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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. D-Space. Deformation Space: A Generalization of Configuration Space.

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## Planning for Deformable Parts

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**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.**D-Space**Deformation Space: A Generalization of Configuration Space. Based on Finite Element Mesh.**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**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 Nominal mesh configuration q Deformed mesh configuration**D-Space: Example**y x • Simple example: 3-noded mesh, 2 fixed. • D-Space: 2-dimensional Euclidean Space. • D-Space shows moving node’s position. Physical space D-Space q0**y**x Topological Constraints: DT • Mesh topology maintained. • Non-degenerate triangles only. Physical space DT D-Space**Self Collisions and DT**Allowed deformation Undeformed part Topology violating deformation**y**x D-Obstacles • Collision of any mesh triangle with an object. • Physical obstacle Ai has an image DAi in D-Space. A1 Physical space DA1 D-Space**Physical space**D-Space y x D-Space: Example**Potential Energy Needed to Escape from a Stable Equilibrium**U(q) • Consider: Stable equilibrium qA, Equilibrium qB. • Capture Region: K(qA) Dfree, such that any configuration in K(qA) returns to qA. qA qB q K( qA )**U(q)**qA qB q Potential Energy Needed to Escape from a Stable Equilibrium • 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” UA K( qA )**U(q)**qA qB q Deform Closure**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 Vq Vo 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(V0p) 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**Path Planning Algorithm**• Node Generation • Random rigid-body motions are evaluated for collision-free configurations Rigid body motion is applied**Path Planning Algorithm**• Node Generation • Random rigid-body motions are evaluated for collision-free configurations collision**Path Planning Algorithm**• Node Generation • 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**Path Planning Algorithm**• Node Connection • Distance function should account for rigid body transformation and deformation Distance total = distance transformation + distance deformation**Path Planning Algorithm**• Node Connection • Connections are performed by a deterministic local planner that generates quasi-static paths between pairs of configurations. Edges**Path Planning Algorithm**• Node Connection • Local planner checks for collisions and admissibility Not a valid edge Violates elasticity limits Not a valid edge There 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**Experimental Results**• Bending Plate • 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**Experimental Results**• Bending Plate N = 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 Plate • 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)**Experimental Results**• Bending Plate N = 200 K = 40 M = 100 Avg run time – 4 hrs 12 min Avg # nodes – 33,600**Experimental Results**• Bending Plate 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

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