Planning for Deformable Parts. Holding Deformable Parts How can we plan holding of deformable objects?

Planning for Deformable Parts

Holding Deformable Parts

How can we plan holding of deformable objects?

Deformable parts

• “Form closure” does not apply:

Can always avoid collisions by deforming the part.

• Deformation Space: A Generalization of Configuration Space.

• Based on Finite Element Mesh.

D-Space

Deformable Polygonal parts: Mesh

• Planar Part represented as Planar Mesh.

• Mesh = nodes + edges + Triangular elements.

• N nodes• Polygonal boundary.

D-Space• A Deformation:

Position of each mesh node.

• D-space: Space of all mesh deformations.

• Each node has 2 DOF.

• D-Space: 2N-dimensional Euclidean Space.

30-dimensional D-space

Nominal mesh configuration

Deformed mesh configuration

Deformations

• Deformations (mesh configurations) specified as list of translational DOFs of each mesh node.

• Mesh rotation also represented by node displacements.

• Nominal mesh configuration (undeformed mesh): q0.

• General mesh configuration: q.

q0

q

D-Space: Example

• Simple example:

3-noded mesh, 2 fixed.

• D-Space: 2-dimensional Euclidean Space.

• D-Space shows moving node’s position.

x

y

Phy

sica

l spa

ceD

-Spa

ce

q0

Topological Constraints: DT

x

y

Phy

sica

l spa

ceD

-Spa

ce

• Mesh topology maintained.

• Non-degenerate triangles only.

DT

Topology violating

deformation

Undeformed part

Allowed deformation

Self Collisions and DT

TDq

TDq

D-Obstacles

x

y

Phy

sica

l spa

ceD

-Spa

ce

• Collision of any mesh triangle with an object.

• Physical obstacle Ai has an image DAi in D-Space.

A1

DA1

D-Space: Example

Phy

sica

l spa

ce

x

y

D-S

pace

Potential Energy Needed to Escape from a Stable Equilibrium• Consider:

Stable equilibrium qA, Equilibrium qB.

• Capture Region:

K(qA) Dfree, such that any configuration in K(qA) returns to qA.

qA

qB

q

U(q)

K( qA )

• UA (qA) = Increase in Potential Energy needed to escape from qA.

= minimum external work needed to escape from qA.

• UA: Measure of “Deform Closure Grasp

Quality”

qA

qB

q

U(q)

UA

Potential Energy Needed to Escape from a Stable Equilibrium

K( qA )

Deform Closure

qA

qB

q

U(q)

MOTION PLANNING FOR DEFORMABLE OBJECTS

From slides by Ilknur Kaynar-Kabul

Introduction Algorithms so far

the world was assumed to be made of rigid objects

Why deformable objects? Deformable moving objects

(wires, metal sheets) Deformable obstacles

(e.g., human-body tissue structures)

Need for physical model

First Paper: Planning paths for elastic objects under manipulation constraints (Lamiraux and

Kavraki)

Energy model

Second Paper: Probabilistic Roadmap Motion Planning for Deformable Objects

(O. Burchan Bayazit, Jyh-Ming Lien, Nancy M.Amato)

Planning Paths for Elastic Objects Under

Manipulation Constraints

Florent Lamiraux

Lydia E. Kavraki

Rice University

Int. J. Robotics Research, 2001.

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

Problem Definition

• What objects are considered?• 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)

ACTUATORS constrain the position of a subset of the points

of the object

Problem Definition• In general, the configuration of an elastic

object can be infinite-dimensional and cannot be represented by a vector

• Configuration• Rest configuration q0

• Configuration q corresponds to mapping object through deformation

VoVq

Configuration q0 Configuration q

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 of these positions in M

• For all x in V0p there is a mapping Xm from V0 to

Vq

For a given position m of the actuators, points V0p are moved to Xm(V0

p)The position of the other points of the objects should be such that the elastic energy of the object is minimized.

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 for each point x

• 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• 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

Path Planning Algorithm• Algorithm

• The following steps are repeated until qinit and qgoal are in the same connected component of the roadmap:• Node generation• Node connection• Enhancement

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

Rigid body motion is applied


• Random rigid-body motions are evaluated for collision-free configurations

collision


• 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 is evaluated


• Distance function should account for rigid body transformation and deformation

Distance total = distance transformation + distance deformation


• Connections are performed by a deterministic local planner that generates quasi-static paths between pairs of configurations.

Edges


• Local planner checks for collisions and admissibility

Not a valid edgeViolates elasticity limits

Not a valid edgeThere exist collision

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

Path Planning Algorithm• Enhancement

• 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• Path finding for a given qinit and qgoal

• A graph search can yield a sequence of edges leading from qinit and qgoal

• Concatenation of local paths results in a global path

• We look for a path that minimizes the number of distinct deformations of the nodes of V belonging to the path

-> reduce unnecessary deformations

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

Path Planning Algorithm• Collision Checking

• With the decoupled motions, a standard collision checking algorithm can be applied, the research in this paper used RAPID (OBB-trees)

• 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


• The actuators are along the 2 opposite long edges

• They are always parallel• Actuators constrain the distance d <= L

between the opposite edges• Thus, deformation is one dimensional


N = 200 K = 40 M = 100

Avg run time – 22.7 min

Avg # nodes – 12,500


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


• Manipulation constraints specify both the position and tangent direction of 2 opposite edges of the plate

• One end of the curve is fixed and the other can move freely (translation along x1 and x3, rotation about x2)


N = 200 K = 40 M = 100

Avg run time – 4 hrs 12 min

Avg # nodes – 33,600


Avg run time – 4 hrs 12 min• Space of deformation is of higher

dimension• Large number of minimizations involved

for computing deformation paths• The free space inside the box is

constrained

Experimental Results• Elastic Pipe – one end fixed

• 5 Dimensional problem• All 5 dimensions for deformation


• Manipulation constraints specify both the position and tangent direction of the ends of the pipe.

