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Smooth Spatiotemporal Tube Synthesis for Prescribed-Time Reach-Avoid-Stay Control

Siddhartha Upadhyay, Ratnangshu Das, Pushpak Jagtap

TL;DR

This paper tackles the problem of designing controllers for control-affine nonlinear systems to satisfy prescribed-time reach-avoid-stay (RAS) specifications. It replaces the traditional circumvent-function based STT approach with an adaptive, real-time STT design that smoothly deforms around static unsafe sets, coupled with a closed-form, approximation-free controller $u(x,t) = -\kappa\xi(x,t)\varepsilon(x,t)$ to keep trajectories inside the evolving tube and complete the task within the prescribed time horizon $t_c$. The main contributions are (i) a smooth, adaptive STT construction that avoids abrupt tube changes and reduces control effort, (ii) a closed-form control law guaranteeing PT-RAS for unknown dynamics under bounded disturbances, and (iii) a case study showing significant reductions in control effort compared to circumvent-based methods while achieving PT-RAS in a 2-D robot navigation scenario. This framework has practical impact for safe, energy-efficient prescribed-time planning in safety-critical robotic systems operating in static environments, with future extension to time-varying obstacles and broader nonlinear system classes.

Abstract

In this work, we address the issue of controller synthesis for a control-affine nonlinear system to meet prescribed time reach-avoid-stay specifications. Our goal is to improve upon previous methods based on spatiotemporal tubes (STTs) by eliminating the need for circumvent functions, which often lead to abrupt tube modifications and high control effort. We propose an adaptive framework that constructs smooth STTs around static unsafe sets, enabling continuous avoidance while guiding the system toward the target within the prescribed time. A closed-form, approximation-free control law is derived to ensure the system trajectory remains within the tube and satisfies the RAS task. The effectiveness of the proposed approach is demonstrated through a case study, showing a significant reduction in control effort compared to prior methods.

Smooth Spatiotemporal Tube Synthesis for Prescribed-Time Reach-Avoid-Stay Control

TL;DR

This paper tackles the problem of designing controllers for control-affine nonlinear systems to satisfy prescribed-time reach-avoid-stay (RAS) specifications. It replaces the traditional circumvent-function based STT approach with an adaptive, real-time STT design that smoothly deforms around static unsafe sets, coupled with a closed-form, approximation-free controller to keep trajectories inside the evolving tube and complete the task within the prescribed time horizon . The main contributions are (i) a smooth, adaptive STT construction that avoids abrupt tube changes and reduces control effort, (ii) a closed-form control law guaranteeing PT-RAS for unknown dynamics under bounded disturbances, and (iii) a case study showing significant reductions in control effort compared to circumvent-based methods while achieving PT-RAS in a 2-D robot navigation scenario. This framework has practical impact for safe, energy-efficient prescribed-time planning in safety-critical robotic systems operating in static environments, with future extension to time-varying obstacles and broader nonlinear system classes.

Abstract

In this work, we address the issue of controller synthesis for a control-affine nonlinear system to meet prescribed time reach-avoid-stay specifications. Our goal is to improve upon previous methods based on spatiotemporal tubes (STTs) by eliminating the need for circumvent functions, which often lead to abrupt tube modifications and high control effort. We propose an adaptive framework that constructs smooth STTs around static unsafe sets, enabling continuous avoidance while guiding the system toward the target within the prescribed time. A closed-form, approximation-free control law is derived to ensure the system trajectory remains within the tube and satisfies the RAS task. The effectiveness of the proposed approach is demonstrated through a case study, showing a significant reduction in control effort compared to prior methods.

Paper Structure

This paper contains 12 sections, 3 theorems, 22 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Notation
  4. System Definition
  5. Problem Formulation
  6. Designing Spatiotemporal Tubes
  7. Reachability Tubes
  8. Avoid Specification
  9. Modifying the Tube
  10. Controller Design
  11. Case Studies
  12. Conclusion

Key Result

Theorem 3.3

The STT $\Gamma(t)=\prod_{i=1,..n}[\gamma_{i,L},\gamma_{i,U}]$ as defined in eqn:gamma_U satisfies the following conditions:

Figures (3)

  • Figure 1: An example of STT design for a two-dimensional system $x(t) = [x_1(t), x_2(t)]^\top$, assigned the task: “Starting from $\mathbf{S}$ (blue), reach $\mathbf{T}$ (green), while avoiding $\mathbf{U}$ (red), within $t_c = 45s$ ”. (a) State space with system trajectory (black dotted line). (b) Reachability tube margin along with the unsafe set projection in $x_2$. (c) STT adapted around the unsafe set projection in $x_2$. (d) STT in $x_1$.
  • Figure 2: Simulation results showing the system trajectory (in black solid lines) followed by a robot with omnidirectional dynamics along with the STTs in dimension $x_1$ and $x_2$.
  • Figure 3: Comparing control effort of STT with circumvent function and modified STT framework

Theorems & Definitions (8)

  • Definition 2.1: Prescribed-time RAS Task
  • Definition 2.3
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.3
  • Lemma 3.4
  • Theorem 4.1
  • Remark 4.2