Time Optimal Attitude Control of 3D Pendulum

3D pendulum is a rigid body supported at a fixed pivot with three rotational degrees of freedom. The objective is to maneuver the 3D pendulum from an initial attitude and angular rate to a desired attitude and angular rate in minimum time in presence of uniform gravity, subjected to constraints on the control input. We derive the necessary conditions for time optimality for a 3D pendulum by formulating a discrete time optimal control problem using a Lie group variational integrator. The approach does not use local parameterizations (like Euler angles or quaternion) for attitude representation but necessary conditions for optimality are derived directly on the special orthogonal group. Further, discrete-time, time optimal attitude control satisfying the necessary conditions is computed using geometrically exact technique on special orthogonal group, such that it preserves the geometric properties of a rigid body.