What is the derivative of the inverse of the unit quaternion?

Question

the unit quaternion is a good way to represent the rotation. but does anyone know the following formula is right or not?I am trying to prove it by myself but I am stuck on it. $\dot{(q^{-1})}=-q^{-1}*\dot{q}*q^{-1}$ where q is a unit quaternion,* is the quaternion multiplication.

Answer 1

I assume you are assuming $q(t)$ is a quaternion-valued function of a real variable $t$.

One thing you can check is that the product rule works for quaternion-valued functions. Indeed, the usual proof works, you just have to be careful not to use the commutative proprety.

Then, defining $p(t):=q(t)^{-1}$, we may differentiate $p(t)q(t)=1$ to obtain

$$ \frac{\mathrm{d}p}{\mathrm{d}t}q+p\frac{\mathrm{d}q}{\mathrm{d}t} = 0. $$

Can you finish?