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Probabilistic Roadmap Methods (PRMs)

Probabilistic Roadmap Methods (PRMs). Nancy M. Amato Parasol Lab,Texas A&M University. The Piano Movers Problem. (Basic) Motion Planning : Given a movable object and a description of the environment , find a sequence of valid configurations that moves it from the start to the goal.

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Probabilistic Roadmap Methods (PRMs)

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  1. Probabilistic Roadmap Methods(PRMs) Nancy M. Amato Parasol Lab,Texas A&M University

  2. The Piano Movers Problem (Basic) Motion Planning: Given a movableobject and a description of the environment, find a sequence of valid configurations that moves it from the start to the goal. start The Alpha Puzzle goal obstacles Motion Planning

  3. Applications of PRM-based Motion Planning (Amato et al)

  4. Outline • C-space, Planning in C-space (basic definitions) • Probabilistic Roadmap Methods (PRMs) • PRM variants (OBPRM, MAPRM) • User-Input: Haptically Enhanced PRMs • PRMs for more `complex’ robots • deformable objects • group behaviors for multiple robots • protein folding

  5. C-obst C-obst C-obst C-obst C-obst g b a 3D C-space (a,b,g) Configuration Space (C-Space) C-Space • “robot” maps to a point in higher • dimensional space • parameter for each degree of freedom • (dof) of robot • C-space = set of all robot placements • C-obstacle = infeasible robot placements 3D C-space (x,y,z) 6D C-space (x,y,z,pitch,roll,yaw) 2n-D C-space (f1,y1, f2, y2, . . . , f n, y n)

  6. workspace obstacles transformed to C-obstacles C-obstacles Figure from Latombe’91

  7. simple 2D workspace obstacle =>complicated 3D C-obstacle C-obstacles Figure from Latombe’91

  8. Path is swept volume Path is 1D curve Motion Planning in C-space Simple workspace obstacle transformed Into complicated C-obstacle!! C-space Workspace C-obst C-obst obst obst C-obst C-obst obst obst y x robot robot

  9. The Complexity of Motion Planning Most motion planning problems of interest are PSPACE-hard[Reif 79, Hopcroft et al. 84 & 86] • The best deterministic algorithm known has running time that is exponential in the dimension of the robot’s C-space [Canny 86] • C-space has high dimension - 6D for rigid body in 3-space • simple obstacles have complex C-obstacles impractical to compute explicit representation of freespace for more than 4 or 5 dof • So … attention has turned to randomized algorithms which • trade full completeness of the planner • for probabilistic completeness and a major gain in efficiency

  10. goal 1. Randomly generate robot configurations (nodes) - discard nodes that are invalid 2. Connect pairs of nodes to form roadmap - simple, deterministic local planner (e.g., straightline) - discard paths that are invalid Query processing start 1. Connect start and goal to roadmap 2. Find path in roadmap between start and goal - regenerate plans for edges in roadmap Probabilistic Roadmap Methods (PRMs)[Kavraki, Svestka, Latombe,Overmars 1996] C-space Roadmap Construction (Pre-processing) C-obst C-obst C-obst C-obst C-obst • Primitives Required: • Method for Sampling points in C-Space • Method for `validating’ points in C-Space

  11. goal PRMs: The Good News C-obst 1. PRMs are probabilistically complete 2. PRMs apply easily to high-dimensional C-space 3. PRMs support fast queries w/ enough preprocessing Many success stories where PRMs solve previously unsolved problems C-obst C-obst C-obst C-obst start goal PRMs: The Bad News C-obst C-obst • 1. PRMs don’t work as well for some problems: • unlikely to sample nodes in narrow passages • hard to sample/connect nodes on constraint surfaces • Our work concentrates on improving PRM performance for such problems. C-obst C-obst start PRMs: Pros & Cons

  12. Related Work (selected) • Probabilistic Roadmap Methods • Uniform Sampling (original) [Kavraki, Latombe, Overmars, Svestka, 92, 94, 96] • Obstacle-based PRM (OBPRM) [Amato et al, 98] • PRM Roadmaps in Dilated Free space[Hsu et al, 98] • Gaussian Sampling PRMs[Boor/Overmars/van der Steppen 99] • PRM for Closed Chain Systems[Lavalle/Yakey/Kavraki 99, Han/Amato 00] • PRM for Flexible/Deformable Objects[Kavraki et al 98, Bayazit/Lien/Amato 01] • Visibility Roadmaps[Laumond et al 99] • Using Medial Axis [Kavraki et al 99, Lien/Thomas/Wilmarth/Amato/Stiller 99, 03, Lin et al 00] • Generating Contact Configurations[Xiao et al 99] • Single Shot [Vallejo/Remmler/Amato 01] • Bio-Applications: Protein Folding [Song/Thomas/Amato 01,02,03, Apaydin et al 01,02] • Lazy Evaluation Methods: [Nielsen/Kavraki 00 Bohlin/Kavraki 00, Song/Miller/Amato 01, 03] • Related Methods • Ariadnes Clew Algorithm [Ahuactzin et al, 92] • RRT (Rapidly Exploring Random Trees) [Lavalle/Kuffner 99]

