Given two rigid-body poses given by dual quaternions ${\bf q}_1$ and ${\bf q}_2$, where ${\bf q}_1 = e^{{\bf v} t}$ is some pose along a screw motion. The screw is given by dual vector (Pluecker coordinates) ${\bf v} = {\bf u} (\theta + \epsilon d)$. Here, $|{\bf u}| = 1$, $\theta$ is the half-angle about the screw axis, $d$ is the half-distance along the screw axis, and $\epsilon$ is the dual unit ($\epsilon^2 = 0$).
I'm looking for an analytical solution for finding the real parameter $t$ such that the magnitude of the relative pose ${\bf q}_1^*{\bf q}_2$ is minimized. Since our definition of magnitude is a real number and includes both rotation and translation, we define the magnitude of a dual quaternion ${\bf a} + \epsilon {\bf b}$ as: ${\rm mag}({\bf a} + \epsilon {\bf b}) = c_1| \log {\bf a}|^2 + c_2 | {\bf b} {\bf a}^*|^2$. The constants $c_1$ and $c_2$ are positive values that are chosen arbitrarily to balance angular and linear differences. Let ${\bf r} = {\bf q}_1^*{\bf q}_2 = e^{-{\bf v} t}{\bf q}_2$. We need to minimize $f(t) = {\rm mag}({\bf r})$. This would boil down to finding $t$ for which $\frac{d f(t)}{d t} = 0$. I'm well aware that there may be multiple, if not an infinite number of, solutions. I made an attempt at solving this, but stranded in the trig equations. Not sure if any computer algebra software is capable of solving this.
Secondly, I'm also trying to find the closest screw velocity matching a given velocity (dual vector) ${\bf w}$. Find a real parameter $t$ that minimizes ${\rm mag}({\bf v} t - {\bf w})$, where ${\rm mag}({\bf a} + \epsilon {\bf b}) = c_1 |{\bf a}|^2 + c_2 \frac{({\bf a} \cdot {\bf b})^2}{|{\bf a}|^2}$. This should be easier, since there are no trig functions involved, but still I struggle to find a solution.
Any help is highly appreciated.
PS: Notice that both magnitude definitons are invariant under rigid body transformation of their arguments. This might help in finding a solution.