# Meaning of “Degrees of Freedom” in control system?

I have been searching google for "degrees of freedom," but it shows results relevant to statistics and physics.

I am interested in answers in the context of electrical engineering, especially control systems.

Degrees of freedom (in an electrical context) is related to a motor which can move and rotate in different directions.

In principle there are the following obvious 6 degrees of freedom:

• movement along x axis
• movement along y axis
• movement along z axis
• rotation around x axis
• rotation around y axis
• rotation around z axis

So in principle there can be 0 to 6 degrees of freedom (0 is a bit useless).

See also Jonk's comments below about non electronic degrees of freedom, and other answers (mention more than the 'obvious' ones above).

• Statistics and chemistry also have meanings different than this. And because of my familiarity with atomic and molecular physics, this idea has a far more important to me meaning in areas where thermalization and the distribution of $\frac12 kT$ energy quantums occur. Relates directly to Johnson noise and kT/C noise in electronics. – jonk Oct 29 '19 at 11:45
• @jonk thanks for this addition, I changed the first sentence according to your comment. – Michel Keijzers Oct 29 '19 at 11:49
• Mostly, I wanted to point out that there can be far more than 6 degrees of freedom. A molecule may have rotational, vibrational, and translational degrees of freedom, together with electronic degrees. Just of the top of my head. Under the right circumstances CO2 may have 9, excluding electronic. 4 of them are vibrational. See en.m.wikipedia.org/wiki/…. H2O, having a bent angle, would have just 3 vibrational modes, though. – jonk Oct 29 '19 at 12:05
• @jonk I am talking about electronic devices/motors, and those can have only 6. In principle you can put a vibration motor but I consider that as a separate motor with its own degrees of freedom. In a checmical structure like CO2 there could be more indeed, but since this is an electronicsi forum and the OP explicitly asked for control systems, I base my answer only that. I fully agree in non electric environments there can be different / other amounts of degrees of freedom, as you state. – Michel Keijzers Oct 29 '19 at 12:32
• Okay. Point taken. The OP question makes that clear, too. Most of my thoughts and work relate to the use of DOF in molecular atmospheric thermalization of line by line wavelengths. So I couldn't get that out of head. There are some different, yet good takes on the idea in answers here. Maybe the OP will say which they are most interested in. – jonk Oct 29 '19 at 12:38

As Michel says, there are 6 degrees of freedom for an object in free space. This results in robotic arms having up to 6 joints, to give all 6 degrees of freedom.

But do you need all 6 all the time? Many machines out there have 3, for X, Y, Z, and then a fourth for a spindle head, such as a milling machine. In that application 4 would be enough (while the spindle may only be only on or off, it is still controlled). Then again there are machine such as lathes which would have the lathe spindle, an X axis for position of the tool along the axis, and a Y for going into the item on the lathe. So that would make a good machine with only 3 degrees of freedom.

However, in a theoretical machine, there may be reason for more DoF than 6. If you look at the robotic arms used on some more sensitive, restrictive (you could say exciting) environments, the can be multiple ways to get to a fixed orientation and location in space.

For instance, some robotic arms used to handle dangerous materials have multiple "elbow" joints, I recall, but can't find it now, some reference to a set up with multiple arms each with 8 degrees of freedom to allow them to reach around obstructions.

You can also talk about getting to a fixed point in space, and then attempting to do something at that point. For instance you orientate a set of jaws to an orientation at a location (using 6 DoF), but the action of closing the jaws would be a 7th DoF.

Degrees of freedom also get complicated when talking about a walking robot. For instance, this video talks about a 10 degrees of freedom machine. As there are 10 joints to worry about.

The way I think about degrees of freedom in a robotic application is how many axis do you control. More axis give more freedom and more cost.

"Degrees of freedom" is also used in many contexts to indicate how many independent measurements are available. For example, in a navigation system, you might be able to measure

• acceleration along three independent axes
• rotation rate on three independent axes
• magnetic field magnitude along three independent axes
• barometric pressure

This would be referred to as a "10-DOF" measurement system.

• How you meant 10?3 ,3 for each of "acceleration along three independent axes" "rotation rate on three independent axes" "magnetic field magnitude along three independent axes" and one for barometric pressure? – abtj Oct 30 '19 at 7:21
• Yes, that's what I said. The first three are 3-D (vector) measurements, while the last one is 1-D (scalar). – Dave Tweed Oct 30 '19 at 10:47