Raman Garimella thesis - Final Draft

Estimation of Power and Energy Expenditure from Sensor Data

in Road Biking

Master Thesis presented

by

Garimella, Raman Matriculation Number: 01/928574

MSc Sports Science

[email protected] / [email protected]

at the

University of Konstanz

Department of Sports Science

Evaluated by

1. Prof. Dr. Dietmar Saupe

2. Prof. Dr. Markus Gruber

Constance, 2016

ESTIMATING POWER FROM SENSOR DATA IN ROAD BIKING 2

Acknowledgements

This thesis has my name on it but it is hardly an individual effort. I thank:

- My supervisor Prof. Dr. Dietmar Saupe – for his hand-holding whenever I needed it.

He made time in every stage of the thesis. I admire his attention to detail, sharp

memory, desire to get to the root of problems, and immense experience in cycling.

- His research group (Powerbike Lab) – they opened doors for me to use their facilities

and knowledge freely. They laid down the project in the first place and I am happy

to finish it for them. I specifically thank Stefan Wolf and Alexander Artiga Gonzalez

for their informal and insightful chats and tips over the last few months about this

topic.

- The Sports Science department – Prof. Dr. Markus Gruber for his timely green

signals as co-supervisor of the thesis. Apart from superior knowledge in his area of

study, his light-hearted humour and warmth over the last two years will be

remembered. My friend and colleague Marcello Grassi took a lot of interest in my

work and lent an ear and a hand whenever I needed.

- John Hamann of Velocomp LLC, USA – for letting me in on the inner workings of

the PowerPod device.

This thesis is the final leg of the MSc course. There were challenges inside and outside the

university, but I thoroughly enjoyed my time in Constance, Germany, and Europe. It was

the first time that I lived away from my home country (India). It is not an exaggeration to

say that the last two years have changed me as a person as well as enhanced my skills as a

professional. For this to happen in the first place, the people to thank are my family: mother,

sister, and brother-in-law. They have been nothing but kind and their support means

everything to me.


Contents

Estimation of Power and Energy Expenditure from Sensor Data in Road Biking ................ 6

Background.......................................................................................................................... 10

Power in Training ............................................................................................................ 10

Power Zones ................................................................................................................. 13

Power Meters ................................................................................................................... 13

Direct Force Power Meters (DFPMs) .......................................................................... 14

Opposing Force Power Meters (OFPMs) ..................................................................... 15

Modelling Power .............................................................................................................. 16

The Equation ................................................................................................................ 17

Estimating Parameters. ................................................................................................. 24

Methods ............................................................................................................................... 28

Shortlisted PMs ................................................................................................................ 28

SRM ............................................................................................................................. 28

Powercal ....................................................................................................................... 29

PowerPod ..................................................................................................................... 30

Sigma ............................................................................................................................ 31

Velocomputer ............................................................................................................... 32

Routes .............................................................................................................................. 33

Subject and Setup ............................................................................................................. 34

Experiment ....................................................................................................................... 35

Analysis ............................................................................................................................... 36

Metrics ............................................................................................................................. 37

Average Power and Relative Error............................................................................... 37

Normalized Power ........................................................................................................ 37

Root Mean Square Error .............................................................................................. 38

Relative RMSE ............................................................................................................. 38

Peak Signal-to-Noise-Ratio .......................................................................................... 39

Energy .......................................................................................................................... 39

Estimating Parameters ..................................................................................................... 39

Results ................................................................................................................................. 42

Part 1 ................................................................................................................................ 42

Full Ride ....................................................................................................................... 42

Training Zones ............................................................................................................. 42

Climbing Section .......................................................................................................... 42


Flat Section ................................................................................................................... 43

Normalizing .................................................................................................................. 43

Smoothing .................................................................................................................... 43

Sprints........................................................................................................................... 44

Part 2 ................................................................................................................................ 44

Full Ride ....................................................................................................................... 45

Climbing Section .......................................................................................................... 45

Smoothing .................................................................................................................... 46

Discussion............................................................................................................................ 47

Part 1 ................................................................................................................................ 47

Powercal ....................................................................................................................... 47

PowerPod ..................................................................................................................... 47

Sigma ............................................................................................................................ 49

Notes............................................................................................................................. 50

Part 2 ................................................................................................................................ 52

Exclusion Criterion ...................................................................................................... 52

Sensor Data .................................................................................................................. 52

CdA ............................................................................................................................... 54

Crr ................................................................................................................................. 55

λ .................................................................................................................................... 57

Predicted Power............................................................................................................ 58

Conclusion ........................................................................................................................... 59

Part 1 ................................................................................................................................ 59

Part 2 ................................................................................................................................ 59

References ........................................................................................................................... 60

Tables .................................................................................................................................. 65

Figures ................................................................................................................................. 77


Abstract

Along with scientists and professional cyclists, a number of amateur athletes and hobby

cyclists have started using portable/mobile power meters (PMs). “Direct force” power

meters (DFPMs) are devices that calculate power based on the deformation in bike

components such as pedals, cranks, crank-web, etc. caused by the pedaling forces of the

cyclist. These PMs tend to be expensive. “Opposing force” power meters (OFPMs)

estimate power from data such as speed, road gradient, heart rate (HR), anthropometric

and bike data, etc. These, though not as accurate as DFPMs, are cheaper and hence

attractive options for non-professionals to use. This study is divided into two parts. Part 1

compares three OFPMs for their accuracy against an industry standard DFPM. We field-

tested them under different conditions – submaximal endurance ride, efforts while

climbing, efforts on a flat section, and sprint efforts. PowerPod was the most accurate

OFPM with a root mean square error (RMSE) of 47 W and relative RMSE of 44% for a

75-minute field test. Part 2 aims to estimate parameters for the power modelling equation

from data collected in Part 1. Using those parameters, we wanted to predict power for a

new ride by the same subject. Our estimated parameters agreed with some of those found

in literature. The RMSE of the power predicted by our equation was 46 W and the relative

RMSE was 41%.

Keywords: road cycling, power, power meters, modelling, parameter estimation


Estimation of Power and Energy Expenditure from Sensor Data in Road Biking

Power is a metric in cycling that is increasingly being followed not just by

scientists and elite athletes but also by hobby cyclists and beginners. A sensor that

measures and outputs power is called a power meter (PM). At first, PMs were used for

scientific purposes, and mostly on indoor bicycle ergometers, as they were the only ones

equipped with PMs. But, cycling indoors on an ergometer is different from cycling

outdoors in many ways – unreasonably high heat and dehydration, the cardiovascular drift

phenomenon, use of only certain muscles, etc. – and hence not realistic. With the advent of

mobile PMs, scientists could measure power output in field tests as well (Gardner et al.,

2004). In the early 2000s, professional cyclists started using PMs in their training and

racing. The first time PMs were adopted by a professional bike team was in 1991.

("History,")

PMs have come a long way since then. The PM market has exploded, facilitated by

the entry of the first PMs for the consumer market in 2005 ("History,"). PMs have become

attractive to the general public mainly because of an interest in training and performance

improvement. PMs have become more mobile, they come in all sizes, and can be installed

on most parts of the drivetrain – for instance on cranks, pedals, wheel hub, and even in-

soles ("Power meters: Everything you need to know," 2016).

Advancements in technology, higher scales of production, a competitive market,

high demand and more spending power among people are some reasons that made PMs

affordable for the average consumer. As of mid-2016, the price of a PM ranged from USD

299 to USD 1,500 ("Spring 2016 Power Meter Pricing Wars Update," 2016).

PMs can be divided into two broad categories. The first includes those that directly

measure force applied by the cyclist while pedalling. These are called “direct force power


meters” or DFPMs. The second category includes those that do not measure leg force but

estimate the power output of the cyclist using indirect methods. These are called “opposing

force power meters” or OFPMs.

In the past, indoor cycle ergometers were compared to each other to validate for

accuracy (Abbiss, Quod, Levin, Martin, & Laursen, 2009; W. Bertucci, Duc, Villerius, &

Grappe, 2005; Novak, Stevens, & Dascombe, 2015). Tests were also conducted to

compare portable DFPMs both indoors as well as outdoors (W. Bertucci, Duc, Villerius,

Pernin, & Grappe, 2005; Bouillod, Pinot, Soto-Romero, & Grappe, 2016). It is very rare to

find in the scientific community studies that compare OFPMs. There was one paper that

way back in 2003 that tested an OFPM (G. Millet, Tronche, Fuster, Bentley, & Candau,

2003). There is anecdotal evidence, user ratings, manufacturer claims, blogs, and

magazines online that report on the testing of OFPMs, based on first-hand tests but none

are conducted scientifically. Most of them do not quantify accuracy of instantaneous

power values – the common method is to simply compare average values for long rides or

some intervals, at best ("PowerPod In-Depth Review," 2016; "PowerTap PowerCal In-

Depth Review," 2012; "Reviews," 2016).

We understand that OFPMs tend to lack accuracy; and in the scientific community,

there is a high stress on accuracy (Gardner et al., 2004). This could be a reason why there

has not been much investigation of OFPMs. But, from the point of view of a consumer,

there is a lot of interest. Hence, we wanted to compare OFPMs in the context of a master

thesis backed by a research group. This is Part 1 of the thesis: we shortlisted three OFPMs

to test them against “actual” power from an industry standard DFPM. We conducted three

trials (Rides 1, 2 and 3) but will present one representative trial.


The attempt to understand forces acting on a cyclist and model cycling

performance goes a long way back (Nonweiler, 1956). In 1998, a landmark study (Martin,

Milliken, Cobb, McFadden, & Coggan, 1998) was published that described all the forces

acting on the bike and cyclist system. The study has been cited more than 200 times. There

have been several studies that use the equation provided by them, or one on similar lines

(W. M. Bertucci, Rogier, & Reiser, 2013; Dahmen, 2012; Dahmen, Wolf, & Saupe, 2012;

G. P. Millet, Tronche, & Grappe, 2014; Peterman, Lim, Ignatz, Edwards, & Byrnes, 2015).

We also use an equation adopted from that of Martin and colleagues. But, their study

measured average power, and we are interested in modelling instantaneous power. By

modelling, we mean predicting power based on sensor data. For this, we used data (actual

power, speed, elevation/gradient, air density) collected in Part 1. On that data, we

performed a multivariate analysis to solve a system of linear equations to estimate certain

constants/parameters. With these parameters, a customized formula to predict

instantaneous power was obtained. This formula was put to test in a field test (Ride 4). We

compared it to actual power as well as to the OFPMs that were used in Part 1. This is Part

2 of the thesis. Understanding the working behind some OFPMs can help in predicting

power better.

There are several methods to estimate parameters (elaborated in Background

chapter) and they have all been found to be valid ways to model performance for a number

of applications. But the ultimate validation is always a field test (Henchoz, Crivelli,

Borrani, & Millet, 2010). We also know that a field test like ours is one of the best ways to

estimate parameters (García-López et al., 2008). The length and conditions of our data

collection rides and validation ride are what makes our project unique. We have been

warned that road cycling is difficult to model as compared to track or other controlled

environments (T. Olds, 2001). But we have assurance from some authors that using an


SRM PM (Schoberer Rad Messtechnik GmbH, Germany) and applying linear regression

methods is a valid way to estimate parameters (Debraux, Grappe, Manolova, & Bertucci,

2011). We expect that our formula will predict power better than some of the commercial

OFPMs that we test in Part 1.

This thesis is a part of the Powerbike Lab at the University of Konstanz. The

research group develops methods for data acquisition, analysis, modelling, optimization

and visualization of performance parameters in endurance sports with emphasis on

competitive cycling ("The Powerbike Project,").


Background

Power is the rate of doing work. The SI unit of power is J/s or, more commonly,

Watt (W), named after the Scottish engineer James Watt.

Cycling is an attractive activity for physiologists and exercise scientists because it

allows to monitor a variety of metrics on an indoor cycling ergometer (Stannard,

Macdermaid, Miller, & Fink, 2015). Of these metrics, power is often used to measure

performance. With portable PMs, it is possible to take this monitoring to more realistic

settings in-field.

Power is of interest to cyclists across the spectrum ranging from professional

athletes to beginner cyclists as it provides an objective method to bring structure to

training and racing. Professional cyclists have sworn by PMs for a long time ("More than

you ever wanted to know about power meters in pro cycling," 2015) and now hobby

cyclists are finding sense in investing in them.

Power in Training

To improve at a particular physical task, it is imperative to have practice in that

task. Repetition would bring improvement. To improve in cycling, the activity that would

bring most benefit to one’s cycling performance is cycling. Playing tennis may or may not

have a substantial “transfer effect” to cycling performance. This is the principle of

specificity in the science of training.

Repetition without a plan may bring improvements to a beginner’s performance. If

he started riding his bike for 60 minutes every day for an extended period, he would surely

improve at the beginning, and rather rapidly. But, eventually, his performance would

“plateau” i.e., not increase anymore. It is important to bring variation in training to

maximise results for the work put in. The “variation” here refers to overloading


(alternating with resting) the body in a planned and controlled manner. This is the

principle of overloading.

