# Jacobian Matrix and joint dependecies

I'm working with a robotic arm and needed to compute the jacobian matrix of it in order to send torque commands. THe arm has 6 joints all revolute. After calculating the jacobian matrix from the DH parameters provided in the datasheet, I noticed that the jacobian has dependency only from the first five joints. The sixth joint corresponds to the hand which only rotates over itself. My question here is: Can the jacobian have no dependency from a joint? On which cases can this happen?

Thanks

## 1 Answer

The Jacobian matrix itself can sometimes not contain some joint variables. See, for example, this textbook, p.115, where they derive the Jacobian for a SCARA manipulator.

The thing is that even though the Jacobian itself does not contain, for example, the $6^\text{th}$ joint variable, that joint still can contribute to changes in position and orientation of the end-effector. Consider the equation

$$\begin{bmatrix}v \\ \omega\end{bmatrix} = J(q)\dot{q},$$ where $v$ and $\omega$ are the end-effector linear and angular velocities, $\dot{q}$ is the joint velocities, and $J$ is the Jacobian (in your case it should be a $6 \times 6$ matrix). If you examine closely, you can see that your $6^\text{th}$ joint can still contribute to $v$ and $\omega$ via the last (right-most) column of $J$.

A Jacobian can be seen as how each joint (velocity) contributes to the end-effector velocities. When a Jacobian does not contain some particular joint variable, it means that the joint configuration does not really affect how other joints can contribute to end-effector velocities.

• Yes, that was what I was thinking.. Thanks for the explanation. Apr 3 '17 at 8:29