Time-Optimal Spacecraft Reorientation Using the Uniform Trigonometrization Indirect Method

This study revisits the classic time-optimal reorientation of an axi-symmetric rigid spacecraft with two controls, a benchmark problem known to contain a challenging second-order singular control arc. Although modern direct collocation methods like pseudospectral methods (PSMs) can solve this problem, they often struggle to accurately resolve the singular arc. The PSM produces jittery control solutions that are physically unrealistic to implement and may obscure the true optimal solution structure. On the other hand, traditional indirect methods, though capable of producing high-resolution solutions, are very sensitive, require very good initial guesses and complex multi-stage hybrid schemes to converge. This has left a clear gap, as the problem has not been solved using a modern, standalone indirect method designed to overcome these implementation challenges and numerical sensitivities. This paper fills that gap by applying a modern indirect framework, the Uniform Trigonometrization Method (UTM). This method resolves the primary drawbacks of traditional indirect methods by using trigonometric regularization to transform the non-smooth problem into a smooth one. The UTM converts the complex multi-point boundary-value problem into a simpler two-point boundary-value problem. A comparative analysis reveals that the UTM not only produces a perfectly smooth solution for the singular arc, but also uncovers a more detailed 17-point optimal switching structure. The UTM identifies additional bang-bang solutions that were obscured by numerical chattering in the 9-point direct method solution. This study concludes that UTM is a very simple and efficient framework for this spacecraft reorientation problem. The UTM offers a simpler formulation while achieving a higher-fidelity solution than previously established hybrid or direct methods.