Introduction to Mechanisms
Yi Zhang
with
Susan Finger
Stephannie Behrens
The degrees of freedom (DOF) of a rigid body is defined
as the number of independent movements it has. Figure 41
shows a rigid body in a plane. To determine the DOF of this body
we must consider how many distinct ways the bar can be moved. In
a two dimensional plane such as this computer screen, there are 3 DOF.
The bar can be translated along the x axis, translated
along the y axis, and rotated about its centroid.
Figure 41 Degrees of freedom of a rigid body in a plane
An unrestrained rigid body in space has six degrees of freedom:
three translating motions along the x, y and z
axes and three rotary motions around the x, y and
z axes respectively.
Figure 42 Degrees of freedom of a rigid body in space
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.
The term kinematic pairs actually refers to
kinematic constraints between rigid bodies. The kinematic pairs
are divided into lower pairs and higher pairs, depending on how the two
bodies are in contact.
There are two kinds of lower pairs in planar mechanisms: revolute pairs and prismatic pairs.
A rigid body in a plane has only three independent motions  two
translational and one rotary  so introducing either a revolute pair
or a prismatic pair between two rigid bodies removes two degrees of
freedom.
Figure 43 A planar revolute pair (Rpair)
Figure 44 A planar prismatic pair (Ppair)
There are six kinds of lower pairs under the category of spatial mechanisms. The types are: spherical pair, plane pair,
cylindrical pair, revolute
pair, prismatic pair, and screw pair.
Figure 45 A spherical pair (Spair)
A spherical pair keeps two spherical centers together. Two
rigid bodies connected by this constraint will be able to
rotate relatively around x, y and z axes,
but there will be no relative translation along any of these
axes. Therefore, a spherical pair removes three degrees of freedom in
spatial mechanism. DOF = 3.
Figure 46 A planar pair (Epair)
A plane pair keeps the surfaces of two rigid bodies together.
To visualize this, imagine a book lying on a table where is can move
in any direction except off the table. Two rigid bodies connected by
this kind of pair will have two independent translational motions in
the plane, and a rotary motion around the axis that is perpendicular
to the plane. Therefore, a plane pair removes three degrees of
freedom in spatial mechanism. In our example, the book would not be
able to raise off the table or to rotate into the table. DOF =
3.
Figure 47 A cylindrical pair (Cpair)
A cylindrical pair keeps two axes of two rigid bodies
aligned. Two rigid bodies that are part of this kind of system will
have an independent translational motion along the axis and a relative
rotary motion around the axis. Therefore, a cylindrical pair removes
four degrees of freedom from spatial mechanism. DOF = 2.
Figure 48 A revolute pair (Rpair)
A revolute pair keeps the axes of two rigid bodies
together. Two rigid bodies constrained by a revolute pair have an
independent rotary motion around their common axis. Therefore, a
revolute pair removes five degrees of freedom in spatial
mechanism. DOF = 1.
Figure 49 A prismatic pair (Ppair)
A prismatic pair keeps two axes of two rigid bodies align and
allow no relative rotation. Two rigid bodies constrained by this kind
of constraint will be able to have an independent translational motion
along the axis. Therefore, a prismatic pair removes five degrees of
freedom in spatial mechanism. DOF = 1.
Figure 410 A screw pair (Hpair)
The screw pair keeps two axes of two rigid bodies aligned and
allows a relative screw motion. Two rigid bodies constrained by a
screw pair a motion which is a composition of a translational motion
along the axis and a corresponding rotary motion around the axis.
Therefore, a screw pair removes five degrees of freedom in spatial
mechanism.
Rigid bodies and kinematic constraints are the basic components of
mechanisms. A constrained rigid body system can be a kinematic chain, a mechanism, a structure, or none of these.
The influence of kinematic constraints in the motion of rigid bodies
has two intrinsic aspects, which are the geometrical and physical
aspects. In other words, we can analyze the motion of the constrained
rigid bodies from their geometrical relationships or using Newton's Second Law.
A mechanism is a constrained rigid body system in which one of the
bodies is the frame. The degrees of
freedom are important when considering a constrained rigid body system
that is a mechanism. It is less crucial when the system is a
structure or when it does not have definite motion.
Calculating the degrees of freedom of a rigid body system is straight
forward. Any unconstrained rigid body has six degrees of freedom in
space and three degrees of freedom in a plane. Adding kinematic
constraints between rigid bodies will correspondingly decrease the
degrees of freedom of the rigid body system. We will discuss more on
this topic for planar mechanisms in the next section.
The definition of the degrees of freedom of a mechanism
is the number of independent relative motions among the rigid bodies.
For example, Figure 411 shows several cases of a
rigid body constrained by different kinds of pairs.
Figure 411 Rigid bodies constrained by different kinds of planar pairs
In Figure 411a, a rigid body is constrained by a revolute pair which allows only rotational
movement around an axis. It has one degree of freedom, turning around
point A. The two lost degrees of freedom are translational movements
along the x and y axes. The only way the rigid body can
move is to rotate about the fixed point A.
