New Cone-Rolling Principle and Its Application for the Underactuated Attitude Maneuver Optimization of a Rigid Body

The global optimal solution for the underactuated attitude maneuver plays a significant role in the fields of spacecraft, robotics, and mechanics. This paper investigated the global optimal underactuated attitude maneuver by using angular velocity on a principal plane in the quaternion statement. The integral of angular velocity was considered as the performance criterion, and a correlation vector was introduced based on the coincidence of equations for the state and co-state. According to the Pontryagin minimum principle, it was concluded that the optimal angular velocity has the opposite direction as the correlation vector's projection on the principal plane, thereby indicating that the optimal process is to roll the principal plane around a cone. The solution of the cone was obtained through two geometric constraints. One is that the angle between the correlation vector and the principal plane remains invariant; the other is that the trajectories of the correlation vector's projection have the same length on the principal plane and the cone surface. The solution of the cone led to the analytical solution of the angular velocity and the control torque. For validation, the simulation results indicated that the optimal cost value obtained with the proposed cone-rolling solution, i.e., the integral of angular velocity, is lower than the suboptimal value obtained with the genetic algorithm. That is, the proposed cone-rolling principle–based global optimal analytical solution is reasonable and feasible.