To implement these two concepts effectively, it is important to quantify load/effort

in an objective and repeatable manner. The effective way to train is to vary the duration

and intensity of workouts and plan training blocks by manipulating these two factors. A

training block could be as long as a week, a fortnight (called micro-blocks) or in the range

of a few months or seasons (macro-blocks). This is the concept of periodization.

Distance is an indicator of volume but time is better. A cyclist could ride 50 km

downhill for 1 hour but the same 50 km could take more than 3 h if the cyclist rode uphill.

The same distance could take less or more time based on wind conditions, type of bike,

and air pressure in the tires, to name a few factors.

Previous methods of training involved using speed, rate of perceived effort/exertion

(RPE) and heart rate (HR) as indicators of load. None of these are objective ways to

quantify effort. If two cyclists rode 50 km each at speeds of 20 km/h and 30 km/h, it would

be natural to think that the one who rode at 30 km/h trained at a higher intensity. But, this

assumes that all other conditions (distance, weather, bike type and condition, riding

position, fitness level, etc.) were identical.

A less subjective – but flawed, nevertheless – approach to quantifying intensity is

HR. HR patterns vary across individuals. An elite athlete’s heart could be pumping at 110

beats per minute (bpm) without them breaking into a sweat while an unfit 60-year-old

individual could react very differently for the same value. There are inter-individual

differences due to age, fitness, medical conditions, etc. Therefore, it is common practice to

quantify effort using HR values as a percentage of baseline/threshold/maximum values of

HR. So, if two cyclists exercise at 70% of their maximum HR, to some extent, intensity


can be considered equal. But, HR can vary within an individual too – if a person ate or

consumed caffeine or alcohol, or did not sleep enough, or is anxious, his HR values would

be higher for the same intensity.

The RPE scale is a rating between 1 and 10 – 1 being “a very light intensity” and

10 being “maximal effort”. The perception of intensity could vary not just between

individuals but also within the individual as quickly as on consecutive days. The RPE

approach suggests that one use the subjective marker in their mind to rate how hard or easy

an effort is. This depends on mood, mental fatigue, physical tiredness, time of day, and

personal experiences. A rating of 8 today may feel as hard as a rating of 8 on a different

day but in terms of performance output (load on the cardiovascular, neural and musculo-

skeletal system, for example), they could be vastly different.

In an age where people want to work “smarter” i.e., get the job done in as short a

time as possible and not necessarily “harder”, power training is the modern cyclist's most

scientific tool. Power is an objective way to quantify training load. Some have called

training with power “a game changer” ("Power meters: Everything you need to know,"

2016). A PM measures effort from the output generated and not from input parameters like

HR. A PM is “blind” to external conditions like weather, terrain, mood, mental/physical

fatigue, and equipment. If a cyclist rides at a power output of 200 W for 60 minutes today

and 200 W for 60 minutes two weeks from now, the same two rides may have been over

different distances or weather conditions; one day may have felt harder or more

challenging than the other, but the mechanical output is undeniably 200 W.

Hence, power can be used to be scientific in planning workouts and training plans

and effectively follow all the guidelines laid down by the principles of overloading,


specificity (even for specific skills within the sport of cycling), and periodization. Power is

also great as a pacing tool/marker in races.

Power Zones

Just like with HR, the absolute power numbers must be normalized to the

individual’s ability in order to be able to use power as a meaningful training tool or to

compare individuals. A common metric in coaching science is the Functional Threshold

Power (FTP) (Andy R. Coggan, 2016b). FTP is the power output a cyclist can generate at

lactate threshold (LT). There are some short protocols that allow for a good estimate of

FTP. A popular one is the “20-minute FTP test”. In its simplest form, FTP is approximated

as 95% of the average power output during a 20-minute maximal test. Power zones are

then established with this value as the baseline.

Power zones are demarcated based on the energy system demanded and the

physiological/neurological adaptations to those training intensities. Of the many charts

outlining power zones, the one in Table 1 is borrowed from (Andy R. Coggan, 2016a).

Power Meters

The PM market is driven mainly due to the high interest in training and amateur

racing. This, in turn, has created jobs in the coaching industry. Cycling coaches can now –

without ever having to meet the athlete in person – prescribe and monitor training plans of

athletes by looking at a range of metrics, all based on power measured by portable PMs.

This boom also facilitated books such as ("A Power Primer – Cycling with Power 101,"

2009), co-written by the aforementioned author. All these advances have been possible

only because of mass production of portable PMs, pioneered by SRM a couple of decades

ago.


Direct Force Power Meters (DFPMs)

DFPMs usually consist of a set of strain gauges and an accelerometer. The strain

gauges calculate force produced by the cyclist by measuring the deformation caused in the

bike component where the PM is located and transform this mechanical force into an

electrical output. Power is defined as the product of torque and angular velocity. The

accelerometers measure angular velocity. The product of torque and angular velocity is

calculated several times a second and the final output is averaged – usually over one

second – and transmitted wirelessly to a bike computer, mobile phone or monitor where

power is displayed to the user and/or stored. DFPMs have their flaws in terms of variation

in accuracy and sensitivity (Bouillod et al., 2016) but their accuracy is more than sufficient

for the purpose of training. Devices with higher accuracy, reliability, and product life

naturally cost more. Accuracy among DFPMs varies based on number and quality of strain

gauges, quality of wireless transmission, and method of calculation. For instance, the

PowerTap wheel-hub-based system was reported to have accuracy = -1.2 ± 1.3% (Gardner

et al., 2004), the Stages PM, which measures power from the left crank, has an accuracy =

-8 ± 1% (Hurst, Atkins, Sinclair, & Metcalfe, 2015), and the SRM system – our

benchmark – has an accuracy = 2% (Quod, Martin, Martin, & Laursen, 2010).

As of mid-2016, the cheapest DFPM costs USD 399 but they can be as expensive

as 1,500 USD ("Spring 2016 Power Meter Pricing Wars Update," 2016). All DFPMs need

regular calibration. Manufacturers prescribe the calibration frequency and instructions.

Extreme conditions like heat affect the accuracy and life of the strain gauges and other

components of the PM ("From Power Models to Opposing Force Power Meters and the

iBike Newton+,").


We cannot list all kinds of DFPMs as it is out of the scope of the thesis but there

are very popular guides where thousands of consumers (including us) flock to get their

information on PMs. ("The Power Meters Buyer’s Guide–2015 Edition," 2015)

("Reviews," 2016).

Opposing Force Power Meters (OFPMs)

OFPMs take data such as HR, elevation, gradient, pressure, speed, cadence, etc. as

well as anthropometric data such as height, weight, age, sex, shoulder width, etc. to

estimate the mechanical power output of a cyclist. These are all cheaper than DFPMs – the

most expensive OFPM is comparable to the cheapest DFPM ("Power Meter Pricing Wars:

Let The Games Begin,"). They start as low as USD 49 or can even be in the form of a free

mobile application ("How Strava measures Power," 2016). That is why these are popular

and have a huge commercial potential in the entry-level cyclist market.

The greatest compromise for buying this kind of PM is accuracy. Other

disadvantages include the fact that they are not always “plug and play” i.e., not intuitive to

setup/use and the calibration can be tedious. The installation of some OFPMs can be

cumbersome and may sometimes need an experienced hand. Since the formulae and

algorithms (which are most often a secret) depend heavily on the input of anthropometric

and bike/ride data by the user, extreme care must be taken while inputting as well as

monitoring these data over time.

OFPMs are often the target of harsh criticism from reviewers, bloggers and

magazines that test them. It is understandable, but they must be judged in their context.

They do not compete with DFPMs directly and in that sense, they must be seen as

commodities belonging to a different segment. But the truth is that most OFPMs do work

well under certain conditions. For example, a heart rate based OFPM reportedly works


well when output is relatively steady ("PowerTap PowerCal In-Depth Review," 2012).

Strava – a free mobile application – supposedly has a reliable method to estimate power

accurately during climbing ("How Strava measures Power," 2016).

However flawed and criticized, from time to time, the OFPM industry comes up

with innovative ways to do the guesswork. Polar S710, a PM now out of production

generated interest even in the scientific community (G. Millet et al., 2003). The concept

was that the device would get bike speed from monitoring chain speed and would estimate

force only from chain vibrations.

We will elaborate on three OFPMs that we felt were the best of all the products out

there in the Methods chapter.

Modelling Power

Modelling helps in predicting performance, planning races and training, testing out

strategies, gearing options, etc. This concept is very much in line with the same “train

smart, not hard” philosophy we addressed earlier. The attempt of modelling performance

has been done many times (both in-lab and -field) in the past. Most of the initial studies

were highly simplified and/or their validity can be questioned because they are too old and

we now have better ways to measure as well as estimate power. We will go over these

studies in the course of this report.

Modelling has made it possible to estimate power and/or parameters for historic

rides, when PMs did not exist. One study modelled power for the hour record rides and

compared riders of difference eras. It is a very detailed and interesting read, especially for

cycling fans (Bassett Jr, Kyle, Passfield, Broker, & Burke, 1999). One landmark study

(cited more than 200 times) by Di Prampero & colleagues estimated parameters in the


equation for retardation force for a cyclist and bike system (Di Prampero, Cortili,

Mognoni, & Saibene, 1979).

Knowledge from studies such as the ones above that set guidelines for modelling

performance has carried over to simple calculators on the internet ("Analytic Cycling

Home Page," ; "Interactive cycling power and speed calculator," ; "Speed & Power

Calculator,"). These calculators are generally good starting points for people outside the

scientific community, presented in a user-friendly way with options to play around with

and model performance. CyclingPowerLab has one of the most exhaustive simulators.

They also offer simulating services at a cost for athletes and coaches ("Power & Speed

Models,").

We, in Part 2, will model power from the “demand side” (T. Olds, 2001) i.e., we

use the biomechanical equation proposed by Martin & colleagues. But it needs to be

mentioned that modelling also helps in understanding and linking the “supply side” i.e.

metabolic energetics with the demand side (e.g. linking oxygen consumption with speed,

oxygen consumption with air resistance, etc.) (Capelli et al., 1993; Davies, 1980; Di

Prampero et al., 1979; T. Olds, 2001; T. S. Olds, Norton, & Craig, 1993).

The Equation

We use the following equation borrowed from an unpublished work of a colleague

in the Powerbike Lab (Thorsten Dahmen’s doctoral dissertation, soon to be published on

KOPS, Uni Konstanz) which was originally inspired from the one proposed by (Martin et

al., 1998) – a study still considered the guiding light of all state of the art publications

(Dahmen, Byshko, Saupe, Röder, & Mantler, 2011). Some OFPMs and online services like

Strava also use a simplified form of this equation in their algorithm to predict power

("How Strava measures Power," 2016).


If the cyclist + bike system is taken as an isolated object and a “free-body diagram”

is drawn, the forces would look like this (Figure 1). When the balance of forces is written

in an equation, it would look be written like this:

𝐹𝑡𝑜𝑡𝑎𝑙 = 𝐹𝐾𝐸 + 𝐹𝑃𝐸 + 𝐹𝑟𝑜𝑙𝑙 + 𝐹𝑎𝑒𝑟𝑜 + 𝐹𝑏𝑒𝑎𝑟𝑖𝑛𝑔

Equation 1

where FKE is the force linked to the change in velocity (kinetic energy); FPE is the

force of gravity; Froll is the frictional losses mainly between the bike and road surface and,

to some extent, between the inner tube and tire; Faero is the aerodynamic drag force; Fbearing

is frictional losses in the bearings of the wheels of the bike.

Since power is the product of force and velocity, for a bike travelling at vG (ground

velocity i.e., bike speed), Equation 1, when multiplied by vG on both sides would result in

the “power equation”:

𝑣𝐺𝐹𝑡𝑜𝑡𝑎𝑙 = (𝐹𝐾𝐸 + 𝐹𝑃𝐸 + 𝐹𝑟𝑜𝑙𝑙 + 𝐹𝑎𝑒𝑟𝑜 + 𝐹𝑏𝑒𝑎𝑟𝑖𝑛𝑔)𝑣𝐺

Equation 2

Bicycles are human powered machines, and no machine is 100% efficient in

reality. So, to account for losses, we introduce the λ term here on the left-hand-side of

Equation 2. This gives us:

𝜆𝑃𝑡𝑜𝑡𝑎𝑙 = 𝑃𝐾𝐸 + 𝑃𝑃𝐸 + 𝑃𝑟𝑜𝑙𝑙 + 𝑃𝑎𝑒𝑟𝑜 + 𝑃𝑏𝑒𝑎𝑟𝑖𝑛𝑔

Equation 3

The terms in Equation 3 are explained in the sections below.


Potential Energy

𝑃𝑃𝐸 = 𝑚𝑔𝑣𝐺𝑐𝑜𝑠𝜃

Equation 4

𝜃 = 𝑡𝑎𝑛−1(𝑆)

Equation 5

𝑆 =ℎ2 − ℎ1

𝑑2 − 𝑑1

Equation 6

The PE term in Equation 3 refers to the rate of work done against the force of

gravity. Force due to gravity of an object of mass m is mg. Work done against gravity to

lift that object to a height h is mgh. Rate of work done against gravity is mgv, where v is

the velocity with which the object moves in the direction perpendicular to the ground. In

our case, that is given by vG cos θ, where θ is the angle of inclination of the road (Figure

16).

