# How to find the Adjoint matrix of multiple twists

So let's say I have a three degrees-of-freedom robot with twists ${\xi}_{1}$, ${\xi}_2$, and ${\xi}_3$. The spatial Jacobian is given by $$J = \begin{bmatrix}\xi_1 & Ad_{g1}{\xi}_2 & Ad_{g12}{\xi}_3\end{bmatrix}$$

I know that $$Ad_{g1} = \begin{bmatrix}R_1 & p \times R_1\\ 0 & R_1\end{bmatrix}$$

However I am not sure how to calculate $Ad_{g12}$. Do I multiply $Ad_{g1} *Ad_{g2}$ or do I get the Transformation matrix of $\xi_1$ and $\xi_2$ and then use the formula for the adjoint?

You want to use the product of exponentials to calculate the transformation of $\zeta_1$ and $\zeta_2$ for $\theta_1$ and $\theta_2$.
To be more clear, using your notation of $g_{12}$: $$g_{12} = e^{\hat{\zeta_1} \theta_1} \cdot e^{\hat{\zeta_2} \theta_2}$$
Where $\hat{\zeta_i}$ is the skew-symmetric matrix representation of the twist. Then you want to use the adjoint transformation of that resulting transformation matrix.
• I found that ${Ad}_{g1} * {Ad}_{g2}$ will give you the same result as the method you described. I was lost between my incomplete understanding of the method and some bugs in my code. Thanks for your answer, I appreciate it. – Dimis Jul 25 '17 at 13:59