For other meanings, see Degrees of freedom or Degree

In mechanical engineering, aeronautical engineering, robotics, and structural engineering, degrees of freedom (DOF) are the set of independent displacements that specify completely the displaced or deformed position of the body or system. For example, a particle that moves in space has three translational displacement components as DOFs, while a rigid body would have at most six DOFs including three rotations. The translations represent the ability to move in each of three dimensions, while the rotations the ability to change angle around the three perpendicular axes. When the body is restrained in a certain manner, its number of true DOFs is reduced accordingly; however, the term DOF is also often used without regard to the actual constraints.

A deformable body or system has an infinite number of DOFs unless it is idealized or approximated by a discrete system or a finite DOF system. When motion involving large displacements is the main objective of study, a deformable body may be approximated as a rigid body in order to simplify the analysis. However, for analysis of the detailed stress distribution in a body, that body must generally be considered deformable, and methods of structural analysis can be used.

In mechanical engineering, aeronautical engineering, and robotics, the six DOFs of a rigid body have special names:

File:DOF example.gif
  1. Moving up and down (heaving);
  2. Moving left and right (swaying);
  3. Moving forward and backward (surging);
  4. Tilting up and down (pitching);
  5. Turning left and right (yawing);
  6. Tilting side to side (rolling).

See also: Euler angles.

A mechanism or linkage containing a number of connected rigid bodies may have more than six degrees of freedom.

In robotics, degrees of freedom is often used to describe the number of directions that a robot can pivot or move a joint. A human arm is considered to have 7 DOF. A shoulder gives pitch, yaw and roll, an elbow allows for pitch, and a wrist allows for pitch, yaw and roll. Only 3 of those movements would be necessary to move the hand to any point in space, but people would lack the ability to grasp things from different angles or directions. A robot (or object) that has mechanisms to control all 6 physical DOF is said to be holonomic. An object with fewer controllable DOF than total DOF is said to be non-holonomic, and an object with more controllable DOF than total DOF (such as the human arm) is said to be redundant.

In electrical engineering, degrees of freedom is often used to describe the number of directions in which a phased array antenna can either form beams or nulls. It is equal to one less than the number of elements contained in the array, as one reference element is used as a reference against which either constructive or destructive interference may be applied using each of the remaining antenna elements. Applications exist for the concept in both radar practice as well as for communication link practice, with beam steering being more prevalent for radar applications and null steering being more prevalent for interference suppression in communication links.