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
CONTENTS ..................................................................................................................... iii
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
GLOSSARY ..................................................................................................................... vi
LIST OF SYMBOLS AND ABBREVIATIONS ........................................................... viii
ACKNOWLEDGMENT .................................................................................................. ix
ABSTRACT ...................................................................................................................... x
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
Kinematic Modeling......................................................................................... 11
3. Results & Discussion ................................................................................................. 18
3.1
Abaqus FEM Kinematic Model Results .......................................................... 18
3.2
Expanding the DH Kinematic Model ............................................................... 24
4. Conclusions................................................................................................................ 27
5. References.................................................................................................................. 28
6. Appendix A: Body Mass Segment Calculations ...................................................... 30
7. Appendix B: Joint Acceleration Calculations........................................................... 34
8. Appendix C: Kinematic Analysis ............................................................................. 37
iii
LIST OF TABLES
Table 1: Planar Two Bar Mechanism DH Parameters...................................................... 8
Table 2: Abaqus Steps .................................................................................................... 15
Table 3: FEM Inertial/Mass Properties (As calculated in Appendix A) ........................ 16
Table 4: FEM Input Loads .............................................................................................. 16
Table 5: FEM Boundary Conditions (As calculated in Appendix B) ............................. 17
Table 6: Five Degrees of Freedom Upper Limb Model ................................................. 25
Table 7: Five Degrees of Freedom Upper Limb Model DH Parameters ........................ 25
Table 8: Body Segment Mass in grams [12] ..................................................................... 31
Table 9: Body Segment Mass in lbm .............................................................................. 32
Table 10: Body Segment Mass of Average NFL quaterback ......................................... 32
Table 11: Abaqus FEM Body Segment Masses ............................................................. 32
Table 12: Joint Angles & Times at Various Points of Interest ....................................... 35
Table 13: Change in Joint Angles at Various Points of Interest ..................................... 35
Table 14: Joint Velocities at Various Points of Interest ................................................. 36
Table 15: Joint Velocities at Various Points of Interest ................................................. 36
iv
LIST OF FIGURES
Figure 1: Throwing Motion Phase Diagram [5]................................................................ 1
Figure 2: NFL Quarterback in the Pre Pass Triangle Phase [6] ........................................ 2
Figure 3: Primary Muscles Used During The Force Producing Movement Phase [7] [8] ... 3
Figure 4: Primary Muscles Used During The Extension Phase [7] [9] .............................. 3
Figure 5: NFL Quarterback in the Follow Through Phase [7] .......................................... 4
Figure 6: DH Parameters ................................................................................................. 5
Figure 7: Kinematic Model Points of Interest ................................................................. 6
Figure 8: Planar 2 Bar Mechanism .................................................................................. 7
Figure 9: Abaqus Finite Model ...................................................................................... 11
Figure 10: Abaqus Connector Representation ............................................................... 12
Figure 11: Joint Angles When Lead Foot First Contacts Ground ................................. 12
Figure 12: Joint Angles at Max External Rotation ........................................................ 13
Figure 13: Joint Angles at Ball Release......................................................................... 14
Figure 14: Joint Angles When Lead Foot First Contacts Ground ................................. 18
Figure 15: Joint Angles at Max External Rotation ........................................................ 19
Figure 16: Joint Angles at Ball Release......................................................................... 19
Figure 17: Joint Angles at Follow Through................................................................... 20
Figure 17: Wrist X position as a function of Time ........................................................ 21
Figure 18: Five Degree of Freedom Upper Limb Model .............................................. 24
v
GLOSSARY
Term
Definition
Bicep
Large flexor muscle of the front of the upper arm
Deltoid Muscles
Muscle forming the rounded contour of the shoulder.
DH Parameters
The Denavit-Hartenberg parameters are the four parameters
associated with a particular convention for attaching reference
frames to the links of spatial kinematic chains.
Elbow Joint
The juncture of the long bones in the middle portion of the upper
extremity. The bone of the arm (humerus) meets both the ulna (the
inner bone of the forearm) and radius (the outer bone of the
forearm) to form a hinge joint at the elbow.
Humerus
Long bone in the arm or forelimb that runs from the shoulder to
the elbow.
Infraspinatus
One of the four muscles of the rotator cuff, the main function of
the infraspinatus is to externally rotate the humerus and stabilize
the shoulder joint.
