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I have a problem with normalization of the Jacobian matrix.

There seems to be no clear method for doing it: in some literature, it has been normalized by using some characteristic length, which is mathematically correct, but the problem is what this characteristic length should be.

(Mostly, in robotics, the characteristic length is the distance from the base coordinates to the platform (end effector) coordinates.)

It is confusing.

My Jacobian matrix is $6 \times 6$, with the first three columns having units of $\mathrm{rad}/\mathrm{L}$ and the last three columns being dimensionless.

My question is, how can I make the entire Jacobian matrix dimensionless?

Is there a way to normalize this matrix according to some mathematical relations (theory)?

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I edited your post to try to make it grammatical and understandable, but I found some parts of it rather hard to make sense of. Please check that I didn't change your intended meaning by mistake. – Ilmari Karonen Mar 8 '12 at 19:52
What exactly do you mean by Jacobian matrix here? – Mariano Suárez-Alvarez Mar 8 '12 at 19:59
Dimensions (I assume you mean units) are not an inherently mathematical concept. (Okay, I lie slightly, but in general dimensions are not something mathematicians think about.) – Willie Wong Mar 9 '12 at 12:07

You can't.

The Jacobian matrix performs the change of variables for you. If your starting variables have different units from the resulting variables, then your Jacobian matrix must not be dimensionless.

For example, simply consider the difference between polar and cartesian coordinates in the plane. The cartesian coordinates measure two lengths. The polar coordinates measure an angle and a length. Were the Jacobian matrix dimensionless, you would have that angles and lengths are measured by the same unit, which on a euclidean geometry is absurd.

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