In Figure 411b, a rigid body is constrained by a prismatic pair which allows only
translational motion. In two dimensions, it has one degree of
freedom, translating along the x axis. In this example, the
body has lost the ability to rotate about any axis, and it cannot move
along the y axis.
In Figure 411c, a rigid body is constrained by a higher pair. It has two degrees of
freedom: translating along the curved surface and turning about the
instantaneous contact point.
In general, a rigid body in a plane has three degrees of freedom.
Kinematic pairs are constraints on rigid bodies that reduce the
degrees of freedom of a mechanism. Figure 411 shows the three kinds
of pairs in planar mechanisms. These
pairs reduce the number of the degrees
of freedom. If we create a lower pair
(Figure 411a,b), the degrees of freedom are reduced to 2. Similarly,
if we create a higher pair (Figure
411c), the degrees of freedom are reduced to 1.
Figure 412 Kinematic Pairs in Planar Mechanisms
Therefore, we can write the following equation:
(41)
Where
 F = total degrees of freedom in the mechanism
 n = number of links (including
the frame)
 l = number of lower pairs
(one degree of freedom)
 h = number of higher pairs
(two degrees of freedom)
This equation is also known as Gruebler's equation.
Look at the transom above the door in Figure 413a. The opening and
closing mechanism is shown in Figure 413b. Let's calculate its
degree of freedom.
Figure 413 Transom mechanism
n = 4 (link 1,3,3 and frame 4), l = 4 (at A, B, C, D), h = 0
(42)
Note: D and E function as a same prismatic pair, so they only
count as one lower pair.
Example 2
Calculate the degrees of freedom of the mechanisms shown in Figure 414b.
Figure 414a is an application of the mechanism.
n = 4, l = 4 (at A, B, C, D), h = 0
(43)
Example 3
Calculate the degrees of freedom of the mechanisms shown in Figure 415.
Figure 415 Degrees of freedom calculation
For the mechanism in Figure 415a
n = 6, l = 7, h = 0
(44)
For the mechanism in Figure 415b
n = 4, l = 3, h = 2
(45)
Note: The rotation of the roller does not influence the
relationship of the input and output motion of the mechanism. Hence,
the freedom of the roller will not be considered; It is called a
passive or redundant degree of freedom.
Imagine that the roller is welded to link 2 when counting the degrees
of freedom for the mechanism.
The number of degrees of freedom of a mechanism
is also called the mobility of the device. The
mobility is the number of input parameters (usually pair
variables) that must be independently controlled to bring the device
into a particular position. The Kutzbach criterion,
which is similar to Gruebler's equation,
calculates the mobility.
In order to control a mechanism, the number of independent input
motions must equal the number of degrees of freedom of the mechanism.
For example, the transom in Figure 413a
has a single degree of freedom, so it needs one independent input
motion to open or close the window. That is, you just push or pull rod 3
to operate the window.
To see another example, the mechanism in Figure
415a also has 1 degree of freedom. If an independent input is
applied to link 1 (e.g., a motor is mounted on joint A to drive
link 1), the mechanism will have the a prescribed motion.
Finite transformation is used to describe the motion of a point on
rigid body and the motion of the rigid body itself.
Figure 416 Point on a planar rigid body rotated through an angle
Suppose that a point P on a rigid body goes through a rotation
describing a circular path from P_{1} to
P_{2} around the origin of a coordinate system. We can
describe this motion with a rotation operator
R_{12}:
(46)
where
(47)
Figure 417 Point on a planar rigid body translated through a distance
Suppose that a point P on a rigid body goes through a
translation describing a straight path from P_{1} to
P_{2} with a change of coordinates of (x, y). We can describe this
motion with a translation operator T_{12}:
(48)
where
(49)
Figure 418 Concatenation of finite planar displacements in space
Suppose that a point P on a rigid body goes through a rotation
describing a circular path from P_{1} to
P_{2}' around the origin of a coordinate system, then
a translation describing a straight path from P_{2}' to
P_{2}. We can represent these two steps by
(410)
and
(411)
We can concatenate these motions to get
(412)
where D_{12} is the planar general displacement operator
:
(413)
We have discussed various transformations to describe the
displacements of a point on rigid body. Can these operators be
applied to the displacements of a system of points such as a rigid
body?
We used a 3 x 1 homogeneous column matrix to describe a vector
representing a single point. A beneficial feature of the planar 3 x 3
translational, rotational, and general displacement matrix operators
is that they can easily be programmed on a computer to manipulate a 3
x n matrix of n column vectors representing n points of a rigid body.
