A Kinematic and Dynamic Analysis of the
American Football Overhead Throwing Motion
by
Anthony Beeman
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
Master of Engineering in Mechanical Engineering
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
December 06, 2015
© Copyright 2015
by
Anthony Beeman
All Rights Reserved
ii
CONTENTS
A Kinematic and Dynamic Analysis of the ........................................................................ i
American Football Overhead Throwing Motion ................................................................ i
LIST OF TABLES ............................................................................................................ iv
IST OF FIGURES.............................................................................................................. v
GLOSSARY ..................................................................................................................... vi
LIST OF SYMBOLS AND ABBREVIATIONS ............................................................ vii
ACKNOWLEDGMENT ................................................................................................ viii
ABSTRACT ..................................................................................................................... ix
1. Introduction and Background ...................................................................................... 1
1.1
Overview of Football Throwing Motion ............................................................ 1
2. Theory and Methodology ............................................................................................ 5
2.1
Denavit-Hartenberg Method .............................................................................. 5
2.2
Planar-Two Bar Mechanism Kinematic Model ................................................. 6
2.3
Dynamic Modeling........................................................................................... 10
2.4
Kinematic Modeling......................................................................................... 11
3. Results........................................................................................................................ 17
3.1
Expanding the DH Kinematic Model ............................................................... 17
4. Discussion and Conclusion ........................................................................................ 18
5. References.................................................................................................................. 19
iii
LIST OF TABLES
Table 1: Planar Two Bar Mechanism DH Parameters...................................................... 7
Table 2: Abaqus Steps .................................................................................................... 15
Table 3: Abaqus Boundary Conditions........................................................................... 16
Table 4: Body Segment Length, Mass, and Izz for Abaqus ModelError! Bookmark not
defined.
iv
IST OF FIGURES
Figure 1: Throwing Motion Phase Diagram [e] ................................................................ 1
Figure 2: NFL Quarterback in the Pre Pass Triangle Phase [f] ........................................ 2
Figure 3: Primary Muscles Used During The Force Producing Movement Phase [g] [h] ... 3
Figure 4: Primary Muscles Used During The Extension Phase [g] [i] ............................... 3
Figure 5: NFL Quarterback in the Follow Through Phase [g] .......................................... 4
Figure 6: DH Parameters ................................................................................................. 5
Figure 7: Kinematic Model Points of Interest ................................................................. 6
Figure 8: Planar 2 Bar Mechanism .................................................................................. 7
Figure 9: Joint Angles When Lead Foot First Contacts Ground ................................... 12
Figure 10: Joint Angles at Max External Rotation ........................................................ 13
Figure 11: Joint Angles at Release ................................................................................ 14
v
GLOSSARY
Term
Definition
vi
LIST OF SYMBOLS AND ABBREVIATIONS
Symbol/Abbreviation
Definition
vii
ACKNOWLEDGMENT
TBD
viii
ABSTRACT
The purpose of throwing a football is to generate a high velocity pass that maintains
precision. This is achieved with proper biomechanics that can be broken up into four
phases.
These phases include the pre pass triangle phase, the force producing
movement phase, the extension phase, and the follow through phase.
Over the past
several years there have been increased interest in overhead throwing mechanics.
Overhead throwing places extremely high stresses on the shoulder joint. These high
stresses are repeated many times and can be lead to a wide range of overuse injuries.
The purpose of this paper is to utilize research conducted in Reference (b) to model
proper throwing mechanics using DH parameterization. Next, A kinematic model shall
be constructed in Abaqus
ix
1. Introduction and Background
Over the past several years there have been increased interest in overhead throwing
mechanics. Overhead throwing places extremely high stresses on the shoulder joint.
These high stresses are repeated many times and can be lead to a wide range of overuse
injuries. Therefore, a better understanding of dynamics of the football pass can provide
sports medical professionals useful information in prevention, treatment, and
rehabilitation of football-related football injures. Additionally, a better understanding of
throwing mechanics can lead to improved performance by the athlete.
Today's
quarterbacks are not trained in proper throwing mechanics. As a result, poor throwing
mechanics are repeated throughout their high school, collegiate, and possible
professional careers.
1.1
Overview of Football Throwing Motion
The purpose of throwing a football is to generate a high velocity pass that maintains
precision. This is achieved with proper biomechanics that can be broken up into four
phases.
