Maya Katherine Nielan - Quantifying Exertion for American Football Linemen via Force, Acceleration, and Heart Rate Measurements

Quantifying Exertion for American Football Linemen
 via Force, Acceleration, and Heart Rate
 Measurements
 by
 Maya Katherine Nielan
 B.S. Engineering as Recommended by the Department of Mechanical
 Engineering & Electrical Engineering and Computer Science,
 Massachusetts Institute of Technology (2022)
 Submitted to the Department of Electrical Engineering and Computer
 Science
 in partial fulfillment of the requirements for the degree of
 Master of Engineering in Electrical Engineering and Computer Science
 at the
 MASSACHUSETTS INSTITUTE OF TECHNOLOGY
 May 2022
 © Massachusetts Institute of Technology 2022. All rights reserved.

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 Department of Electrical Engineering and Computer Science
 May 6, 2022
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 Anette (Peko) Hosoi
 Neil and Jane Pappalardo Professor of Mechanical Engineering
 Thesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 Katrina LaCurts
 Chair, Master of Engineering Thesis Committee

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Quantifying Exertion for American Football Linemen via
 Force, Acceleration, and Heart Rate Measurements
 by
 Maya Katherine Nielan

 Submitted to the Department of Electrical Engineering and Computer Science
 on May 6, 2022, in partial fulfillment of the
 requirements for the degree of
 Master of Engineering in Electrical Engineering and Computer Science

Abstract
Understanding exertion during exercise helps athletes prevent injuries and train at
an optimal level. Currently, there exist metrics to determine exertion levels that
are specific to individual activities that are mostly dynamic in nature. American
football linemen spend most of their energy maintaining static loads; thus, they are
in need of a new exertion metric. To design this metric, acceleration, force, and
heart rate data is recorded over different weight lifting, running, and football-specific
activities. From this data, a dimensionless external load value is calculated as =
 ( )2 + ( * ( ))2 and an internal load or exertion value is calculated as
 1 ∫︀ 
 * 0
 = 0 ( ) − . These external and internal load values are compared
 ∫︀

within the football specific activity experiments and across all experiments of different
activities. The relationship between these values is represented through this power
fit = 795.5 * 0.1708
 − 197.8, suggesting that the relative change in external load
gives rise to a proportional relative change in the body’s exertion levels.

Thesis Supervisor: Anette (Peko) Hosoi
Title: Neil and Jane Pappalardo Professor of Mechanical Engineering

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Acknowledgments
Thank you to Dr. Anette (Peko) Hosoi for the guidance and positivity while provok-
ing thoughtful discussions in every meeting. In times when I was unsure about the
direction of the project and worried about finishing, I left every meeting with you
excited and inspired about the project. Though I am no longer a student athlete, I
will take the learnings from this project into my future athletic endeavors.
 Thank you to Dr. Sarah Fay for contributing to the model inspiration with Peko
and for answering even the smallest of questions. It was amazing having two female
mentors interested in engineering and athletics to guide me through this project.
 Thank you Anantha for the excitement in the project and early guidance. While
this project may have shifted focus, your support was still meaningful. Go 9ers!
 Thank you to Dr. Hughey and the rest of the 2.671 staff for providing me the
sensors to collect data with and working through the initial plan for data collection.
 Thank you AJ Jurko for helping collect data, making sense of parts of my analysis,
listening to how "cool" all my findings were, and for the endless love and support.
 Thank you to my friends, especially to Eva Anderson and Sarah Moseson, for
convincing me to do the MEng in the first place, for staying an extra semester with
me, for being the best teammates and friends, and for holding me together during
one of my toughest semesters at MIT. Thanks to Emily for reminding me that college
is short and that it is important to take a break from studying.
 Finally, thank you to my family for providing love and support all the way from
California. To my parents – you provided the foundation for my interest in science
and engineering to develop and always believed I could attend MIT. To my brother –
you kicked off and supported my love for sports, being the best coach, fan and sibling.
I am looking forward to joining you all back in the Bay Area in September.

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Contents

1 Introduction 13

2 Related Work 15
 2.1 General Understanding of Muscle Fatigue . . . . . . . . . . . . . . . . 15
 2.2 Energy Production during Short Bursts of Intense Activity . . . . . . 17
 2.3 Load Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
 2.3.1 External Load Monitoring . . . . . . . . . . . . . . . . . . . . 18
 2.3.2 Internal Load Monitoring . . . . . . . . . . . . . . . . . . . . 19
 2.4 Existing Athlete Load Metrics . . . . . . . . . . . . . . . . . . . . . . 20
 2.4.1 Player Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
 2.4.2 Critical Power . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
 2.4.3 Wingate Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
 2.4.4 One Rep Max . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
 2.4.5 VO2 Max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
 2.4.6 Strava Relative Effort . . . . . . . . . . . . . . . . . . . . . . . 24
 2.5 Existing Loading Models . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Football Background Information 27
 3.1 Lineman Free Body Diagram . . . . . . . . . . . . . . . . . . . . . . . 27
 3.2 Relevant Time Scales in Football . . . . . . . . . . . . . . . . . . . . 28

4 Improved Load Metric Model 31

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5 Methods 33
 5.1 Running Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
 5.2 Football Specific Experiment . . . . . . . . . . . . . . . . . . . . . . . 34
 5.2.1 Simulated Football Stance . . . . . . . . . . . . . . . . . . . . 35
 5.2.2 Football Specific Time Scale Data . . . . . . . . . . . . . . . . 35
 5.3 Strength Training Data Collection . . . . . . . . . . . . . . . . . . . . 36

6 Results and Discussion 39
 6.1 Single Trial Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
 6.1.1 Running . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
 6.1.2 Football Specific . . . . . . . . . . . . . . . . . . . . . . . . . 40
 6.1.3 Strength Training . . . . . . . . . . . . . . . . . . . . . . . . . 41
 6.2 Football Specific Work:Rest Ratio Analysis . . . . . . . . . . . . . . . 42
 6.3 Comparing Loads of Different Activities . . . . . . . . . . . . . . . . 45
 6.4 Football Relevance and Next Steps . . . . . . . . . . . . . . . . . . . 47

