Table of contents
Problem
For specified a pair of start and final positions and orientations, what is the timeoptimal trajectory to link them for a mobile robot with its dynamic and kinematic constraints in the unobstructed plane?
The motivations for studying this problem are both for practical applications and for theoretical challenges. For practical applications, an effective algorithm or the knowledge on structures of the fastest paths can help for the effective design of motion planning and control in the obstructed environment.
For theoretical interest, the analysis of this problem was a significant example for the development of modern geometric optimal control theory Sussmann91. Moreover, from studying other types of robots, does there exist a generic mechanism or technique for the determination of the time optimal trajectories for such these systems? What are the characteristics of the systems
which is able to be addressed by this generic mechanism and
technique?
Methodology
 Before using Pontryagin Maximum Principle (PMP)
 Basic methods
 Local reasoning: by modern geometric optimal control theory= Pontryagin Maximum Principle(PMP)+Lie algebra
the sufficient family: the set contains all optimal trajectories types between any two configurations.
cf.Sussmann91
 Global reasoning: by the partition of the configuration space
optimal synthesis: the particular optimal trajectory type between two specified configurations.
cf.Souères96
 Local reasoning: by modern geometric optimal control theory= Pontryagin Maximum Principle(PMP)+Lie algebra
 More techniques
But the algorithms for determination of the exact optimal trajectory between two specified configurations need more techniques.

 For differential drive robots, given start and final configurations, the optimal trajectory type can be easily identified, and then the corresponding exact optimal trajectory can be determined easily. But for another wheeled robots, the optimal trajectory type cannot be easily determined, even when the optimal trajectory type is known, the corresponding exact optimal trajectory is not easily determined. cf.Balkcom02
 For steered carlike robots, differential drive robots and omnidirectional robots, the optimal trajectories can be described by motion of the robot relative to a switching line in the plane. But how to determine this switching line is an open problem. cf.Balkcom06
 For steered carlike robots, local reasoning focuses on the rotation rules for a defined switchingrelated vector in a special coordinate system which is built according to properties of the switching structure. A proposition is derived: if the start and final positions and rotation directions of this vector are known, then the exact optimal trajectory is uniquely determined directly. Does there exist such proposition for other wheeled robots? If such proposition exists, is it easy or feasible to determine the start and final switching vectors? cf.Wang07
Subproblems
Completed problems
An algorithm has been achieved to determine an exactly timeoptimal trajectory to link any specified start and final configurations.
Such algorithms only are presented for steered cars and differential drive robots with kinematic constraints and bounded velocities.
Link to:
 Timeoptimal trajectories for a differential drive robot
 Timeoptimal trajectories for a carlike robot
Wheeled robots  Admissible control  
Differential drive robots
The locomotion system of differential drive robots constitutes by two parallel driving wheels each being controlled independently. 

Steered Carlike robots
The steered carlike robot has the steering wheels which control the direction of vehicle. Timeoptimal trajectories for a carlike robot Left are four kinds of the steered carlike robots with different admissible controls which are bounded. Dubins, RS and Simple car have a bound on its angle of steering. RS, Simple car and CRS have the equivalent timeoptimal 

Problems under consideration
 The timeoptimal trajectories for omnidirectional robots: a complete and minimal classification of the optimal trajectories are presented. Future: What is the timeoptimal trajectories for specified start and final configurations.cf.Balkcom06
 The timeoptimal trajectories for Hilarelike mobile robots: cf.Soueres98
 The timeoptimal trajectories for the Dubins airplane: cf.Chitsaz07
Open problems
 Are there techniques, which combined with modern geometric optimal control theory, can obtain the timeoptimal trajectories for the wheeled robots with different constrains.
 If such techniques exist, what are the characteristics of a system such that these techniques can be used to achieve the optimal trajectories?
References
L. E. Dubins. On curves of minimal length with a constraint on average curvature and with prescribed initial and terminal positions and tangents. American Journal of Mathematics, 79:497516, 1957.
J. A. Reeds and L. A. SheppOptimal paths for a car that goes both forwards and backwards. Pacific Journal of Mathematics, 145(2):367393, 1990.
H. Sussmann, G. Tang,Shortest Paths for the ReedsShepp Car: a Worked Out Example of the Use of Geometric Techniques in Nonlinear Optimal Control, Technical Report SYNCON 9110, Dept. of Mathematics, Rutgers University, Piscataway, NJ, 1991.
P. Souères and J.P. Laumond,
Shortest Paths Synthesis for a CarLike Robot. IEEE Transactions on Automatic Control, vol. 41, no. 5, 1996, pp 672688.
P. Souères and J.D. Boissonnat. Optimal trajectories for nonholonomic mobile robots. In J.P. Laumond, editor, Robot Motion Planning and Control, pages 93170. Springer, 1998.
Subproblems
D. Balkcom and M. Mason.
Time optimal trajectories for differential drive vehicles.International Journal of Robotics Research, 21(3):199217, 2002.
D. Balkcom and M. Mason.
Extremal trajectories for bounded velocity mobile robots. In IEEE International Conference on Robotics and Automation, 2002.
D. Balkcom and M. Mason.
Time optimal trajectories for bounded velocity differential drive robots. In IEEE International Conference on Robotics and Automation, 2000.
D. Balkcom and M. Mason.Timeoptimal Trajectories for an Omnidirectional Vehicle,The International Journal of Robotics Research, Vol. 25, No. 10, 985999 (2006)
H. Chitsaz and S. M. LaValle.
Timeoptimal paths for a Dubins airplane. In Proceedings IEEE Conference Decision and Control, New Orleans, LA, USA, Dec. 1214, 2007 pp:23792384.
H.Wang, Y. Chen and P. Souères, An Efficient Algorithm Involving the Canonical Switching Structure to Compute Minimumlength Trajectories for a Car, submitted to IEEE Transactions on Robotics.
H. Wang, Y. Chen and P. Souères, An Efficient Geometric Algorithm to Compute TimeOptimal Trajectories for a CarLike Robot, Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, LA, USA, Dec. 1214, 2007 pp:53835388.
Discussions and comments
 So far we only consider the wheeled robots. What are the
problems of the fastest paths for other kinds of robots?  Please give your opinions and
advice on the considering problems and open problems.
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