# What are degrees of freedom?

This post’s subject is degrees of freedom (DOF). The concept in mechanics and how to calculate DOF in a mechanical system.

### Definition in mechanics

Degrees of freedom are the number of variables that indicates a mechanism’s position. Each axis of coordinates x, y, and z is a translation degree of freedom.

And each rotational motion perpendicular to one of the axis is a rotational degree of freedom.

Therefore, a body that moves in a three-dimensional space has 6 degrees of freedom.

The figure below shows multiple systems of object pairs and its degrees of freedom (DOF) between parenthesis.

Is it possible for a mechanical system to have more than 6 DOF? The answer is yes, human arm has 7.

This humanoid robot has 9 DOF.

### Determining degrees of freedom

How about more complex mechanical systems? How to calculate the number of necessary actuators for mechanisms like this?

#### Grübler equation

This is the equation to calculate mobility.

$DOF=m(N-1)-\sum_{i=1}^{J}(m-f_{i})$

• $N$: Number of links, A link is a part which can connect to two or more parts. Also can be the surface or carcass.
• $J$: Number of pair or joints, which connects two links.
• $m$: Mobility of a body. As shown before, 6 in a three-dimensional space and 3 in a plane.
• $f_{i}$: Number of degrees of freedom in pair i.

Using this system as example. Have 4 links $N=4$ and 4 joints $J=4$. $m=3$ because it’s in a plane.

Two bodies separated in plane have DOF=6. When these bodies are linked by a joint, they lose 2 and get DOF=4, therefore $f_{i}=1$. Calculating mobility.

$DOF=3(4-1)-\sum_{i=1}^{4}(3-1)$

$DOF=3(4-1)-4\cdot (2)=9-8=1$

How many DOF have this machine with a double joint and a slider? $N=8$ and $m=3$.

The double joint count as 2, therefore $J=10$ and all pairs have 1 mobility degree $f_{i}=1$.

$DOF=3(8-1)-\sum_{i=1}^{10}(3-1)$

$DOF=24-3-2\cdot 10=1$

And this one with a wheel? $m=3$ and $N=4$.

The pair wheel-surface has 2 degrees of freedom, by that, $f_{4}=2$. Therefore, equation is:

$DOF=3(4-1)-[(3-1)+(3-1)+(3-1)+(3-2)]$

$DOF=12-3-[3(3-1)+(3-2)]=9-3\cdot 2-1=2$

How many degrees of freedom there are in the mechanism showed before? $m=3$, $N=8$ (including surface), $J=9$ e $f_{i}=1$.

[WPGP gif_id=”19460″ width=”600″]

$DOF=3(8-1)-\sum_{i=1}^{9}(3-1)$

$DOF=24-3-9\cdot 2=24-3-18=3$

To apply for three-dimensional systems, considerate $m=6$ and count DOF of all pairs. This Grübler equation can be simplified for two dimensions.

$DOF=3(N-1)-2J_{1}-J_{2}$

• $J_{1}$: joints with 1 DOF.
• $J_{2}$: joints with 2 DOF.

And for three dimensions.

$DOF=6(N-1)-5J_{1}-4J_{2}-3J_{3}-2J_{4}-J_{5}$

These simplifications are called Kutzbach equation. The number of necessary actuators to activate a mechanism is equal to degrees of freedom.

### Degrees of freedom lower or equal than zero

When the degree of freedom or mobility is equal to zero, it means that is a structure and not a mechanism. If it is lower than 0, the structure is preloaded and has a force accumulation which generates mechanical tensions. 