Kinematic chain

Last updated January 12, 2024

The arms, fingers, and head of the JSC Robonaut are modeled as kinematic chains. JSC2001-01725.jpg — The arms, fingers, and head of the JSC Robonaut are modeled as kinematic chains.

In mechanical engineering, a kinematic chain is an assembly of rigid bodies connected by joints to provide constrained motion that is the mathematical model for a mechanical system.^[1] As the word chain suggests, 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 model for a typical robot manipulator.^[2]

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, often called lower pairs and the surface contact joints critical to cams and gearing, called higher pairs. These joints are generally modeled as holonomic constraints. 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.^[3]^[4]

Mobility formula

The degrees of freedom, or mobility, of a kinematic chain is the number of parameters that define the configuration of the chain.^[2]^[5] A system of $n$ rigid bodies moving in space has $6 n$ 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 $M = 6(N - 1)$ , where $N = n + 1$ 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 $c$ 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 kinematic chain formed from $n$ moving links and $j$ joints each with freedom $f i$ , $i = 1, 2, \dots, j$ , is given by

M=6n-\sum _{i=1}^{j}(6-f_{i})=6(N-1-j)+\sum _{i=1}^{j}f_{i}

Recall that $N$ 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 $[Z]$ to characterize the relative movement allowed at each joint and separate rigid transformations $[X]$ 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 $n$ links connected in series has the kinematic equations,

[T]=[Z_{1}][X_{1}][Z_{2}][X_{2}]\cdots [X_{n-1}][Z_{n}],\!

where $[T]$ 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.^[6]

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: a serial chain, a parallel manipulator, a tree structure, or a graph.
Its geometrical 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.^[5]

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.^[7]

Perhaps the most developed formulation of kinematic synthesis is for four-bar linkages, which is known as Burmester theory.^[8]^[9]^[10]

Ferdinand Freudenstein 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 systems.^[7]

This work has been generalized to the synthesis of spherical and spatial mechanisms.^[2]

Related Research Articles

A machine is a physical system that uses power to apply forces and control movement to perform an action. The term is commonly applied to artificial devices, such as those employing engines or motors, but also to natural biological macromolecules, such as molecular machines. Machines can be driven by animals and people, by natural forces such as wind and water, and by chemical, thermal, or electrical power, and include a system of mechanisms that shape the actuator input to achieve a specific application of output forces and movement. They can also include computers and sensors that monitor performance and plan movement, often called mechanical systems.

Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies 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 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, not kinematics. For further details, see analytical dynamics.

In computer animation and robotics, inverse kinematics is the mathematical process of calculating the variable joint parameters needed to place the end of a kinematic chain, such as a robot manipulator or animation character's skeleton, in a given position and orientation relative to the start of the chain. Given joint parameters, the position and orientation of the chain's end, e.g. the hand of the character or robot, can typically be calculated directly using multiple applications of trigonometric formulas, a process known as forward kinematics. However, the reverse operation is, in general, much more challenging.

In robotics, robot kinematics applies geometry to the study of the movement of multi-degree of freedom kinematic chains that form the structure of robotic systems. The emphasis on geometry means that the links of the robot are modeled as rigid bodies and its joints are assumed to provide pure rotation or translation.

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 configured so the links move in parallel planes, and the assembly is called a planar four-bar linkage. Spherical and spatial four-bar linkages also exist and are used in practice.

A mechanical linkage is an assembly of systems connected 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 example, and are called joints. A linkage modeled as a network of rigid links and ideal joints is called a kinematic chain.

In physics, the degrees of freedom (DOF) of a mechanical system is the number of independent parameters that define its configuration or state. It is important in the analysis of systems of bodies in mechanical engineering, structural engineering, aerospace engineering, robotics, and other fields.

<span class="mw-page-title-main">Overconstrained mechanism</span> Moveable linkage with zero mobility

In mechanical engineering, an overconstrained mechanism is a linkage that has more degrees of freedom than is predicted by the mobility formula. The mobility formula evaluates the degree of freedom of a system of rigid bodies that results when constraints are imposed in the form of joints between the links.

Multibody system is the study of the dynamic behavior of interconnected rigid or flexible bodies, each of which may undergo large translational and rotational displacements.

In robot kinematics, forward kinematics refers to the use of the kinematic equations of a robot to compute the position of the end-effector from specified values for the joint parameters.

In classical mechanics, a kinematic pair is a connection between two physical objects that imposes constraints on their relative movement (kinematics). German engineer Franz Reuleaux introduced the kinematic pair as a new approach to the study of machines that provided an advance over the motion of elements consisting of simple machines.

There are many conventions used in the robotics research field. This article summarises these conventions.

