Control Barrier Functions for Prescribed-time Reach-Avoid-Stay Tasks using Spatiotemporal Tubes

TL;DR

This work addresses PT-RAS tasks by developing a spatiotemporal tubes (STT) framework to construct time-varying control barrier functions (TV-CBFs). By defining STTs and a normalized error-based TV-CBF $b(t,x)$, it guarantees forward invariance of a safe set ${\mathcal{C}}(t)$ within the prescribed horizon $t_c$, ensuring reach to ${\mathbf{T}}$ while avoiding ${\mathbf{U}}$. A quadratic program computes the minimal-control input that preserves safety, and higher-order CBFs extend the method to systems with relative degree $\eta>1$. Case studies on a 2D mobile robot and a multi-UAV payload system demonstrate PT-RAS satisfaction with reduced control effort and scalability compared to STT and learning-based CBF approaches. The STT-CBF framework thus offers a rigorous, computationally efficient pathway to safe, time-constrained control in obstacle-rich environments.

Abstract

Prescribed-time reach-avoid-stay (PT-RAS) specifications are crucial in applications requiring precise timing, state constraints, and safety guarantees. While control carrier functions (CBFs) have emerged as a promising approach, providing formal guarantees of safety, constructing CBFs that satisfy PT-RAS specifications remains challenging. In this paper, we present a novel approach using a spatiotemporal tubes (STTs) framework to construct CBFs for PT-RAS tasks. The STT framework allows for the systematic design of CBFs that dynamically manage both spatial and temporal constraints, ensuring the system remains within a safe operational envelope while achieving the desired temporal objectives. The proposed method is validated with two case studies: temporal motion planning of an omnidirectional robot and temporal waypoint navigation of a drone with obstacles, using higher-order CBFs.

Figures (5)

• Figure 1: Barrier function for a two dimensional case with $a_1 = a_2 = 2$. At a given time instance $t$, the superlevel set of $b(t,x)$, ${\mathcal{C}}(t)$ is a subset of the hyperrectangle enclosed by the tubes.
• Figure 2: (a),(b) Presents the spatiotemporal tubes in $x_1$ and $x_2$ dimensions respectively, (c) Controlled trajectory, (d) $b(t,x) > 0$ for all time,(e) Evolution of $b(t,x)$ with time, where the superlevel set ${\mathcal{C}}(t)$ is depicted in black solid line.
• Figure 3: (a) STTs, (b) Barrier function $b(t,x)$, (c) Controlled trajectory, for the mobile robot case
• Figure 4: (a) STTs, (b) Barrier function $b(t,x)$ and Thrust, (c), (d) Controlled trajectory, for the multi-UAV case.
• Figure 5: Comparing control effort of STT with STT-CBF.

Theorems & Definitions (11)

• Definition 2.1: Prescribed-Time Reach-Avoid-Stay Task
• Definition 2.2: Candidate Control Barrier Function CBF_STL
• Definition 2.3: Forward Invariant Set HOCBF
• Definition 2.4: Valid Control Barrier Function
• Definition 3.1: Spatiotemporal Tubes for PT-RAS
• Theorem 3.2
• Proof 3.3
• Theorem 3.4
• Proof 3.5
• Remark 3.6