Layer, Lie algebraic method of motion planning for nonholonomic systems. (English) Zbl 1254.93064
Summary: In this paper a layer, Lie algebraic method of motion planning for nonholonomic systems is presented. It plans locally a motion towards a goal by searching for optimal directions in equi-cost spaces. The spaces are easy to determine via exploiting Lie algebraic properties of vector fields that define the controlled system. The method is illustrated on the unicycle robot and the inverted pendulum.
MSC:
| 93B25 | Algebraic methods |
| 93C15 | Control/observation systems governed by ordinary differential equations |
| 93C85 | Automated systems (robots, etc.) in control theory |
Keywords:
algebraic method of motion planning for nonholonomic systems; Lie algebraic properties of vector fields; optimal directions in equi-cost spaces; unicycle robot; inverted pendulumReferences:
| [1] | Brockett, R. W., Control theory and singular Riemannian geometry, (Hilton, P. J.; Young, G. S., New Directions in Applied Mathematics (1981), Springer: Springer Berlin), 11-27 · Zbl 0483.49035 |
| [2] | Chow, W. L., Uber Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Mathematische Annalen, 117, 1, 98-105 (1939) · Zbl 0022.02304 |
| [3] | Divelbiss, A. W.; Wen, J. T., Nonholonomic path planning with inequality constraints, (Proceedings of the IEEE Conference on Robotics and Automation (1994)), 52-57 |
| [4] | Duleba, I., On a computationally simple form of the Generalized Campbell-Baker-Hausdorff-Dynkin formula, Systems and Control Letters, 34, 191-202 (1998) · Zbl 0909.93021 |
| [5] | Duleba, I., Locally optimal motion planning of nonholonomic systems, Journal of Robotic Systems, 14, 11, 767-788 (1997) · Zbl 0893.70006 |
| [6] | I. Duleba, W. Khefifi, Pre-control form of the gCBHD formula and its application to nonholonomic planning motion, in: Proceedings of the IFAC Symposium on Robot Control, Wroclaw, 2003, pp. 1780-1785.; I. Duleba, W. Khefifi, Pre-control form of the gCBHD formula and its application to nonholonomic planning motion, in: Proceedings of the IFAC Symposium on Robot Control, Wroclaw, 2003, pp. 1780-1785. |
| [7] | Duleba, I.; Khefifi, W., Pre-control form of the gCBHD formula for affine nonholonomic systems, Systems and Control Letters, 55, 2, 146-157 (2006) · Zbl 1129.93353 |
| [8] | Fernandes, C.; Gurvits, L.; Li, Z., Optimal nonholonomic motion planning for falling cat, (Li, Z.; Canny, J., Nonholonomic Motion Planning (1993), Kluwer Academic Publ.), 379-421 · Zbl 0795.92005 |
| [9] | Fliess, M.; Rouchon, P.; Lévine, J.; Martin, P., Flatness, motion planning and trailer systems, (Proceedings of the Conference on Decision Control (1993)), 2700-2705 |
| [10] | Ghommam, J.; Hehrjerdi, H.; Mnif, F.; Saad, M., Cascade design for formation control of nonholonomic systems in chained form, Journal of the Franklin Institute, 348, 6, 973-998 (2011) · Zbl 1222.93154 |
| [11] | Hermann, R.; Krener, A., Nonlinear controllability and observability, IEEE Transactions on Automatic Control, 22, 5, 728-740 (1977) · Zbl 0396.93015 |
| [12] | Hermes, H., Lie algebras of vector fields and local approximation of attainable sets, SIAM Journal on Control and Optimization, 16, 6, 715-728 (1978) · Zbl 0388.49025 |
| [13] | Hermes, H., On the synthesis of a stabilizing feedback control via Lie algebraic method, SIAM Journal on Control and Optimization, 18, 6, 352-361 (1980) · Zbl 0477.93046 |
| [14] | Jurdjevic, V., Geometric Control Theory (1997), Cambridge University Press · Zbl 0940.93005 |
| [15] | G. Lafferriere, H. Sussmann, Motion Planning for Controllable Systems without Drift, in: Proceedings of the IEEE Conference on Robotics and Automation, Sacramento, 1991, pp. 1148-1153.; G. Lafferriere, H. Sussmann, Motion Planning for Controllable Systems without Drift, in: Proceedings of the IEEE Conference on Robotics and Automation, Sacramento, 1991, pp. 1148-1153. |
| [16] | Laumond, J. P., A motion planner for nonholonomic mobile robots, IEEE Transactions on Robotics and Automation, 10, 5, 577-593 (1994) |
| [17] | Laumond, J. P., Robot Motion Planning and Control (1998), Springer Verlag |
| [18] | Li, Z.; Canny, J. F., Nonholonomic Motion Planning (1993), Kluwer Academic Publ. |
| [19] | Lian, K.; Wang, L.; Fu, L., Controllability of spacecraft systems in a central gravitational field, IEEE Transactions on Automatic Control, 39, 12, 2426-2440 (1994) · Zbl 0825.93069 |
| [20] | Lynch, K.; Black, C. K., Recurrence, controllability, and stabilization of juggling, IEEE Transactions on Robotics and Automation, 17, 2, 113-124 (2001) |
| [21] | Murray, R. M.; Sastry, S., Nonholonomic motion planning: steering using sinusoids, IEEE Transactions on Automatic Control, 38, 5, 700-716 (1993) · Zbl 0800.93840 |
| [22] | Pomet, J. B., Explicit design of time-varying stabilizing control laws for a class of controllable systems without drift, Systems and Control Letters, 18, 147-158 (1992) · Zbl 0744.93084 |
| [23] | Rahideh, A.; Shaheed, M. H., Stable model predictive control for a nonlinear system, Journal of the Franklin Institute, 348, 8, 1983-2004 (2011) · Zbl 1231.93031 |
| [24] | Schutte, A. D., Permissible control of general constraint mechanical systems, Journal of the Franklin Institute, 347, 1, 208-227 (2010) · Zbl 1298.93065 |
| [25] | Serre, J. P., Lie Algebras and Lie Groups (1965), Benjamin: Benjamin New York · Zbl 0132.27803 |
| [26] | Spong, M.; Vidyasagar, M., Robot Dynamics and Control (1989), MIT Press: MIT Press Cambridge |
| [27] | Sussmann, H. J.; Jurdjevic, V., Controllability of Nonlinear System (1977), Springer-Verlag: Springer-Verlag New York |
| [28] | Strichartz, R. S., The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations, Journal of Functional Analysis, 72, 320-345 (1987) · Zbl 0623.34058 |
| [29] | Tachwali, Y.; Refai, H. H., System prototype for vehicle collision avoidance using wireless sensors embedded at intersections, Journal of the Franklin Institute, 346, 5, 488-499 (2009) · Zbl 1167.93306 |
| [30] | Wai, R-J.; Liu, C-M.; Lin, Y-W., Design of switching path-planning control for obstacle avoidance of mobile robot, Journal of the Franklin Institute, 348, 4, 718-737 (2011) · Zbl 1227.93087 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