7 Conclusion 51

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List of Figures

 2-1 Neuromuscular Fatigue Effects (modified from Zahir et al. [28]) . . . 16
 2-2 Critical power profile taken from cpsinmotion.com [1] where the dashed
 line represents critical power and the blue line represents the anaerobic
 work capacity. As time grows larger, the amount of power that can be
 sustained approaches the critical power. . . . . . . . . . . . . . . . . . 22

 3-1 Football Lineman FBD . . . . . . . . . . . . . . . . . . . . . . . . . . 27
 3-2 Relevant Football Time Scales and Predicted Heart Rate Response . 30

 5-1 Running Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . 34
 5-2 Simulated Football Static Loading Stance . . . . . . . . . . . . . . . . 35
 5-3 Strength Training Movement Data Collection with Barbell Squat, Mod-
 ified Knee Push-up, and Elbow Plank . . . . . . . . . . . . . . . . . . 37

 6-1 50m Run Time Series Data . . . . . . . . . . . . . . . . . . . . . . . . 39
 6-2 200m Run Time Series Data . . . . . . . . . . . . . . . . . . . . . . . 40
 6-3 Football Play with 3W:1R Ratio Time Series . . . . . . . . . . . . . . 40
 6-4 Football Drive Time Series . . . . . . . . . . . . . . . . . . . . . . . . 41
 6-5 Squat Time Series Data . . . . . . . . . . . . . . . . . . . . . . . . . 41
 6-6 Push-up Time Series Data . . . . . . . . . . . . . . . . . . . . . . . . 41
 6-7 Plank Time Series Data . . . . . . . . . . . . . . . . . . . . . . . . . 42
 6-8 Average Pushing Force without Normalization . . . . . . . . . . . . . 43
 6-9 Average Pushing Force Normalizing to Athlete 1 . . . . . . . . . . . . 43
 6-10 Scatter plot of Different Work:Rest Ratios . . . . . . . . . . . . . . . 44

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6-11 Interpolated Map of Predicted Work:Rest Ratios . . . . . . . . . . . . 45
6-12 Activity Data Internal Load vs External Load . . . . . . . . . . . . . 46
6-13 Log Fit as Compared to a Power Fit for All Activity Data . . . . . . 46
6-14 Exertion Comparison of Two Activities with Equivalent External Load-
 ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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List of Tables

 2.1 Common Load Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . 21

 5.1 Play level Work:Rest Ratios . . . . . . . . . . . . . . . . . . . . . . . 36

 6.1 Statistics for Log Fit vs Power Fit for Activity Data . . . . . . . . . . 47

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Chapter 1

Introduction

Estimating an athlete’s fatigue level is helpful to prevent injuries and to allow coaches
and athletes to make educated decisions about activity load during training and
competition. For sports where the main mode of exertion is dynamic movement and
changing direction (ie: soccer, basketball, tennis), load is usually calculated using
differences in acceleration. However, for sports where the main mode of exertion is
maintaining a static or pseudo-static load, via holding or moving more than one’s
own body weight, (ie: football, wrestling, weightlifting) another way of measuring
exertion is needed.
 Football linemen provide excellent candidates in need of load estimation in their
sport. They frequently undergo tackles in which they are holding the weight and
receiving the force of their opponent while they use the stabilizing ability of their
muscles to hold their position and stay in the same location. They run and accelerate
relatively little compared to their teammates like receivers and quarterbacks; thus,
their main mode exertion comes from maintaining a static or pseudo-static load.
 This thesis seeks to understand the relative fatigue levels of an athlete during
different types of activity across the movement spectrum, from dynamic efforts to
static efforts.

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Chapter 2

Related Work

2.1 General Understanding of Muscle Fatigue

Producing force with muscle involves a sequence of events from cortical excitation
to motor unit activation to excitation–contraction coupling. These events ultimately
lead to muscle activation. Muscle fatigue is defined as a decrease in maximal force
or power production in response to contractile activity [27]. According to Wan et al.,
muscle fatigue "can originate at different levels of the motor pathway, which is usually
divided into central and peripheral components. Peripheral fatigue is produced by
changes at the neuromuscular junction. Central fatigue originates at the central
nervous system (CNS) which is responsible for exciting neurons up to the muscle
motor unit" [27].
 Differences at any level in the pathway can reduce generated force and thus lead
to increased muscle fatigue. The changes in the nervous, ion, vascular, and energy
systems are summarized below.

 • Nervous System: Central neurotransmitters, especially 5-HT, DA and NA
 are needed to begin excitation from the CNS. Exercise induced fatigue causes
 changes in these levels which can lead to decrease motor neuron firing. In
 addition to the molecules listed above, there exists even more that are involved
 in the CNS, sympathetic nervous system, endocrine system, and innate immune

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Figure 2-1: Neuromuscular Fatigue Effects (modified from Zahir et al. [28])

 system that are impacted due to bodily stress from exercise induced fatigue.
 Some specific reactants include cortisol, catecholamine, IL-6 and HSPs.

• Ion Balance: Ca+ at correct levels is needed for cross bridge cycling, the
 process where muscle filaments to slide past each other and contract. Fatigue
 can disrupt these levels.

• Vascular System: Blood flow brings O2 for aerobic ATP production and
 removes other excess by-products explained in the energy system section. It
 should be noted that oxygen critical for moderate work below one’s VO2 Max.

• Energy Generation: It is well established that ATP energy is needed for mus-
 cle contraction. ATP can be produced during an aerobic process from glycogen
 and other energy stores. Without oxygen readily available, anaerobic ATP pro-
 duction takes place, and ATP as well as other molecules such as H+ and lactate
 as well as creatine phosphate (CrP) and Pi interact to successfully contract a
 muscle. Increased Pi and increased H+ substantially impair myofibrillar per-

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formance resulting in impaired muscle force.

