Motion Planning for Deformable Robots

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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 AM 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 AM University
• Constraint-based
• Gayle et al. UNC at Chapel Hill
• Conclusion

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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

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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 ? evaluated at ?-1(v)
• The energy of ? is

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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

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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

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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

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

• Different approaches
• Physically-based
• Anshelevich et al. Rice University
• Geometry-based
• Bayazit et al. Texas AM University
• Constraint-based
• Gayle et al. UNC at Chapel Hill
• Conclusion

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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

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Overview

• Critical steps in the algorithm
• Roadmap construction
• Querying and deformation

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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 factors 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

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Roadmap Construction

• Shrinkable robot
• Penetration
• Swept volume of the path found

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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

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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

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Deformations
Geometric deformation a) Colliding
configuration b) Intersecting polygonsc)
Deformed version
Bounding-box deformation
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Results
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Results
(for narrow scenario)
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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

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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

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Outline

• Different approaches
• Physically-based
• Anshelevich et al. Rice University
• Geometry-based
• Bayazit et al. Texas AM University
• Constraint-based
• Gayle et al. UNC at Chapel Hill
• Conclusion

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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

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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

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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

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Overall Architecture
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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

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Simulating Deformation

• Represent deformation as a list of constraints
• Considerations
• Continuum representation
• Energy minimization
• Volume preservation
• Interaction with the environment

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Continuum Representation

• Uses a simple Mass-Spring framework
• Computationally inexpensive
• Simple implementation and relatively easy
  interaction with the environment

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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

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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

Force due to pressure on a surface
Ideal Gas Law
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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
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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

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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

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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

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Advantages

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

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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

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Results

• Ball in cup

• Many spheres

Cup - 500 Polygons Robot 320 Polygons
Spheres - 3200 Polygons Robot 320 Polygons
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Results

• Walls with holes

Walls - 216 Polygons per wall Robot 720
Polygons
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Results

• Tunnel

Tunnel - 72 Polygons Robot 720 Polygons
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Performance Results
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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

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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

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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

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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

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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

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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

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CD Speedup
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Catherization scenarioCatheter 10K triangles
Arteries 90K triangles
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Results
53
Video
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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

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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

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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.