A Measure Theoretical Approach for Path Planning Problem of Nonlinear Control Systems

.
DOI: 10.4236/ica.2011.22017   PDF   HTML     4,128 Downloads   6,781 Views   Citations

Abstract

This paper presents a new approach to find an approximate solution for the nonlinear path planning problem. In this approach, first by defining a new formulation in the calculus of variations, an optimal control problem, equivalent to the original problem, is obtained. Then, a metamorphosis is performed in the space of problem by defining an injection from the set of admissible trajectory-control pairs in this space into the space of positive Radon measures. Using properties of Radon measures, the problem is changed to a measure-theo- retical optimization problem. This problem is an infinite dimensional linear programming (LP), which is approximated by a finite dimensional LP. The solution of this LP is used to construct an approximate solution for the original path planning problem. Finally, a numerical example is included to verify the effectiveness of the proposed approach.

Share and Cite:

A. Jajarmi, H. Ramezanpour, M. Nayyeri and A. Kamyad, "A Measure Theoretical Approach for Path Planning Problem of Nonlinear Control Systems," Intelligent Control and Automation, Vol. 2 No. 2, 2011, pp. 144-151. doi: 10.4236/ica.2011.22017.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] J. M. Athans and P. L. Falb, “Optimal Control: An Introduction to the Theory and Its Applications,” McGraw- Hill, New York, 1996.
[2] A. T. Hasan, A. M. S. Hamouda, N. Ismail and H. M. A. A. Al-Assadi, “A New Adaptive Learning Algorithm for Robot Manipulator Control,” Proceedings of the Institution of Mechanical Engineers Part I-Journal of Systems and Control Engineering, Vol. 221, No. 4, 2007, pp. 663-672. doi:10.1243/09596518JSCE321
[3] A. S. Rana and A. M. S. Zalzala, “Collision-Free Motion Planning of Multi-Arm Robots Using Evolutionary Algorithms,” Proceedings of the Institution of Mechanical Engineers Part I-Journal of Systems and Control Engineering, Vol. 211, No. 5, 1997, pp. 373-384. doi:10.1243/0959651971539902
[4] C. L. Chen and C. J. Lin, “Motion Planning of Redundant Robot Manipulators Using Constrained Optimization: A Parallel Approach,” Proceedings of the Institution of Mechanical Engineers Part I-Journal of Systems and Control Engineering, Vol. 212, No. 4, 1998, pp. 281-292. doi:10.1243/0959651981539460
[5] Y. Wang, D. M. Lane and G. J. Falconer, “Two Novel Approaches for Unmanned under Water Vehicle Path Planning: Constrained Optimization and Semi-Infinite Constrained Optimization,” Robotica, Vol. 18, No. 2, 2000, pp. 123-142. doi:10.1017/S0263574799002015
[6] J. C. Latombe, “Robot Motion Planning,” Kluwer Academic Publishers, Boston, 1991.
[7] L. Kavarki, P. Svestka, J. Latombe and M. Overmars, “Probabilistic Road Maps for Path Planning in High Dimensional Configuration Space,” IEEE Transactions on Robotics and Automation, Vol. 12, No. 4, 1996, pp. 566-580. doi:10.1109/70.508439
[8] G. C. Luh and W. W. Liu, “Motion Planning for Mobile Robots in Dynamic Environments Using a Potential Field Immune Network,” Proceedings of the Institution of Mechanical Engineers Part I-Journal of Systems and Control Engineering, Vol. 221, No. 7, 2007, pp. 1033-1045. doi:10.1243/09596518JSCE400
[9] L. Kavarki, M. Kolountzakis and J. Latombe, “Analysis of Probabilistic Road Maps for Path Planning,” IEEE Transactions on Robotics and Automation, Vol. 14, No. 1, 1998, pp. 166- 171. doi:10.1109/70.660866
