −0.0280 | 0.0234 | |
0.0010 | −0.0024 | 0.0039 | −0.0056 | −0.0091 | |
0.0005 | −0.0040 | 0.0130 | −0.0140 | −0.0596 | |
−0.0002 | 0.0018 | −0.0044 | 0.0171 | 0.0211 | |
0.0014 | 0.0015 | −0.0039 | −0.0561 | 0.0165 | |
0.0001 | 0.0006 | 0.0011 | −0.0122 | −0.0091 | |
0.0004 | 0.0012 | −0.0043 | 0.0186 | 0.0209 | |
−0.0006 | −0.0025 | 0.0105 | 0.0258 | −0.0482 | |
0.0003 | −0.0009 | −0.0007 | 0.0138 | 0.0099 | |
0.0000 | −0.0007 | 0.0031 | 0.0103 | −0.0135 | |
0.0017 | −0.0004 | −0.0020 | −0.0150 | 0.0156 | |
0.0000 | 0.0015 | −0.0051 | 0.0110 | 0.0268 | |
0.0001 | 0.0007 | −0.0032 | 0.0230 | 0.0172 | |
0.0007 | −0.0005 | 0.0013 | −0.0408 | −0.0049 | |
0.0001 | −0.0004 | 0.0028 | 0.0180 | −0.0150 | |
−0.0006 | −0.0016 | 0.0069 | 0.0105 | −0.0330 | |
−0.0002 | −0.0011 | 0.0059 | −0.0004 | −0.0299 | |
−0.0002 | 0.0014 | −0.0002 | −0.0134 | −0.0047 | |
0.0007 | −0.0001 | 0.0012 | −0.0241 | −0.0066 | |
−0.0002 | 0.0005 | 0.0022 | 0.0165 | −0.0159 | |
0.0014 | 0.0001 | −0.0026 | −0.0285 | 0.0161 | |
0.0001 | −0.0016 | 0.0058 | −0.0034 | −0.0265 | |
0.0002 | −0.0003 | −0.0026 | 0.0176 | 0.0192 | |
0.0015 | −0.0002 | −0.0002 | −0.0190 | 0.0045 | |
0.0000 | −0.0012 | 0.0029 | 0.0382 | −0.0112 | |
−0.0003 | 0.0016 | −0.0049 | 0.0244 | 0.0246 | |
0.0006 | 0.0002 | 0.0004 | 0.0038 | −0.0036 | |
0.0000 | −0.0007 | 0.0045 | 0.0032 | −0.0234 | |
0.0003 | 0.0000 | 0.0009 | 0.0164 | −0.0063 | |
0.0001 | −0.0019 | 0.0112 | −0.0188 | −0.0582 | |
−0.0001 | 0.0007 | −0.0039 | 0.0014 | 0.0199 | |
−0.0011 | −0.0005 | 0.0070 | −0.0042 | −0.0393 | |
0.0006 | 0.0012 | −0.0068 | 0.0142 | 0.0366 | |
−0.0012 | 0.0015 | −0.0022 | −0.0212 | 0.0074 | |
0.0004 | 0.0013 | −0.0089 | −0.0123 | 0.0480 | |
−0.0004 | −0.0002 | 0.0001 | −0.0175 | 0.0014 | |
−0.0003 | −0.0011 | 0.0010 | 0.0295 | −0.0016 |
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López-Muñoz, R.; Lopez-Pacheco, M.A.; Maya-Rodriguez, M.C.; Vega-Alvarado, E.; Corona-Ramírez, L.G. Inverse Kinematics: Identifying a Functional Model for Closed Trajectories Using a Metaheuristic Approach. Mathematics 2025, 13, 1847. https://doi.org/10.3390/math13111847
López-Muñoz R, Lopez-Pacheco MA, Maya-Rodriguez MC, Vega-Alvarado E, Corona-Ramírez LG. Inverse Kinematics: Identifying a Functional Model for Closed Trajectories Using a Metaheuristic Approach. Mathematics. 2025; 13(11):1847. https://doi.org/10.3390/math13111847
Chicago/Turabian StyleLópez-Muñoz, Raúl, Mario A. Lopez-Pacheco, Mario C. Maya-Rodriguez, Eduardo Vega-Alvarado, and Leonel G. Corona-Ramírez. 2025. "Inverse Kinematics: Identifying a Functional Model for Closed Trajectories Using a Metaheuristic Approach" Mathematics 13, no. 11: 1847. https://doi.org/10.3390/math13111847
APA StyleLópez-Muñoz, R., Lopez-Pacheco, M. A., Maya-Rodriguez, M. C., Vega-Alvarado, E., & Corona-Ramírez, L. G. (2025). Inverse Kinematics: Identifying a Functional Model for Closed Trajectories Using a Metaheuristic Approach. Mathematics, 13(11), 1847. https://doi.org/10.3390/math13111847