The Basics of Analytical Mechanics, Optimization, Control and Estimation

Abstract

This chapter reviews the basics of analytical mechanics as applied to orbit theory, optimization, control, and estimation. The chapter starts by presenting some notions in Lagrangian and Hamiltonian mechanics, including canonical transformations. Lagrangian and Hamiltonian mechanics are two main approaches to mechanics that constitute a generalization of the classical Newtonian mechanics. The rationale behind the Lagrangian and Hamiltonian formalisms is related to variational principles. Hamiltonian and Lagrangian mechanics is discussed in the chapter for better understanding of the perturbation modeling, relative motion modeling, mean-to-osculating element conversion using the Brouwer transformation, and averaging. The chapter proceeds with a discussion on Delaunay variables and their applications in Brouwer's satellite theory. The chapter explains static optimization, control, and filtering methods and presents some key tools of analytical mechanics, optimization, control, and estimation. The optimization theory, including the concept of bordered Hessians, is discussed for developing optimal formation-keeping maneuvers. The chapter discusses the role of linear quadratic regulators and control Lyapunov functions in controlling spacecraft formations. Application of Kalman filtering in problem of relative navigation is also discussed.