Conditional Timed Partial Orders: An Expressive and Interpretable Framework for Robot Task Specification and Planning

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.

Efficient task allocation for large-scale Heterogeneous Multi-Robot Systems (HMRS) is critical, yet dealing with complex temporal logic tasks in partially known environment (PKE) remains a computational bottleneck. Existing approaches often struggle to balance exploring uncertain regions and exploiting known resources, while also suffering from exponential computational complexity. To address these issues, this paper presents a robust planning framework that simultaneously handles high-level logical constraints and environmental uncertainty without sacrificing scalability. We formulate the problem as a min-max regret optimization, proposing a Region-Binding Atomic Proposition (RbAP) to capture resource uncertainty within the automaton structure. To solve this, we propose the Extended Planning Decision Tree (E-PDT) equipped with a novel Regret-based Branch-and-Bound (BnB) strategy. Unlike traditional methods that rely on prior probabilities or worst-case analysis, our approach dynamically prunes suboptimal policies, effectively balancing the need for information gathering (exploration) and task completion (exploitation). Theoretical analysis confirms the feasibility and completeness of our approach. Extensive numerical and physical experiments demonstrate that the proposed framework achieves near-linear scalability with respect to the number of robots and types, significantly outperforming MILP-based baselines in both solution quality and computational efficiency.

Large language models (LLMs) remain limited in multi-agent planning because independently generated plans can create coordination failures such as spatial collisions, resource contention, and temporal deadlocks. We introduce Tensor-Coord, a multilinear algebra framework that represents the joint plan of N agents as a third-order tensor $T \in R^{N \times H \times A}$ over agents, timesteps, and actions. Canonical Polyadic (CP) and Tucker decompositions are used to identify latent coordination structure. The minimal epsilon-approximate CP rank R* defines a computable coordination complexity measure, with $CC(Pi)=(R*-N)/N$. We prove that R*=N is necessary and sufficient for plan independence. The residual $E=T-T_{R*}$ defines a conflict score over agent pairs, timesteps, and actions, localizing failures without domain-specific rules. Tucker factors provide interpretable agent roles, temporal phases, and action clusters that are converted into natural language constraints for iterative LLM replanning. Experiments on multi-robot delivery tasks across Easy (2 agents, 5x5 grid), Medium (3 agents, 5x5 grid), and Hard (4 agents, 5x5 grid) settings show convergence to conflict-free plans in 100% of 2-agent cases within 1.4 iterations on average, 80% of 3-agent cases within 3.2 iterations, and 60% of 4-agent cases within 4.0 iterations. CP rank scaled approximately linearly as $R*(N) = 3.9N + 0.5$, supporting its use as a predictor of coordination complexity.