Data-Driven Predictive Control of Nonholonomic Robots Based on a Bilinear Koopman Realization: Data Does Not Replace Geometry

TL;DR

The study tackles the problem of achieving high-precision predictive control for nonholonomic differential-drive robots using data-driven surrogate models learned with EDMD on a Koopman framework. It advances a bilinear, control-affine surrogate that integrates into model predictive control and demonstrates, through extensive simulations and hardware experiments, that incorporating sub-Riemannian geometric costs is essential for stability, especially with actuator dynamics. The key contributions include experimental validation of a purely data-driven MPC for both kinematic and second-order dynamics robots, evidence of data efficiency with few samples, and clear guidance on dictionary design and reprojection strategies. Overall, the work shows that data alone cannot replace the geometry of nonholonomic systems; a geometry-informed, data-driven approach yields reliable control and practical insights for deploying such methods on real robotic platforms.

Abstract

Advances in machine learning and the growing trend towards effortless data generation in real-world systems has led to an increasing interest for data-inferred models and data-based control in robotics. It seems appealing to govern robots solely based on data, bypassing the traditional, more elaborate pipeline of system modeling through first-principles and subsequent controller design. One promising data-driven approach is the Extended Dynamic Mode Decomposition (EDMD) for control-affine systems, a system class which contains many vehicles and machines of immense practical importance including, e.g., typical wheeled mobile robots. EDMD can be highly data-efficient, computationally inexpensive, can deal with nonlinear dynamics as prevalent in robotics and mechanics, and has a sound theoretical foundation rooted in Koopman theory. On this background, this present paper examines how EDMD models can be integrated into predictive controllers for nonholonomic mobile robots. In addition to the conventional kinematic mobile robot, we also cover the complete data-driven control pipeline - from data acquisition to control design - when the robot is not treated in terms of first-order kinematics but in a second-order manner, allowing to account for actuator dynamics. Using only real-world measurement data, it is shown in both simulations and hardware experiments that the surrogate models enable high-precision predictive controllers in the studied cases. However, the findings raise significant concerns about purely data-centric approaches that overlook the underlying geometry of nonholonomic systems, showing that, for nonholonomic systems, some geometric insight seems necessary and cannot be easily compensated for with large amounts of data.

Figures (12)

• Figure 1: Schematic sketch and photograph of the employed type of custom-built differential-drive robot.
• Figure 2: Simulation. Closed-loop trajectories for parking the kinematic mobile robot \ref{['eq:kinematic_model']} to the origin with different predictive controllers. Left: Planar plot. Right: Value functions.
• Figure 3: Simulation. Left: Closed-loop trajectories for parking the kinematic mobile robot \ref{['eq:kinematic_model']} with EDMD-based predictive controllers subject to quadratic and mixed-exponents costs. Right: Corresponding value functions.
• Figure 4: Simulation. Empirical cumulative distribution functions (CDFs) of the deviation after $\unit[10]{s}$ in the hard-to-control $x_2$-direction when parking the kinematic mobile robot \ref{['eq:kinematic_model']} with EDMD-based predictive controllers subject to different costs and reprojection configurations.
• Figure 5: Hardware results. Experimental closed-loop results for parallel parking the kinematic mobile robot with different predictive controllers. Left: Scene stills of the EDMD-based controller utilizing the mixed-exponents cost. Middle: Deviation in the hard-to-control $x_2$-direction. Right: Value function of the underlying OCPs.