  13. Other Sampling Strategies • Obstacle Sensitive Sampling Strategies • Goal: Sample nodes in Narrow Passages • OBPRM: Obstacle-Based PRM • sampling around obstacles • MAPRM: Medial-Axis PRM • sampling on the medial axis • Repairing/Improving approximate paths • transforming invalid samples into valid samples

  14. goal goal OBPRM Roadmap C-obst C-obst C-obst C-obst C-obst C-obst C-obst C-obst start start • Idea: Can we sample nodes near C-obstacle surfaces? • we cannot explicitly construct the C-obstacles... • we do have models of the (workspace) obstacles... OBPRM: An Obstacle-Based PRM[IEEE ICRA’96, IEEE ICRA’98, WAFR’98] • To Navigate Narrow Passages we must sample in them • most PRM nodes are where planning is easy (not needed) PRM Roadmap

  15. 2 3 4 5 1 OBPRM: Finding Points on C-obstacles Basic Idea (for workspace obstacle S) 1. Find a point in S’s C-obstacle (robot placement colliding with S) 2. Select a random direction in C-space 3. Find a free point in that direction 4. Find boundary point between them using binary search (collision checks) Note: we can use more sophisticated heuristics to try to cover C-obstacle C-obst

  16. PRM vs OBPRM Roadmaps • PRM • 328 nodes • 4 major CCs • OBPRM • 161 nodes • 2 major CCs

  17. OBPRM for Paper Folding Box: 12 => 5 dof Periscope: 11 dof • Roadmap Statistics • 218 nodes, 3 sec • Roadmap Statistics • 450 nodes, 6 sec

  18. Medial axis MAPRM: Medial-Axis PRMSteve Wilmarth, Jyh-Ming Lien, Shawna Thomas [IEEE ICRA’99, ACM SoCG’99, IEEE ICRA’03] • Key Observation: We can efficiently retract almost any configuration, free or not, onto the medial axis of the free space without computing the medial axis explicitly. C-obstacle

  19. repaired path approximate path Repairing Approximate Paths • Create initial roadmap • Extract approximate path P • Repair P (push to C-free) • Focus search around P • Use OBPRM-like techniques • Problem: Even with the best sampling methods, roadmaps may not contain valid solution paths • may lack critical cfgs in narrow passages • may contain approximate paths that are ‘nearly’ valid Repairing/Improving Approximate Paths C-obstacle C-obstacle

  20. PHANToM Hybrid Human/Planner System[IEEE ICRA’01, IEEE ICRA’00, PUG’99] • 1. User collects approximate path using haptic device • User insight identifies critical cfgs • User feels when robot touches obstacles and adjusts trajectory • 2. Approximate path passed to planner and it fixes it • Current Applications • Intelligent CAD Applications • Molecule Docking in drug design • Animation w/ Deformable Models

  21. Hybrid Human/Planner SystemHaptic Hints Results 0.85 0.95 1 Flange Problem

  22. Hybrid Human/Planner SystemHaptic Hints Results 0.85 0.95 1 Flange Problem

  23. Applications • Same Method: Diverse range of problems • Deformable Objects • Simulating Group Behaviors • homing, covering, shepherding • Modeling Molecular Motions • Protein folding, RNA folding, protein/ligand binding

  24. Deformable Objects[IEEE ICRA’02] • Problem: Find a path for a deformable object that can deform to avoid collision with obstacles • deformable objects have infinite dof! • Our Approach: • Build roadmap relaxing collision free requirement (allow penetration) • Extract approximate path for a rigid version of robot (select path with smallest collision/penetration) • Follow path and deform the robot to avoid the collisions

  25. Deformed ChainMail Bounding Box Apply FFD ChainMail Box Obstacle Deformable ObjectsRepairing Path usingBounding Box Deformation • Bounding Box Deformation • build a 3D voxel bounding box. • Convert it to ChainMail bounding box (3D grid of springs) • Deform ChainMail Bounding box. • Deform objects using Free Form Deformation based on • deformation of the bounding box.

  26. Approximate Solution The approximate path returned is not the shortest path but the energetically most feasible path (less deformation required) Query

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