Apparatus and Method for Lower Body Power Output Estimation

Apparatus and Method for Lower Body Power Output Estimation

PER SEFASTSSON

Master of Science Thesis

Apparatus and Method for Lower Body Power Output Estimation

Per Sefastsson

Master of Science Thesis MMK 2011:56 MDA 413 KTH Industrial Engineering and Management

Machine Design

SE-100 44 STOCKHOLM

Master of Science Thesis MMK 2011:56 MDA 413

Apparatus and Method for Lower Body Power Output Estimation

Per Sefastsson

Approved

2011-month-day

Examiner

Mats Hanson

Supervisor

Bengt Eriksson

Commissioner

RF Bosön

Contact person

Daniele Cardinale

Abstract

One of the Swedish sports federation RF’s ongoing goals is to find new and better methods of testing athletes. Therefore a project, with the goal of investigating which method to use for measuring lower body muscle power output during free barbell exercises, in order to give training recommendations, was undertaken. This had already been done in a study done by Cormie et al 2007 [9]. In this study, the muscle power output is estimated using methods relying on knowledge of the position of the barbell and the athlete’s reaction forces acting on the floor.

The goals of this thesis was to make a test apparatus capable of repeating the test methods used in Cormie et al 2007 [9], but also explore the possibility of finding other methods of muscle power estimation and to implement these.

In order to reach these goals, first a literature study was carried out, looking at other previous studies, technologies and commercial systems. Then models of the mechanics and the test setup were developed. Lastly, the test apparatus was developed, using linear position transducers for determining the position of the barbell, along with a force plate for measuring reaction forces.

The data was then collected and analyzed using a PC-system developed in LabVIEW. As a final step, a limited trial was conducted in order to compare results with Cormie et al 2007 [9].

The resulting apparatus determines the position of the barbell’s center of mass and its orientation in space, measuring the position of two points on the barbell using a total of five linear position transducers, three on one side and two on the other. The reaction forces are measured using a force plate position under the athlete. All of this, in 3D.

One new method of estimating muscle power output is also presented in this thesis. This method accounts for the lifter and the barbell as two separate bodies, able to move independently, whereas the previous method of choice only accounts for them as one stiff body.

The trial showed good coherence with Cormie et al 2007 [9], comparing the same methods. Also the new method gave results that seem reasonable in comparison with the old methods’ results.

The conclusion and recommendation of this report is to, in the future, use this new method. This

new method should, in theory, be able to capture more of the reality of free barbell exercises than

the old method. The conclusion of this report is therefore that this method should be used.

Examensarbete MMK 2011:56 MDA 413

Apparatur och metod för uppskattning av underkropsseffektutveckling

Per Sefastsson

Godkänt

2011-mån-dag

Examinator

Mats Hanson

Handledare

Bengt Eriksson

Uppdragsgivare

RF Bosön

Kontaktperson

Daniele Cardinale

Sammanfattning

Riksidrottsförbundet har som stående mål att ständigt förbättra metoderna med vilka man testar idrottare. Därför åtogs ett projekt med målet att undersöka vilken metod för att mäta

muskeleffektutveckling under fria skivstångsövningar som ska användas för att kunna ge träningsråd. En sådan studie har tidigare gjorts av Cormie et al 2007 [9]. I denna studie uppskattades muskeleffektutvecklingen med hjälp av vetskap om skivstångens position samt idrottarens reaktionskrafter mot golvet.

Målet med den här rapporten var att utforma en testapparatur som kan upprepa de testmetoder som är använda i Cormie et al 2007 [9], samt undersöka möjligheten till andra metoder för uppskattning av muskeleffektutveckling och implementera dessa.

För att uppnå dessa mål genomfördes först en litteraturstudie där tidigare studier, teknik och existerande system undersöktes. Sedan utvecklades modeller av mekaniken samt av själva testuppställningen. Sist utformades själva apparaten bestående linjärpositionsgivare för

positionsbestämning av skivstången samt en kraftplatta för mätning av reaktionskrafterna. Data samlades sedan in m.h.a. en PC och ett därför utvecklat gränssnitt gjort i LabVIEW, där beräkningarna utförs. Som ett sista steg utfördes en begränsad försöksstudie för att kunna jämföra med Cormie et al 2007 [9].

Den resulterande testapparaturen bestämmer positionen och orienteringen för stångens masscentrum genom att mäta positionerna för två punkter på stången med totalt fem

linjärpositionsgivare. Reaktionskrafterna mäts m.h.a. en kraftplatta positionerad direkt under utövaren. Allting i 3D.

I den här rapporten lägg även en ny metod för effektuppskattning fram. Den nya metoden redogör för systemet atlet-skivstång som två kroppar som kan röra sig fritt oberoende av varandra, där tidigare metoder enbart använt sig av en enkroppsmodell.

Försöksstudien uppvisade god överstämmelse med Cormie et al 2007 [9] vid jämföring av de tidigare metoderna. Även den nyutvecklade metoden visade rimliga resultat i jämförelse med de tidigare metoderna. Den nya metoden kan, i teorin, fånga mer av skivstångsrörelsernas natur.

Därför är rekommendationen i den här rapporten att denna metod i framtiden bör användas.

Nomenclature

Abbreviations

1RM One Repetition Maximum, a persons maximal training load A/D Analog to Digital Converter

BNC Bayonet Neill–Concelman connector, a connector type for coaxial cable.

