Multi-Agent Planning with Spatio-Temporal and Topological Constraints using STL-GO

Spatio-Temporal Logic with Graph Operators (STL-GO) extends Signal Temporal Logic (STL) to multi-agent systems via graph operators that count neighboring agents satisfying a property, together with multi-agent quantifiers. While Boolean semantics for STL-GO are well-defined, quantitative semantics have not yet been developed and existing quantitative semantics for spatio-temporal logics such as STREL cannot capture the counting constraints in STL-GO's graph operators. We develop quantitative semantics for STL-GO as a layered algebraic construction that separates temporal aggregation from graph-operator aggregation (governed by an abstract accumulator with a monotone fold and readout). We prove that soundness and completeness reduce to monotonicity conditions on these components. We implement the framework and evaluate it on two multi-agent environments: a 2D bounded region with stochastic Dubins-car dynamics and a 3D Earth-satellite system, under four semantic instantiations (Boolean, min-max, signed-deficit, and a hybrid), demonstrating the tradeoffs between accumulator choices and reporting scalability in the number of agents and time horizon.

Spatiotemporal motion planning, especially in multi-robot settings, requires robots to reason about collision-free regions that change over time, which is challenging in continuous spaces when feasible regions are transient and geometrically constrained. We present an algorithmic framework based on graphs of space-time convex sets (ST-GCSs), where collision-free regions are represented as convex sets in space-time and trajectories correspond to paths on the graph together with continuous motions within the selected sets. We formulate time-optimal planning on ST-GCSs as a graph-search problem over path-indexed states and develop a best-first search solver that evaluates partial paths via continuous trajectory optimization, guided by admissible heuristics and dominance checks. We further present an Exact Convex Decomposition (ECD) scheme to reserve trajectory occupancies in space-time, enabling unified handling of dynamic obstacles and multi-robot interactions. For multi-robot motion planning, we integrate ST-GCS planning and ECD into prioritized planning methods and introduce a windowed coordination scheme to improve efficiency. Extensive experiments on single-robot and multi-robot problems demonstrate substantial speedups over various planners while maintaining high solution quality, particularly in environments with narrow and transient feasible regions. Large-scale demonstrations further show that the proposed multi-robot motion planner can solve instances with up to $100$ robots within only a few minutes. Project homepage: https://sites.google.com/view/stgcs

This paper investigates continuous-time motion planning under Signal Temporal Logic (STL) specifications. The goal is to generate smooth robot trajectories that satisfy high-level logical and timing requirements while respecting low-level motion constraints. To this end, we propose an efficient framework that combines timed-automata reasoning with graphs of convex sets (GCS). An STL specification is first represented by a timed automaton, which is then coupled with a convex decomposition of the configuration space to form a joint transition system encoding both task progress and region occupancy. Based on this joint transition system, the STL motion-planning problem is reformulated as a shortest-path problem over a GCS, whose solution induces a smooth Bézier-spline trajectory satisfying the STL specification, smoothness requirements, and velocity bounds. We establish the soundness of the proposed formulation and analyze its computational complexity, showing that, once the timed automaton and convex decomposition are fixed, the convex relaxation scales polynomially with the configuration-space dimension and the Bézier degree. We further develop a compact timed-automaton construction for an expressive STL fragment using dedicated templates and Boolean composition. Numerical experiments on low-dimensional benchmarks, a $3$-D quadrotor, a $30$-DoF humanoid, and a hardware experiment on a UR-3 robot arm demonstrate that the proposed method efficiently solves complex STL motion-planning problems and produces smooth executable trajectories.