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

**Definition in mechanics**

Degrees of freedom or mobility are the number of variables which 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 axis is a rotational degree of freedom.

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

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

Is it possible 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?

[WPGP gif_id=”19460″ width=”600″]

**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 actuator to activate a mechanism is equal to degrees of freedom.

**Degrees of freedom lower or equal than zero**

When 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, structure is preloaded and has a force accumulation which generates mechanical tensions.