BW Body Weight CA Charge Amplifier COM Center of Mass

CSD Center for Sports Development FP Force Plate

GRF Ground Reaction Force GUI Graphical User Interface I/O Input / Output

JS Jump Squat

LPE Linear Position Encoder LPT Linear Position Transducer NI National Instruments

RF The Swedish sports federation

TDMS Technical Data Management System-file format

Constants Unit

g Gravitational acceleration, 9,82 m/s²

lb Distance between wire connections on the barbell, 1,27 m

h Time between samples, 0,001 s

Notations Unit

¯

e Unit vector N/A

F¯ Force vector N

¯

p Position vector m

¨¯

p Acceleration vector m/s²

˙¯

p Velocity vector m/s

l Length m

m Mass kg

T Height Threshold m

x, y, z Cartesian coordinates N/A

List of Figures

2.1 The three types of human muscle tissue [1] . . . 4

2.2 Squat sequence front/side view [3]. . . 8

2.3 JS sequence front/side view [3]. . . 8

2.4 Geometric overview of trilateration . . . 11

2.5 The idealized mechanical system . . . 14

2.6 Free drawing of the idealized system in Figure 2.5 on page 14 . . 15

3.1 The test setup . . . 19

3.2 Above and front view (opposite of the athlete’s view) of the LPT:s and coordinate system placement in relation to the position of the barbell . . . 23

3.3 Trilateration used to determine the position of the wire connec- tion points on the barbell . . . 24

3.4 The model from which the LPT retraction force was calculated . 26 3.5 Flow chart of the recording-process. . . 29

3.6 Barbell vertical position signal with triggering, buffers and actual recording . . . 30

3.7 File structure of the measurements file . . . 32

3.8 Screenshot of the user interface . . . 33

4.1 The peak and average power with standard deviation within the trial group, measured during the concentric phase of the JS . . . 36

Foreword

I

would first like to thank my contact at RF, Daniele Cardinale, for his support and never ending optimism. He has made it fun to come to work no matter how large the setbacks were or how far behind schedule I was. His personal commitment to his field has been a constant inspiration.

Secondly I would like to thank my tutor Bengt Eriksson and my examiner Mats Hanson for their help respectively, both when it comes to purely technical questions and for their constructive criticism.

I would also like to thank all other employees at Bosön that I have meet for their warm welcoming and their patience for my questions and inquiries.

A last and special thanks to the athletes whom have both participated and let their routines be interrupted to make this project possible.

Per Sefastsson, Stockholm 2011-06-16

Contents

1 Introduction 1

1.1 Background . . . 1

1.2 Purpose . . . 1

1.3 Delimitation . . . 2

1.4 Method . . . 2

1.5 Chapter overview . . . 2

2 Theory 4 2.1 Physiology . . . 4

2.1.1 Skeletal muscle . . . 5

2.1.2 Strength and power . . . 5

2.1.3 Training for enhanced muscle power output . . . 6

2.2 Weight lifting . . . 7

2.2.1 Movements . . . 8

2.3 Existing systems . . . 9

2.3.1 Sensors in use . . . 9

2.3.2 Commercial systems . . . 10

2.4 Mathematics . . . 11

2.4.1 Trilateration . . . 11

2.4.2 Error analysis . . . 13

2.5 Mechanics . . . 13

2.5.1 Methods for power estimation . . . 15

3 Design 18

3.1 Lab setup . . . 18

3.1.1 Equipment . . . 20

3.1.2 Positioning of the LPT:s . . . 21

3.1.3 Localization of the barbell . . . 24

3.1.4 Velocity and acceleration . . . 26

3.1.5 LPT retraction force . . . 26

3.2 Data collection and conditioning . . . 27

3.2.1 Triggering . . . 30

3.2.2 Filtering . . . 30

3.2.3 Phase intervals . . . 31

3.2.4 Saving . . . 31

3.3 PC Interface . . . 32

3.3.1 Key values . . . 34

4 Trial 35 5 Results and Discussion 37 5.1 Commercial potential . . . 37

5.2 Power calculation methods . . . 38

5.3 Suggestions for improvements of the application . . . 38

A Matlab code for error propagation 43 A.1 Main.m . . . 43

A.2 PosError.m . . . 44

A.3 PropError.m . . . 46

A.4 LPTSim.m . . . 47

Chapter 1

Introduction

1.1 Background

In sports physiology, muscle power output is of great interest since this gives a good indication of physical performance relevant to most sports. Therefor a large effort is put on trying to find training methods to improve this performance and also to test it. There have been many studies conducted on trying to find the optimal training load, eliciting maximum muscle power output, for training for power improvement, but the results are so far contradicting [18, 9, 8].

One major reason is the difficulty, and varying methods used, to measure muscle power output. Also the previous reports conducted trying to find the optimal training load [24, 6, 7, 19, 5, 11, 18, 9], even though being explicit when it comes to the physiology part, are fuzzy when it comes to technical- and methodological details, making it hard to reproduce results.

Bosön at Lidingö, Sweden, is the Swedish Sports Federation’s (RF) Center for Sports Development (CSD). It houses a physiological laboratory, with equip- ment for a wide range of different sport performance tests, with an emphasis on aerobic and anaerobic endurance testing. They also have some equipment for testing explosive muscular power. This equipment is fairly rudimentary and limited in its scope of use. To improve its capabilites on this area, CSD launched a project aiming to validate a previous study concerning lower body dynamic power estimation and investigate the possibility of improving these methods.

1.2 Purpose

The purpose of this thesis is to design and construct a test apparatus for evalua- tion of athlete muscle power output during free barbell exercises. This apparatus

should be able to repeat the testing procedures and methods (see Section 2.5.1) described in Cormie et al [9]. Other methods of power estimation should also be analyzed and, if beneficial, be implemented. The main goal is to develop a system, making it possible to perform a new validation study of the same nature as Cormie et al [9].