In mechanical engineering, the Denavit–Hartenberg parameters are the four parameters associated with a particular convention for attaching reference frames to the links of a spatial kinematic chain, or robot manipulator.

In kinematics, cognate linkages are linkages that ensure the same coupler curve geometry or input-output relationship, while being dimensionally dissimilar. In case of four-bar linkage coupler cognates, the Roberts–Chebyshev Theorem, after Samuel Roberts and Pafnuty Chebyshev, states that each coupler curve can be generated by three different four-bar linkages. These four-bar linkages can be constructed using similar triangles and parallelograms, and the Cayley diagram.

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.

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:

Kinematics equations are the constraint equations of a mechanical system such as a robot manipulator that define how input movement at one or more joints specifies the configuration of the device, in order to achieve a task position or end-effector location. Kinematics equations are used to analyze and design articulated systems ranging from four-bar linkages to serial and parallel robots.

In mechanical engineering, kinematic 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 word synthesis refers to combining parts to form a whole. Hartenberg and Denavit describe kinematic synthesis as

...it is design, the creation of something new. Kinematically, it is the conversion of a motion idea into hardware.

In kinematics, a five-bar linkage is a mechanism with two degrees of freedom that is constructed from five links that are connected together in a closed chain. All links are connected to each other by five joints in series forming a loop. One of the links is the ground or base. This configuration is also called a pantograph, however, it is not to be confused with the parallelogram-copying linkage pantograph.

In robotics, Cartesian parallel manipulators are manipulators that move a platform using parallel-connected kinematic linkages ('limbs') lined up with a Cartesian coordinate system. Multiple limbs connect the moving platform to a base. Each limb is driven by a linear actuator and the linear actuators are mutually perpendicular. The term 'parallel' here refers to the way that the kinematic linkages are put together, it does not connote geometrically parallel; i.e., equidistant lines.

References

↑ Reuleaux, F., 1876 The Kinematics of Machinery, (trans. and annotated by A. B. W. Kennedy), reprinted by Dover, New York (1963)
1 2 3 J. M. McCarthy and G. S. Soh, 2010, Geometric Design of Linkages, Springer, New York.
↑ Larry L. Howell, 2001, Compliant mechanisms, John Wiley & Sons.
↑ Alexander Slocum, 1992, Precision Machine Design, SME
1 2 J. J. Uicker, G. R. Pennock, and J. E. Shigley, 2003, Theory of Machines and Mechanisms, Oxford University Press, New York.
↑ J. M. McCarthy, 1990, Introduction to Theoretical Kinematics, MIT Press, Cambridge, Massachusetts.
1 2 R. S. Hartenberg and J. Denavit, 1964, Kinematic Synthesis of Linkages, McGraw-Hill, New York.
↑ Suh, C. H., and Radcliffe, C. W., Kinematics and Mechanism Design, John Wiley and Sons, New York, 1978.
↑ Sandor, G.N., and Erdman, A.G., 1984, Advanced Mechanism Design: Analysis and Synthesis, Vol. 2. Prentice-Hall, Englewood Cliffs, NJ.
↑ Hunt, K. H., Kinematic Geometry of Mechanisms, Oxford Engineering Science Series, 1979

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[Reuleaux1876-1] Reuleaux, F., 1876 The Kinematics of Machinery, (trans. and annotated by A. B. W. Kennedy), reprinted by Dover, New York (1963)

[McCarthy2010-2] 1 2 3 J. M. McCarthy and G. S. Soh, 2010, Geometric Design of Linkages, Springer, New York.

[3] Larry L. Howell, 2001, Compliant mechanisms, John Wiley & Sons.

[4] Alexander Slocum, 1992, Precision Machine Design, SME

[Uicker2003-5] 1 2 J. J. Uicker, G. R. Pennock, and J. E. Shigley, 2003, Theory of Machines and Mechanisms, Oxford University Press, New York.

[6] J. M. McCarthy, 1990, Introduction to Theoretical Kinematics, MIT Press, Cambridge, Massachusetts.

[Hartenberg1964-7] 1 2 R. S. Hartenberg and J. Denavit, 1964, Kinematic Synthesis of Linkages, McGraw-Hill, New York.

[8] Suh, C. H., and Radcliffe, C. W., Kinematics and Mechanism Design, John Wiley and Sons, New York, 1978.

[9] Sandor, G.N., and Erdman, A.G., 1984, Advanced Mechanism Design: Analysis and Synthesis, Vol. 2. Prentice-Hall, Englewood Cliffs, NJ.

[10] Hunt, K. H., Kinematic Geometry of Mechanisms, Oxford Engineering Science Series, 1979

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