These phases include the pre pass triangle phase, the force producing
movement phase, the extension phase, and the follow through phase. Each of the four
phases are illustrated in Figure 1 below.
6"
45 ͦ
90 ͦ
Phase 1
Phase 2
Phase 3
Figure 1: Throwing Motion Phase Diagram [e]
1
Phase 4
Phase 1- Pre Pass Triangle Phase
The kinetic chain in the arm starts in the pre pass triangle position. The triangle position
provides a power position to launch the football and reduces the tendency to internally
rotate the arm and naturally aligns the arm to the force producing movement phase.
Figure 2 illustrates an NLF quarterback in the pre pass triangle phase.
Figure 2: NFL Quarterback in the Pre Pass Triangle Phase [f]
Phase 2- Force Producing Movement Phase
The next position in the kinetic chain during the throw is the force producing movement
phase.
This is achieved by the infrasprinatus and terses minor muscles externally
rotating the arm back into an approximate 90 degree angle in order to elongate the
suprasprinatus and subscaturits rotator cup muscles. This prepares the deltoid muscles to
propel the elbow through the extension phase. Figure 3 illustrates an NFL quarterback in
the force producing movement phase and shows the rotator cuff muscles being utilized
during this phase of the throwing motion. Improper biomechanics in the force producing
movement phase can result in increased strain on the rotator cup which over time can
lead to injury.
2
Figure 3: Primary Muscles Used During The Force Producing Movement Phase [g] [h]
Phase 3- Extension Phase
The next phase in the kinetic chain results in the elbow moving above and in front of the
shoulders. This phase is responsible for consistent power and accuracy on the throw.
The deltoid muscle is used to force the elbow above and ahead of the shoulders until it
reaches the "zero position". The zero position is defined as the location where there is
zero strain on the rotator cuff muscles.
This is achieved by placing the elbow
approximately 6 inches above the shoulder and 45 degrees above the transverse plane
which load the tricep muscle in preparation of the follow through phase. Improper
biomechanics in the extension phase can result in additional strain on the rotator cup
which over time can lead to injury.
Figure 4: Primary Muscles Used During The Extension Phase [g] [i]
3
Phase 4- Follow Through Phase
During the follow through phase the triceps transfers energy from the elbow, wrist,
fingertips, and finally to the ball. The follow through phase is responsible for the release
of the football which will determine the final trajectory and velocity of the ball.
Figure 5: NFL Quarterback in the Follow Through Phase [g]
4
2. Theory and Methodology
In order to establish a systematic method for biomechanically modeling the overhead
throwing motion it is necessary to establish a convention for representing links and
joints. The human arm can be represented as a sequence of rigid body links which are
connected by the shoulder and elbow joints.
2.1
Denavit-Hartenberg Method
In 1955 Denavit and Hartenberg developed a systematic method, DH method, for
describing the position of orientation of successive links. The DH method is based upon
characterizing the configuration of link i with respect to link i-1 with the use of four
parameters which include d, θ, α, and a. Figure 6 illustrates two successive links and the
corresponding DH parameters.
Figure 6: DH Parameters
From Figure 6 the DH parameter di is defined as the distance from Xi-1 to Xi measured
along Zi-1, θi is the angle from Xi-1 to Xi measured about Zi-1, αi is the angle from Zi-1 to
Zi measured about Xi, and ai is the distance from Zi-1 to Zi measured along Xi.
Assigning successive reference frames using the DH method can be completed by
following three simple rules. Rule 1 states that that Zi-1 must be the axis of actuation of
joint i. This will result in the axis of revolution for a revolute joint or an axis of
translation for a prismatic joint. Next, Rule 2 states that the axis Xi must be set such that
it is perpendicular to and intersects Zi-1. Finally, Rule 3 states that the direction of Yi
must derived from Xi and Zi in accordance with the right hand rule.
