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Trajectory Planning with Signal Temporal Logic Costs using Deterministic Path Integral Optimization

Patrick Halder, Hannes Homburger, Lothar Kiltz, Johannes Reuter, Matthias Althoff

TL;DR

Addressing trajectory planning under Signal Temporal Logic (STL) costs in a deterministic, discrete-time setting, the paper formulates a finite-horizon STL-augmented OCP and solves it with a path-integral (PI) based solver. It introduces a state-augmentation technique to cast the problem into Bolza form and derives an exact deterministic solution by progressively shrinking a temperature-like parameter $\beta$ within a Monte Carlo MPPI framework, enabling parallel sampling and convergence to the true minimizer. The method mitigates non-differentiable STL terms and nesting, achieving competitive or superior results on three benchmark problems, including challenging complex-planning scenarios where other solvers struggle. The work demonstrates a scalable, real-time-capable approach for STL-constrained trajectory planning and lays groundwork for future extensions to MPC with STL and prioritized specifications, potentially leveraging GPU acceleration for additional speedups.

Abstract

Formulating the intended behavior of a dynamic system can be challenging. Signal temporal logic (STL) is frequently used for this purpose due to its suitability in formalizing comprehensible, modular, and versatile spatiotemporal specifications. Due to scaling issues with respect to the complexity of the specifications and the potential occurrence of non-differentiable terms, classical optimization methods often solve STL-based problems inefficiently. Smoothing and approximation techniques can alleviate these issues but require changing the optimization problem. This paper proposes a novel sampling-based method based on model predictive path integral control to solve optimal control problems with STL cost functions. We demonstrate the effectiveness of our method on benchmark motion planning problems and compare its performance with state-of-the-art methods. The results show that our method efficiently solves optimal control problems with STL costs.

Trajectory Planning with Signal Temporal Logic Costs using Deterministic Path Integral Optimization

TL;DR

Addressing trajectory planning under Signal Temporal Logic (STL) costs in a deterministic, discrete-time setting, the paper formulates a finite-horizon STL-augmented OCP and solves it with a path-integral (PI) based solver. It introduces a state-augmentation technique to cast the problem into Bolza form and derives an exact deterministic solution by progressively shrinking a temperature-like parameter within a Monte Carlo MPPI framework, enabling parallel sampling and convergence to the true minimizer. The method mitigates non-differentiable STL terms and nesting, achieving competitive or superior results on three benchmark problems, including challenging complex-planning scenarios where other solvers struggle. The work demonstrates a scalable, real-time-capable approach for STL-constrained trajectory planning and lays groundwork for future extensions to MPC with STL and prioritized specifications, potentially leveraging GPU acceleration for additional speedups.

Abstract

Formulating the intended behavior of a dynamic system can be challenging. Signal temporal logic (STL) is frequently used for this purpose due to its suitability in formalizing comprehensible, modular, and versatile spatiotemporal specifications. Due to scaling issues with respect to the complexity of the specifications and the potential occurrence of non-differentiable terms, classical optimization methods often solve STL-based problems inefficiently. Smoothing and approximation techniques can alleviate these issues but require changing the optimization problem. This paper proposes a novel sampling-based method based on model predictive path integral control to solve optimal control problems with STL cost functions. We demonstrate the effectiveness of our method on benchmark motion planning problems and compare its performance with state-of-the-art methods. The results show that our method efficiently solves optimal control problems with STL costs.

Paper Structure

This paper contains 19 sections, 22 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Optimization with STL
  4. MPPI Control
  5. Contributions
  6. Preliminaries
  7. System Definition and Projection Operator
  8. Signal Temporal Logic
  9. Classical MPPI
  10. Problem Statement
  11. Solution
  12. Problem Transformation
  13. Exact Solution for Deterministic Problems with MPPI
  14. Algorithm
  15. Numerical Experiments
  16. ...and 4 more sections

Figures (3)

  • Figure 1: Example of our PI-based solution for solving an optimal control problem with STL costs. The objective is to reach the blue box eventually while ensuring that the gray circular obstacle is avoided at all times. Four intermediate solutions are presented.
  • Figure 2: STL-OCP solution of our PI solver for problem I. Also, the sampled trajectories of four iterations are presented.
  • Figure 3: Solutions for problem III. Note that we evaluate the robustness function only at discrete time steps.