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RRT* Based Optimal Trajectory Generation with Linear Temporal Logic Specifications under Kinodynamic Constraints

Saksham Gautam, Ratnangshu Das, Pushpak Jagtap

TL;DR

A novel RRT*-based strategy for generating kinodynamically feasible paths that satisfy temporal logic specifications that integrates a robustness metric for Linear Temporal Logics with the system's motion constraints, ensuring that the resulting trajectories are both optimal and executable.

Abstract

In this paper, we present a novel RRT*-based strategy for generating kinodynamically feasible paths that satisfy temporal logic specifications. Our approach integrates a robustness metric for Linear Temporal Logics (LTL) with the system's motion constraints, ensuring that the resulting trajectories are both optimal and executable. We introduce a cost function that recursively computes the robustness of temporal logic specifications while penalizing time and control effort, striking a balance between path feasibility and logical correctness. We validate our approach with simulations and real-world experiments in complex environments, demonstrating its effectiveness in producing robust and practical motion plans. This work represents a significant step towards expanding the applicability of motion planning algorithms to more complex, real-world scenarios.

RRT* Based Optimal Trajectory Generation with Linear Temporal Logic Specifications under Kinodynamic Constraints

TL;DR

A novel RRT*-based strategy for generating kinodynamically feasible paths that satisfy temporal logic specifications that integrates a robustness metric for Linear Temporal Logics with the system's motion constraints, ensuring that the resulting trajectories are both optimal and executable.

Abstract

In this paper, we present a novel RRT*-based strategy for generating kinodynamically feasible paths that satisfy temporal logic specifications. Our approach integrates a robustness metric for Linear Temporal Logics (LTL) with the system's motion constraints, ensuring that the resulting trajectories are both optimal and executable. We introduce a cost function that recursively computes the robustness of temporal logic specifications while penalizing time and control effort, striking a balance between path feasibility and logical correctness. We validate our approach with simulations and real-world experiments in complex environments, demonstrating its effectiveness in producing robust and practical motion plans. This work represents a significant step towards expanding the applicability of motion planning algorithms to more complex, real-world scenarios.

Paper Structure

This paper contains 23 sections, 2 theorems, 32 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries and Problem Formulation
  3. Notation
  4. System Definition
  5. Linear Temporal Logic (LTL)
  6. Problem Formulation
  7. Cost Function
  8. Control Cost
  9. TLTL Cost Computation through Robustness Semantics
  10. Overall Cost Function
  11. Optimal Trajectory
  12. Linear System Representation
  13. Optimal Control for Fixed Final State and Time
  14. Optimal Arrival Time
  15. Optimal Trajectory between Two States
  16. ...and 8 more sections

Key Result

Theorem 1

Given initial state $x(0) = \mathsf{x}_{0}$ and fixed final state $\mathsf{x}_{1}$ at a specified time $\tau_1$, i.e., $x(\tau_1) = \mathsf{x}_{1}$, the optimal control policy $u(t)$ is given by: where $G(0,\tau_1)$ is the weighted controllability Gramian: where $R > 0$ is a control-weighting matrix as defined in eq:cost_kd. The term $\hat{x}(\tau_1)$ represents the state evolution at time $\tau

Figures (4)

  • Figure 1: Real-world demonstration of optimal, kinodynamically feasible trajectories, meeting LTL specifications.
  • Figure 2: Simulation results for mobile robot under Specification $\psi_1$.
  • Figure 3: Comparison of simulation results with and without kinodynamic (KD) constraints for mobile robot under Specification $\psi_2$.
  • Figure 4: Navigation of UAV under Specification $\psi_3$.

Theorems & Definitions (5)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • proof
  • proof