The existing and, if found, new methods for power estimation are then to be implemented using a PC-user interface. This application shall be a daily tool for the researchers and test leaders at the sports physiology laboratory.

The test equipment is to be installed in a permanent way at CSD, as an exten- sion of their previous platform for explosive power analysis and testing.

1.3 Delimitation

Maximum of five linear position transducers (LPT) should be used for determin- ing the position of the barbell. Also a “fake” LPT, without sensory, is avaliable to balance the retraction forces from the LPT:s. To measure kinetic data, a force plate (FP) should be used. For the data collection, a data acquisition sys- tem from National Instruments (NI) and the program developing software LabVIEW, also from NI, should be used. All equipment to be used for the test setup was already avaliable at CSD, which, in combination with a limited budget, led to the above limitations.

1.4 Method

First a literature study was carried out. During this study, previous studies within the field of sports physiology and technical reports with connection to the project were acquired by searching the internet ant various article databases.

Other technology suitable for the task was also looked at, such as existing com- ercial systesms, other suitable sensors etc..

After the literature study, calculations and models of the test setup and the mechanics were made, the test equipment was configured and finally the test software was developed.

Finally, a limited trial study, similiar to the one in Cormie et al [9], was carried out for testing the methodology and test setup. In this study nine division 1 junior male hockey players performed lifts at six different training loads in order to compare the methods and apparatus with the results of Cormie et al [9].

1.5 Chapter overview

Chapter 1 - Introduction Covers the background, purpose and delimita- tion of the project. It briefly describes the chosen method which is thor-

oughly covered in Chapter 3.

Chapter 2 - Theory Contains necessary theory to understand the design pro- cess and the underlying purpose of the project. In this chapter, existing methods, previous studies on the subject and technology relevant to this are covered. Also the physiology needed is discussed, along with the used mathematical methods.

Chapter 3 - Design Describes the working process and models, calculations, procedures, processes and equipment used in the apparatus.

Chapter 4 - Trial Describes the trial study conducted in order to compare the apparatus and methods to previously performed studies.

Chapter 5 - Results and Discussion Presents and discusses the results from Chapter 3 and 4 and suggests future work and improvement.

Chapter 2

Theory

2.1 Physiology

Physiology is the science of how living organisms function. In sports, the main interest within physiology is the mechanical performance of humans.

The human body consists of many different organs, with different functions. The organs that gives the body its ability to move (and thereby mechanically perform in sports) are the muscles. Muscles are divided into three different groups;

skeletal, cardiac and smooth (see Figure 2.1 on page 4) [13]. Cardiac muscle tissue is found only in the heart, and the smooth muscle tissue is controlled by the autonomic nerve system. Skeletal muscles are the ones that gives us the ability to move our extremities and perform controlled motions.

Figure 2.1: The three types of human muscle tissue [1]

2.1.1 Skeletal muscle

Skeletal muscle is probably the type of which most people think of when they hear muscle. These muscle are all consciously controllable and are, at their ends, connected via ligaments to the skeleton. Skeletal muscles are always connected to two different skeleton parts that are in turn connected via a joint. Hence the contraction of the muscle results in a displacement in the joint.

All muscle consists of fibers. These fibers are the cells in the muscle performing the actual mechanical work. The fibers contain myofibrils, which are bundles of filamentary proteins [20]. These proteins have the ability to “climb” on one another, hence creating a contraction. This reaction can only occur in one direction, which means that a muscle can only contract on its own. This phase of the movement of a muscle is called the concentric part, while movement in the other direction (against the direction in which it can contract on its own) is called the eccentric part. To make it possible to move limbs in both directions, every muscle is paired with an antagonist. In every movement, the antagonist is also involved to some degree.

There are different types of muscle fibers specialized on different tasks. Some of them are fast contracting, explosive and have a high momentary power output but without great endurance (Type II), while others are more economic during continuos strain but with lower momentary power output (Type I)[15]. All skeletal muscle consist of all types of fibers, but the percentages vary.

It has been shown that muscles and the nervous system adapts to specific train- ing [7, 10, 15], which constitutes the very rational of strength training for en- hancing sports performance.

2.1.2 Strength and power

Strength is a somewhat ambiguous word. In some context it may mean only the maximum force that an individual can evoke. In other cases it may mean the muscle power output. E.g. in bench press, strength usually means the ability to move a weight a short distance with the same muscles during the whole movement. This means that here, only a large force output is needed, since the barbell can move slowly without negative impact on the goal of the discipline.

In Olympic lifting, however, all lifts are combinations of movements in sequence, involving different muscle groups. Therefore it is important to give the barbell as much kinetic energy as possible with the muscle groups that are able to create the most force. This means that the velocity of the barbell is higher in these types of movements, which makes mechanical power more important in Olympic lifting than in power lifting [23] (which is a somewhat misleading term).

However in most sports, the result is often depending on the force, but also on the speed at which it is exerted and thus making power a better indicator of actual performance than sheer force output [31].

As key values for measuring muscle power output, peak and average power (W ) for the concentric phase of the movement are most often used. The mutual importance of these values are thought to vary as indicators of performance amongst different sport and movement.

2.1.3 Training for enhanced muscle power output

For impulse-like explosive movements, such as throwing etc. Olympic lifting is a commonly recommended form of training [14]. These movements are however technically complex, and may take years to properly learn. Therefore there exists a lot of different simpler and less complex, partial movements, such as jump squat (JS), power clean, power jerk, bench press throw etc.

Because of the technical aspect of the exercise, in most cases, it might be better to asses the athlete’s performance based on one of these simpler movements to avoid the influence of lacking technique on the test results. E.g. for lower body strength and power performance, the JS is widely considered the standard indi- cator. The use of simpler movements for all athletes also allows for comparison of results between different disciplines and sports.

