  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.
