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Spatiotemporal Tubes for Probabilistic Temporal Reach-Avoid-Stay Task in Uncertain Dynamic Environment

Siddhartha Upadhyay, Ratnangshu Das, Pushpak Jagtap

TL;DR

The paper addresses safe control for nonlinear systems operating in dynamic, uncertain environments where obstacle centers are uncertain and modeled probabilistically. It extends the Spatiotemporal Tube (STT) framework to Probabilistic Temporal Reach-Avoid-Stay (PrT-RAS) by deriving a real-time, approximation-free, model-free, and optimization-free closed-form controller that confines the system within a time-varying tube with probabilistic safety guarantees. Key contributions include a formal PrT-RAS formulation, online tube center and radius adaptation using non-central chi-square-based collision probabilities, finite-time convergence to the target, and extensive validation through 2D hardware experiments, 3D UAV simulations, and a 7-DOF manipulator with disturbances. The framework demonstrates scalable, real-time probabilistic safety in cluttered, uncertain environments without relying on exact dynamics, promising practical impact for autonomous robots in safety-critical tasks. Future directions include distributionally robust extensions and incorporating probabilistic temporal logic specifications.

Abstract

In this work, we extend the Spatiotemporal Tube (STT) framework to address Probabilistic Temporal Reach-Avoid-Stay (PrT-RAS) tasks in dynamic environments with uncertain obstacles. We develop a real-time tube synthesis procedure that explicitly accounts for time-varying uncertain obstacles and provides formal probabilistic safety guarantees. The STT is formulated as a time-varying ball in the state space whose center and radius evolve online based on uncertain sensory information. We derive a closed-form, approximation-free control law that confines the system trajectory within the tube, ensuring both probabilistic safety and task satisfaction. Our method offers a formal guarantee for probabilistic avoidance and finite-time task completion. The resulting controller is model-free, approximation-free, and optimization-free, enabling efficient real-time execution while guaranteeing convergence to the target. The effectiveness and scalability of the framework are demonstrated through simulation studies and hardware experiments on mobile robots, a UAV, and a 7-DOF manipulator navigating in cluttered and uncertain environments.

Spatiotemporal Tubes for Probabilistic Temporal Reach-Avoid-Stay Task in Uncertain Dynamic Environment

TL;DR

The paper addresses safe control for nonlinear systems operating in dynamic, uncertain environments where obstacle centers are uncertain and modeled probabilistically. It extends the Spatiotemporal Tube (STT) framework to Probabilistic Temporal Reach-Avoid-Stay (PrT-RAS) by deriving a real-time, approximation-free, model-free, and optimization-free closed-form controller that confines the system within a time-varying tube with probabilistic safety guarantees. Key contributions include a formal PrT-RAS formulation, online tube center and radius adaptation using non-central chi-square-based collision probabilities, finite-time convergence to the target, and extensive validation through 2D hardware experiments, 3D UAV simulations, and a 7-DOF manipulator with disturbances. The framework demonstrates scalable, real-time probabilistic safety in cluttered, uncertain environments without relying on exact dynamics, promising practical impact for autonomous robots in safety-critical tasks. Future directions include distributionally robust extensions and incorporating probabilistic temporal logic specifications.

Abstract

In this work, we extend the Spatiotemporal Tube (STT) framework to address Probabilistic Temporal Reach-Avoid-Stay (PrT-RAS) tasks in dynamic environments with uncertain obstacles. We develop a real-time tube synthesis procedure that explicitly accounts for time-varying uncertain obstacles and provides formal probabilistic safety guarantees. The STT is formulated as a time-varying ball in the state space whose center and radius evolve online based on uncertain sensory information. We derive a closed-form, approximation-free control law that confines the system trajectory within the tube, ensuring both probabilistic safety and task satisfaction. Our method offers a formal guarantee for probabilistic avoidance and finite-time task completion. The resulting controller is model-free, approximation-free, and optimization-free, enabling efficient real-time execution while guaranteeing convergence to the target. The effectiveness and scalability of the framework are demonstrated through simulation studies and hardware experiments on mobile robots, a UAV, and a 7-DOF manipulator navigating in cluttered and uncertain environments.
Paper Structure (15 sections, 5 theorems, 35 equations, 5 figures)

This paper contains 15 sections, 5 theorems, 35 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Notation
  4. System Definition
  5. Problem Formulation
  6. Designing Spatiotemporal Tubes
  7. Controller Design
  8. Case Studies
  9. 2D Omnidirectional Robot
  10. UAV
  11. 7-DOF Manipulator
  12. Discussion
  13. Conclusion
  14. Proof of Proposition \ref{['prop:1']}
  15. Non-central chi squared distribution

Key Result

Proposition 3.1

The probability that the center ${\mathsf{\textbf{c}}}(t)$ avoids the unsafe set in eqn:prob_q can be rewritten as follows: where $\hat{Z}^{(j)}(t)$ follows a non-central chi-square distribution with degrees of freedom equal to the dimension of the state space and non-centrality parameter $\lambda^{(j)}({\mathsf{\textbf{c}}}(t),t)=\frac{\lVert {\mathsf{\textbf{c}}}(t)-\mu^{(j)}(t)\rVert^2}{(\sigm

Figures (5)

  • Figure 1: A schematic representation of the probabilistic avoidance mechanism corresponding to an uncertain obstacle $O_p^{(j)}(t)\sim\mathcal{N}(\mu^{(j)}(t),\Sigma^{(j)}(t))$.
  • Figure 2: Hardware demonstration of a omnidirectional robot in three different dynamic uncertain environment, https://youtu.be/-SaFbrn9BJM
  • Figure 3: Simulation of an omnidirectional mobile robot in a 2D uncertain environment, along with a 3D plot showing tube synthesis with time $t$, https://youtu.be/-SaFbrn9BJM
  • Figure 4: UAV simulation in a 3-D uncertain environment, where obstacles with higher uncertainty or alternatively higher $(\sigma^{(j)})^2$ values are depicted as dark red spheres, and those with lower uncertainty appear in lighter shades, https://youtu.be/-SaFbrn9BJM
  • Figure 5: Snapshot of hardware experiment at two different time stamps of a 7-DOF manipulator assigned with the task of pick and place in two different uncertain environments, https://youtu.be/-SaFbrn9BJM

Theorems & Definitions (11)

  • Definition 2.1: Time-varying Uncertain Unsafe Set
  • Remark 2.2
  • Definition 2.3: Probabilistic Temporal Reach-Avoid-Stay
  • Definition 2.5
  • Remark 2.6
  • Proposition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Remark 3.4
  • Theorem 4.1
  • ...and 1 more