Kinetic Energy

𝑃𝐾𝐸 = 𝑚′𝑎𝑣𝐺

Equation 7

where

𝑚′ = 𝑚 (1 +𝐼𝑊

𝑟2)

Equation 8


If vG is constant, the kinetic energy (KE) is constant and hence work done to

increase the KE is 0. Everything else being constant and steady, if the velocity of the bike

needs to be decreased, negative work needs to be done in the form of braking or natural

declaration. The braking would “absorb” the KE. This component of the power equation is

given by PKE. a is the acceleration at the instant. m' is the effective mass because “there is

additional KE stored in the rotating wheels (KE = ½Iw2), where I is the moment of inertia

of the wheels and w is the angular velocity of the wheels. The angular velocity of the

wheels is proportional to VG as w = vG/r, where r is the outside radius of the tire.

Therefore, the KE stored in the wheels can be expressed as KE = ½VG2/r2” (Martin et al.,

1998). Moment of Inertia of the wheels Iw can be measured or assumed. If it cannot be

measured, m' can be taken as the total mass (rider mass + bike mass) + the tire mass + rim

mass + 1/3 spokes’ mass (Wilson & Papadopoulos, 2004).

Rolling Resistance

𝑃𝑟𝑜𝑙𝑙 = 𝐶𝑟𝑟𝑚𝑔𝑣𝐺𝑐𝑜𝑠𝜃

Equation 9

Frictional force of an object at rest is proportional to the force normal to the

surfaces in contact i.e., on level ground, the weight of the object. Hence, this force can be

expressed as µmg for an object of mass m. The “coefficient of friction” µ depends on the

properties of the two surfaces.

In the same way, rolling resistance in cycling has been proven to be proportional to

the weight of the bike and rider system (Di Prampero et al., 1979). Rolling resistance is the

biggest of all resistive forces for speeds < 3 m/s (Wilson & Papadopoulos, 2004) in still air

on level ground. Proll also depends on the ground velocity of the bike, tire pressure, width


of tires, size of wheels, type of road surface, tread count, rubber type, etc. All of this is

represented in the Crr term named coefficient of rolling resistance.

“Tyre width, tread pattern, tread count and tyre inflation pressure influence

performance. Tyre pressure affects the contact surface between the tyre and the ground.

When the tyre is under inflated, the rolling resistance increases (Grappe et al., 1999)”

(Steyn & Warnich, 2014). Hence, high pressure reduces Crr. Mass does not affect it as

much as pressure does, according to the same authors.

But where does Proll go? “The main cause of this loss of energy is the deformation

of the tyre, the deformation of the terrain surface and the movement below the surface”

(Karlsson, Hammarström, Sörensen, & Eriksson, 2011).

Rolling resistance is a bigger issue in mountain terrain bikes (MTBs) because of

larger area in contact with the road, lower wheel diameter and rubber thickness. In road

bikes, this force is 2-3 times lower (W. M. Bertucci et al., 2013). Also, since speeds are

faster in road cycling, rolling resistance is not the main focus – aerodynamic drag is.

Schwalbe, a leading manufacturer of tires and tubes, summarizes this force: “1. the

higher the inflation pressure, the lower the tire deformation and thus rolling resistance; 2.

tires with a smaller diameter have a higher rolling resistance, because a small diameter tire

is flattened more and is ‘less round’; 3. a narrower tire deflects more and so deforms more;

4. tire construction: by using less material, less material can be deformed. The more

flexible the material is, the less energy is lost through deformation. " ("Rolling

Resistance,")


The effect of tire pressure and total mass on Crr is a nonlinear relationship (Grappe

et al., 1999). These authors also claim that Crr diminishes performance up to as much as

11.4% for an increase of 15 kg in mass of the system.

Bearing Losses

𝑃𝑏𝑒𝑎𝑟𝑖𝑛𝑔 = (𝛽0 + 𝛽1𝑣𝐺) 𝑣𝐺 = 𝛽0𝑣𝐺+𝛽1𝑣𝐺2

Equation 10

where β0 and β1 are constants.

Losses in the form of rotation and friction of the bearings in the wheels and other

components is proportional to the speed of the bike. The corresponding power, then, is

proportional to the square of the velocity. But this forms a negligible part of the equation

and several studies neglect this (Di Prampero et al., 1979; Meyer, Kloss, & Senner, 2016;

Wilson & Papadopoulos, 2004). We will also neglect these losses in our study. There are

tests online on bearing losses as well as losses in the rest of the drivetrain. Some

companies also sell the findings of these tests for a price ("Friction Facts,").

Aerodynamic Drag Force

𝑃𝑎𝑒𝑟𝑜 =1

2𝜌𝐶𝑑𝐴𝑣𝑊

2 𝑣𝐺

Equation 11

where vW is the speed of the bike relative to wind. The rest of the terms are

explained shortly:

In the field of modelling, in pro cycling, in product development, and so on,

aerodynamic drag force is the most studied and scrutinized topic. This is natural because


drag force accounts for 50-90% of the forces that a cyclist overcomes depending on speed

and position (for MTBs it is less) (Chowdhury & Alam, 2012) (Peterman et al., 2015).

Drag force in fluid dynamics is directly proportional to density of the medium ρ (in

our case, density of air), area A of the object (frontal area of cyclist in head-wind

condition), geometry and properties of the surface of the object represented by ‘drag

coefficient’ Cd, and square of the relative velocity between the object and the fluid vW

(speed of bike relative to wind). This force is given by:

𝐹𝑑𝑟𝑎𝑔 =1

2𝜌𝐶𝑑𝐴𝑣𝑊

2

Cd and A, when taken separately, are completely independent quantities. A refers to

the size of rider, and Cd depends on geometric shape of rider (and clothes) (Grappe,

Candau, Belli, & Rouillon, 1998). Cd depends on Reynolds number but in the range of

speeds applicable to road cycling, it can be taken as a constant. (Di Prampero et al., 1979).

But, for simplicity’s sake, we take them to be one constant CdA, commonly referred to as

“drag area”.

Air resistance is of two forms: bluff body (normal to the surface of the rider’s body)

and skin friction (tangential to surface) (Wilson & Papadopoulos, 2004). Some authors call

it “viscous drag” and “form drag”. The former is reduced by lowering surface roughness

(Cd component), and the latter by position (A component) (Defraeye, Blocken, Koninckx,

Hespel, & Carmeliet, 2010a). Area is the most important factor against drag (Debraux et

al., 2011).

Majority of drag force is attributed to the cyclist and not the bike (Barry, Burton,

Sheridan, Thompson, & Brown, 2014). So, where does the Paero go? Much of air resistance

goes into the eddies in the wake (behind the rider) i.e., the KE of air molecules. Beyond a


vG of 7 m/s, aerodynamic drag force is almost 100% of the forces that the cyclist has to

overcome (Wilson & Papadopoulos, 2004).

Drivetrain Efficiency

No bike is 100% efficient. Apart from the bearing losses, we also know that some

energy from the cyclist’s legs also goes into hysteresis losses in the drivetrain (chain,

jockey wheels, other components) as well as bike frame itself. This phenomenon would be

factored-in in the efficiency constant λ. We will assume that this value is constant

throughout the ranges of velocity and power of our experiments. Naturally, λ is a positive

quantity, and cannot be greater than 1. Quoting from Martin & colleagues, “Frictional

losses occur in the drive chain and are related to the power transmitted. Since this loss

occurs between the crank and the rear wheel, it can be viewed as a chain efficiency factor.

Therefore, the net estimated power must be divided by the chain efficiency.” (Martin et al.,

1998).

Estimating Parameters.

Although we have to estimate three parameters for our simplified equation, we will

focus on two: drag area (CdA) and coefficient of rolling resistance (Crr). Drivetrain

efficiency does not account for much of the cyclist’s power in comparison to these two

components.

Drag Area (CdA)

One method of calculating area is weighing photographs of a cyclist by a sensitive

scale. The photographs would have to be taken without parallax error and there would

have to be a reference object with known area in the photograph for equating the weight of

the pictures to frontal area (de Groot, Sargeant, & Geysel, 1995). Once digital cameras

came into the fray, this technique was emulated on computers. Instead of weighing


pictures in grams, the pixel count of the images was taken as the measure. Naturally, this

was a much less labour intensive method. (Chowdhury & Alam, 2012) (Debraux et al.,

2011) (Peterman et al., 2015).

Cd according to one author is given by the expression: 4.45 x m-0.45 where m is the

mass of the subject (Heil, 2002). Some authors have used this equation as well as similar

ones linking surface area and height and mass (Bassett Jr et al., 1999). These approaches,

obviously, cannot be relied on because of the large variation in individuals, types of

jerseys, types of cycling positions and not to mention, the sensitivity of the terms. A study

in 2008 warns that arriving at surface area from body mass is wrong (García-López et al.,

2008).

“Towing tests” are a valid way to estimate aero-parameters in the field (Capelli et

al., 1993). The concept is that a cyclist is towed by a car or motorbike joined by a rope and

a force transducer in series. Usually this is done when wind conditions are relatively still

and the ground is level. An indoor velodrome or hallway is an ideal location for these tests.

Density is calculated from barometric pressure readings at the location. Aerodynamic

forces can be calculated after removing rolling resistance and other components of the

force from the force transducer. To simulate actual riding conditions, cyclists may “soft-

pedal” without actually applying force on the a transmission chain “to reproduce air

turbulence induced by moving legs during actual cycling” (Capelli et al., 1993).

A purely lab-based method is Computational Fluid Dynamics (CFD). CFD is used

to estimate aerodynamic properties and forces. This, though inexpensive, may be

inaccurate (Defraeye et al., 2010a). But, opinion on this is divided. Some say that CFD

allows researchers to validate as well as improve results from wind tunnel tests. “CFD will

provide the ability to compute many output forces and identify exact components which


cause the most drag, a result which is extremely hard to achieve in wind tunnel testing. It

will however be a number of years before CFD can be used to solve accurately flow

around complex moving geometries, such as a pedalling rider.” (Lukes, Chin, & Haake,

2005)

Wind tunnels are a semi-lab/semi-field test. Bikes and cyclists are “mounted” on a

platform in a tunnel with wind blowing from turbines simulating outdoor conditions. They

are used widely for aerospace applications and automobile industry and now in several

other fields including cycling. Though impressive and seemingly state-of-the-art (perhaps

due to the costs associated), wind tunnels have also been criticized: discrepancies were

reported (Grappe et al., 1998); they overestimate Cd (Defraeye et al., 2010a); wind tunnel

tests do not put cyclists under stress because of soft-pedalling i.e., some resistance should

be there to make it realistic (García-López et al., 2008); the lateral movements we see

outdoors are not present indoors while soft-pedalling (Candau et al., 1999); the continuous

variation of wind speed and wind direction is not accounted for; turbulence level is usually

very low compared to outdoor environment, mostly “static” cyclists are considered, i.e.

without pedalling and rotating wheels; the boundary layer on the wind-tunnel floor is not

present in reality (Defraeye et al., 2010a). An alternative option is to go for a scale model

in a smaller wind tunnel, saving costs (Defraeye, Blocken, Koninckx, Hespel, &

Carmeliet, 2010b).

Coefficient of Rolling Resistance (Crr)

Towing tests are also used to estimate Crr. These tests are conducted outdoors (Di

Prampero et al., 1979). They are conducted in exactly the same way as explained above for

CdA estimation.


Another method similar to towing tests is the coasting/deceleration tests (Grappe et

al., 1999). In the simplest example, if a bike (with the same rider, same posture, same

clothes, same tire pressure) with two different tires is released from the top of an incline

and “coasts” down the incline, the distance covered or the time taken can be measured in

order to find out which tire “rolls” better i.e., we can get a ratio of the Crrs of both tires.

A lab test which has been found online is done by

www.bicyclerollingresistance.com ("Tire Test - Continental Grand Prix," 2014). The

website publishes results of tires pumped to different pressures and tested on their

indigenously made rig. It consists of a drum on which the tire rolls – the drum’s surface is

made to resemble the road surface. The speed and power provided to the drum are known

and the power lost in tire deformation can be found out from the number of revolutions

completed by the tire. The load on one tire corresponds to 42.5 kg, which means that for

two tires, the total load would be ~85 kg, which is a realistic “typical” was of a rider and

cyclist system.

http://www.bicyclerollingresistance.com/


Methods

We shortlisted four OFPMs based on reviews and customer ratings from popular

websites ("CycleOps PowerCal review," 2013; "First Ride with new $299 PowerPod

Power Meter," 2015; "Hands on with a $200 Bluetooth Smart power meter and precision

distance/speed sensor," 2012; "PowerPod In-Depth Review," 2016; "PowerTap PowerCal

In-Depth Review," 2012; "Sigma Introduces the ROX 10.0 Ant+-Enabled GPS Cycling

Computer," 2013): PowerPod (Velocomp LLC, USA), Powercal (Powertap, USA), Sigma

Rox 10.0 (Sigma Sport GmbH, Germany) and Velocomputer (Velocomputer, Canada). Of

these, Velocomputer was excluded for reasons explained in the next section.

