Higher-order sensitivity matrix method for probabilistic solution to uncertain Lambert problem and reachability set problem

This paper presents a derivative-free method for computing approximate solutions to the uncertain Lambert problem (ULP) and the reachability set problem (RSP) while utilizing higher-order sensitivity matrices. These sensitivities are analogous to the coefficients of a Taylor series expansion of the deterministic solution to the ULP and RSP, and are computed in a derivative-free and computationally tractable manner. The coefficients are computed by minimizing least squared error over the domain of the input probability density function (PDF), and represent the nonlinear mapping of the input PDF to the output PDF. A non-product quadrature method known as the conjugate unscented transform is used to compute the multidimensional expectation values necessary to determine these coefficients with the minimal number of full model propagations. Numerical simulations for both the ULP and the RSP are provided to validate the developed methodology and illustrate potential applications. The benefits and limitations of the presented method are discussed.