CIRP Encyclopedia of Production Engineering

Living Edition
| Editors: The International Academy for Production Engineering, Sami Chatti, Tullio Tolio


  • Gabor StepanEmail author
Living reference work entry



Kinematic Chain

An assemblage of links (considered as rigid bodies) and joints, interconnected in a way to provide a controlled output motion of one or more links in response to input motions supplied by actuators.


A kinematic chain in which at least one link is “grounded,” or attached, to the frame of reference.

Theory and Application

The major goal of designing a mechanism is to have a working part (body) of the mechanism that undergoes a desired motion (Norton 2004). In the easiest case, the desired motion is only several fixed points and/or orientations in the (3D or 2D) space that the working part needs to pass through. In a complicated case, the complete shape of the path of the working part is given. Moreover, the velocity or acceleration characteristics can be prescribed along the path, too. If the mechanism cannot provide the exact desired motion, then the goal is to provide a motion that is close to the desired motion.

Most of the engineering examples can be realized by 4-bar or 6-bar kinematic chains. It is important to predesign the mechanism geometrically: the types of the linkage connections and the topology of the linkage.


Links are considered rigid bodies (RB) in the mechanical model of the mechanisms.

Degree of freedom (DOF) of a link (RB) is defined as the number of independent scalar functions that uniquely determine the position and orientation of the link in the space.

Joint is a connection between two or more links (at their nodes), which allows some motion between the connected links. Joints are also called as kinematic pairs. Different types of joints are shown in Table 1
Table 1

Different types of ideal joints that can appear in a kinematic chain


1DOF (full joints)

2DOF (half joints)

1- or 2DOF


Pin joint

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Link against a plane

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May roll, may slide

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Pin slider

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Spherical joint

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Helical joint

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Planar joint

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Mobility of mechanism (or DOF of the mechanism) is the number of inputs, which are needed to provide the prescribed motion. The general coordinates q(t) are the independent scalar functions that uniquely describe the motion of the mechanism. The number of the general coordinates is equal to the mobility of the mechanism. In practice, the mobility gives the necessary number of actuators that are needed to build in the mechanism. In planar case, the mobility of a mechanism can be calculated as (not considering special geometries and paradoxes)
$$ M=3\left(L-1\right)-2\left({J}_1-{J}_2\right). $$
  • L is the number of links in the mechanism,

  • J1 is the number of full joints in the mechanism, and

  • J2 is the number of half joints in the mechanism.

Linkages can be classified into open mechanisms and closed mechanisms. Open mechanisms have link that has at least one open node (neither grounded nor linked). Closed mechanisms have no nodes that are not linked or grounded.


All the parts of the mechanisms are considered as rigid bodies, that is, the deformations of the links during the motion are neglected. Knowing the position, the velocity, and the acceleration inputs (at the actuators), the position, velocity, and acceleration state of any part of the mechanisms can be determined.

Position Analysis

Position analysis is the determination of the position and the orientation of each link as function of the general coordinates. The position analysis of open mechanisms is direct and explicit, while the position analysis of closed mechanisms usually leads to a system of nonlinear equations that can be generally solved numerically. After the position analysis, the position vectors r between any nodes of any links can be given with respect to the general coordinates q(t) analytically or numerically, i.e.,
$$ {\mathbf{r}}_{XY}(t)={\mathbf{r}}_{XY}\left(\mathbf{q}(t)\right), $$
where X and Y are two subsequent nodes on a selected link of the mechanism. The position w.r.t. the frame of reference can be expressed as
$$ {\mathbf{r}}_Y(t)={\mathbf{r}}_X(t)+{\mathbf{r}}_{XY}(t), $$

Velocity Analysis

The velocity of a given point Y of a link can be derived as
$$ {\mathbf{v}}_Y(t)={\dot{\mathbf{r}}}_Y(t), $$
where dot denotes derivation w.r.t. time. Since the links are rigid bodies, the velocity state of a link at time instant t is said to be known if the velocity vector at a point Y and the angular velocity vector ω of the link are known. The velocity state can be given by the following pair
$$ {\left[\omega, {\mathbf{v}}_Y\right]}_Y. $$
of vectors (see “Screw Theory”). The velocity of another point of the link can be calculated from vY by means of the reduction formula as
$$ {\mathbf{v}}_X={\mathbf{v}}_Y+{\mathbf{v}}_{XY}={\mathbf{v}}_Y+\boldsymbol{\upomega} \times {\mathbf{r}}_{YX}. $$

