1. 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

2. Collection of
survey papers

3. Dubins´ car

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

5. Classical homicidal chauffeur game,

6. Reinforced homicidal chauffeurP , :

7. Report by
R. Isaacs, 1951

8. Book by
R.Isaacs, 1965

9. Isaacs´ transformationx h P y

10. Time-limited reachable sets, ,
Wave front M M

11. Level sets of the optimal time function

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

13. Computation scheme F3 F2 F1 G0 M

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

15. Reachable sets for reinforced car
Isaacs´ |
transform. ,

16. Dubins´ car

17. 683

22. Reeds and Shepp´s car

24. Reachable sets at a given time for a > 0

26. M

27. M

28. M

29. Time-limited reachable sets,

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

31. M

32. M

33. Computation scheme for two usable parts

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

35. Families of semipermeable curves for reinforced car1 a 1 -1 a -1 a

36. Barrier lines for Dubins´ cary -1 1 x -1 1 -1 1 -1 1

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