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Abstract. This paper concerns the dynamics of a rigid body of finite extent moving under the influence of a central gravitational field. A principal motivation behind this paper is to reveal the hamiltonian structure of the n-body problem for masses of finite extent and to understand the approximation inherent to modeling the system as the motion of point masses. To this end, explicit account is taken of effects arising because of the finite extent of the moving body. In the spirit of Arnold and Smale, exact models of spin-orbit coupling are formulated, with particular attention given to the underlying Lie group framework. Hamiltonian structures associated with such models are carefully constructed and shown to be non-canonical. Special motions, namely relative equilibria, are investigated in detail and the notion of a non-great circle relative equilibrium is introduced. Non-great circle motions cannot arise in the point mass model. In our analysis, a variational characterization of relative equilibria is found to be very useful.The reduced hamiltonian formulation introduced in this paper suggests a systematic approach to approximation of the underlying dynamics based on series expansion of the reduced hamiltonian. The latter part of the paper is concerned with rigorous derivations of nonlinear stability results for certain families of relative equilibria. Here Arnold's energyCasimir method and Lagrange multiplier methods prove useful.

A constrained filtering method is proposed to deal with the filtering problems for nonlinear systems with constraints. The problem is convened to a sequence of recursive estimation problems in which the system equations and constraint conditions are treated as pscudo-measurements. To resolve the singularity problem arising from the constraints, a modified maximum-likelihood method for nonlinear systems is developed. The simulation results from the application of the proposed scheme to the target tracking problem shows that the constrained filtering method can enhance the performance of filter design significantly.