Since the distance of each particle of a rigid body from every other
point of the rigid body is constant, the vectors locating each point
of a rigid body must undergo the same transformation when the rigid
body moves and the proper axis, angle, and/or translation is specified
to represent its motion. (Sandor
& Erdman 84). For example, the general planar transformation
for the three points A, B, C on a rigid body can be represented
by
(414)
We can describe a spatial rotation operator for the rotational
transformation of a point about an unit axis u passing through the
origin of the coordinate system. Suppose the rotational angle of the point
about u is ,
the rotation operator will be expressed by
(415)
where
 u_{x}, u_{y}, u_{z} are the othographical
projection of the unit axis u on x, y, and
z axes, respectively.
 s =
sin
 c =
cos
 v = 1 
cos
Suppose that a point P on a rigid body goes through a
translation describing a straight path from P_{1} to
P_{2} with a change of coordinates of (x, y, z), we can describe this
motion with a translation operator T:
(416)
Suppose a point P on a rigid body rotates with an angular
displacement about an unit axis u passing through the origin of
the coordinate system at first, and then followed by a translation
Du along u. This composition of this rotational
transformation and this translational transformation is a screw
motion. Its corresponding matrix operator, the screw
operator, is a concatenation of the translation operator in Equation 47 and the rotation operator in Equation 49.
(417)
For a system of rigid bodies, we can establish a local Cartesian
coordinate system for each rigid body. Transformation matrices are
used to describe the relative motion between rigid bodies.
For example, two rigid bodies in a space each have local coordinate
systems x_{1}y_{1}z_{1} and
x_{2}y_{2}z_{2}. Let point P be
attached to body 2 at location (x_{2}, y_{2},
z_{2}) in body 2's local coordinate system. To find the
location of P with respect to body 1's local coordinate system,
we know that that the point x_{2}y_{2}z_{2}
can be obtained from x_{1}y_{1}z_{1} by
combining translation L_{x1} along the x axis and
rotation z about z
axis. We can derive the transformation matrix as follows:
(418)
If rigid body 1 is fixed as a frame, a
global coordinate system can be created on this body. Therefore, the
above transformation can be used to map the local coordinates of a
point into the global coordinates.
The transformation matrix above is a specific example for two
unconstrained rigid bodies. The transformation matrix depends on the
relative position of the two rigid bodies. If we connect two rigid
bodies with a kinematic constraint, their
degrees of freedom will be decreased. In other words, their relative
motion will be specified in some extent.
Suppose we constrain the two rigid bodies above with a revolute pair as shown in Figure 419. We can
still write the transformation matrix in the same form as Equation 418.
Figure 419 Relative position of points on constrained bodies
The difference is that the L_{x1} is a constant
now, because the revolute pair fixes the origin of coordinate system
x_{2}y_{2}z_{2} with respect to coordinate system
x_{1}y_{1}z_{1}. However, the rotation
z
is still a variable. Therefore, kinematic constraints specify the
transformation matrix to some extent.
DenavitHartenberg notation (Denavit & Hartenberg 55) is
widely used in the transformation of coordinate systems of linkages and robot mechanisms. It can be
used to represent the transformation matrix between links as shown in
the Figure 420.
Figure 420 DenavitHartenberg Notation
In this figure,
The transformation matrix will be T_{(i1)i}
(419)
The above transformation matrix can be denoted as T(a_{i},
_{i}, _{i}, d_{i})
for convenience.
A linkage is composed of several constrained rigid bodies. Like a
mechanism, a linkage should have a frame. The matrix method can be
used to derive the kinematic equations of the linkage. If all the
links form a closed loop, the concatenation of all of the
transformation matrices will be an identity matrix. If the mechanism
has n links, we will have:
T_{12}T_{23}...T_{(n1)n} = I (420)
Complete Table of Contents
 1 Introduction to Mechanisms
 2 Mechanisms and Simple Machines
 3 More on Machines and Mechanisms
 4 Basic Kinematics of Constrained Rigid Bodies
 4.1 Degrees of Freedom of a Rigid Body
 4.1.1Degrees of Freedom of a Rigid Body in a Plane
 4.1.2 Degrees of Freedom of a Rigid Body in Space
 4.2 Kinematic Constraints
 4.2.1 Lower Pairs in Planar Mechanisms
 4.2.2 Lower Pairs in Spatial Mechanisms
 4.3 Constrained Rigid Bodies
 4.4 Degrees of Freedom of Planar Mechanisms
 4.4.1 Gruebler's Equation
 4.2.2 4.4.2 Kutzbach Criterion
 4.5 4.5 Finite Transformation
 4.5.1 Finite Planar Rotational Transformation
 4.5.2 Finite Planar Translational Transformation
 4.5.3 Concatenation of Finite Planar
Displacements
 4.5.4 Planar RigidBody Transformation
 4.5.5 Spatial Rotational Transformation
 4.5.6 Spatial Translational Transformation
 4.5.7 Spatial Translation and Rotation Matrix for
Axis Through the Origin
 4.6 Transformation Matrix Between Rigid Bodies
 4.6.1 Transformation Matrix Between two Arbitray
Rigid Bodies
 4.6.2 Kinematic Constraints Between
Two Rigid Bodies
 4.6.3 DenavitHartenberg Notation
 4.6.4 Application of Transformation Matrices
to Linkages
 5 Planar Linkages
 6 Cams
 7 Gears
 8 Other Mechanisms
 Index
 References
sfinger@ri.cmu.edu
