Calculate Torques & Forces From Jacobian Matrix

Dear Robotics Stack Exchange,

I am trying to calculate the torques and forces for a 2DOF robot.

I’m used the partial differentiation method to find the Jacobian matrix. I managed to find the Transformation martix to find the Jacobian matrix in as shown in the figure but, I have hit a stopping block when I’ve been calculating the torques and forces. I’ve shown my working out below just as a reference.

I was wondering if anyone can shed light into finding the torques as im very close into gathering the equations.

Your computations of the transformation matrices are incorrect. DH transformation matrices are computed as a multiplication of four matrices: rotate about z by $$\theta$$, translate along z by $$d$$, translate along x by $$a$$, and rotate about x by $$\alpha$$. More simply: you are missing the $$\theta$$ terms in the rightmost columns of your transformation matrices.