Exponentially Stable Projector-based Control of Lagrangian Systems with Gaussian Processes

TL;DR

This work addresses robust trajectory tracking for uncertain Euler-Lagrange systems by marrying physics-informed Lagrangian-Gaussian Processes (L-GPs) with a structure-preserving projector-based controller. It introduces covariance-driven, uncertainty-adaptive gains to balance feedforward and feedback while preserving nonlinear impedance, and proves probabilistic exponential stability with explicit rate $\alpha(t)$ and radius $\varrho(t)$. The approach rests on a contraction-theory–informed Lyapunov metric and a energy-structuring GP model that preserves kinetic and potential energy forms. Numerical experiments on a two-link manipulator and a planar soft robot demonstrate improved tracking accuracy and robustness, validating both the theoretical guarantees and practical efficacy. Overall, the method offers a principled framework for safe, data-driven control of complex mechanical systems with explicit stability guarantees and energy-based interpretability.

Abstract

Designing accurate yet robust tracking controllers with tight performance guarantees for Lagrangian systems is challenging due to nonlinear modeling uncertainties and conservative stability criteria. This article proposes a structure-preserving projector-based tracking control law for uncertain Euler-Lagrange (EL) systems using physically consistent Lagrangian-Gaussian Processes (L-GPs). We leverage the uncertainty quantification of the L-GP for adaptive feedforward-feedback balancing. In particular, an accurate probabilistic guarantee for exponential stability is derived by leveraging matrix analysis results and contraction theory, where the benefit of the proposed controller is proven and shown in the closed-form expressions for convergence rate and radius. Extensive numerical simulations not only demonstrate the controller's efficacy based on a two-link and a soft robotic manipulator but also all theoretical results are explicitly analyzed and validated.

Figures (9)

• Figure 1: Block scheme of the proposed projector-based L-GP control with uncertainty-adaptive feedback: $\hat{\bm{M}}$, $\hat{\bm{C}}$, $\hat{\bm{g}}_{\bm{q}}$ and $\hat{\bm{d}}_{\dot{\bm{q}}}$ denote the learned inertia, Coriolis matrix, gravity and friction, respectively, of the system with uncertain Lagrangian $L$.
• Figure 2: Tracking performances of the standard parametric and proposed, structure-preserving or L-GP-based, PD+ controllers.
• Figure 3: Reference trajectory in the cartesian workspace of the two-link (left) and tracking performance comparison of the standard parametric with the proposed variance-adaptive and structure-preserving, L-GP-based, PD+ controller (right).
• Figure 4: Two-link: Lyapunov function and bounds for trajectories of the L-GP nat-PD+ controller with initial conditions $\bm{q}_0\!\sim\!\mathcal{N}(\bm{0},\sigma_0^2\bm{I})$ and $\dot{\bm{q}}_0\!\sim\!\mathcal{N}(\pi/2\bm{1},\sigma_0^2\bm{I})$, where $\sigma_0\!=\!\pi/3$. Solid lines indicate trajectories, shaded areas respective functional or error norm regions from Theorem \ref{['lgp_nat_pdp_theo1']} and Lemma \ref{['lemma_met']} with the bounds \ref{['prob_theo1']} and \ref{['lmi_bounds_metric']} given by the dashed lines.
• Figure 5: Exponential convergence behaviors of the proposed natural structure-preserving controllers evaluated on the two-link for randomly drawn initial conditions as in Fig. \ref{['two_link_nat_pdp_V_and_norm_ede']}. Solid lines indicate the variance-adaptive (L-GP var-nat-PD+) controller, dashed the static gain (L-GP nat-PD+) variant.

Theorems & Definitions (6)

• Remark 1
• Lemma 1
• Theorem 1: Natural $\bm{\Sigma}$-adaptive PD+
• Remark 2
• Lemma 2
• Corollary 1