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## Considering Context when Determining Degrees of Freedom

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By Tyler Lucas August 31, 2017 - 2:22pm

Does a pebble really have 6 degrees of freedom of movement? It's a rock. It doesn't move. According to the definition in Mataric (pg. 39), it does: "... a free body in 3D space has a total of six DOF". The following definition is given:

A degree of freedom (DOF) is any of the minimum number of coordinates required to completely specify the motion of a mechanical system.

Without a contextual limit on what constitutes the "mechanical system", any system, regardless of its construction, has at least 6 DOF for simply existing in 3-dimensional space. A coordinate system can be constructed to include or exclude a great number of possible movements. Consider a stake driven into the ground, its movement described relative to an independent celestial coordinate system such as the supergalactic coordinate system. Suddenly we have 6 DOF. A stake doesn't move, it has zero DOF, according to our common reference of the Earth near us because we have constructed a coordinate system in our heads (or formally) around the Earth that factors out its movement with respect to other frames of reference. You've heard the idea that 'an ant lives in 2-dimensional space', referring to the ant's frame of reference excluding any sense of height, which works out well for the ant. In the same way, we exclude those descriptions of position and movement that are not relevant to our goals.

The question becomes 'What is a valid frame of reference when determining degrees of freedom?' Herein lies the debate. It depends on context; on what you're trying to achieve; on why you need to count the DOF. A mobile crane with stabilizers that has 4 DOF (swing, boom up/down, extend/retract, and winch up/down) does not need to include its ability to travel from one location to another when using a count of its degrees of freedom to help develop software that limits its maximum loads, as it does not travel while using the crane*. On the other hand, a carry-deck with the same crane does need to consider this ability, as it is designed to travel with a suspended load (giving it at least 2 more DOF). These two machines have identical joints and types of movement, but it doesn't make sense to count their DOF in the same way.

We can come up with some guidelines as to when to apply this or that frame of reference, keeping in mind that it depends on the end goal for which the degrees of freedom need be determined just as much as it depends on the system configuration.

How many DOF does a pebble have? None, when it's embedded in a quartz deposit in the crust; 6, when I'm skipping it across a pond.

All this came about when defending my count of DOF in the human hand in the Food for Thought section of Unit 2 (Chapter 4, pg. 44): https://landing.athabascau.ca/blog/view/2690961/human-hand-degrees-of-freedom.

*Note: Actually, most mobile cranes and similar machines do have a limited ability to travel with a suspended load, but let's ignore that for the sake of argument.