Isomonodromic Connections on Principal Bundles as a Framework for Robotic Control Systems

Introduction. The geometric study of control systems on manifolds provides tools for analyzing controllability, stability, and optimality of trajectories in mechanical systems with non-trivial configuration spaces. This work addresses the problem of designing robust controllers for systems operating on curved spaces, including robots subject to nonholonomic constraints, by exploiting the geometry of connections on principal bundles.

Methods. A correspondence is established between isomonodromic deformations of logarithmic connections on principal G-bundles over Riemann surfaces and G-equivariant control systems on the total space of the bundle. The monodromy representation of the connection is used as the central invariant, and the Schlesinger equations governing isomonodromic deformations are analyzed to derive conditions on the residues of the connection.

Results. The controllability of the system is equivalent to the condition that the residues of the logarithmic connection generate the full Lie algebra under the Lie bracket operation. Optimal control trajectories minimizing a quadratic cost functional correspond to geodesics on the Riemann surface with respect to a metric determined by the monodromy data. The isomonodromic property guarantees robustness, as small perturbations of the trajectory preserve the monodromy representation, ensuring recovery from disturbances without deviation from the optimal path.

Conclusions. This framework provides a systematic approach for controlling mechanical systems on non-flat configuration spaces, including wheeled robots and underwater vehicles. Extensions to quantum control and manifold-based machine learning are also identified as directions for further application.