2.2 Energy Production during Short Bursts of In-
 tense Activity
As explained in the prior section, ATP can be generated aerobically or anaerobically.
During aerobic ATP generation, glycolysis occurs with oxygen and produces ATP,
water, and carbon dioxide as outputs. Anaerobic ATP generation can happen in two
ways: by glycolysis without oxygen producing ATP and lactic acid as outputs, or
by the phosphagen system in which creatine phosphate (CrP) provides Pi to create
ATP [8]. During very intense efforts lasting seconds (such as throws, jumps or 100-
to 400-m sprints) or during intermittent game activities and field sports (such as
American football), most ATP is derived anaerobically [19]. In extremely short bursts
of effort (0-5s), a majority of the ATP comes from the phosphagen system, a little from
and anaerobic glycolysis, and a very small amount from aerobic glycolysis. As the
work time increases and is still under 3 minutes, anaerobic glycolysis takes over with
some contibution from the phosphagen system, and little contribution from aerobic
glycolysis. As work time increases past 3 minutes, aerobic ATP production dominates
[19]. Thus, on the time scale that is relevant in football (this time scale will be further
explained in section 3), the phosphagen system dominates energy production [24].
Replenishing the store of creatine phosphate happens up to 70% within 3 minutes,
and up to 100% within 15 minutes [24]. To do this, high levels of aerobic cellular
respiration (for which heart rate and breathing levels must increase to supply enough
oxygen) occurs, and the ATP produced provides high energy phosphates to replenish
the creatine phosphate.

2.3 Load Monitoring
Athlete load monitoring can be separated into two different approaches: internal load
or external load monitoring. External load is defined as "the work completed by

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the athlete measured independently of his or her internal characteristics" [26]. An
example for cyclists would include measuring the power output over a certain amount
of time. Internal load is defined as the "relative physiological stress imposed on the
athlete" [26] and is usually estimated by measuring biological signals. Athlete fatigue
can be thought of as the relationship between internal and external loads [4].

2.3.1 External Load Monitoring

External load measurement can be thought of as answering the question of "How
many/far/fast/heavy exercises can the athlete complete?". While there are many
ways this type of load estimation can be achieved, they are usually separated into the
categories below:

Power

In endurance sports like cycling and rowing, power is used as a training metric as
devices like the bike and the ergometer can directly calculate power using gear /
resistance and rotations per minute (rpm) or stroke rate.

Time Motion Analysis

In many team and certain individual sports, GPS and acceleration are used via small
wearable devices on each player. Even if the players don’t directly wear a device,
these metrics can often be inferred from video analysis. Companies like PlayerTek
and Catapult [12] have focused on the issue of "player tracking" and have designed
small devices and algorithms to determine athlete acceleration.

Neuromuscular Function

Measures of neuromuscular function such as jump tests (ie: maximum height squat
jump), sprint performance (ie: 50 meter sprint) and isokinetic and isoinertial dy-
namometry (ie: one-rep max) are also used to measure external load. These tests

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are popular because they are easy to conduct and introduce a minimal amount of
additional fatigue.

2.3.2 Internal Load Monitoring

Perception of Effort

Rating of perception of effort (RPE) involves retrospectively asking athletes to rate
how hard they perceived an exercise to be (often on a scale of 1-10). According to
Borresen and Lambert [16], "RPE correlates well with heart rate during steady-state
exercise" and "high-intensity interval cycling training, but not well during short-
duration high intensity soccer drills". Variations of this approach can be used by
multiplying the RPE score by the duration of the effort in an approach called session
RPE.

Heart Rate

Measuring heart rate is one of the most popular ways to estimate internal load.
According to Hopkins [2], "the use of heart rate monitoring during exercise is based
on the linear relationship between heart rate and oxygen uptake and the intensity of
steady-state exercise". Percent of maximum heart rate and heart rate recovery (the
rate at which heart rate declines after exercise has stopped) are other methods that
use heart rate to measure internal load. One other method that involves heart rate,
Training Impulse (TRIMP) [10], separates heart rate into different zones and factors
in the amount of time spent in each zone to determine total athlete load. One thing
to note about heart rate measurement is that there can be daily variation (up to
6.5%) attributed to hydration, environment, and medication [4].

Lactate Concentration

Lactate is a by-product produced by the body during normal metabolism and exercise.
Specifically, it is created when a high energy demand exceeds the aerobic capacity of
muscle cells and anaerobic energy production is used. Lactate concentration increases

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when the rate of lactate production exceeds the rate of lactate removal. Blood lactate
levels can then serve as an indirect marker for biochemical events such as fatigue
within an exercising muscle.

Oxygen Consumption

Measuring oxygen consumption can give insight on the aerobic fitness levels, or the
body’s ability to deliver oxygen to muscles, during exercise. This can be done by
performing a cardiopulmonary exercise test which involves wearing a mask over the
face while running on a treadmill or riding a bike and measuring the volume and the
composition of carbon dioxide and oxygen moved with each breath of air. Another
way to measure oxygen consumption is by measuring oxygen saturation (SmO2) levels
in the capillaries of muscle tissue. Wearable devices use near-infrared light to measure
oxygenated and deoxygenated hemoglobin to get an estimate of oxygen consumption
and how it changes over the course of an exercise.

Biochemical / Immunilogical / Hormonal

It is possible to measure the concentration of various biological markers such as crea-
tine kinase, urea, free testosterone, and insulin linke growth factor via blood or saliva
to assess the internal load of athletes, though it is not practical for everyday training
application [20].

