Many sampling-based methods can be adapted from to without much difficulty. The time dependency of obstacle models must be taken into account when verifying that path segments are collision-free; the techniques from Section 5.3.4 can be extended to handle this. One important concern is the metric for . For some algorithms, it may be important to permit the use of a pseudometric because symmetry is broken by time (going backward in time is not as easy as going forward).

For example, suppose that the C-space is a metric space, . The metric can be extended across time to obtain a pseudometric, , as follows. For a pair of states, and , let

Using , several sampling-based methods naturally work. For example, RDTs from Section 5.5 can be adapted to . Using for a single-tree approach ensures that all path segments travel forward in time. Using bidirectional approaches is more difficult for time-varying problems because is usually not a single point. It is not clear which should be the starting vertex for the tree from the goal; one possibility is to initialize the goal tree to an entire time-invariant segment. The sampling-based roadmap methods of Section 5.6 are perhaps the most straightforward to adapt. The notion of a

Steven M LaValle 2020-08-14