Session 2793
Development of Solid Models and Multimedia
Presentations of Kinematic Pairs
Scott Michael Wharton, Dr. Yesh P. Singh
The University of Texas at San Antonio, San Antonio, Texas
Abstract
Understanding of complex 3D motion of kinematic pairs with 1 to 5 degrees of freedom is a
difficult task to grasp for students enrolled in introductory course in kinematics. In this paper,
the development of solid models and multimedia presentations of kinematic pairs is presented.
Through the use of commercially available computer programs, Solidworks 99 and Photoworks,
detailed three-dimensional models of kinematic pairs were developed. Form-closed and forceclosed variants of each kinematic pair were modeled for a total of 24 models. The models were
then animated to show the relative motion between the two bodies that make up the kinematic
pair. Each animation was processed into a Windows audio/video interleave (AVI) file, allowing
the viewing of the animation either on the Internet or in the classroom through the use of
multimedia screens. A summary of all kinematic pairs is provided in Table 1, it will serve as a
useful handout for students in reviewing the classification, degrees of freedom, name, and
symbol of kinematic pairs. Table 2 presents captured screen shots from the AVI movies for each
of the 12 kinematic pairs, both form-closed and force-closed are shown.
Introduction
For students, the visual understanding of complex three-dimensional motion is a difficult task to
master. In the study of biomechanics, it is highly important to understand the relative motion
between two bodies. To replace the knee or elbow joint on the human body, an understanding on
how the relative motion between the bones in the knee and elbow joint must be investigated.
Connections that allow constrained relative motion are called kinematic joints, also referred to as
kinematic pairs 1. Essentially, a kinematic pair consists of two rigid bodies that are kept in
contact such that a constrained motion can occur between the two bodies. Kinematics is “the
study of motion of mechanisms and methods of creating them” 2.
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Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition
Copyright  2001, American Society for Engineering Education
A kinematic pair can permit 1 to 5 degrees of freedom of motion between two contacting bodies.
Degrees of freedom can be defined as the number of independent parameters needed to specify
the relative positions of the two bodies in contact 3. An unconstrained rigid body has six degrees
of freedom, three translations and three rotations about the three orthogonal axes.
Kinematic pairs are divided into five different classes based on the degrees of freedom that the
kinematic joint possesses 1. A class I pair has one degree of freedom and the class II pair has
two degrees of freedom. The classification stops at class V because beyond five degrees of
freedom the rigid bodies no longer have a contact constrained motion between them.
Reuleaux introduced another classification of kinematic pairs based on type of contact between
the two bodies 1. In this classification system, kinematic pairs are placed in one of the two
groups, lower kinematic pairs and higher kinematic pairs. For a kinematic joint to be classified
as a lower kinematic pair, the two rigid bodies have either area or surface contact. Higher
kinematic pairs have either line or point contact.
Form-closed and force-closed terminology helps to define the appearance of kinematic pairs.
Form-closed kinematic joints use the surfaces of one body to constrain the motion of the other
body in the pair. No other bodies or forces are necessary to constrain the motion of the moving
body. Force-closed kinematic joints require an additional force to help constrain the motion of
the moving body. The additional force may include gravity or springs which help to keep the
two bodies in contact with one another.
For force-closed kinematic pairs, a simple algebraic equation can be used to determine the
degrees of freedom that the kinematic joint possesses. The algebraic equation states that when
the number of point contacts, nc, between the kinematic pairs is subtracted from six times the
difference between the number of bodies, nL, and one, the difference is the number of degrees of
freedom that the pair possesses. The number six comes from the fact that an unconstrained rigid
body has six degrees of freedom. The force-closed equation is shown below.
DFspatial = 6(n L − 1) − nC
(1)
Since there are only two rigid bodies in a kinematic pair, Equation (1) simplifies to the
following.
DFspatial = 6 − nC
(2)
To help identify kinematic pairs, the kinematic joints have been given names and symbols 1. For
the simple case of a one-degree of freedom kinematic joint that permits rotation only, the name
revolute is given. The corresponding symbol for the revolute joint is the capital letter R.
Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition
Copyright  2001, American Society for Engineering Education
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Without a complete understanding of kinematic pairs and their motions, a student cannot begin
to understand the more complex motions of biomedical joints. For students having trouble
visualizing complex three-dimensional motion of kinematic pairs by sketches and descriptions
given in textbooks, the use of multimedia animations can be of significant help in understanding
the relative motion.
