Optimal Control of Foldable Drones and Its Geometric Integration

We study optimal control problems for mechanical systems with control-dependent inertia evolving on Lie groups, with application to the attitude dynamics of foldable multirotor unmanned aerial vehicles (UAVs). In this setting, internal controls modify the inertia tensor of the system, producing geometric effects that are fundamentally different from those of external force inputs. We derive the necessary conditions for optimality using a presymplectic formulation on an extended phase space, which intrinsically accounts for the interplay between internal and external controls. Since the configuration space is the Lie group SO(3), we construct variational integrators that preserve the geometric structure of the optimal control flow, including the manifold structure and symplecticity, by discretizing the variational principle rather than the equations of motion directly. We prove that the resulting discrete scheme satisfies a regularity condition analogous to that of the continuous-time problem. Numerical simulations for a foldable quadrotor validate the proposed framework and illustrate the structure-preserving properties of the integrator in comparison with standard numerical methods.