Jacobian Matrix
Matrix of all first-order partial derivatives of a vector valued
function.
Kinematics
Branch of classical mechanics which describes the motion of
points, bodies (objects), and systems of bodies (groups of objects)
without consideration of the causes of motion. Kinematics as a
field of study is often referred to as the "geometry of motion".
Prismatic Joint
A prismatic joint provides a linear sliding movement between two
bodies, and is often called a slider, as in the slider-crank linkage.
Revolute Joint
A revolute joint is a one-degree-of-freedom kinematic pair used in
mechanisms. Revolute joints provide single-axis rotation function
used in many places such as door hinges, folding mechanisms, and
other uni-axial rotation devices.
Rotator Cuff
A capsule with fused tendons that supports the arm at the shoulder
joint and is often subject to athletic injury.
vi
Shoulder Joint
The joint connecting an upper limb or forelimb to the body. It is a
ball-and-socket joint in which the head of the humerus fits into the
socket of the scapula.
Terses Minor
The teres minor is a narrow, elongated muscle of the rotator cuff.
Tricep
The muscle that extends (straightens) the forearm.
vii
LIST OF SYMBOLS AND ABBREVIATIONS
Symbol/Abbreviation
Definition
𝑇𝑖𝑖−1
DH parameter homogeneous transformation matrix that
relates the position of joint i to the joint i-1 reference frame.
α [rad]
d[in]
Angle from Zi-1 to Zi measured about Xi
Distance from Xi-1 to Xi measured along Zi-1
θ [rad]
Angle from Xi-1 to Xi measured about Zi-1
a [in]
Distance from Zi-1 to Zi measured along Xi
𝜃1 [rad]
Rotation angle of the shoulder joint about the Z1 axis
𝜃2 [rad]
Rotation angle of the elbow joint about the Z2 axis
L1 [in]
Length of the upper arm
L2 [in]
Length of the forearm
𝑇10
Transformation matrix that relates the position of joint 1
(elbow) to the joint 0 reference frame (shoulder).
𝑇21
Transformation matrix that relates the position of joint 2
(wrist) to the joint 1 reference frame (elbow).
𝑇20
Transformation matrix that relates the position of joint 2
(wrist) to the joint 0 reference frame (shoulder).
𝑟21
Position of joint 2 (wrist) relative to the joint 1 reference
frame (elbow)
𝑟10
Position of joint 1 (elbow) relative to the joint 0 reference
frame (shoulder)
0
𝑤𝑟2
Position of joint 2 (wrist) relative to the joint 0 reference
frame (shoulder)
𝐽𝑤
Jacobian Matrix; Matrix of all first-order partial derivatives
of a vector valued function.
𝐽𝑤̇
Matrix of all first-order partial derivatives of the Jacobian
matrix.
𝑋̇
Forward velocity of the planar two bar mechanism
𝑋̈𝑤
Forward acceleration of the planar two bar mechanism
viii
ACKNOWLEDGMENT
TBD
ix
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 (2) to model
proper throwing mechanics using DH parameterization. Next, A kinematic model shall
be constructed in Abaqus
x
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 [5]
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 [6]
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 infraspinatus and terses minor muscles externally rotating
the arm back into an approximate 90 degree angle in order to elongate the suprasprinatus
and subscaturits rotator cuff 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 cuff which over time can lead to injury.
2
Figure 3: Primary Muscles Used During The Force Producing Movement Phase [7] [8]
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 cuff
which over time can lead to injury.
Figure 4: Primary Muscles Used During The Extension Phase [7] [9]
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 [7]
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
2.2
𝑐𝜃
𝑠𝜃
=[
0
0
−𝑐𝛼 ∗ 𝑠𝜃
𝑐𝛼 ∗ 𝑐𝜃
𝑠𝛼
0
𝑠𝛼 ∗ 𝑠𝜃
−𝑠𝛼 ∗ 𝑐𝜃
𝑐𝛼
0
𝑎 ∗ 𝑐𝜃
𝑎 ∗ 𝑠𝜃
] Eq. (1)
𝑑
1
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 7
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, then the arm is propelled forward
until 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.
