From Dubins´ car to Reeds and Shepp´s mobile robot Valerii S. Patsko Institute of Mathematics and Mechanics Ural Branch of the Russian Academy of Sciences, Ekaterinburg Varvara L. Turova Center of Advanced European Studies and Research, Bonn 7th World Congress on Computational Mechanics, Los Angeles, July 16-22, 2006

Collection of survey papers

Dubins´ car

a is a fixed parameter of the problem, Reeds and Shepp´s car P a is a fixed parameter of the problem,

Classical homicidal chauffeur game ,

Reinforced homicidal chauffeur P , :

Report by R. Isaacs, 1951

Book by R.Isaacs, 1965

Isaacs´ transformation x h P y

Time-limited reachable sets , , Wave front M M

Level sets of the optimal time function

Construction of time-limited reachable sets Step-by-step procedure Usable part

Computation scheme F3 F2 F1 G0 M

Construction of a wave front Local-convexity Local-concavity 1. 2.

Reachable sets for reinforced car Isaacs´ | transform. ,

Dubins´ car

683

Reeds and Shepp´s car

Reachable sets at a given time for a > 0

M

M

M

Time-limited reachable sets,

Level sets of the value function for the reinforced homicidal chauffeur Isaacs´ transform. , More complicated structure of barrier lines and wave fronts

M

M

Computation scheme for two usable parts

Roots of the Hamiltonian. Semipermeable directions , _ + _ +

Families of semipermeable curves for reinforced car 1 a 1 -1 a -1 a

Barrier lines for Dubins´ car y -1 1 x -1 1 -1 1 -1 1

Hamilton-Jacobi equation for optimal time function in case of reinforced car Hamilton-Jacobi equation for value function of the reinforced homicidal chauffeur game