Shortlisted PMs

SRM

The benchmark PM we used is the SRM device (SRM Science Power Meter,

Schoberer Rad Messtechnik GmbH, Germany). The specific model (called “Science”)

consists of a series of 16 strain gauges with an advertised accuracy of ±0.5% ("SRM

Store,"). But according to literature, the accuracy is around 2% (Quod et al., 2010). In

general, though, SRM is widely accepted as the benchmark and gold standard for portable

PMs, which was validated using the benchmark Monark ergometers (Jones & Passfield,

1998)

The PM is in the crank-web i.e., between the bottom bracket and the crank (Figure

3). It was calibrated as directed by the manufacturer before start of each ride. Here,

calibration means eliminating the zero offset and must not be confused for a longer

calibration process that involves “recalibrating” the PM in rigs with known weights across

force and angular velocity ranges. We are only referring to the simple process of


unclipping from the pedals while the bike computer resets the offset for an unloaded-crank

condition. We assume that this is enough to retain validity of the SRM.

SRM is used for validating indoor ergometers as well, replacing the Monark

ergometers. It can be used in parallel with any ergometer since it is mounted on the

crankset. For example, the result of this study: (W. Bertucci, Duc, Villerius, & Grappe,

2005) states that a certain ergometer should not be used for scientific purposes. SRM has

that level of authority in the academic circles now. It was also used to test an indigenously-

made pedal force measurement system (Bini, Hume, & Cerviri, 2011). It is also the

benchmark for other PM manufacturers.

But the SRM PMs are not without flaws: One paper in recent years reports that the

PM cannot record power < 25 W accurately (Abbiss et al., 2009). Another is that it is not

easily portable between bikes. It needs to be compatible with the bottom bracket of the

bike, and removing and installing the PM is a long process.

Powercal

Powercal uses HR values to guess power. It is simply an HR strap (Figure 4) with

an internal algorithm that somehow converts HR data into watts. It does not have a

calibration or baseline measurement process. Calibration used to exist in an earlier version

of the product, but not anymore, as the manufacturers felt that this extra step did not add

value in terms of outputting accurate results. ("PowerTap PowerCal In-Depth Review,"

2012). The device does not take the input of any anthropometric parameters either. It

wirelessly transmits heart rate as well as power numbers to a bike computer or a mobile

phone that must be within the range of communication of the computer/mobile phone at all

times. The manufacturer offers two models that differ in the way they transmit data –

either via Bluetooth Light Energy (BLE) or ANT+ protocol. We chose the BLE version


(Figure 14) and used a mobile phone (Apple iPhone 5, Apple Inc., USA) to record data.

The BLE version works in conjunction with a mobile application (“PowerTap Mobile”)

that is available to use only on Apple Inc. products. The mobile application takes the input

of age, sex, weight and type of bike but this is only to estimate caloric energy expenditure.

For calculating the actual power numbers, there is no input taken. The mobile phone was

placed in the bag mounted on the top tube of the bike (Figure 11).

Users on online forums criticize this product. It must be noted that the

manufacturer of this product makes the more famous and successful PowerTap DFPMs.

For them, Powercal is the lowest-priced product, understandably aimed at beginners. They

present this as just a rough way to quantify training effort, and nothing more ("FAQs,").

From one review, we learned that Powercal fails in sprints, and that is expected. HR values

react with a lag when it comes to short, maximal efforts. But the review points that

Powercal performs well in long, steady efforts and that the average values are close to

actual power data. For indoor rides, though, it is reported to not be as accurate. This makes

sense because cardiovascular drift i.e., an increase in heart rate values over time is a

common phenomenon when the body is not cooled quick enough. Some reviews were

even more critical and rated it 1 on a scale of 5, saying “the Powercal isn’t a power meter,

and shouldn’t be presented as one.” ("CycleOps PowerCal review," 2013)

PowerPod

“PowerPod uses a combination of an accelerometer, air pressure, and barometric

sensors to measure the total forces opposing your motion on the bike to provide pro-level

accuracy,” claim the manufacturers. ("PowerPod power meter for cycling fitness," 2015).

The manufacturer of this device claims to be in business for 12 years.


It must be used with wireless speed and cadence sensors. It weighs 32 g (Figure 5).

It must be mounted firmly on the handlebar of the bike (Figure 12), with the pressure

sensor facing forward. PowerPod does not have a display unit of its own, but can be paired

wirelessly with a computer to display and/or store data. We did not use a monitor to view

the power numbers of the PowerPod during the experiment. PowerPod has a simple

calibration process that lasts around five minutes. The user is instructed to ride at any

speed while the device records inclination from pressure sensors and accelerometers and

synchronizes these two sensors.

Only one blogger by alias “DCRainmaker” talked about PowerPod in detail, with

glowing reviews, recommending it to beginners of the sport ("PowerTap PowerCal In-

Depth Review," 2012). There appears to be a conflict of interest here, though, but it is

declared. He is one of the backers of the project, getting a discount in exchange for

investing in the product. The author does not report accuracy figures for instantaneous

power – only focusing on average values. Instead, graphs are presented for visual

inspection. The same author also claims that it has become more popular ("PowerPod rolls

out ANT+/Bluetooth Smart version, improved road surface algorithms," 2016).

Sigma

Sigma Sport makes bike computers, among other cycling accessories. In their most

expensive bike computer (as of April 2016), the Sigma Sport ROX 10.0 (Figure 6), there is

a formula for power based on speed, elevation, and anthropometric data. The bike

computer can display power in case a PM is present, is GPS-enabled and contains a

barometric altimeter. In the absence of a PM, the computer uses its own formula to

estimate it. The formula is not revealed. “A calculated power estimation is based on factors

like bike weight, rider height and weight, shoulder width, speed, cadence, and incline, but


it can’t calculate external factors like headwinds—or motor-pacing! For this reason,

calculated power data becomes more accurate when there’s less wind and when terrain is

steeper,” reports one review ("Sigma Introduces the ROX 10.0 Ant+-Enabled GPS Cycling

Computer," 2013).

The anthropometric and bike data that this device asked for can be seen in Table 2.

This computer was also mounted on the handlebars of the bike, with live feedback to the

rider (Figure 11). The same ANT+ speed and cadence sensors (Garmin Ltd., USA) as for

PowerPod were also paired with this device (Figure 13). The computer estimated speed

from GPS data when the sensors were not available. There was no calibration process.

Velocomputer

Velocomputer was a PM introduced four years ago ("Hands on with a $200

Bluetooth Smart power meter and precision distance/speed sensor," 2012) with the claim

that it could estimate power from bike speed and accelerometer data. The product was a

magnet-based speed and cadence sensor with a built-in accelerometer to detect road

inclination. The device could only be used with a mobile phone application (called

“Velocomputer”). The production and support of the device was discontinued, as we

learned just before the start of the experiment. It supposedly made use of mass,

acceleration, a constant CdA, Crr and used vG as vW in the power equation.

This device was discontinued and, in its place came a magnet-less speed and

cadence sensor set that could pair with any number of computers or mobile phones via

Bluetooth and/or ANT+ protocols, including the proprietary mobile application

Velocomputer. We purchased this as well. We faced the following issue: power values

were clearly wrong or sometimes even negative. We guess that this is because the formula


for power was designed for an earlier version of the PM where elevation data was also

taken into account. There were more issues:

the mobile phone would lose connection with the speed and cadence

sensors multiple times and, hence, continuous data could not be recorded

the new version of the mobile application would not sync with the old

product, and the old version of the application was no longer available

there was no support offered from the manufacturers even after repeated

efforts to contact them. The last update to their mobile application was on

5th January, 2016 ("Velocomputer,")

The company has moved on from being a self-proclaimed PM to being a

manufacturer of speed and cadence sensors. They received several poor reviews from

consumers ("Velocomputer,"). This reinforced our belief that the product was unreliable.

Attempts to understand the reasons behind these issues by way of reaching out to

consumers on online forums also yielded no results either.

Routes

For Part 1, we chose a 33-km round-trip route in the Thurgau canton of

Switzerland (Figure 7). This stretch of road had hardly any traffic in the evenings. This

helped in maintaining a near-non-stop ride. Being a round-trip, the ride started and ended

at the same point, with a net elevation gain and loss of 333 m. It fulfilled our conditions of

having a climb section (5.9 km long) and a flat section (3.6 km long). The surface was

asphalt and uniform for the most part. The ride took roughly 75 minutes. We recorded

three rides on this track for Part 1.

For Part 2, we chose a shorter (15.5 km) but similar track to the one in Part 1

(Figure 8). It was in the outskirts of the German town of Constance. It was also a round-


trip, had a climbing section and was on near-uniform asphalt roads. The total elevation

gained and lost was 148 m.

Subject and Setup

The subject (Figure 9) was the author himself – a 27-year-old moderately-trained

male (1.72 m, 68 kg – mass includes clothes and accessories just before the start of the

rides of the experiment) cyclist with more than 5 years of experience in road cycling, with

an average of 5 h of cycling per week in the last 2 years. From experience, the subject

knew that weight loss through dehydration is at a rate of 0.5 kg/hour for summer

conditions for a submaximal ride. Hence, the mass of the system was deducted at the rate

of 1 kg/h Part 2. Other anthropometric data can be found in Table 2.

Clipless pedals (Shimano PD-R540 SPD SL Sport Pedals, Shimano Inc., Japan)

and cleated shoes (Shimano RP2 SPD-SL, Shimano Inc., Japan) were used. The PMs were

mounted on the road racing bike of the lab (Radon GmbH, Germany). The bike frame

material was an aluminium alloy. The bike was one size larger (56 cm) than the subject

was used to but was comfortable to ride. The tires used on the bike were standard road

bike training tires (Continental Grand Prix, Continental AG, Germany) and were used at a

tire pressure of 110 psi (7.58 bar or 758 kPa). They were pumped using a manual floor

pump and by visual inspection of a dial gauge (LifeLine pumps, Wiggle Ltd., UK). The

weight of the bike – including bottles, water, and accessories – was 13 kg for all rides of

the experiment (Figure 15).

The computers and mobile phones were all charged fully and calibrated if, and

when required before the start of every ride. The 3 rides on the longer track were

conducted within 11 days of each other. The bike set up was not changed in all this time.


Experiment

Part 1: The constraint for the subject was to ride in the hoods position so as to

maintain a constant frontal area (Figure 10). Hoods are not a racer/professional type of

position – it is a position that most beginners adopt. No other conditions of choice of

gearing, or cadence were imposed. The next instruction was to ride in a range of power

(~200 W) – not a constant effort, but a steady effort – for the climbing and flat sections.

The value of 200 W is well below the FTP of 240 W of the subject. At all other times, the

instruction was to ride “normally”, with no regard to the constant pedalling, breaks,

braking, and so on. Except for three sprints in the end, the effort was submaximal

throughout.

Part 2: This ride (Ride 4) was seven weeks after Ride 1 of Part 1. The same

instructions as for Part 1 applied for this ride.

The SRM PM was paired with a bike computer (Garmin Edge 510, Garmin Ltd.,

USA). This computer was mounted on the handlebars to always have visual feedback of

power. The computer stored the power averaged over one second.

We think that this duration and style of both parts best mimics amateur riders – the

target consumers of the OFPM companies. The clothes and shoes used for the study were

similar for all four rides – each piece of garment was of the same level of tightness – but

not identical.


Analysis

Sampling Frequency: All power data was stored at a frequency of ~1 data point per

second. The sampling frequency of each PM was higher than 1 Hz, but it is common to get

the averaged power over one second.

Software:

MS Excel (Microsoft Corporation, USA) was used for organizing data,

simple arithmetic operations and producing graphs.

MATLAB (MathWorks, USA) was used for the least square analysis of the

parameter estimation of Part 2. We used the lsqlin function. This function

calculates the solution to the set of equations such that the sum of squares

of the estimated power and actual power is the least.

Golden Cheetah, an open source software, was used to convert file types to

.csv format.

Sigma Software (Sigma Sport, Germany) was used to export the workout

files stored by the Sigma.

Isaac Software (Velocomp, USA) was used to view and export the

PowerPod files to .csv format.

For the Powercal, it was the online software VirtualTraining (PowerTap,

USA).

Garmin Connect (Garmin Ltd., USA) was used to upload files from the

Garmin device for power data from SRM, elevation and ground speed. We

downloaded the workout file to a personal laptop computer (Lenovo Group,

China).

File types:


1. SRM: Garmin Connect was used to export files to .fit or .tcx file type.

Golden Cheetah was used to convert it to a .csv file

2. Powercal: The HR and power numbers were downloaded in .tcx file format.

Golden Cheetah was used to convert this file to the .csv file format

3. PowerPod: The file format of the recorded workout is .ibr. “Isaac Software”

exports the files to a .csv file format

4. Sigma: The file can be exported to the .tcx file format from Sigma’s

interface. Golden Cheetah was used to convert to a.csv file format

Metrics

Average Power and Relative Error

Average power is the arithmetic mean of all the data points recorded by the

respective PMs. Standard Deviation (SD) is also mentioned. Both are expressed in Watts.

