# Dynamic Model of a Manipulator

I'm stuck on equation 4.30 of page 176 in
http://www.cds.caltech.edu/~murray/books/MLS/pdf/mls94-complete.pdf

This equation:
$\frac {\partial M_{ij}} {\partial \theta_k} = \sum_{l=\max(i,j)}^n \Bigl( [A_{ki} \xi_i, \xi_k]^T A_{lk}^T {\cal M}_l' A_{lj} \xi_j + \xi_i^T A_{li}^T {\cal M}_l' A_{lk} [A_{kj} \xi_j, \xi_k] \Bigr)$

seems impossible to process because it requires adding a 2x1 to a 1x2 matrix. going by ROWSxCOLUMNS notation. Matrices M and A are 6x6 and $\xi$ is a 6x1, so how does this addition statement fit the rules of matrix addition? This must be my mistake, I just don't see how.

The Lie Bracket is 6x1, not 6x2, so both terms should be 1x1.

• please explain Lie Bracket. Are your saying $[A_{ki} \xi_i, \xi_k]$ is not a concatenation of two 6x1 matrices? How does it work? @SteveO
– Drew
Commented Apr 19, 2016 at 4:17
• I think your text explains it quite well in section 3.3, "Robot dynamics and the product of exponentials formula" just prior to equation 4.30. The Lie Bracket is a sort of gradient measure. The mathematical purists will hate that description! Commented Apr 19, 2016 at 4:23
• And yes, it is not a concatenation. Commented Apr 19, 2016 at 4:24
• Oh shoot, Lie brackets are explained in the preceding page. Ok, thanks SteveO
– Drew
Commented Apr 19, 2016 at 4:24