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
7
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 1 a kinematic model can be created to
represent the planar two bar mechanism. Using the homogeneous transformation matrix,
Equation 2, 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
L1 ∗ 𝑐𝜃1
L1 ∗ 𝑠𝜃1
]
0
1
Eq. (2)
which reduces to:
𝑐𝜃1
𝑠𝜃
𝑇10 = [ 1
0
0
−𝑠𝜃1
𝑐𝜃1
0
0
0 L1 ∗ 𝑐𝜃1
0 L1 ∗ 𝑠𝜃1
] Eq. (3)
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
0
1
0
L2 ∗ 𝑐𝜃2
L2 ∗ 𝑠𝜃2
]
0
1
Eq. (4)
Multiplying Equations 3 and 4 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.
8
𝑇20 = 𝑇10 𝑇21
𝑐(𝜃1 + 𝜃2 ) −𝑠(𝜃1 + 𝜃2 )
𝑠(𝜃 + 𝜃2 ) 𝑐(𝜃1 + 𝜃2 )
𝑇20 = [ 1
0
0
0
0
Eq. (5)
0 L2 ∗ 𝑐(𝜃1 + 𝜃2 ) + L1 ∗ 𝑐𝜃1
0 L2 ∗ 𝑠(𝜃1 + 𝜃2 ) + L1 ∗ 𝑠𝜃1
] Eq. (6)
1
0
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
Eq. (7)
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
Eq. (8)
It should be noted that equations 7 and 8 are simply the 4th column of Equations 4 and 3.
In order to relate the wrist (frame 2) with respect to the shoulder (frame 0) coordinate
system one can use the derived rotation matrix (equation 9).
L2 ∗ 𝑐(𝜃1 + 𝜃2 ) + L1 ∗ 𝑐𝜃1
L2 ∗ 𝑠(𝜃1 + 𝜃2 ) + L1 ∗ 𝑠𝜃1 ]
0
0 1
𝑤𝑟2 = 𝑇1 𝑟2 = [
0
1
Eq. (9)
The Jacobian matrix, represents the differential relationship between the joint
displacements and the resulting wrist motion. For a planar two bar mechanism the
Jacobian matrix can be expressed as:
9
δ 𝑥𝑒 (𝜃1 ,𝜃2 )
δ 𝑥𝑒 (𝜃1 ,𝜃2 )
δ𝜃1
δ𝜃2
δ 𝑦𝑒 (𝜃1 ,𝜃2 )
δ 𝑦𝑒 (𝜃1 ,𝜃2 )
δ𝜃1
δ𝜃2
𝐽𝑤 = [
]
Eq. (10)
Next, one can take the partial differential of the wrist position to form the Jacobian
matrix, 𝐽𝑤 . The Jacobian matrix for the two bar planar mechanism is noted below:
−𝐿 𝑠𝜃 − 𝐿2 𝑠(𝜃1 + 𝜃2 ) −𝐿2 𝑠(𝜃1 + 𝜃2 )
𝐽𝑤 = [ 1 1
]
𝐿1 𝑐𝜃1 + 𝐿2 𝑐(𝜃1 + 𝜃2 )
𝐿2 𝑐(𝜃1 + 𝜃2 )
Eq. (11)
The forward velocity of the planar two bar mechanism can then be represented with the
following equation:
𝜃̇
̇
[𝑋] = 𝐽𝑤 [ 1 ]
𝑌̇
𝜃̇2
Eq. (12)
Next, the partial derivative of the Jacobian matrix can be obtained as shown below:
𝐽̇
𝐽𝑤̇ = [ 𝑤11
̇
𝐽𝑤21
̇
𝐽𝑤12
]
̇
𝐽𝑤22
Eq. (13)
where:
̇
𝐽𝑤11
= (−𝐿1 𝑐(𝜃1 ) − 𝐿2 𝑐(𝜃1 + 𝜃2 ))𝜃̇1 − 𝐿2 𝑐(𝜃1 + 𝜃2 )𝜃̇2
̇
𝐽𝑤12
= −𝐿2 𝑐(𝜃1 + 𝜃2 )𝜃̇1 − 𝐿2 𝑐(𝜃1 + 𝜃2 )𝜃̇2
Eq. (15)
̇
𝐽𝑤21
= (−𝐿1 𝑠(𝜃1 ) − 𝐿2 𝑠(𝜃1 + 𝜃2 ))𝜃̇1 − 𝐿2 𝑠(𝜃1 + 𝜃2 )𝜃̇2
̇
𝐽𝑤22
= −𝐿2 𝑠(𝜃1 + 𝜃2 )𝜃̇1 − 𝐿2 𝑠(𝜃1 + 𝜃2 )𝜃̇2
Eq. (14)
Eq. (16)
Eq. (17)
With the wrist velocity and acceleration Jacobian matrix known the equations of motion
with respect to the shoulder (frame 0) can be expressed as:
𝑋̇ = 𝐽𝑤 𝑞̈ + 𝐽𝑤̇ 𝑞̇
10
Eq. (18)
[
2.3
𝑋̈𝑤
𝜃̈1
−𝐿 𝑠𝜃 − 𝐿2 𝑠(𝜃1 + 𝜃2 ) −𝐿2 𝑠(𝜃1 + 𝜃2 )
−𝐿 𝑐𝜃
][
]=[ 1 1
]+[ 1 1
̈𝑌𝑤
𝐿1 𝑐𝜃1 + 𝐿2 𝑐(𝜃1 + 𝜃2 )
𝐿2 𝑐(𝜃1 + 𝜃2 ) 𝜃1̈ + 𝜃2̈ )
−𝐿1 𝑠𝜃1
2
𝐿2 𝑐(𝜃1 + 𝜃2 )
𝜃1̇
][
]
𝐿2 𝑠(𝜃1 + 𝜃2 ) (𝜃̇ 1 + 𝜃2̇ )2
Eq. (19)
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 9. Stationary
