Multiple parts

The manipulation planning framework nicely generalizes to multiple parts, ${\cal P}_1$, $\ldots$, ${\cal P}_k$. Each part has its own C-space, and ${\cal C}$ is formed by taking the Cartesian product of all part C-spaces with the manipulator C-space. The set ${{\cal C}_{adm}}$ is defined in a similar way, but now part-part collisions also have to be removed, in addition to part-manipulator, manipulator-obstacle, and part-obstacle collisions. The definition of ${{\cal C}_{sta}}$ requires that all parts be in stable configurations; the parts may even be allowed to stack on top of each other. The definition of ${{\cal C}_{gr}}$ requires that one part is grasped and all other parts are stable. There are still two modes, depending on whether the manipulator is grasping a part. Once again, transitions occur only when the robot is in ${{\cal C}_{tra}}= {{\cal C}_{sta}}\cap {{\cal C}_{gr}}$. The task involves moving each part from one configuration to another. This is achieved once again by defining a manipulation graph and obtaining a sequence of transit paths (in which no parts move) and transfer paths (in which one part is carried and all other parts are fixed). Challenging manipulation problems solved by motion planning algorithms are shown in Figures 7.17 and 7.18.

Other generalizations are possible. A generalization to $k$ robots would lead to $2^k$ modes, in which each mode indicates whether each robot is grasping the part. Multiple robots could even grasp the same object. Another generalization could allow a single robot to grasp more than one object.