Integrator Forwading Design for Unicycles with Constant and Actuated Velocity in Polar Coordinates

TL;DR

The paper leverages a polar-coordinate representation of the unicycle to overcome nonholonomic stabilization challenges and develops a modular CLF framework that enables smooth static feedback. It introduces two forwarding-based controllers, Global Forwarding (GloFo) and BoFo, each equipped with explicit CLFs and GAS proofs, and demonstrates how forwarding can be extended to actuated forward velocity. It also shows that fixing the forward speed reduces the problem to a feedforward form for the Dubins car and yields deadbeat parking via integrator forwarding with finite-time stability guarantees. Overall, the work reveals a deep connection between backstepping and forwarding in nonholonomic control, provides a modular toolkit for inverse optimal design, and offers practical smooth steering laws with provable stability across unicycle and Dubins car parking problems.

Abstract

In a companion paper, we present a modular framework for unicycle stabilization in polar coordinates that provides smooth steering laws through backstepping. Surprisingly, the same problem also allows the application of integrator forwarding. In this work, we leverage this feature and construct new smooth steering laws together with control Lyapunov functions (CLFs), expanding the set of CLFs available for inverse optimal control design. In the case of constant forward velocity (Dubins car), backstepping produces finite-time (deadbeat) parking, and we show that integrator forwarding yields the very same class of solutions. This reveals a fundamental connection between backstepping and forwarding in addressing both the unicycle and, the Dubins car parking problems.

Figures (4)

• Figure 1: Unicycle orientation $(x,y,\theta)$ with respect to the goal state $(0,0,0)$, along with the polar coordinate transformation $(x,y,\theta)\mapsto (\rho,\delta,\gamma)$.
• Figure 2: Simulation with the BoFo control law \ref{['eq:BoFo']} along with \ref{['eq-basic-v-control']} and \ref{['eq-omega-general']} from Theorem \ref{['thm:CLF_BoFo']} with $(k_1,k_2,k_3,k_4) = (1,3,2,1.5)$.
• Figure 3: Simulations with the steering control in Theorem \ref{['thm:Dubins-FT-stabilize6']}, with $c_1 = 2.05$, $c_2 = 2.1$, and $v = 0.5$. Since the forward velocity is constrained to a nonzero constant, the controllers must be turned off once $\rho(T) = 0$. To mitigate numerical inaccuracies near the boundary of the interval of existence, we impose a cutoff on both control inputs if $\rho(t) \leq 0.01$.
• Figure 4: Simulation with the steering control law in Theorem \ref{['thm:Dubins-FT-stabilize7']} with $c_1 = 0.7$, $c_2 = 1.3$ 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.

Theorems & Definitions (13)

• Definition 1
• Definition 2: CLF for the unicycle \ref{['eq:unicycle_polar_closed_loop-Gv-1']}
• Theorem 1
• proof
• Corollary 1
• Theorem 2
• proof
• Lemma 1: Krstic2025_Dubins
• Lemma 2: Krstic2025_Dubins
• Theorem 3