Academy transaction note “closed form solution for a minimum deviation magnetically controllable satellite angular trajectory”

Highlights

• Magnetically controllable attitude trajectory is constructed.

• Stabilization accuracy near unstable gravitational equilibrium is optimized.

• Quadratic programming problem is formulated with linear constraint.

• Control torque is perpendicular to the induction vector.

• Solution is found in a closed form using the Moore-Penrose pseudo inverse matrix.

Abstract

Purely magnetic satellite attitude control is considered. Special reference motion is constructed which provides local controllability. The angular trajectory is found in the neighborhood of unstable gravity gradient torque equilibrium. Direct solution for the minimization of the quadratic cost function representing the deviation from the equilibrium is derived. Linear equality type constraint on the torque direction is imposed. This solution is compared to the one found by the particle swarm optimization.

Introduction

The magnetic attitude control system suffers from the problem of local uncontrollability which was identified in initial findings [1,2]. Although the controllability is proven in general [3,4], significant effort is required to achieve it in practice. Specific control gain tuning procedure for a feedback law [[5], [6], [7], [8], [9]], sliding control modifications [[10], [11], [12]], model predictive control [13,14], and oscillating controls [15] were designed for this purpose. More detailed survey of the developments in the three axis magnetic control is provided in Refs. [16,17].

This note further develops a two-step approach suggested in Ref. [18]. First, a reference angular trajectory is constructed in the neighborhood of the required angular position using particle swarm optimization (PSO) algorithm [[19], [20], [21]]. Second, when an acceptable trajectory is found, the control gains of a feedback law that ensure asymptotic stability are obtained using another PSO routine.

The first step is reduced to a quadratic programming problem using linearization in the present note. The trajectory is derived in a closed form. It is not optimal for the nonlinear case. However, it is relatively close to the PSO-derived trajectory. Small accuracy loss is balanced by significant saving in the computational time.