Effects of Attitude Parameterization Methods on Attitude Controller Performance

Abstract

This paper intends to compare the effect of the attitude parameterization method on the performance of the attitude controller designed with the direct feedback of corresponding attitude states. To achieve this, the proportional-derivative controllers using the finite rotation angles and the modified Rodrigues parameters are designed with the same structure and equivalent performance of the direct quaternion-feedback control. It has been demonstrated through a series of comparative analyses that three different parameterization methods can be commonly used in designing their direct feedbacks for the spacecraft’s attitude control. Next, the effects of the nonlinear transform relations among three parameterizations are thoroughly investigated through the controller-response analyses for an eigen-axis rest-to-rest maneuver with varying the initial angular position. The controller designed with the finite rotation angles shows a consistent performance regardless of the initial angular position. Whereas, those using other two methods show large variations in their response characteristics. From the results, it can be concluded that the direct feedback controller designed with the finite rotation angles outperforms those using the modified Rodrigues parameters or the quaternion, especially when the spacecraft experiences an aggressive maneuver or has a wide operating range of the attitude.

Appendix

This appendix intends to show how accurate kinematics associated with three-parameter finite rotation angles can be obtained without singularity when the attitude $\theta$ approaches zero. For this purpose, function values used in the kinematics, such as ${\mathbf{f}}(\theta )$ and ${\mathbf{g}}(\theta )$, are approximated using the Taylor series expansion of trigonometric functions, as shown in Eqs. (29) and (30) [23]. Less than four terms are enough to achieve the machine precision when $\theta$ becomes less than 1 degree.

$$\sin \theta \cong \theta - \frac{{\theta^{3} }}{3!} + \frac{{\theta^{5} }}{5!} - \frac{{\theta^{7} }}{7!},$$

(29)

$$\cos \theta \cong 1 - \frac{{\theta^{2} }}{2!} + \frac{{\theta^{4} }}{4!} - \frac{{\theta^{6} }}{6!}.$$

(30)

Applying the above equations to Eqs. (4) and (10), the functions used in the kinematics can be approximated using (31) to accurately estimate ${\mathbf{T}}$ and ${\dot{\mathbf{\theta }}}$ when the angular displacement is small enough.

$$\begin{gathered} g(\theta ) = \frac{\sin \theta }{\theta } \cong 1 - \frac{{\theta^{2} }}{3!} + \frac{{\theta^{4} }}{5!} - \frac{{\theta^{6} }}{7!}, \hfill \\ f(\theta ) = \frac{1 - \cos \theta }{{\theta^{2} }} \cong \frac{1}{2!} - \frac{{\theta^{2} }}{4!} + \frac{{\theta^{4} }}{6!}, \hfill \\ h(\theta ) = \frac{\theta - \sin \theta }{{\theta^{3} }} \cong \frac{1}{3!} - \frac{{\theta^{2} }}{5!} + \frac{{\theta^{4} }}{7!}. \hfill \\ \end{gathered}$$

(31)