Three-Dimensional Modeling
Through the use of SolidWorks 99, twelve different kinematic pairs were modeled 4. For each of
the kinematic pairs, force-closed and form-closed models were developed. A total of 24 solid
models are created.
Each kinematic pair has two rigid bodies, a moving body and a fixed body. Each rigid body was
modeled using Solidworks 99. Each solid model was detailed to allow for clearances and fillets.
Color-coding of the kinematic pair components was used to help distinguish the fixed and
moving component. The moving component of each model was colored green and the fixed
rigid body was colored gray.
Upon completion of the two rigid bodies, an assembly drawing was created using the moving
and fixed models. The fixed model was first imported into the assembly drawing and
constrained to the assembly drawing’s coordinate system. This constraint forced the rigid body
to be fixed in the assembly space of the drawing. The moving rigid body was then imported into
the assembly drawing.
Through the use of Solidwork’s mating reference system, the moving body was mated with the
fixed body to complete the assembly of the kinematic pairs. The mating process required that
the moving body still be allowed all degrees of freedom that would be exhibited by the kinematic
pair. For example, in the form-closed revolute joint, the z-axis of the fixed body was mated with
the z-axis of the moving body. This mating allowed the moving body to exhibit two degrees of
freedom, rotation about the z-axis and translation along the z-axis. Since the revolute joint has
only one degree of freedom a second mating reference was needed to complete the mating
process. The second mating reference was the point origin of the fixed body must be coincident
with the point origin of the moving body. The two mating references together constrained the
revolute assembly to exhibit only the rotation about the z-axis of the assembly. A similar
process of mating was carried out for each of the remaining 23 assembly models.
The coordinate system for each body was added to the assembly drawing to help with
visualization of translation(s) and/or rotation(s). The global axes were fixed in space and these
correspond to the fixed rigid body. The local coordinate system was then constrained with the
moving body to reproduce all the degrees of movement that the moving body produced. If the
moving body rotated about its x-axis the local axis would also rotate about its x-axis. Each
coordinate system was color coded for ease of visualization. The global axes were colored blue
and the local axes were colored in red.
Model Rendering
Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition
Copyright  2001, American Society for Engineering Education
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Through the use of Photoworks 99, each model was rendered with shadow effects to help define
the three-dimensional nature of each model 5. Photoworks 99 is an add-on program that works
within the Solidworks program. Photoworks 99 allows the user to define the surface appearance
of the models. Aesthetically pleasing surfaces were chosen not to detract from the model. Each
rendered model was then saved in a JPG picture format. The JPG picture format is a digital
graphic image that is suitable to be used on the World Wide Web and many other computer
programs.
Kinematic Pairs
The following 12 figures show the kinematic joints that were modeled using the Solidworks 99
solid modeler. Each figure was rendered using Photoworks 99 and shown in the JPG format.
The axis systems have been removed from the pictures for better viewing of the kinematic pair
models.
A revolute joint model is shown in figure 1 given below. Figure 1(a) shows the form-closed
version of the revolute joint and Fig. 1(b) shows the force-closed revolute joint. Counting the
number of contact points of the force-closed model shows there are five contact points. Three of
the contact points are between the pyramid that is recessed in the fixed rigid body and the sphere
of the moving body. The other two contact points are between the spheres of the moving body
and the flat surface of the fixed rigid body. Using equation (2), it can be shown that the forceclosed revolute joint has one degree of freedom. Knee and elbow joints of the human body are
sometimes modeled as a revolute joint for simplicity.
Fig. 1 Revolute Joint, R
DOF = 1
Figure 2 shows the form-closed and force-closed models of the prism joint. The prism joint can
only translate along one axis and therefore has one degree of freedom. The capital letter P is the
given symbol for the prism joint.
Fig. 2 Prism Joint, P
DOF = 1
Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition
Copyright  2001, American Society for Engineering Education
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A helical joint is shown in figure 3. It is different from the revolute and prism joint that it
appears to have two degrees of freedom, one rotation and one translation. The reason the helical
joint is considered to have one degree of freedom is that the rotation and translation are coupled;
one cannot exist with out the other. For a given amount of rotation, the moving body will
translate a fixed amount. A nut and bolt is a good example of a helical joint. The capital letter H
is the symbol used for the helical joint.