2.4 Existing Athlete Load Metrics

Table 2.1 summarizes many commonly used load metrics. Each metric can be rated
on a number of different attributes: where it is best for anaerobic or aerobic exercises,
whether it use internal or external load monitoring, whether it requires a specific test
procedure or if the metric can be calculated during exercise in real time. The last
row of the table lists the necessary characteristics of a metric that is to be employed
by football linemen. Ideally this metric is optimal for mostly anaerobic efforts, uses

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internal load monitoring or unimpededly measures external loads during live football
play, can be evaluated in real-time, and doesn’t require extra testing.
Metric Name Description Common Uses Aerobic or Anaerobic Internal or External Load Monitoring Test or Real Time
Player Load Sum of instantaneous rate of change in acceleration in three planes Soccer, basketball, hockey Mostly Aerobic / Some Anaerobic External Real Time
Critical Power Power that can be maintained at steady state exercise Cycling, running, rowing Aerobic External Test
Wingate Test 30 s all out sprint on stationary bike against predetermined torque Cycling Anaerobic External Test
One Rep Max Maximum weight that can be lifted for one repetition (squat, bench press, etc.) Weightlifting, football, hockey Anaerobic External Test
VO_2 Max Maximum oxygen consumption Endurance sports Aerobic Internal Real Time / Test
Strava Relative Effort heart rate zones * time spent in each zone * weighting factor Multi-sport, endurance focused Mostly Aerobic / Some Anaerobic Internal Real Time
? Measures relative effort of football related exercises Football Mostly Anaerobic / Some Aerobic Internal/External Real Time

 Table 2.1: Common Load Metrics

2.4.1 Player Load

Player load is an estimate of physical demand calculated by combining the instanta-
neous rate of change in acceleration in three planes: forward/backward X, side/side Y,
and up/down Z [17]. Summing these instantaneous rates of change over the duration
of the exercise results in accumulated player load.

 = 
 ∑︁ √︁
 = ( = − = −1 )2 + ( = − = −1 )2 + ( = − = −1 )2
 =0
 (2.1)
This metric uses external load to estimate athlete fatigue. It is most commonly
used in sports where athletes exert a majority of their energy changing directions
and speeds. Example sports where this metric is relevant include soccer, hockey,
basketball, and certain football positions such as running back or receiver. Catapult
Sports is a company that has commercialized this metric and many other studies have
built off of it [12].

2.4.2 Critical Power

Critical power (CP) is a metric used to determine aerobic capacity. This metric
measures external loads via power output over different time intervals to determine
a model for overall athlete load at the aerobic-anaerobic junction. Specifically, it is
defined as the amount of power that can be sustained "infinitely". While we know
that no exercise can actually be sustained infinitely, we can ask athletes to perform
an exercise at certain power levels for as long as they can and plot the time vs. power

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curve (as seen in figure 2-2). The amount of power that can be sustained approaches
a constant as time approaches infinity. That constant is the critical power. CP
can also be defined as the "greatest metabolic rate that results in ’wholly-oxidative’
energy provision" [22]. Said another way, this asymptote represents the required
power at which aerobic work is not enough and anaerobic work is needed to perform
the desired amount of work. A similar metric, Functional Threshold Power (FTP),

Figure 2-2: Critical power profile taken from cpsinmotion.com [1] where the dashed
line represents critical power and the blue line represents the anaerobic work capacity.
As time grows larger, the amount of power that can be sustained approaches the
critical power.

is the maximum average power that you can sustain for an hour [3]. This metric is
commonly used in cycling. CP is a lower bound of FTP.

 Another metric built off of FTP is Training Stress Score (TSS)[13]. TSS is defined
as = ( * * )/( * 3600) * 100. NP is defined as normalized power
and IF is the intensity factor. NP and IF are developed by TrainingPeaks and more
information can be found on their website [9].

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2.4.3 Wingate Test

As mentioned in figure 2-2, the blue rectangle under the curve represents the amount
of anaerobic work available to the athlete. To calculate peak power and this decreasing
power curve, different anaerobic fitness tests can be performed. The most popular
test, the Wingate Test, involves 30 seconds of maximum effort exercise on a cycle
ergometer or arm-crank ergometer set at a prescribed resistance (can differ between
males & females and by body weight). From this test, Peak Power (PP), Anaerobic
Fatigue (AF) and Anaerobic Work (AW) can be calculated. PP is the highest power
ideally measured within the first 5 seconds of the test. Fatigue is calculated as =
 − 
 where LP is lowest power. Anaerobic capacity is represented as where
 ∑︀ = 
 =0

P is the power sample collected at a certain moment in time [14]. As we know from
prior sections, anaerobic activity produces by-products that need to eventually be
removed in order for activity to continue. This area under the curve and above the
critical power level can be thought of as the anaerobic work capacity that is finite
and can be drawn from during intense activity.

2.4.4 One Rep Max

A one repetition maximum or 1RM is the heaviest weight you can lift using maximal
effort. A 1RM can be used in any exercise, though the most commonly measured
exercises are squat, deadlift, and bench press.

2.4.5 VO2 Max

VO2 Max is a metric that measures the maximum amount of oxygen an athlete can
consume. This is important for endurance sports as oxygen consumption dictates
how fast energy can be produced aerobically. VO2 can be calculated directly using
a respirometer (a device that measures oxygen and carbon dioxide percentages and
volumes in breath during exercise) while riding a bike or running and increasing
 *% 2 
exertion levels. Absolute VO2 can be calculated as 2 = *% 2 
 where Q 
represents the quantity in mL of air inhaled and % represents the percent of oxygen

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in the quantity inhaled. Q and % are the same as above but for the exhaled
breath. A derivation, relative VO2 , divides absolute VO2 by the athlete’s body weight
and can be a better metric to compare the fitness levels of different athletes [11].
Alternatively, VO2 Max can be calculated using heart rate. A crude estimate can
be calculated by dividing your maximum heart rate by your resting heart rate and
multiplying the result by 15.3 [7].

2.4.6 Strava Relative Effort

Strava recently came out with a new metric called "Relative Effort" [6] used to quan-
tify effort exerted by a single athlete and between athletes over different exercises.
It is based on the TRIMP method explained above. It works by separating heart
rate into fives zones that approximate different levels of intensity. It then weights the
amount of time spent in each zone to calculate the relative effort score. In order to
get these weightings, Strava engineers first looked at running data between a large
subset of athletes giving roughly the same effort and optimized to minimize variance.
Then, they extended the model to biking, and noticed that a non-weight bearing
sport usually has heart rate values 10-15 BPM lower than that of running. Finally,
to extend the relative effort score to many different sports, the engineers divided each
sport into a weight-bearing or non-weight-bearing category, and assigned running or
biking weights accordingly. This metric works by using monitoring the internal load
of the athlete via heart rate.

