1. Degree of Freedom (DOF) :
Degree of freedom of a point in space :
Degree of freedom (DOF) of a point is the number of indeoendent coordinates required to define the position and orientation of a point.
Thus a point in space has only three degrees of freedom or three coordinates are required to define its position. This ia as shown in Fig The point P can be uniquely defined by specifying its three co-ordinates x, y and z.
(b) Degree of freedom (DOF) of a rigid body :
Consider a rigid body moving in space as shown in Fig This rigid body have possesses following independent motion.
Translation motions along any three perpendicular axis x, y and z.

2. Three rotational motions about the axis x, y and z.
Thus degree of freedom of rigid body have six degree of freedom in free space.
(c) Degree of freedom (DOF) of a mechanism :
A mechanism can also have several degree of freedom (DOF).
The degree of freedom (DOF) of a mechanism is decided by the degree of freedom (DOF) of the links constituting that mechanism. Accordingly, we can classify mechanism under two general categories as follows :
Spatial mechanism.
Planner mechanism.
These two types of mechanisms are described in detail here under:

3. Spatial mechanism :
In this type of mechanisms, the complete motions cannot be represented in a single plane. That is, to describe the motion of such mechanisms, more than one plane would be required. They have three dimensional motion paths.
Examples : Robot arm, Cranes, Hookes Joint.
Planer mechanism :
These are much simpler mechanisms.
The complete motion paths of the mechanism could be represented on a single plane. Or in other words, the entire mechanism could be represented to a scale on a sheet of paper.
During the cource of this study majority of the mechanisms which we will come across would be planer mechanisms.

4. It could be seen that the planner mechanism has lesser DOF
It could be seen that the planner mechanism has lesser DOF. Let us see how many DOF a planer mechanism can have.
Every link of the mechanism can have two translatory motion in the X and Y axis. In addition they can have one rotation about the Z-axis.
Hence the DOF for a planer mechanism will reduce to only three DOF. Due to this less number of DOF, the analysis of planer mechanism will be much simpler as compared to a spatial mechanism.

5. Mobility and Degree of Freedom (DOF) :
Mobility and DOF are assentially the same with very little diffAAterence.
DOF is the number of independent co-ordinates required to define the position of each link, in a mechanism, while mobility is the number of independent input parameters that are to be controlled so that the mechanism can take up a particular position. e.g. Consider a sliding pair consisting of links 1 and 2 as shown in Fig
Since this is a planar mechanism, it can have a maximum of three DOF only. But the slider is connected to the connecting rod, its rotation about an axis perpendicular to plane of the paper is removed.

6. Further, the slider is confined within the cylinder, its motion along the Y-axis is also restrained. Thus it is left with only one DOF that is along the line of stroke and only one parameter needs to be controlled to given any position of the mechanism. In this case, the crank angle θ. Hence its mobility is also one.
Number of inputs :
We can also state that the number of inputs required to produce the constrained motion of a mechanism is called the Degree of Freedom. So if only one inputs is required to produce the constrained motion of a mechanism, them its DOF is 1. As we have seen in the case of a single slider crank mechanism

7. Classification od mechanisms based on number of general restraints: Mechanisms are classified into different orders based on the number of restraints common to the various links as follows :
Zero order mechanism
First order mechanism
Second order mechanism
Third order mechanism
Fourth order mechanism
Fidth order mechanism
No general restraint.
One general restraint.
Two general restraint.
Three general restraint.
Four general restraint.
Fifth general restraint.

8. Kutzbach Criterion :
Degrees of freedom (DOF) of a mechanism in space can be determined as follows :
Let, n = Total number of links in a mechanism
f = Degree of freedom of a mechanism
In a mechanism one link should be fixed. Therefore total number of movable links are in mechanism is (n – 1).
Therefore, total number of degree of freedom of (n – 1) movable links is,
f = 6 (n - 1)
Let, p1 = Number of pairs having 1 DOF ;
p2 = Number of pairs having 2 DOF ;

9. p3 = Number of pairs having 3 DOF ;
We know that,
Any pair having 1 DOF will impose 5 restrains on the mechanism, which reduces its total degree of freedom by 5 p1.
Any pair having 2 DOF will impose 4 restrains on the mechanism, which reduces its total degree of freedom by 4 p2.
Similarly for pair having 3 DOF, 4 DOF and 5 DOF will reduces its total degree of freedom by 3 p3, 2 p4 and 1 p5 respectively and for pair having 6 DOF will impose zero restrains on mechanism, which reduces its total degree of freedom by zero.

10. Therefore, in a mechanism if we consider the links having 1 to 6 DOF, the total number of degree of freedom of the mechanism considering all restrains will becomes,
f = 6 (n – 1) – 5 p1 – 4 p2 – 3 p3 – 2 p4 – 1 p5 – 0 p6
The above equation is the general form of Kutzbach criterion. This is applicable to any type of mechanism including a spatial mechanism.

11. Grubler’s Criterion :
If we apply the kutzbach criterion to planer mechanism, then Equation (1.5) will be modified and that modified equation is known as Grubler’s Criterion for planer mechanism.
As we know that, in a planer mechanism, the maximum DOF possible is only 3 i.e. two translations and one rotational DOF. Therefore, total number of degree of freedom of (n – 1) movable link is,
f = 3 (n – 1)
Let, p1 = Number of pairs having 1 DOF or Number of lower pairs
p2 = Number of pairs having 2 DOF or Number of higher pairs
p3 = Number of links having 3 DOF

12. Any pair in planer mechanism having 1 DOF will impose 2 restrains on the mechanism, which reduces it’s total degree of freedom by 2 p1. any pair in planer mechanism having 2 DOF will impose 1 restrains on the mechanism, which reduces its total degree of freedom by 1 p2.
And any pair in planer mechanism having 3 DOF will impose zero restrains on mechanism, which reduces its total degree of freedom by zero. Therefore in a planer mechanism if we consider the links having 1 to 3 DOF, the total number of degree of freedom of the mechanism considering all restrains will becomes,
f = 3 (n – 1) – 2 p1 – 1 p2 – 0 p3 ;
f = 3 (n – 1) – 2 p1 – 1 p2 ;
The above equation is known as Grubler’s Criterion for planer mechanism.