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Inverse Kinematics Computation -- why are alpha angle values not included?

Robotics Asked by hey on January 6, 2022

Given a desired transform matrix of the end effector relevant to the base frame of the P560:
transform

John J. Craig, in his book, Introduction to Robotics
Mechanics and Control, computes the inverse kinematic solutions of a Puma 560, with (correct me if wrong) Modified DH parameters and gets the following set of equations for theta angles:
transform

and I noticed that there are no alpha angles in these calculations.

So, my question is why aren’t the alpha angle values not used in the calculation for the desired pose with the given end effector transform? Why is it independent of the axes twist angles of the robot?

One Answer

Check out Equation 3.6:

$$ ^{i-1}_iT = left[ begin{array}{c} ctheta_i & -stheta_i & 0 & r_{i-1} \ stheta_icalpha_{i-1} & ctheta_icalpha_{i-1} & -salpha_{i-1} & -salpha_{i-1}d_i \ stheta_isalpha_{i-1} & ctheta_isalpha_{i-1} & calpha_{i-1} & calpha_{i-1}d_i \ 0 & 0 & 0 & 1 end{array} right] $$

and Figure 3.21, "Link parameters of the PUMA 560":

$$ begin{array}{c} i & alpha_{i-1} & r_{i-1} & d_i & theta_i \ \ 1 & 0 & 0 & 0 & theta_1 \ 2 & -90^{circ} & 0 & 0 & theta_2 \ 3 & 0 & a_2 & d_3 & theta_3 \ 4 & -90^{circ} & a_3 & d_4 & theta_4 \ 5 & 90^{circ} & 0 & 0 & theta_5 \ 6 & -90^{circ} & 0 & 0 & theta_6 \ end{array} $$

Here I've used $r$ instead of $a$ to prevent any confusion between $alpha$ and $a$. In Equation 3.6, $s$ means $sin$ and $c$ means $cos$.

To your question, $alpha$ is more related to the orientation of two joints relative to each other as opposed to the motion of the joints relative to each other. That is, $alpha$ is generally fixed by physical construction and thus, as a parameter, isn't time-dependent.

So hopefully you can see that, from Figure 3.21, all of the $alpha$ values are either +/- 90 degrees or zero. Equation 3.6 then uses it only in sine and cosine terms, meaning that they can all be reduced to +/-1 or zero.

I really, really hate when authors jump through simplifications like this without a heads up, so I totally understand the confusion.

Answered by Chuck on January 6, 2022

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