Normal Form Theory Of Periodic Hamiltonian Systems With Multiple Degrees Of Freedom And Its Application To Attitude Stability Of Spacecraft In Elliptical Orbits

Attitude stability has been an important topic in nonlinear studies of spacecraft attitude dynamics and control.However,current studies have mainly focused on the spacecraft in torque-free states or circular orbits,but rarely on elliptical orbits.In some space missions such as military reconnaissance,communications coverage,and deep space exploration,spacecraft need to operate in elliptical orbits with large eccentricities.The differential equations for the perturbed motion of a spacecraft in an elliptical orbit are characterized by a strong nonlinearity and time-varying behavior,which belong to a class of non-autonomous systems.For non-autonomous systems,there is no efficient way to construct Lyapunov functions.Fortunately,this kind of system can be transformed into Hamiltonian systems with a period of 2π by the transformation between time and true anomaly.Therefore,the stability problem of which can be solved on the basis of the normalization method of Hamiltonian and KAM theory.However,the classical normalization method is difficult to apply to multiple degrees of freedom systems due to its computational overhead.This paper aims to extend the stability theory of Hamiltonian systems to the case of multiple degrees of freedom and to provide an efficient algorithm for the stability analysis of the periodic Hamiltonian systems with multiple degrees of freedom,and apply it to the study of the attitude stability of spacecraft in elliptical orbits.The main content of the paper is as follows:First,the stability of multiple degrees of freedom linear periodic Hamiltonian system with small parameters has been studied in the context of a spacecraft in a nearly circular orbit,based on the normal form theory of Hamiltonian systems and Lyapunov’s first method.On account of the anti-symmetry of symplectic space,it was proved that a linear periodic Hamiltonian system can be decomposed into a linear superposition of resonant and non-resonant systems in the complex field.It was proved that the non-resonant system is stable,by analyzing its structure of the solution,and thus the stability of the system depends on the resonant system.An analytical criterion for the instability of resonant systems was established using the multi-scale method to analyze the integral curves of the resonant system in the neighborhood of the trivial solution of the system.Based on the above results,an efficient method has been proposed to analyze the stability of such linear periodic Hamiltonian systems with multiple degrees of freedom.Second,in the context of a spacecraft in an elliptical orbit with arbitrary eccentricity,the theoretical problems related to the efficient calculation of the coefficients of the normal form of the nonlinear periodic Hamiltonians with multiple degrees of freedom have been studied.Based on Jacobi’s theorem and the fundamental theory of symplectic geometry,it has been proven that for a nonlinear periodic Hamiltonian system with multiple degrees of freedom,there exists a symplectic mapping from the neighborhood of the equilibrium point to itself.The normal form of the symplectic mapping was derived,the initial value problem of the Hamilton-Jacobi equation was reduced to a boundary value problem using the coefficients of the normalized symplectic mapping,and the analytical solution was obtained based on the idea of the trial function method.Using the analytical solution,an explicit analytical functional relationship between the coefficients of the normalized Hamiltonian and the symplectic mapping was established,which greatly improves the efficiency of calculating the coefficients of the normal form of the periodic Hamiltonian function with multiple degrees of freedom and guaranties sufficient computational accuracy.In addition,the above results on the stability of Hamiltonian systems have been applied to the attitude stability analysis of spacecrafts in elliptical orbits.The unique advantage of this approach is that there is no need to explicitly write down the specific form of the differential equations of the perturbed motion in the Hamiltonian framework,but the Hamilton functions are normalized directly on the basis of the normal form theory of Hamiltonian systems.This allows the system to be reduced to the simplest form while maintaining its dynamical properties and symplectic structure.In this way,the stability can be studied directly based on the results of the KAM theory,and it is beneficial to reveal the key parameters that affect the stability of the system.As an application of the results of the above studies,the stability of the cylindrical precession and resonant rotation of a gyrostat satellite in elliptical orbit has been studied,and the influence of the gyroscopic moment,orbital eccentricity and mass distribution of the system on its stability were revealed.Cylindrical precession is a type of regular precession in which the axis of symmetry of the system is perpendicular to the orbital plane.The resonant rotation is a planar periodic motion such that the body completes one rotation in inertial space during two orbital revolutions of its center of mass.The results of the stability of the cylindrical precession indicated that,in the case of elliptical orbits,even a large gyroscopic moment can destabilize the cylindrical precession,whereas in the case of fourth-order resonances,a large gyroscopic moment can stabilize the cylindrical precession.The increasing orbital eccentricity is harmful to the stability of the cylindrical precession,while the closer the inertial ellipsoid is to a sphere,the more favorable it is to the stability of the cylindrical precession.The stability analysis of the resonant rotation has been shown that the gyroscopic moment perpendicular to the orbital plane cannot enlarge the linear stable region of the resonant rotation and may even lead to the instability of resonant rotation.For nonlinear systems,the gyroscopic moment can stabilize or destabilized the resonant rotation.The influence of gyroscopic moments on the stability of resonant rotation depends on the orbital eccentricity and the magnitude and direction of the gyroscopic moment.Finally,the effect of the connection stiffness and the mass distribution of the system on the attitude stability of a flexible connected two-body satellite in an elliptical orbit has been studied.It was shown that there are two types of periodic motions,namely,Mercury resonance and resonant rotation,and the necessary conditions for the existence of periodic motions were determined.Mercury resonance is a planar periodic motion such that the body completes three rotations in inertial space during the two orbital revolutions of its center of mass.The results of the stability analysis indicated that the linear stable region enlarges with the increasing connection stiffness.Furthermore,there are several third-and fourth-order resonance curves in the linear stable regions.In the linear stability region,the periodic motions can be destabilized in the presence of some third-and fourth-order resonances,while they are stable in non-resonant cases and other resonant cases.The results presented in this paper provide an efficient algorithm for the normalization of multiple degrees of freedom periodic Hamiltonian systems for the stability analysis,make a useful attempt at studying the stability theory of Hamiltonian systems with multiple degrees of freedom.In addition,the effects of the orbital eccentricity,the gyroscopic moment,the mass distribution of the system,and the connection stiffness on the attitude stability of the spacecraft in elliptical orbits have been revealed,which have potential values for spacecraft design and attitude control.