Kinematics is a subfield of
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 ...
, developed in
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, that describes the
motion
In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and m ...
of points,
bodies
Bodies may refer to:
* The plural of body
* ''Bodies'' (2004 TV series), BBC television programme
* Bodies (upcoming TV series), an upcoming British crime thriller limited series
* "Bodies" (''Law & Order''), 2003 episode of ''Law & Order''
* ...
(objects), and systems of bodies (groups of objects) without considering the
forces that cause them to move.
Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of
mathematics. A kinematics problem begins by describing the geometry of the system and declaring the
initial conditions
In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). For ...
of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within
kinetics
Kinetics ( grc, κίνησις, , kinesis, ''movement'' or ''to move'') may refer to:
Science and medicine
* Kinetics (physics), the study of motion and its causes
** Rigid body kinetics, the study of the motion of rigid bodies
* Chemical ki ...
, not kinematics. For further details, see
analytical dynamics
In classical mechanics, analytical dynamics, also known as classical dynamics or simply dynamics, is concerned with the relationship between Motion (physics), motion of bodies and its causes, namely the force (physics), forces acting on the bodies ...
.
Kinematics is used in
astrophysics to describe the motion of
celestial bodies and collections of such 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, an ...
,
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 integrate ...
, and
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 ...
kinematics is used to describe the motion of systems composed of joined parts (multi-link systems) such as an
engine
An engine or motor is a machine designed to convert one or more forms of energy into mechanical energy.
Available energy sources include potential energy (e.g. energy of the Earth's gravitational field as exploited in hydroelectric power ...
, a
robotic arm
A robotic arm is a type of mechanical arm, usually programmable, with similar functions to a human arm; the arm may be the sum total of the mechanism or may be part of a more complex robot. The links of such a manipulator are connected by join ...
or the
human skeleton.
Geometric transformations, also called
rigid transformations, are used to describe the movement of components in a
mechanical system, simplifying the derivation of the equations of motion. They are also central to
dynamic analysis.
Kinematic analysis is the process of measuring the
kinematic quantities used to describe motion. In engineering, for instance, kinematic analysis may be used to find the range of movement for a given
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 ...
and, working in reverse, using
kinematic synthesis
In mechanical engineering, kinematic synthesis (also known as mechanism synthesis) determines the size and configuration of mechanisms that shape the flow of power through a mechanical system, or machine, to achieve a desired performance. The wo ...
to design a mechanism for a desired range of motion.
[J. M. McCarthy and G. S. Soh, 2010]
''Geometric Design of Linkages,''
Springer, New York. In addition, kinematics applies
algebraic geometry to the study of the
mechanical advantage
Mechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system. The device trades off input forces against movement to obtain a desired amplification in the output force. The model for ...
of a
mechanical system or mechanism.
Etymology of the term
The term kinematic is the English version of
A.M. Ampère's ''cinématique'', which he constructed from the
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
''kinema'' ("movement, motion"), itself derived from ''kinein'' ("to move").
Kinematic and cinématique are related to the French word cinéma, but neither are directly derived from it. However, they do share a root word in common, as cinéma came from the shortened form of cinématographe, "motion picture projector and camera", once again from the Greek word for movement and from the Greek ''grapho'' ("to write").
Kinematics of a particle trajectory in a non-rotating frame of reference
Particle kinematics is the study of the trajectory of particles. The position of a particle is defined as the coordinate vector from the origin of a coordinate frame to the particle. For example, consider a tower 50 m south from your home, where the coordinate frame is centered at your home, such that east is in the direction of the ''x''-axis and north is in the direction of the ''y''-axis, then the coordinate vector to the base of the tower is r = (0 m, −50 m, 0 m). If the tower is 50 m high, and this height is measured along the ''z''-axis, then the coordinate vector to the top of the tower is r = (0 m, −50 m, 50 m).
