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.
![translation DOF](https://www.electricalelibrary.com/wp-content/uploads/2020/03/translations-diagram-1.jpg)
And each rotational motion perpendicular to one of the axis is a rotational degree of freedom.
![rotational degree of freedom](https://www.electricalelibrary.com/wp-content/uploads/2020/03/rotations-diagram-1.jpg)
Therefore, a body that moves in a three-dimensional space has 6 degrees of freedom.
![6 DOF](https://www.electricalelibrary.com/wp-content/uploads/2020/03/6DOF.svg_-1-1024x972.png)
The figure below shows multiple systems of object pairs and its degrees of freedom (DOF) between parenthesis.
![](https://www.electricalelibrary.com/wp-content/uploads/2020/03/fig-2.7-1.png)
Is it possible for a mechanical system to have more than 6 DOF? The answer is yes, human arm has 7.
![human arm](https://www.electricalelibrary.com/wp-content/uploads/2020/03/The-seven-principal-degrees-of-freedom-of-the-human-armadapted-from-4-1.jpg)
This humanoid robot has 9 DOF.
![humaniod robot](https://www.electricalelibrary.com/wp-content/uploads/2020/03/1383327558-3-750x750-1.jpg)
Determining degrees of freedom
How about more complex mechanical systems? How to calculate the number of necessary actuators for mechanisms like this?
![](https://www.electricalelibrary.com/wp-content/uploads/2020/03/moving-triangle-1.gif)
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.
![](https://www.electricalelibrary.com/wp-content/uploads/2020/03/Four-Link-Mechanism_1-1.png)
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.
![](https://www.electricalelibrary.com/wp-content/uploads/2020/03/fig-2.10-1.png)
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.
![](https://www.electricalelibrary.com/wp-content/uploads/2020/03/Three-Link-Mechanism-with-higher-pair-1.png)
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.