Computational Geometry

1. Piyush Kumar (Lecture 10: Robot Motion Planning) Computational Geometry Welcome to CIS5930

2. Reading Chapter 13 in Textbook David Mount’s Lecture notes. Slides Sources Lecture notes from Dr. Bayazit Dr. Spletzer, Dr. Latombe. Pictures from the web and David Mount’s notes.

3. Robots Fiction Real World

4. Robot Motion Planning Obstacles Free space

5. Robot Motion Planning Work Space : Environment in which robot operates Obstacles : Already occupied spaces of the world. Free Space : Unoccupied space of the world.

6. Configuration Space OR C-space Helps in determining where a robot can go. Modelling a robot Configuration: Values which specify the position of a robot Geometric shape description

7. Motion Planning start goal obstacles Given a robot, find a sequence of valid configurations that moves the robot from the source to destination.

8. Configuration Space Configuration: Specification of the robot position relative to a fixed coordinate system. Usually a set of values expressed as a vector of positions/orientations. Configuration Space: is the space of all possible robot configurations.

9. Configuration Space Example workspace robot reference direction  y reference point x • 3-parameter representation: q = (x,y,) • In 3D? 6-parameters - (x,y,z,)

10. Configuration Space Y A robot which can translate in the plane C-space: 2-D (x, y) X Euclidean space: Y A robot which can translate and rotate in the plane C-space: 3-D (x, y, ) Y X x Courtesy J.Xiao

11. q2 q1 C-Space q = (q1,q2,…,q10)

12. Configuration Space /Obstacles Circular Robot

13. C-Obstacles • Convex polygonal robot

14. Minkowski Sum A B = { a+b | a  A, b  B }  

15. Minkowski Sums • 3D Minkowski sum difficult to compute • Many Applications • Configuration Space Computation • Offset • Morphing • Packing and Layout • Friction model

16. Configuration Obstacle • Only for robots in 2d that can translate. • CP = { p | R(p) ∩ P ≠ Null }

17. C-Obstacles • Lemma : CP = P  (-R) • Proof: Show that R(q) intersects P iff q є P  (-R). • q є P  (-R) iff there exists p є P and (-r) є (-R) such that q = p – r • R(q) intersects P iff there exists r є R and p є P such that r+q = p. Equivalent

18. An illustration

19. Computing Minkowski sum • For a given convex polygonal obstacle (with n vertices) and a convex footprint robot (with m vertices), how fast can we compute the CP? O(m + n) Idea: Walk.

20. Complexity of Minkowski Sums? • Can we bound the complexity of the minkowski sum of disjoint convex obstacles with n vertices in the plane? • Naïve bound? • Triangulate the obstacles : O(n) edges. • Minkowski sum of R with triangles = O(nm) • Complexity of the union? O((nm)2)?

21. Pseudodisks: Defn. • A set of convex objects {o1,o2,…,on} is called a collection of pseudodisks if for any two distinct objects oi and oj both of the set theoretic differences oi\oj and oj\oi are connected.

22. Lemma 1 • Given a set of convex objects {T1, T2,…, Tn,} with disjoint interiors and convex R, the set {Ti  R | i = 1..n } is a collection of psedodisks. Proof: On the chalkboard 

23. Lemma 2 • Given a collection of pseudodisks with n vertices, the complexity of their union is O(n). • Why?

24. Planning approaches in C-space • Roadmap Approach: • Visibility Graph Methods • Cell Decomposition Approach • Potential Fields • Many other Algorithms…

25. goal start Visibility Graph in C-space Each path in c-space from s to t represents a viable move from s to t of the Robot in the original space.

26. goal start Visibility Graph in C-space Computation time? Each path in c-space from s to t represents a viable move from s to t of the Robot in the original space.

27. Vis Graph in higher dimensions? Will it work?

28. 13 12 10 11 9 7 6 5 4 2 3 1 8 Cell Decomp: Trapezoidal Decomp. GOAL START 1) Decompose Region Into Cells

29. 13 12 11 10 9 7 8 6 4 2 3 1 5 Cell Decomp: Trapezoidal Decomp. GOAL START 1) Decompose Region Into Cells 2) Construct Adjacency Graph

30. Cell Decomp: Trapezoidal Decomp. GOAL START

31. Cell Decomposition: Other approaches Uniform Quadtree

32. Potential field approach • 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

33. Next Class • Final Review: Q&A session.