The need and ambition to improve athletes’ physical abilities has given rise to several published studies with the goal of finding the exercise load that elicits maximum muscle power output. Different means of measuring kinetics and kinematics and different methods for calculating muscle power output has lead to that most of the results are inconclusive (see Table 2.1 on page 7).

Table 2.1: Comparison of previous studies aiming to determine optimal load for lower body power development[9]

Study Sensors (see

2.3) Method

2.5.1)(see

Exercise

(see 2.2) Optimal (%1RM)load

Baker et al. 2001 1-LPT Ib JS 55-59%

Borque &

Sleivert 2003 1-LPT Ia cJS 0-30%

Esliger &

Sleivert 2003 1-LPT Ib cJS 63%

McBride et al.

1999 1-LPT, FP III JS 30%

Sleivert &

Taingahue 2004 Accelerometer Ia cJS 60%

Stone et al. 2003 V-scope Ia JS & cJS 10%

Izquierdo et al.

1999 1-LPT Ia S 60-70%

Siegel et al. 2002 1-LPT Ia S 60%

Cormie, McBride

& McCaulley 2007

2-LPT, FP III JS, S 0%, 56%

1-LPT, FP III JS, S 0%, 71%

FP II JS, S 0%, 71%

2-LPT Ia JS, S 0%, 0%

1-LPT Ia JS, S 0%, 0%

1-LPT Ib JS, S 42%, 71%

S = Squat, JS = Jump Squat, cJS = Concentric JS, 1RM = One Repetition Maximum

2.2 Weight lifting

Weight lifting is a group of physical activities with different purposes. These include injury prevention, force and power output enhancement, rehabilitation, esthetic reasons, muscle endurance etc. It is not surprising that these different purposes have different means of achievement [29]. In this report only the parts of weight lifting associated with sports performance and more specifically explosive muscular power is covered.

2.2.1 Movements

Squat

Figure 2.2: Squat sequence front/side view [3].

The squat is widely considered the premier exercise for development of leg strength. It is performed with the barbell resting on the shoulders/upper back.

From a standing position, the lifter squats down into a position where the thighs are at least parallel to the floor or deeper. From this position the lifter pushes up until reaching the starting position again.

Jump squat

Figure 2.3: JS sequence front/side view [3].

The JS is commonly used for testing lower body power performance [9] and also for enhancing lower body muscle power output. It is always, in contrast to the squat, done in an explosive fashion.

It starts in the turning point position of the normal squat (often not as deep).

From here an explosive jump is performed. While in the air the legs are kept straight and on landing the lifter returns to the starting position of the normal squat and in the same motion absorbs the impact of the landing.

2.3 Existing systems

There exists a variety of systems for measuring different aspects of barbell weight lifting exercises. Many are one-of-a-kind systems stationed at sports physiology research centers (most systems used in Table 2.1 on page 7), while others are commercial and available for sports clubs and associations for purchase.

2.3.1 Sensors in use

Linear position transducer

A linear position transducer (LPT) is a device that measure a linear distance.

In sports applications, because of the three dimensional movement of the test subject, the most common LPT used is basically a spring loaded cable, with the cable drum shaft connected to a potentiometer. A LPT gives only the absolute distance from the LPT to its connection on the subject, not direction.

Linear position encoder

A linear position encoder (LPE) is similar to a LPT in the sense that it can be used in place for an LPT in all applications. But in contrast to a LPT, the signal from a LPE is quantized using an encoder with finite resolution whereas a LPT has a theoretical infinite resolution.

Most LPE:s used in sports physiology applications uses the increment of a rotary encoder for determining distance.

Video motion analysis

Knowledge of distances, scales and camera parameters makes it possible to extract information about an objects motion from the analysis of video.

Single camera By using some knowledge about the surroundings or the ob- ject of interest, such as scales, distances, ratios etc. it is possible to get information of an objects 3D motion using only the 2D projection given by a normal camera.

Stereo vision By using stereo vision, it is possible to derive a objects position in 3D without knowledge about the surroundings or the object of interest.

The configuration of the cameras need, however, to be known.

There are several difficulties using video motion analysis, such as occlusion, lighting, camera models, camera resolution, distortion and the way objects are recognized.

Ultrasonic

By using a transmitter on the object to which distance is wanted and a transpon- der at a fixed point and knowing the speed of ultrasonic sound in the current medium, it is possible to calculate distance between the two. This technology can be used in a similar way as a LPT.

Force plate

A force plate (FP) measures the ground reaction force. This is done mainly by using one of two techniques; strain gage or piezoelectricity. Strain gage uses the varying resistance of a conductor during deformation while piezoelectricity sensory uses the fact that a piezoelectric material builds up a electrical charge when under mechanical stress.

Accelerometer

An accelerometer is basically a mass-spring system, where the dislocation of the mass is measured and used to calculate the corresponding acceleration. There exists many ways of measuring the dislocation, such as piezoelectric, capacitive and optical methods.

2.3.2 Commercial systems

Ballistic measurement system

The Ballistic Measurement System is a 1D system from Innervations consisting of a PC-interface and a LPT that is connected to a PC via USB[2].

The LPT is placed underneath the barbell, hence approximating the vertical displacement of the barbell. The system is also expandable with a FP to directly measure the GRF. The software then calculates power, force and displacement and from this derives key values such as peak, average, etc.

The system is also compatible with Innervations’ barbell braking system Bal- listic Breaking System.

