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Jacobians in 7 DoF serial manipulators

Anyone here particularly familiar with Jacobians and their use?

I am working on a 7 degree of freedom system. I believe/recall from classes that Jacobians are limited to containing 6 degrees of freedom, that with any [6 x n] system, n<7 so that the Jacobian can never exceed [6 x 6] and still be useful. I recall the reasoning being issues dimension mismatching with any matrix mathematics involving matrices of [6 x n], where n>6. I also recall that something like a [6 x7] matrix would have issues with its rank, and so would run into singularities more frequently - but I am far less sure of this point.

Instead, I recall that when you have a 7-DoF system, one must use some kind of kinematics decoupling and arrange their joints in such a way that the decoupling is possible.

But someone I know, who is quite knowledgeable in robotics, is questioning whether this is true or not. They are not sure one way or another, but they do question whether a standard Jacobian has an upper limit on containing 6 individual joints. Anyone know the answer to this debate, or where I might be able to learn more?

I’ve been re-reading Spong 2006, but have never been a fan on its chapter about Jacobians (does a good job covering how to calculate a Jacobian, less-so when it comes to their exact usage, limitations, and general context of their usage)

The Jacobian is just a matrix of derivatives that determines the gradient of a function. There is no limit to its size. Different considerations come into play if it is square, or not.

No idea what “Spong 2006” is, but you might do some more general reading on the various uses of the Jacobian. One place to start is of course Wikipedia: https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant

For the special case of a robotic arm, the number “6” would come into play if the output of the arm consists of six values, (x,y,z) position and three orientation angles of the tip (in an idealized system).

If you have more than six degrees of freedom in the arm, there will be more than one way to use those degrees of freedom to attain a particular set of six output values. The (idealized) system is then indeterminate.

thankyou my issue has been solved