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physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, the degrees of freedom (DOF) of a mechanical system is the number of independent
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s that define its configuration or state. It is important in the analysis of systems of bodies in
mechanical engineering Mechanical engineering is the study of physical machines that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, and ...
,
structural engineering Structural engineering is a sub-discipline of civil engineering in which structural engineers are trained to design the 'bones and muscles' that create the form and shape of man-made structures. Structural engineers also must understand and cal ...
, aerospace engineering,
robotics Robotics is an interdisciplinary branch of computer science and engineering. Robotics involves design, construction, operation, and use of robots. The goal of robotics is to design machines that can help and assist humans. Robotics integrat ...
, and other fields. The position of a single railcar (engine) moving along a track has one degree of freedom because the position of the car is defined by the distance along the track. A train of rigid cars connected by hinges to an engine still has only one degree of freedom because the positions of the cars behind the engine are constrained by the shape of the track. An automobile with highly stiff suspension can be considered to be a rigid body traveling on a plane (a flat, two-dimensional space). This body has three independent degrees of freedom consisting of two components of translation and one angle of rotation. Skidding or drifting is a good example of an automobile's three independent degrees of freedom. The position and
orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building de ...
of a rigid body in space is defined by three components of
translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
and three components of rotation, which means that it has six degrees of freedom. The
exact constraint Exact may refer to: * Exaction, a concept in real property law * ''Ex'Act'', 2016 studio album by Exo * Schooner Exact, the ship which carried the founders of Seattle Companies * Exact (company), a Dutch software company * Exact Change, an Ameri ...
mechanical design method manages the degrees of freedom to neither underconstrain nor overconstrain a device.


Motions and dimensions

The position of an ''n''-dimensional
rigid body In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
is defined by the rigid transformation, 'T''nbsp;=  'A'', ''d'' where ''d'' is an ''n''-dimensional translation and ''A'' is an ''n'' × ''n'' rotation matrix, which has ''n'' translational degrees of freedom and ''n''(''n'' − 1)/2 rotational degrees of freedom. The number of rotational degrees of freedom comes from the dimension of the rotation group 
SO(n) In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
. 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. The degree of freedom of a system can be viewed as the minimum number of coordinates required to specify a configuration. Applying this definition, we have: #For a single particle in a plane two coordinates define its location so it has two degrees of freedom; #A single particle in space requires three coordinates so it has three degrees of freedom; #Two particles in space have a combined six degrees of freedom; #If two particles in space are constrained to maintain a constant distance from each other, such as in the case of a diatomic molecule, then the six coordinates must satisfy a single constraint equation defined by the distance formula. This reduces the degree of freedom of the system to five, because the distance formula can be used to solve for the remaining coordinate once the other five are specified.


