# For the second Paden-Kahan Sub-problem, how is 'r' determined?

When using Paden-Kahan Sub-problems to solve the inverse kinematics of manipulators, 'r' is described as the intersection point between the first and second twist axes. But how is this r actually found?

Referencing Murray (here), on page 122.

• Definition is actually on page 119 of the PDF, or page 101 of the document. – Chuck Apr 10 '17 at 14:26

The subproblem starts with the assumption that $\xi_1$ (axis one) and $\xi_2$ (axis two) intersect.
Each axis has a unit normal, $\omega_1$ and $\omega_2$, that points in the direction of their respective axes.
If $\omega_1 \times \omega_2 \neq 0$, then there is an intersection.