2.5 Existing Loading Models
Other authors have measured internal loads during static loading, including Amini
[15], who studied heart rate response while lifting boxes of varying weights, dura-
tions, and frequencies with an application to American workers. Another study by
Iridiastadi & Nussbaum [21] focused on performing shoulder abductions at varying
duty cycles and contraction levels while EMG and RPE levels were recorded.
 Chen and Lee [18] also defined a simplified heart rate recovery cost model and

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validated it under simultaneous dynamic and static loading by asking participants to
use a cycle ergometer at varied speeds while carrying a backpack with varied weights.
This study found that change in heart rate and recovery time don’t account for all
physiological stress, though they can be used to make a simplified model.
 Separately, existing research has tried to model muscle recovery times [15] in
different areas of the body.

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Chapter 3

Football Background Information

3.1 Lineman Free Body Diagram

 Figure 3-1: Football Lineman FBD

 I will focus on measuring the fatigue of player 1 (right). First, I will assume

 27

player 1 is a rigid body. In red, we can see all the forces acting on player one: gravity,
friction, the normal force from the turf, and the force from the other player. In blue,
we can see the force exerted by player 1 onto player 2. If player 1 is stationary, using
Newton’s second law, the sum of the forces in the horizontal direction should equal
zero, so ℎ * ( ) = where represents the angle between ℎ and
the horizontal axis aligned with . We will assume for the remainder of this
analysis that max( ) will always be larger than ℎ , so player 1 will never slide
backwards. Player 1 is also not moving vertically, so = − ℎ * ( ).
Next, using Newton’s third law and assuming both players are stationary, ℎ =
 . It would also be advantageous for player 1 if > ℎ , causing player
2 to move backwards. Finally, in order to maintain this rigid body assumption,
player 1’s muscles must work to stabilize his joints so his body doesn’t collapse (ie:
elbows, hips, and knees don’t bend). During this stabilization time, the muscles get
fatigued, and through this project, we hope to find a measurement of how fatigued
those muscles are.

3.2 Relevant Time Scales in Football

According to Rhea et. al [23], the average high school football play lasts 5.6 seconds,
the average college play lasts 5.1 seconds, and the average National Football League
(NFL) play lasts 5.2 seconds. During the active play duration, offensive linemen are
constantly receiving and applying loads to the opposing linemen as they protect their
quarterback or create space for other members of their team to run forward with the
ball. Similarly, defensive linemen are receiving and applying loads to the opposing
offensive linemen as they try to get past them and tackle the player with the ball.
Thus, the first time scale that is interesting is that of a single play, which is on the
order of a couple seconds. The ratio of work to rest is about equal, each period of
rest or work lasting a second or less as linemen constantly bump and push against
each other during the total duration of the play.
 After a single play is over, the same team continues their possession if they still

 28

have remaining downs, or that team scores or turns over the ball and exits the field for
another group of players to come on. In the case that the team keeps possession, they
continue to execute plays until their possession comes to an end. These consecutive
plays during a single team’s possession make up a drive. In the NFL, the maximum
time between plays during a drive is 40 seconds (unless the clock stops for various
reasons), due to the play clock rules. In reality, there may be additional time in
between plays due to commercial breaks (if professional or college), injury, or different
end-of-game rules that occur. In the NFL, the average number plays per drive in 2022
was 6.04 [5]. Combining this information with the play clock rules, the average drive
should be at most (5.1 + 40)*6 seconds (without another reason for stoppage), or
around 4.5 minutes. Thus, the macro-level time scale that is interesting is that of a
single drive, which is on the order of a few minutes. The work to rest ratio in this
time scale that is interesting is at most 5s:40s or 1:8. In reality (sometimes the ball
is snapped before the play clock expires), this ratio is 1:6.2 for NFL games, 1:6.1 for
college games, and 1:5.5 for high school games [23].

 Figure 3-2 sketches the expected heart rate results guided by initial experimenta-
tion. Work time can be defined as and similarly, rest time can be defined as .
In the activities that are relevant football, work periods of length are followed by
rest periods of and this pattern is repeated over the course of the active exercise
duration. In the bottom left plot, and are both on the order of seconds, and
the responding heart rate response I predict to increase almost linearly up to a certain
level. This is what I expect for a single play. In the top left, is on the order of
a minute and is on the order of a second. I would predict that heart rate would
increase, though at a faster rate than that of on the order of minutes. The reason
I don’t predict heart rate dips during rest periods on the order of seconds is because
heart rate has a delayed response, so by the time the rest is over, heart rate will not
have enough time to start to decrease.

 Conversely, the bottom right plot represents what I expect the heart rate response
of a drive to look like. is on the order of seconds, while is on the order of
minutes. I expect heart rate to recover to near starting levels during this time, In the

 29

top right plot, and are both on the order of minutes. I expect heart rate to
begin to decrease, but to have that increased work time have an effect over time and
prevent heart rate from returning to before exercise levels.
 However, during initial testing, I found heart rate levels were not measurably
impacted during single trials of 5 seconds and the simulated football set-up made it
very difficult to perform and on the order single seconds or less than seconds.
Thus, in the methods section described later on, I will use a longer simulated play
time and slightly increased and . Further studies should be designed with
this in mind to create a more realistic data collection procedure that is relevant for
football.

 Figure 3-2: Relevant Football Time Scales and Predicted Heart Rate Response

 30

Chapter 4

Improved Load Metric Model

I will investigate whether heart rate is a reasonable metric to estimate athlete load
and if it can be used to compare the load of static and dynamic exercises. First, I
will define external load, , as the sum of the squared output generated over the
duration of an exercise, normalized by the athlete’s critical power and mass. As
explained in the related work section, critical power is rate of work you can sustain
for extended amounts of time, and should be divided out to try to calculate how much
of the anaerobic capacity an exercise uses. Dividing by critical power also makes the
external load unitless. The athlete’s generated output can take the form of exerted
force if performing a static exercise, or of the athlete’s mass (or whatever mass the
athlete is moving) multiplied by the athlete’s acceleration.

