1. Manipulation Planning

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

3. The problem:

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

5. 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.

6. The composite C-space CS:

10. 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.

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

12. 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.

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

14. 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:

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

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

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

18. 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.

19. 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.

20. 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.

21.  Building the manipulation graph: Building the nodes  Nodes of the graph is the set GRASP ∩ PLACEMENT

22. GRASP: PLACEMENT: GRASP ∩ PLACEMENT:

23.  Building the manipulation graph: Building the nodes  Nodes of the graph is the set GRASP ∩ PLACEMENT

24.  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.

25.  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 ).

26. Transit State:

27.  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 ).

28.  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,_).

29. Transfer State:

30.  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,_).

31. The manipulation graph looks like :

33. Finally graph search is done to find a manipulation path.

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

35. 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.

36. To end with: