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Premise: I am by no means an expert on Robotics, I have to deal with this project where we use a robotic arm (franka emika), so feel free to assume that I got the basics wrong!

I have this arm ( https://frankaemika.github.io/docs/control_parameters.html ) and I have its DH parameters for each joint. I need to compute each x-y-z joint positions.

I understand from here that you can compute, for each joint, a $ [^{n-1}T_n]$ matrix which represents a "change of frame" from the previous frame; I also understand that the product of the first n matrixes gives the change-of-frame from the initial frame to the n-th joint frame. But, how do you use this T matrix to compute the x-y-z positions?

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By combining all the matrices, you'll end up having an homogeneous transformation $T \in SE\left(3\right)$ that can be expressed as: $$ T= \left( \begin{matrix} \mathbf{R} & \mathbf{p} \\ 0 & 1\end{matrix} \right), $$ where the matrix $\mathbf{R} \in \mathbb{R}^{3 \times 3}$ is symmetric and defines how the final frame is rotated with respect to the root, whereas the vector $\mathbf{p} \in \mathbb{R}^{3 \times 1}$ accounts for the translational part.

Indeed, it comes out that: $$ \mathbf{p}=\left( \begin{matrix} x \\ y \\ z \end{matrix} \right). $$

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