Rigid bodies

A single
rigid body In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
has at most six degrees of freedom (6 DOF) ''3T3R'' consisting of three translations ''3T'' and three rotations ''3R''. See also Euler angles. For example, the motion of a ship at sea has the six degrees of freedom of a rigid body, and is described as:
    Translation and rotation:
# Walking (or surging): Moving forward and backward; # Strafing (or swaying): Moving left and right; # Elevating (or heaving): Moving up and down; # Roll rotation: Pivots side to side; # Pitch rotation: Tilts forward and backward; # Yaw rotation: Swivels left and right; For example, the trajectory of an airplane in flight has three degrees of freedom and its attitude along the trajectory has three degrees of freedom, for a total of six degrees of freedom. * For rolling in flight and ship dynamics, see
roll (aviation) An aircraft in flight is free to rotate in three dimensions: '' yaw'', nose left or right about an axis running up and down; ''pitch'', nose up or down about an axis running from wing to wing; and ''roll'', rotation about an axis running from ...
and
roll (ship motion) Ship motions are defined by the six degrees of freedom that a ship, boat or any other craft can experience. Reference axes The '' vertical/Z axis'', or ''yaw axis'', is an imaginary line running vertically through the ship and through it ...
, respectively. ** An important derivative is the roll rate (or roll velocity), which is the angular speed at which an aircraft can change its roll attitude, and is typically expressed in degrees per second. * For pitching in flight and ship dynamics, see
pitch (aviation) An aircraft in flight is free to rotate in three dimensions: '' yaw'', nose left or right about an axis running up and down; ''pitch'', nose up or down about an axis running from wing to wing; and ''roll'', rotation about an axis running from ...
and
pitch (ship motion) Ship motions are defined by the six degrees of freedom that a ship, boat or any other craft can experience. Reference axes The '' vertical/Z axis'', or ''yaw axis'', is an imaginary line running vertically through the ship and through its ...
, respectively. * For yawing in flight and ship dynamics, see
yaw (aviation) An aircraft in flight is free to rotate in three dimensions: '' yaw'', nose left or right about an axis running up and down; ''pitch'', nose up or down about an axis running from wing to wing; and ''roll'', rotation about an axis running from ...
and
yaw (ship motion) Ship motions are defined by the six degrees of freedom that a ship, boat or any other craft can experience. Reference axes The '' vertical/Z axis'', or ''yaw axis'', is an imaginary line running vertically through the ship and through its ...
, respectively. ** One important derivative is the yaw rate (or yaw velocity), the angular speed of yaw rotation, measured with a
yaw rate sensor A yaw-rate sensor is a gyroscopic device that measures a vehicle's yaw rate, its angular velocity around its vertical axis. The angle between the vehicle's heading and velocity is called its slip angle, which is related to the yaw rate. Types T ...
. ** Another important derivative is the yawing moment, the angular momentum of a yaw rotation, which is important for
adverse yaw Adverse yaw is the natural and undesirable tendency for an aircraft to yaw in the opposite direction of a roll. It is caused by the difference in lift and drag of each wing. The effect can be greatly minimized with ailerons deliberately designed to ...
in aircraft dynamics.


Lower mobility

Physical constraints may limit the number of degrees of freedom of a single rigid body.  For example, a block sliding around on a flat table has 3 DOF ''2T1R'' consisting of two translations ''2T'' and 1 rotation ''1R''.  An XYZ positioning robot like
SCARA The SCARA is a type of industrial robot. The acronym stands for Selective Compliance Assembly Robot Arm or Selective Compliance Articulated Robot Arm. By virtue of the SCARA's parallel-axis joint layout, the arm is slightly compliant in the X ...
has 3 DOF ''3T'' lower mobility.


Mobility formula

The mobility formula counts the number of parameters that define the configuration of a set of rigid bodies that are constrained by joints connecting these bodies.J. J. Uicker, G. R. Pennock, and J. E. Shigley, 2003, Theory of Machines and Mechanisms, Oxford University Press, New York. Consider a system of ''n'' rigid bodies moving in space has 6''n'' degrees of freedom measured relative to a fixed frame. In order to count the degrees of freedom of this system, include the fixed body in the count of bodies, so that mobility is independent of the choice of the body that forms the fixed frame. Then the degree-of-freedom of the unconstrained system of ''N'' = ''n'' + 1 is : M=6n=6(N-1), \! because the fixed body has zero degrees of freedom relative to itself. Joints that connect bodies in this system remove degrees of freedom and reduce mobility. Specifically, hinges and sliders each impose five constraints and therefore remove five degrees of freedom. It is convenient to define the number of constraints ''c'' that a joint imposes in terms of the joint's freedom ''f'', where ''c'' = 6 − ''f''. In the case of a hinge or slider, which are one degree of freedom joints, have ''f'' = 1 and therefore ''c'' = 6 − 1 = 5. The result is that the mobility of a system formed from ''n'' moving links and ''j'' joints each with freedom ''f''''i'', ''i'' = 1, ..., j, is given by : M = 6n - \sum_^j\ (6 - f_i) = 6(N-1 - j) + \sum_^j\ f_i Recall that ''N'' includes the fixed link. There are two important special cases: (i) a simple open chain, and (ii) a simple closed chain. A single open chain consists of ''n'' moving links connected end to end by ''n'' joints, with one end connected to a ground link. Thus, in this case ''N'' = ''j'' + 1 and the mobility of the chain is : M = \sum_^j\ f_i For a simple closed chain, ''n'' moving links are connected end-to-end by ''n'' + 1 joints such that the two ends are connected to the ground link forming a loop. In this case, we have ''N'' = ''j'' and the mobility of the chain is : M = \sum_^j\ f_i - 6 An example of a simple open chain is a serial robot manipulator. These robotic systems are constructed from a series of links connected by six one degree-of-freedom revolute or prismatic joints, so the system has six degrees of freedom. An example of a simple closed chain is the RSSR spatial four-bar linkage. The sum of the freedom of these joints is eight, so the mobility of the linkage is two, where one of the degrees of freedom is the rotation of the coupler around the line joining the two S joints.