[10] Y. Koren and J. Borenstein, “Potential Field Methods and Their Inherent Limitations for Mobile Robot Navigation,” Proceedings of the IEEE Conference on Robotics and Automation, Sacramento, 1991. doi:10.1109/ROBOT.1991.131810
[11] P. C. Zhou, B. R. Hong and J. H. Yang, “Chaos Genetic Algorithm Based Path Planning Method for Mobile Robot,” Journal of Harbin Institute of Technology, Vol. 36, No. 7, 2004, pp. 880-883.
[12] Y. O. Qin, D. B. Sun, N. Li and Y. G. Cen, “Path Planning for Mobile Robot Using the Particle Swarm Optimization with Mutation Operator,” Proceedings of the 3rd International Conference on Machine Learning and Cybernetics, Shanghai, 2004.
[13] T. Cecil and D. E. Marthaler, “A Variational Approach to Path Planning in Three Dimensions Using Level Set Methods,” Journal of Computational Physics, Vol. 211, No. 1, 2006, pp. 179-197. doi:10.1016/j.jcp.2005.05.015
[14] M. G. Earl and R. Danderia, “Modelling and Control of a Multi-Agent System Using Mixed Integer Linear Programming,” Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, 10-13 December 2002. doi:10.1109/CDC.2002.1184476
[15] A. Richards and J. How, “Aircraft Trajectory Planning with Collision Avoidance Using Mixed Integer Programming,” Proceedings of the IEEE American Control Conference, Anchorage, Alaska, 2002.
[16] M. Gachpazan and A. V. Kamyad, “Solving of Second Order Nonlinear PDE Problems by Using Artificial Controls with Controlled Error,” Korean Journal of Computational & Applied Mathematics, Vol. 15, No. 1-2, 2004, pp. 173-184.
[17] S. A. Alavi, A. V. Kamyad and M. Gachpazan, “Solving of Nonlinear Ordinary Differential Equations as a Control Problem by Using Measure Theory,” Scientia Iranica, Vol. 7, No. 1, 2000, pp. 1-7.
[18] M. Gachpazan, A. Kerayechian and A. V. Kamyad, “A New Method for Solving Nonlinear Second Order Differential Equations,” Korean Journal of Computational & Applied Mathematics, Vol. 7, No. 2, 2000, pp. 333-345.
[19] A. Jajarmi, M. Gachpazan and A. V. Kamyad, “Open- loop Control of Nonlinear Systems via a Sequence of Nonlinear Programming Problems,” Proceedings of the 17th Iranian Conference of Electrical Engineering, Tehran, 13-15 August 2009.
[20] J. E. Rubio, “Control and Optimization, the Linear Treatment of Non-linear Problems,” Manchester University Press, Manchester, 1986.
[21] A. V. Kamyad, J. E. Rubio and D. A. Wilson, “An Optimal Control Problem for the Multidimensional Diffusion Equation with a Generalized Control Variable,” Journal of Optimization Theory and Applications, Vol. 75, No. 1, 1992, pp. 101-132. doi:10.1007/BF00939908
[22] A. V. Kamyad and A. H. Borzabadi, “Strong Controllability and Optimal Control of the Heat Equation with a Thermal Source,” Korean Journal of Computational & Applied Mathematics, Vol. 7, No. 3, 2002, pp. 555-568.
[23] A. H. Borzabadi, A. V. Kamyad and M. H. Farahi, “Optimal Control of the Heat Equation in an Inhomogeneous Body,” Applied Mathematics and Computation, Vol. 15, No. 1-2, 2004, pp. 127-146.
[24] D. Hinrichsen and A. J. Pritchard, “Mathematical System Theory I: Modeling, State Space Analysis, Stability and Robustness,” Springer, Berlin, 2005.
[25] A. V. Kamyad and H. H. Mehneh, “A Linear Programming Approach to the Controllability of Time-Varying Systems,” International Journal of Engineering Science, Vol. 14, No. 8, 2003, pp. 143-151.

  
comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.