Virtual nonlinear nonholonomic constraints from a symplectic point of view
Efstratios Stratoglou, Alexandre Anahory Simoes, Anthony Bloch, Leonardo Colombo
TL;DR
The paper develops a geometric characterization of virtual nonlinear nonholonomic constraints within a symplectic framework on $TQ$, showing that under a transversality condition there exists a unique feedback law that enforces the constraint manifold $\mathcal{M}$ and yields a closed-loop dynamics that satisfy a Chetaev-type equation. The main idea is to express dynamics via the symplectic form $\omega_L$ induced by a Lagrangian $L$ and the almost-tangent structure $J$, leading to a projection interpretation where the constrained motion is the projection of the uncontrolled dynamics onto $T\mathcal{M}$. A key result is that the closed-loop vector field obeys $i_{\Gamma}\omega_L - dE_L \in \flat_{\mathcal{G}^c}(\mathcal{F}^V)$, ensuring invariant virtual constraints, with explicit application to a controlled double pendulum showing energy dissipation and constraint behavior under simulation. This framework provides a rigorous geometric foundation for designing and analyzing virtual nonlinear nonholonomic constraints in Lagrangian control systems, linking variational structure to constraint satisfaction and controllable dynamics.
Abstract
In this paper, we provide a geometric characterization of virtual nonlinear nonholonomic constraints from a symplectic perspective. Under a transversality assumption, there is a unique control law making the trajectories of the associated closed-loop system satisfy the virtual nonlinear nonholonomic constraints. We characterize them in terms of the symplectic structure on $TQ$ induced by a Lagrangian function and the almost-tangent structure. In particular, we show that the closed-loop vector field satisfies a geometric equation of Chetaev type. Moreover, the closed-loop dynamics is obtained as the projection of the uncontrolled dynamics to the tangent bundle of the constraint submanifold defined by the virtual constraints.