parts such as the head, left arm, and lower body were modeled for information but
motion was restricted for this analysis.
Figure 9: Abaqus Finite Model
11
Insert FEA Picture HERE
Figure 10: Abaqus Connector Representation
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
Figure 11 through Figure 13.
y2
w
x2
y0
y1
θ2
x1
x0
s
e
Figure 11: Joint Angles When Lead Foot First Contacts Ground
Shoulder Angle θ1= 0 ͦ
Elbow Angle θ2= 90 ͦ
Time 𝑡𝑖 =0 (Video Position 10:25)
12
The first point of interest occurs when the leading foot first contacts the ground (force
producing movement phase). From the Kinovea's angle measurement tool it can be seen
that elbow is rotated 90 degrees (θ2) with respect to the shoulder coordinate system. The
force producing movement phase will be the beginning of the finite element analysis.
Therefore, Figure 11 illustrates the position of the arm at when time equals zero.
Figure 12: Joint Angles at Max External Rotation
Shoulder Angle θ1= 152 ͦ
Elbow Angle θ2= 100 ͦ
Time 𝑡1 = 0.50 (position=11:00)
13
The next phase in the kinetic chain, maximum external rotation, results in the elbow
moving above and in front of the shoulders. This phase is responsible for consistent
power and accuracy on the throw.
From the Kinovea's angle measurement tool, as
shown in Figure 12, it can be seen that shoulder has rotated 152 degrees (θ1) with
respect to its initial position and the elbow has rotated 100 degrees (θ2) with respect to
the shoulder coordinate system. The point at which maximum external rotation occurs
shall be the second step in the finite element analysis.
Figure 13: Joint Angles at Ball Release
Shoulder Angle θ1= 156 ͦ
Elbow Angle θ2= 70 ͦ
Time 𝑡𝑓 = 0.75 (position = 12:00)
14
The next point of interest occurs during ball release.
From the Kinovea's angle
measurement tool, as shown in Figure 13, it can be seen that shoulder has rotated 156
degrees (θ1) with respect to its initial position and the elbow has rotated 70 degrees (θ 2)
with respect to the shoulder coordinate system. The point at which maximum external
rotation occurs shall be the third step in the finite element analysis. With key angles and
times known for the various points of interest, the Finite element model can now be
constructed in a series of steps.
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.
Each step was
modeled as static general step with non-linear geometry turned on. It is important to
note that non-linear geometry was turned on in the FEM because large displacements
take place.
This ensures the FEM accurately determines the final position of the
elements after large displacements occurs. Table 2 illustrates each of the four steps
created in the FEM and the time for each step.