Acceleration Analysis

The acceleration of a given point Y of a link can be derived as
$$ {\mathbf{a}}_Y(t)={\dot{\mathbf{v}}}_Y(t)={\ddot{\mathbf{r}}}_Y(t), $$
The acceleration vector can be separated to tangential and normal components along the local path at Y
$$ {\mathbf{a}}_Y={\mathbf{a}}_{Y,t}+{\mathbf{a}}_{Y,n}, $$

The acceleration state of a link in a mechanism is given if the acceleration vector aY at point Y and the angular acceleration vector ε are known assuming that the velocity state of the link is already determined.

The acceleration of another point of the link can be calculated from aY by means of the reduction formula as
$$ {\displaystyle \begin{array}{c}{\mathbf{a}}_X={\mathbf{a}}_Y+{\mathbf{a}}_{XY,t}+{\mathbf{a}}_{XY,n}\\ {}={\mathbf{a}}_Y+\boldsymbol{\upvarepsilon} \times {\mathbf{r}}_{YX}+\boldsymbol{\upomega} \times \left(\boldsymbol{\upomega} \times {\mathbf{r}}_{\mathbf{YX}}\right).\end{array}} $$


Dynamic analysis gives the equation of motions of the entire mechanism (Ginsberg 1995). Once the accelerations and the internal forces in the structure are known, then the dimensions of the links can be designed.

Classical Formalism

Newton-Euler Equations (Newton’s Second Law)

The equation of motion is derived using free-body diagrams (FBDs) for each rigid body. The FBDs contain kinematical (acceleration, angular acceleration, angular velocity) and dynamical (external/reaction forces, moments) variables. The Newton-Euler equations consist of two parts, the translational part and the rotational part.

The translational part (for the ith body) is
$$ {\dot{\mathbf{I}}}_i=\sum \limits_j{\mathbf{F}}_{i,j}. $$
  • Ii = mivi,C is the linear momentum of the ith RB with C being the center of gravity and

  • Fi,j is an external or internal (reaction) force acting on the ith RB.

The rotational part (for the ith body) is
$$ {\mathbf{D}}_{i,X}=\sum \limits_j{\mathbf{r}}_{i, Xj}\times {\mathbf{F}}_{i,j}+\sum \limits_k{\mathbf{M}}_{i,k}, $$
$$ {\mathbf{D}}_{i,X}={\dot{\mathbf{L}}}_{i,X}+{\mathbf{v}}_{i,X}\times {\mathbf{I}}_{i,X} $$
is the kinetic momentum of the ith RB w.r.t. the general point X. If X is a permanently steady point or X is the center of gravity, the following simplification holds:
$$ {\mathbf{D}}_{i,X}={\dot{\mathbf{L}}}_{i,X}={\Theta}_{i,X}{\boldsymbol{\upvarepsilon}}_i+{\omega}_i\times {\mathbf{L}}_{i,X} $$
  • Li,X = Li,C + ri,XC × Ii is the angular momentum of the ith RB w.r.t. point X.

  • Li,C = Θi,Cωi is the angular momentum w.r.t. the center of gravity C.

  • Θi,C is the mass moment of inertia matrix w.r.t. the center of gravity C.

  • ri,XC is the spatial vector that points to the point of application of Fi,j.

  • Mi,k is the reaction moment or external torque acting on the ith RB.

  • ωi is the angular velocity of the ith link.

  • εi is the angular acceleration of the ith link.

Virtual Power-Based Formulization

Lagrange Equation of the Second Kind
This equation provides the equations of motion of a holonomic (having only geometrical constraints) mechanical system (mechanism) in a kth-dimensional ODE form. Note that the inner forces are excluded from the equations. The following formula is the so-called Routh-Voss equation that is the Lagrange equation of the second kind extended to kinematical constraints, too:
$$ \frac{d}{dt}\frac{\partial L}{\partial \dot{\mathbf{q}}}-\frac{\partial L}{\partial \mathbf{q}}+\frac{\partial D}{\partial \dot{\mathbf{q}}}=\mathbf{Q}+{\mathbf{A}}^{\mathrm{T}}\boldsymbol{\upmu} $$
  • L = TU is the Lagrange function that contains the kinetic energy T and the potential function U,