Relative error is given by the formula:

𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝐸𝑟𝑟𝑜𝑟 (%) = 100 ∗(𝑃𝑜𝑤𝑒𝑟𝑂𝐹𝑃𝑀 − 𝑃𝑜𝑤𝑒𝑟𝑆𝑅𝑀

)

𝑃𝑜𝑤𝑒𝑟𝑆𝑅𝑀

Equation 12

Normalized Power

Normalized Power (NP) is a popular metric in coaching and is described as “In

essence, it is an estimate of the power that you could have maintained for the same

physiological "cost" if your power output had been perfectly constant" (Andy R. Coggan).

NP is calculated as follows: take the 30 second average of power; raise these to the power

of 4; take the average of these values; take the 4th root of this average. It is given by


𝑁𝑃 (𝑊) = √∑ 𝑃𝑜𝑤𝑒𝑟

30𝑖4𝑇

𝑖=30

𝑇

4

Equation 13

Root Mean Square Error

Average power and NP are interesting to note but as sports scientists and athletes, we are

more interested in the accuracy of instantaneous power. Hence, we want to track the Root

Mean Square Error (RMSE), expressed in Watts, which is given by

𝑅𝑀𝑆𝐸 (𝑊) = √∑ (𝑃𝑜𝑤𝑒𝑟𝑆𝑅𝑀𝑖 − 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑𝑃𝑜𝑤𝑒𝑟𝑖)2𝑇

𝑖=30

𝑇

Equation 14

It is the square root of the average of squares of difference between actual and

estimated power at every second.

Relative RMSE

If we want to quantify the error in instantaneous power relative to actual power, we

introduce relative RMSE, given by

𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑅𝑀𝑆𝐸 (%) = 100 ∗√∑ (

𝑃𝑜𝑤𝑒𝑟𝑆𝑅𝑀𝑖 − 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑 𝑃𝑜𝑤𝑒𝑟𝑖

𝑃𝑜𝑤𝑒𝑟𝑆𝑅𝑀𝑖)

2𝑇𝑖=1

𝑇

Equation 15

Naturally, this is valid only for non-zero power. We went a step further and ignored

data where actual power was < 25 W because for small values of power, the above formula

tends to large values. SRM PMs are also known to be inaccurate for power values below

25 W (Abbiss et al., 2009).


Peak Signal-to-Noise-Ratio

Signal-to-noise-ratio (SNR) essentially compares the useful part of a signal with

the noise in a signal. Peak Signal to Noise Ratio (PSNR) has different formulae. This is

one formula, expressed in decibels:

𝑃𝑒𝑎𝑘 𝑆𝑖𝑔𝑛𝑎𝑙 𝑡𝑜 𝑁𝑜𝑖𝑠𝑒 𝑅𝑎𝑡𝑖𝑜 (𝑑𝐵) = 20 𝑙𝑜𝑔10 𝑃𝑒𝑎𝑘𝑃𝑜𝑤𝑒𝑟𝑆𝑅𝑀 − 20𝑙𝑜𝑔10𝑅𝑀𝑆𝐸

Equation 16

Energy

“The total mechanical energy expenditure is calculated by integrating the power

measurements over the entire training session.” (Vogt et al., 2006). In our case, data was

recorded in intervals of 1 s. So, Energy is the sum of all the data points, expressed in

Joules, given by

𝐸𝑛𝑒𝑟𝑔𝑦 (𝐽) = ∑ 𝑃𝑜𝑤𝑒𝑟𝑖

𝑇

𝑖=1

Equation 17

Estimating Parameters

For estimating power based on our own formula, we first had to calculate the

parameters (λ, CdA and Crr) from the rides in Part 1. In total, we had a total of ~3 x 75 =

225 minutes of power, speed and elevation data. From this, we removed data with power <

25 W and cadence 0 RPM.

Substituting in Equation 3 and rearranging, we get:

𝑃𝑆𝑅𝑀 =1

𝜆(𝑚′��𝑣𝐺 + 𝑚𝑔𝑣𝐺𝑠𝑖𝑛𝜃) +

𝐶𝑟𝑟

𝜆(𝑚𝑔𝑣𝐺𝑐𝑜𝑠𝜃) +

𝐶𝐷𝐴

2𝜆(𝜌𝑉𝐺

3)

Equation 18


Equation 18 now takes the form

𝑦 = 𝑘1𝑥1 + 𝑘2𝑥2 + 𝑘3𝑥3

Equation 19

Where k1, k2, and k3 are three unknowns and yi, x1i, x2i, x3i are the knowns.

𝑦𝑖 = 𝑃𝑆𝑅𝑀𝑖 = 𝑘1𝑥1𝑖 + 𝑘2𝑥2𝑖 + 𝑘3𝑥3𝑖

Equation 20

𝑘1 =1

𝜆 𝑘2 =

𝐶𝑟𝑟

𝜆 𝑘3 =

𝐶𝐷𝐴

2𝜆

Equation 21

𝑥1𝑖 = 𝑚𝑖′��𝑖 + 𝑚𝑖𝑔𝑣𝐺𝑖

𝑠𝑖𝑛𝜃𝑖 𝑥2𝑖 = 𝑚𝑖𝑔𝑣𝐺𝑖𝑐𝑜𝑠𝜃𝑖 𝑥3𝑖 = ��𝑖𝑣𝐺𝑖3

Equation 22

In theory, we only need three equations to find out these unknowns given above.

But we have thousands of data points in order to calculate the “best fit” for these

unknowns.

We measured moment of inertia Iw for our bike another study (unpublished work of

Thorsten Dahmen) and its value is 0.18 kgm2. That made m’ = 82.6 kg at the start of the

ride. Average density was found to be 1.12 kg/m3, 1.15 kg/m3, 1.13 kg/m3 and 1.13 kg/m3

for Rides 1-4, respectively and we used these values in Equation 18. We used 9.81 m/s2 for

the value of g. We do linear regression – reported to be a good way to estimate parameters

to “provide best match of parameters” (Dahmen et al., 2012). We set upper and lower

limits for the values that these parameters can assume, based on physical constraints and

values obtained from literature (Table 3). Since two of the three PMs in Part 1 use bike


speed in their formula (instead of wind speed, which is the correct method), we decided

that to be able to have a fair comparison between our model and those PMs, we must use

bike speed as well.

When we finish the lsqlin algorithm, we get parameters that are put back in the

formula. Note that this formula also returns negative values for power – as that is the

nature of the algorithm. Since it is not possible to have negative power from a cyclist’s

legs, we take these instances as 0 power instances. This mostly happens when the road is

sloping down too much or when there is a sharp deceleration (in the form of braking,

perhaps).


Results

Part 1

Full Ride

We provide results for Ride 3, which was representative of Rides 1-3 and all the

tests we conducted.

The average power recorded by SRM for the full ride was 156 W, which reflects

that it was a sub-maximal ride. To get a visual idea of the trends of the OFPMs, Figure 17

shows power vs time of each PM averaged over 30s. It is clear that, on average, Powercal

overestimates power, Sigma underestimates it while PowerPod is close to the actual

power. The average power values (Table 5) confirms this suspicion. PowerPod's average is

indeed very close to real values, as foretold by reviews. In all metrics, PowerPod stands

out from the other two. NP is higher for PowerPod, suggesting that there were errors in

estimating low power but it is 10 times more accurate than the other two PMs. While

average values are impressive, the RMSE and relative RMSE and both high in absolute

terms – 47 W and 44%, respectively. Once again, it is better than the other two by a

margin.

Training Zones

We split the based on time spent in power zones described in Background chapter.

Figure 22 shows the histograms for each OFPM. This gives a visual idea of the intensity

of the ride – that it is submaximal. A maximal ride would be between zones 3 and 4.

Climbing Section

In this section, which was 3.6 km long and 13.5 min long with an elevation gain of

147 m, similar trends were observed for PowerPod and Sigma as for the full ride – the

former was close to actual power while the latter was underestimating. Powercal, on the


other hand failed to show any real correlation, at least visually (Figure 18). PowerPod got

slightly worse in terms of average values (Table 5). RMSE and relative RMSE dropped

heavily in this section - probably because of fewer data points. Once again, PowerPod was

most accurate of all.

Flat Section

Flat section was the only time that Sigma was closely correlated to SRM in certain

sections within the flat section (Figure 19). This also happened to be the time that speed

was relatively steady and unchanging (see Discussion). Powercal was overestimating

power once again, and PowerPod looked to be the same. Sigma was better than the other

two in estimating NP, with an error of -5.26% (Table 5). But in the other errors, PowerPod

turned out to be the most accurate.

Normalizing

After noticing a steady trend of over- and under-estimating power by Powercal and

Sigma, it was suggested (by Alexander Artiga Gonzalez) to study the data after subtracting

the respective means from the datasets. Perhaps the power differed by a constant? There

are indeed reductions in the RMSE and PSNR for Powercal and Sigma (Table 6), but still

not enough to be better than the PowerPod. Notice also that the error values for PowerPod

did not change even after this transformation.

Smoothing

To give a benefit of doubt to the OFPMs, we decided to observe the trends in errors

when the data is smoothed i.e., when a moving average of power numbers is taken. We

averaged the data over 3s, 5s and 30s. Beyond 30s, we do not think that athletes,

consumers or scientists would be interested. The expectation was that the errors would


drop with progressive smoothing, and it was true (Figure 20). The PowerPod continued to

be the best of all PMs.

Sprints

Figure 21 shows the unsmoothed curves for the power output of the PMs for the 3

sprints. Each sprint was studied for 30s. If at all, only the PowerPod came close to the

SRM, with Sigma and Powercal not coming close to estimating the max power. We even

allowed for a lag of at least 10s just in case the Powercal decides to react late. This did not

happen. Table 7 summarizes the max power recorded in the 30s window and the PowerPod

did impressively well in one trial with an accuracy of 3%.

Part 2

During Ride 2 of the three rides in Part 1, we noticed that the parameters estimated

were slightly off (Table 4). when investigated, it turned out that the reason this could be is

the weather conditions. When the difference between vW and vG is taken, if it is positive,

we can consider it as headwind i.e., Table 8 shows the duration for which there was

headwind and tailwind. Clearly, Ride 2 was in conditions that were different from the other

rides. This is reflected in the parameters too. Hence, we decided to exclude Ride 2. We

took the average of the parameters obtained from Rides 1 and 3, and substituting for k1, k2

and k3 in equation 19, we get

𝑃𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑𝑖 = 1.0101(𝑚′𝑎��𝑣𝐺𝑖 + 𝑚𝑔𝑣𝐺𝑖𝑠𝑖𝑛𝜃𝑖) + 0.0095(𝑚𝑔𝑣𝐺𝑖𝑐𝑜𝑠𝜃𝑖) + 0.1429(𝜌𝑉𝐺𝑖3 )

Equation 23

Hence, our final equation is as follows

𝑃𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑𝑖 = 83.43𝑎��𝑣𝐺𝑖 + 802.64𝑣𝐺𝑖𝑠𝑖𝑛𝜃𝑖 + 7.55𝑣𝐺𝑖𝑐𝑜𝑠𝜃𝑖 + 0.16𝑉𝐺𝑖3

Equation 24


According to our predicted power formula, 40% of the power went in overcoming

aerodynamic forces, 30% to rolling resistance and the rest to changes in KE, overcoming

gravity, bearing losses and drivetrain losses.

Full Ride

Total distance was 15.5 km, net elevation gain was 148 m and time taken was ~35

min. Figure 23 shows the graph for the smoothed power output vs time (time not indicated

on x-axis). Visually, we can tell that our prediction grossly underestimates power in more

than a couple of sections. Towards the end of the ride, when speed was slightly high and

power output was more than average, our prediction overestimated power. This could be

due to the fact that there was a tailwind, and our formula cannot account for it. The graph

also shows the other OFPMs during the same ride. Our prediction is second best, visually

speaking.

Table 9 shows the summary of the metrics tracked. Predicted power fared well in

the RMSE, with a value of 45.71 W, which is better than that of PowerPod. In most other

metrics, PowerPod fared better but only by a small margin. These are promising results.

The trend was that predicted power underestimated power by a small factor. NP was very

high, suggesting many 0s in our power prediction. This can be attributed to the fact that

when Pprediction was negative, we took it to be 0. This is the result of using the least square

fit algorithm to solve for the system of linear equations.

Climbing Section

Total distance was 1.6 km, time taken was 6.5 min and total elevation gain/loss was

81 m/11 m. Figure 24 shows the graph of smoothed power vs time for the climbing

section. There is an underestimation of power followed by an overestimation, but they

seem to be within acceptable limits because these differences do not show in the numbers,


and in comparison, predicted power is the best. Table 9 shows the summary of metrics.

Predicted power was most accurate in quite a few of the metrics, except of average values

once again. This is again attributed to the fact that we end up with negative power values,

which we truncate to 0.

Smoothing

Figure 25 shows the effect of progressively smoothing data on the RMSE. With a

healthy error for instantaneous power, the error does not reduce at a rate fast enough when

compared to the other OFPMs. But, RMSE for instantaneous power is far better for

predicted power when compared to Powercal and Sigma.


