Nonlinear nonholonomic constraints stop bending to noisy data: Chetaev geometry enters the Gaussian-process prior, so predictions meet the acceleration constraint exactly

When learning the dynamics of mechanical systems under nonlinear nonholonomic constraints, the model can make predicted accelerations lie strictly in the directions the constraints allow, rather than being pulled back to a feasible set at the output. Earlier constraint-preserving Gaussian processes mainly handled linear nonholonomic distributions by writing a linear projector directly into a matrix-valued kernel; in the nonlinear case admissible velocities do not form a vector space, and second-order tangency gives an affine constraint on accelerations, so the earlier projection construction does not carry over directly. On a mechanical benchmark of three-dimensional directional flocking, adding perturbations normal to the admissible directions to the same noisy acceleration observations leaves the relative field error of the nonholonomic Gaussian process unchanged, while the standard and post-projected Gaussian processes degrade as contamination grows.

The approach puts the constraint geometry into the Gaussian-process prior instead of first fitting an unconstrained regression and projecting the result. Post-projection only guarantees that the final prediction lies in the constraint set; observation components incompatible with the constraints that have already propagated into the posterior through conditioning cannot be removed. With kernel and noise hyperparameters held fixed, a perturbation of the stacked observations that satisfies the constraint condition leaves the posterior mean of the constrained prior unchanged, whereas the corresponding change in the post-projected estimator is generally nonzero; the difference is still observed as the number of agents rises to 50 and the redundancy of the constraint representation increases.

The framework still holds when the constraints are given by constraint functions that need not be independent, and the result does not depend on which set of constraint equations is chosen. The natural cross-product description of parallel-velocity constraints in flocking contains redundant scalar equations whose Jacobian is rank-deficient at a coherently aligned state, so solving the multiplier equations directly becomes singular. The Moore–Penrose pseudoinverse yields an affine decomposition independent of the constraint representation; for the same set of agents, every connected interaction graph induces the same tangent space and kinetic-energy-metric projector on the coherently aligned manifold, and the experiments cover 4, 8, 20 and 50 agents.

The demonstrated setting is a mechanical benchmark of three-dimensional directional flocking: equal-mass particles driven by a Morse potential under a nonlinear parallel-velocity constraint, with connected interaction graphs and agent counts from 4 to 50, using controlled normal-direction perturbations to emulate constraint-incompatible observations. The source proposes learning away from the constraint manifold, time-varying interaction topologies and broader classes of nonlinear nonholonomic systems as next steps.

The source states that the larger-agent cases are structural scalability tests rather than sample-complexity comparisons, because the number of training states decreases as the number of agents grows. The constraint-preservation guarantee assumes the state lies on the constraint manifold, and the case away from the manifold is left to future work.