# Definition for the number of degrees of freedom

From Wikipedia:

The degree of freedom of a system can be viewed as the minimum number of coordinates required to specify a configuration

From Modern Robotics, definition 2.1:

The configuration of a robot is a complete specification of the position of every point of the robot. The minimum number n of real-valued coordinates needed to represent the configuration is the number of degrees of freedom (dof) of the robot

Those definitions feel intuitive, but however bijections can be made bewteen $$\mathbb{R}$$ and $$\mathbb{R}^n$$. Intuitively, we can interleave the decimals (plus some tricks). Thus, the "minimum number $$n$$ of real-valued coordinates" doesn't seem sufficient, since we could always reduce this number to 1, except if we add some extra condition about the topology of our coordinate system.

Am I missing something, or do we need to "fix" those definitions ?

The point is: you need n independent values to describe the robot configuration, and represent those n values in one way or another.

Whether you define these n independent values as "a set of n unrelated real values", or as "a combination of one real value and a specific choice for a bijection that unambiguously relates this one value to n separate values" is just a choice of how to represent those n independent values.

But it is incorrect to state that the configuration would be fully described by just one real value. This is only correct if you also specify your specific choice for the bijection.

• Obviously, it does not necessarily have to be real values either. E.g. in case of stepper motors with a limited set of stable positions, or encoders with a limited resolution, a discrete representation could be more appropriate. But, imo., this is outside of the scope of the definition of Degrees of Freedom, which is rather about the minimum number of values one needs to describe the system.
– JRTG
Commented Sep 4 at 9:23
• I believe there are other arguments, for example if you interleave decimals, the physical displacement direction of the robot for a small configuration variation is ill-defined. Having the configuration being a proper manifold would be more satisfying, but is it always the case ? Commented Sep 5 at 10:29
• To asses 'variation' between values implies choosing a metric for 'distance' between two values, i.e. some norm. My backround is mechanical engineering rather than theoretical mathematics, but it seems obvious that the typical norms for real numbers are irrelevant, if the real number does not represent one value but is rather some sort of concatenation of n separate, unrelated values.
– JRTG
Commented Sep 5 at 14:35
• Also note the following: a robot manipulator configuration (i.e. end effector pose) is in general described by a position and an orientation, the former typically described by three values (e.g. X,Y,Z), the latter by 3 or more values (e.g. Euler angles, a quaternion or a rotation matrix). These values do not form a vector space. So there is no trivial notion of 'distance' in task space, it also need some choice for a norm that in one way or another defines and combines position and orientation differences.
– JRTG
Commented Sep 5 at 14:52
• @Gregwar I do believe that the configuration space of an articulated rigid-body mechanical system is always a manifold. There's a lot of work on geometric mechanics that addresses this. Commented Sep 7 at 15:32

I think the disconnect arises in defining "real-valued coordinates".

In this robotics Q&A site, we deal with physical things, whose irascible dimensions, behavior, and coordinates are ill approximated by infinitely interleaved decimals or anything else fancy enough to interest Georg Cantor or math.stackexchange.com. Our "real values" are more like distributions

But we can still learn from the mathematician-but-physicist-wannabe Lagrange, whose "generalized coordinates" map pretty precisely to degrees of freedom. And it turns out that he would agree with the Modern Robotics reference.

p.s. I can never keep straight bi- in- and surjections. Is there a useful mnemonic?