[Paper Review] Robust Motion Planning employing Signal Temporal Logic

Lars Lindemann, Dimos V. Dimarogonas|arXiv (Cornell University)|Mar 6, 2017

Formal Methods in Verification4 citations

TL;DR

This paper introduces Discrete Average Space Robustness (DASR), a novel robust semantics for Signal Temporal Logic (STL), integrated into a linear Model Predictive Control (MPC) framework to enable robust motion planning. By directly maximizing DASR in a convex Linear Program, the method ensures improved average satisfaction of temporal specifications and enhanced noise resilience, demonstrated in simulations for a planar robot navigating multiple regions while avoiding obstacles.

ABSTRACT

Motion planning classically concerns the problem of accomplishing a goal configuration while avoiding obstacles. However, the need for more sophisticated motion planning methodologies, taking temporal aspects into account, has emerged. To address this issue, temporal logics have recently been used to formulate such advanced specifications. This paper will consider Signal Temporal Logic in combination with Model Predictive Control. A robustness metric, called Discrete Average Space Robustness, is introduced and used to maximize the satisfaction of specifications which results in a natural robustness against noise. The comprised optimization problem is convex and formulated as a Linear Program.

Motivation & Objective

• To address the limitations of traditional motion planning by incorporating temporal logic specifications that capture complex, time-dependent behaviors.
• To overcome the state explosion problem in LTL-based methods by using Signal Temporal Logic (STL) with quantitative robustness metrics.
• To develop a computationally efficient, robust control synthesis method that maximizes specification satisfaction under noise and uncertainty.
• To enable practical motion planning for robotic systems by combining STL with convex optimization via linear programming.

Proposed method

• Proposes Discrete Average Space Robustness (DASR), a new robustness metric that evaluates average satisfaction across time and space, improving upon worst-case-focused Space Robustness (SR).
• Integrates DASR directly into the cost function of a linear MPC formulation, enabling direct optimization of robustness against noise and disturbances.
• Models the system dynamics as a discrete-time linear time-invariant (LTI) system with state, output, and input vectors, and uses affine predicate mappings to express STL formulas.
• Encodes the robustness computation as a linear program (LP), ensuring convexity and computational efficiency compared to mixed-integer or non-convex formulations.
• Uses the infinity norm to define rectangular regions in the workspace, enabling linear representation of predicates for tractable optimization.
• Employs recursive feasibility and past satisfaction concepts to ensure robustness and stability in the MPC framework.

Experimental results

Research questions

• RQ1 Can a new robustness metric based on average satisfaction improve control performance compared to worst-case robustness in STL-based control?
• RQ2 Can DASR be efficiently integrated into a linear MPC framework to achieve robustness without resorting to mixed-integer programming?
• RQ3 How does the proposed method perform under process noise and model uncertainty in a motion planning context?
• RQ4 Can the method ensure robust satisfaction of complex STL specifications involving temporal and spatial constraints in a planar workspace?

Key findings

• The proposed method achieves robust motion planning by directly maximizing Discrete Average Space Robustness (DASR), resulting in improved resilience to noise and disturbances.
• The optimization problem is formulated as a convex Linear Program (LP), enabling low computational cost and real-time applicability compared to non-convex MILP approaches.
• Simulations show that the MPC controller maintains specification satisfaction even under significant noise, with a Signal-to-Noise Ratio (SNR) of 16.23 dB, demonstrating robustness.
• The method successfully guides a robot to visit three designated regions within a 5–25 second window while respecting safety constraints, as verified in both noise-free and noisy scenarios.
• The use of DASR leads to better average performance than worst-case-focused robustness metrics, as it avoids over-conservatism in control design.
• The framework supports complex STL specifications involving multiple temporal and spatial requirements, such as visiting multiple regions and maintaining safety, in a computationally efficient manner.