Real-Time Online Learning for Model Predictive Control using a Spatio-Temporal Gaussian Process Approximation

Lars Bartels; Amon Lahr; Andrea Carron; Melanie N. Zeilinger

eess.SYMar 18, 2026

Real-Time Online Learning for Model Predictive Control using a Spatio-Temporal Gaussian Process Approximation

Authors: Lars Bartels, Amon Lahr, Andrea Carron, Melanie N. Zeilinger

Organizations: Institute for Dynamic Systems and Control, ETH Zürich, CH-8092 Zürich, Switzerland.

Abstract

Learning-based model predictive control (MPC) can enhance control performance by correcting for model inaccuracies, enabling more precise state trajectory predictions than traditional MPC. A common approach is to model unknown residual dynamics as a Gaussian process (GP), which leverages data and also provides an estimate of the associated uncertainty. However, the high computational cost of online learning poses a major challenge for real-time GP-MPC applications. This work presents an efficient implementation of an approximate spatio-temporal GP model, offering online learning at constant computational complexity. It is optimized for GP-MPC, where it enables improved control performance by learning more accurate system dynamics online in real-time, even for time-varying systems. The performance of the proposed method is demonstrated by simulations and hardware experiments in the exemplary application of autonomous miniature racing.

Figures & tables

Fig. 1: High-speed miniature racing despite a time-varying steering perturbation: Online learning using spatio-temporal Gaussian process approximations enables real-time adaptation to time-varying disturbances. The car aims to race along the track centerline (blue) while staying within track boundaries (red) and therefore plans its future trajectory (yellow) including uncertainty estimates based on the GP model of the residual dynamics.

Fig. 2: Steering perturbation mapping parametrized by the neutral steering offset $\delta_{0}$ (red dot). The shaded area illustrates the set of all employed perturbation mappings, with magenta indicating positive values and green indicating negative values of $\delta_{0}$ .

Fig. 3: Experimental GP-MPC solve times using different GP models with thick lines representing a rolling average of the last 100 solves. Left: Exact GP with a SoD approximation using the 400 most recent data points (blue), compared to simulation results using the full dataset (light blue). Right: Proposed approximate spatio-temporal GP model with 80 spatial inducing points based on hardware experiments (green).

Fig. 4: One-step MPC prediction error over experiment runtime without a GP (red) and with online learning by an approximate spatio-temporal GP (blue) with a 2- $\sigma$ confidence interval to model the residual dynamics. The bar up top illustrates the time-varying steering perturbation present in the system with the neutral steering offset $\delta_{0}$ transitioning from zero through positive (green) and negative (magenta) values and back to zero.

Fig. 5: Evolution of the most recent lap times over the experiment runtime under a time-varying steering perturbation for nominal MPC (red squares) and GP-MPC with an exact GP model (SoD) (orange diamonds), a spatial conventional inducing-point GP model (SoD) (purple triangles), and an approximate spatio-temporal GP model (blue circles). The exact GP and spatial inducing-point GP employ a subset of data approximation and are conditioned on the most recent 400 data points. Lap times are recorded and plotted at the moment the car has completed a lap. The bar up top illustrates the time-varying steering perturbation present in the system with green corresponding to a positive neutral steering offset $\delta_{0}$ and magenta conversely indicating a negative value.

Fig. 6: Trajectory of racing car on track under a time-varying steering perturbation controlled by nominal MPC without GP (left) and approximate spatio-temporal GP-MPC with online learning (right).

Model Predictive Control

Model Predictive Control optimizes future actions with a system model, repeatedly applying the first action as new information arrives.

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Nonplanar Model Predictive Control for Autonomous Vehicles with Recursive Sparse Gaussian Process Dynamics

Ahmad Amine, Kabir Puri, Viet-Anh Le, Rahul Mangharam

Department of Electrical & Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA

Model Predictive Control Residual Dynamics Learning Gaussian Process Regression

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This paper proposes a nonplanar model predictive control (MPC) framework for autonomous vehicles operating on nonplanar terrain. To approximate complex vehicle dynamics in such environments, we develop a geometry-aware modeling approach that learns a residual Gaussian Process (GP). By utilizing a recursive sparse GP, the framework enables real-time adaptation to varying terrain geometry. The effectiveness of the learned model is demonstrated in a reference-tracking task using a Model Predictive Path Integral (MPPI) controller. Validation within a custom Isaac Sim environment confirms the framework's capability to maintain high tracking accuracy on challenging 3D surfaces.

Abstract

Model Predictive Control

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