Half-Global Deadbeat Parking for Dubins Vehicle

TL;DR

This work addresses finite-time stabilization (deadbeat parking) of a Dubins vehicle under steering-only actuation by treating the distance to a stationary target as a time-like variable, enabling a backward/forward design in polar coordinates. A backstepping control law with a Lyapunov function $V = \delta^2 + \zeta^2$ and the update $\zeta = \tan\gamma + c_1 \delta$ yields $\dot V$ that decreases proportionally to $V/\rho$, giving finite-time convergence while keeping the input bounded. The authors extend the basic result with safety-critical extensions: parking without crossing in front of the target, deceleration towards the target, and curb-violation avoidance using a nonovershooting framework, with numerical demonstrations. The framework generalizes to missile guidance and pursuit problems and provides insights applicable to broader nonholonomic systems represented by the Dubins model.

Abstract

This paper presents a framework for stabilizing the Dubins vehicle model to zero in finite time (deadbeat parking) by interpreting distance as a time-like variable. We develop control laws that bring the system to a desired position and orientation even when the forward velocity cannot be directly actuated. While the controllers employ inverse-distance gains, we show that the control input remains bounded for all time. In addition to basic deadbeat parking, we incorporate safety considerations by proposing algorithms that prevent the vehicle from crossing in front of the target, enforce deceleration as it approaches the target, and guarantee parking without curb violations. The resulting methods are well-suited for missile guidance and fixed-wing pursuit, but are broadly applicable to physical systems that are represented by the Dubins vehicle model.

Figures (5)

• Figure 1: Simulation with the steering control law \ref{['eq2-control-omega']} with $c_1 = 1.01$ and $c_2 = 5$ and $v = 0.5$.
• Figure 2: Simulation with the steering control law \ref{['eq2-control-omega']} with $c_1 = c_2 = 1.2$ and $v = 0.5$. The trajectory in the $xy$-plane is nearly identical to that shown in Fig. \ref{['fig:trajectory_thrm1']}, differing only in minor details, and is therefore omitted.
• Figure 3: Trajectory comparison between the steering control \ref{['eq2-control-omega']} with $v = 0.2$, $c_1 = c_2 = 0.5$ (red) and \ref{['eq-third-controller']} with $v = 0.2$, $c_1 = 1$, $c_2 = 15$ (blue), where the latter successfully avoided crossing in front of the target.
• Figure 4: Simulation with the control law derived in Theorem \ref{['thm:Dubins-FT-stabilizeslow1']} with $c_0 = 0.5$, $c_1 = c_2 = 1.2$, and $n=2$, which makes for a smooth velocity $v(t)$. The difference is seen in $v(t)$ where the forward velocity gradually decayed to zero when approaching the target. The trajectory in the $xy$-plane is nearly identical to that shown in Fig. \ref{['fig:trajectory_thrm1']}, differing only in minor details, and is therefore omitted.
• Figure 5: Simulation with the control law derived in Theorem \ref{['thm:nonov2']} with $c_1$ chosen as \ref{['eq:nonov2_c1']}, $c_2 = 1$, and $v = 0.5$.

Theorems & Definitions (12)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5