Planning Paths for Elastic Objects Under Manipulation Constraints

Florent Lamiraux Lydia E. Kavraki

Rice University

Presented by: Michael Adams

Outline

• Introduction

• Related Work

• Problem Definition

• Path Planning Algorithm

• Experimental Results

• Conclusions

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

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

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)

Problem Definition

• Configuration– Rest configuration q0

– Rest volume V0 in R3

– Configuration q corresponds to changing volume from V0 to Vq in R3

VoVq

Configuration q0 Configuration q

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

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)

Problem Definition

• Manipulation Constraints– Actuators constrain a subset of points V0

p 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

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

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

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

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

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)

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

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

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

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

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

xryr

zr

xd

zd

ydr

d

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

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

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

Experimental Results

• Bending Plate– 7 Dimensional problem– 6 for placement– 1 for deformation

Experimental Results

• Bending PlateN = 200 K = 40 M = 100

Avg run time – 22.7 min

Avg # nodes – 12,500

Experimental Results

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

Experimental Results

• Bending PlateN = 200 K = 40 M = 100

Avg run time – 4 hrs 12 min

Avg # nodes – 33,600

Experimental Results

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

Experimental Results

• Elastic Pipe – one end fixedNodes = 200 K = 40 M = 0

Avg run time – 14.2

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