5
With rules established for defining successive coordinate systems using the DH
parameters can be completed with the following homogeneous transformation matrix
𝑇𝑖𝑖−1
𝑐𝜃
𝑠𝜃
=[
0
0
−𝑐𝛼 ∗ 𝑠𝜃
𝑐𝛼 ∗ 𝑐𝜃
𝑠𝛼
0
𝑠𝛼 ∗ 𝑠𝜃
−𝑠𝛼 ∗ 𝑐𝜃
𝑐𝛼
0
𝑎 ∗ 𝑐𝜃
𝑎 ∗ 𝑠𝜃
]
𝑑
1
The homogenous transformation matrix in
2.2
Planar-Two Bar Mechanism Kinematic Model
In order to analyze the overhead throwing motion of a quarterback a kinematic models
were developed to determine the position of the wrist as a function of time. Figure XX
illustrates the 3 points of interest. The first point of interest occurs when the leading foot
first contacts the ground (force producing movement phase). Next, the arm is cocked
back and the external max rotation of the body occurs, Next the arm is propelled
forward to the ball release.
tf
ti=0
t1>ti
Figure 7: Kinematic Model Points of Interest
6
A planar two bar mechanism was selected to model the kinematic systems to simplify
the problem by constraining the motion to two degrees of freedom. The planar two bar
mechanism is composed of two rigid bodies, the upper arm and fore arm, which are
connected to a ground. Each link is connected with revolute joints and is free to rotate
about the z axis.
Insert 2 Bar Planar Mechanism Picture HERE
Figure 8: Planar 2 Bar Mechanism
DH parameterization method shall be used to model the over head throwing motion
represented as a planar two bar mechanism. The DH transformation matrix includes
rotations and translations and is a function of four parameters which relate the coordinate
frames i and i-1
Link
α
d
θ
a
[rad] [in] [rad] [in]
1
0
0
𝜃1
L1
2
0
0
𝜃2
L2
Table 1: Planar Two Bar Mechanism DH Parameters
With the DH parameters provided in Table X a kinematic model can be created to
represent the planar two bar mechanism. Using the homogeneous transformation matrix,
Equation X, the elbow (frame 1) can be related to the shoulder (frame 0) with the
following expression:
𝑐𝜃1
𝑠𝜃
𝑇10 = [ 1
0
0
−𝑐𝛼1 ∗ 𝑠𝜃1
𝑐𝛼1 ∗ 𝑐𝜃1
𝑠𝛼1
0
𝑠𝛼1 ∗ 𝑠𝜃1
−𝑠𝛼1 ∗ 𝑐𝜃1
𝑐𝛼1
0
which reduces to:
7
L1 ∗ 𝑐𝜃1
L1 ∗ 𝑠𝜃1
]
0
1
𝑐𝜃1
𝑠𝜃
𝑇10 = [ 1
0
0
−𝑠𝜃1
𝑐𝜃1
0
0
0 L1 ∗ 𝑐𝜃1
0 L1 ∗ 𝑠𝜃1
]
1
0
0
1
similarly, the wrist (frame 2) can be related to the elbow (frame 1) with the following
expression:
𝑐𝜃2
𝑠𝜃
𝑇21 = [ 2
0
0
−𝑠𝜃2
𝑐𝜃2
0
0
0 L2 ∗ 𝑐𝜃2
0 L2 ∗ 𝑠𝜃2
]
1
0
0
1
Multiplying Equations X and X results in a global transformation matrix that locates
frame 2 with respect to frame 0. It should be noted that the XYZ location of the wrist
(frame 2) with respect to the shoulder (frame 0) is indicated in the 4th column of the 𝑇20
transformation matrix
𝑇20 = 𝑇10 𝑇21
𝑐(𝜃1 + 𝜃2 )
𝑠(𝜃 + 𝜃2 )
𝑇20 = [ 1
0
0
−𝑠(𝜃1 + 𝜃2 )
𝑐(𝜃1 + 𝜃2 )
0
0
0
0
1
0
L2 ∗ 𝑐(𝜃1 + 𝜃2 ) + L1 ∗ 𝑐𝜃1
L2 ∗ 𝑠(𝜃1 + 𝜃2 ) + L1 ∗ 𝑠𝜃1
]
0
1
The wrist (frame 2) can be located relative to the shoulder joint (frame 1) with the
following expression:
L2 ∗ 𝑐𝜃2
L ∗ 𝑠𝜃2
𝑟21 = [ 2
]
0
1
Similarly, the elbow (frame 1) can be located relative to the shoulder joint (frame 0) with
the following expression.
