# Jacobian determinant of three degree of freedom robot

I want to determine Jacobian determinant of spherical wrist structure, but my Jacobian is 6x3, so it is not square. How can I get it?

If $J(q)$ isn't square, you're looking for some alternate metric to take its place. You can take the determinant of $J(q)J^T(q)$. per this tutorial or any number of other pseudo-inverse methods (from Introduction to Inverse Kinematics with Jacobian Transpose, Pseudoinverse and Damped Least Squares methods, by Samuel Buss).

:EDIT:

As highlighted by @SteveO and @PetchPuttichai in the comments, it would appear that the $J(q)J^T(q)$ only when $J(q)$ has full row rank. In any other case, use $J^T(q)J(q)$.

• @mariposa - If this answered your question, please mark it as accepted by clicking the check mark between the up and down arrows to the left of the answer. This flags the question as answered, which marks it as resolved internally and helps future visitors find the answers they're looking for faster. – Chuck Dec 21 '17 at 21:11
• Shouldn't you use $J(q)^TJ(q)$ instead of $J(q)J(q)^T$ so that the result is 3x3 instead of 6x6? – SteveO Dec 21 '17 at 21:30
• @SteveO - The 6 rows physically represent the six degrees of freedom the (typically) end effector. I believe the point of checking the determinant of the Jacobian (haven't personally done this before) is to see if there is one or more axes that can't be controlled. As you would want to check this for all six degrees of freedom, you would want a result that returns a 6x6 matrix. – Chuck Dec 21 '17 at 21:36
• @Chuck - I understand that, but a spherical wrist only has 3 DOF. I think the 6x6 would always have a determinant of 0. – SteveO Dec 21 '17 at 21:46
• @Chuck - Actually whenever $A$ has more rows than $B$, it is always true that $\det(AB) = 0$ (provided that $AB$ is a square matrix). So in this case, it should be $J^TJ$. In the material you provide, $JJ^T$ is used when $J$ has full row rank. – Petch Puttichai Dec 22 '17 at 3:05

In order to answer your question it would be interesting why you need the determinandt of the Jacobian... maybe there is another way for your problem.

• i want to compute a manipulability and plot it in matlab – mariposa Dec 22 '17 at 11:25