• No twisting of pipe


Nodes = 200 K = 40 M = 0

Avg run time – 14.2

MOVIES

Conclusions• Very general algorithm, easily

tunable• Improved energy minimization would

be very helpful• Many representations from graphics

do not conserve surface area or volume

Disadvantages and Future Work• Planner is not suitable for real-time

use• expensive operations for solving

mechanical models• generating collision detection data

structures• Has so far only been applied to

simple objects, such as a sheet of metal or a pipe-like robot

Probabilistic Roadmap Motion Planning for Deformable

ObjectsO. Burchan Bayazit

Jyh-Ming LienNancy M. Amato

ICRA2002

Outline• Introduction• Problem Definition• Roadmap construction

• Shrink factor• Penetration depth

• Query processing• Bounding box deformation• Geometric deformation

• Experimental Results• Conclusions

Algorithm• Motion planning for deformable

objects• Framework is based on a PRM model• Difference from typical motion

planning• The robot is allowed to change its

shape (deform) to avoid collisions as it moves along the path

2 Stage Algorithm• An approximate path is found for

the rigid version of the robot• Might contain collisions

• Collisions on this path are corrected by deforming the robot

Overview of algorithm

1. Build a feasible roadmap2. while (a valid path is not found)3. find a feasible path4. foreach path configuration5. if (there is collision)6. deform robot7. if (not deformable)8. remove path segment from roadmap 9. endwhile

Roadmap

If there exist collision -> Deform Robot

If robot cannot be deformed

If robot cannot be deformed -> Delete edge

Roadmap Construction• Roadmap may include some colliding

configurations• Weight the edges to denote the

expected difficulty of deforming that edge

• Weights are selected to denote the expected deformation energy required to deform the edge

• Computing deformation energy is difficult

Roadmap Construction• 2 strategies for constructing the

roadmap• Use reduced-scale version of the rigid

robot until it is collision-free• Use the amount of penetration by the

original robot and check it is less than some acceptable bound

• In both cases goal is to build a roadmap containing paths that are • Already valid OR• Can be made valid by deforming the robot

Shrinkable Robots• ‘shrink’ factor required to obtain a free

robot in a particular configuration is some indication of deformation energy

robot

Shrinkable Robots• Can be computed relatively inexpensively

• Pre-compute a set of scaled models• Construct by scaling all vertices and

maintaining the robot topologyrobot

Shrinkable Robotsrobot

Scaled models are used for

collision detection in

node generation and

connection

Node generation and construction• For each node

• Begin with the largest robot and progressively reduce its scale until a collision free robot size is found

• The roadmap is connected as with traditional PRM.

• An edge weight is set as the sum of the shrink factors of the two scaled models that are the edge’s endpoints.• A larger(smaller) robot model has a

smaller(larger) shrink factor

• A roadmap generated with shrinkable robots

Roadmap configurations are shown in their

accepted shrink size with wired surfaces

Penetration• Another estimate of deformation

energy• The deeper the robot penetrates

inside an obstacle the harder it will be to deform the robot into a collision free shape

• Unfortunately, penetration depth computation is hard to compute

Penetration• C-space penetration depth is used

Allowable

Penetration depth

Penetration• If any of vectors is collision free,

then accept configuration

Allowable

Penetration depth

Node generation and connection• Collision detection is replaced by a

test for allowable penetration• Advantage: we can use a standard

PRM, including all node generation and connection methods

A roadmap generated with penetrating robots

QUERY• Rigid body -> roadmap construction

Deformable body -> query processing

• The query process must check for collision

• If collisions are found, use a deformation method to try to deform the offending configurations into collision-free configurations

PATH QUERY

• The swept volume of a path found• Although the shortest path is on the right

side of ball, the planner selected a longer path which had a smaller deformation energy

DEFORMATION• Goal is to produce realistic looking

deformations fast• Not physically complete• Precise computation of energy is not

done

DEFORMATION• 2 objects participate in deformation

• Deformable object• Deformer

• The deformer pushes (a portion of) the deformable object towards a collision-free state

• Deformable object changes shape according to these external forces

BOUNDING BOX DEFORMATION• Deforms the given model

hierarchically• First, the bounding box of the model

is deformed, and then the model itself is deformed according to the deformation computed for the bounding box

BOUNDING BOX DEFORMATION• Combination of Chainmail

deformation and Free-form deformation is used

• First, the bounding box of the model is deformed by Chainmail deformation

• Then, this model is deformed by FFD according to shape of its bounding box

GEOMETRIC DEFORMATION• Continuously moves the colliding

polygons of the robot until they are outside the obstacle

• Which direction to move?• Sampling strategy to identify the

correct direction1. Normals of colliding polygons on the

robot

2. Normals of the colliding polygons on the obstacle

GEOMETRIC DEFORMATION

GEOMETRIC DEFORMATION• Problems

• If the robot does not fit in a narrow passage, then this method will fail to find a pushing direction unless we use a reduced scale version of the robot

• Can produce unrealistic deformations because the resulting polygons may be quite large• Apply constraints• Smooth the polygons

MOVIES

Documents

Planning for Deformable Parts. Holding Deformable Parts How can we plan holding of deformable objects?