Computing Minimum-Power Dipole Solutions for Interdipole Forces Using Nonlinear Constrained Optimization with Application to Electromagnetic Formation Flight

Abstract

Electromagnetic formation flight (EMFF) denotes a method of formation flight control in which a cluster of spacecraft are equipped with controllable magnetic dipoles for coordination of their relative positions using interdipole forces. We present a method for finding a minimum-power dipole solution for a given set of desired interdipole forces. We approach this nonlinear constrained optimization problem using sequential quadratic programming, which requires a Jacobian relating changes in the dipoles to changes in forces, as well as the gradient and Hessian of a Lagrangian function. We derive compact analytic solutions for all three of these quantities, using linear-algebraic representations and vector calculus, which can be implemented numerically with a small set of simple functions. Our approach does not rely on arbitrary parameterizations as have prior approaches, and the structure enables further analysis of numerical conditioning and convergence. We conduct numerical simulations, using a number of configurations relevant to EMFF, to verify the method and characterize its performance when numerical routines are randomly initialized, which can serve as a benchmark against which future improvements can be quantified. The method presented may have other uses beyond EMFF, including being applied to new classes of modular magnetic systems.