Nonlinear Method For Orbit Targeting And Uncertainty Propagation

As a classical boundary-value problem and an extension to the initial-value problem,the orbit maneuvering and the orbital uncertainty propagation are the fundamentally theoritical problems which are widely and continuly studied in the area of orbit mechanics.Moreover,they are also the basic techniques in the missions of space operation and space conjunction analysis.In previous research efforts,these problems were solved under some simplified assumptions,e.g.unperturbed dynamics or linearized dynamics.Consequently,many practical factors such as perturbations and uncertainties were not considered by previous studies,thus the obtained solutions might be inaccuracy.As an improvement,this paper investigates the orbit targeting and uncertainty propagation problems by accounting for the practical perturbations and uncertainties.Several nonlinear semi-analytical or analytical algorithms are developed,and the main achievements are summarized as follows.A homotopic perturbed Lambert algorithm is developed for the absolute orbit transfer problem.1)For the long-duration,perturbed orbit transfer problem,a perturbed multiple-revolution Lambert algorithm using the homotopy method is developed to avoid the divergence of previous orbit targeting techniques;2)A feasible iteration optimization approach for solving the multiple-impulse perturbed orbit transfer problem is formulated,in which the homotopic perturbed Lambert algorithm is used to handle the nonlinear equality constrants;and 3)The proposed method can effectively solve the long-duration,far-range orbit transfer problem and provide more practical solution by considering perturbations.A nonlinear analytic solution for satellites relative motion under J2perturbation is derived and a nonlinear orbital targeting algorithm is formulated for relative orbit transfer.1)By using the geometric method that avoids directly solving the complex relative-motion differential equations,a second-order analytic solution for relative motion in J₂-perturbed elliptic orbits is derived;2)Based on the nonlinear relatie-motion equations,an analytic orbit targeting algorithm that includes the effects of J₂ and second-order terms is developed;and 3)In comparison to the previous linear or nonlinear analytic results,the proposed method provides more accurate solution for satellites relative motion with relatively large separation distance and relatively large orbital eccentricity.A hybrid uncertainty propagation method combining the state transition tensors and the Gaussian mixture model is developed.1)Considering the nonlinear,non-Gaussian characteristics of uncertainty propagation in practical dynamical systems,the Gaussian mixture model?GMM?is used to approximate the probability density function?PDF?of an arbitrarily distributed uncertainty,and the method to split and merge a GMM is presented;2)For the absolute orbit motion and the relative orbit motion,a semi-analytic and an analytic state transition tensors?STT?are derived respectively,and these STT are further extended to consider the effects of abrupt state jumps,e.g.a trajectory with impusive maneuvers;3)Combining the STT with the covariance analysis technique,the nonlinear propagation of the statistical moments of uncertainties is developed;and 4)Combining the STT with the GMM,the nonlinear propagation of the PDF of uncertainties is further developed,in which the STT are used to propagate the mean and covariance matrix of each Gaussian mixture kernel.A nonlinear robust trajectory optimization method is developed by accounting for uncertainties.1)Considering the effects of practical uncertainties and the orbit re-planning process,a set of robust performance indices are defined to quantify the robustness of trajectory;2)Combining the the minimum of?v and the minimum of trajectory robustness,a nonlinear robust trajectory optimization considering orbit re-planning is developed;and 3)A trade-off solution which can balance the?v-cost and the trajectory robustness is obtained by solving the robust trajectory optimization model.This new method can markedly improve the orbital precision in space rendezvous mission.Considering the effects of perturbations and uncertaitnies,a set of nonlinear methods for orbit targeting,uncertainty propagation and robust trajectory optimization are systematically developed.The hybrid STT and GMM method has been proposed for accurately propagating orbital uncertainties.The perturbed orbit targeting algorithm and the robust trajectory optimization method have also been developed,which can afford some innovative solutions to spacecraft orbit design and control problems.