Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lyapunov-Based $\mathcal{L}_2$-Stable PI-Like Control of a Four-Wheel Independently Driven and Steered Robot

Branimir Ćaran, Vladimir Milić, Bojan Jerbić

TL;DR

The paper tackles robust trajectory tracking for a four-wheel robot with independent steering and drive under modelling mismatch and disturbances. It develops a velocity-space dynamic model with explicit stability structure, derives a Lyapunov function and $\mathcal{L}_2$-gain bounds, and designs a PI-like inner loop with feedforward that preserves passivity. A virtual kinematic controller provides reference trajectories while a diagonal, positive-definite gain set ensures robust performance; the analysis yields explicit conditions on $\lambda_{\min}(\mathbf{K}_P)$ and related terms to guarantee $\mathcal{L}_2$ stability and finite gain from disturbances to tracking errors. Experimental validation on horizontal and vertical surfaces demonstrates stable tracking, robustness to gravity, adhesion, and contact uncertainties, and confirms the practical feasibility of the approach for real-time robotic control.

Abstract

In this letter, Lyapunov-based synthesis of a PI-like controller is proposed for $\mathcal{L}_2$-stable motion control of an independently driven and steered four-wheel mobile robot. An explicit, structurally verified model is used to enable systematic controller design with stability and performance guarantees suitable for real-time operation. A Lyapunov function is constructed to yield explicit bounds and $\mathcal{L}_2$ stability results, supporting feedback synthesis that reduces configuration dependent effects. The resulting control law maintains a PI-like form suitable for standard embedded implementation while preserving rigorous stability properties. Effectiveness and robustness are demonstrated experimentally on a real four-wheel mobile robot platform.

Lyapunov-Based $\mathcal{L}_2$-Stable PI-Like Control of a Four-Wheel Independently Driven and Steered Robot

TL;DR

The paper tackles robust trajectory tracking for a four-wheel robot with independent steering and drive under modelling mismatch and disturbances. It develops a velocity-space dynamic model with explicit stability structure, derives a Lyapunov function and -gain bounds, and designs a PI-like inner loop with feedforward that preserves passivity. A virtual kinematic controller provides reference trajectories while a diagonal, positive-definite gain set ensures robust performance; the analysis yields explicit conditions on and related terms to guarantee stability and finite gain from disturbances to tracking errors. Experimental validation on horizontal and vertical surfaces demonstrates stable tracking, robustness to gravity, adhesion, and contact uncertainties, and confirms the practical feasibility of the approach for real-time robotic control.

Abstract

In this letter, Lyapunov-based synthesis of a PI-like controller is proposed for -stable motion control of an independently driven and steered four-wheel mobile robot. An explicit, structurally verified model is used to enable systematic controller design with stability and performance guarantees suitable for real-time operation. A Lyapunov function is constructed to yield explicit bounds and stability results, supporting feedback synthesis that reduces configuration dependent effects. The resulting control law maintains a PI-like form suitable for standard embedded implementation while preserving rigorous stability properties. Effectiveness and robustness are demonstrated experimentally on a real four-wheel mobile robot platform.
Paper Structure (16 sections, 3 theorems, 31 equations, 4 figures, 1 table)

This paper contains 16 sections, 3 theorems, 31 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical model and properties
  3. Kinematics
  4. Dynamics
  5. Controller design based on Lyapunov stability analysis
  6. Control problem formulation
  7. Virtual kinematic controller
  8. Dynamic controller
  9. Stability analysis
  10. Residual dynamics
  11. Derivation of a Lyapunov function candidate
  12. $\mathcal{L}_2$ stability conditions
  13. Experimental validation
  14. Experimental set-up
  15. Experimental results
  16. ...and 1 more sections

Key Result

Corollary 1

Let the storage function be the Lyapunov function on the left-hand side of lyap_eq3: By Property property_2 and $\mathbf{K}_I = \mathbf{K}_I^{\text{T}} > \mathbf{0}$, for all admissible $\mathbf{q}$ the following bounds hold $V \geq (a_1 \|\mathbf{e}_v\|^2 + \lambda_{min}\{ \mathbf{K}_I \} \|\bm{\eta}\|^2)/2$ and $V \leq (a_2 \|\mathbf{e}_v\|^2 + \lambda_{max}\{ \mathbf{K}_I \} \|\b

Figures (4)

  • Figure 1: Mobile robot schematics.
  • Figure 2: Block diagram of the overall control system for the considered mobile robot.
  • Figure 3: Experimental setup of the mobile robot on vertical surface (left) and horizontal surface (right)
  • Figure 4: Experimental results for trajectory tracking: (a) horizontal surface, (b) vertical surface. Green arrows indicate robot orientation.

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Remark 3
  • Corollary 1: Quadratic bounds on the storage function
  • Proposition 1: Negative semidefiniteness of the Lyapunov function derivative
  • Remark 4
  • Remark 5
  • Proposition 2: Output strict passivity and finite-gain $\mathcal{L}_2$ stability
  • Remark 6
  • Remark 7