How to plot quaternion constraint equation vs time

Question

If K = q1^2 + q2^2 + q3^2 + q4^2 and K0 = q1^2 + q2^2 + q3^2 + q4^2 = 0 + 0 + 0 + 1 = 1, how to plot K-K0 vs time graph. K is the quaternion constraint equations where the quaternions are changing with time. K0 is K at initial time. What equations do I need to find quaternions as a function of time? I have angular velocity and the dynamic differential equations (angular velocity dot). Can I find K as a function of time with that information?

Answer 1

What equations do I need to find quaternions as a function of time? I have angular velocity

This reference page 14 and page 16 give the equation describing the derivative of the quaternion in terms of angular velocity $\omega$. This equation can be integrated to get the quaternion at any given time instant.

The equations are $$ \dot{q}_{(t)} = \frac{1}{2}\ \omega_{(t)}\otimes q_{(t)}\quad \text{or}\\ \dot{q}_{(t)} = \frac{1}{2}\ \ q_{(t)} \otimes \omega_{\mathcal{B}(t)} $$ Be careful

Can I find K as a function of time with that information?

Yes. Solve for $q_{(t)}$ first. Then compute K from the result.