Constrained single-axis path planning of underactuated spacecraft

The single-axis maneuver planning problem is investigated for underactuated spacecraft with one unactuated axis fixed in a certain direction of the spacecraft in the presence of a single conical forbidden pointing zone. Specifically, a nonnominal Euler rotation scenario is formulated by imposing admissible constraints on rotation axes, and it is proven that the singular sensitive axis can be pointed along an arbitrary inertial direction outside the forbidden zone by at most two principal rotations. Given this fact, an exact solution is obtained by numerically solving a quartic trigonometric function equation about the first rotation angle, while enforcing the rotation axes tangent to the forbidden cone for the shorter path. Furthermore, an approximate but computationally efficient solution is analytically developed by Ferrari's formulas for the linearized quartic trigonometric function equation around the first tangent direction, where the smallest real one is selected from the four candidate roots. The solvability of the two-step planning schemes is also assured by the corresponding accessible regions. Finally, comprehensive numerical simulations are conducted to corroborate the effectiveness of both the exact and approximate planning schemes.