Aerodynamic Promptness of Tilting-Propeller Multirotors

This work develops a geometric theory of aerodynamic promptness for multirotor platforms whose propeller directions depend on a smooth orientation-mechanism configuration. The actuator state consists of the actual signed propeller speeds and the mechanism configuration, and the nonlinear actuator map retains both the signed-quadratic propeller law and the configuration-dependent thrust directions. Constant direction maps represent fixed propellers, while shared dependence on mechanism coordinates represents mechanically coupled tilts. A Riemannian metric on the product actuator manifold compares propeller accelerations and orientation-mechanism rates through physically interpretable scales. Composing the actuator-map differential, the musical isomorphism induced by this metric, and the dual differential defines a task-rate authority map from wrench covectors to wrench-rate vectors. In fixed task coordinates, the square root of the determinant of its matrix defines aerodynamic promptness. Positive promptness is equivalent to local wrench-rate surjectivity, while the associated spectrum resolves directional authority and conditioning. The same construction yields the metric horizontal inverse, the orthogonal projector onto wrench-fiber directions, and task-preserving internal motions that regulate promptness without changing the instantaneous wrench. The construction is related intrinsically to dynamic feedback linearization and specialized to the independently tilting, radially arranged Holocopter and Voliro architectures, clarifying how their established allocation and rank constructions fit within the same actuator-manifold geometry.