In the most general case, a three-dimensional coordinate system is used to define the position of a particle. However, if the particle is constrained to move within a plane, a two-dimensional coordinate system is sufficient. All observations in physics are incomplete without being described with respect to a reference frame.
The position vector of a particle is a
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
drawn from the origin of the reference frame to the particle. It expresses both the distance of the point from the origin and its direction from the origin. In three dimensions, the position vector
can be expressed as
where
,
, and
are the
Cartesian coordinates and
,
and
are the unit vectors along the
,
, and
coordinate axes, respectively. The magnitude of the position vector
gives the distance between the point
and the origin.
The
direction cosines of the position vector provide a quantitative measure of direction. In general, an object's position vector will depend on the frame of reference; different frames will lead to different values for the position vector.
The ''trajectory'' of a particle is a vector function of time,
, which defines the curve traced by the moving particle, given by
where
,
, and
describe each coordinate of the particle's position as a function of time.
Velocity and speed
The
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
of a particle is a vector quantity that describes the magnitude as well as direction of motion of the particle. More mathematically, the rate of change of the position vector of a point with respect to time is the velocity of the point. Consider the ratio formed by dividing the difference of two positions of a particle by the time interval. This ratio is called the average velocity over that time interval and is defined as
where
is the change in the position vector during the time interval
. In the limit that the time interval
approaches zero, the average velocity approaches the instantaneous velocity, defined as the time derivative of the position vector,
where the dot denotes a derivative with respect to time (e.g.
). Thus, a particle's velocity is the time rate of change of its position. Furthermore, this velocity is
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
to the particle's trajectory at every position along its path. Note that in a non-rotating frame of reference, the derivatives of the coordinate directions are not considered as their directions and magnitudes are constants.
The
speed
In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (ma ...
of an object is the magnitude of its velocity. It is a scalar quantity:
where
is the arc-length measured along the trajectory of the particle. This arc-length must always increase as the particle moves. Hence,
is non-negative, which implies that speed is also non-negative.
Acceleration
The velocity vector can change in magnitude and in direction or both at once. Hence, the acceleration accounts for both the rate of change of the magnitude of the velocity vector and the rate of change of direction of that vector. The same reasoning used with respect to the position of a particle to define velocity, can be applied to the velocity to define acceleration. The
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
of a particle is the vector defined by the rate of change of the velocity vector. The average acceleration of a particle over a time interval is defined as the ratio.
where Δv is the difference in the velocity vector and Δ''t'' is the time interval.
The acceleration of the particle is the limit of the average acceleration as the time interval approaches zero, which is the time derivative,
or
Thus, acceleration is the first derivative of the velocity vector and the second derivative of the position vector of that particle. Note that in a non-rotating frame of reference, the derivatives of the coordinate directions are not considered as their directions and magnitudes are constants.
The magnitude of the
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
of an object is the magnitude , a, of its acceleration vector. It is a scalar quantity:
Relative position vector
A relative position vector is a vector that defines the position of one point relative to another. It is the difference in position of the two points.
The position of one point ''A'' relative to another point ''B'' is simply the difference between their positions
which is the difference between the components of their position vectors.
If point ''A'' has position components
If point ''B'' has position components
then the position of point ''A'' relative to point ''B'' is the difference between their components:
Relative velocity
The velocity of one point relative to another is simply the difference between their velocities
which is the difference between the components of their velocities.
If point ''A'' has velocity components
and point ''B'' has velocity components
then the velocity of point ''A'' relative to point ''B'' is the difference between their components:
Alternatively, this same result could be obtained by computing the time derivative of the relative position vector r
B/A.
In the case where the velocity is close to the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
''c'' (generally within 95%), another scheme of relative velocity called
rapidity, which depends on the ratio of ''v'' to ''c'', is used in
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws ...
.
Relative acceleration
The acceleration of one point ''C'' relative to another point ''B'' is simply the difference between their accelerations.
which is the difference between the components of their accelerations.
If point ''C'' has acceleration components
and point ''B'' has acceleration components
then the acceleration of point ''C'' relative to point ''B'' is the difference between their components:
Alternatively, this same result could be obtained by computing the second time derivative of the relative position vector r
B/A.
Assuming that the initial conditions of the position,
, and velocity
at time
are known, the first integration yields the velocity of the particle as a function of time.
A second integration yields its path (trajectory),
Additional relations between displacement, velocity, acceleration, and time can be derived. Since the acceleration is constant,
can be substituted into the above equation to give:
A relationship between velocity, position and acceleration without explicit time dependence can be had by solving the average acceleration for time and substituting and simplifying
where
denotes the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
, which is appropriate as the products are scalars rather than vectors.
The dot product can be replaced by the cosine of the angle between the vectors (see
Geometric interpretation of the dot product for more details) and the vectors by their magnitudes, in which case:
In the case of acceleration always in the direction of the motion and the direction of motion should be in positive or negative, the angle between the vectors () is 0, so
, and
This can be simplified using the notation for the magnitudes of the vectors
where
can be any curvaceous path taken as the constant tangential acceleration is applied along that path, so
This reduces the parametric equations of motion of the particle to a Cartesian relationship of speed versus position. This relation is useful when time is unknown. We also know that
or
is the area under a velocity–time graph.
We can take
by adding the top area and the bottom area. The bottom area is a rectangle, and the area of a rectangle is the
where
is the width and
is the height. In this case
and
(note that the
here is different from the acceleration
). This means that the bottom area is
. Now let's find the top area (a triangle). The area of a triangle is
where
is the base and
is the height. In this case,
and
or
. Adding
and
results in the equation
results in the equation
. This equation is applicable when the final velocity is unknown.
Particle trajectories in cylindrical-polar coordinates
It is often convenient to formulate the trajectory of a particle r(''t'') = (''x''(''t''), ''y''(''t''), ''z''(''t'')) using polar coordinates in the ''X''–''Y'' plane. In this case, its velocity and acceleration take a convenient form.
Recall that the trajectory of a particle ''P'' is defined by its coordinate vector r measured in a fixed reference frame ''F''. As the particle moves, its coordinate vector r(''t'') traces its trajectory, which is a curve in space, given by:
where ''i'', ''j'', and ''k'' are the unit vectors along the ''X'', ''Y'' and ''Z'' axes of the
reference frame
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathe ...
''F'', respectively.
Consider a particle ''P'' that moves only on the surface of a circular cylinder ''r''(''t'') = constant, it is possible to align the ''Z'' axis of the fixed frame ''F'' with the axis of the cylinder. Then, the angle ''θ'' around this axis in the ''X''–''Y'' plane can be used to define the trajectory as,
where the constant distance from the center is denoted as ''R'', and ''θ'' = ''θ''(''t'') is a function of time.
The cylindrical coordinates for r(''t'') can be simplified by introducing the radial and tangential unit vectors,
and their time derivatives from elementary calculus:
Using this notation, r(''t'') takes the form,
In general, the trajectory r(''t'') is not constrained to lie on a circular cylinder, so the radius ''R'' varies with time and the trajectory of the particle in cylindrical-polar coordinates becomes:
Where ''R'', ''θ'', and ''z'' might be continuously differentiable functions of time and the function notation is dropped for simplicity. The velocity vector v
''P'' is the time derivative of the trajectory r(''t''), which yields:
Similarly, the acceleration a
''P'', which is the time derivative of the velocity v
''P'', is given by:
The term
acts toward the center of curvature of the path at that point on the path, is commonly called the centripetal acceleration. The term
is called the Coriolis acceleration.
