1. Motion Planning for Deformable Robots Serhat Tekin 11/7/2006

2. Motivation  Motion planning is a classical problem  Mostly for rigid or articulated robots  Deformable variants are recent Massive configuration space even for simple cases Existing methods not directly applicable

3. Motivation  Why need deformable robots?  Applications in Industry CAD and virtual prototyping Computer generated animation Bioinformatics Computer-aided surgery

4. Outline  Different approaches Physically-based  Anshelevich et al. Rice University Geometry-based  Bayazit et al. Texas A&M University Constraint-based  Gayle et al. UNC at Chapel Hill  Conclusion

5. Outline  Different approaches Physically-based  Anshelevich et al. Rice University Geometry-based  Bayazit et al. Texas A&M University Constraint-based  Gayle et al. UNC at Chapel Hill  Conclusion

6. Physically-based Approach  Builds upon a similar framework introduced for elastic plates Lamiraux et al. Rice University  An extension to PRM that takes deformation energy into account  Volume deformations represented by a mass-spring lattice

7. Continuous Mechanical Model  Uses the linear elastic physical model For a point v, energy density is defined by F is the matrix of partial derivatives of the deformation function γ evaluated at γ -1 (v) The energy of γ is

8. Discrete Spring Model  Approximates the continuous model  Two types of springs between the masses Straight springs Angular springs  Constant is picked according to type  Discritized energy function

9. Volume Deformation  Things to consider Grasp/Manipulation constraints on volume  Restricts positions on some parts of the volume  i.e. fix the positions of some point masses Energy minimization

10. Path Planning  Uses PRM for path planning  Local planner Interpolates between manipulation constraints to form a sequence of intermediate constraints  Elasticity limits Constants to prevent unnatural deformations  Plane strain limit: How much the material stretches locally  Curvature limit: How much the material bends locally

11. Results (on an SGI R10000) Deformable cable with fixed end (32x3x3 lattice) 14.5 mins (average) Elastic pipe through a cube with an L-shaped hole (21x3x3 lattice) 8h 39mins

12. Outline  Different approaches Physically-based  Anshelevich et al. Rice University Geometry-based  Bayazit et al. Texas A&M University Constraint-based  Gayle et al. UNC at Chapel Hill  Conclusion

13. Geometry-based Approach  PRM extension, similar to the first method  Deformations are not represented by physical means  Aims at a reasonable-time limit with plausible deformations, rather than physical correctness

14. Overview  Critical steps in the algorithm Roadmap construction Querying and deformation

15. Roadmap Construction  Need to estimate the edge weights  Two different heuristics Shrinkable robots  Use rigid robots with different scales  Edge weight is the sum of “shrink factor”s of endpoints Allowing penetration  Work in C-space to estimate penetration depth  Sample n different C-space vectors (empirically, n = 20)  If any sample is collision free, accept  Minimum depth is used for the edge weight

16. Roadmap Construction a)Shrinkable robot b)Penetration c)Swept volume of the path found

17. Query  Deformable robot used in the query phase Collisions must be avoided by deformations Configuration accepted if deformation energy below threshold Edges with higher weights are likely to fail, so test them first

18. Deformations  Bounding-box deformation Deformer pushes object into collision-free condition ChainMail deformation  Similar to FFD  Deforming boundary box vertex affects neighbors Free-form deformation (FFD)  Only used for visualization  Geometric deformation Deform the colliding portion directly  Translate along surface normals

19. Deformations Geometric deformation: a) Colliding configuration b) Intersecting polygons c) Deformed version Bounding-box deformation

20. Results

21. (for “narrow” scenario)

22. Summary  Both methods are PRM extensions  Differ in the way they handle roadmaps and deformations Physically-based  Deformation taken into account during roadmap construction  Deformations are physical simulations Geometry-based  Robot treated as rigid during roadmap construction  Deformations are geometric

23. Advantages/Disadvantages  (+) Both methods offer a generalized framework to the problem Same deformation scheme can be used with a different randomized planner  (-) Can handle only simple robots and environments First approach is computationally expensive Second one is not physically accurate

24. Outline  Different approaches Physically-based  Anshelevich et al. Rice University Geometry-based  Bayazit et al. Texas A&M University Constraint-based  Gayle et al. UNC at Chapel Hill  Conclusion

25. Constraint-Based Motion Planning  M. Garber and M. Lin, Constraint-based motion planning using Voronoi diagrams. Proc. Fifth International Workshop on Algorithmic Foundations of Robotics (WAFR), 2002Constraint-based motion planning using Voronoi diagrams  Reformulate the planning problem as a boundary value problem (BVP) Builds on similarity between BVP and Motion planning Map initial and goal configuration to boundary values Map motion into a constrained dynamics function  Solvable through dynamical simulation

26. CBMP Goal  To find a (near) minimal set of constraints which are sufficient to solve the problem  Example: A 2D rigid robot in a simple environment The robot must:  Reach a goal  Avoid obstacles

27. Constraints  Hints at how the object should move Hard constraints: Must be satisfied at each step  No penetration or intersection with obstacles  Robot must stay within boundaries  Articulated links must stay together  Joint limits must be satisfied Soft constraints: Encourage a certain behavior  Robot should follow a guiding path  Robot should move towards the goal configuration  Robot should avoid nearest obstacles

28. Overall Architecture

29. CBMP for Deformable Robots  DPlan: Builds upon CBMP Represent deformation as a list of constraints Represent energy minimization as a constraint  Two stage approach Off-line roadmap generation  Simple PRM for a point robot  Possibly contains collisions Runtime path query by constrained dynamic simulation  Performs deformation and local adjustments to path

30. Simulating Deformation  Represent deformation as a list of constraints  Considerations Continuum representation Energy minimization Volume preservation Interaction with the environment

