Linear Equations of Relative Motion

This chapter presents a variety of linear differential equations for modeling relative motion under the two-body assumptions. The system of equations for the description of relative motion is classified according to the coordinate system utilized—Cartesian or curvilinear. There are also issues of the choices of the independent variable—time or an angle variable and the space in which the linearization is carried out. The focus in this chapter is on the physical coordinate description of relative motion. This chapter begins with a discussion of the derivation of the linearized equations of relative motion with respect to a circular reference orbit, also known as the Clohessey–Wiltshire (CW) equations. The CW equations are used extensively for the analysis, design, and control of formations. A brief description of these applications is given in the chapter. A derivation of the CW equations is presented from the perspective of analytical mechanics, via the Lagrangian and the Hamiltonian. The chapter presents a comparison and error analysis of the linearized equations obtained by using Cartesian and curvilinear coordinates. Tschauner–Hempel equations for elliptic reference orbits are then presented. Brief descriptions of several approaches to the derivation of the two-body relative motion State Transition Matrix (STM) are also included. The chapter concludes with a discussion of the application of the elliptic reference orbit STM to obtain initial conditions for preventing secular drift.