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I find that in some robotic toolboxs(e.g. FROST https://ayonga.github.io/frost-dev/index.html) they use

enter image description here

to calculate time derivative of jacobian matrix.

Here is my guess about the reason of using this formula:

enter image description here

if this is right, could anyone please tell me how to prove this part which shows that derivative and partial derivative can exchange? enter image description here

Thanks in advance!!

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If you use the partial differentiation you can get to that expression as

\begin{align} \frac{dJ}{dt} &= \frac{\partial J}{\partial q}\frac{\partial q}{\partial t} \\ \frac{dJ}{dt} &= \frac{\partial J}{\partial q}\frac{d q}{d t}\\ \frac{dJ}{dt} &= \frac{\partial (J \dot q)}{\partial q} \end{align}

In the last line $\dot q$ is a constant that can enter the partial differentiation part.

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  • $\begingroup$ Hi jdios, thanks a lot for your quick reply,now I understand the formula of calculating dJ/dt! I still have a question which is: Can time derivative and partial derivative exchange? (the formula in the third image) Thanks in advance again. $\endgroup$ – BrP Al Dec 10 '20 at 4:33
  • $\begingroup$ @BrPAl If the Jacobian is differentiable w.r.t. time then Schwarz' theorem should be sufficient. $\endgroup$ – Tobias Dec 10 '20 at 5:50
  • $\begingroup$ Thanks a lot Tobias, I will check it later. $\endgroup$ – BrP Al Dec 12 '20 at 12:54

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