Constant radius
If the trajectory of the particle is constrained to lie on a cylinder, then the radius ''R'' is constant and the velocity and acceleration vectors simplify. The velocity of v
P is the time derivative of the trajectory r(''t''),
Planar circular trajectories
A special case of a particle trajectory on a circular cylinder occurs when there is no movement along the ''Z'' axis:
where ''R'' and ''z''
0 are constants. In this case, the velocity v
''P'' is given by:
where
is the
angular velocity of the unit vector around the ''z'' axis of the cylinder.
The acceleration a
''P'' of the particle ''P'' is now given by:
The components
are called, respectively, the ''radial'' and ''tangential components'' of acceleration.
The notation for angular velocity and
angular acceleration is often defined as
so the radial and tangential acceleration components for circular trajectories are also written as
Point trajectories in a body moving in the plane
The movement of components of a
mechanical system are analyzed by attaching a
reference frame
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathe ...
to each part and determining how the various reference frames move relative to each other. If the structural stiffness of the parts are sufficient, then their deformation can be neglected and rigid transformations can be used to define this relative movement. This reduces the description of the motion of the various parts of a complicated mechanical system to a problem of describing the geometry of each part and geometric association of each part relative to other parts.
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
is the study of the properties of figures that remain the same while the space is transformed in various ways—more technically, it is the study of invariants under a set of transformations. These transformations can cause the displacement of the triangle in the plane, while leaving the vertex angle and the distances between vertices unchanged. Kinematics is often described as applied geometry, where the movement of a mechanical system is described using the rigid transformations of Euclidean geometry.
The coordinates of points in a plane are two-dimensional vectors in R
2 (two dimensional space). Rigid transformations are those that preserve the
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
between any two points. The set of rigid transformations in an ''n''-dimensional space is called the special
Euclidean group on R
''n'', and denoted
SE(''n'').
Displacements and motion
The position of one component of a mechanical system relative to another is defined by introducing a reference frame, say ''M'', on one that moves relative to a fixed frame, ''F,'' on the other. The rigid transformation, or displacement, of ''M'' relative to ''F'' defines the relative position of the two components. A displacement consists of the combination of a
rotation and a
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 ...
.
The set of all displacements of ''M'' relative to ''F'' is called the
configuration space of ''M.'' A smooth curve from one position to another in this configuration space is a continuous set of displacements, called the
motion
In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and m ...
of ''M'' relative to ''F.'' The motion of a body consists of a continuous set of rotations and translations.
Matrix representation
The combination of a rotation and translation in the plane R
2 can be represented by a certain type of 3×3 matrix known as a homogeneous transform. The 3×3 homogeneous transform is constructed from a 2×2
rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix
:R = \begin
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\en ...
''A''(''φ'') and the 2×1 translation vector d = (''d
x'', ''d
y''), as:
These homogeneous transforms perform rigid transformations on the points in the plane ''z'' = 1, that is, on points with coordinates r = (''x'', ''y'', 1).
In particular, let r define the coordinates of points in a reference frame ''M'' coincident with a fixed frame ''F''. Then, when the origin of ''M'' is displaced by the translation vector d relative to the origin of ''F'' and rotated by the angle φ relative to the x-axis of ''F'', the new coordinates in ''F'' of points in ''M'' are given by:
Homogeneous transforms represent
affine transformations. This formulation is necessary because a
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 ...
is not a
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
of R
2. However, using projective geometry, so that R
2 is considered a subset of R
3, translations become affine linear transformations.
Pure translation
If a rigid body moves so that its
reference frame
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathe ...
''M'' does not rotate (''θ'' = 0) relative to the fixed frame ''F'', the motion is called pure translation. In this case, the trajectory of every point in the body is an offset of the trajectory d(''t'') of the origin of ''M,'' that is:
Thus, for bodies in pure translation, the velocity and acceleration of every point ''P'' in the body are given by:
where the dot denotes the derivative with respect to time and v
''O'' and a
''O'' are the velocity and acceleration, respectively, of the origin of the moving frame ''M''. Recall the coordinate vector p in ''M'' is constant, so its derivative is zero.
