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

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.

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.

Cite this paper

nullA. 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.

nullA. Jajarmi, H. Ramezanpour, M. Nayyeri and A. Kamyad, "A Measure Theoretical Approach for Path Planning Problem of Nonlinear Control Systems,"

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.

[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.