1. NUS CS5247 Motion Planning for Car- like Robots using a Probabilistic Learning Approach --P. Svestka, M.H. Overmars. Int. J. Robotics Research, 16:119-143, 1997. Presented by: Li Yunzhen

2. NUS CS5247 Papers Motivation & Organization Motivation build a non-redundant of milestones (randomized), apply non-holonomic constraints for car-robot to do multi-query processing Organization 1.Two types of Car Robots and nonholonomic constraints 2.Probabilistic Roadmap 3.Application of Forest uniform Sampling in General Car-like Robot 4.Application of Directed Graph uniform Sampling in Forward Car- like Robot 5 Summary

3. NUS CS5247 1.Car-Like Robots: Configuration Configuration Space: Front point F Rear point R Maximal steering angle configuration

4. NUS CS5247 1.Car-Like Robots Translational motion: along main axis Rotational motion: around a point on As perpendicular axis. Rotational angle is decided by forward and backward motion

5. NUS CS5247 1. Holonomic Constraints--Free flying robot Its motions are of a holonomic nature infinitesimal motion in Cfree-space can be achieved Thus, path independent

6. NUS CS5247 1 Nonholonomic Constraints the number of degrees of freedom of motion is less than the dimension of the configuration space Path dependent (collision-free path not always feasible)

7. NUS CS5247 1.Nonholonomic Constraints Forward car-like Robot Start Not possible for forward Car-like Robot Path Dependent

8. NUS CS5247 1. Car-Like Robot 1. Nonholonomic Car-Like Roboty x L q = (x,y, ) q= dq/dt = (dx/dt,dy/dt,d /dt) dx sin – dy cos = 0 is a particular form of f(q,q)=0 A robot is nonholonomic if its motion is constrained by a non- integrable equation of the form f(q,q) = 0 dx/dt = v cos dy/dt = v sin d dt = (v/L) tan | < dx sin – dy cos = 0

9. NUS CS5247 1. Car-Like Robot 1. Nonholonomic Car-Like Roboty x L Upper bound turning angle =>Lower-bounded turning radius Rmin = Lctg dx/dt = v cos dy/dt = v sin d dt = (v/L) tan | < dx sin – dy cos = 0

10. NUS CS5247 1.Two Types of car-like Robots under Non-Holonomic Constraints Normal Car-like Robot: Move Forwards & Backwards, (Bounded) turn, cannot move sidewise Forwards Car-like Robot: Move Forwards, (Bounded) turn, cannot move sidewise

11. NUS CS5247 2. Probabilistic Roadmap Learning Phase: Local Method: used to compute a feasible path for connection of 2 nodes. deterministic & terminative Metric: determine the distance of 2 nodes Edge adding Methods: Cycle detection & try to connect nodes within maximum dist to avoid failure Query Phase: start from start position and goal position, do random walk For Holonomic Constraints, Local method can return any path as long as it does not intersects with obstacles. (Local method returns line-segments in Lecture notes)

12. NUS CS5247 2.Forest Uniform Sampling Non-redundant Property: From one node to another node, there is only one or no path

13. NUS CS5247 2. Directed Graph uniform sampling Similar to Forest Sampling. Redundant Checking: An edge e=(a,b) in a Graph G=(V,E) is redundant iff there is a directed path from a to b in the graph G=(V,E-e).

14. NUS CS5247 3.Apply Undirected graph to general car-like robot Link method: constructs a path connecting its argument configurations in the absence of obstacles, and then test whether this path intersects any obstacles. RTR path: concatenation of an extreme rotational path, a translational path, and another extreme rotational path.

15. NUS CS5247 3.Apply Undirected graph to general car-like robot Two RTR paths for a triangular car-like robot, connecting configurations a,b RTR link method: given two argument configurations a and b, if the shortest RTR path connecting a to b intersects no obstacles, return the path, else return failure. RTR metric (DRTR): distance between two configurations is defined as the length of the shortest RTR path connecting them.

16. NUS CS5247 3.Apply Undirected graph to general car-like robot---Query phase Nw: maximal number of walks Lw: maximal length of the walk( used for upper bound of RTR metric) Use these two constraints to upper-bound the random walk

17. NUS CS5247 3.General car-like robot: Node Adding Strategy Random Node Adding Non-Random Node Adding: guiding the node adding by the geometry of the workspace

18. NUS CS5247 3.General car-like robot: guiding the node adding by the geometry of the workspace Random Node adding strategy 1.Compute Geometry Configurations at important position, e.g. along edges, next to vertices of obstacles. Each edge and convex vertex defines two such geo- configurations.

19. NUS CS5247 3.General car-like robot: guiding the node adding by the geometry of the workspace 2. Add configurations from Geo-Configuration set (just computed) in a random order to the graph, but discard those are not free. 3. Learning Process can be continued by adding random nodes.

20. NUS CS5247 3.General car-like robot: Experiments(1) Experimental Set up: Random Walk parameter: Nw=10 Lw=0.05 So time spend on per query is bounded by 0.3 s. Minimal turning radius: Rmin = 0.1 Neighborhood size: Maxdist =0.5 The percentage number in the table shows how many percent of trials of query is solved.

21. NUS CS5247 3.General car-like robot: Experiments(1) The lower left table gives results for geometric node adding, the table at the lower right for random node adding.

22. NUS CS5247 3.General car-like robot: Experiments(2) The lower left table gives results for geometric node adding, the table at the lower right for random node adding

23. NUS CS5247 3.General car-like robot: Experiments(3) The lower left table gives results for geometric node adding, the table at the lower right for random node adding

24. NUS CS5247 3.General car-like robot: Experiments(4) Parking with large minimal turning radii. In the left case r min is 0.25 and in the right case 0.5

25. NUS CS5247 4.Forward car-like robot RTR forward path: the concatenation of extreme forward rotational path, a forward translational path and another extreme forward rotational path. RTR forward link method: RTR link method + direction Metric (RTR forward metric): RTR metric+direction

26. NUS CS5247 4.Forward car-like robot Why do we need to build directed graph? The red RTR path does not suitable for forward car-like. So directed edge refers to directed RTR path.

27. NUS CS5247 4.Forward car-like robot The table gives result for random node adding

28. NUS CS5247 4.Forward car-like robot The table gives result for geometric adding

29. NUS CS5247 5.Summary Apply Non-redundant Graph roadmap for the motion of car-like robots. Why not build redundant graph roadmap? --After smoothing, redundant graph and non- redundant graph will general similar results.

30. NUS CS5247 Q&A ?