 ∫︁ 
 1
 = ( )2 + ( * ( ))2 (4.1)
 * 0

 Next, I will define the internal load in this study, L , as the body’s response to the
external load placed upon it, and I will use heart rate as the biological signal that I
monitor. Mathematically, it is defined as the sum of the heart rate with the constant
base heart rate subtracted out, over the duration of the exercise.

 ∫︁ 
 = ( ) − (4.2)
 0

 31

If heart rate is a reasonable metric to estimate external load endured by the
athlete, then external load should be related to the internal over the course of the
exercise. The experiments defined in the next section hope to figure out that a
relationship exists between these two metrics, and determine an estimate for what
that relationship is.
 Furthermore, it is possible to separate the above two equations into a relationship
between heart rate and force for static loading, as see in equation 4.3 and a relationship
between heart rate and acceleration as see in 4.4.

 ∫︁ ∫︁ 
 1 ?
 ( )2 = ( ) − (4.3)
 * 0 0

 ∫︁ ∫︁ 
 1 ?
 2
 ( * ( )) = ( ) − (4.4)
 * 0 0

 The remainder of this thesis will measure force, acceleration, and heart rate to
determine if and how heart rate can be used as a metric to evaluate athlete load
during static and dynamic exercises.

 32

Chapter 5

Methods

Data was collected during dynamic activity and static activity while heart rate data
and either force or acceleration data was collected. Data was collected by two different
athlete users. Athlete 1 is a 22-year-old female. Athlete 2 is a 25-year-old male.

5.1 Running Experiment
Two separate running experiments were conducted while heart rate and acceleration
were measured at a rate of 200 Hz using Vernier Exercise Heart Rate Monitor and
Vernier LabQuest Mini 2. After 15 seconds of data collection, a heart rate in beats
per minute (BPM) is estimated every 5 seconds until data collection ends. Both
experiments were conducted by athlete 1.

 1. 50 meter Sprint: This exercise consisted 1 minute of data collection. First, the
 subject stood in place for 15 seconds while heart rate data collection calibrated.
 Then, the user sprinted 50 meters in a straight line. The user quickly came to
 a stop and rested upright for the remainder of the minute.

 2. 200 meter Run: This exercise consisted of 2 minutes of data collection. First,
 the subject stood in place for 15 seconds while heart rate data collection was
 calibrated. Then, the user ran 200 meters around an indoor track as fast as
 they could. The user came to a stop and rested upright for the remainder of

 33

Figure 5-1: Running Data Collection

 the two minutes.

Between each trial, the user rested until heart rate returned to below 80 BPM. Five
50 meter sprint trials were collected along with three 200 meter runs.

5.2 Football Specific Experiment

Static and pseudo-static (some movement moving a load larger than that of normal
body weight) loading data was collected by pushing in a simulated football stance and
by performing common strength training movements: squat, push-ups, and plank.

 34

Figure 5-2: Simulated Football Static Loading Stance

5.2.1 Simulated Football Stance

Heart Rate (BPM) and force (N) readings were recorded at a sample rate of 250 Hz
for each trial using the Vernier Heart Rate Monitor and Vernier Hand Dynamometers.

5.2.2 Football Specific Time Scale Data

 1. Drive-Level Data Collection: As explained in chapter 4, a normal football
 drive is 4.5 minutes long. To simulate this, two long data collections of 5
 minutes were recorded. During the first experiment, plays of T = 5 seconds
 were performed with rest between plays T = 25 seconds. This sequence was
 repeated 10 times for a total of 10 plays. During the second experiment, plays
 of T = 7.5 seconds were performed with rest between plays T = 35 seconds.
 Athlete 1 conducted all trials.

 2. Play-Level Data Collection: As explained in chapter 4, a normal football
 play is on the order of 5 seconds; however, that short of a period of pushing

 35

T (sec) T (sec) T : T Ratio Athlete
 1 1 1 1
 2 1 2 1
 3 1 3 1
 4 1 4 1
 5 1 5 1
 3 2 1.5 2
 4 2 2 2
 6 2 3 1
 7 2 3.5 2
 6 4 1.5 2
 2 2 1 1
 3 3 1 1
 1 2 0.5 1
 2 4 0.5 1

 Table 5.1: Play level Work:Rest Ratios

 wasn’t enough time to see a significant heart rate response in this simulated
 setting. Thus, data was collected for 1 minute, using the first 20 seconds for
 the heart rate monitor to calibrate. The remaining 40 seconds cycle between
 seconds of all-out pushing against the force sensors on the wall, and of
 upright resting. In table 5.1 you can see the various work and rest times test.
 Each : pair was tested in two different trials. User 2 conducted 7W:2R,
 3W:2R, 4W:2R, 6W:4R ratio trials. Athlete 1 conducted the remainder.

5.3 Strength Training Data Collection

In order to test the maximum heart rate response that could be measured during
a static exercise, additional common strength training movements were recorded on
video (and then acceleration was derived via LoggerPro video analysis software) while
heart rate was measured using the same Vernier Heart Rate Monitor in prior exper-
iments. The movements included variations on squats, push-ups and planks. Due
to the up and down movement that occurs during squats and push-ups, these exer-
cises are not truly static, though they require the athlete to move more than just

 36

the normal body weight for those muscle groups. We will call the squat and push-up
exercises "pseudo-static". Athlete 1 conducted all trials for each experiment.

Figure 5-3: Strength Training Movement Data Collection with Barbell Squat, Modi-
fied Knee Push-up, and Elbow Plank

 1. Squats: I continued to repeat back squats until failure with 65 lbs (29.48 kg).
 One trial was recorded.

 2. Knee Push-ups: I completed push-ups on knees until failure. One trial was
 recorded.

 3. Elbow Plank: I held a plank until failure. Three trials were recorded.

 37

38

Chapter 6

Results and Discussion

6.1 Single Trial Results
Below are time series examples of the raw data collected in a single trial for each
experiment. In all experiments, critical power is set to a value of 150 W.