Step 0
Variable
Step 1
Step 2
Step 3
Step 4
Maximum
Initial
Foot
External
Follow
Step
Contact
Rotation
Ball Release
Through
Time (sec)
0
1.0
0.5
0.25
0.25
Increment size
0
0.10
0.05
0.05
0.05
Number of
-
10
10
5
5
Output Database
(ODB) frames
Table 2: Abaqus Steps
The time between the initial step and foot contact is arbitrary as it simply rotates the arm
from the zero position, θ1= θ2=0 the foot contact position. Alternatively, the time
between foot contact and MER, MER and ball release, and ball release and follow
15
through represent the times obtained from the Kinovea software and were determined
with equation 20.
𝑡𝑛 = 𝑡𝑛 − 𝑡𝑛−1
Eq. (20)
The increment set as a constant value as illustrated in Table 2. This was completed in
order to ensure that several Output Database (ODB) frames were available in the Abaqus
results file. Each frame represents a snapshot in time when going from one step to the
next. With four unique steps clearly defined the input loads could then be added to the
analysis.
FEM Input Loads
The FEM input loads used in this analysis consisted of adding point masses at the CG of
the upper arm (L1) and fore arm (L2); as shown in Table 3. It should be noted that the
forearm mass includes the mass of the football, forearm, and hand.
Link
Mass [lbm]
Upper Arm (L1)
5.9
Fore Arm (L2)
11.63
Table 3: FEM Inertial/Mass Properties (As calculated in Appendix A)
Additionally, a gravity load was added which acts in the -y direction. The gravity load,
as shown in Table 4, is constant throughout the analysis and is consistent with the in-seclbm unit convention established in the Abaqus FEM.
Step 0
Load
Gravity [in/s2]
Step 1
Step 2
Step 3
Step 4
Maximum
Initial
Foot
External
Follow
Step
Contact
Rotation
Ball Release
Through
-386.089
-386.089
-386.089
-386.089
-386.089
Table 4: FEM Input Loads
16
Abaqus Boundary Conditions
The Abaqus FEM consisted of three unique boundary condition. The first boundary
condition titled ground, is an Encastre boundary condition that restrains the two bar
planar mechanism in all six degrees of freedom (u1=u2=u3=ur1=ur2=ur3=0). Next, a
velocity connector boundary condition is applied to the shoulder joint. It should be
noted that Abaqus defines the hinge axis as a local x coordinate system. Therefore, in
order to articulate the shoulder the boundary condition is applied about the local x, UR1,
axis.
Similar to the shoulder joint, the Elbow joint's axis of rotation is defined as a local x
coordinate system. Therefore, the elbow is articulated by applying a rotation about the
local x, UR1, axis.
Table 5 provides the various boundary conditions applied for each step. For each step
the shoulder and elbow velocity is a function of the change in angle and the time delta
between each analysis step, as shown in Equation 21.
𝜔𝑛 =
Step 0
𝜃𝑛 −𝜃𝑛−1
𝑡𝑛 −𝑡𝑛−1
Step 1
Boundary
Condition
Ground
Shoulder Joint
Eq. (21)
Step 2
Step 3
Step 4
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) [rad/s]
Elbow Joint
(Δω2) [rad/s]
Table 5: FEM Boundary Conditions (As calculated in Appendix B)
17
3. Results & Discussion
3.1
Abaqus FEM Kinematic Model Results
The first point of interest occurs when the leading foot first contacts the ground (force
producing movement phase). From the Abaqus FEM the rotational displacement can be
output with the UR keyword. Figure 14 illustrates that the elbow has been rotated 1.571
radians (θ2) with respect to the shoulder coordinate system.
Figure 14: Joint Angles When Lead Foot First Contacts Ground
The next phase in the kinetic chain, maximum external rotation, results in the elbow
moving above and in front of the shoulders. This phase is responsible for consistent
power and accuracy on the throw.
From the Abaqus UR field output, as shown in
Figure 15, it can be seen that shoulder has rotated 2.653 radians (θ1) with respect to its
initial position and the elbow has rotated 0 radians (θ2). Step 2 of the FEM, ball release,
occurs at t=1.75.
18
Figure 15: Joint Angles at Max External Rotation
The next point of interest occurs during ball release. the Abaqus UR field output, as
shown in Figure 16, it can be seen that shoulder has rotated 2.723 radians (θ1) with
respect to its initial position and the elbow has rotated 0 radians (θ2). Step 3 of the FEM,
Maximum External Rotation, occurs at t=1.5.
Figure 16: Joint Angles at Ball Release
19
The final point of interest occurs after the arm follow through phase. From the Abaqus
UR field output, as shown in Figure 17, it can be seen that shoulder has rotated 3.316
radians (θ1) with respect to its initial position and the elbow has rotated 0 radians (θ2).