V-scope

The V-scope from Sport Expert is a system based on ultrasonic distance measurement. By using one or more transmitters (for one or several measured points) and three transponders, the location in 3D is derivable using trilateration (see 2.4.1). From this, the kinematics of the movement are derived and key values are calculated [17].

MuscleLab Power

MuscleLab Power from Ergotest is a 1D system similar to the Ballistic Measurement System, with the difference that it uses a LPE instead of a LPT [12].

2.4 Mathematics

2.4.1 Trilateration

The solution to the problem of finding a point for which only the distances from three known points are known equals finding the intersection of three spheres with centers in these points. This method of localization is called trilateration and is widely used in geodesy and in particular within the satellite positioning system GPS [30].

A geometric description of the problem is given in Figure 2.4 on page 11, where

¯ p1=



 x1

y1

z1



 , ¯p2=



 x2

y2

z2



 , ¯p3=



 x3

y3

z3



 , ¯p =



 x y z





and using the euclidean norm, the three equations

(x− x¹)²+ (y− y¹)²+ (z− z¹)² = l²₁ (x− x²)²+ (y− y²)²+ (z− z²)² = l²₂ (x− x³)²+ (y− y³)²+ (z− z³)² = l²₃ can be formulated.

z

y x

p3 p2 p1 l1 p

l3 l2

x' y' z'

Figure 2.4: Geometric overview of trilateration

To simplify the calculations, a new coordinate system (x^!, y^!, z^!)with its origin in ¯p1 is introduced. The x^!-axis of this system points in the direction ¯p2 and the y^!-axis is perpendicular to the x’-axis pointing in the direction of ¯p3. In this new coordinate system, the three known positions are

¯

p^!₁ = (0, 0, 0)^T

¯

p^!₂ = (d, 0, 0)^T

¯

p^!₃ = (i, j, 0)^T

This gives the simplified system in the new coordinate system x^!2+ y^!2+ z^!2 = l²₁

(x^!− d)²+ y^!2+ z^!2 = l²₂ (x^!− i)²+ (y^!− j)²+ z^!2 = l²₃

d = "¯p²− ¯p¹"

i = ¯ex^!· (¯p³− ¯p¹) j = ¯ey^!· (¯p³− ¯p¹) which has the solution

x^!= r₁²− r²2+ d² 2d

y^!= r₁²− r²3+ i²+ j²

2j −i(r₁²− r2²+ d²)

2dj (2.1)

z^!=±

% r₁²−

&r₁²− r2²+ d² 2d

'²

−

&r₁²− r²3+ i²+ j²

2j −i(r₁²− r2²+ d²) 2dj

'²

The coordinate transformation back to the original system (see Figure 2.4 on page 11) is then given by the basis vectors of the introduced system expressed in the original coordinate system

¯

ex^! = p¯2− ¯p¹

"¯p²− ¯p¹"

¯

ey^! = p¯3− ¯p¹− i · ¯e^x!

"¯p³− ¯p¹− i · ¯e^x!"

¯

ez^! = e¯x^!× ¯e^y!

And a position ¯p^! in the introduced coordinate system is translated into a posi- tion ¯p in the room-fixed coordinate system through

¯ p =(

¯

ex^! e¯y^! e¯z^! )

¯ p^!+ ¯p1

2.4.2 Error analysis

Given a arbitrary function, depending on several input variables, Y = f (X1, X2, . . . , Xn)

the Taylor expansion of the first order around the point Y0 is

Y ≈ Y⁰+ ∂f

∂X1

X1+ ∂f

∂X2

X2+ . . . + ∂f

∂Xn

Xn

This means that given an error in the input variable X, called %X, Y = ˆY + ∂f

∂X1%X¹+ ∂f

∂X2%X²+ . . . + ∂f

∂Xn%Xⁿ

,where ˆY denotes the correct value and Y the observed value. This means that the error in Y called ∆Y can be written

∆Y = ∂f

∂X1%X¹+ ∂f

∂X2%X²+ . . . + ∂f

∂Xn%Xⁿ

Given independent errors in X of gaussian distributions, the error in Y is written [16]

%Y²=

**

** ∂f

∂X1

**

**

2

%X1²+

**

** ∂f

∂X2

**

**

2

%X2²+ . . . +

**

** ∂f

∂Xn

**

**

2

%Xn² (2.2) An error in X can be be split up into two parts, one describing the systematic errors that are constant throughout all readings ∆SX and one for the random error ∆RX of noise added to the systematic error giving the total error

∆X = ∆SX + ∆RX

2.5 Mechanics

The system consisting of the athlete and the barbell is idealized as shown in Figure 2.5 on page 14. The athlete is modeled as a continuos elastic body, for which only the mass is known. The barbell is modeled as a point mass. There are four outer forces acting on the system as shown in Figure 2.6 on page 15; the gravity acting on the two masses respectively, the GRF acting on the athlete ( ¯Fg) and, if used, the LPT retraction force acting on the barbell ( ¯Fr). The force connecting the two bodies is the one exerted by the athlete in order to move the barbell ( ¯Fb).

z

y x

pa pc

ma mb

Barbell Athlete

Figure 2.5: The idealized mechanical system

Newton’s first law gives

map¨¯a = F¯g− ¯Fb− m^ag¯ez (2.3) mbp¨¯c = F¯r+ ¯Fb− m^bg¯ez (2.4) By twice differentiating ¯pc, which is measured, ¨¯pc is derived. By plugging that into (2.4), the force

F¯b= mbp¨¯c+ mbg¯ez− ¯Fr

is calculated and thereby the right hand side in (2.3) is known. The athlete velocity can now be calculated

˙¯

pa= ˆ

¨¯

padt + C (2.5)

In this particular application, the athlete velocity is assumed to be zero at the beginning of the motion, hence meaning that C = 0.

ma

mb -mbgez -magez

F g

-Fb

Fb F r

p ¨ a

p ¨ c

Figure 2.6: Free drawing of the idealized system in Figure 2.5 on page 14

2.5.1 Methods for power estimation

The existing methods that have been used in previous studies (Table 2.1 on page 7) can be divided into three sub-categories;

• Purely kinematic methods

• Purely kinetic methods

• Methods using a mixture of kinetics and kinematics.

