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Here is a top view of the robot, and the end effector velocity if $\dot{\theta}_1 > 0$:

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In Lagrangian mechanics, you're required to deal with the generalized forces and the concept of virtual work $\delta W$. In our case we have: $$\delta W_{F_x} = F_x \cdot \delta q_1 + F_x \left( -l_2s_2\right) \cdot \delta q_2,$$ where the coefficients of the terms $\delta q_i$ are given by $F_x \frac{\partial x_m}{\partial q_i}$. By contrast, $y_m$ doesn'...

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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,...

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In robotics, the Degrees of Freedom (DoF) typically refers to the dimension of the Joint Space. In this case, DoF = k. In the planar case, 3 is the maximum dimension of the Task Space, instead. When k > 3, we basically deal with a redundant planar manipulator. Here's a quite illustrative series of slides on the topic of Kinematic Redundancy.

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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 ...

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