I would like to compute the variance in a quantity derived from a unit quaternion. The variance-covariance matrix of the quaternion is known because the quaternion forms part of a Kalman filter's state. Specifically, I am looking for the variance in the inclination and azimuth derived from the quaternion (formulas given below).
Some background
The unit quaternion is defined as $\mathbf{q}=(q_0,q_1,q_2,q_3)^T$, $|\mathbf{q}|=1$ where the scalar component is $q_0$. The global coordinate system has $\hat{x}$ north, $\hat{y}$ east, and $\hat{z}$ down. The quaternion represents the transformation from the global coordinate system to a rotated body coordinate system.
The variance-covariance matrix for $\mathbf{q}$ is the 4x4 matrix $\mathbf{P}$.
Derived values
Two of the values derived from the quaternion are:
- $\theta=\cos^{-1}(q^2_0-q^2_1-q^2_2+q^2_3)$
- $\phi=\tan^{-1}\frac{q_2q_3-q_0q_1}{q_0q_2+q_1q_3}$
Given the trig functions, some of these values will be indeterminate (e.g. for inclination $\theta=0$ or $\pi$, azimuth $\phi=\tan^{-1}\frac{0}{0}$). I imagine that some linearization of the inverse trig functions would be in order, but I am completely in the dark about how to derive Var$[\theta]$ or Var$[\phi]$ given $\mathbf{q}$, $\mathbf{P}$, and the above expressions.
It looks like this question cannot be answered as posed. Without expected-value information about the elements of the quaternion (much less the higher powers of the quaternion elements that result when the trig functions are expanded), there is not enough information to derive a solution.