Stabilizing the coupled orbit–attitude dynamics of a rigid body in a J2 gravity field using Hamiltonian structure

Abstract The gravitationally coupled orbit–attitude dynamics of a rigid body in a J2 gravity field is a high-precision model for the spacecraft in the close proximity of a small spheroid celestial body, since the gravitational orbit–attitude coupling of the spacecraft is naturally taken into account in this model. A Hamiltonian structure-based feedback control law is proposed to stabilize relative equilibria of the rigid body in the J2 gravity field. The proposed stabilization control law is consisted of three parts: potential shaping, momentum control, and energy control. The potential shaping is used to modify the gravitational potential artificially so that the relative equilibrium is a minimum of the modified Hamiltonian. It is shown that the unstable relative equilibrium can always be stabilized in the Lyapunov sense by the potential shaping with sufficiently large feedback gains. Then the momentum control leads the motion to the same invariant manifold with the relative equilibrium, which is actually a momentum level set. The energy control introduces energy dissipation into the system and the motion will asymptotically converge to the minimum of the modified Hamiltonian on the invariant manifold, i.e., the relative equilibrium. The feasibility of the proposed stabilization control law is validated through numerical simulations. The control law can be implemented by the attitude control system and low-thrust engines onboard and the fuel consumption is reasonable. This Hamiltonian structure-based stabilization approach is applicable to non-canonical Hamiltonian systems commonly existing in the astrodynamics. Since the natural dynamical behaviors of the system is fully utilized, the proposed control law is very simple and is easy to implement autonomously with little computation in the space applications.

[1]  J. Marsden,et al.  Geometric Mechanics and the Dynamics of Asteroid Pairs , 2004 .

[2]  R. Broucke,et al.  Numerical integration of periodic orbits in the main problem of artificial satellite theory , 1994 .

[3]  J. A. Vera.  Eulerian equilibria of a triaxial gyrostat in the three-body problem: Rotational Poisson dynamics in Eulerian equilibria with oblateness , 2009 .

[4]  Symmetry, reduction and relative equilibria of a rigid body in the J2 problem , 2013, 1306.4140.

[5]  J. Marsden,et al.  Introduction to mechanics and symmetry , 1994 .

[6]  J. A. Vera,et al.  Hamiltonian Dynamics of a Gyrostat in the N-Body Problem: Relative Equilibria , 2006 .

[7]  Naomi Ehrich Leonard,et al.  Potential shaping and the method of controlled Lagrangians , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[8]  A. J. Schaft,et al.  Stabilization of Hamiltonian systems , 1986 .

[9]  Christopher D. Hall,et al.  RELATIVE EQUILIBRIA OF A RIGID SATELLITE IN A CIRCULAR KEPLERIAN ORBIT , 1998 .

[10]  Andrzej J. Maciejewski,et al.  Reduction, relative equilibria and potential in the two rigid bodies problem , 1995 .

[11]  J. L. Guirao,et al.  Lagrangian relative equilibria for a gyrostat in the three-body problem: bifurcations and stability , 2010 .

[12]  Shijie Xu,et al.  Hamiltonian structures of dynamics of a gyrostat in a gravitational field , 2012 .

[13]  Shijie Xu,et al.  Gravitational Orbit-Rotation Coupling of a Rigid Satellite around a Spheroid Planet , 2014 .

[14]  Shijie Xu,et al.  On the existence of the relative equilibria of a rigid body in the J2 problem , 2014 .

[15]  Shijie Xu,et al.  Stability of relative equilibria of the full spacecraft dynamics around an asteroid with orbit–attitude coupling , 2014 .

[16]  H. Schaub,et al.  J2 Invariant Relative Orbits for Spacecraft Formations , 2001 .

[17]  Shijie Xu,et al.  Relative equilibria of full dynamics of a rigid body with gravitational orbit-attitude coupling in a uniformly rotating second degree and order gravity field , 2014 .

[18]  Shijie Xu,et al.  Stability of the classical type of relative equilibria of a rigid body in the J2 problem , 2013, 1304.6867.

[19]  Jerrold E. Marsden,et al.  Stabilization of relative equilibria , 2000, IEEE Trans. Autom. Control..

[20]  Shijie Xu,et al.  On the nonlinear stability of relative equilibria of the full spacecraft dynamics around an asteroid , 2014 .

[21]  Ming Xu,et al.  On the existence of J2 invariant relative orbits from the dynamical system point of view , 2012 .

[22]  G. B. Sincarsin,et al.  Gravitational orbit-attitude coupling for very large spacecraft , 1982 .

[23]  Anthony M. Bloch,et al.  Stability and stabilization of relative equilibria of dumbbell bodies in central gravity , 2005 .

[24]  J. L. Guirao,et al.  Dynamics of a gyrostat on cylindrical and inclined Eulerian equilibria in the three-body problem ☆ , 2010 .