Degree of Freedom | Definition, Formula, Questions

By Mohit Uniyal|Updated : May 19th, 2022

A mechanical system's degree of freedom (DOF) is the number of independent characteristics that describe its configuration or state. It is useful in mechanical engineering, structural engineering, aerospace engineering, robotics, and other domains when analyzing body systems. The number of independent movements done by the robot wrist in three-dimensional space, relative to the robot's base, is defined as the degree of freedom for a robot.

In mechanics, degrees of freedom are particular, defined modes in which a mechanical device or system can move. The total number of independent displacements or features of motion equals the number of degree of freedom. A machine with more than three degrees of freedom can operate in two or three dimensions. The phrase is commonly used to describe robot motion capabilities.

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What is the Degree of Freedom?

The set of independent displacements and/or rotations that specify completely the displaced or deformed position and orientation of the body or system are known as degrees of freedom (DOF) in mechanics.

Degree of Freedom Definition

“A system's degree of freedom can be conceived of as the minimum number of coordinates needed to specify a configuration.”

Using this definition, we get the following:

  • An unconstrained object in space has six degrees of freedom: three translational and three rotational.
  • An unconstrained object in a plane has three degrees of freedom: two translational and one rotational.

Degree of Freedom of a Mechanism

“It is the minimum number of independent inputs required to get a constrained output from a mechanism.” 

If any mechanism is having two degrees of freedom then two independent inputs are required to get a constrained output from the mechanism. Example: Epicyclic gear train.

The Formula of Degree of Freedom

Kutzback was a scientist who gave the relation to determine the degree of freedom of a 2-D planar mechanism.

DOF = 3(L-1) – 2j – h


L = number of links

j = number of joints

h = number of higher pairs

When one or more links of a mechanism may be moved without causing the rest of the mechanism's links to move, this is referred to as a redundant degree of freedom. The planar mechanism's degree of freedom formula has been updated as a result of this.

DOF = 3(L-1) – 2j – h – Fr 


Fr = redundant motion

Now, Grubler was another scientist who used the Kutzback equation and came up with his own formulation, in which he employed the equation and set the degree of freedom to 1 and the higher pair to zero for a kinematic chain. The Grubler criterion was an extension of the Kutzback equation, and it was written as

DOF = 3(L-1) – 2j – h

1 = 3(L-1) – 2j – 0

3L – 2j – 4 = 0  (derived equation of Grubler’s criterion.)

According to Grubler's criterion, the number of links required to hold the equation should be even, i.e. the minimal number of links required is four.

Degrees of Freedom Questions

Question 1: What is the number of degrees of freedom for a rigid body?

Solution: A rigid body's position and orientation in space are determined by three components of translation and three components of rotation, giving it six degrees of freedom.

Question 2: What is the number of degrees of freedom in a four-bar linkage?

Solution: Grubler's criterion determines the degree of freedom (F) for a simple mechanism:

F = 3 (n - 1) - 2j - h

Here,  j = number of revolute joints, n = number of links, h = number of higher pairs

Given: n = 4, j = 4, h = 0

F = 3 (4 - 1) - 2 × 4 - 0

F = 9 - 8 = 1

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FAQs on Degree of Freedom

  • If the motion of all the links of a mechanism is in the same plane, the mechanism is called a planar mechanism. In a planar mechanism, only two sorts of lower kinematic pairs are possible: a turning pair and a sliding pair. As a result, in a planner mechanism, any lower pair can only have one degree of freedom.

  • The Degree of freedom for a mechanism is defined as the number of coordinates or variables required to be specified such that the position and orientation of all the members of the mechanism can be stated as a function of time. In order to calculate the degree of freedom of a mechanism, we use the kutzback criterion.

  • A screw pair is a set of two pieces that work together to create simultaneous rotation and axial translation. Both the motions of the screws in a screw pair are interdependent. Because of this, it only has one degree of freedom.

  • A structure is an arrangement of resistant bodies (known as members) with no relative motion between them. A structure is designed to carry weight. Because the members do not move relative to one another, as a result, the structure has zero degree of freedom.

  • There may be one or more links in a mechanism that do not put any additional limitations on the mechanism. This type of link is known as a redundant link. If there are any redundant linkages in the mechanism, they should not be counted when calculating the degree of freedom.


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