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Degrees of freedom (engineering)
Encyclopedia
In mechanics
, degrees of freedom (DOF) are the set of independent displacement
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
, aeronautical engineering
, robotics
, structural engineering
, etc.
It can also be defined as minimum number of coordinates required to specify the position of a particle or system of particles.
A rigid body that moves in three dimensional space has three translation
al displacement components as DOFs, while a rigid body would have at most six DOFs including three rotations. Translation
is the ability to move without rotating, while rotation
is angular motion about some axis.
If you were to think of an automobile as a rigid body traveling on a plane (a flat, two-dimensional space), it has three independent degrees of freedom: translation along or across the plane, and rotation to point in any direction or heading. Skidding or drifting is a good example of an automobile's 3 independent DOFs. By contrast, a train moves along a track so that the heading of the train is determined by its position on the track. Thus, the train is restricted
to only one degree of freedom: position along the track.
The Exact constraint mechanical design method manages the degrees of freedom to neither underconstrain nor overconstrain a device.
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 frame is unrestricted, except that it has to have the same scale as the original—so it has (d-1) DOFs. The second axis has to be orthogonal to the first, so it has (d-2) DOFs. Proceeding in this way, we get d(d-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:
See also: Euler angles
.
As defined above one can also get degree of freedom using minimum number of coordinates required to specify a position.Applying it:
1.For a single particle we need 2 coordinates in 2-D plane to specify it's position and 3 coordinates in 3-D plane.Thus it's degree of freedom in 3-D plane is 3.
2.For a body consisting of 2 particles(ex.diatomic molecule) in 3-D plane with constant distance between them(let's say d) we can show it's degree of freedom to be 5.
Let's say a particle in this body has coordinate (x1,y1,z1) along with x-coordinate(x2) and y-coordinate(y2)of second particle.Then using distance formula of distance between two coordinates we have distance d=sqrt(((x1-x2)2+(y1-y2)2+(z1-z2)2))
We have one equation with one unknown where we can solve for z2.
Note:Here any one of x1,x2,y1,y2,z1,z2 can be unknown.
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
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
, where a set of rigid links are connected at joint
s; a joint may provide one DOF (hinge/sliding), or two (cylindrical). Such chains occur commonly in robotics
, biomechanics
, 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 roll, and a wrist allows for pitch and yaw. 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 DOFs than total DOFs is said to be non-holonomic, and an object with more controllable DOFs than total DOFs (such as the human arm) is said to be redundant.
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. A fixed-wing aircraft, with 3–4 control DOFs (forward motion, roll, pitch, and to a limited extent, yaw) in a 3-D space, is also non-holonomic, as it cannot move directly up/down or left/right.
A summary of formulas and methods for computing the degrees-of-freedom in mechanical systems has been given by Pennestri, Cavacece, and Vita.
degrees of freedom is often used to describe the number of directions in which a phased array
antenna
can form either beams or nulls
. It is equal to one less than the number of elements contained in the array, as one 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 and communication link practice, with beam steering being more prevalent for radar applications and null steering being more prevalent for interference suppression in communication links.
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, degrees of freedom (DOF) are the set of independent displacement
Displacement (vector)
A displacement is the shortest distance from the initial to the final position of a point P. Thus, it is the length of an imaginary straight path, typically distinct from the path actually travelled by P...
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 a discipline of engineering that applies the principles of physics and materials science for analysis, design, manufacturing, and maintenance of mechanical systems. It is the branch of engineering that involves the production and usage of heat and mechanical power for the...
, aeronautical engineering
Aerospace engineering
Aerospace engineering is the primary branch of engineering concerned with the design, construction and science of aircraft and spacecraft. It is divided into two major and overlapping branches: aeronautical engineering and astronautical engineering...
, robotics
Robotics
Robotics is the branch of technology that deals with the design, construction, operation, structural disposition, manufacture and application of robots...
, structural engineering
Structural engineering
Structural engineering is a field of engineering dealing with the analysis and design of structures that support or resist loads. Structural engineering is usually considered a specialty within civil engineering, but it can also be studied in its own right....
, etc.
It can also be defined as minimum number of coordinates required to specify the position of a particle or system of particles.
A rigid body that moves in three dimensional space has three translation
Translation (physics)
In physics, translation is movement that changes the position 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. Translation
Translation (physics)
In physics, translation is movement that changes the position of an object, as opposed to rotation. For example, according to Whittaker:...
is the ability to move without rotating, while rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
is angular motion about some axis.