In all these methods, the barbell and athlete are accounted for as one uniformly moving system, (except for method IV on page 16, which is a new method as of this report). This, however, is not true as the athlete change pose during the exercise, resulting in a displacement of his/her center of mass (COM). To compensate for this, praxis is that 90% (since the feet and lower extremities are almost stationary) of the athlete’s body weight (BW) and 100% of the barbell weight are included in the system mass.

Kinematic methods

These methods rely only on knowledge of kinematic variables, such as position, speed etc. The typical way used in most of the kinematic studies in Table 2.1 on page 7, is the use of one or more LPT:s to measure the position of the barbell.

Ia) By differentiation of the position data, the velocity and acceleration of the barbell are attained and with knowledge about the system mass, the power output is estimated. The power output is calculated

P (t) = ˙¯p(t)· (¨¯p(t) + ¯e^z· g) · m

Ib) A simplified version of method 1a) also exists where the upward force is accounted for as only the gravity acting on the barbell, therefor making it constant. In these studies the power output was calculated

P (t) = ˙¯p(t)· ¯e^z· g · m

Kinetic methods

II) These methods rely purely on the knowledge of the athlete’s ground re- action force (GRF). By knowledge of the system mass and integration of the acceleration, the velocity for the total system’s COM is attained. From this the power is calculated as

P (t) = ¯Fg(t)

t

ˆ

0

F¯g(t)

m − ¯e^z· g dt

Kinetic-Kinematic methods

III) The studies in Table 2.1 on page 7 that include both a kinematic and a kinetic mean of measurement uses the velocity measured from the kinematic sensors and the force measured from the kinetic sensors. The power is accounted for as

P (t) = ˙¯p(t) ¯Fg(t)

IV)¹ All outer forces on the system (the GRF, the LPT retraction force and the forces of gravity) in Figure 2.6 on page 15 are directly observable, the force between the two sub-systems is calculated from (2.4), and the velocities of the two bodies are either observable or implicitly known from (2.3). The power of the whole system can be divided into two parts, one for each sub-system, Pa

and Pb.

For the barbell, since it is considered a point mass, the power has only a transla- tive component. For the athlete, on the other hand, there are internal forces, torques, velocities and rotations that, with the model described in Section 2.5, are unobservable. The only thing that is possible to derive is the acceleration of

1This method is not used in any of the studies refered to in Table 2.1 on page 7 and is new as of this report.

the athlete’s COM, and hence the velocity of the athlete’s COM is derivable by integration as seen in (2.5). Therefor the only part of the power of the athlete that can be calculated is the translative power of the COM, giving

Pa(t) = p˙¯a(t)· ( ¯Fg(t)− ¯Fb(t)) Pb(t) = p˙¯c(t)· ¯Fb(t)

This gives the total power

P (t) = Pa(t) + Pb(t) = ˙¯pa(t)· ( ¯Fg(t)− ¯Fb(t)) + ˙¯pc(t)· ¯Fb(t) (2.6) Under the assumption that ˙¯pa= ˙¯pc= ˙¯p, this generates

P (t) = ˙¯p(t)· ¯Fg(t)

which means that method III) only is viable under the assumption that the system as a whole has the same velocity and that this velocity either can be derived from the barbell position or the GRF.

Chapter 3

Design

3.1 Lab setup

The purpose of the lab setup (see Figure 3.1 on page 19) is to be able to capture and measure the kinematics and kinetics of an athlete performing a barbell exercise. For this purpose five LPT:s (plus one without sensor, used for the symmetry of the test set up) and one FP was available.

To be able to implement the existing methods described in 2.5.1, the LPT:s need to be used to measure the barbell’s position while the FP need to be used to measure the GRF from the athlete’s feet. The charge signal from the FP is transduced into a voltage via the charge amplifier (CA) and both this signal and the LPT signals are converted into digital signals through the analog to digital converter (A/D) and thereafter fed to a PC where the calculations are carried out.

FP

LPT

CA

A/D

DistanceForce

PC

(a) Schematic view of the test setup

(b) Photo of the test setup in place at CSD

Figure 3.1: The test setup

3.1.1 Equipment

LPT - Celesco PT5A

The PT5A is a LPT based on a plastic-hybrid precision potentiometer [4]. The range and zero of the output is configurable by two trimmers, thus making it possible to optimize the voltage output depending on the stroke length of the application. The characteristics of the PT5A is presented in Table 3.1 on page 20.

Table 3.1: Stated characteristics of the PT5A from Celesco[4]

Range 0 . . . 3.81 m

Accuracy ±0.0069 m

Repeatability ±0.00076 m

Input Voltage 30 VDC

Output Voltage −10 . . . 10 VDC

FP - Kistler 9281E

The 9281E is a piezoelectric FP with four sensors placed in a rectangular pat- tern. It measures GRF in all three dimensions and gives a charge output linearly proportional to the applied force. The characteristics of the 9281E is given in Table 3.2 on page 20.

