# Torque required to achieve a desired quaternion

I was hoping someone can either explain or direct me towards a source that can help me with the following problem (not for homework, more of a hobby).

Given an object with a current quaternion $$q_c$$ and initial angular velocity $$\vec{\omega}_i$$, find a torque $$\vec{T}$$ that when applied over an impulse time $$\tau_{imp}$$, will result in the object reaching the desired quaternion $$q_d$$ in a total time $$\tau_{tot}$$. Of course, we can disregard any constraint on the final angular velocity as I feel that would over constrain the problem.

I find the problem much easier when there is no initial angular momentum. When that is the case, one can simply find the delta quaternion ($$q_{delta} = q_d q_c^{-1}$$) and convert it into axis-angle, which can then be used to find the Torque vector (taking into account that $$\tau_{imp}$$ may be on the same order as $$\tau_{tot}$$).

Unfortunately, it gets a little more complicated when dealing with an initial angular velocity. One can find the quaternion change due to the initial angular velocity by following $$\dot{q} = \frac{1}{2}q_cw_i$$ but it's not as simple since we will be dealing with two concurrent rotations: rotation from the initial angular velocity and rotation due to an imparted torque (as opposed to the analogous translational case where the desired delta position can be a linear combination of position change due to the initial velocity and due to an ~impulsive force).

I look forward to hearing your thoughts/advice (as I have only been looking into quaternions for a month or so). I will try to address any issues in clarity if brought up as well.

Thank you!

• What is "an object with a current quaternion"? What physical properties does it have? We need more context. – Somos Feb 26 at 18:56