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Motion Planning for Deformable RobotsSerhat Tekin11/7/2006
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
Motivation Why need deformable robots? Applications in
Industry CAD and virtual prototyping Computer generated animation Bioinformatics Computer-aided surgery
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
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
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
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
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
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
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
Results (on an SGI R10000)
Deformable cable withfixed end (32x3x3 lattice)
14.5 mins (average)
Elastic pipe through a cube withan L-shaped hole (21x3x3 lattice)
8h 39mins
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
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
Overview Critical steps in
the algorithm Roadmap
construction Querying and
deformation
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
Roadmap Construction
a) Shrinkable robotb) Penetrationc) Swept volume of the path found
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
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
Deformations
Geometric deformation:a) Colliding configurationb) Intersecting polygonsc) Deformed version
Bounding-box deformation
Results
Results
(for “narrow” scenario)
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
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
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
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), 2002
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
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
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
Overall Architecture
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
Simulating Deformation Represent deformation as a list of
constraints Considerations
Continuum representation Energy minimization Volume preservation Interaction with the environment
Continuum Representation Uses a simple Mass-Spring framework
Computationally inexpensive Simple implementation and relatively easy
interaction with the environment
FKxxCxM
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
j
jjs Ldk
XE 2)(2
)(
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
V
TnRgP Ideal Gas Law
nn
p PAF
Force due to pressure on a surface
Adjusting Pressure Internal pressure constant defines the robot
behavior The RHS constant of Ideal Gas Law (nRgT) is
assigned by trial-and-error
Low Pressure Medium Pressure High Pressure
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
Deformation Step Perform collision detection Handle collisions to enforce non-penetration constraints Accumulate spring forces Fs
Compute the volume V of the object Set P = nRgT / V For each face f on the geometry
Set Fp = PA For each vertex v of f
Find the pressure forces on v by adding Fp divided by the number of faces incidental to v
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
Advantages Allows for complex robots Computes physically-plausible deformations Performs sampling in low-degree of
freedom space (i.e. workspace)
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
Results Ball in cup Many spheres
Cup - 500 Polygons Robot – 320 Polygons
Spheres - 3200 Polygons Robot – 320 Polygons
Results Walls with holes
Walls - 216 Polygons per wall Robot – 720 Polygons
Results
Tunnel - 72 Polygons Robot – 720 Polygons
Tunnel
Performance Results
Scenario Obstacle (tris)
Robot (tris) Path Est. Time (sec)
Total Sim Time (sec)
Avg. Step Time (sec)
Ball In Cup 500 320 1.0 41.5 0.015
Many Spheres
3200 320 1.0 333.16 0.077
Walls with Holes
216 720 48 608.958 0.037
Tunnel 72 720 575 833.24 0.068
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
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
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
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
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
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
CD Speedup
Catherization scenarioCatheter: ~10K triangles Arteries: ~90K triangles
Results
Video
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
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
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
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.