Table 3.2: Stated characteristics of the 9281E from Kistler[22]

Range x, y −10 . . . 10 kN Range z −10 . . . 20 kN

Accuracy ±50 mN

Linearity < 0.2 % Sensitivity x, y −7.5 pC/N Sensitivity z −3.8 pC/N

CA - Kistler 9865E

The 9865E is a 8-channel CA from Kistler. It is build for use with the 9281E, but since it has only 8 channels, and the 9281E has 12 output channels, the x- and y-channels are merged into groups of two [21]. Since the input signal is a charge, the output will slowly diverge from the initial value due to the fact that the cables and circuits in the 9865E and the test setup are not perfectly insulated, leading to a charge drift. The impact of this drift is however small when relatively short dynamical forces are measured.

To be able to optimize the output from the CA, the 9281E has four different output ranges that are selectable for the z-channels as a group and for the x, y- channels as a group. The characteristics for the 9865E with the range in use (±2631 N) is presented in Table 3.3 on page 21.

Table 3.3: Stated characteristics of the 9865E from Kistler for both channels on range 3 [21]

Range −10 . . . 10 nC

Accuracy <±0.1 V

Noise <±0.02 V (rms.)

Drift < 70 µV/s

Output Voltage −10 . . . 10 V

A/D - NI 9215 & NI cDAQ 9178

The cDAQ 9178 form NI is a USB-connected chassis with 8 slots for use with NI’s CSeries input/output (I/O) modules[26]. These modules come in a variety of configurations for different types of input and output.

The CSeries I/O module used in this application is the NI 9215. The NI 9215 is a four channel A/D module. It comes in two versions, one with Bayonet Neill–Concelman connector (BNC) and one with screw terminal connection. It allows for simultaneous sampling of all four channels at 100 kHz[25]. Other characteristics are given in Table 3.4 on page 21.

A total of four NI 9215 are used in the test apparatus.

Table 3.4: Stated characteristics of the NI 9215 module from NI [25]

Range −10 . . . 10 V

Resolution 16 bit

Offset error 0.0028 V

Gain error 0.02 % of reading

LabVIEW

LabVIEW is a graphical programming language from NI [27]. It is developed for data acquisition and measurement and supports a large number of data input and output devices, making it suitable for laboratory use.

3.1.2 Positioning of the LPT:s

Because five LPT:s are avaliable, it is possible to find two positions on the barbell using trilateration (see Section 2.4.1), hence making it possible to not

only find the position of the barbell’s COM but also its orientation. This could be of interest for future add-ons to the equipment and was therefor implemented instead of just measuring the barbell’s COM directly using three LPT:s.

To minimize the effect of the retraction forces from the LPT:s pulling the barbell, the LPT:s were positioned symmetrically around the center of the FP, over which the barbell is intended to mainly move.

To minimize the risk, both of the athlete tripping on wires and injuring him/herself and of damaging the equipment, the LPT:s were positioned in the roof of the laboratory.

To avoid the unintentional release of the LPT wires (free release of the wires can harm the LPT:s since the retraction force is large to avoid slacking of the wires), the wires were mounted in a permanent fashion to the barbell. Because of this, it was not possible to attach the wires at the end points of the barbell, which from a geometrical perspective is preferable, in order to be able to add and remove weight plates. The LPT can also not be mounted in such a fashion that the wires at any time during a lift touches the athlete or other objects.

Therefore all the LPT:s were mounted on an area between the weight plates seen from above. The chosen configuration is shown in Figure 3.2 on page 23 and the positions of the LPT:s are given in Table 3.5 on page 23.

Barbell LPT

2

1

3 5

Fake

x

4

1, 3 2 Fake 4, 5

y

x z Above

Front

Figure 3.2: Above and front view (opposite of the athlete’s view) of the LPT:s and coordinate system placement in relation to the position of the barbell

Table 3.5: The LPT positions Position

LPT x y z

1 0.000 0.000 3.020 2 0.666 0.595 3.020 3 0.000 1.150 3.020 4 1.364 0.000 3.020 5 1.364 1.157 3.020 Fake 0.698 0.585 3.020

3.1.3 Localization of the barbell

z y

x p

_c

l1

l3 l2

p

_l

l4 l5 lb

p

_r

Figure 3.3: Trilateration used to determine the position of the wire connection points on the barbell

On the athlete’s right hand side connection point on the barbell (connected to LPT:s 1, 2 and 3), the procedure of localization is straight forward following the method of trilateration described in Section 2.4.1. Three LPT:s are connected with known positions and wire lengths and this is simply plugged into (2.1).

On the left side, however, only two real LPT:s are connected. Here the fact that the distance between connection points lb is known is used as a third LPT.

The calculated position of the right hand side connection point is used as the origin and the distance between connection points is used as the wire length of a third, fictive, LPT.

When both side’s connection points, ¯pr and ¯pl are known, the center of the barbell can be calculated as

¯

pc= p¯l+ ¯pr

2 Error analysis

The position of the center of the barbell is a function

¯

pc= f (¯p1, ¯p2, ¯p3, l1, l2, l3)

It is assumed that the random fluctuating errors of the input variables are eliminated through the filtering process described in 3.2.2. This leaves only the systematic errors (measuring the positions of the LPT:s and the calibration of the range and length of the LPT wires). The systematic errors in these variables

are assumed to be of Gaussian distribution with the following approximated errors

%¯pⁱ =



 0.005 0.005 0.003





%lⁱ = 0.002

The error propagation method in Section 2.4.2 gives the values for the errors presented in Table 3.6 on page 25. These errors were calculated in Matlab and the code used is found in Appendix A.

