Small angle approximation for quaternions

For small angles $\theta$ (see quaternion definition), we can use the approximations:

$$\sin (\theta /2)\approx \theta /2$$ $$\cos (\theta /2)\approx 1-\frac{{\theta }^2}{8}$$

Substituting these into the quaternion representation:

$$q\approx \left[\begin{array}{c}1-\frac{{\theta }^2}{8}\\ \mathbf{u}\cdot \frac{\theta }{2}\end{array}\right]\approx \left[\begin{array}{c}1\\ \mathbf{u}\cdot \frac{\theta }{2}\end{array}\right]$$