A Universal Approach For Solving The Ultra-Revolution Lambert's Problem

Lambert's problem, a cornerstone of orbital mechanics, is crucial in the design and planning of space missions, from satellite rendezvous to deep-space trajectory optimization. This paper presents a universal approach (valid across all conics) to solving Lambert’s Problem, specializing in the ultra-revolution problem. By employing matrix exponential solutions to the Sundman transform, the transfer time equation--constructed of universal functions--is evaluated through exponential functions, resulting in two key advantages. The first advantage is the removal of all trigonometric functions and associated inverses when evaluating the transfer time function, reducing computation cost. The second is a unified transfer time formulation across elliptic and hyperbolic regimes, reducing algorithm complexity. Furthermore, the multi-revolution term is removed from the transfer time equation, reducing the complexity of transfer time derivatives and improving the numerical stability of multi-revolution solutions. A simple-but-accurate initial guess scheme that leverages the transfer angle is employed, resulting in accurate initial guesses and solutions even at 100,000 revolutions. Simulations indicate computation rates similar to state-of-the-art methods. Tests across a wide range of representative cases confirm that the solver achieves 10-digit velocity accuracy within two iterations for most transfer geometries and one iteration for near-circular cases, regardless of revolution count.