EXAMPLES OF PRACTICAL ALGORITHMS FOR PATH PLANNING

1. EXAMPLES OF PRACTICAL ALGORITHMS FOR PATH PLANNING

2. Bug Algorithms

3. Bug 2

4. Hj Lj Bugs 2 qtarget M-line New leave point condition: d<d(Hj,Target) qinit

5. qtarget M-line Hj Lj • From point Lj-1 move along M-line until: • Target is reached. Stop • An obstacle is hit at Hj. Goto 2 • Turn left and follow the boundary until: • Target is reached. Stop • M-line met at distance d from target such that: • d < dist(Hj,qtarget) • Define Lj, set j=j+1, and goto 1. • If we return to Hj without meeting the M-line, stop. The target is trapped. New leave point condition: d<d(Hj,Target) qinit

6. Bug 2 Algorithm • From point Lj-1 move along M-line until: • Target is reached. Stop • An obstacle is hit at Hj. Goto 2 • Turn left and follow the boundary until: • Target is reached. Stop • M-line met at distance d from target such that: d < dist(Hj,qtarget) Define Lj, set j=j+1, and goto 1. • If we return to Hjwithout meeting the M-line, stop. The target is trapped.

7. A Hard Example for Bug 2

8. A Hard Example for Bug 2

9. Bug + Noncontact Sensors • Suppose the robot is provided with info within a disk of radius r • E.g.: vision, infra-red, ultrasonic • The robot reconstructs at each point, the path that it would generate with no info • Result: smooth version of previous paths

10. Bug + Non Contact Sensor

11. Probabilistic Roadmaps

12. Probabilistic Roadmaps • Path planning algorithm are expensive. Search on small grid cannot be used. • If #DOFs’ is large, deterministic algorithms are not effective • Solution: use a probabilistic roadmap planner (PRM). • PRM are complete in a probabilistic sense • An heuristic planner: • potential fields. • Disadvantage: • local minima. • Solution: • randomize to escape local minima

13. Description of Probabilistic Roadmap Algorithm • Roadmap = undirected graph R = (N, E ) • N : (nodes) set of selected configurations in Cfree • E : (edges) collection of simple paths. The Local Paths • Local paths are computed by the fast but not powerful local planner • Idea: connect qinitand qgoalwith qinit’and qgoal’ in N • Search Rfor a path

14. Preprocessing Phase Three stages • Roadmap construction. Objectives: • Obtain reasonable connected graph • Be sure “difficult regions contain a few nodes • Roadmap expansion. Objectives: Improve graph connectivity by selecting nodes of R which lie in (heuristic) difficult regions and adding nodes there • Roadmap Component reduction. Optional. Attempts to simplify the graph

15. Roadmap Construction • Initially R is empty • Repeatedly, generate & add a free configuration to R • For every new q, select some nodes in R and try to connect them to q using local planner • On success, add the edge (q, q’ ) to E : Cfree× Cfree {0,1} is a local function returned by local planner D: pseudo metric in Cfree

16. Create random configurations AS a first step we create random configurations in the space. The algorithm creates the points one at a time, but we’re not going to do that with the slides. As configurations are created we try to connect to already existing nodes in the graph (if they are close enough they will get connected)

17. Update Neighboring Nodes’ Edges As edges are added to the graph we start forming connected regions.

18. Connected nodes End of Construction Step At the end of the construction step we end up with one or more connected components. We can probably improve connectivity by adding more nodes in the expansion stage at the difficult areas of the space.

19. Expansion Step The nodes are added and will be connected again using the local planner on the closest nodes.

20. End of Expansion Step At the end of the expansion step we end up with the same number, or fewer connected components.

21. Select start and goal Start Goal We represent our start and end goal

22. Connect Start and Goal to Roadmap Start Goal Find the nodes in the graph that are closest to the start and goal and connect then with edges.

23. Find the Path from Start to Goal Start Goal Now simply search for a path.

24. Roadmap construction algorithm • Nø , Eø • Loop • q  randomly selected free configuration • Nqa set of candidate neighbors of q from N • N = N  {q} •  q’ Nq in order of increasing D(q,q’ ), do • If q, q’ are not in the same connected comp. and(q,q’) =1, then EE  {(q, q’ )} • Update R’s connected components How? Meaning? Select Algorithm?

25. Creation of Random Configurations • Nodes of R = uniform random sampling of Cfree • Obtain q by drawing each of its coordinates. • From allowed interval for the corresponding DOF • Check q for collisions • If q is collision-free, add it to N. Otherwise discard

26. Ideas for the Local Planner • Deterministic vs. Non-deterministic planner • Powerful planer = slower planner • Empirical results: use deterministic and simple but very fast planner • Example: connect two configurations by a straight line in configuration space • Check for collisions by bisection

27. Distance Function D • D is used for constructing & sorting Nq • Hopefully, D reflect the chance that the local planner will fail • Simple selection that works in practice: D(q,q’)=maxxRobot||x(q)-x(q’)||

28. Roadmap Expansion • N should be a fair enough covering of Cfree • Roften consists of a few large components and several small ones • Expansion = add nodes to form a large component with as many nodes as possible • Try to cover the “difficult” parts of Cfree • Define weight w(q). Large wq is in a difficult region • Add M nodes to the collection

29. End of slides