Motion Planning with Temporal Logic Precedence Specifications in Dynamic Environments via Spacetime Augmented Graphs of Convex Sets

We extend the augmented graph of convex sets GCS framework for motion planning with temporal logic precedence specifications to dynamic environments in which obstacles, keys, and doors undergo piecewise-linear translation and/or dilation over time. The main insight is that piecewise-linear motion of polytopic sets in R d lifts exactly to a static polytopic partition of spacetime R d X R>=0. We show that the main result from the static case carries over to spacetime essentially without modification: a shortest path in the spacetime augmented GCS provides an exact optimal solution to the motion planning problem with precedence specifications. Moreover, the layered subgraph lattice that encodes signal temporal logic (STL) precedence specifications retains precise correspondence with the Bellman--Held--Karp (BHK) algorithm, and the full library of restricted but expressive STL specifications generalizes to the dynamic setting. Further, the framework allows time-gated keys and doors, collectible or passable only within a temporal window. The geometric side of the framework requires two modifications: a causality constraint (the time coordinate of a trajectory must be non-decreasing) and a spacetime speed constraint (bounded physical velocity), both of which are convex in the Bézier control points.