GMPC: Geometric Model Predictive Control for Wheeled Mobile Robot Trajectory Tracking

TL;DR

This work tackles trajectory tracking for wheeled mobile robots under nonholonomic constraints by respecting the $SE(2)$ geometry. It introduces a geometric model predictive control framework (GMPC) that derives continuous-time error dynamics on $SE(2)$ and convexifies the control problem via Lie-algebra linearization, enabling efficient QP-based optimization. Compared to Euler-angle-based NMPC and FBC, GMPC yields smoother trajectories and faster compute times, validated through simulations on two WMRs and physical experiments, plus an open-source Python simulator. The key contributions are the continuous-time error dynamics, two linearization schemes with empirical evidence favoring the linear-twist form, and a complete convex MPC formulation for real-time deployment.

Abstract

The configuration of most robotic systems lies in continuous transformation groups. However, in mobile robot trajectory tracking, many recent works still naively utilize optimization methods for elements in vector space without considering the manifold constraint of the robot configuration. In this letter, we propose a geometric model predictive control (MPC) framework for wheeled mobile robot trajectory tracking. We first derive the error dynamics of the wheeled mobile robot trajectory tracking by considering its manifold constraint and kinematic constraint simultaneously. After that, we utilize the relationship between the Lie group and Lie algebra to convexify the tracking control problem, which enables us to solve the problem efficiently. Thanks to the Lie group formulation, our method tracks the trajectory more smoothly than existing nonlinear MPC. Simulations and physical experiments verify the effectiveness of our proposed methods. Our pure Python-based simulation platform is publicly available to benefit further research in the community.

Figures (8)

• Figure 1: Difference between Lie group and Euler angle representation: The mapping from variable $x$ to Euler angles exhibits discontinuity, whereas the mapping from $x$ to the special orthogonal group $SO(2)$ (an isomorphism of the complex circle group $e^{ix}$) remains continuous.
• Figure 2: Wheeled mobile robot platforms.
• Figure 3: Trajectory tracking scenarios.
• Figure 4: Circular trajectory tracking with Turtlebot 3 using different linearization schemes. The initial position of the robot is $[-0.06, -0.06]^{\top}$, and the initial orientation is $0$.
• Figure 5: Monte Carlo tests of tracking a circular trajectory with Turtlebot 3. The initial position of the robot is randomly selected between $[-0.2, 0]^{\top}$ and $[-0.2, 0]^{\top}$, and the initial orientation is randomly selected between $[-\frac{\pi}{6},0]$.

Theorems & Definitions (1)

• Remark 1