Table of Contents
ISRN Aerospace Engineering
Volume 2014 (2014), Article ID 136315, 15 pages
Research Article

Satellite Attitude Control Using Analytical Solutions to Approximations of the Hamilton-Jacobi Equation

University of Toronto Institute for Aerospace Studies, 4925 Dufferin Street, Toronto, ON, Canada M3H 5T6

Received 25 October 2013; Accepted 11 December 2013; Published 20 February 2014

Academic Editors: A. Desbiens, C. Meola, and S. Simani

Copyright © 2014 Stefan LeBel and Christopher J. Damaren. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The solution to the Hamilton-Jacobi equation associated with the nonlinear control problem is approximated using a Taylor series expansion. A recently developed analytical solution method is used for the second-, third-, and fourth-order terms. The proposed controller synthesis method is applied to the problem of satellite attitude control with attitude parameterization accomplished using the modified Rodrigues parameters and their associated shadow set. This leads to kinematical relations that are polynomial in the modified Rodrigues parameters and the angular velocity components. The proposed control method is compared with existing methods from the literature through numerical simulations. Disturbance rejection properties are compared by including the gravity-gradient and geomagnetic disturbance torques. Controller robustness is also compared by including unmodeled first- and second-order actuator dynamics, as well as actuation time delays in the simulation model. Moreover, the gap metric distance induced by the unmodeled actuator dynamics is calculated for the linearized system. The results indicated that a linear controller performs almost as well as those obtained using higher-order solutions for the Hamilton-Jacobi equation and the controller dynamics.