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Well, the answer is quite straight forward. With the given equation $ T^n_0 = \ldots $ you calculate the transformation from the $n^\mathrm{th}$ coordinate frame (attached to the end effector) to the inertial coordinate frame. Just imagine you want to know the Jacobian from the joint $n-1$, you do the exact same procedure. Calculate $T^{n-1}_0 = \ldots $ ...


I believe you refer to first order partial derivatives. It is as easy as to take the forward kinematics from T0n , where n is the point to where you want to take the partial jacobian matrix and do the same of the first order partial derivatives.


I'm unable to comment due to reps shortage, so, I'm adding it as an answer. This might answer your question. In your new attempt, the transformation of axes from origin-1 (i.e. $x_1, y_1, z_1$) to origin-2 ($x_2, y_2, z_2$) is incorrect as $x_2 \not\perp z_1$ and $x_2 \not\cap z_1$, violating DH convention. PS: It would be better if you could update your ...


There are 2 main rules in assigning frames following DH convention: source: Robot Modeling and Control, Spong et. al $x_i \perp z_{i-1}$ $x_i$ intersects $z_{i-1}$ In your attempts, Your first attempt is incorrect as your $z_2$ axis is not in the direction of actuation of the prismatic joint. I'm not clear on your second attempt as $x_1$ appears to be ...

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