If you were to think of an automobile as a rigid body traveling on a plane (a flat, two-dimensional space), it has three independent degrees of freedom: translation along or across the plane, and rotation to point in any direction or heading. Skidding or drifting is a good example of an automobile's 3 independent DOFs. By contrast, a train moves along a track so that the heading of the train is determined by its position on the track. Thus, the train is restricted
Derailment
A derailment is an accident on a railway or tramway in which a rail vehicle, or part or all of a train, leaves the tracks on which it is travelling, with consequent damage and in many cases injury and/or death....
to only one degree of freedom: position along the track.
The Exact constraint mechanical design method manages the degrees of freedom to neither underconstrain nor overconstrain a device.
Motions and dimensions
Rigid body
In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...
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 frame is unrestricted, except that it has to have the same scale as the original—so it has (d-1) DOFs. The second axis has to be orthogonal to the first, so it has (d-2) DOFs. Proceeding in this way, we get d(d-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);
- Moving forward and backward (surging);
- Tilting forward and backward (pitchFlight dynamicsFlight dynamics is the science of air 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 .Aerospace engineers develop control systems for...
ing); - Turning left and right (yawFlight dynamicsFlight dynamics is the science of air 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 .Aerospace engineers develop control systems for...
ing); - Tilting side to side (rollFlight dynamicsFlight dynamics is the science of air 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 .Aerospace engineers develop control systems for...
ing).
See also: Euler angles
Euler angles
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body. To describe such an orientation in 3-dimensional Euclidean space three parameters are required...
.
As defined above one can also get degree of freedom using minimum number of coordinates required to specify a position.Applying it:
1.For a single particle we need 2 coordinates in 2-D plane to specify it's position and 3 coordinates in 3-D plane.Thus it's degree of freedom in 3-D plane is 3.
2.For a body consisting of 2 particles(ex.diatomic molecule) in 3-D plane with constant distance between them(let's say d) we can show it's degree of freedom to be 5.
Let's say a particle in this body has coordinate (x1,y1,z1) along with x-coordinate(x2) and y-coordinate(y2)of second particle.Then using distance formula of distance between two coordinates we have distance d=sqrt(((x1-x2)2+(y1-y2)2+(z1-z2)2))
We have one equation with one unknown where we can solve for z2.
Note:Here any one of x1,x2,y1,y2,z1,z2 can be unknown.
Systems of bodies
Linkage (mechanical)
A mechanical linkage is an assembly of bodies connected together to manage forces and movement. The movement of a body, or link, is studied using geometry so the link is considered to be rigid. The connections between links are modeled as providing ideal movement, pure rotation or sliding for...
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.-Classification:...
s; a joint may provide one DOF (hinge/sliding), or two (cylindrical). Such chains occur commonly in robotics
Robotics
Robotics is the branch of technology that deals with the design, construction, operation, structural disposition, manufacture and application of robots...
, biomechanics
Biomechanics
Biomechanics is the application of mechanical principles to biological systems, such as humans, animals, plants, organs, and cells. Perhaps one of the best definitions was provided by Herbert Hatze in 1974: "Biomechanics is the study of the structure and function of biological systems by means of...
, 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 roll, and a wrist allows for pitch and yaw. 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 DOFs than total DOFs is said to be non-holonomic, and an object with more controllable DOFs than total DOFs (such as the human arm) is said to be redundant.
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. A fixed-wing aircraft, with 3–4 control DOFs (forward motion, roll, pitch, and to a limited extent, yaw) in a 3-D space, is also non-holonomic, as it cannot move directly up/down or left/right.
A summary of formulas and methods for computing the degrees-of-freedom in mechanical systems has been given by Pennestri, Cavacece, and Vita.
Electrical engineering
In electrical engineeringElectrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...
degrees of freedom is often used to describe the number of directions in which a phased array
Phased array
In wave theory, a phased array is an array of antennas in which the relative phases of the respective signals feeding the antennas are varied in such a way that the effective radiation pattern of the array is reinforced in a desired direction and suppressed in undesired directions.An antenna array...
antenna
Antenna (radio)
An antenna is an electrical device which converts electric currents into radio waves, and vice versa. It is usually used with a radio transmitter or radio receiver...
can form either beams or nulls
Beamforming
Beamforming is a signal processing technique used in sensor arrays for directional signal transmission or reception. This is achieved by combining elements in the array in a way where signals at particular angles experience constructive interference and while others experience destructive...
. It is equal to one less than the number of elements contained in the array, as one 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 an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...
practice and communication link practice, with beam steering being more prevalent for radar applications and null steering being more prevalent for interference suppression in communication links.