Quasi-Periodic Gaussian Process Predictive Iterative Learning Control

TL;DR

The paper addresses slow convergence and sensitivity to time-varying disturbances in iterative learning control by embedding a Quasi-Periodic Gaussian Process (QPGP) into predictive ILC (PILC). The approach yields two prediction schemes—element-wise and block-based—whose inference scales as $\mathcal{O}(p^3)$ and remains independent of the number of past iterations, enabling continual online GP learning. Theoretical contraction-based stability guarantees show mean error convergence and bounded error covariance under reasonable gain and kernel bounds. Empirically, QPGP-PILC outperforms standard ILC and GP-based PILC across vehicle tracking, a 3-link planar manipulator, and a Stretch robot, with faster convergence and lower computational overhead, demonstrating practical applicability to repetitive dynamical systems.

Abstract

Repetitive motion tasks are common in robotics, but performance can degrade over time due to environmental changes and robot wear and tear. Iterative learning control (ILC) improves performance by using information from previous iterations to compensate for expected errors in future iterations. This work incorporates the use of Quasi-Periodic Gaussian Processes (QPGPs) into a predictive ILC framework to model and forecast disturbances and drift across iterations. Using a recent structural equation formulation of QPGPs, the proposed approach enables efficient inference with complexity $\mathcal{O}(p^3)$ instead of $\mathcal{O}(i^2p^3)$, where $p$ denotes the number of points within an iteration and $i$ represents the total number of iterations, specially for larger $i$. This formulation also enables parameter estimation without loss of information, making continual GP learning computationally feasible within the control loop. By predicting next-iteration error profiles rather than relying only on past errors, the controller achieves faster convergence and maintains this under time-varying disturbances. We benchmark the method against both standard ILC and conventional Gaussian Process (GP)-based predictive ILC on three tasks, autonomous vehicle trajectory tracking, a three-link robotic manipulator, and a real-world Stretch robot experiment. Across all cases, the proposed approach converges faster and remains robust under injected and natural disturbances while reducing computational cost. This highlights its practicality across a range of repetitive dynamical systems.

Figures (7)

• Figure 1: Our key insight is that errors in iterative learning control evolve in a quasi-periodic manner. Incorporating predictive modeling exploiting this structure this into ILC (QPGP-PILC) results in noticeably reduced error magnitude and improved convergence behaviour over standard ILC.
• Figure 2: Comparison of error convergence for vehicle trajectory tracking for Standard ILC, GP-PILC, Sparse GP-PILC, and QPGP-PILC (block and element-wise). Element-wise QPGP-PILC converges fastest, followed by block-based QPGP-PILC. GP-PILC shows similar convergence, while Sparse GP-PILC converges slightly slower. QPGP-PILC consistently achieves the most accurate tracking.
• Figure 3: Comparison of vehicle trajectories at the 50th iteration under Standard ILC and QPGP-PILC controllers. The QPGP-PILC trajectory closely follows the reference, whereas the Standard ILC trajectory shows visible deviations.
• Figure 4: Comparison of error convergence for manipulator between Standard ILC, QPGP-PILC, and GP-PILC. QPGP-PILC achieves the fastest convergence, followed by GP-PILC and then Standard ILC. In addition to its superior convergence rate, QPGP-PILC is computationally more efficient, as it requires only the most recent iteration’s errors for predictions.
• Figure 5: Response of the manipulator to a mid-iteration disturbance under Standard ILC and QPGP-PILC. Standard ILC reacts slowly, while QPGP-PILC anticipates and corrects deviations, recovering to pre-disturbance error levels much faster.

Theorems & Definitions (2)

• Theorem 1
• proof