Symmetry-Reduced Optimal Control of Quadrotor Dynamics via Noether’s Theorem

Modern quadrotor unmanned aerial vehicles (UAVs) rely extensively on numerical trajectory optimization and path-planning algorithms for navigation in complex environments. While these approaches emphasize feasibility, collision avoidance, and control constraints, they often overlook underlying physical symmetries in the governing dynamics, leading to redundant computations. This work introduces a symmetry-aware trajectory optimization framework that employs principles from classical mechanics, specifically Noether’s theorem, to exploit the rotational invariance of quadrotor dynamics about the yaw-axis. By identifying yaw as a cyclic coordinate in the Lagrangian formulation, a conserved quantity corresponding to angular momentum about the inertial vertical axis is derived using Noether's theorem and enforced as a constraint. This eliminates two redundant input dimensions in the system's state vector, replacing yaw-dynamics with algebraic expressions that leverage symmetry-conserved constants and omit yaw-heading rotations. This yields a reduced-order model that preserves the behavior of the system. To evaluate the proposed framework, physics-based models are implemented using CasADi, where both full-order and symmetry-reduced models are formulated as nonlinear optimal control problems and then solved using the Acados multiple-shooting OCP solver. CasADi is then used for solution verification against physical constraints. A multi-objective cost function is considered, incorporating energy expenditure, control effort, and total travel time. Comparative analyses are performed to assess solution quality and computational efficiency across both formulations versus existing solutions. Comparative results indicate that the proposed framework achieves improved computational efficiency relative to conventional formulations, operating within memory requirements of approximately 1–2 GB of RAM. The symmetry-reduced model is expected to improve algorithmic speed by reducing system dimensionality and eliminating redundant dynamics. A practical consideration of the proposed framework is the reduction of direct control over heading, making the approach most suitable for scenarios where yaw-tracking is not a primary requirement. Extensions of the framework could incorporate hybrid strategies to resolve this constraint.