High-Order Method For Nonlinear Orbital Uncertainty Propagation Based On Differential Algebra

The uncertainty propagation problem is a basic problem involved in many typical aerospace dynamics studies such as orbit determination,high-precision orbit prediction,and space situational awareness.In the past decade,orbital uncertainty propagation based on some nonlinear high-order algorithms is a hot issue in aerospace dynamics research.Differential algebra is an automatic differentiation technique which is extended to any or-der and can automatically expand complex nonlinear functions into Taylor polynomials at an arbitrary order.Based on differential algebra technology,this thesis studies the high-order uncertainty propagation methods of nonlinear orbits.Compared with the existing methods,this thesis focuses on the method of accurately propagating the overall probabil-ity density function,effectively capturing the non-Gaussian characteristics of distribution,and using it to evaluate the risk of space debris collision.The main research results of the thesis are as follows:A high-order state transition polynomial method for propagation of orbital state deviation and time deviation is proposed.1)A high-order state transition polynomial model based on differential algebra is established,which is a high-order Taylor polyno-mial of the final state about the initial state deviation,and a method for a priori estimating the accuracy and convergence range of the state transition polynomial is provided.2)Con-sidering the deviation of the propagation time,the state transition polynomial is extended to the time dimension,which can directly give the continuous trajectory near the nominal final time.3)The proposed method can accurately predict the final state corresponding to given state and time deviations,and has potential on-board online application capability.A high-order mapping method for calculating the probability density function at a final point is proposed.1)Based on the principle of conservation of probability and the Fokker-Planck equation,the transfer equation of the probability density function in general nonlinear mapping is derived.2)Establish a high-order mapping method from the initial or final state to the corresponding probability density in the differential alge-bra framework.3)The proposed method can accurately calculate the probability density function corresponding to any state within a certain range.The transition of probability denstiy in orbit transformation and propagation process is analyzed.A Gaussian mixture model high-order propagating method for overall uncer-tainty propagation is proposed.1)For an initial Gaussian distribution,the method of calculating the final mean vector and the covariance matrix by high-order state transition polynomial is given.2)Establish a high-order propagation algorithm for Gaussian mix-ture model.Use a Gaussian mixture model to fit the state distribution,and propagate each Gaussian distribution by a high-order state transition polynomial to obtain the Gaussian mixture model of the overall final distribution.3)The obtained final Gaussian mixture model can fit the final state distribution well and give the overall probability density dis-tribution.A method for distribution evolution and threat analysis of space collision debris is proposed based on uncertainty propagation.1)Use the Gaussian mixture model high-order propagating method to predict the distribution of space collision debris within several months,and analyze the influence of time,J₂ perturbation,initial deviations of orbital elements on the debris distribution.2)All the proposed uncertainty propagation algorithms are used to analyze the debris collision risk to an on-orbit target.The expected number of debris and the corresponding dangerous moments in every day is given,and the potential nominal collision velocity and velocity distribution are calculated at the moment when the collision threat is large.All in all,based on the differential algebra technology,the thesis establishes a set of methodologies for high-order propagation of nonlinear orbital uncertainty,which has significant academic value;The proposed method is used to analyze the problem of space collision debris evolution and threat analysis.It can also be applied to the solution of many other practical engineering problems,and improve its accuracy and efficiency.It has potential engineering application value.