Lyapunov-Based Physics-Informed Deep Neural Networks with Skew Symmetry Considerations
Rebecca G. Hart, Wanjiku A. Makumi, Rushikesh Kamalapurkar, Warren E. Dixon
TL;DR
This work introduces SS-LbPINN, a physics-informed, Lyapunov-based DNN controller for uncertain Euler-Lagrange dynamics that explicitly enforces skew-symmetry in the inertia $M(q)$ and centripetal-Coriolis $C(q,\dot{q})$ terms. By modeling unknown dynamics with a family of DNNs and embedding a skew-symmetric prediction error into the update laws, the method achieves real-time weight adaptation with stability guarantees. A Lyapunov function incorporating tracking errors, prediction error, and weight estimation error leads to asymptotic convergence of $e$, $r$, and $\widetilde{E}$ under appropriate gain conditions. Simulations on a two-link planar robot show comparable tracking performance to a baseline method while delivering substantial improvements in the accuracy of $M$, $C$, and $F$ estimates (19.87% overall function-approximation improvement and 4.84%, 21.78%, 2.52% improvements for $M$, $C$, and $F$ respectively). These results highlight the value of physics-informed, symmetry-aware learning for robust, real-time robotic control.
Abstract
Deep neural networks (DNNs) are powerful black-box function approximators which have been shown to yield improved performance compared to traditional neural network (NN) architectures. However, black-box algorithms do not incorporate known physics of the system and can yield results which are physically implausible. Physics-informed neural networks (PINNs) have grown in popularity due to their ability to leverage known physical principles in the learning process which has been empirically shown to improve performance compared to traditional black-box methods. This paper introduces the first physics-informed DNN controller for an Euler-Lagrange dynamic system where the adaptation laws are designed using a Lyapunov-based stability analysis to account for the skew-symmetry property of the inertia matrix and centripetal-Coriolis matrix. A Lyapunov-based stability analysis is provided to guarantee asymptotic convergence of the tracking error and the skew-symmetric prediction error. Simulations indicate that the developed update law demonstrates improvement in individual and overall function approximation capabilities when compared to a physics-informed adaptation law which does not incorporate knowledge of system symmetries.