Administration

1. Administration • Feedback on assignment • Late Policy • Some Matlab Sample Code • makeRobot.m, moveRobot.m

2. Introduction to Motion Planning CSE350/450-011 16 Sep 03

3. Class Objectives • Cell decomposition procedures for motion planning • Review exact approaches from last week • Introduce approximating approaches • A* Algorithm • Introduction to potential field approaches to motion planning

4. Supporting References • Motion Planning • “Motion Planning Using Potential Fields”, R. Beard & T. McClain, BYU, 2003 *** PRINT THIS OUT FOR NEXT TIME *** • “Robot Motion Planning”, J.C. Latombe, 1991

5. The Basic Motion Planning Problem Given an initial robot position and orientation in a potentially cluttered workspace, how should the robot move in order to reach an objective pose?

6. Planning Approaches • Roadmap Approach: • Visibility Graph Methods • Cell Decomposition: • Exact Decomposition • Approximate: Uniform discretization & quadtree approaches • Potential Fields

7. The Visibility Graph Method GOAL START

8. The Visibility Graph Method (cont’d) GOAL START

9. GOAL START The Visibility Graph Method (cont’d)

10. GOAL START The Visibility Graph Method (cont’d)

11. The Visibility Graph Method (cont’d) We can find the shortest path using Dijkstra’s Algorithm GOAL Path START

12. 13 12 10 11 9 7 6 5 4 2 3 1 8 Exact Cell Decomposition Method GOAL START 1) Decompose Region Into Cells

13. 13 12 10 11 9 7 6 5 4 2 3 1 8 Exact Cell Decomposition Method (cont’d) GOAL START 2) Construct Adjacency Graph

14. 10 12 1 2 7 6 9 Exact Cell Decomposition Method (cont’d) GOAL START • Construct Channel from shortest cell path • (via Depth-First-Search)

15. Exact Cell Decomposition Method (cont’d) GOAL START 4) Construct Motion Path P from channel cell borders

16. Exact Cell Decomposition Method Using Euclidean Metric GOAL START

17. Exact Cell Decomposition Method Using Euclidean Metric GOAL START

18. Perform cell tessellation of configuration space C • Uniform or quadtree • Generate the cell graph G(V,E) • Each cell in Cfree corresponds to a vertex in V • Two vertices vi,vj V are connected by an edge eij if they are adjacent (8-connected for exact) • Edges are weighted by Euclidean distance • Find the shortest path from vinit to vgoal Approximate Cell Decomposition

19. Cell Decomposition Uniform Quadtree

20. Continuity of trajectory a function of resolution • Computational complexity increases dramatically with resolution • Inexactness. Is this cell an obstacle or in Cfree? r Neighbors Cell Decomposition Issues

21. Explores G(V,E) iteratively by following paths originating at the initial robot position vinit • For each OPEN node, only keep track of its minimum weight path from vinit • The set of all such paths forms a spanning treeT  Gof the vertices OPEN so far • Define a cost function f(v) = g(v) + h(v) where • g(v) is the distance from vinit to v • h(v) is a heuristic estimate of the distance from v to vgoal • Define a cost function k(v1,v2) = g(v1) + distance(v1,v2) So how do we find the shortest path?A* Algorithm [Hart et al, 1968]

22. Define a list OPEN with the following operations • insert(OPEN, v) inserts node v into the list • delete(OPEN, v) removes node v from the list • minF(OPEN) returns the node v of minimum weight f(v) from OPEN, and removes it from the list • isMember(OPEN, v) returns TRUE if v is in the list, false otherwise • isEmpty(OPEN) returns TRUE if the list is empty, false otherwise • Define the following operations over our spanning tree T • insert(T, v2, v1) inserts node v2 into the tree with ancestor v1 • delete(T, v) removes node v from the tree A* Algorithm (cont’d)

23. A* - The Algorithm (Latombe, 1991) function A*(G, vinit, vgoal, h, k) insert(T, vinit, nil); % vinit is the tree root insert(OPEN, vinit); while isEmpty(OPEN)==FALSE v = minF(OPEN); % optimal node from f(v) metric if v==vgoal break; end for vPlus  adjacent(v) % All of the cells adjacent to cell v if vPlus is NOT visited insert(T, vPlus, v); % Expansion of spanning tree insert(OPEN, vPlus); else if g(vPlus)>g(v)+k(v,vPlus) % Better way to reach vPlus if true delete(T, vPlus); insert(T, vPlus, v); % Change ancestor to reflect better path if isMember(OPEN, vPlus) delete(OPEN, vPlus); end insert(OPEN, vPlus); % Change f(v) value to reflect better path end end end % End while loop if v==vgoal return the path constructed from vgoal to vinit in T else return failure; % We didn’t find a valid path end

24. The algorithm requires an admissible heuristich(v) where h*(v) is the optimal path distance from v to vgoal • For admissible h(v) A* is guaranteed to generate a minimum cost path from vinit to vgoal for the given cell decomposition • QUESTION: What would be a reasonable function choice for h(v)? • When h(v)=0 (a “non-informed” heuristic), A* reduces to our friend Dijkstra’s Algorithm The Optimality of A*

25. A* Simulations No Obstacles Single Obstacle

26. A* Simulations (cont’d) Multiple Obstacles No Path

27. PROS • Applicable to general obstacle geometries • With A*, provides shorter paths than exact decomposition • CONS • Performance a function of discretization resolution (δ) • Inefficiencies • Lost paths • Undetected collisions • Number of graph vertices |V| grows with δ2 and A* runs in O(|V|2) time -> O(δ4) running time Approximate Cell Summary

28. The Potential Field Approach

29. Introduced by Khatib [1986] initially as a real-time collision avoidance module, and later extended to motion planning • Robot motion is influenced by an artificial potential field – a force field if you will – induced by the goal and obstacles in C • The field is modeled by a potential functionU(x,y) over C • Motion policy control law is akin to gradient descent on the potential function • A shortcoming of the approach is the lack of performance guarantees: the robot can become trapped in local minima • Koditschek’s extension introduced the concept of a “navigation function” - a local-minima free potential function Some Background…

30. Example Application: Target Tracking with Obstacle Avoidance

31. The Idea… GOAL

32. In 2D, the gradient of a function f is defined as • The gradient points in the direction where the derivative has the largest value (the greatest rate of increase in the value of f) • The gradient descent optimization algorithm searches in the opposite direction of the gradient to find the minimum of a function • Potential field methods employ a similar approach Flashback: What is the Gradient?

33. Potential fields can live in continuous space • No cell decomposition issues • Local method • Implicit trajectory generation • Prior knowledge of obstacle positions not required • The bad: Weaker performance guarantees Some Characteristics of Potential Field Approaches

34. Generating potential functions for motion control • PRINT OUT COPY OF POTENTIAL FIELD NOTES by R. Beard (they’re on the web page). Next Time…