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Model Predictive Control of Non-Holonomic Vehicles: Beyond Differential-Drive

Mario Rosenfelder, Henrik Ebel, Jasmin Krauspenhaar, Peter Eberhard

TL;DR

The paper addresses the challenge of stabilizing driftless non-holonomic vehicles with model predictive control, where conventional quadratic costs often fail. It develops a constructive design framework that combines homogeneous nilpotent approximations with privileged coordinates to create geometry-aware, non-quadratic stage costs, guaranteeing local asymptotic stability for sufficiently long horizons. The contributions include a general condition showing the insufficiency of quadratic costs, a practical MPC design procedure, and application to unicycle, kinematic car, and trailer systems with simulations and hardware experiments. This work provides a playable blueprint for deploying MPC on a broad class of non-holonomic platforms, enabling more reliable autonomous parking and maneuvering in practice.

Abstract

Non-holonomic vehicles are of immense practical value and increasingly subject to automation. However, controlling them accurately, e.g., when parking, is known to be challenging for automatic control methods, including model predictive control (MPC). Combining results from MPC theory and sub-Riemannian geometry in the form of homogeneous nilpotent system approximations, this paper proposes a comprehensive, ready-to-apply design procedure for MPC controllers to steer controllable, driftless non-holonomic vehicles into given setpoints. It can be ascertained that the resulting controllers nominally asymptotically stabilize the setpoint for a large-enough prediction horizon. The design procedure is exemplarily applied to four vehicles, including the kinematic car and a differentially driven mobile robot with up to two trailers. The controllers use a non-quadratic cost function tailored to the non-holonomic kinematics. Novelly, for the considered example vehicles, it is proven that a quadratic cost employed in an otherwise similar controller is insufficient to reliably asymptotically stabilize the closed loop. Since quadratic costs are the conventional choice in control, this highlights the relevance of the findings. To the knowledge of the authors, it is the first time that MPC controllers of the proposed structure are applied to non-holonomic vehicles beyond very simple ones, in particular (partly) on hardware.

Model Predictive Control of Non-Holonomic Vehicles: Beyond Differential-Drive

TL;DR

The paper addresses the challenge of stabilizing driftless non-holonomic vehicles with model predictive control, where conventional quadratic costs often fail. It develops a constructive design framework that combines homogeneous nilpotent approximations with privileged coordinates to create geometry-aware, non-quadratic stage costs, guaranteeing local asymptotic stability for sufficiently long horizons. The contributions include a general condition showing the insufficiency of quadratic costs, a practical MPC design procedure, and application to unicycle, kinematic car, and trailer systems with simulations and hardware experiments. This work provides a playable blueprint for deploying MPC on a broad class of non-holonomic platforms, enabling more reliable autonomous parking and maneuvering in practice.

Abstract

Non-holonomic vehicles are of immense practical value and increasingly subject to automation. However, controlling them accurately, e.g., when parking, is known to be challenging for automatic control methods, including model predictive control (MPC). Combining results from MPC theory and sub-Riemannian geometry in the form of homogeneous nilpotent system approximations, this paper proposes a comprehensive, ready-to-apply design procedure for MPC controllers to steer controllable, driftless non-holonomic vehicles into given setpoints. It can be ascertained that the resulting controllers nominally asymptotically stabilize the setpoint for a large-enough prediction horizon. The design procedure is exemplarily applied to four vehicles, including the kinematic car and a differentially driven mobile robot with up to two trailers. The controllers use a non-quadratic cost function tailored to the non-holonomic kinematics. Novelly, for the considered example vehicles, it is proven that a quadratic cost employed in an otherwise similar controller is insufficient to reliably asymptotically stabilize the closed loop. Since quadratic costs are the conventional choice in control, this highlights the relevance of the findings. To the knowledge of the authors, it is the first time that MPC controllers of the proposed structure are applied to non-holonomic vehicles beyond very simple ones, in particular (partly) on hardware.
Paper Structure (17 sections, 2 theorems, 34 equations, 4 figures)

This paper contains 17 sections, 2 theorems, 34 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Quadratic costs and driftless, input-affine non-holonomic vehicles
  3. Homogeneous approximation of non-holonomic systems
  4. Preliminaries from sub-Riemannian geometry
  5. Privileged coordinates
  6. Homogeneous nilpotent approximation
  7. Tailored MPC based on homogeneous approximation
  8. Application to non-holonomic vehicles
  9. Differentially driven mobile robot (unicycle)
  10. Kinematic car
  11. Differential drive with an attached trailer
  12. Differential drive with two attached trailers
  13. Conclusions
  14. Insufficiency of quadratic costs for considered non-holonomic vehicles
  15. Kinematic car
  16. ...and 2 more sections

Key Result

Lemma 1

Assume that the prediction horizon $T\in (0,\infty)$ and weighting matrices $\bm{Q}=\bm{Q}^{\mkern-1.5mu\mathsf{T}} \succ \bm{0}$, $\bm{R}=\bm{R}^{\mkern-1.5mu\mathsf{T}} \succ \bm{0}$ are given. Then, if the initial condition $\bm{x}_0\in\mathbb{R}^{n_x}\setminus \lbrace \bm{0} \rbrace$ satisfies the control input trajectory $\bm{u}^{\star}(\cdot\,\vert\, 0)\equiv \bm{0}$, which is constantly ze

Figures (4)

  • Figure 1: Considered non-holonomic vehicles.
  • Figure 2: Comparison of the experimental (orange line) and simulative (blue dash-dotted line) result for the unicycle.
  • Figure 3: Simulations of the kinematic car and the two-trailer system utilizing the proposed non-quadratic costs.
  • Figure 4: Photograph of the experimental one-trailer system (left) and corresponding parallel parking scenario (right).

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Theorem 1
  • proof