## partial derivatives of TauHat

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Joined: Wed Sep 14, 2016 11:41 pm

### partial derivatives of TauHat

For simEngine3d track...

I understand that for this class, Our applied forces and torques are constants or, at most, a function of time. They are certainly not a function of position nor velocity.

Regarding Slide 33 from lecture 10/17/16:
I can easily see that F_r, F_p, F_rdot, and F_pdot will be zero matrices, because of the way F is defined (slide 26, lecture 10/05/16).

I can see that TauHat_r and TauHat_rdot will be zero matrices.

But from the definition of TauHat (slide 26, lecture 10/05/16), I see there are other terms that are functions of p and pdot (euler parameters).

I am struggling to understand how TauHat_p, TauHat_pdot are defined. Slide 33 doesn't explain it.

— Samuel

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Joined: Wed Sep 03, 2008 12:24 pm

### Re: partial derivatives of TauHat

Samuel - take a look at slide 26 of http://sbel.wisc.edu/Courses/ME751/2016/Documents/lecture1005.pdf.
You can see the expression of \tau_hat there (or page 454, equation 11.3.29 in Haug's book). The only term that will give you partial derivates wrt p and pdot is the therm 8 times G_dot^T J_bar G_dot p.
You will have to compute the partial derivates wrt to p and pdot.
The partial wrt p is simply the term that multiplies the p term, since the quantity 8 times G_dot^T J_bar G_dot p is linear in p.
The partial derivative wrt p_dot: take a look at slide 8 of http://sbel.wisc.edu/Courses/ME751/2016/Documents/lecture1019.pdf. You follow the same steps. We didn't derive this in class, but if you look at the bottom of the slide, that partial derivative is essentially identical to what you need to compute. Do you see why?
I hope this helps.
Dan

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Joined: Wed Sep 14, 2016 11:41 pm

### Re: partial derivatives of TauHat

Thanks. I walked through the slide you mentioned and was able to derive the expression I needed.
- Samuel

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