Planar and spherical movement

It is common practice to design the linkage system so that the movement of all of the bodies are constrained to lie on parallel planes, to form what is known as a ''planar linkage''. It is also possible to construct the linkage system so that all of the bodies move on concentric spheres, forming a ''spherical linkage''. In both cases, the degrees of freedom of the links in each system is now three rather than six, and the constraints imposed by joints are now ''c'' = 3 − ''f''. In this case, the mobility formula is given by :M = 3(N- 1 - j)+ \sum_^j\ f_i, and the special cases become * planar or spherical simple open chain, M = \sum_^j\ f_i, * planar or spherical simple closed chain, M = \sum_^j\ f_i - 3. An example of a planar simple closed chain is the planar
four-bar linkage In the study of mechanisms, a four-bar linkage, also called a four-bar, is the simplest closed-chain movable linkage. It consists of four bodies, called ''bars'' or ''links'', connected in a loop by four joints. Generally, the joints are config ...
, which is a four-bar loop with four one degree-of-freedom joints and therefore has mobility ''M'' = 1.


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 Mechanism may refer to: * Mechanism (engineering), rigid bodies connected by joints in order to accomplish a desired force and/or motion transmission *Mechanism (biology), explaining how a feature is created *Mechanism (philosophy), a theory that ...
or
linkage Linkage may refer to: * ''Linkage'' (album), by J-pop singer Mami Kawada, released in 2010 *Linkage (graph theory), the maximum min-degree of any of its subgraphs *Linkage (horse), an American Thoroughbred racehorse * Linkage (hierarchical cluster ...
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. It is also defined in context of the configuration space, task space and workspace of a robot. A specific type of linkage is the open kinematic chain, where a set of rigid links are connected at
joint A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
s; a joint may provide one DOF (hinge/sliding), or two (cylindrical). Such chains occur commonly in
robotics Robotics is an interdisciplinary branch of computer science and engineering. Robotics involves design, construction, operation, and use of robots. The goal of robotics is to design machines that can help and assist humans. Robotics integrat ...
,
biomechanics Biomechanics is the study of the structure, function and motion of the mechanical aspects of biological systems, at any level from whole organisms to organs, cells and cell organelles, using the methods of mechanics. Biomechanics is a branch 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 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 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. Although keep in mind that it is not redundant in the human arm because the two DOFs; wrist and shoulder, that represent the same movement; roll, supply each other since they can't do a full 360. The degree of freedom are like different movements that can be made. 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 engineering ''degrees of freedom'' is often used to describe the number of directions in which a phased array
antenna Antenna ( antennas or antennae) may refer to: Science and engineering * Antenna (radio), also known as an aerial, a transducer designed to transmit or receive electromagnetic (e.g., TV or radio) waves * Antennae Galaxies, the name of two collid ...
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.
Radar Radar is a detection system that uses radio waves to determine the distance ('' ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, we ...
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.


See also

* * * *


References

{{reflist Mechanics Robot kinematics Rigid bodies