Step 4 of the FEM, follow through step, occurs at t=2.0.
Figure 17: Joint Angles at Follow Through
20
Figure 18: Wrist X position as a function of Time
Figure 19: Wrist Y position as a function of Time
21
Figure 20: Shoulder & Elbow X Force as Function of Time
Figure 21: Shoulder & Elbow Y Force as Function of Time
22
Figure 22: Shoulder & Elbow Torque as Function of Time
23
3.2
Expanding the DH Kinematic Model
The DH parameter method is a suitable model for addressing the motion of human
kinematics that are arranged in series. This is accomplished by creating a series of
rotational or prismatic joints. Sections 2.2-2.4 illustrated that DH parameterization can
accurately represent a planar two bar mechanism with two degrees of freedom. This
section shall discuss the methodology for expanding the DH parameters to a 5 degree of
freedom model.
Figure 23 illustrates a five degree of freedom upper limb model. This model consists of
two joints which include the shoulder and elbow joint. The shoulder joint is represented
with a series of three prismatic joints which control the flexion/extension,
abduction/adduction, and internal/external rotations of the shoulder. Additionally, the
elbow is represented with a series of two prismatic joints which control the
flexion/extension of the fore arm and the pronation/supination angle of the wrist and
forearm.
Figure 23: Five Degree of Freedom Upper Limb Model
24
A summary of each joints physical location and functionality are provided in Table 6
below.
Joint
Location
Functionality
0
Shoulder Center
Ground location of the throwing arm
1
Shoulder
Flexion/Extension angle of the upper arm
2
Shoulder
Abduction/adduction of the upper arm
3
Shoulder
Internal/External rotation of the upper arm
4
Elbow
Flexion/Extension of the fore arm
5
Elbow
Pronation/Supination angle of the wrist and forearm
6
Wrist
Wrist location
Table 6: Five Degrees of Freedom Upper Limb Model
The DH parameters for the five degrees of freedom upper limb model is provided in
Table 7.
Link
α
d
θ
a
[rad] [in] [rad] [in]
1
0
0
𝜃1
0
2
-1.57
0
𝜃2
0
3
1.57
L1
𝜃3
0
4
-1.57
0
𝜃4
0
5
1.57
L2
𝜃5
0
6
0
0
𝜃6
0
Table 7: Five Degrees of Freedom Upper Limb Model DH Parameters
With the DH parameters known one can determine the wrist location, Equation 22, with
the use of the homogeneous transformation matrix; Equation 1.
𝑇𝑖𝑖−1
𝑐𝜃
𝑠𝜃
=[
0
0
−𝑐𝛼 ∗ 𝑠𝜃
𝑐𝛼 ∗ 𝑐𝜃
𝑠𝛼
0
𝑠𝛼 ∗ 𝑠𝜃
−𝑠𝛼 ∗ 𝑐𝜃
𝑐𝛼
0
25
𝑎 ∗ 𝑐𝜃
𝑎 ∗ 𝑠𝜃
] Eq. (1)
𝑑
1
𝑇60 = 𝑇10 𝑇21 𝑇32 𝑇43 𝑇54 𝑇65
Eq. (22)
Next, one can obtain the velocity and acceleration Jacobian matrix in order to obtain the
equations of motion of the wrist with respect to the shoulder (frame 0) as shown in the
equation below:
𝑋̇ = 𝐽𝑤 𝑞̈ + 𝐽𝑤̇ 𝑞̇
26
Eq. (18)
4. Conclusions
TBD
27
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 Cuff 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
28
[12]
Drillis, Rudolfs. Contini, Renato. Bluestein, Maurice. Body Segment Parameters:
A Survey of Measurement Techniques.
29
6. Appendix A: Body Mass Segment Calculations
30
Purpose: The purpose of the following calculation is to determine the total mass of the
forearm and upper arm body segments. All segment masses have been obtained from
Reference 12 and were then converted to lbm to match the unit convention used in the
Abaqus Finite Element Model.
