Constricting Tubes for Prescribed-Time Safe Control

Abstract

We propose a constricting Control Barrier Function (CBF) framework for prescribed-time control of control-affine systems with input constraints. Given a system starting outside a target safe set, we construct a time-varying safety tube that shrinks from a relaxed set containing the initial condition to the target set at a user-specified deadline. Any controller rendering this tube forward invariant guarantees prescribed-time recovery by construction. The constriction schedule is bounded and tunable by design, in contrast to prescribed-time methods where control effort diverges near the deadline. Feasibility under input constraints reduces to a single verifiable condition on the constriction rate, yielding a closed-form minimum recovery time as a function of control authority and initial violation. The framework imposes a single affine constraint per timestep regardless of state dimension, scaling to settings where grid-based reachability methods are intractable. We validate on a 16-dimensional multi-agent system and a unicycle reach-avoid problem, demonstrating prescribed-time recovery with bounded control effort.

Figures (5)

• Figure 1: Geometry of the constricting tube framework. The initial condition $x(0)$ lies outside the target set $\mathcal{C} = \{x : h(x) \geq 0\}$. The schedule $r(x(0),t)$ inflates $\mathcal{C}$ into a tube $\tilde{\mathcal{C}}(t) = \{x : h(x) + r(x(0),t) \geq 0\}$ containing $x(0)$ at $t=0$. As $r(x(0),t) \to 0$, the tube boundary constricts through the level sets of $h$ until $\tilde{\mathcal{C}}(T) = \mathcal{C}$. Any controller rendering $\tilde{\mathcal{C}}(t)$ forward invariant guarantees $x(T) \in \mathcal{C}$.
• Figure 2: Barrier authority $\sigma(x)$ for the controlled pendulum targeting the upright equilibrium $(\pi,0)$ (dot) with $u_{\max}=1.5$. Blue: $\sigma > 0$; red: $\sigma < 0$. The red lobes correspond to states where the pendulum is displaced from upright and moving further away, so gravitational drift and velocity compound to reduce $\sigma_{\min}$.
• Figure 3: Prescribed-time recovery for $N=8$ agents with joint barrier $h(X) = c - \|X\|^2$, $c = 0.5$, and deadline $T = 6.4$ s. (Left) Phase portrait of agent trajectories under our method (solid) and KG-EA-DP:2020_convergenceCLF (dashed). The shaded disk is $\mathcal{C}$. Both methods achieve prescribed-time recovery. (Middle) Joint barrier $h(X(t))$. Our method tracks the constricting tube floor $-r(t)$ (solid blue), entering $\mathcal{C}$ exactly at $T$. (Right) Control effort $\|U(t)\|$. The minimum norm controller stays well below the saturation limit $\sqrt{N}u_{\max} \approx 28.28$, while KG-EA-DP:2020_convergenceCLF saturates persistently.
• Figure 4: Higher order constricting tubes for the double integrator with relative degree 2, $T = 25\,\mathrm{s}$. (Left) Position trajectory: the HOCBF controller (blue) enters $\mathcal{C}$ by $T$; the nominal PD controller (red) does not. (b) Barrier values $\psi_0(x,t)$ (tube) and $h(x(t))$ (target set): $\psi_0 \geq 0$ is maintained throughout and $h(x(T)) \geq \delta > 0$ at the deadline.
• Figure 5: Prescribed-time reach-avoid planning with the unicycle. (Left) The planner navigates around the obstacle (red) and enters the target set (green) by the deadline. (Right) Barrier value vs. time: Prescribed-time reach CBF $h(x(t))$ (blue, solid) rises from $-24.75$ to $+0.233 > \delta$ at $t = T$, staying above the constricting tube floor $-r(x(0),t)$ (blue, dotted) throughout. Collision avoidance CBF $h_{\mathrm{obs}}(x(t))$ (red, dash-dotted) stays positive.

Theorems & Definitions (8)

• Definition 1: Constricting CBF
• Theorem 1: Prescribed-time recovery via tube invariance
• Example 1: Barrier authority for inverted pendulum
• Theorem 2: Feasibility and minimum recovery time
• Corollary 3: Closed-form $T_{\min}$ for linear systems
• Remark 1: Extension to higher relative degree
• Remark 2: Relation to HJ reachability
• Remark 3: Recursive feasibility