An exponentially fast attitude tracking controller on the rotation group

In this paper a continuous attitude tracking control law is derived directly on the rotation group SO(3). The proposed control law is shown to reduce the closed-loop attitude dynamics to a linear oscillator description of the eigen-axis error, without the need for a small angle approximation. The main practical benefit of this is that the gains can be easily tuned to drive this eigen-axis error to zero exponentially fast and with a damped response without oscillations. The approach uses geodesic error metrics on the rotation group and the angular velocity to construct a Lyapunov function. The time-derivative of this Lyapunov function is control dependent and a continuous control is selected to guarantee asymptotic tracking of the reference motion. Furthermore, the closed-loop system, with this rotation-matrix based feedback control applied, is converted to its quaternion form and further reduced to an eigen-axis error description of the dynamics. This reduction reveals a simple method for tuning the control which involves only one parameter which can be selected to obtain the fastest convergence to the reference motion. However, this control suffers the problem related to the exponential coordinates; the control is not defined globally. This paper shows that by converting the control to quaternions and augmenting the control a globally defined exponentially fast tracking control law can be defined. The proposed control is applied in simulation to the attitude control of a small spacecraft and shows a settling time performance enhancement, for given actuator constraints, compared to a conventional quaternion tracking controller.