Rotation of a body around a fixed axis
Rotational or angular kinematics is the description of the rotation of an object.
In what follows, attention is restricted to simple rotation about an axis of fixed orientation. The ''z''-axis has been chosen for convenience.
Position
This allows the description of a rotation as the angular position of a planar reference frame ''M'' relative to a fixed ''F'' about this shared ''z''-axis. Coordinates p = (''x'', ''y'') in ''M'' are related to coordinates P = (X, Y) in ''F'' by the matrix equation:
where
is the rotation matrix that defines the angular position of ''M'' relative to ''F'' as a function of time.
Velocity
If the point p does not move in ''M'', its velocity in ''F'' is given by
It is convenient to eliminate the coordinates p and write this as an operation on the trajectory P(''t''),
where the matrix
is known as the angular velocity matrix of ''M'' relative to ''F''. The parameter ''ω'' is the time derivative of the angle ''θ'', that is:
Acceleration
The acceleration of P(''t'') in ''F'' is obtained as the time derivative of the velocity,
which becomes
The description of rotation then involves these three quantities:
* Angular position: the oriented distance from a selected origin on the rotational axis to a point of an object is a vector r(''t'') locating the point. The vector r(''t'') has some projection (or, equivalently, some component) r
(''t'') on a plane perpendicular to the axis of rotation. Then the ''angular position'' of that point is the angle ''θ'' from a reference axis (typically the positive ''x''-axis) to the vector r
).
* Angular velocity: the angular velocity ''ω'' is the rate at which the angular position ''θ'' changes with respect to time ''t'':
The angular velocity is represented in Figure 1 by a vector Ω pointing along the axis of rotation with magnitude ''ω'' and sense determined by the direction of rotation as given by the
.
* Angular acceleration: the magnitude of the angular acceleration ''α'' is the rate at which the angular velocity ''ω'' changes with respect to time ''t'':
The equations of translational kinematics can easily be extended to planar rotational kinematics for constant angular acceleration with simple variable exchanges:
are, respectively, the initial and final angular velocities, and ''α'' is the constant angular acceleration. Although position in space and velocity in space are both true vectors (in terms of their properties under rotation), as is angular velocity, angle itself is not a true vector.
and acceleration of points in a moving body as they trace trajectories in three-dimensional space. This is particularly important for the center of mass of a body, which is used to derive equations of motion using either
.
In order to define these formulas, the movement of a component ''B'' of a mechanical system is defined by the set of rotations
This notation does not distinguish between P = (X, Y, Z, 1), and P = (X, Y, Z), which is hopefully clear in context.
This equation for the trajectory of ''P'' can be inverted to compute the coordinate vector p in ''M'' as:
This expression uses the fact that the transpose of a rotation matrix is also its inverse, that is:
The velocity of the point ''P'' along its trajectory P(''t'') is obtained as the time derivative of this position vector,
The dot denotes the derivative with respect to time; because p is constant, its derivative is zero.
This formula can be modified to obtain the velocity of ''P'' by operating on its trajectory P(''t'') measured in the fixed frame ''F''. Substituting the
where the vector ''ω'' is the angular velocity vector obtained from the components of the matrix
is the position of ''P'' relative to the origin ''O'' of the moving frame ''M''; and
is the velocity of the origin ''O''.
The acceleration of a point ''P'' in a moving body ''B'' is obtained as the time derivative of its velocity vector:
where ''α'' is the angular acceleration vector obtained from the derivative of the angular velocity matrix;
is the relative position vector (the position of ''P'' relative to the origin ''O'' of the moving frame ''M''); and
is the acceleration of the origin of the moving frame ''M''.
