How to define the degrees of freedom of a flexible manipulator?

Question

Consider a cable-driven flexible manipulator which has 3 cables. Consider the 3 cables to be actuated by 3 motors. Assume that by pulling the three cables in different configurations, we can span the entire 3D space in the workspace of the manipulator.

The manipulator is made up of a single flexible backbone and has disks mounted on it as shown in the figure..

I read on some sources as the manipulator is flexible it has infinite degrees of freedom. Is that correct?

How to define the degrees of freedom of the manipulator in task space and in configuration space?

could you replace the space between any two disks with two servos in XY orientation and maintain functionality? — jsotola, Jan 27 '20 at 18:28
You have to keep in mind your system is actually a 2 stage system with independent control from each other...hence “section 1&2” — DrMrstheMonarch, Jan 27 '20 at 22:19
@jsotola i didn't understand what you meant by replace the space with servos in XY... Could you please elaborate? — Omkar Paranjape, Jan 29 '20 at 03:21
@morbo agreed. So what would be the degrees of freedom of each section? — Omkar Paranjape, Jan 29 '20 at 03:22

score 1 · Answer 1 · answered Jan 31 '20 at 01:03

Your diagram has 6 cables and your question refers to 3 cables, so I'll just consider the simpler case.

The distal end (effector) moves in 3D space (a volume) - task space. Whilst the accessible space is limited, it does require 3 axes to describe its terminal point.
The input is 3 cables, so you only have 3 variables to play with to get a particular (accessible) distal end position. I suspect that is what you have called the configuration space. That looks like 3 dof to me.
The mapping of 3 inputs to 3 outputs is possible, and that transformation has no superfluous components (meaning you have enough, but not too much, information).
A little change to any input makes a little change to the output; there are no discontinuities in the transformation. In this sense, the manipulator can, theoretically, move infinitely small distances.

In a jointed manipulator (like an arm, with a hand), there can be lots of servos. However, at the end of the first finger, we still only need 3 numbers to describe its position in space, even if we chose to move 17 servos to get it there. Our 17-dof manipulator still only achieves the position in 3D, needing only 3 numbers to define its position.

(Note that we have ignored the orientation of the manipulated object. Yes, there's lots more to think about, yet.)

I agree that it has 3 controllable variables (length of control cables) and as long as it is a deterministic or linear system, this would mean it's a 3DOF system. The other 3 cables (tendon cables) are not controllable. — Gürkan Çetin, May 30 '20 at 13:48

How to define the degrees of freedom of a flexible manipulator?

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