Shortest Dubins paths through three points

The 3-Point Dubins Problem (3PDP) consists of steering a Dubins vehicle through three consecutive points with prescribed heading orientations at initial and final points so that the resulting path is the shortest. Characterizing the path of the 3PDP is important because (1) it provides insightful views on the solution paths of the Dubins Traveling Salesman Problem (DTSP) and the Curvature-Constrained Shortest-Path Problem (CCSPP) as they are natural extensions of the 3PDP and (2) some algorithms in the literature for solving the DTSP and the CCSPP require efficient methods for solving the 3PDP. In this paper, Pontryagin’s maximum principle is used to show that the path of 3PDP must lie in a sufficient family of 18 types. Moreover, a formula in terms of the parameters of the 3PDP for all the 18 types is established, and this formula reveals the relationship between the unknown orientation angle at mid point and known parameters. By observing that the formula can be converted into some polynomials, the 3PDP can therefore be efficiently solved by finding zeros of those polynomials. Finally, numerical simulations illustrate the developments by comparing with the straightforward discretization-based method.

[1]  L. Dubins.  On Curves of Minimal Length with a Constraint on Average Curvature, and with Prescribed Initial and Terminal Positions and Tangents , 1957 .

[2]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems , 1998, JACM.

[3]  Pramod P. Khargonekar,et al.  An Approximation Algorithm for a Shortest Dubins Path Problem , 2016, ArXiv.

[4]  Emilio Frazzoli,et al.  On the Dubins Traveling Salesman Problem , 2012, IEEE Transactions on Automatic Control.

[5]  Sung Yong Shin,et al.  Approximation of Curvature-Constrained Shortest Paths through a Sequence of Points , 2000, ESA.

[6]  Tal Shima,et al.  Discretization-Based and Look-Ahead Algorithms for the Dubins Traveling Salesperson Problem , 2017, IEEE Transactions on Automation Science and Engineering.

[7]  Eugene L. Lawler,et al.  The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization , 1985 .

[8]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[9]  Stephen L. Smith,et al.  On efficient computation of shortest Dubins paths through three consecutive points , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[10]  Nicholas J. Higham,et al.  INVERSE PROBLEMS NEWSLETTER , 1991 .

[11]  Tal Shima,et al.  Motion planning algorithms for the Dubins Travelling Salesperson Problem , 2015, Autom..

[12]  Ümit Özgüner,et al.  Motion planning for multitarget surveillance with mobile sensor agents , 2005, IEEE Transactions on Robotics.

[13]  R. Bellman.  Dynamic programming. , 1957, Science.

[14]  C. Yalçın Kaya,et al.  Markov–Dubins path via optimal control theory , 2017, Comput. Optim. Appl..

[15]  R. Takei,et al.  A practical path-planning algorithm for a simple car: a Hamilton-Jacobi approach , 2010, Proceedings of the 2010 American Control Conference.

[16]  Jean-Daniel Boissonnat,et al.  Shortest path synthesis for Dubins non-holonomic robot , 1994, Proceedings of the 1994 IEEE International Conference on Robotics and Automation.

[17]  Andrey V. Savkin,et al.  A method for guidance and control of an autonomous vehicle in problems of border patrolling and obstacle avoidance , 2011, Autom..

[18]  Xavier Goaoc,et al.  Bounded-Curvature Shortest Paths through a Sequence of Points Using Convex Optimization , 2013, SIAM J. Comput..

[19]  Xin Yu,et al.  A genetic algorithm for the Dubins Traveling Salesman Problem , 2012, 2012 IEEE International Symposium on Industrial Electronics.

[20]  Stephen J. Wright.  Coordinate descent algorithms , 2015, Mathematical Programming.

[21]  Xiang Ma,et al.  Receding Horizon Planning for Dubins Traveling Salesman Problems , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[22]  Rogelio Lozano,et al.  Dubins path generation for a fixed wing UAV , 2014, 2014 International Conference on Unmanned Aircraft Systems (ICUAS).

[23]  George M. Siouris,et al.  Applied Optimal Control: Optimization, Estimation, and Control , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

[24]  Ryo Takei,et al.  Optimal Trajectories of Curvature Constrained Motion in the Hamilton–Jacobi Formulation , 2013, J. Sci. Comput..

[25]  Tal Shima,et al.  On the discretized Dubins Traveling Salesman Problem , 2017 .

[26]  Kevin M. Lynch.  Optimal control of the thrusted skate , 2003, Autom..

[27]  Tal Shima,et al.  Integrated task assignment and path optimization for cooperating uninhabited aerial vehicles using genetic algorithms , 2011, Comput. Oper. Res..

[28]  Raja Sengupta,et al.  A Resource Allocation Algorithm for Multivehicle Systems With Nonholonomic Constraints , 2007, IEEE Transactions on Automation Science and Engineering.