Optimal stabilization of spacecraft in an inertial coordinate system based on a strapdown inertial navigation system

We consider the optimal control problem for spacecraft motion during correction of its position in an inertial coordinate system by means of control torques. Control torques arise from the acceleration of inertial flywheels of a strapdown inertial navigation system. We investigate optimal control, which ensures a smooth change in the spacecraft orientation. This smooth corrective motion is described as the motion along the shortest path in the configuration space of a special orthogonal group $SO(3)$. The shortest path coincides with the large circle arc of the unit hypersphere $S^3$. We also consider a control algorithm using the original procedure of nonlinear spherical interpolation of quaternions. Four inertial flywheels are used as the main executive bodies for orientation of the dynamic control loop of the strapdown inertial navigation system when solving the optimal control problem. Three flywheels are oriented along the axes of the spacecraft. The fourth flywheel is oriented along the bisector. The simulation results are presented. We consider examples for corrective motion in which the spacecraft has zero velocity and acceleration at the beginning and end of the maneuver. We give an animation of the corrective movement of the spacecraft. The proposed formalism can be extended to control the spacecraft motion at an initial angular velocity different from zero, as well as in the orbital coordinate system.