Fig. 3 Helical Joint, H
DOF = 1
The first of the class II kinematic pairs is the slotted spheric joint shown in figure 4. The slotted
spheric joint has two degrees of freedom. The degrees of freedom consist of two rotations. The
symbol used for the slotted spheric joint is SL.
Fig. 4 Slotted Spheric Joint, SL
DOF = 2
A cylinder joint is shown in Figure 5. It has two degrees of freedom, one rotation and one
translation. Unlike the helical joint, the rotation and translation in the cylinder joint are
independent of one another. For a kinematic joint to be called a cylinder joint, both the rotation
and translation must occur upon the same axis. The capital letter C is used as the symbol for the
cylinder joint.
Fig. 5 Cylinder Joint, C
DOF = 2
Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition
Copyright  2001, American Society for Engineering Education
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Figure 6 illustrates a cam joint. Like the cylinder joint, the cam joint has two degrees of
freedom, one rotation, and one translation. Unlike the cylinder joint, the cam joint’s rotation and
translation occur upon different axis. The symbol for the cam joint is Ca.
Fig. 6 Cam Joint, Ca
DOF = 2
The spheric pair joint shown in Figure 7 is a class III kinematic pair. The spheric pair joint has
three rotations, one about each of its principle axis directions. For biomedical joints, the hip
joint is modeled as a spheric pair. The spheric pair is also known as a ball and socket joint. The
capital letter S is used as the symbol for the spheric pair joint.
Fig. 7 Spheric Pair Joint, S
DOF = 3
The sphere slotted cylinder pair has two rotations and one translation. Returning to equation 2,
one can see that the sphere slotted cylinder has 3 degrees of freedom. The contact points are one
between the flat plane of the fixed rigid body and the sphere of the moving body and two contact
points exist between the v-groove in the fixed body and the sphere of the moving body. The
symbol SS is given to the sphere slotted cylinder joint. The sphere slotted cylinder is shown in
figure 8.
Fig. 8 Sphere Slotted Cylinder Joint, SS
DOF = 3
Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition
Copyright  2001, American Society for Engineering Education
Page 6.381.6
The two translations and the one rotation make the plane pair joint a class III kinematic pair.
The symbol for the plane pair joint is PL. Using the Reuleaux classification system, the plane
pair joint is classified as a lower kinematic pair. Viewing of the form-closed model shows that
the surfaces of the moving body are in contact with the surface of the fixed body. Figure 9
illustrates the plane pair joint.
Fig. 9 Plane Pair Joint, PL
DOF = 3
The first kinematic pair in the class IV classification is the sphere groove joint. Similar to the
sphere slotted cylinder joint, the sphere groove permits 3 rotations and one translation. Sg is the
symbol for the sphere groove joint. The sphere groove joint is shown in figure 10.
Fig. 10 Sphere Groove Joint, Sg
DOF = 4
The cylinder plane pair is pictured in Figure 11. Using equation (2) and counting the two point
contacts between the flat plane of the fixed rigid body and the two spheres of the moving body, it
can be shown that the cylinder plane has four degrees of freedom. Viewing the form-closed
model, it can be shown that the cylinder plane pair is a higher kinematic pair based on the
Reuleaux classification system. There exists only a line contact between the cylinder of the
moving body and the surface of the fixed rigid body. The symbol given to the cylinder plane
pair is Cp.
Fig. 11 Cylinder Plane Pair Joint, CP
DOF = 4
Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition
Copyright  2001, American Society for Engineering Education
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A class V kinematic pair is the sphere-plane joint. An example of a sphere plane joint is a ball
on a flat table. The ball can perform three rotations, one about each of its principle axis
directions and two translations along the surface of the table. The sphere plane joint symbol is
SP. Figure 12 shows the form-closed and force-closed models of a sphere-plane joint.
Fig. 12 Sphere Plane Joint, SP
DOF = 5
Animation of Kinematic Pairs
To animate each model, animating capabilities of Solidworks 99 software were used 6. Each
degree of freedom for every model was simulated using the software and saved as a Windows
AVI movie. An AVI movie is a combination of still pictures with small delays between each
picture to simulate animation.
