Two important subsets of are used in the manipulation planning problem. See Figure 7.15. Let denote the set of stable part configurations, which are configurations at which the part can safely rest without any forces being applied by the manipulator. This means that a part cannot, for example, float in the air. It also cannot be in a configuration from which it might fall. The particular stable configurations depend on properties such as the part geometry, friction, mass distribution, and so on. These issues are not considered here. From this, let be the corresponding stable configurations, defined as
We must always ensure that either or . Therefore, let , to reflect the subset of that is permissible for manipulation planning.
The mode space, , contains two modes, which are named the transit mode and the transfer mode. In the transit mode, the manipulator is not carrying the part, which requires that . In the transfer mode, the manipulator carries the part, which requires that . Based on these simple conditions, the only way the mode can change is if . Therefore, the manipulator has two available actions only when it is in these configurations. In all other configurations the mode remains unchanged. For convenience, let denote the set of transition configurations, which are the places in which the mode may change.
Using the framework of Section 7.3.1, the mode space, , and C-space, , are combined to yield the state space, . Since there are only two modes, there are only two copies of , one for each mode. State-based sets, , , , and , are directly obtained from , , , and by ignoring the mode. For example,
The task can now be defined. An initial part configuration, , and a goal part configuration, , are specified. Compute a path such that and . Furthermore, the action trajectory must be specified to indicate the appropriate mode changes whenever . A solution can be considered as an alternating sequence of transit paths and transfer paths, whose names follow from the mode. This is depicted in Figure 7.16.
Steven M LaValle 2020-08-14