Kinematic constraints are constraints on the movement of components of a mechanical system. Kinematic constraints can be considered to have two basic forms, (i) constraints that arise from hinges, sliders and cam joints that define the construction of the system, called
, and (ii) constraints imposed on the velocity of the system such as the knife-edge constraint of ice-skates on a flat plane, or rolling without slipping of a disc or sphere in contact with a plane, which are called
. The following are some common examples.
exactly constrains all 6 degrees of freedom.
of its angular velocity with a vector from the point of contact to the center of mass:
.
This is the case where bodies are connected by an idealized cord that remains in tension and cannot change length. The constraint is that the sum of lengths of all segments of the cord is the total length, and accordingly the time derivative of this sum is zero.
. Another example is a drum turned by the pull of gravity upon a falling weight attached to the rim by the inextensible cord.
An ''equilibrium'' problem (i.e. not kinematic) of this type is the
.
s. He distinguished between higher pairs which were said to have line contact between the two links and lower pairs that have area contact between the links. J. Phillips shows that there are many ways to construct pairs that do not fit this simple classification.
A lower pair is an ideal joint, or holonomic constraint, that maintains contact between a point, line or plane in a moving solid (three-dimensional) body to a corresponding point line or plane in the fixed solid body. There are the following cases:
* A revolute pair, or hinged joint, requires a line, or axis, in the moving body to remain co-linear with a line in the fixed body, and a plane perpendicular to this line in the moving body maintain contact with a similar perpendicular plane in the fixed body. This imposes five constraints on the relative movement of the links, which therefore has one degree of freedom, which is pure rotation about the axis of the hinge.
* A prismatic joint, or slider, requires that a line, or axis, in the moving body remain co-linear with a line in the fixed body, and a plane parallel to this line in the moving body maintain contact with a similar parallel plane in the fixed body. This imposes five constraints on the relative movement of the links, which therefore has one degree of freedom. This degree of freedom is the distance of the slide along the line.
* A cylindrical joint requires that a line, or axis, in the moving body remain co-linear with a line in the fixed body. It is a combination of a revolute joint and a sliding joint. This joint has two degrees of freedom. The position of the moving body is defined by both the rotation about and slide along the axis.
* A spherical joint, or ball joint, requires that a point in the moving body maintain contact with a point in the fixed body. This joint has three degrees of freedom.
* A planar joint requires that a plane in the moving body maintain contact with a plane in fixed body. This joint has three degrees of freedom.
Generally speaking, a higher pair is a constraint that requires a curve or surface in the moving body to maintain contact with a curve or surface in the fixed body. For example, the contact between a cam and its follower is a higher pair called a ''cam joint''. Similarly, the contact between the involute curves that form the meshing teeth of two gears are cam joints.
s''.
and robots are examples of kinematic chains. The
of a kinematic chain is computed from the number of links and the number and type of joints using the
. This formula can also be used to enumerate the
of kinematic chains that have a given degree of freedom, which is known as ''type synthesis'' in machine design.
s assembled from ''N'' links and ''j'' hinges or sliding joints are:
* ''N'' = 2, ''j'' = 1 : a two-bar linkage that is the lever;
* ''N'' = 4, ''j'' = 4 : the
. This must have two links ("ternary links") that support three joints. There are two distinct topologies that depend on how the two ternary linkages are connected. In the
, the two ternary links do not have a common joint and are connected by binary links.
* ''N'' = 8, ''j'' = 10 : eight-bar linkage with 16 different topologies;
* ''N'' = 10, ''j'' = 13 : ten-bar linkage with 230 different topologies;
* ''N'' = 12, ''j'' = 16 : twelve-bar linkage with 6,856 topologies.
For larger chains and their linkage topologies, see R. P. Sunkari and
, "Structural synthesis of planar kinematic chains by adapting a Mckay-type algorithm", ''Mechanism and Machine Theory'' #41, pp. 1021–1030 (2006).
(1913) D.H. Delphenich translator
from the University of New South Wales.
featuring movies and photos of hundreds of working models of mechanical systems at
of classic texts on mechanical design and engineering.