Analytic Solution Of Generating Function Of Linear Hamiltonian System And Its Application In Spacecraft Formation Reconfiguration

Spacecraft formation flight is a new space operation mode,which has the advantages of low cost,strong system reliability and adaptability.It has become a hot spot in the research of space distributed mission,representing the future development trend of space technology,and has been widely valued by scholars in the field of space at home and abroad.There are many problems in the dynamics,control and application of spacecraft formation flying technology to be studied.The linear quadratic(LQ)optimal control problem is one of the classical problems in the optimal control theory.By introducing a costate variable,an optimal control problem can be reduced to the calculation of the state trajectory of the Hamiltonian system.The traditional method of calculating the trajectory of the Hamiltonian system is equivalent to solving the Hamiltonian Jacobian equation of a value function.In particular,for Hamiltonian systems,since the optimal control problem with a fixed boundary condition over a finite time interval is reduced to a two-point boundary value problem of an ordinary differential equation,the generating function can be used to solve a two-point boundary value problem of an ordinary differential equation to deal with an optimal control problem.If the generating function is found,a series of optimal trajectories under different boundary conditions can be obtained,while the traditional optimal control method can only obtain one trajectories by solving the Hamilton Jacobian equation of a value function.At present,the generating function method is only used to solve the linear optimal control problem of spacecraft formation flight.In this paper,the linear relative motion model between the main spacecraft and the accompanying spacecraft of spacecraft formation flying and the energy optimal control problem based on this linear Hamiltonian system are given firstly.Secondly,according to the analytical solution of the matrix Riccati equation,the solution of the Hamiltonian equation of the linear Hamiltonian system of Spacecraft Formation flying,i.e.the analytical solution form of the generating function is derived.A method of calculating the relative trajectories and optimal control laws of the main and accompanying spacecraft is obtained by using the number method combined with the double generating function.An example of solving the two-dimensional form of the generating function is given.According to the results obtained by the analytical method,the coefficient matrix of the generating function is consistent with the results obtained by the numerical method.It is verified that the method of solving the coefficient of the generating function of the analytical solution is feasible when the eigenvalues of the Hamiltonian matrix are all real numbers.Finally,this paper uses the double generating function numerical method to solve the relative motion of two spacecraft.The energy optimal control problem is solved,the optimal control and relative motion trajectory are calculated,and a simulation example of formation reconfiguration of a spacecraft formation flight is solved by the double generating function method.The influence of different initial position and process time on the optimal control law and optimal trajectory is analyzed.