31. Continuum Representation  Uses a simple Mass-Spring framework  Computationally inexpensive  Simple implementation and relatively easy interaction with the environment

32. Energy Minimization  Robot energy function (defined by springs):  k is spring constant, d is current distance, L is rest length  Relax the case, i.e. allow small changes to the volume

33. Volume Preservation  Relaxation Measures internal pressure variations Computes a pressure constraint force to adapt to changes in pressure Uses a simplified model based on the Ideal Gas Law Ideal Gas Law Force due to pressure on a surface

34. Adjusting Pressure  Internal pressure constant defines the robot behavior  The RHS constant of Ideal Gas Law (nR g T) is assigned by trial-and-error Low PressureMedium PressureHigh Pressure

35. Interaction  Hard constraints for interaction with environment Bounding-volume collision detection  Assumes collision if robot is within a tolerance to an obstacle  Applies impulses and repulsion forces at the affected masses  Soft constraints for global behavior Path following

36. Deformation Step  Perform collision detection  Handle collisions to enforce non-penetration constraints  Accumulate spring forces F s  Compute the volume V of the object  Set P = nR g T / V  For each face f on the geometry Set F p = PA For each vertex v of f  Find the pressure forces on v by adding F p divided by the number of faces incidental to v

37. Summary  Builds upon CBMP Adds constraints for deformation, path following, and interaction with the environment Uses a simplified global path to help escape local minima while using CBMP to make local adjustments to ensure a collision-free path

38. Advantages  Allows for complex robots  Computes physically-plausible deformations  Performs sampling in low-degree of freedom space (i.e. workspace)

39. Limitations  Does not ensure a path will be found  Cannot guarantee accurate deformations  Restricted ability to represent robots with sharp edges  Applicable only to closed robots  Limited scalability

40. Results  Ball in cup  Many spheres Cup - 500 Polygons Robot – 320 Polygons Spheres - 3200 Polygons Robot – 320 Polygons

41. Results  Walls with holes Walls - 216 Polygons per wall Robot – 720 Polygons

42. Results Tunnel - 72 Polygons Robot – 720 Polygons  Tunnel

43. Performance Results ScenarioObstacle (tris) Robot (tris)Path Est. Time (sec) Total Sim Time (sec) Avg. Step Time (sec) Ball In Cup5003201.041.50.015 Many Spheres 32003201.0333.160.077 Walls with Holes 21672048608.9580.037 Tunnel72720575833.240.068

44. Improving Performance  Support for complex environments  FlexiPlan: Path Planning for Deformable Robots in Complex Environments (FlexiPlan)  Builds upon DPlan by improving primary bottlenecks Guiding path improvements Simulation improvements

45. Improvements  More optimal global guiding path Samples along the medial axis of the workspace to create a path (Medial Axis PRM) Generalized Voronoi Diagram is another possibility  Computed efficiently with GPUs  Simulation improvements Mass-Spring simulation  More stable (Semi-Implicit Verlet integration)  Supports angular springs to counteract shearing Better collision detection scheme

46. Collision Detection  Dominating factor in running time  DPlan only uses a bounding volume to remove unnecessary checks  BVH is not a viable option Robot is often too close to obstacles in most scenarios BVH would not eliminate most tests and incur an update cost  Speed up collision tests by 2.5D overlap test Set-based computation

47. 2.5D Overlap Test  Based on CULLIDE Choose a viewing direction Check whether R is fully visible with respect to O along that direction Utilize GPU occlusion query

48. Reliable GPU Check  Might miss overlaps due to pixel precision  To prevent this Determine the size of a pixel Compute Minkowski sum of the obstacles and robot with a pixel Conservative, since may include pixels from geometry which does not overlap

49. Set-Based Computation  Maintains a PCS (Potentially Colliding Set) throughout computation Initially everything is in the PCS Uses overlap tests to remove obstacles from the PCS  Do exact collision detection on the PCS If number of primitives is small, test all pariwise combinations Else, use bounding boxes for speed-up

50. CD Speedup

51. Catherization scenario Catheter: ~10K triangles Arteries: ~90K triangles

52. Results

53. Video

54. Summary  Guiding Path Follows the medial axis of the workspace  Spring-Mass Support for larger systems  Catherization scenario has over 100,000 springs Greater stability  Collision Detection GPU-based culling and set partitioning

55. Summary  Advantages Scales better to complex scenes Introduces a planning specific CD algorithm  Limitations Same planning restrictions as DPlan  No definite path  May not have accurate deformation  Restricted to closed objects Setting constants for the simulation Requires a high-end graphics card

56. Conclusion  The area itself is relatively new The first two approaches can be thought as pioneering work The last one takes novel approaches to planning and collision detection  Current methods need to be extended for Complex robot shapes Articulated deformable robots Deformable obstacles Dynamic environments

57. References  F. Lamiraux, L. Kavraki. Path planning for elastic objects under manipulation constraints. International Journal of Robotics Research, 20(3):188-208, 2001.  E. Anshelevich, S. Owens, F. Lamiraux, L. Kavraki. Deformable volumes in path planning applications.IEEE Int. Conf. Robot. Autom. (ICRA), pp. 2290-2295, 2000.  O. B. Bayazit, H. Lien, and N. Amato. Probabilistic roadmap motion planning for deformable objects. IEEE Int. Conf. Robot. Autom. (ICRA), 2002.  R. Gayle, M. C. Lin, D. Manocha. Constraint-Based Motion Planning of Deformable Robots. International Conference of Robotics and Automation, 2005.  R. Gayle, W. Segars, M. C. Lin, D. Manocha. Path Planning for Deformable Robots in Complex Environments. Robotics: Systems and Science, 2005.