A new look at the uncontrollable linearized quaternion dynamics with implications to LQR design in underactuated systems

Abstract

© 2018 European Control Association (EUCA).Quaternion algebra is frequently employed for spacecraft attitude description due to its convenient numerical properties when compared to minimal formulations. In parallel, Linear Quadratic Control (LQR)-based attitude controllers are often applied to underactuated vehicles due to its intuitive tuning process and satisfactory stability robustness properties. However, nonlinear quaternion differential equations of motion linearization yields non-stabilizable systems. Thus, LQR techniques cannot be directly employed since the associated algebraic Riccati equation is ill-posed. The commonplace solution resorts to a reduced quaternion model where only three out of four quaternion coordinates are exploited. The present work shows that such choice exhibits numerically unstable regions that impedes solving the LQR problem for all possible operating points. Additionally, we propose two methods to obtain wellposed LQR problems over all operating points. The first is based on the reduced quaternion model with an appropriate change of coordinates. The second is to append a virtual stabilizing input (VSI) to the nonlinear system to attain controllable linearized systems. The VSI direction should be appropriately chosen to not disturb the controllable modes of the system. Finally, we show that a class of constant angular velocity tracking problems is time-invariant under an appropriate change of variables such that time-invariant LQR techniques are applicable.