Fast Constraint Extraction for Corrective Control under STL Specifications via Logical Dependency Tracking
Authors: Antoine Besset, Joris Tillet, Chuchu Fan, Julien Alexandre dit Sandretto
Abstract
Ensuring the satisfaction of Signal Temporal Logic (STL) specifications under uncertainty is challenging, as reachability-based monitoring provides guarantees but does not indicate how to restore satisfaction when it becomes indeterminate. A key difficulty is identifying which uncertain components actually affect global satisfaction, especially for nested formulas. This paper introduces a logical dependency tracking framework that propagates uncertainty through the STL structure and captures the causal contribution of reachable sets to satisfaction. By associating markers to uncertain predicates and propagating them via three-valued semantics, we extract in milliseconds a compact Disjunctive Normal Form (DNF) of sufficient constraints, avoiding combinatorial enumeration. As an application, we formulate control correction as a minimum-effort optimization problem. Using zonotopic reachability, the derived constraints are enforced via linear programming, yielding corrections that guarantee STL satisfaction under bounded uncertainty and provide certified probabilistic bounds in the stochastic case. We demonstrate the approach on a nonlinear system with nested STL specifications, showing that dependency tracking enables efficient and formally guaranteed correction. The tracking implementation is available at https://github.com/Antoine-Bst/STL-Three-Valued-Clause-Filtering/.
Signal Temporal Logic (STL), has recently seen extensive development, owing to its rich expressivenes for autonomous planning and control. Nevertheless, existing verification and control synthesis methods are limited with respect to the complexity and degree of nesting of the formulae. In this work, we propose a novel approach to STL based on an operator acting on reachability value functions. This constitutes a new theoretical framework for handling complex multi-nested formulae while at the same time providing tools for on-line control synthesis. In contrast to focusing on the design of STL-based reachability (or control barrier) functions, we develop operator-based nesting rules directly. Our method's expressiveness is demonstrated both theoretically, where necessary and sufficient conditions for STL formula satisfaction are extracted, as well as in simulations with complex fragments.
We present a unified trajectory planning and control framework for the satisfaction of Signal Temporal Logic (STL) specifications defined over convex predicates. At the planning layer, STL tasks are encoded as time-varying convex sets in configuration space, specifically designed so that forward invariance of the system with respect to these sets implies satisfaction of the specification with a prescribed robustness margin. This representation is then lifted to the joint time--configuration space and combined with the Graphs of Convex Sets (GCS) framework, yielding a shortest-path formulation of the planning problem over convex spatio-temporal sets. Trajectories are parameterized by B-splines, which enable continuous-time enforcement of STL satisfaction, collision avoidance, and smoothness constraints. At the control layer, the same time-varying sets used for planning are exploited to design a feedback controller that tracks the planned trajectory while prioritizing satisfaction of the STL specification during execution in the presence of tracking errors and model mismatch. We validate the proposed approach in simulation and in real-world experiments on space robotic platforms.
Nicola De Carli, Gregorio Marchesini, Dimos V. Dimarogonas
We study feedback motion planning for continuous-time stochastic nonlinear systems under signal temporal logic (STL) specifications. We propose a framework that synthesizes control policies for chance-constrained STL trajectory optimization problems, with the goal of ensuring that the closed-loop stochastic system satisfies a given STL formula with high probability (e.g., 99.99%). Our approach is based on a predicate erosion strategy that transforms the intractable stochastic problem into a deterministic STL trajectory optimization problem with tightened STL formula constraints. The amount of erosion is determined by a probabilistic reachable tube (PRT) that bounds the deviation between the stochastic trajectory and an associated nominal trajectory. To compute such bounds, we leverage contraction theory and feedback design, and develop several tracking controllers. This yields a complete feedback motion planning pipeline which can be implemented by numerical optimizations. We demonstrate the efficacy and versatility of the proposed framework through simulations on several robotic systems and through experiments on a real-world quadrupedal robot, and show that it is less conservative and achieves higher specification satisfaction probability than representative baselines.