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Manipulation Planning

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In 1995 Alami, Laumond and T. Simeon proposed to solve the problem by building and searching a ‘manipulation graph’.

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The problem:

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Constraints: The movable objects cannot move by itself. The movable objects cannot be left alone in unstable positions.

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The problem: We consider the composite configuration space of the robot and all movable objects: CS = CSR x (CSA x CSB x…) Where CSR is the configuration space of the robot and CSA, CSB,… are the configuration spaces of the movable objects.

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The composite C-space CS:

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We obtain a ‘manipulation-path’ (solid blue line) between two configurations in the composite C-space. Not any path (like the dotted ones) in the free space is a manipulation path.

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A manipulation path: Satisfies the physical constraints of the problem Consists of alternation of: Transit Path Transfer Path

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Transit Path: The robot moves from its current configuration, to any configuration that enables the part to grasping an object. Transfer Path: The robot carries the grasped object to its desired goal configuration.

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A manipulation path: Satisfies the physical constraints of the problem Consists of alternation of: Transit Path Transfer Path

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Few more definitions… PLACEMENT: the subspace of free CS containing all valid placements for all objects, i.e. placements which respect the physical constraints. In PLACEMENT: Not in PLACEMENT:

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Few more definitions… G-connectivity: 2 configurations of free(CS) are g-connected if they are connected by a transfer path.

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Few more definitions… GRASP: the subspace of free(CS) containing the configurations which are g-connected with a configuration of PLACEMENT.

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The case of discrete PLACEMENTS and GRASPS for several movable objects:

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The case of discrete PLACEMENTS and GRASPS for several movable objects: Consider a robot R and two moveable objects A and B. Let p 1 A, p 2 A,… є CSA and p 1 B, p 2 B,… є CSB the valid placements for A and B respectively.

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The case of discrete PLACEMENTS and GRASPS for several movable objects: Consider a robot R and two moveable objects A and B. Let p 1 A, p 2 A,… є CSA and p 1 B, p 2 B,… є CSB the valid placements for A and B respectively. Let G A 1, G A 2,… and G B 1, G B 2,… be finite number of possible grasps for A and B.

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The case of discrete PLACEMENTS and GRASPS for several movable objects: For a given start and goal position in the composite C-space CS = CR x CA x CB, obtain a manipulation path by building and searching a manipulation graph.

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Building the manipulation graph: Building the nodes Nodes of the graph is the set GRASP ∩ PLACEMENT

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GRASP: PLACEMENT: GRASP ∩ PLACEMENT:

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Building the manipulation graph: Building the nodes Nodes of the graph is the set GRASP ∩ PLACEMENT

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Building the manipulation graph: Building the nodes Nodes of the graph is the set GRASP ∩ PLACEMENT Building the edges Nodes are linked by transit edges or transfer edges.

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Building the manipulation graph: Building the nodes Nodes of the graph is the set GRASP ∩ PLACEMENT Building the edges Nodes are linked by transit edges or transfer edges. A transit edge exists between two nodes if they belong to the same transit state C(_, p A i, p B j ).

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Transit State:

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Building the manipulation graph: Building the nodes Nodes of the graph is the set GRASP ∩ PLACEMENT Building the edges Nodes are linked by transit edges or transfer edges. A transit edge exists between two nodes if they belong to the same transit state C(_, p A i, p B j ).

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Building the manipulation graph: Building the nodes Nodes of the graph is the set GRASP ∩ PLACEMENT Building the edges Nodes are linked by transit edges or transfer edges. A transit edge exists between two nodes if they belong to the same transit state C(_, p A i, p B j ). A transfer edge can be build between two nodes if they belong to the same transfer state C(G A i, _, p B j ) or C(G B i, p A j,_).

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Transfer State:

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Building the manipulation graph: Building the nodes Nodes of the graph is the set GRASP ∩ PLACEMENT Building the edges Nodes are linked by transit edges or transfer edges. A transit edge exists between two nodes if they belong to the same transit state C(_, p A i, p B j ). A transfer edge can be build between two nodes if they belong to the same transfer state C(G A i, _, p B j ) or C(G B i, p A j,_).

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The manipulation graph looks like :

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Finally graph search is done to find a manipulation path.

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Implementation: Manipulation Task Planner Builds the manipulation graph and searches a manipulation path using A* Algorithm. Motion Planner Computes the edges of the graph.

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Extension from discrete to infinite set of grasps: Now the discrete configurations of the set GRASP ∩ PLACEMENT is replaced by connected components of GRASP ∩ PLACEMENT obtained by its cell decomposition.

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To end with:

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