A Linear MPC with Control Barrier Functions for Differential Drive Robots

TL;DR

This paper tackles safe autonomous navigation for a two-wheeled differential-drive robot under state and obstacle-avoidance constraints. It introduces a Safety-Critical Model Predictive Control framework based on Dynamic Feedback Linearization (SCMPCDFL) that runs Linear MPC on a linear extended model derived from the nonlinear unicycle dynamics and enforces safety via a Control Barrier Function, reducing online computational burden to a QCQP. The authors derive a mapping between the nonlinear model and a linear extended model, provide stability and recursive feasibility proofs, and validate the approach through simulations showing robust obstacle avoidance and real-time solvability. The proposed method offers a practical, theoretically grounded path to safe, real-time navigation for differential-drive robots with underactuated dynamics.

Abstract

The need for fully autonomous mobile robots has surged over the past decade, with the imperative of ensuring safe navigation in a dynamic setting emerging as a primary challenge impeding advancements in this domain. In this paper, a Safety Critical Model Predictive Control based on Dynamic Feedback Linearization tailored to the application of differential drive robots with two wheels is proposed to generate control signals that result in obstacle-free paths. A barrier function introduces a safety constraint to the optimization problem of the Model Predictive Control (MPC) to prevent collisions. Due to the intrinsic nonlinearities of the differential drive robots, computational complexity while implementing a Nonlinear Model Predictive Control (NMPC) arises. To facilitate the real-time implementation of the optimization problem and to accommodate the underactuated nature of the robot, a combination of Linear Model Predictive Control (LMPC) and Dynamic Feedback Linearization (DFL) is proposed. The MPC problem is formulated on a linear equivalent model of the differential drive robot rendered by the DFL controller. The analysis of the closed-loop stability and recursive feasibility of the proposed control design is discussed. Numerical experiments illustrate the robustness and effectiveness of the proposed control synthesis in avoiding obstacles with respect to the benchmark of using Euclidean distance constraints. Keywords: Model Predictive Control, MPC, Autonomous Ground Vehicles, Nonlinearity, Dynamic Feedback Linearization, Optimal Control, Differential Robots.

Figures (8)

• Figure 1: (a) Differential drive robots safe navigation task where $\{e_{1},e_{2}\}$ is the body fixed frame and $\{e_{B1},e_{B2}\}$ is the global frame. (b) differential drive robots model coordinates.
• Figure 2: SCMPCDFL control scheme for the unicycle ground robot.
• Figure 3: In (a), $c_{safe}$ and $c_{unsafe}$ are the sets of safe and unsafe points in $\mathbb{R}^{3}$ respectively and $\mathcal{H}(x_{k})$ is the proposed CBF in \ref{['eq:100']}. In (b), $c_{k}$ is the level set of the CBF defined in \ref{['eq:101']}. at a given time $k.$
• Figure 4: Part (a) shows the Output performance of the SCMPCDFL and SCNMPC Control (presented in \ref{['eq:pr3']}-\ref{['eq:pr7']}) schemes with different values of $\gamma=0.1,0.3,0.5,0.7,0.9,1$ and $N=8$. Part (b) illustrates the performance with different values of $\gamma$ and $N$. Part (c) compares between the performance of the proposed SCMPCDFL utilizing CBF against the benchmark of using Euclidean distance with the same prediction horizon $N=8$. Part (D) depicts a comparison between the output performance of the proposed scheme (demonstrated in Blue line) with and without output distance by a Gaussian noise of zero mean and 0.05 variance.
• Figure 5: The linear velocity of the robot with different values of $N$ and $\gamma.$

Theorems & Definitions (7)

• Lemma 1
• Lemma 2
• Definition 1
• Lemma 3
• proof
• Theorem 1
• proof