Complete Kepler Propagator Including Second-Order Sensitivities

Motivated by second-order optimization and estimation applications, the Kepler initial value problem is revisited with the goals of fast runtime, robust convergence, and accurate computation of the state transition matrix and tensor. Goodyear’s universal time equation and Pitkin’s sensitivity formulation are extended with computational improvements, including a new recursive formula for arbitrary-order partial derivatives of Kepler’s time equation with respect to the universal variable. A new initial guess algorithm, applicable to other formulations, is combined with a ninth-order correction step to globally converge the root solve in 1 or 2 iterations. A new mapping of partial derivatives between independent variables leads to both time-fixed and universal variable-fixed sensitivities. The latter is new and particularly useful for regularized applications. The former provides a quadratic, synchronous relative motion model that is functionally equivalent to Pitkin’s. Problem regions in the parameter space are identified to mitigate numerical difficulties. The resulting root solver and state and sensitivity propagators are extensively validated over the complete domain through numerical self-consistency checks and benchmarking against existing solvers. The new propagators demonstrate strong robustness and consistency in terms of both speed and accuracy, especially in the problematic regions of the parameter space. The accompanying code is open source.