Optimal attitude maneuvers in the presence of prohibited directions

We consider the problem of realigning a space telescope from one observation target to the next in the presence of prohibited directions. More specifically, we want to steer the space telescope – modelled as a gyrostat – from an initial attitude at rest to a final attitude at rest within a fixed time interval. During the motion, the line of sight of the telescope must be kept away from forbidden directions towards bright objects like the sun, moon or earth due to power or thermal requirements. The kinematics of the spacecraft motion are governed by a differential equation on the rotation group SO(3). Treating the angular velocities as control variables, this equation takes the form of a controlled dynamical system. To ensure reorientation maneuvers satisfying these pointing constraints, we introduce a cost functional penalizing proximity of the line of sight of the telescope to any of the forbidden directions. Furthermore, we include penalty terms which provide a smooth motion of the satellite and ensure the execution of a rest-to-rest maneuver. The chosen cost functional is minimized over all possible trajectories of the controlled dynamical system between the prescribed initial and target attitudes, which leads to an optimal control problem on SO(3), which is solved by applying a version of Pontryagin's Maximum Principle tailor-made for optimal control problems on Lie groups. Parametrizing SO(3) in terms of Cardan angles, the solution is formulated as a boundary-value problem on Euclidean space and hence can be solved numerically by conventional methods. The existence of two first integrals is established and exploited to reduce the computational effort. The applicability of this approach is shown in concrete examples.