  • q(t) contains the general coordinates,

  • D is the dissipative potential that can introduce viscous dampers in the equations of motions,

  • Q contains the general forces, and

  • μ contains the Lagrange multipliers

$$ \mathbf{A}\left(\mathbf{q},t\right)\dot{\mathbf{q}}+\mathbf{b}\left(\mathbf{q},t\right)=0 $$
describes the kinematical constraints.
Lagrange Equation of the First Kind
With this equation, both geometrical and kinematical constrains can be considered, and it usually leads to a mixed differential and algebraic equation (DAE) as the governing equation of motions. The links (RB) of the mechanism can be considered as a specially distributed, equivalent, system of material particles (of mass mi) that describes the mass inertia of the original system, too, with the so-called natural coordinates (ri). In this way, the definitions of angular velocities and angular accelerations are excluded because of the difficulties in the definition of the angular coordinates. Then, the Lagrange equation of the first kind has the form
$$ {\displaystyle \begin{array}{c}{m}_i{\ddot{\mathbf{r}}}_i={\mathbf{F}}_i+{\boldsymbol{\Phi}}_{{\mathbf{r}}_i}^{\mathrm{T}}\boldsymbol{\uplambda} +{\mathbf{A}}^{\mathrm{T}}\boldsymbol{\upmu}, \\ {}\quad \boldsymbol{\varphi} \left({\mathbf{r}}_i,t\right)=\mathbf{0},\\ {}\mathbf{A}\left({\mathbf{r}}_i,t\right){\dot{\mathbf{r}}}_i+\mathbf{b}\left({\mathbf{r}}_i,t\right)=\mathbf{0},\end{array}} $$
  • Fi are the external forces,

  • φ contains the geometric constraints defined in the mechanism, and

$$ {\boldsymbol{\Phi}}_{{\mathbf{r}}_i}=\frac{\partial \boldsymbol{\varphi}}{\partial {\mathbf{r}}_i} $$

λ and μ are the Lagrange multipliers.

Multibody Formulations
Both abovementioned formulizations can be described by the following common form that is usually used nowadays to describe the so-called multibody systems:
$$ \left.\begin{array}{c}\quad \mathbf{M}\;\ddot{\mathbf{q}}+\mathbf{C}\left(\mathbf{q},\dot{\mathbf{q}}\right)+{\boldsymbol{\Phi}}_{\mathbf{q}}^{\mathrm{T}}\boldsymbol{\uplambda} +{\mathbf{A}}^{\mathrm{T}}\boldsymbol{\upmu} =\mathbf{Q}\\ {}\quad\quad \boldsymbol{\upphi} \left(\mathbf{q},t\right)=0\\ {}\quad\quad \mathbf{A}\left(\mathbf{q},t\right)\;\dot{\mathbf{q}}+\mathbf{b}\left(\mathbf{q},t\right)=0\ \end{array}\right\}, $$
  • M is the mass matrix,

  • q is the vector of the chosen coordinates (can be redundant),

  • Q is the load vector, and

  • C is a nonlinear function that contains the system damping and stiffness.

In order to avoid the difficult DAE form, one can use the time derivatives of the constraint equations, and stabilization techniques (e.g., Baumgarte stabilization) can be applied in order to avoid possible numerical problems.


Several machining centers with open and closed loop mechanisms are listed in the review paper of Weck and Staimer (2002).



  1. Ginsberg JH (1995) Advanced engineering dynamics. Cambridge University Press, New YorkCrossRefGoogle Scholar
  2. Norton RL (2004) Design of machinery, an introduction to the synthesis and analysis of mechanisms and machines, 3rd edn. McGraw-Hill, New YorkGoogle Scholar
  3. Weck M, Staimer D (2002) Parallel kinematic machine tools-current state and future potentials. CIRP Ann Manuf Technol 51(2):671–683CrossRefGoogle Scholar

Copyright information

© CIRP 2018

Authors and Affiliations

  1. 1.Department of Applied MechanicsBudapest University of Technology and EconomicsBudapestHungary

Section editors and affiliations

  • Hans-Christian Möhring
    • 1
  1. 1.Institut für WerkzeugmaschinenUniversität StuttgartStuttgartGermany