It is well known that for non-linear Hamiltonian systems there exist ordered regions with quasi-periodic orbits and regions with chaotic orbits. Usually, these regions are distributed in the phase space in very compli...It is well known that for non-linear Hamiltonian systems there exist ordered regions with quasi-periodic orbits and regions with chaotic orbits. Usually, these regions are distributed in the phase space in very complicated ways, which often makes it very difficult to distinguish between them, especially when we are dealing with many degrees of freedom. Recently, a new, very fast and easy to compute indicator of the chaotic or ordered nature of orbits has been introduced by Zotos （2012）, the so-called ＂Fast Norm Vector Indicator （FNV1）＂. Using the double pendulum system, in the paper we present a detailed numerical study comporting the advantages and the drawbacks of the FNVI to those of the Smaller Alignment Index （SALI）, a reliable indicator of chaos and order in Hamiltonian systems. Our effort was focused both on the traditional behavior of the FNVI for regular and fully developed chaos but on the ＂sticky＂ orbits and on the quantitative criterion proposed by Zotos, too.展开更多
Rendezvous in circular or near circular orbits has been investigated in great detail, while rendezvous in arbitrary eccentricity elliptical orbits is not sufficiently explored. Among the various optimization methods p...Rendezvous in circular or near circular orbits has been investigated in great detail, while rendezvous in arbitrary eccentricity elliptical orbits is not sufficiently explored. Among the various optimization methods proposed for fuel optimal orbital rendezvous, Lawden＇s primer vector theory is favored by many researchers with its clear physical concept and simplicity in solu- tion. Prussing has applied the primer vector optimization theory to minimum-fuel, multiple-impulse, time-fixed orbital ren- dezvous in a near circular orbit and achieved great success. Extending Prussing＇s work, this paper will employ the primer vec- tor theory to study trajectory optimization problems of arbitrary eccentricity elliptical orbit rendezvous. Based on linearized equations of relative motion on elliptical reference orbit （referred to as T-H equations）, the primer vector theory is used to deal with time-fixed multiple-impulse optimal rendezvous between two coplanar, coaxial elliptical orbits with arbitrary large ec- centricity. A parameter adjustment method is developed for the prime vector to satisfy the Lawden＇s necessary condition for the optimal solution. Finally, the optimal multiple-impulse rendezvous solution including the time, direction and magnitudes of the impulse is obtained by solving the two-point boundary value problem. The rendezvous error of the linearized equation is also analyzed. The simulation results confirmed the analyzed results that the rendezvous error is small for the small eccentric- ity case and is large for the higher eccentricity. For better rendezvous accuracy of high eccentricity orbits, a combined method of multiplier penalty function with the simplex search method is used for local optimization. The simplex search method is sensitive to the initial values of optimization variables, but the simulation results show that initial values with the primer vector theory, and the local optimization algorithm can improve the rendezvous accuracy effectively with fast convergence, because the optimal results obtained by the primer vector theory are already very close to the actual optimal solution.展开更多
文摘It is well known that for non-linear Hamiltonian systems there exist ordered regions with quasi-periodic orbits and regions with chaotic orbits. Usually, these regions are distributed in the phase space in very complicated ways, which often makes it very difficult to distinguish between them, especially when we are dealing with many degrees of freedom. Recently, a new, very fast and easy to compute indicator of the chaotic or ordered nature of orbits has been introduced by Zotos （2012）, the so-called ＂Fast Norm Vector Indicator （FNV1）＂. Using the double pendulum system, in the paper we present a detailed numerical study comporting the advantages and the drawbacks of the FNVI to those of the Smaller Alignment Index （SALI）, a reliable indicator of chaos and order in Hamiltonian systems. Our effort was focused both on the traditional behavior of the FNVI for regular and fully developed chaos but on the ＂sticky＂ orbits and on the quantitative criterion proposed by Zotos, too.
基金supported by the National Natural Science Foundation of China(Grant Nos. 10832004 and 11072122)
文摘Rendezvous in circular or near circular orbits has been investigated in great detail, while rendezvous in arbitrary eccentricity elliptical orbits is not sufficiently explored. Among the various optimization methods proposed for fuel optimal orbital rendezvous, Lawden＇s primer vector theory is favored by many researchers with its clear physical concept and simplicity in solu- tion. Prussing has applied the primer vector optimization theory to minimum-fuel, multiple-impulse, time-fixed orbital ren- dezvous in a near circular orbit and achieved great success. Extending Prussing＇s work, this paper will employ the primer vec- tor theory to study trajectory optimization problems of arbitrary eccentricity elliptical orbit rendezvous. Based on linearized equations of relative motion on elliptical reference orbit （referred to as T-H equations）, the primer vector theory is used to deal with time-fixed multiple-impulse optimal rendezvous between two coplanar, coaxial elliptical orbits with arbitrary large ec- centricity. A parameter adjustment method is developed for the prime vector to satisfy the Lawden＇s necessary condition for the optimal solution. Finally, the optimal multiple-impulse rendezvous solution including the time, direction and magnitudes of the impulse is obtained by solving the two-point boundary value problem. The rendezvous error of the linearized equation is also analyzed. The simulation results confirmed the analyzed results that the rendezvous error is small for the small eccentric- ity case and is large for the higher eccentricity. For better rendezvous accuracy of high eccentricity orbits, a combined method of multiplier penalty function with the simplex search method is used for local optimization. The simplex search method is sensitive to the initial values of optimization variables, but the simulation results show that initial values with the primer vector theory, and the local optimization algorithm can improve the rendezvous accuracy effectively with fast convergence, because the optimal results obtained by the primer vector theory are already very close to the actual optimal solution.
