Degrees of freedom (engineering)

Degrees of Freedom (engineering)

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Degrees of freedom (engineering)

In mechanics

Classical mechanics

Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies....
, degrees of freedom (DOF) are the set of independent displacement

Displacement (vector)

In physics, displacement is the vector that specifies the change in position of a point or a particle in reference to a previous position. When the previous point is the origin, this is better referred to as a position vector....
s and/or rotations that specify completely the displaced or deformed position and orientation of the body or system. This is a fundamental concept relating to systems of moving bodies in mechanical engineering

Mechanical engineering

Mechanical Engineering is an engineering discipline that involves the application of physics#branches of physics for analysis, design, manufacturing, and maintenance of machine....
, aeronautical engineering

Aerospace engineering

Aerospace engineering is the branch of engineering behind the design, construction and science of aircraft and spacecraft. Aerospace engineering has broken into two major and overlapping branches: Aeronautics engineering and Astronautics engineering....
, robotics

Robotics

Robotics is the science and technology of robots, and their design, manufacture, and application. Robotics has connections to electronics, mechanics, and software....
, structural engineering

Structural engineering

Structural engineering is a field of engineering dealing with the analysis and design of structures that support or resist structural loads. Structural engineering is usually considered a specialty within civil engineering, but it can also be studied in its own right....
, etc.

A particle that moves in three dimensional space has three translation

Translation (physics)

In physics, translation is movement that changes the displacement of an object, as opposed to rotation. For example, according to Whittaker:...
al displacement components as DOFs, while a rigid body would have at most six DOFs including three rotations.

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Encyclopedia

In mechanics

Classical mechanics

Displacement (vector)

Mechanical engineering

Aerospace engineering

Robotics

Robotics is the science and technology of robots, and their design, manufacture, and application. Robotics has connections to electronics, mechanics, and software....
, structural engineering

Structural engineering

Translation (physics)

Motions and Dimensions

In general, a rigid body in d dimensions has d(d + 1)/2 degrees of freedom (d translations and d(d -1)/2 rotations). One line of reasoning for the number of rotations goes that rotational freedom is the same as fixing a coordinate frame. Now, the first axis of the new 1)/2 rotational DOFs in d dimensions. In 1-, 2- and 3- dimensions then, we have one, three, and six degrees of freedom.

A non-rigid or deformable body may be thought of as a collection of many minute particles (infinite number of DOFs); this is often approximated by a finite DOF system. When motion involving large displacements is the main objective of study (e.g. for analyzing the motion of satellites), a deformable body may be approximated as a rigid body (or even a particle) in order to simplify the analysis.

In three dimensions, the six DOFs of a rigid body are sometimes described using these nautical names:
Moving up and down (heaving);
Moving left and right (swaying
Sway

Things and places commonly known as Sway or sway include:...
);
Moving forward and backward (surging
Surge

Surge was a citrus soft drink first introduced in Norway, under the name Urge , by the Coca-Cola Company to compete with Pepsi's Mountain Dew....
);
Tilting forward and backward (pitch
Flight dynamics

Flight dynamics is the science of aircraft and spacecraft vehicle orientation and control in three dimensions. The three critical flight dynamics parameters are the angles of rotation in three dimensions about the vehicle's center of mass, known as pitch, roll and yaw ....
ing);
Turning left and right (yaw
Flight dynamics

Flight dynamics is the science of aircraft and spacecraft vehicle orientation and control in three dimensions. The three critical flight dynamics parameters are the angles of rotation in three dimensions about the vehicle's center of mass, known as pitch, roll and yaw ....
ing);
Tilting side to side (roll
Flight dynamics

Flight dynamics is the science of aircraft and spacecraft vehicle orientation and control in three dimensions. The three critical flight dynamics parameters are the angles of rotation in three dimensions about the vehicle's center of mass, known as pitch, roll and yaw ....
ing).
See also: Euler angles
Euler angles

The Euler angles were developed by Leonhard Euler to describe the orientation of a rigid body in dimension Euclidean space. To give an object a specific orientation it may be subjected to a sequence of three rotations described by the Euler angles....
.

Systems of Bodies

A system with several bodies would have a combined DOF that is the sum of the DOFs of the bodies, less the internal constraints they may have on relative motion. A mechanism or linkage
Linkage (mechanical)

A mechanical linkage is a series of rigid links connected with joints to form a closed chain, or a series of closed chains. This is created by two or more levers that are put together....
containing a number of connected rigid bodies may have more than the degrees of freedom for a single rigid body. Here the term degrees of freedom is used to describe the number of parameters needed to specify the spatial pose of a linkage.

A specific type of linkage is the open kinematic chain
Kinematic chain

A kinematic chain is the assembly of several kinematic pairs connecting rigid body segments. The complexity of the chain is determined by the following factors:...
, where a set of rigid links are connected at joint
Joint

A joint is the location at which two or more bones make contact. They are constructed to allow movement and provide mechanical support, and are classified structurally and functionally....
s; a joint may provide one DOF (hinge/sliding), or two (cylindrical). Such chains occur commonly in robotics
Robotics

Robotics is the science and technology of robots, and their design, manufacture, and application. Robotics has connections to electronics, mechanics, and software....
, biomechanics
Biomechanics

Biomechanics is the application of mechanical principles to living organisms. This includes bioengineering, the research and analysis of the mechanics of living organisms and the application of engineering principles to and from biological systems....
and for satellites and other space structures. A human arm is considered to have seven DOFs. 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
Holonomic

In mathematics, the term holonomic may occur with several different meanings....
. 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
Redundancy (total quality management)

In total quality management, TQM, redundancy in quality or redundant quality means quality which exceeds the required quality level. Engineering_tolerance may be too accurate, for example, creating unnecessarily high costs of production....
.

In mobile robotics, a car-like robot can reach any position and orientation in 2-D space, so it needs 3 DOFs to describe its pose, but at any point, you can move it only by a forward motion and a steering angle. So it has two control DOFs and three representational DOFs - i.e. it is non-holonomic. An airplane, with 3-4 control DOFs (forward motion, roll, pitch - and to a limited extent, yaw) in a 3-D space, is also non-holonomic.

In electrical engineering
Electrical engineering

Electrical engineering, sometimes referred to as electrical and electronic engineering, is a field of engineering that deals with the study and application of electricity, electronics and electromagnetism....
, degrees of freedom is often used to describe the number of directions in which a phased array

Phased array

This article is about general theory and electromagnetic phased array.'For the ultrasonic and medical imaging application, see phased array ultrasonics....
antenna

Antenna (radio)

An 'antenna' is a transducer designed to transmitter or receive Electromagnetic radiations. In other words, antennas convert electromagnetic waves into electrical currents and vice versa....
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

Radar

Radar is a system that uses electromagnetic radiation waves to identify the range, altitude, direction, or speed of both moving and fixed objects such as aircraft, ships, motor vehicles, weather formations, and terrain....
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.