Discussion

Part 1

Powercal

This study (Vogt et al., 2006) found that energy estimated from HR data is over- as

well as under-estimated depending on the power zone. This happens even when adjusted

for individuals. Hence, using HR to estimate power is flawed in the first place. And when

there is no distinction between individuals, it is worse. Such a device can surely be

expected to fail under many conditions: it would fail for sudden changes in speed, for

instance. In downhill periods when HR is still high and the cyclist is not pedalling, it

would output a non-zero power value (since it anyway does not take cadence into account).

The reviews said that in long workouts with a steady output, power numbers are estimated

well, but we did not find this to be true. Powercal presumably takes not just the absolute

value of HR but also the changes in HR. Any such pattern or correlation is very hard to see

in the plot of HR and power over time (Figure 18; Figure 19).

Does Powercal account for cardiovascular drift i.e., the increase in HR over time

for the same power output? Probably not. This is designed as a plug & play device, i.e.

theoretically, a user could be 3 h into their workout and strap this on and still get a power

reading without knowing any history (neither of the workout, nor of the athlete) or asking

for input. Even the calibration process that once existed was done away with (see

Methods) because the manufacturers thought that it was not worth the effort for the user.

PowerPod

The average power readings were excellent and were surely in the advertised range

between -2.89% and +2.77% off actual power for submaximal rides between 2 and 3 hours

("PowerPod power meter for cycling fitness," 2015).


PowerPod takes CdA as a constant value of 0.332 m2. This value was probably

arrived at from tabulated data from literature based on the anthropometric data of the rider,

bike type and riding position.

Algorithms that want to estimate power based on ground speed go wrong (as we

see with the Sigma) because wind is not accounted for. For modelling in still air, this

approach works. But for real-world tracks, estimating power based on ground speed would

not work. PowerPod passes the first test. The second test, that of yaw angle is still a

challenge to be met. Wind speed also comes with a yaw angle (Figure 26). PowerPod’s

dynamic pressure sensor only measures wind that is in the direction perpendicular to the

sensor i.e., in the direction of the ride. This is a flaw because it neglects cross-winds. In

heavy cross-wind conditions, the PowerPod can be expected to drop in its accuracy

because the high drag forces in the actual direction of the wind would be neglected (C. R.

Kyle & Burke, 1984) and the effective frontal area, that changes with yaw angle, is also

measured wrongly (Bassett Jr et al., 1999).

The device has two pressure sensors – a static pressure sensor and a dynamic

pressure sensor. The dynamic pressure (1

2𝜌𝑣𝑊

2 ) is directly plugged into their equation, thus

avoiding errors in measuring density and velocity separately. This is one of the strengths of

the device. The other major strength is the way the device uses accelerometer, discussed in

Part 2.

Data from the static pressure sensor and thermometer is used to measure density,

which is then plugged into dynamic pressure to “back calculate” speed with respect to the

wind, but this is only done to provide this data to the user. The device and algorithm

actually do not need this information to predict power.


Sigma

Sigma generally underestimates power. In the flat sections, it sometimes

overestimated power probably because of the high bike speed (power is proportional to the

cube of speed).

The only consolation for the Sigma was that it worked better than the other two

OFPMs under one special condition: when speed/power was steady and terrain was flat

(Figure 19) with RMSE = 17 W for a 200 W effort over 120 s.

The review, based on which we shortlisted the device also says that “Calculated

power is based on a formula that measures a rider’s progress over time; it’s a best

estimation of power output displayed in watts."("Sigma Introduces the ROX 10.0 Ant+-

Enabled GPS Cycling Computer," 2013). This means that the Sigma device perhaps had an

algorithm that “learns” and adapts to the rider. How it does so is not revealed, and neither

is our study long enough to investigate this aspect of the algorithm. For the record, for the

few rides that we recorded and the several trials that we conducted in preparation for the

experiments, the Sigma did not improve dramatically in its accuracy (we are talking about

a period of three months).

We noticed that when the Sigma Software gave us the file, the time entry was in

increments of 0.9-1.1 seconds and not necessarily in 1 second intervals. When Golden

Cheetah was used to convert it to .csv file type, it shows us data in increments of 1 s. We

suspect that this kind of rounding up/down done by Golden Cheetah could have accounted

for an error. But it is not easy to say to what extent this rounding process affects it.


Notes

These OFPMs cannot be used for scientific purposes. As explained in the

Background chapter, accuracies expected for scientific PMs are much higher than what we

found.

In studies comparing PMs, power over a range of different cadences is recorded for

validation. But, for OFPMs, no such thing needs to be considered. In fact, high cadence,

which leads to high HR, will only result in higher power values in Powercal. Other than

that absurdity, both Sigma and PowerPod do not use cadence in their formula, except for

the fact that when cadence is 0, power is automatically taken to be 0. They only use

cadence as a check.

More importantly, most studies comparing PMs verify accuracy of power

measurement across a range of power values. This is one of the limitations of Part 1. We

used only one subject for comparing PMs. It is common practice to recruit only one

subject (Bouillod et al., 2016) (W. Bertucci, Duc, Villerius, Pernin, et al., 2005). Rather

than recruiting more subjects, we should have checked for validity and accuracy of the

PMs across different power brackets, and not just submaximal as we did.

We chose to work on the road bike because: 1. Powerbike lab works with road

bikes, 2. most research is conducted on road bikes (except for some rare cases (W. M.

Bertucci et al., 2013)), 3. It is far less complicated to work with a relatively stiff road bike

frame on a relatively smooth surface, both being near ideal-conditions where modelling is

highly simplified. Perhaps most research is conducted on road bikes because road cycling

is the sport that is very obsessed with getting faster for the same power output, due to the

high aerodynamic resistance (as a percentage of total resistance). In Mountain Biking, it is


less to do with cutting corners in technology or position on the bike and more to do with

skill, technique, experience, etc.

It must be stressed again that the Sigma Rox 10.0 is not the flagship product of the

parent company. Neither was Powercal for their parent company. In some sense, it was

perhaps an unfair comparison to make between Sigma, Powercal and PowerPod because

the parent company of PowerPod does nothing else but make the device, and have been

improving over the last 12 years.

One of the reasons we chose to study OFPMs is because they are affordable for the

beginner. But prices of DFPMs have only been dropping over the last few years ("Power

Meter Pricing Wars: Let The Games Begin,"). In fact, the only company to have ever

hiked the price of a PM was the company that produced the cheapest PM. They hiked their

price from USD 350 to USD 399. Other than this, every single PM has dropped its price

when compared to three years ago. We guess that the reasons for this are the following:

better production techniques, cheaper sourcing of material, economies of scale, highly

competitive market, and a high demand for PMs driving a high supply, and a combination

of these reasons driving each other. Does this mean that eventually we will see a day when

DFPMs become so affordable that the one USP (“Unique Selling Point”) that OFPMs have

i.e., affordability, will be lost? Only time will tell.

As a closing note and as a final word of advice purely from a consumer point of

view, I would like to quote Rob Kitching of CyclingPowerLab: “Certainly no power meter

delivers the last word in accuracy; all accuracy is relative to the extent that the prominent

goal is often consistency. And certainly, all power meters are absolutely dependent on

good calibration. It’s fair to say that the opposing force approach does require more and

continuous attention to certain calibration inputs to deliver good and consistent numbers...


It’s really down to how you interpret the potential limitations of each device, the

characteristics you value and how much money you want to spend.” ("From Power Models

to Opposing Force Power Meters and the iBike Newton+,"). Knowing under what special

conditions an OFPM works well is the recommendation for consumers.

Part 2

Exclusion Criterion

We excluded Ride 2 because it was too windy (Table 8). It is common for scientists

to suspend or exclude data when conditions are windy (W. M. Bertucci et al., 2013;

Peterman et al., 2015). These studies focused on measuring parameters in still air. While

avoiding wind altogether was not our aim, it was physically impossible that parameters

estimated from Ride 2 in the context of the parameters obtained from Rides 1 and 3.

Sensor Data

Ground Speed from ANT+ sensor: The principle of the sensor is as follows: it is

mounted on one of the wheel hubs (front or rear) of the bike. The in-built accelerometer

counts the number of revolutions. For every revolution completed, the sensor transmits the

information to the bike computer. The bike computer, in turn, knows the wheel size (i.e.,

circumference) that was manually input by the user – 2105 mm in our case. This value is

displayed and stored as speed for the user in the workout file. While it is accurate enough

for the needs of a cyclist and/or coach, there may be an issue when we try to use this speed

data to calculate acceleration and use that for modelling power.

Acceleration: We computed the average bike speed over 3 s. Acceleration was

calculated as the rate of change of this average velocity. This provided a smoothing of the

acceleration data, and helped in lessening the erratic nature of predicted power.


Elevation: The products in the earlier generation of PowerPod’s history used

accelerometers with sampling frequency of 800 Hz ("From Power Models to Opposing

Force Power Meters and the iBike Newton+,"). PowerPod’s five minute calibration

process exists to calibrate this same accelerometer to the pressure sensors to get baseline

elevation values and to account for any “out of level mounting issues.” ("PowerPod power

meter for cycling fitness," 2015). This way, gradient and elevation data are calculated far

more accurately than by barometric altimeters that are found in bike computers. For our

tests, we used this accelerometer gradient data for two of the rides (Ride 3 and Ride 4)

when it was available. When it was missing (due to false calibration perhaps), we used

smoothed elevation data from the Garmin device. The illustration and method of

calculating slope is given in Figure 10. We used height and distance difference between

points 15 seconds apart in order to get smoother and realistic slope values. This helped in

reducing the erratic pattern in the predicted power. Using instantaneous elevation data

from the barometric altimeter of the Garmin Edge 510 cycling computer records elevation

data with a lag – slower than what we need for instantaneous power.

Density: A number of authors (Bassett Jr et al., 1999; Martin, Gardner, Barras, &

Martin, 2006; Meyer et al., 2016; T. Olds, 2001; Peterman et al., 2015) did not take the

standard value of density of dry air at sea level as 1.22 kg/m3. They all stress on the fact

that since aerodynamic drag is usually the most important resistance to overcome in road

cycling, this term in the equation is sensitive and that density must be calculated at the

location for the specific weather conditions (mainly temperature, humidity, pressure). Most

often, they calculated density from peto tube measurements at the spot, or used

information from local weather stations. We also did not use the default value, since we

have density data from the PowerPod device during each ride, which it calculates from the

static pressure port present inside. This is the only information we used to get an


“advantage” over the OFPMs that used a default value of air density. Average density was

found to be 1.12 kg/m3, 1.15 kg/m3, 1.13 kg/m3 and 1.13 kg/m3 for Rides 1-4, respectively.

Drag Area (CdA)

All methods - coasting, towing, wind tunnel etc. have all been found to be valid

ways to model power. Table 10 shows results obtained for CdA estimation in literature

from authors in the past few decades using various methods. Of these, only two studies

were within 0.01 m2 of our CdA (0.283 m2).

There are many reasons for such a wide range of values (0.2-0.4 m2) as seen in the

table: there is large variation in estimating CdA because of anthropometric differences, bike

type, accessories, static and dynamic conditions (Defraeye et al., 2010a); jersey design

affects aerodynamics in cycling i.e., lose and tight clothing and even choice of fabric plays

a crucial role in CdA (Oggiano, Troynikov, Konopov, Subic, & Alam, 2009); helmet

design also affects drag area (Barry et al., 2014); it is difficult to exactly reproduce

position and obtain same CdA even within individuals (García-López et al., 2008); even for

one position, there are significant differences between individuals in drag area. (Lim et al.,

2011) Hence, mean values like those calculated by some authors listed in the table mean

nothing.

CyclingPowerLab says 0.361 m2 is CdA for a subject of our anthropometric data

and riding style. But we know from all the above evidence that we must be wary of using

empirical data.

For further validation, hiring a much smaller wind tunnel to test scale models

would be one recommendation (Defraeye et al., 2010b). CFD analysis would also be a

recommendation but the accuracy of modelling the shape and surface of human beings in

motion (even a cyclic, so-called “predictable” motion like pedalling) is questionable and


needs to be investigated. The topic is quite vast and the author of this thesis has put

together a proposal for doctoral research on this very topic in recent months.

For some reason, if we want to know Cd for each ride, we could imitate what these

authors did: Step 1: estimate CdA from field test; Step 2: Divide drag area by frontal area

i.e., Cd = CdA/A. They found using this method that there are individual differences in drag

coefficients (Peterman et al., 2015).

PowerPod used CdA as 0.332 m2 for our subject. Isaac Software has a feature to do

CdA using actual power data. We did not go into this feature as the method itself is not new

to us – they would use linear regression as well. Also, the main use of PowerPod was

thought to be limited to Part 1 and not for parameter estimation. The next step and

recommendation would be to use PowerPod (or a similar device) in conjunction with a

DFPM to better estimate parameters. With the resources and powerful tools of the

Powerbike lab, we are certain that much use can be made of this device. The idea and

product are cheap, already available in the lab, do not require much computing time, and

support from the manufacturer is guaranteed.

