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. Apr 10, 2017 at 14:26

1 Answer

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.

You are told that the two axes intersect, but you can verify that for yourself by checking that the cross product of the two axes is nonzero.

If $\omega_1 \times \omega_2 \neq 0$, then there is an intersection.

From there, you can find the intersection of the two axes using an intersection test.