L1 ∗ 𝑐𝜃1
L ∗ 𝑠𝜃1
𝑟10 = [ 1
]
0
1
8
It should be noted that equations X and X are simply the 4th column of Equations X and
X.
In order to relate the wrist (frame 2) with respect to the shoulder (frame 0) coordinate
system one can used the derived rotation matrix (equation X).
L2 ∗ 𝑐(𝜃1 + 𝜃2 ) + L1 ∗ 𝑐𝜃1
L2 ∗ 𝑠(𝜃1 + 𝜃2 ) + L1 ∗ 𝑠𝜃1 ]
0
0 1
𝑤𝑟2 = 𝑇1 𝑟2 = [
0
1
The Jacobian Matrix, represents the differential relationship between the joint
displacements and the resulting wrist motion.
δ 𝑥𝑒 (𝜃1 , 𝜃2 ) δ 𝑥𝑒 (𝜃1 , 𝜃2 )
δ𝜃1
δ𝜃2
𝐽𝑤 =
δ 𝑦𝑒 (𝜃1 , 𝜃2 ) δ 𝑦𝑒 (𝜃1 , 𝜃2 )
[
]
δ𝜃1
δ𝜃2
𝐽𝑤 = [
−𝐿1 𝑠𝜃1 − 𝐿2 𝑠(𝜃1 + 𝜃2 )
𝐿1 𝑐𝜃1 + 𝐿2 𝑐(𝜃1 + 𝜃2 )
−𝐿2 𝑠(𝜃1 + 𝜃2 )
]
𝐿2 𝑐(𝜃1 + 𝜃2 )
Next, one can take the differential of the wrist Jacobian matrix, 𝐽𝑤 , in order to obtain the
first derivative Jacobian matrix.
𝐽𝑤̇ = [
−𝐿1 𝑐𝜃1
−𝐿1 𝑠𝜃1
−𝐿2 𝑐(𝜃1 + 𝜃2 )
𝜃1̇
][
]
−𝐿2 𝑠(𝜃1 + 𝜃2 ) 𝜃1̇ + 𝜃2̇
With the wrist velocity and acceleration Jacobian matrix known the equations of motion
with respect to the shoulder (frame 0) can be expressed as:
𝑋̈𝑤 = 𝐽𝑤 𝑞̈ + 𝐽𝑤̇ 𝑞̇
9
[
𝜃̈1
𝑋̈𝑤
−𝐿 𝑠𝜃 − 𝐿2 𝑠(𝜃1 + 𝜃2 ) −𝐿2 𝑠(𝜃1 + 𝜃2 )
]=[ 1 1
][
]
̈
𝐿1 𝑐𝜃1 + 𝐿2 𝑐(𝜃1 + 𝜃2 )
𝐿2 𝑐(𝜃1 + 𝜃2 ) 𝜃1̈ + 𝜃2̈ )
𝑌𝑤
−𝐿 𝑐𝜃
+[ 1 1
−𝐿1 𝑠𝜃1
2.3
2
−𝐿2 𝑐(𝜃1 + 𝜃2 )
𝜃1̇
][
]
−𝐿2 𝑠(𝜃1 + 𝜃2 ) (𝜃̇ 1 + 𝜃2̇ )2
Dynamic Modeling
10
2.4
Kinematic Modeling
Abaqus, was used to create the Finite Element Model and perform the kinematic
analysis. The Finite Element Model was constructed utilizing a series of hinge and beam
connector elements.
Inertial mass properties have been included in the model by
separating the beam elements into two equal segments. Additionally, display bodies
were included in order to provide a physical representation of the human arm as it
transitions from each of the four phases of the throwing movement. The series of
Abaqus connector representing the throwing arm are illustrated in Figure X. Stationary
parts such as the head, left arm, and lower body were modeled for information but
motion was restricted for this analysis.
Since the models under analysis in this paper pertain to the elbow and shoulder; these
key angles were extracted by using Kinovea’s angle measurement tool and are defined in
Figures X-X.