6.1.1 Running

 Figure 6-1: 50m Run Time Series Data

 While running acceleration data was collected using a 3-axis accelerometer, only
the x-axis is used in analysis. The x-axis was aligned in the direction that the ath-
lete moves forward in. The y-axis was aligned with vertical axis of the body and
picked up acceleration due to footfalls and movement from the insufficient attach-
ment mechanism to the waist of the athlete (it was secured with medical tape and
slightly bounced up and down against the abdomen). The z-axis recorded left to

 39

Figure 6-2: 200m Run Time Series Data

right movement data, which isn’t relevant because the athlete was moving forward
and backward, not laterally.
 As seen in figure 6-2, the athlete started running prior to heart rate data being
collected, so it is likely that heart rate increase slightly more than recorded. Many
running trials had to be eliminated due to errors in data collection (device just stopped
recording or produced errant heart rate values that didn’t agree with Apple Watch
data). More running data should be collected to confirm the validity of the results.
Additionally, further efforts to design an attachment mechanism to the midline of the
athlete should be performed so y-axis data can also be incorporated to the external
load calculation.

6.1.2 Football Specific

 Figure 6-3: Football Play with 3W:1R Ratio Time Series

 Figure 6-3 shows the pushing data from a 3W:1R experiment trial. It is interesting
to analyze the force profile over the course of a single work period. It seems as though
the force is higher when the athlete first starts pushing, decreases over the remainder

 40

Figure 6-4: Football Drive Time Series

of the effort slightly, then increases at the very end as the athlete stops pushing.
The increased initial force is probably from the impulse when the body first starts
to push at a maximum, then declines over time. The final increase is likely because
the athlete slightly leans against the wall when he/she is pushing, and has to push
harder to counteract gravity and stand upright. Further analysis could examine the
profiles of different work:rest ratios.

6.1.3 Strength Training

 Figure 6-5: Squat Time Series Data

 Figure 6-6: Push-up Time Series Data

 41

Figure 6-7: Plank Time Series Data

 Squat and push-up time acceleration data were both derived via manual video
analysis. It is interesting to see that the magnitude of acceleration for push-up was
much larger than that of a squat. There may be some error accumulated from the
manual video analysis. If this experiment were to be repeated, athletes should wear
accelerometers instead of relying on video analysis. To calculate the external work in
both of these movements, I used equation 4.1 and set = ( + % ℎ *
 ) * ( + ). For squats, the external mass, , was equal to the weight of
the barbell, which was 65 pounds or 29.5 kg, and for push-ups there was no external
weight, so = 0. % ℎ represents the constant corresponding to the
percent of the athlete’s body that is being move or lifted in the exercise. For squats,
 % ℎ = 0.66 and for knee push-ups, % ℎ = 0.49. For planks, % ℎ 
= 0.75. These percentages were derived from a study conducted at Purdue [25]. 
= athlete 1’s body weight, so 150 lbs or 68.0 kg. Accelerations were only used from
the y-axis, or in the up and down directions. In the plank experiment, the athlete
was not accelerating, so the only contribution is gravity.

6.2 Football Specific Work:Rest Ratio Analysis

Figure 6-8, shows the average pushing force for each : ratio. The colors corre-
spond to equivalent work to rest period ratios. For instance, the 1W:1R, 2W:2R, and
3W:3R are all the same color. When examining the four largest average pushing force
magnitudes in this plot, those were trials 7W:2R, 3W:2R, 4W:2R, 6W:4R. Those were
also the trials completed by athlete 2, a male athlete. To address this, the maximum

 42

force was capped at 82 newtons, the average pushing value conducted by athlete 1
(a female athlete) across all experiment trials. Figure 6-9 shows the average pushing
force during periods of work with this normalization. The bars with a black X on top
in figure 6-9 are those trials that were normalized.

 Figure 6-8: Average Pushing Force without Normalization

 Figure 6-9: Average Pushing Force Normalizing to Athlete 1

 Figure 6-10 shows the external load, , and the resulting internal load, for
each : ratio experiment averaged across all trials. As a reminder from chapter
4, the external load is defined here as = ( )2 + ( * ( ))2 and
 1 ∫︀ 
 * 0

the body’s response, or internal load is defined here as =
 ∫︀ 
 0 ( ) − .
As expected, it appears that as the rest to work ratio increases, the athlete exerts
a larger external load and in result, a larger internal load (or body response) is
also measured. To say this another way, as working time increases and resting time
decreases, more external work is completed. When more external work is completed,

 43

the body responds accordingly and a larger internal load can measured from the
increases in heart rate.

 Figure 6-10: Scatter plot of Different Work:Rest Ratios

 Figure 6-11 uses the points from figure 6-10 and interpolates between each point
to determine the likely rest to work ratio at a certain (external load, internal load)
pair. It is clear that the higher work:rest ratios are predicted to occur when external
loads and resulting internal loads are very high. Likewise, the lower work:rest ratios
are predicted to occur when external loads are very low and seem nearly independent
of internal load, though there are few data points in the low high regime.
This is because there are likely few exercises that can be performed that fall into this
range that have a low external load (little exercise completed) but large internal load
(high body response) or impact on the athlete’s body, so this regime is unrealistic
and should be ignored.