Coefficient of Rolling Resistance (Crr)

Table 11 shows a snapshot of Crr data recorded by several authors in the past. Of

these, our value of 0.0094 came close to those obtained by four authors. This validation,

based purely on comparison, is more promising than the CdA estimation, which was

comparable to only two studies. Since some of them lump the Crr as a comprehensive

coefficient for other forms of losses, it is hard to compare Crr across studies (Zhan, Chang,

Chen, & Terzis, 2012).

The problem with using results from coasting or towing tests is that the results are

limited to the particular surface in question, for the particular pressure and that particular


mass of the bike and cyclist system. This limitation will always be the problem. This

limitation exists for our findings as well. We determined Crr for only one type of surface

i.e., smooth tarmac. Once again, these experimental conditions are far from real conditions

where riders change and experiment with pressure, tubes, brands, and models all the time.

The testing methodology itself is not flawed, but the approach is impractical.

The issue with using results from lab tests on rollers is that they are not being

tested on a realistic riding surface, so the deformations caused cannot be the same.

In any case, the rolling resistance term in the power equation contributes only 10-

30% of the total resistance. Only at speeds of less than 3 m/s (Wilson & Papadopoulos,

2004) in the conditions of level road and still air is the rolling resistance the highest force

to overcome.

A very interesting concept of using a dynamic i.e., a non-constant Crr has been

introduced by two parties: ("PowerPod rolls out ANT+/Bluetooth Smart version, improved

road surface algorithms," 2016) and (Meyer et al., 2016). The concept uses vibrations

measured at the handlebar level in the z-direction (i.e., perpendicular to the riding surface)

by the PowerPod or by a smartphone, respectively, to come up with an ever-changing Crr.

Using accelerometers of the calibre of 800 Hz sampling frequency once again, the

PowerPod claims improved results from this approach. PowerPod device, which was not

recommended for use on shaky surfaces like cobbled streets can now be used anywhere if

they are to be believed. At the moment, Isaac Software takes a default value of 0.005 based

on the pressure input by the user. But it would be interesting to note how far off this value

the “actual” dynamic Crr is. We could not get into this aspect of the product because the

upgrade in the device came after our experiments had already been conducted: Sep 5,

2016.


Tire pressure can and does change during a long ride (G. P. Millet et al., 2014). We

did not measure pressure at end of our rides, and this could be a limitation. This also goes

in line with a “non-constant” Crr approach – either account for pressure change and take a

constant Crr, or account for a changing Crr and take pressure constant.

A crucial point that several studies do not list is bump losses. When vibrations and

bumps are encountered on a ride, they are most likely absorbed by the rider, and no simple

formulae exist to quantify this (Wilson & Papadopoulos, 2004). The authors of this

reference say that it is also a function of how well the rider handles the bike, which is a

function of experience in some cases.

Drivetrain Efficiency (λ)

97.6% drivetrain efficiency was assumed by (Barry et al., 2014) in their work. In a

way, we also assumed a value just by decided to set upper bounds in the MATLAB

program. The best results were obtained for whatever upper bound we entered in our

iterations: between 95% and 99%. We got a 99% efficiency since this was the upper bound

we chose.

97.7% was the chain efficiency that Martin & colleagues took for both their studies

more than a decade apart. One was for modelling average power and the other was for

modelling sprint performance. It is hard to believe that the same value can be taken given

how many leaps and bounds bike technology has made in terms of frame stiffness and

efficient components in this period. The only explanation here that we find convincing is

that the losses are negligible. For our value of 99% efficiency, and Martin & colleagues’

value of 97.7%, for a ride with average power ~150 W, we are only looking at an error of

1.95 W.


Once again, it is common for authors to routinely club drivetrain losses, bearing

losses and perhaps even rolling losses into one lumped coefficient. This makes us wonder

how small the losses actually are. In a way, we also lumped the bearing losses as well as

chain and frame efficiency into one coefficient. Such a lumping resulting in only 1% loss

is likely not possible physically. But we accept that our power prediction is based on very

simple ideas.

Predicted Power

The final step of validating parameters is a field test. Our field test and results

amply proved that the parameters were satisfactorily estimated within reasonable limits.

Predicted power was far better than that from OFPMs which use ground speed for

predicting power, and marginally better than the OFPM that uses wind speed. Given the

limitations of: 1. using ground speed instead of speed relative to wind; 2. hardly

accounting for bump/bearing/drivetrain losses; 3. using not-so-ideal elevation data from

bike-computer altimeters; 4. erratic acceleration calculated from speed data from sensors,

we would say that the parameter estimation was successful and validated. Of course, there

are several limitations addressed in this section as well as the previous sections. Correcting

or accounting for those will surely improve accuracy of predicting power. Given that we

used a very simple set of linear equations, constants and sample space of data, the results

are impressive.


Conclusion

Part 1

1. The comparison of OFPMs showed that PowerPod was closest to actual

power with a root mean square error of 47 W.

2. Sigma ROX 10.0 worked best when elevation and speed were unchanging

with a root mean square error of 17 W.

3. Powercal did not fare well, with a poor showing in all the tests.

4. These PMs are not recommended for scientific use as PMs.

Part 2

1. The parameters we estimated agreed with those found in literature.

2. The predicted power in a field test based on the parameters obtained from

rides in Part 1 had a smaller RMSE (46 W) than that of the OFPMs tested

in Part 1.

3. Parameter estimation from field tests yields better results than commercially

available OFPMs, for almost no cost.


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Tables

Table 1

Training zones by Dr. Andrew Coggan

This table summarizes zones based on average power values as a percentage of FTP and

HR values as a percentage of LTHR. RPE is on a scale of 1-10. A larger table (from which

this table is an excerpt) describes the physiological adaptations from trianing within these

zones.

Zone Name Average Power Average HR RPE

1 Active Recovery <55% <68 <2

2 Endurance 56-75% 69-83% 2-3

3 Tempo 76-90% 84-94% 3-4

4 Lactate

Threshold

91-105% 95-105% 4-5

5 VO2 Max 106-120% >106% 6-7

6 Anaerobic

Capacity

>121% N/A >7

7 Neuromuscular

Power

N/A N/A Maximal


Table 2

Summary of anthropometric and bike data

Powercal PowerPod Sigma

Sex X Male Male

Height X 172 cm 172 cm

Rider Weight X 68 kg 68 kg

Riding position X Hoods Hoods

Shoulder width X x 51 cm

FTP X 240 W x

Bike weight X 13 kg 13 kg

Tire pressure X 110 psi x

Type of Bike X Road Bike Road Bike

Wheel Circumference x 2105 mm 2105 mm


Table 3

Limits for parameters λ, CdA and Crr

These values were chosen such that they made sense physically and were not exaggerated.

We set broad limits based on some information from literature. Based on these limits, the

minimum and maximum values of the constants k1, k2, and k3 were calculated

Min

Max

0.1000 λ 0.9900

0.1000 CdA 1.0000

0.0001 Crr 0.0100

Min Max

1.0101 k1 10.0000

0.0001 k2 0.1000

0.0505 k3 5.0000


Table 4

Parameters from rides in Part 1

Clearly, Ride 2 had vastly different results from the other rides. We ommitted this ride for

reasons explained in the Discussion chapter. The “Final” parameters were the average

values of λ, Crr and CdA from Ride 1 and Ride 3. The “Final” constants were calculated

from these parameters.

Ride 1 Ride 2 Ride 3 Final

k1 1.0101 1.0101 1.0101 1.0101

k2 0.0097 0.0036 0.0094 0.0095

k3 0.1551 0.2332 0.1307 0.1429

Ride 1 Ride 2 Ride 3 Final

λ 0.9900 0.9900 0.9900 0.9900

Crr 0.0096 0.0036 0.0093 0.0094

CdA 0.3071 0.4618 0.2587 0.2829


Table 5

Results for Part 1

The first third shows results for the full ride (~74 min and ~33 km)

The second third lists results for the climbing section (13.5 min and 3.6 km)

The last third shows results for the flat section (12 min and 5.9 km)

Full Ride SRM Powercal PowerPod Sigma

Mean Power (W) 156.36 208.75 155.72 116.78

Standard Deviation (W) 88.12 97.72 96.08 71.74

Relative Error (%)

33.51 -0.41 -25.31

Normalized Power (W) 189.94 234.84 194.75 150.15

Relative Error (%)

23.64 2.53 -20.95

Root Mean Square (RMS) Error (W)

89.42 47.16 88.67

Relative RMS Error (%)

77.5 44.15 60.42

Peak Signal to Noise Ratio (dB)

17.27 22.83 17.34

Energy (kJ) 695.02 927.89 692.18 519.09

Climbing SRM Powercal PowerPod Sigma

Mean Power (W) 203.09 244.45 194.69 146.18


Relative Error (%)

20.37 -4.14 -28.02


Relative Error (%)

20.19 -3.24 -24.72

RMS Error (W)

67.78 28.85 34.30

Rel RMS Error (%)

43.14% 18.23% 49.23%

Peak SNR (dB)

12.23 19.65 18.14

Energy (kJ) 164.50 198.01 157.70 118.41

Flat

SRM Powercal PowerPod Sigma

Mean Power (W) 157.75 203.83 146.43 136.87


Relative Error (%) 29.21 -7.17 -13.23


Relative Error (%) 26.70 -5.33 -5.26

Root Mean Square Error (W) 80.75 44.70 68.44

Relative RMS Error (%) 64.38 42.52 51.03

Peak Signal-to-Noise Ratio (dB) 13.06 18.19 14.49

Energy k(J) 113.58 146.76 105.43 98.55


Table 6

RMSE and PSNR for the OFPMs after normalizing

After normalizing data by subtracting the respective mean values from the series, Powercal

and Sigma improved in their errors, suggesting an “offset” that needs correction

(investigating which is not in the scope of our project). PowerPod did not change,

suggesting a “spot on” estimation in the first place

Powercal PowerPod Sigma

Before After Before After Before After

Root Mean Square Error (W) 89.42 72.46 47.16 47.16 88.67 79.34

Peak Signal to Noise Ratio (PSNR) 17.27 19.1 22.82 22.83 17.34 18.31


Table 7

Results: Sprints

The table shows maximum power recorded and the respective relative errors for the three

OFPMs. PowerPod’s error was lesser than the least error from the other two PMs.

Sprints

SRM Powercal PowerPod Sigma

Max Power Sprint 1 (W) 608 469 625 305

Relative Error (%) -23 3 -50






Table 8

Wind conditions

This table shows for how long each ride was ridden against the direction of the wind. Ride

2 was clearly different from the other rides in that it was a very windy day. This table

quantifies our reason for excluding the ride, just like some other authors (listed in

Discussion chapter) do in their studies.

Headwind (s) Tailwind (s) Total (s) Headwind (%) Tailwind (%)

Ride 1 3510 883 4446 78.95 19.86

Ride 2 4366 134 4500 97.02 2.98

Ride 3 3095 1351 4455 69.47 30.33

Ride 4 1190 832 2023 58.82 41.13


Table 9

Results Part 2

The first half of the table gives a summary of results for the full ride of Ride 4 (~33.5 min

and ~15.46 km)

The second half shows the same metrics from the Climbing Section of Ride 4 (6.5 min and

1.6 km )

Full Ride

SRM Prediction Powercal PowerPod Sigma

Mean Power (W) 162.22 160.16 219.07 160.75 126.39

Standard Deviation (W) 72.76 83.70 75.66 81.61 63.92

Relative Error (%) -1.27 35.05 -0.91 -22.09

Normalized Power (W) 185.20 195.14 232.14 187.75 155.64

Relative Error (W) 5.37 25.35 1.38 -15.96

Root Mean Square Error (W) 45.71 84.76 46.80 73.39

Relative RMS Error (%) 41.22 80.46 39.93 57.35

Peak Signal to Noise Ratio (dB) 23.26 17.89 23.05 19.14

Climbing Section

SRM Prediction Powercal PowerPod Sigma

Mean Power (W) 196.22 192.16 236.60 196.87 142.28

Standard Deviation (W) 32.71 36.85 45.05 29.36 37.69

Relative Error (%) -2.07 20.58 0.33 -27.49

Normalized Power (W) 203.45 201.97 244.40 202.54 154.04

Relative Error (W) -0.73 20.13 -0.45 -24.29

Root Mean Square Error (W) 22.97 54.67 23.92 68.59

Relative RMS Error (%) 17.19 32.62 20.76 40.41

Peak Signal to Noise Ratio (dB) 20.74 13.20 20.38 11.23


Table 10

Summary of CdA values found in literature dating back to 1975

CdA Authors Method Comments

0.219 (Defraeye et al.,

2010a)

CFD Analysis Cyclist CdA was obtained by

subtracting bike CdA from total CdA.

