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Signal Temporal Logic Planning with Time-Varying Robustness

Yating Yuan, Thanin Quartz, Jun Liu

TL;DR

This letter aims to generate a continuous-time trajectory consisting of piecewise Bézier curves that satisfy signal temporal logic (STL) specifications with piecewise time-varying robustness, which enables more effective tracking in practical applications.

Abstract

This letter aims to generate a continuous-time trajectory consisting of piecewise Bézier curves that satisfy signal temporal logic (STL) specifications with piecewise time-varying robustness. Our time-varying robustness is less conservative than the real-valued robustness, which enables more effective tracking in practical applications. Specifically, our continuous-time trajectories account for dynamic feasibility, leading to smaller tracking errors and ensuring that the STL specifications can be met by the tracking trajectory. Comparative experiments demonstrate the efficiency and effectiveness of the proposed approach.

Signal Temporal Logic Planning with Time-Varying Robustness

TL;DR

This letter aims to generate a continuous-time trajectory consisting of piecewise Bézier curves that satisfy signal temporal logic (STL) specifications with piecewise time-varying robustness, which enables more effective tracking in practical applications.

Abstract

This letter aims to generate a continuous-time trajectory consisting of piecewise Bézier curves that satisfy signal temporal logic (STL) specifications with piecewise time-varying robustness. Our time-varying robustness is less conservative than the real-valued robustness, which enables more effective tracking in practical applications. Specifically, our continuous-time trajectories account for dynamic feasibility, leading to smaller tracking errors and ensuring that the STL specifications can be met by the tracking trajectory. Comparative experiments demonstrate the efficiency and effectiveness of the proposed approach.
Paper Structure (17 sections, 2 theorems, 11 equations, 2 figures, 1 table)

This paper contains 17 sections, 2 theorems, 11 equations, 2 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. Preliminary
  3. Bézier Curves
  4. Signal Temporal Logic
  5. Problem Statement
  6. Trajectory Construction
  7. Continuity Constraints
  8. Dynamic Constraints
  9. Encoding STL Satisfactions with Bézier Curves
  10. Encoding STL Satisfactions with Linear Constraints
  11. The Soundness of STL Encoding
  12. Experiments
  13. Mission Scenarios
  14. Comparison with Other Algorithms
  15. Run time
  16. ...and 2 more sections

Key Result

Proposition 1

For $t\in [t_k, t_{k+1}]$, if an $n$-degree Bézier curve $B_k(t)$ satisfies eq.predicate, then $B_k \subset \text{Poly}(H,b)$. Similarly, if $B_k$ satisfies eq.negation, then $B_k \cap \text{Poly}(H,b) = \emptyset$.

Figures (2)

  • Figure 1: A simple reach-avoid task: (a) presents a reference trajectory (blue solid line) with a real-valued robustness tube (blue shadow) and its tracking result (red dashed line); (b) shows a trajectory intersecting obstacles between time steps; (c) illustrates that the Bézier curves (blue solid line) violate the specifications due to incorrectly chosen control points. The red dashed lines are linear segments between the end points of the Bézier curves and the blue frames are the bounding boxes of the curves.
  • Figure 2: Benchmarks and Results. The first row presents the planning outcomes, where solid lines depict the Bézier curves generated by the proposed method, and dashed lines represent the actual trajectory followed by the vehicle as it tracks these curves. Circles mark the starting points of the trajectories, while stars indicate the goal locations. The second row shows the tracking errors, with red lines illustrating the time-varying robustness and blue lines representing the distance tracking errors.

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • proof
  • Theorem 1
  • proof