 44

Figure 6-11: Interpolated Map of Predicted Work:Rest Ratios

6.3 Comparing Loads of Different Activities

Figure 6-12 shows each activity’s external load plotted against the body’s correspond-
ing internal load. The football pushing activities are responsible for the lower part of
the regime. It seemed that for these activities, heart rate increased as work increased,
though the increase in total work performed during different work to rest trials was
marginal in comparison to the change in heart rate. Thus, these trials all appear in
the same external load column. The running activities performed were "harder" in
terms of the external and internal loads as compared to the football pushing. For
the strength training exercises, the plank seemed to be in a similar regime as the
football pushing, which could make sense because those are both completely static
exercises. The push-up was in a similar regime to the running data, and the squat
exercise produced the largest external load with similar internal load to that of the

 45

Figure 6-12: Activity Data Internal Load vs External Load

push-up. The relationship between internal load and external load for all the activity
data can be best fitted with a line using a power law relationship:

 = 795.5 * 0.1708
 − 197.8 (6.1)

While ultimately choosing a line of best fit using a power curve, I first predicted that
a log fit may represent the data best. Though when examining the relevant statistics
and residual plots as seen in figure 6-13 for a log fit vs. a power fit, it is clear that a
power fit represents the data the best.

 Figure 6-13: Log Fit as Compared to a Power Fit for All Activity Data

 46

Log Fit Power Fit
 SSE 1.446e+06 1.364e+06
 R-square 0.8929 0.8990
 Adjusted R-square 0.8886 .8905
 RMSE 240.5 238.4

 Table 6.1: Statistics for Log Fit vs Power Fit for Activity Data

 Ultimately, this power relationship shows that relative change in external load
gives rise to a proportional relative change in the body’s internal load. This relation-
ship also shows that as the body continues to perform external work, the rate at which
heart rate responds quickly increases, then eventually levels off around the athlete’s
maximum heart rate. This relationship also shows where various static to dynamic
activities fit on this exertion line and how they compare. It seems that static activities
like the simulated football push and plank produced less external work and were less
taxing on the athlete’s body than dynamic activities like running when performed
on a similar time scale. The pseudo-static activities like squats produced the highest
external load and resulting internal load over the same time period. Further work
should examine the differences in internal load when exactly the same external load
is produced over a course of different activities. It should be noted that while experi-
ments included in this analysis all occurred on similar time scales (under 80 seconds),
they were not exactly the same duration of active effort. Further experimentation
should hold exercise time length as a constant to validate these results.

6.4 Football Relevance and Next Steps

This analysis is relevant in football because it shows how typical movements in football
games and practices compare to other training activities the athletes may perform,
such as running and weight lifting. Additionally, it shows how different work:rest
ratios of static loading similar to live football play affect the body.
 Once again, this study proposed a metric to compare internal and external loads
between different activities and explored the results. These initial results proved

 47

promising and showed that this model should be further investigated. First off, addi-
tional work needs to be done to collect better data. Specifically, during the running
trials, a better securing system for the accelerometer needs to be designed. For the
strength training trials, accelerometers should be worn by the athletes instead of
doing video analysis. Finally, a better system to simulate football linemen activity
should be created and all trials should be collected with multiple different athletes.
 Next, more trials should be conducted to confirm the statistical significance of this
project. Expanding on that work, more experiments should be designed including
holding the time periods of each active period within an exercise constant, while
conducting the same experiments at a range of time scales. Also, a larger range of
activities other than the experiments proposed in this study should be investigated.
 In addition to the next steps listed above, it would also be helpful to create a
comparison as done in figure 6-14, where two activities’ internal loads are compared
when their external loads are held equivalent. This would show the relative exertion
levels when the same amount of work is performed. Two activities with equivalent
exertion levels would have a line of y=x on this plot.
 Finally, to move past collecting data in the simulated experiments as performed
in this study, force sensing gloves, accelerometers, and heart rate monitors could be
worn by the linemen in games and practices (with an appropriate form factor) to
gauge actual force and acceleration values.

 48

Figure 6-14: Exertion Comparison of Two Activities with Equivalent External Load-
ing

 49

50

Chapter 7

Conclusion

Understanding exertion levels during exercise helps athletes prevent injuries and train
at an optimal level. Currently, there exist metrics to determine exertion levels that
are specific to individual activities. These metrics are especially well developed for
sports where dynamic activity produces most of the fatigue, like running, cycling,
soccer, and basketball. However, there is not a well defined metric for sports in which
static or psuedo-static activity is the main mode of exertion. While weightlifters
undergo predictable and measurable amounts of static and psuedo-static loading,
American football linemen experience extremely unpredictable amounts of static and
pseudo-static loading as opposing team members push against them. Thus, they are
in need of an exertion monitoring system. Additionally, many athletes cross-train
with multiple exercises that are on the spectrum between purely static and purely
dynamic. A metric for exertion is also needed that is able to compare activities with
differing modes of exertion.
 To achieve both of these needs, this thesis presents a model that predicts ex-
ertion from force and acceleration measurements, using heart rate as an evaluation
for exertion levels. This model can take in data from multiple types of activities,
ranging from purely static to purely dynamic activities. The relationship between
internal load (exertion) and external load can be represented via the power law
 = 795.5 * 0.1708
 − 197.8. This relationship shows that as external load increases,
internal load increases at a decreasingly proportional rate. This makes sense because

 51

activities over short time periods with relatively low external loads quickly increase
heart rate from resting to that of active exercise. These activities can then be com-
pared to those of larger external loads, such as sprinting or middle-distance running,
where heart rate quickly increases, but then approaches its maximum value and stays
there for the remainder of the exercise. Thus, the slope of the change in heart rate
is steeper for activities with low external loads as compared to that of activities with
larger external loads.
 This model provides a proof of concept that heart rate can be used as a measure
between exertion levels over different types of activities. However, many next steps
should be taken to evaluate the robustness of this model, including and not limited
to (1) collecting more reliable data, (2) collecting more trials of each activity, (3)
collecting a larger range of activities, (4) holding the time periods of each exercise
constant and (5) comparing the internal loads of pairs of activities where their re-
sulting external loads are equal. Additionally, different biological signals (such as any
of those listed in the related work section) should be evaluated to compare different
types of activities.
 This exploration has taken the first steps in creating an exertion metric agnostic
to mode of exertion. It specifically focuses on evaluating the effects of static loading
in the context of American football linemen. This exertion information helps athletes
train and compete at their optimal levels. Better informed athletes produce better
performances, which is the ultimate goal of any competitor.

 52

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