This is oversimplified

0.255 (Capelli et al.,

1993)

Towing Subject did not wear a helmet

0.270 (Defraeye et al.,

2010a)

CFD

0.299 (Grappe et al.,

1998)

Field test (indoor

velodrome = level

ground, still air)

Tested only one subject who was

similar in height and weight to our

subject

0.332 (Candau et al.,

1999)

Coasting Tested only one subject who was

similar in height and weight to our

subject

0.343 (Barry et al.,

2014)

Wind tunnel Assumed Crr and drivetrain

efficiency to be 0.005 and 97.6%,

respectively

0.355 (Candau et al.,

1999)

Deceleration

0.360 (de Groot et al.,

1995)

Field test (flat

asphalt road, still

air)

Subject wore a jacket

0.260-

0.380

(Zdravkovich,

Ashcroft,

Chisholm, &

Hicks, 1996)

Wind tunnel Mean values

0.299-

0.390

(García-López et

al., 2008)

Wind tunnel Mean values


0.300 (de Groot et al.,

1995)

Field test (flat

asphalt road, still

air)

7 subjects. Area calculated from

photographs, density assumed 1.2

kg/m3

0.387 (C. Kyle &

Edelman, 1975)

Deceleration test Touring bike: bike geometry is

slightly different from road bikes

0.390 (Gross, Kyle, &

Malewicki, 1983)

Deceleration test


Table 11

Summary of Crr values found in literature dating back to 1995

Crr Reference Method & Surface Comment

0.0030 (Grappe et al., 1998) Treadmill test Tubeless tires

0.0038 (de Groot et al., 1995) Field test (flat

asphalt)

Tubeless tires

0.0040 (Grappe et al., 1998) Field test velodrome

(flat, still air)

Linoleum floor, tubeless tires

0.0043 ("Tire Test -

Continental Grand

Prix," 2014)

Lab test on a

rotating drum

110 psi pressure, load = 42.5

kg for one tire (85 kg for

two)

0.0056 (Candau et al., 1999) Coasting Indoor level hallway.

Subject’s mass was similar to

ours = 66.2 kg

0.0050 (Barry et al., 2014) Assumed This value yielded an

accurate value of drag area

0.0080 (Pugh, 1974) Field test on runway Tubular tires, 630kPa (~90

psi). Subject mass was 86 kg

0.0122 (Meyer et al., 2016) Asphalt

They used

vibrations measured

on the handlebar

using a smartphone

4-wheeled e-bike. “Results

may vary for other bikes”

0.0072 (Grappe et al., 1999) Using the formula

Crr = 0.1071xP-0.477

23 mm tires on clincher

wheels, subject mass = 66 kg

0.0700-

1.1500

(Zhan et al., 2012) Lumped coefficient

with other losses

Average of seven bikes


Figures

Figure 1: Free body diagram

This of a cyclist in motion (idea borrowed from Thorsten Dahmen’s yet to be published

dissertation) illustrates the forces acting on the bike and rider system when isolated. Ftot is

the force exerted by the cyclist in order to counter all the other forces. Bicycle efficiency λ

is not represented in this figure. Icon from www.iconfinder.com

Fbearing

Faero

FKE

Froll

FPE

Ftot

http://www.iconfinder.com/


Figure 2: Illustration of the relative areas in contact with the ground

For mountain bike tires (above) and thinner tires (below); a wider tire has more rubber in

contact with the ground, resulting in greater rolling resistance, everything else being equal.

Image from www.schwalbe.com

http://www.schwalbe.com/


Figure 3: The SRM Science PM

The strain gauges, accelerometer and circuitry are located in the crankset i.e. between the

crank and the bottom bracket. Image from www.Store.SRM.de

Figure 4: The Powercal, manufactured by Powertap, USA

It is a strap to be worn on the chest after installing the battery. Image from

www.powertap.com

http://www.store.srm.de/

http://www.powertap.com/


Figure 5: A perspective view of the PowerPod

In this view we see a grey squarish button (to start/reset the device) and an LED (to

indicate start/reset). At the near end we see the micro-USB cable to upload workout files to

a computer. In much the same way as a DFPM, the device does not have a display of its

own because it aspires to be more like a traditional DFPM than a computer. Image from:

www.PowerPodsports.com

Figure 6: The Sigma ROX 10.0 bike computer. Image from: www.Sigma-rox.com

http://www.powerpodsports.com/

http://www.sigma-rox.com/


Figure 7: The map (bottom) and elevation profile (top) of the route in Switzerland for the

rides in Part 1. Distance ~ 33 km and net elevation gain ~ 333 m. Pictures from

www.gpsies.com

http://www.gpsies.com/


Figure 8: The map (bottom) and elevation profile (top) of the ride in Germany for Part 2.

Distance ~ 15.5 km and net elevation gain ~ 148 m. Pictures from www.gpsies.com

http://www.gpsies.com/


Figure 9: The subject

This image shows all the PMs and accessories on the rider and bike.

Figure 10: An illustration of the typical hoods position on a road bike

“Hoods” refers to the part of the handlebar called “brake hoods”. It is one of the three most

common positions to adopt while cycling – the other two being “upright” and “drops” in

which the hands go on the top of the handlebar, and in the dropped part, respectively.

Picture from www.ilovebicycling.com

http://www.ilovebicycling.com/


Figure 11: Top view: The mobile phones (Powercal and Velocomputer) are placed in a

pouch on the top tube, the Sigma device and Garmin bike computer are mounted on the

stem and left half of the handlebar, respectively. The PowerPod was firmly secured on the

right half of the handlebar, in a location where the brake and gear cables do not obstruct air

flow into the wind sensors.

Figure 12: A side view of our cockpit.


Figure 13: Speed & Cadence Sensors

The cadence sensor (left) was strapped to the left crank. The accelerometers in this sensor

calculate the number of revolutions completed by the sensor. The sensor wirelessly

transmits the count of revolutions per minute to the bike computer by ANT+ protocol.

Since the units of cadence most commonly used is RPM, it only registers the count of

revolutions and does not need to know the length of the crank.

The speed sensor (right) was mounted on the front wheel. This works on the same

principle as the cadence sensor, except for the fact that the bike computer or the user has

no use for the count of revolutions per minute. When multiplied by the circumference of

the wheel (2105 mm), the count of revolutions can be converted to speed. This

circumference was taken uniform for all devices that required this input.


Figure 14: The Powercal was worn by the user just like a heart rate strap, to be in direct

contact with skin with some moisture (provided by the sweaty skin). It can be worn in any

orientation. It does not require any initial setup except for the insertion of a battery. To

give credit where it is due, this device was the easiest to “install”.

Figure 15: A side view of the test bike with all the PMs mounted on it, along with full

water bottles and some tools in the bag under the saddle. The combined weight of

everything shown in this picture was taken as the mass of the bike (13 kg).


𝐴𝐵 = ℎ2 − ℎ1

𝐴𝐶 = 𝑑2 − 𝑑1

𝐵𝐶 = √𝐴𝐶2 − 𝐴𝐵2 ≈ 𝐴𝐶

𝑆 =𝐴𝐵

𝐵𝐶=

ℎ2 − ℎ1

𝑑2 − 𝑑1

𝑆 = 𝑡𝑎𝑛𝜃 =𝐴𝐵

𝐵𝐶

Figure 16: Calculating slope from elevation data

Using simple trigonometry, the angle of inclination can be approximated very well from

elevation and distance readings. Icon from www.queensu.ca

θ

C B

A

http://www.queensu.ca/


Figure 17: Results Part 1: Full Ride

Graphs showing the 30s smoothed values of power vs time (time not indicated on axis) for

the full duration of the 74 minute ride in Part 1. Powercal (top) seems to overestimate,

Sigma (bottom) seems to underestimate while PowerPod seems to be the most accurate

among the three, from visual inspection.

0

100

200

300

400

500P

ow

er (

W)

Powercal 30s Average SRM 30s Average

0

100

200

300

400

500

Po

wer

(W

)

SRM 30s Average Powerpod 30s Average

0

100

200

300

400

500

Po

wer

(W

)

SRM 30s Average Sigma 30s Average


Figure 18: Results Part 1: Climbing Section

Graphs showing the power output (W) vs time (time not indicated on axis) for the climbing

section of the ride in Part 1. The section lasted 13.5 min and 3.6 km for a total elevation

gain of 147 m. Top: HR values are also shown and there does not seem to be much

correlation with power. There is no apparent pattern that is being followed. Middle: Once

again, PowerPod seems to be doing the job well. Bottom: Sigma consistently

underestimated power even when wind was not the major force to overcome

155

165

175

0

200

400

He

art

Rat

e (

bp

m)

Po

we

r (W

)

SRM Powercal HR

0

100

200

300

400

Po

we

r (W

)

SRM Powerpod

0

100

200

300

400

Po

we

r (W

)

SRM Sigma


Figure 19: Results Part 1: Flat Section

Graphs showing the power output (W) vs time (time not indicated on axis) for the flat

section of the ride in Part 1. The section lasted 12 min and 5.9 km for a total elevation

gain/loss of 5 m/17 m. Top: HR values are also shown. There is some pattern in terms of

how the power reacts to changes in HR but it is not in the scope of our thesis to investigate

this. Middle: PowerPod seems erratic but on average seems to be predicting power well.

Bottom: Sigma worked well when speed was steady and there was no gradient in a brief

segment. This two-minute segment will be explored in the Discussion chapter.

120

140

160

0

200

400

Hea

rt R

ate

(bp

m)

Po

wer

(W

)

SRM Powercal HR

0

200

400

Po

wer

(W

)

SRM Powerpod

10

20

30

40

0

200

400

Spee

d (

km\h

)

Po

wer

(W

)

SRM Sigma Speed


Figure 20 Results Part 1: RMSE

Bar graphs showing the drop in RMSE of all the three PMs in all three “conditions” (top:

full ride; middle: climbing section; bottom: flat section) with progressive

averaging/smoothing of data. The relative positions of the PMs in the “ranking” does not

change. The order of accuracy is always – PowerPod, Sigma and Powercal

0

20

40

60

80

100

Powercal Powerpod Sigma

RM

SE (

W)

0

20

40

60

80

100


RM

SE (

W)

0

20

40

60

80

100


RM

SE (

W)

RMS RMS3 RMS5 RMS30


Figure 21: Results Part 1: Sprints

Graphs showing the sprint power output from the four PMs in the “sprint” intervals. Each

sprint was studied in a window of 30 seconds (time not indicated on axis). These OFPMS

are poor at estimating power for short, sudden, maximal efforts.

0

100

200

300

400

500

600

700

Po

wer

(W

)

0

100

200

300

400

500

600

700

Po

wer

(W

)

0

100

200

300

400

500

600

700

Po

wer

(W

)

SRM Powercal

Powerpod Sigma


Figure 22: Results Part 1: Zones

This is a graph of the duration spent (in seconds) by the rider in the training zones

discussed in the Background chapter according to the four PMs in the ride of Part 1. The

black line signifies actual power from the SRM device.

0

500

1000

1500

2000

2500

Zone 1 Zone 2 Zone 3 Zone 4 Zone 5 Zone 6+

Tim

e (s

)

Powercal Powerpod Sigma SRM


Figure 23: Results Part 2: Full Ride

Top: Graphs showing the 30s smoothed values of power vs time (time not indicated on

axis) for the full duration of the 35 minute ride in Part 2. The predicted power over- as well

as under-estimates power in some segments

Bottom: The same graph as above, with 30 s-averaged power from the three OFPMs

during the same ride. The trends of all three OFPMs are the same as in Part 1.

0

50

100

150

200

250

300

350

400

450

500P

ow

er (

W)

SRM 30s Avg Prediction 30s Avg

0

50

100

150

200

250

300

350

400

450

500

Po

wer

(W

)

SRM 30s Avg Powercal 30s Avg Powerpod 30s Avg

Sigma 30s Avg Prediction 30s Avg


Figure 24: Results Part 2: Climbing Section

Top: Graphs showing the 30s smoothed values of power vs time (time not indicated on

axis) for the climbing section lasting 6.5 min and 1.6 km and total elevation gained/lost 81

m/11 m. The predicted power seems erratic but on average gets the value right. Bottom:

The same graph as above, with 30 s-averaged power from the three OFPMs during the

same segment. The trends of Powercal and Sigma are the same as in Part 1. SRM,

PowerPod and our predicted power are almost indistinguishable

50

100

150

200

250

300P

ow

er (

W)

SRM 30s Average Predicted Power 30s Average

50

100

150

200

250

300

Po

wer

(W

)

SRM 30s Avg Powercal 30s Avg Powerpod 30s Avg

Sigma 30s Avg Prediction 30s Avg


Figure 25: Error trend with progressive smoothing

As in Part 1, the averaging/smoothing of data reduces the error in measurement. But, this

time, predicted power no longer stays the best of the OFPMs. PowerPod does a better job

in the 30-second averaged power.

0

20

40

60

80

100

Prediction Powercal Powerpod Sigma

RM

SE (

W)

RMS RMS3 RMS5 RMS30


Figure 26: Vector diagram of speeds: A diagram illustrating relative velocity

In real conditions, it is very rare that wind while riding a bike is always in the direction

exactly opposite to that of the bike’s velocity. Wind can be split into two components –

head/tail wind (i.e., wind along the direction of the ride) and crosswind (i.e., perpendicular

to the direction of the ride). The disadvantage of the PowerPod is explained in this visual.

The PowerPod only takes the component of velocity that is in the direction of the rider

(vhead), effectively neglecting the drag forces resulting from the crosswind component.

Picture from: www.wing-light.de

http://www.wing-light.de/

Documents

Raman Garimella thesis - Final Draft