Body Segment
Head
Upper Trunk
Lower Trunk
Upper Arm
Forearm
Hand
Thigh
Shank
Foot
Both upper
extremities
Both lower
extremities
Total Weight
Absolute Weight [grams]
Cadaver 1 Cadaver 2 Average
4555
3747
4151
23055
17779
20417
6553
4868
5710.5
2070
1448
1759
1160
795.5
977.75
540
383.6
461.8
7165
5887
6526
2800
2247.5 2523.75
1170
985.2
1077.6
7540
5254
6397
22270
78878
18239.4
61634.2
20254.7
70256.1
Table 8: Body Segment Mass in grams [12]
Body Segment
Head
Upper Trunk
Lower Trunk
Upper Arm
Forearm
Hand
Thigh
Shank
Foot
Both upper
extremities
Both lower
extremities
Total Weight
Absolute Weight [lbm]
Cadaver 1 Cadaver 2 Average
10.0
8.3
9.2
50.8
39.2
45.0
14.4
10.7
12.6
4.6
3.2
3.9
2.6
1.8
2.2
1.2
0.8
1.0
15.8
13.0
14.4
6.2
5.0
5.6
2.6
2.2
2.4
16.6
11.6
14.1
49.1
173.9
40.2
135.9
44.7
154.9
31
Table 9: Body Segment Mass in lbm
Next, the body segment mass has been scaled to obtain a total body weight of
approximately 225 lbm. This is the average quarterback mass in the NFL.
Body Segment
Head
Upper Trunk
Lower Trunk
Upper Arm
Forearm
Hand
Thigh
Shank
Foot
Both upper
extremities
Both lower
extremities
Total Weight
Absolute Weight [lbm]
Cadaver 1 Cadaver 2 Average
13.1
10.7
11.9
66.1
51.0
58.5
18.8
14.0
16.4
5.9
4.1
5.0
3.3
2.3
2.8
1.5
1.1
1.3
20.5
16.9
18.7
8.0
6.4
7.2
3.4
2.8
3.1
21.6
15.1
18.3
63.8
226.1
52.3
176.6
58.1
201.4
Table 10: Body Segment Mass of Average NFL quaterback
Noted below is the total forearm and upper arm mass that will be used in the Abaqus
finite element analysis. It should be noted that the fore arm mass shall include the mass
of the football, hand, and forearm.
Body Segment
Football mass
Forearm
Hand
Upper Arm
Abaqus Forearm Mass
Abaqus Upper Arm
Mass
Weight
[lbm]
0.93
3.3
1.5
5.9
11.63
5.9
Table 11: Abaqus FEM Body Segment Masses
32
33
7. Appendix B: Joint Acceleration Calculations
34
Purpose: The purpose of the following calculation is to determine the joint velocities
for each Abaqus step. All angles and times were extracted from the Kinovea software.
0
90
1
Follow Through
0
0
0
Ball Release
θ1
θ2
t
Maximum External
Rotation
Foot Contact
initial
Parameters
Shoulder Angle [Deg]
Elbow Angle [Deg]
Time [s]
152
100
1.5
156
70
1.75
180
0
2
Table 12: Joint Angles & Times at Various Points of Interest
0
90
1
152
-180
0.5
Follow Through
0
0
0
Ball Release
Δθ1
Δθ2
t
Maximum External
Rotation
Foot Contact
initial
Parameters
Change in Shoulder Angle
[Deg]
Change in Elbow Angle [Deg]
Time [s]
4
-30
0.25
Table 13: Change in Joint Angles at Various Points of Interest
35
24
-70
0.25
0
90
1
Follow Through
0
0
0
Ball Release
ω1
ω2
t
Maximum External
Rotation
Foot Contact
initial
Parameters
Shoulder Velocity [Deg/s]
Elbow Velocity [Deg/s]
Time [s]
304.0
-360.0
0.5
16
-120
0.25
96
-280
0.25
Table 14: Joint Velocities at Various Points of Interest
Noted below is the joint angle velocities for each Abaqus analysis step. It should be
noted that the units have been converted to radians/second in order to match that FEM
units.
0.000
1.571
1.000
5.306
-6.283
0.500
0.279
-2.094
0.250
Table 15: Joint Velocities at Various Points of Interest
36
Follow Through
0.000
0.000
0.000
Ball Release
ω1
ω2
t
Maximum External
Rotation
Foot Contact
initial
Parameters0
Shoulder Velocity [Rad/s]
Elbow Velocity [Rad/s]
Time [s]
1.676
-4.887
0.250
8. Appendix C: Kinematic Analysis
37