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From the question, it seems the loading is located at the end effector but oriented according to the {s} frame. I do prefer to resolve everything on the world inertial frame which makes all the coordinate transformations unnecessary. For example, if the end effector is located at $$\boldsymbol{r}_b = \pmatrix{ 2 + \tfrac{1}{\sqrt{2}} \\ \tfrac{1}{\sqrt{2}} \... 0 Here is a top view of the robot, and the end effector velocity if \dot{\theta}_1 > 0: 2 It seems there are two types of Jacobian matrix which are the geometric and analytic ones. They are not the same but related. The one I've provided in my question is the geometric Jacobian expressed in the spatial frame. The same one expressed in the body frame is$$ \begin{align} \mathcal{B}_1 &= [0,0,1,0,L_1+L_2,0]^T \\ \mathcal{B}_2 &= [0,0,1,0,...
For this case shouldn't the "space Jacobian" $J_s$ be $6 \times 2$ while the "Jacobian" $J$ is $2 \times 2$? It seems like the "space Jacobian" operates on the joint velocities to deliver the end-effector space twist, while the "Jacobian" operates on the joint velocities to deliver the end-effector velocity in the ...