Reach-Avoid-Stabilize Using Admissible Control Sets

TL;DR

This work addresses safe mission planning for systems requiring sequential targets, obstacle avoidance, and eventual stabilization by leveraging admissible control sets derived from Hamilton-Jacobi reachability. It avoids introducing new value functions and instead uses forward propagation with ACS to under-approximate the Reach-Avoid-Stabilize set under time-series constraints. Key contributions include formal ACS definitions for stabilization and reach/avoid, a sampling-based admissible control synthesis, and a Multi_ACS algorithm that computes a safe RAS under multiple targets/obstacles with guaranteed safety. The approach demonstrates computational efficiency and provides practical validation on a 2D double integrator and a 3D Dubins car, illustrating applicability to complex robotic systems through a principled, safety-guaranteed framework.

Abstract

Hamilton-Jacobi Reachability (HJR) analysis has been successfully used in many robotics and control tasks, and is especially effective in computing reach-avoid sets and control laws that enable an agent to reach a goal while satisfying state constraints. However, the original HJR formulation provides no guarantees of safety after a) the prescribed time horizon, or b) goal satisfaction. The reach-avoid-stabilize (RAS) problem has therefore gained a lot of focus: find the set of initial states (the RAS set), such that the trajectory can reach the target, and stabilize to some point of interest (POI) while avoiding obstacles. Solving RAS problems using HJR usually requires defining a new value function, whose zero sub-level set is the RAS set. The existing methods do not consider the problem when there are a series of targets to reach and/or obstacles to avoid. We propose a method that uses the idea of admissible control sets; we guarantee that the system will reach each target while avoiding obstacles as prescribed by the given time series. Moreover, we guarantee that the trajectory ultimately stabilizes to the POI. The proposed method provides an under-approximation of the RAS set, guaranteeing safety. Numerical examples are provided to validate the theory.

Figures (5)

• Figure 1: The ACSs that guarantee asymptotically stabilizability (left) and exponential stabilizability with rate 0.3 (middle and right). The blue and green lines denote the upper and lower bounds for ACS, respectively. Since the inequality in \ref{['eqn:SCS_CLVF2']} is strict, the control cannot take the value on the bounds. When $\hat{\gamma} = \gamma$, there are some states whose upper and lower bounds of the ACS are the same (middle plot, near -1 and 1), and by definition, the ASC is empty for those states. However, this does not mean there exists no control that stabilizes the system with $\hat{\gamma}$ rate. As shown in the right, when picking $0.3 = \hat{\gamma} < \gamma = 0.5$, the ACS is non-empty for all $x \in (-1,1)$. Though we cannot prove that $\mathcal{U}_{p}^s (x ; \gamma_1, \hat{\gamma}) \subset \mathcal{U}_{p}^s (x ; \gamma_2, \hat{\gamma})$ for $\gamma_1 < \gamma_2$, this example implies that when computing the R-CLVF, using a larger $\gamma$ will allow more flexibility for the ACS.
• Figure 2: The RAS set from Alg. \ref{['algo:Multi_ACS']}, shown in green. The target (obstacle) and reach (avoid) sets are shown in the blue (red) line and blue (red) dashed line, and the reach-avoid set from HJR is shown in blake dashed line. One trajectory with initial state $[-1;1]$ (black pentagram) is shown in black stars. The location of the obstacle and the dynamics of the system make it impossible for the system to hit the obstacle after reaching the target in $[0,1]$ and before entering $\mathcal{I}_M$. The total iteration is 144 and takes 39.13s with 401*401 initial states.
• Figure 3: The RAS set from Alg. \ref{['algo:Multi_ACS']}, shown in green. Target 1 (target 2, obstacle) and reach set1 (reach set2, avoid set) are shown in the blue (black, red) lines and blue (black, red) dashed lines. One trajectory with initial state $[-1;1]$ (black pentagram) is shown in black stars. From Remark \ref{['remark:reach_bdry']}, at $s = t_{r,1}$, all the states will reach exactly the boundary of target 1. Therefore, the states that can then reach target 2 are the states that reach the intersection of reach set2 and target 1 at $s = t_{r,1}$. The total iteration is 580 and takes 149.40s with 401*401 initial states.
• Figure 4: The RAS set from Alg. \ref{['algo:Multi_ACS']}, shown in green. Target 1 (target 2, obstacle) and reach set1 (reach set2, avoid set) are shown in the blue (black, red) lines and blue (black, red) dashed lines. One successful trajectory with initial state $[-0.6;0.6]$ (black pentagram) is shown in black stars, and two failed trajectories with initial states $[-0.96,0.86],[-0.49;0.78]$ (red pentagram) are shown in red stars. The failed trajectory either hits the avoid set or does not enter the intersection of the reach set 2 and target 1. The total iteration is 235 and takes 153.34s with 401*401 initial states.
• Figure 5: The RAS set from Alg. \ref{['algo:Multi_ACS']}, shown in green. Target 1, target 2, obstacle are shown in the blue, black, and red lines, and $\mathcal{I}_m$ is shown in magenta. Two successful trajectories with initial states $[-1;5;0], [-2.7;-0.4;1.9]$ (black pentagram) are shown in black stars: they first reach target 1, then reach target 2, and finally stabilize to $\mathcal{I}_m$. Left, the original 3D plot. Right, the projected plots in $x_1-x_2$ plane. The total iteration is 132 and takes 189.78s with 71*71*51 initial states.

Theorems & Definitions (13)

• Definition 1: RAS Set
• Proposition 1
• proof
• Proposition 2
• proof
• Remark 1
• Proposition 3
• proof
• Lemma 1
• proof