1. Planning Paths for Elastic Objects Under Manipulation Constraints Florent Lamiraux Lydia E. Kavraki Rice University Presented by: Michael Adams

2. Outline Introduction Related Work Problem Definition Path Planning Algorithm Experimental Results Conclusions

3. Introduction Goal: Plan paths for elastically deformable objects in a static environment What is hard? –Representing the shape of an object with a possibly infinitely dimensional configuration space –Computing object shapes under actuator loading conditions –Collision checking for a shape-changing object

4. Related Work Paper draws from other fields including: –High dimensional robot planning – random path planning –Mechanics – energy modeling of deformation shapes –Geometric modeling – representation of infinitely dimensional configuration space –Graphics – physically based models of deformable objects

5. Problem Definition What objects are we looking at? –Elastically deformable objects constrained by two actuators –Shape is determined by the lowest energy state for a given configuration of the actuators –Only the actuators are responsible for deformations (object cannot touch obstacles)

6. Problem Definition Configuration –Rest configuration q 0 –Rest volume V 0 in R 3 –Configuration q corresponds to changing volume from V 0 to V q in R 3 VoVo VqVq Configuration q 0 Configuration q

7. Problem Definition Local Deformation Field –Object deformation is defined by a field of local deformations over the volume of the object –Local deformation at a point x is defined as: e(x) = ½(U|V – u|v) –Where u&v are two vectors at x before deformation and U&V are the two vectors after deformation and (M|N) is the inner product of M&N

8. Problem Definition Elasticity and Energy –Reversibility of deformation due to restoring force –Characterize elasticity by the density (  ) of elastic energy at every point x –E el (q) =  V0 (  (x,e(x))dx –This paper considers homogeneous isotropic linear elastic material –  (e)

9. Problem Definition Manipulation Constraints –Actuators constrain a subset of points V 0 p in V 0 –Denote M as set of possible actuator positions and m is one these positions in M –For all x in V 0 p there is a mapping X m from V 0 to V q

10. Problem Definition Stable Equilibrium Configurations –Motion is slow enough to consider quasi-static paths – only stable equilibrium configurations –Stable equilibrium configurations are shapes at which the elastic energy is minimized Minimum EnergyCannot form this with two actuators

11. Problem Definition Elastically Admissible Configurations –Elastic materials have a range of elastic deformation, large deformations may exceed this range and permanently deform –A range of elastic e(x) is defined –Admissible configurations are those in which e(x) is within the elastic range for all x in V 0

12. Problem Definition In path planning, “collision-free paths” are not enough – other conditions must be met –Manipulability: every point along the path must meet the actuator constraints –Quasi-static equilibrium: every point along the path must be in stable equilibrium (a minimum energy shape) –Elastic admissibility: no points along the path exceed the elastic limits of the material

13. Path Planning Algorithm Geometric Representation –Approximate infinite-dimensional space as some finite-dimensional space –A geometric representation of configuration space is a family, G n, of finite-dimensional subspaces where: Lim n  max d C (q,G n ) = 0 (d C is a distance function) –Most common are polynomial or finite difference representations

14. Path Planning Algorithm Computation of Stable Equilibrium Configurations –Stable equilibrium configurations are found by minimizing elastic energy –Elastic energy is computed by integrating the energy density  over the volume (analytically or numerically depending on geometric representation) Computation of Stable Equilibrium Configurations –Stable equilibrium configurations are found by minimizing elastic energy –Elastic energy is computed by integrating the energy density  over the volume (analytically or numerically depending on geometric representation)

15. Path Planning Algorithm Algorithm –PRM approach is used, similar to conventional planners Initial/Final configurations are chosen Random free stable equilibrium configurations are chosen as nodes in roadmap Nodes are connected by a local planner to form edges Decompose deformation and position of object to save computing time and minimize wear on material

16. Path Planning Algorithm Node Generation –A random manipulator position is chosen and minimum energy shape calculated and admissibility is checked –Random rigid-body motions are evaluated for collision-free configurations –N collision-free configurations are found for the same deformation

17. Path Planning Algorithm Node Connection –Each newly generated node is tested for connection with its K closest neighbors –Distance function should account for rigid body transformation and deformation –Local planner checks for collisions and admissibility

18. Path Planning Algorithm Enhancement –Under the assumption that unconnected nodes are in difficult parts of the configuration space, add more nodes in these difficult areas –Randomly walk away from unconnected nodes in the same configuration for a certain distance, reflecting off obstacles –A total of M enhancement nodes are added

19. Path Planning Algorithm Local Planner –For efficiency, again decouple deformation and position –Each configuration is denoted q = (d,r) –d is deformation and r is position in space of a local reference frame xrxr yryr zrzr xdxd zdzd ydyd r d

20. Path Planning Algorithm Local Planner –First checks the path with rigid body motion of the local frame –Then checks the path considering deformation within the local frame –Saves time by avoiding energy minimizations

21. Path Planning Algorithm Distance Metric –Distance d(p,q) = d d (p,q) + d r (p,q) –d d is deformation distance, defined as the maximum distance a point moves in the local frame during a deformation –d r is rigid body translation and rotation distance, defined as the Euclidean distance in R 6 –Weighting d d and d r has yielded no significant improvements, but using only d d has been reasonable

22. Path Planning Algorithm Collision Checking –With the decoupled motions, a standard collision checking algorithm can be applied, the research in this paper used a method called RAPID –By keeping deformation separate from position, deformations can be stored and reused speeding up collision checking

23. Experimental Results Bending Plate –7 Dimensional problem –6 for placement –1 for deformation

24. Experimental Results Bending Plate N = 200 K = 40 M = 100 Avg run time – 22.7 min Avg # nodes – 12,500

25. Experimental Results Bending Plate –9 Dimensional problem –6 for placement –3 for deformation

26. Experimental Results Bending Plate N = 200 K = 40 M = 100 Avg run time – 4 hrs 12 min Avg # nodes – 33,600

27. Experimental Results Elastic Pipe – one end fixed –5 Dimensional problem –All 5 dimensions for deformation

28. Experimental Results Elastic Pipe – one end fixed Nodes = 200 K = 40 M = 0 Avg run time – 14.2

29. Conclusions Very general algorithm, easily tunable Improved energy minimization would be very helpful Tailored geometric representations with energy and energy gradient calculation in mind would move toward this goal Many representations from graphics do not conserve surface area or volume