Exact Smooth Reformulations for Trajectory Optimization Under Signal Temporal Logic Specifications

2025-11-10

AI Summary

This work addresses motion planning under Signal Temporal Logic (STL) constraints by formulating robustness-driven trajectory optimization as a differentiable nonlinear program. To overcome the fundamental non-differentiability of STL’s max/min logical operators, we propose, for the first time, an exact smooth reconstruction method that introduces no approximation error while preserving the original Boolean and temporal semantics rigorously—thereby achieving full differentiability and mathematical completeness of STL constraints. The resulting formulation enables efficient gradient-based optimization with theoretical soundness and computational tractability. Numerical experiments demonstrate that our framework consistently generates high-quality trajectories satisfying complex spatiotemporal logic specifications, outperforming existing approximate methods in accuracy, convergence reliability, and expressive power for task specification.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Planning, Routing, and Scheduling: Replanning and Plan Repair

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

We study motion planning under Signal Temporal Logic (STL), a useful formalism for specifying spatial-temporal requirements. We pose STL synthesis as a trajectory optimization problem leveraging the STL robustness semantics. To obtain a differentiable problem without approximation error, we introduce an exact reformulation of the max and min operators. The resulting method is exact, smooth, and sound. We validate it in numerical simulations, demonstrating its practical performance.

Problem

• Exact smooth reformulation of STL trajectory optimization
• Differentiable problem without approximation error introduced
• Motion planning under Signal Temporal Logic specifications

Innovation

• Exact reformulation of max and min operators
• Differentiable trajectory optimization without approximation
• Smooth STL synthesis using robustness semantics