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I have a mobile manipulator that moves on the plane. The base moves in SE(2) and the manipulator is also planar with 4 links. The mobile base will be a differential drive robot and its motion is defined as:

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The manipulator is attached to the center of our mobile base. How do I compute the Jacobian of the mobile manipulator?

I know that for the mobile manipulator I have something of the form:

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The middle matrix is what I need to compute.

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This wikibook on robot kinematics does a pretty good job of explaining a 2 link arm example. https://en.wikibooks.org/wiki/Robotics_Kinematics_and_Dynamics

You should be able to extend it to 4 links like this:

$$ J = \begin{bmatrix} -a_1 s_1 - a_2 s_{12} - a_3 s_{123} - a_4 s_{1234} & -a_2 s_{12} - a_3 s_{123} - a_4 s_{1234} & - a_3 s_{123} - a_4 s_{1234} & - a_4 s_{1234} \\ a_1 c_1 + a_2 c_{12} + a_3 c_{123} + a_4 c_{1234} & a_2 c_{12} + a_3 c_{123} + a_4 c_{1234} & a_3 c_{123} + a_4 c_{1234} & a_4 c_{1234}\\ 0 & 0 & 0 & 0 \end{bmatrix} $$

Where: $a_1$ is the link 1 length, $a_2$ is the link 2 length, etc. $s_1$ is $\sin(\theta_1)$, $c_{12}$ is $\cos(\theta_1 + \theta_2)$ etc.

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