Abstract
In this paper we address the problem of designing energy minimizing collision-free maneuvers for multiple agents moving on a plane. We show that the problem is equivalent to that of finding the shortest geodesic in a certain manifold with nonsmooth boundary. This allows us to prove that the optimal maneuvers are C1 by introducing the concept of u-convex manifolds. Moreover, due to the nature of the optimal maneuvers, the problem can be formulated as an optimal control problem for a certain hybrid system whose discrete states consist of different “contact graphs”. We determine the analytic expression for the optimal maneuvers in the two agents case. For the three agents case, we derive the dynamics of the optimal maneuvers within each discrete state. This together with the fact that an optimal maneuver is a C1 concatenation of segments associated with different discrete states gives a characterization of the optimal solutions in the three agents case.
Research supported by NSF and DARPA. The authors would like to thank Ekaterina Lemch for the helpful discussions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
F. Albrecht and I.D. Berg. Geodesics in Euclidean space with analytic obstacle. Proceedings of the American Mathematics Society, 113(1):201–207, 1991.
R. Alexander and S. Alexander. Geodesics in Riemannian manifolds-withboundary. Indiana University Mathematics Journal, 30(4):481–488, 1981.
S.B. Alexander, I.D. Berg, and R.L. Bishop. Cauchy uniqueness in the Riemannian obstacle problem. In Lecture notes in Math., vol.1209, pag.1–7. Springer-Verlag, 1985.
A. Bicchi and L. Pallottino. Optimal planning for coordinated vehicles with bounded curvature. In Proc. Workshop on Algorithmic Foundations of Robotics, Dartmouth, 2000.
M.P. de Carmo. Riemannian geometry. Birkhäuser Boston, 1992.
N. Durand, J.M. Alliot, and J. Noailles. Automatic aircraft conflict resolution using genetic algorithms. In 11th Annual ACM Conf. on Applied Computing, 1996.
M. Erdmann and T. Lozano-Perez. On multiple moving objects (motion planning). Algorithmica, 2(4):477–521, 1987.
E. Frazzoli, Z.H. M ao, J.H. Oh, and E. Feron. Resolution of conflicts involving many aircraft via semidefinite programming. AIAA J. of Guidance, Control and Dynamics. To appear.
K. Fujimura. Motion planning in dynamic environments. Springer-Verlag, 1991.
J. Hu, M. Prandini, and S. Sastry. Optimal coordinated maneuvers for multiple agents moving on a plane. Technical report, Univ. of California at Berkeley, 2001.
J. Hu, M. Prandini, and S. Sastry. Optimal maneuver for multiple aircraft conflict resolution: a braid point of view. In IEEE 39th Conf. on Decision and Control, Sydney, Australia, 2000.
J. Hu and S. Sastry. Geodesics in manifolds with boundary: a case study. In preparation, 2001.
J. Kuchar and L.C. Yang. Survey of conflict detection and resolution modeling methods. In AIAA Guidance, Navig., and Control Conf., New Orleans, LA, 1997.
J.-C. Latombe. Robot motion planning. Kluwer Academic Publishers, 1991.
S.M. LaValle and S.A. Hutchinson. Optimal motion planning for multiple robots having independent goals. IEEE Trans. on Robotics and Aut., 14(6):912–925, 1998.
J. Lygeros, G.J. Pappas, and S. Sastry. An introduction to hybrid systems modeling, analysis and control. In Preprints of the First Nonlinear Control Network Pedagogical School, pag. 307–329, Athens, Greece, 1999.
P.K. Menon, G.D. Sweriduk, and B. Sridhar. Optimal strategies for free-flight air traffic conflict resolution. AIAA J. of Guidance, Control and Dynamics, 22(2):202–211, 1999.
A. Miele, T. Wang, C.S. Chao, and J.B. Dabney. Optimal control of a ship for collision avoidance maneuvers. J. of Optim. Theory and App., 103(3):495–519, 1999.
J.W. Milnor. Morse theory. Annals of mathematics studies, 51. Princeton University Press, 1963. Based on lecture notes by M. Spivak and R. Wells.
Radio Technical Commission for Aeronautics. Minimum aviation system performance standards for automatic dependent surveillance-broadcast (ADS-B). Technical report, RTCA-186, 1997. DRAFT 4.0.
S. Simic, K.H. Johansson, S. Sastry, and J. Lygeros. Towards a geometric theory of hybrid systems. Hybrid Systems: Computations and Control. Third International Workshop, 2000. In Lecture Notes in Computer Science, number 1790.
B. Vronov, M. de Berg, A.F. van der Stappen, P. Svestka, and J. Vleugels. Motion planning for multiple robots. Discrete & Comput. Geometry, 22(4):502–525, 1999.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hu, J., Prandini, M., Johansson, K.H., Sastry, S. (2001). Hybrid Geodesics as Optimal Solutions to the Collision-Free Motion Planning Problem. In: Di Benedetto, M.D., Sangiovanni-Vincentelli, A. (eds) Hybrid Systems: Computation and Control. HSCC 2001. Lecture Notes in Computer Science, vol 2034. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45351-2_26
Download citation
DOI: https://doi.org/10.1007/3-540-45351-2_26
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-41866-5
Online ISBN: 978-3-540-45351-2
eBook Packages: Springer Book Archive