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I am trying to find the kinametic parameters using the J. Craig of the DH parameters for the following 2-DOF robotic manipulator.

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where $\ x,y $ are the axes of the reference frame and $\ q_1 , q_2 $ are the parameters of the revolute and prosmatic joint correspondingly. Until now, I have come up with the following solution:

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My problem is that I am not at all sure if the term $\ θ_1 - 90 $ is correct and if I have correctly placed the frames for the joints and the end effector. Axes $\ x_0 , y_0 , z_0 $ are the axes of the reference frame, axes $\ x_1 , z_1 $ are for the revolute joint, axes $\ x_2,z_2 $ are for the prismatic joint and axes $\ x_e,z_e $ are for the end effector. I have intentionally left out the $\ y $ axes of the joints and the end effector since they are derived from the right hand rule.

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