DH parameters of a PUMA-type manipulator

Question

I'm struggling to find the DH parameters for this PUMA-type manipulator that yield the same results as the author (1):

The way I'm checking if the parameters I have are correct is by comparing the resulting J11, J21 & J22 matrices with the author. These sub-matrices are the constituents of the wrist Jacobian matrix (Jw).

I tried many different combinations of the DH parameters including:

α =[0,90,0,-90,90,-90]
θ =[0,0,0,0,0,0]
a=[0,0,a2,-a3,0,0]
d=[d1,-d2,0,-d4,0,0]

Which result in the same matrices as the author except for some minor differences. The general wrist Jacobian matrix and the sub-matrices obtained by the author are given by:

Whereas the result I got for J11 was:

$$ \left[ \begin{array}{ccc} -d_2 c_1-s_1 (a_2 c_2-a_3 c_{23}+d_4 s_{23}) & c_1 (d_4 c_{23}-a_2 s_2+a_3 s_{23}) & c_1 (d_4 c_{23}+a_3 s_{23}) \\ c_1 (a_2 c_2-a_3 c_{23}+d_4 s_{23})-d_2 s_1 & s_1 (d_4 c_{23}-a_2 s_2+a_3 s_{23}) & s_1 (d_4 c_{23}+a_3 s_{23}) \\ 0 & a_2 c_2-a_3 c_{23}+d_4 s_{23} & d_4 s_{23}-a_3 c_{23} \\ \end{array}\right] $$

And for the J22 matrix I got:

$$ \left[ \begin{array}{ccc} -c_1 s_{23} & c_4 s_1+c_1 c_{23} s_4 & s_1 s_4 s_5-c_1 (c_3 (c_5 s_2+c_2 c_4 s_5)+s_3 (c_2 c_5-c_4 s_2 s_5)) \\ -s_1 s_{23} & c_{23} s_1 s_4-c_1 c_4 & -c_5 s_1 s_{23}-(c_2 c_3 c_4 s_1-c_4 s_2 s_3 s_1+c_1 s_4) s_5 \\ c_{23} & s_{23} s_4 & c_{23} c_5-c_4 s_{23} s_5 \\ \end{array}\right] $$ And the same J12 matrix as the author.

Perhaps the most pronounced difference here is that every Sin [ θ2 + θ3 ] is replaced with Cos [ θ2 + θ3 ] and vice versa, in addition to some sign differences.

Where am I going wrong here?

(1) Wenfu Xu, Bin Liang, Yangsheng Xu, "Practical approaches to handle the singularities of a wrist-partitioned space manipulator".

Answer 1

In looking at your matrices, I would guess that you (or the author!) have made a mistake somewhere in your alpha $\alpha$ terms. Recall the period shift identity that states:

$$ \sin{\left(\theta + \pi/2 \right)} = +\cos{\theta} \\ \cos{\left(\theta + \pi/2 \right)} = -\sin{\theta} \\ $$ So, if you (or again, the author) were off by 90 degrees ($\pi/2$), then that would explain the fact that you appear to have some sines and cosines swapped and some sign errors, too.

I don't have the time at the moment to go through each step by hand and try to evaluate where you've gone wrong, but I would imagine that you should be able to compare individual joint transforms and find the ones that don't match. Try changing the $\alpha$ values for that joint by $\pm \pi/2$ and see if what you have matches what the author has.

Authors can and do make mistakes, and if you're positive the author made a mistake then the professional thing to do would be to contact that author and/or the journal that published the paper and alert them to the mistake. I would just double- and triple-check that the author was incorrect before taking that step, though (run it by your professors, etc.)