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Spatiotemporal Tubes for Differential Drive Robots with Model Uncertainty

Ratnangshu Das, Ahan Basu, Christos Verginis, Pushpak Jagtap

TL;DR

The paper tackles safe, time-bounded navigation for differential-drive robots under model uncertainty by introducing Spatiotemporal Tubes (STTs) with circular cross-sections to create dynamic safe corridors that enforce Temporal Reach-Avoid-Stay (T-RAS). It develops a sampling-based synthesis procedure to construct STTs that originate in the start region, avoid obstacles, and reach the target by a prescribed time, along with a closed-form, approximation-free control law that guarantees the robot remains within the STT despite disturbances. The approach yields robust performance and reduces computation compared to state-of-the-art methods like CBF and MPC, as demonstrated in simulations including cluttered environments and dynamic obstacles. The work provides a principled, scalable framework for robust, time-constrained navigation in the presence of uncertainty, with potential extensions to higher-order dynamics and explicit input constraints.

Abstract

This paper presents a Spatiotemporal Tube (STT)-based control framework for differential-drive mobile robots with dynamic uncertainties and external disturbances, guaranteeing the satisfaction of Temporal Reach-Avoid-Stay (T-RAS) specifications. The approach employs circular STT, characterized by smoothly time-varying center and radius, to define dynamic safe corridors that guide the robot from the start region to the goal while avoiding obstacles. In particular, we first develop a sampling-based synthesis algorithm to construct a feasible STT that satisfies the prescribed timing and safety constraints with formal guarantees. To ensure that the robot remains confined within this tube, we then design analytically a closed-form, approximation-free control law. The resulting controller is computationally efficient, robust to disturbances and {model uncertainties}, and requires no model approximations or online optimization. The proposed framework is validated through simulation studies on a differential-drive robot and benchmarked against state-of-the-art methods, demonstrating superior robustness, accuracy, and computational efficiency.

Spatiotemporal Tubes for Differential Drive Robots with Model Uncertainty

TL;DR

The paper tackles safe, time-bounded navigation for differential-drive robots under model uncertainty by introducing Spatiotemporal Tubes (STTs) with circular cross-sections to create dynamic safe corridors that enforce Temporal Reach-Avoid-Stay (T-RAS). It develops a sampling-based synthesis procedure to construct STTs that originate in the start region, avoid obstacles, and reach the target by a prescribed time, along with a closed-form, approximation-free control law that guarantees the robot remains within the STT despite disturbances. The approach yields robust performance and reduces computation compared to state-of-the-art methods like CBF and MPC, as demonstrated in simulations including cluttered environments and dynamic obstacles. The work provides a principled, scalable framework for robust, time-constrained navigation in the presence of uncertainty, with potential extensions to higher-order dynamics and explicit input constraints.

Abstract

This paper presents a Spatiotemporal Tube (STT)-based control framework for differential-drive mobile robots with dynamic uncertainties and external disturbances, guaranteeing the satisfaction of Temporal Reach-Avoid-Stay (T-RAS) specifications. The approach employs circular STT, characterized by smoothly time-varying center and radius, to define dynamic safe corridors that guide the robot from the start region to the goal while avoiding obstacles. In particular, we first develop a sampling-based synthesis algorithm to construct a feasible STT that satisfies the prescribed timing and safety constraints with formal guarantees. To ensure that the robot remains confined within this tube, we then design analytically a closed-form, approximation-free control law. The resulting controller is computationally efficient, robust to disturbances and {model uncertainties}, and requires no model approximations or online optimization. The proposed framework is validated through simulation studies on a differential-drive robot and benchmarked against state-of-the-art methods, demonstrating superior robustness, accuracy, and computational efficiency.

Paper Structure

This paper contains 14 sections, 3 theorems, 24 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Notation
  4. System Definition
  5. System Specification
  6. Spatiotemporal Tube (STT)
  7. Proof:
  8. Proof:
  9. Controller Synthesis
  10. Proof:
  11. Case Studies
  12. Navigation in a Cluttered Office-Like Environment
  13. Reach-Avoid Task with Time-Varying Unsafe Set
  14. Conclusion and Future Work

Key Result

Lemma 3.2

If the point-to-set distances of two points $y_1$ and $y_2$ from a set ${\mathbf{A}}$ is defined as $\text{dist}(y_1,{\mathbf{A}})$ and $\text{dist}(y_2,{\mathbf{A}})$, then $\text{dist}(y_1,{\mathbf{A}}) - \text{dist}(y_2,{\mathbf{A}}) \leq \| y_1 - y_2 \|$.

Figures (4)

  • Figure 1: Robot inside circular STT
  • Figure 2: The constructed STT and the corresponding vehicle trajectory navigating through an office space.
  • Figure 3: The constructed STT and the corresponding vehicle trajectory in a dynamic environment with time-varying obstacles.
  • Figure 4: Comparison with existing approaches

Theorems & Definitions (7)

  • Definition 2.1: Temporal Reach-Avoid-Stay Task
  • Definition 3.1: STT for T-RAS Specification
  • Lemma 3.2
  • Theorem 3.3
  • Remark 3.4
  • Theorem 4.1
  • Remark 4.2