For a joint with multiple degrees of freedom, each degree of freedom was animated separately to
help identify each movement. During the filming of each degree of freedom, further constraints
were added to each assembly to insure that only one degree of freedom would be produced for
that segment of film. The movies of each individual motion were then combined to make one
movie that showed each degree of freedom in turn. The program used to combine the individual
AVI files was ImageForge by CursurArts 7.
Kinematic names and symbols were then added to the AVI files using ImageForge. Additional
information about each kinematic pair, number of degrees of freedom, form-closed or forceclosed and the magnitude and type of each movement the kinematic pair exhibited, were added
into the AVI movies.
Summary
Through the use of commercially available computer programs, solid models and animation of
kinematic pairs were developed and the complex motions of kinematic pairs are presented. The
Windows AVI files are created that can be placed on the World Wide Web for viewing or
brought to the classroom through the use of multimedia presentation equipment. Table 1
provides a summary of all kinematic pairs. It will serve a useful handout to students to review
the classification, degrees of freedom, name, and symbol of many kinematic pairs. Table 2
shows screen captured pictures from the multimedia AVI movie files. Students, who are having
trouble visualizing complex three-dimensional motion of kinematic pairs by description provided
in textbooks, will find the animations significantly useful.
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Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition
Copyright  2001, American Society for Engineering Education
Bibliography
1. Soni, A.H., 1974, Mechanism Synthesis and Analysis, Robert E. Krieger Publiching Company, Florida.
2. Erdman, Arthur G., and Sandor, George N., 1997, Mechanism Design, Analysis and Synthesis, Volume 1, Third
Edition, Prentice Hall, New Jersey.
3. Wilson, Charles E., and Sadler, J. Peter, 1993, Kinematics and Dynamics of Machinery, Second Edition, Harper
Collins College Publishers, New York.
4. Solidworks, 1999, Solidworks 99 User’s Guide, Solidworks Corporation, Massachusetts.
5. Lightworks Design Limited, 1999, Photoworks Help, Lightworks Design Limited., United Kingdom.
6. Immersive Design, Inc., 1999, Solidworks Animator Help Topics, Immersive Design, Inc. Massachesett.
7. ImageForge, 2000, ImageForge Help Topics, CursorArts Company, Oregon.
SCOTT MICHAEL WHARTON
Scott Wharton received his master’s degree in Mechanical Engineering for the University of Texas at San Antonio
in the December 2000. Before returning to graduate school, Scott worked for Exponent, Inc. in Houston as a
laboratory technician and with C&S Metal Fabricators in Houston as the factory supervisor. Scott received his B.S.
in Engineering Technology from Texas A&M University in May 1995.
Dr. YESH P. SINGH
Yesh P. Singh is an Associate Professor of Mechanical Engineering at the University of Texas at San Antonio
(UTSA). He also serves as Chair of ME Graduate Program and Director of the Engineering Machine Shop. He
joined Mechanical Engineering at UTSA in September 1985 after 23 years of broad-based hands-on Mechanical
Design experience in industries in USA, formal USSR, and India. He was elected to ASME Fellow grade in 1992.
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Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition
Copyright  2001, American Society for Engineering Education
KINEMATIC PAIRS
Table 1
Class
Degrees
of
Freedom
Name and Symbol
Diagram
FormClosed
Revolute – R
I
1
Prism – P
R
P
H
Helical – H
ForceClosed
R
P
H
FormClosed
Slotted Spheric - SL
II
2
Cylinder – C
SL
C
Ca
Cam - Ca
ForceClosed
SL
C
Ca
Spheric Pair – S
III
3
Sphere Slotted
Cylinder - SS
FormClosed
S
SS
PL
ForceClosed
Plane Pair - PL
S
SS
PL
FormClosed
Sphere Groove - Sg
IV
4
Sg
CP
Cylinder Plane Pair CP
ForceClosed
Sg
CP
FormClosed
V
5
Sphere Plane - SP
SP
SP
Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition
Copyright  2001, American Society for Engineering Education
Page 6.381.10
ForceClosed
KINEMATIC PAIRS MULTIMEDIA MOVIE SCREEN CAPTURES
Table 2
Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition
Copyright  2001, American Society for Engineering Education
Page 6.381.11
KINEMATIC PAIRS MULTIMEDIA MOVIE SCREEN CAPTURES
Table 2 Continued
Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition
Copyright  2001, American Society for Engineering Education
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