11
y2
w
x2
y0
y1
θ2
x1
x0
s
e
Figure 9: Joint Angles When Lead Foot First Contacts Ground
θ1= 0 ͦ
θ2= 90 ͦ
𝑡𝑖 =0 (Video Position 10:25)
12
Figure 10: Joint Angles at Max External Rotation
θ1= 152 ͦ
θ2= 100 ͦ
𝑡1 = 0.50 (position=11:00)
13
Figure 11: Joint Angles at Release
θ1= 156 ͦ
θ2= 70 ͦ
𝑡𝑓 = 0.75 (position = 12:00)
14
Abaqus Steps
The Abaqus Finite Element analysis is comprised of four unique steps in order to
simulate the kinematics of throwing a football. These steps included foot contact,
maximum external rotation, ball release, and follow through step.
Step 0
Variable
Time (sec)
Step 1
Step 2
Step 3
Step 4
Maximum
Initial
Foot
External
Follow
Step
Contact
Rotation
Ball Release
Through
0
1.0
0.5
0.25
0.25
0
0.10
0.05
0.05
0.05
Step 3
Step 4
Table 2: Abaqus Steps
Abaqus Loads
Step 0
Load
Gravity [in/s2]
Step 1
Step 2
Maximum
Initial
Foot
External
Follow
Step
Contact
Rotation
Ball Release
Through
-386.089
-386.089
-386.089
-386.089
-386.089
Step 3
Step 4
Table 3: Abaqus Load
Abaqus Boundary Conditions
Step 0
Step 1
Boundary
Condition
Ground
Shoulder Joint
Step 2
Maximum
Initial
Foot
External
Follow
Step
Contact
Rotation
Ball Release
Through
Encastre
Encastre
Encastre
Encastre
Encastre
ur1=0
ur1=0
ur1=5.306
ur1=0.279
ur1=1.676
ur1=0
ur1=1.571
ur1=-6.283
ur1=-2.094
ur1=-4.887
(Δω1)
Elbow Joint
(Δω2)
15
Table 4: Abaqus Boundary Conditions
u1=u2=u3=ur1=ur2=ur3=0
Abaqus Inertial Mass Properties
16
3. Results
TBD
3.1
Expanding the DH Kinematic Model
Talk about adding DOF Using DH Method
17
4. Discussion and Conclusion
TBD
18
5. References
[1]
Chapman, Arthur E. Biomechanical Analysis of Fundamental Human
Movements. 2008. Human Kinetics Publishers.
[2]
Rash, Gregory. Shapiro, Robert. A Three-Dimensional Dynamic Analysis of the
Quarterback's Throwing Motion in American Football. 1995. Human Kinetics
Publishers, Inc. Journal of Applied Biomechanics 1995, Volume 11, Pg. 443459.
[3]
Dillman, Charles. Fleisig, Glenn. Andrews, James. Biomechanics of Pitching
with Emphasis upon Shoulder Kinematics. August 1993. Journal of Orthopedic
& Sports Physical Therapy. Volume 18, Number 2.
[4]
Elliot, Bruce. Takahashi, Kotaro. Marshall, Robert. Internal Rotation of the
Upper Arm: The Missing Link in the Kinematic Chain.
[5]
Verduzco, Mario. The biomechanics of the quarterback position: a kinematic
analysis and integrative approach. 1991. San Jose State University
[6]
Chase. Chris. (April 26, 2011) The best No. 6 selection ever? Choosing best
picks by draft order. Retrieved from
http://sports.yahoo.com/nfl/blog/shutdown_corner/post/The-best-No-6-selectionever-Choosing-best-pic?urn=nfl-wp1241
[7]
Wells, Brad. (May 25, 20011) Peyton Manning's Neck Injury Likely Did
Happen During 2010 Season. Retrieved from
http://www.stampedeblue.com/2011/5/25/2188594/peyton-mannings-neckinjury-likely-did-happen-during-2010-season
[8]
(August 12, 2013) Medline Plus Medical Encyclopedia: Rotator Cup Muscles.
Retrieved from
https://www.nlm.nih.gov/medlineplus/ency/imagepages/19622.html
[9]
(Summer 2001) Functional Electrical Stimulation News Letter. Retrieved from
http://www.salisburyfes.com/sept2001.htm
[10]
Herman, Irving. Physics of the Human Body. 2007. Springer.
[11]
Drawing Skeleton. Retrieved from http://mostviewsvideo.com/drawingskeleton.html/humanskeleton1
19