Planning Paths for Elastic Objects Under Manipulation Constraints

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)

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Problem Definition

• Configuration
• Rest configuration q0
• Rest volume V0 in R3
• Configuration q corresponds to changing volume
  from V0 to Vq in R3

Vq
Vo
Configuration q0
Configuration q
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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(UV uv)
• Where uv are two vectors at x before deformation
  and UV are the two vectors after deformation and
  (MN) is the inner product of MN

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
• Eel(q) ?V0(?(x,e(x))dx
• This paper considers homogeneous isotropic linear
  elastic material ?(e)

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Problem Definition

• Manipulation Constraints
• Actuators constrain a subset of points V0p in V0
• Denote M as set of possible actuator positions
  and m is one these positions in M
• For all x in V0p there is a mapping Xm from V0 to
  Vq

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 Energy
Cannot form this with two actuators
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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 V0

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, Gn, of finite-dimensional subspaces
  where
• Limn?? max dC(q,Gn) 0 (dC 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

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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

d
zd
zr
yd
r
yr
xr
xd
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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) dd(p,q) dr(p,q)
• dd is deformation distance, defined as the
  maximum distance a point moves in the local frame
  during a deformation
• dr is rigid body translation and rotation
  distance, defined as the Euclidean distance in R6
• Weighting dd and dr has yielded no significant
  improvements, but using only dd 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

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Experimental Results

• Bending Plate
• N 200 K 40 M 100
• Avg run time 22.7 min
• Avg nodes 12,500

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Experimental Results

• Bending Plate
• 9 Dimensional problem
• 6 for placement
• 3 for deformation

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Experimental Results

• Bending Plate
• N 200 K 40 M 100
• Avg run time 4 hrs 12 min
• Avg nodes 33,600

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Experimental Results

• Elastic Pipe one end fixed
• 5 Dimensional problem
• All 5 dimensions for deformation

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Experimental Results

• Elastic Pipe one end fixed
• Nodes 200 K 40 M 0
• Avg run time 14.2

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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