In mechanical engineering, a kinematic chain is an assembly of
rigid bodies connected by
joints to provide constrained (or desired) motion that is the
mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
for a
mechanical system.
[ Reuleaux, F., 187]
''The Kinematics of Machinery,''
(trans. and annotated by A. B. W. Kennedy), reprinted by Dover, New York (1963) As in the familiar use of the word
chain, the rigid bodies, or links, are constrained by their connections to other links. An example is the simple open chain formed by links connected in series, like the usual chain, which is the
kinematic
Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fie ...
model for a typical robot
manipulator.
[J. M. McCarthy and G. S. Soh, 2010]
''Geometric Design of Linkages,''
Springer, New York.
Mathematical models of the connections, or joints, between two links are termed
kinematic pairs. Kinematic pairs model the hinged and sliding joints fundamental to
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 ...
, often called ''lower pairs'' and the surface contact joints critical to
cams and
gear
A gear is a rotating circular machine part having cut teeth or, in the case of a cogwheel or gearwheel, inserted teeth (called ''cogs''), which mesh with another (compatible) toothed part to transmit (convert) torque and speed. The basic ...
ing, called ''higher pairs.'' These joints are generally modeled as
holonomic constraints
In classical mechanics, holonomic constraints are relations between the position variables (and possibly time) that can be expressed in the following form:
:f(u_1, u_2, u_3,\ldots, u_n, t) = 0
where \ are the ''n'' generalized coordinates that d ...
. A
kinematic diagram is a schematic of the mechanical system that shows the kinematic chain.
The modern use of kinematic chains includes compliance that arises from flexure joints in precision mechanisms, link compliance in
compliant mechanisms and
micro-electro-mechanical systems, and cable compliance in cable robotic and
tensegrity systems.
Mobility formula
The
degrees of freedom, or ''mobility,'' of a kinematic chain is the number of parameters that define the configuration of the chain.
[J. J. Uicker, G. R. Pennock, and J. E. Shigley, 2003, Theory of Machines and Mechanisms, Oxford University Press, New York.]
A system of rigid bodies moving in space has degrees of freedom measured relative to a fixed frame. This frame is included in the count of bodies, so that mobility does not depend on link that forms the fixed frame. This means the degree-of-freedom of this system is , where is the number of moving bodies plus the fixed body.
Joints that connect bodies impose constraints. Specifically, hinges and sliders each impose five constraints and therefore remove five degrees of freedom. It is convenient to define the number of constraints that a joint imposes in terms of the joint's freedom , where . In the case of a
hinge
A hinge is a mechanical bearing that connects two solid objects, typically allowing only a limited angle of rotation between them. Two objects connected by an ideal hinge rotate relative to each other about a fixed axis of rotation: all other ...
or
slider, which are one-degree-of-freedom joints, have and therefore .
The result is that the mobility of a kinematic chain formed from moving links and joints each with freedom , , is given by
:
Recall that includes the fixed link.
Analysis of kinematic chains
The constraint equations of a kinematic chain couple the range of movement allowed at each joint to the dimensions of the links in the chain, and form
algebraic equations that are solved to determine the configuration of the chain associated with specific values of input parameters, called
degrees of freedom.
The constraint equations for a kinematic chain are obtained using
rigid transformations to characterize the relative movement allowed at each joint and separate rigid transformations to define the dimensions of each link. In the case of a serial open chain, the result is a sequence of rigid transformations alternating joint and link transformations from the base of the chain to its end link, which is equated to the specified position for the end link. A chain of links connected in series has the kinematic equations,
:
where is the transformation locating the end-link—notice that the chain includes a "zeroth" link consisting of the ground frame to which it is attached. These equations are called the
forward kinematics equations of the serial chain.
Kinematic chains of a wide range of complexity are analyzed by equating the kinematics equations of serial chains that form loops within the kinematic chain. These equations are often called ''loop equations''.
The complexity (in terms of calculating the
forward and
inverse kinematics) of the chain is determined by the following factors:
* Its
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
: a serial chain, a
parallel manipulator, a
tree
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...
structure, or a
graph.
* Its
geometrical
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 c ...
form: how are neighbouring
joints spatially connected to each other?
Explanation
Two or more rigid bodies in space are collectively called a rigid body system. We can hinder the motion of these independent rigid bodies with kinematic constraints. Kinematic constraints are constraints between rigid bodies that result in the decrease of the degrees of freedom of rigid body system.
Synthesis of kinematic chains
The constraint equations of a kinematic chain can be used in reverse to determine the dimensions of the links from a specification of the desired movement of the system. This is termed ''kinematic synthesis.''
[R. S. Hartenberg and J. Denavit, 1964, ''Kinematic Synthesis of Linkages,'' McGraw-Hill, New York.]
Perhaps the most developed formulation of kinematic synthesis is for
four-bar linkages, which is known as
Burmester theory
In kinematics, Burmester theory comprises geometric techniques for synthesis of linkages. It was introduced in the late 19th century by Ludwig Burmester (1840–1927). His approach was to compute the geometric constraints of the linkage direct ...
.
[Hunt, K. H., Kinematic Geometry of Mechanisms, Oxford Engineering Science Series, 1979]
Ferdinand Freudenstein
Ferdinand Freudenstein (12 May 1926 – 30 March 2006) was an American physicist and engineer known as the "Father of Modern Kinematics." Freudenstein applied digital computation to the kinematic synthesis of mechanisms. In his Ph.D. dissertati ...
is often called the father of modern kinematics for his contributions to the kinematic synthesis of
linkages beginning in the 1950s. His use of the newly developed computer to solve ''Freudenstein's equation'' became the prototype of
computer-aided design
Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve co ...
systems.
This work has been generalized to the synthesis of spherical and spatial mechanisms.
See also
*
Assur group
*
Denavit–Hartenberg parameters
*
Chebychev–Grübler–Kutzbach criterion
The Chebychev–Grübler–Kutzbach criterion determines the number of degrees of freedom of a kinematic chain, that is, a coupling of rigid bodies by means of mechanical constraints. These devices are also called linkages.
The Kutzbach criteri ...
*
Configuration space
*
Machine (mechanical)
*
Mechanism (engineering)
In engineering, a mechanism is a device that transforms input forces and movement into a desired set of output forces and movement. Mechanisms generally consist of moving components which may include:
* Gears and gear trains;
* Belts and chai ...
*
Six-bar linkage
In mechanics, a six-bar linkage is a mechanism with one degree of freedom that is constructed from six links and seven joints. An example is the Klann linkage used to drive the legs of a walking machine.
In general, each joint of a linkage ...
*
Simple machines
A simple machine is a mechanical device that changes the direction or magnitude of a force. In general, they can be defined as the simplest mechanisms that use mechanical advantage (also called leverage) to multiply force. Usually the term ref ...
*
Six degrees of freedom
*
Superposition principle
References
{{DEFAULTSORT:Kinematic Chain
Computer graphics
3D computer graphics
Computational physics
Robot kinematics
Virtual reality
Mechanisms (engineering)
Diagrams
Classical mechanics