Table 3.6: The offset error for the barbell center position

Position [m] Error [m]

x y z %x %y %z

0.4350 0.3950 0.3000 0.0176 0.0098 0.0039 0.4350 0.3950 1.4000 0.0245 0.0111 0.0066 0.4350 0.3950 2.5000 0.0279 0.0111 0.0206 0.4350 0.5950 0.3000 0.0177 0.0098 0.0043 0.4350 0.5950 1.4000 0.0247 0.0111 0.0076 0.4350 0.5950 2.5000 0.0322 0.0110 0.0258 0.4350 0.7950 0.3000 0.0180 0.0098 0.0049 0.4350 0.7950 1.4000 0.0253 0.0111 0.0088 0.4350 0.7950 2.5000 0.0376 0.0111 0.0319 0.6350 0.3950 0.3000 0.0189 0.0099 0.0031 0.6350 0.3950 1.4000 0.0281 0.0118 0.0044 0.6350 0.3950 2.5000 0.0288 0.0113 0.0111 0.6350 0.5950 0.3000 0.0190 0.0099 0.0036 0.6350 0.5950 1.4000 0.0284 0.0118 0.0057 0.6350 0.5950 2.5000 0.0339 0.0113 0.0157 0.6350 0.7950 0.3000 0.0194 0.0099 0.0042 0.6350 0.7950 1.4000 0.0290 0.0118 0.0071 0.6350 0.7950 2.5000 0.0402 0.0113 0.0206 0.8350 0.3950 0.3000 0.0218 0.0102 0.0031 0.8350 0.3950 1.4000 0.0352 0.0129 0.0046 0.8350 0.3950 2.5000 0.0369 0.0118 0.0122 0.8350 0.5950 0.3000 0.0219 0.0101 0.0035 0.8350 0.5950 1.4000 0.0353 0.0129 0.0058 0.8350 0.5950 2.5000 0.0424 0.0118 0.0163 0.8350 0.7950 0.3000 0.0222 0.0102 0.0040

3.1.4 Velocity and acceleration

The velocity and acceleration are derived using finite differentiation, giving

˙¯

pc(t + h) = p¯c(t + h)− ¯p^c(t)

h (3.1)

where h is the sampling interval. By using the same principle, the acceleration can be written as

¨¯

pc(t + h) = p˙¯c(t + h)− ˙¯p^c(t)

h = p¯c(t + h)− 2¯p^c(t) + ¯pc(t− h)

h² (3.2)

Error analysis

Because it is assumed that there are no random fluctuating errors, the error between two samples of the position is approximated to be the same, since h is very small in relation to the speed of the movement and the error only depend on systematic errors in the input variables of ¯pc.

%¯p^c(t + h)≈ %¯p^c(t) Equation (3.1) with this approximation gives

˙¯

pc(t + h) = p¯c(t + h) + ∆¯pc(t + h)− ¯p^c(t)− ∆¯p^c(t + h)

h =p¯c(t + h)− ¯p^c(t) h

The same is true for the acceleration, meaning that the acceleration also is not affected by the systematic error in ¯pc.

3.1.5 LPT retraction force

LPT

l

m

R

mg

Figure 3.4: The model from which the LPT retraction force was calculated

The LPT:s each contribute with a pulling retraction force in the direction of the wire from the barbell. This retraction force is modeled as a linear spring

R = k· l + R⁰ (3.3)

The constants k and R0 were determined using a test setup as shown in Figure 3.4 on page 26. By releasing a known weight m and deriving its acceleration ¨l as a function of l, the retraction force is calculated

R(l) = m(g− ¨l(l)) where ¨l(l) is fitted as a linear function.

From this the constants in (3.3) are calculated.

The resulting force on the barbell’s COM was calculated by, for each LPT, finding the unit vector ¯eci pointing from the barbell’s COM towards the LPT.

The total retraction force was then calculated F¯r=+

i

¯

eci· (k · lⁱ+ Ri0)

The constants for the retraction force from the LPT:s calculated with this method is presented in Table 3.7 on page 27.

Table 3.7: The retraction force constants for each LPT Constants

LPT k R0

1 1.103 6.6 2 1.103 7.3 3 1.103 6.6 4 1.103 6.6 5 1.103 6.6 Fake 1.103 7.3

3.2 Data collection and conditioning

The sensors in Figure 3.1a on page 19 are all sampled at 1 kHz.

For the test application to be both user friendly and the test recordings to be standardized, the capturing of the signal needs to be automized. This is done using triggers for data capturing followed by extraction of a part of that data following some criteria.

A continuos circular pre-recording buffer runs in the recording state of the soft- ware. This assures that 0.9 s before the triggered movement is collected. After

the second triggering, another 0.9 s are saved to a post-recording buffer. These buffert have a large safety marginal, and only a small part of the bufferts are in general used. This is shown in Figure 3.6 on page 30.

Before the filtering the pre- and post-recording buffers are merged with the actual recording. After the filtering, part of the pre- and post-recording buffers are discarded to eliminate boundary issues.

To get a standardized beginning and end of a movement, another trimming is performed. This trimming procedure looks from the ends in the remaining pre- and post-buffers for where the vertical absolute velocity becomes smaller than 0, 1m/s (see example in Figure 3.6 on page 30). If no such point is found the whole remaining buffer is included in the recording.

The procedure of capturing and conditioning the signals is described in Figure 3.5 on page 29.

START

RECIEVE SAMPLES

RECIEVE SAMPLES START TRIGGERED?

RECORD POST- RECORDING

BUFFER

UPDATE PRE-RECORDING

BUFFER

ADD TO RECORDING

MERGE RECORDING AND BUFFERS

FILTER AND CUT OUT DATA

PERFORM CALCULATIONS

STOP TRIGGERED?

YES NO

NO

YES SAVE DATA

TO FILE

Figure 3.5: Flow chart of the recording-process.

3.2.1 Triggering

t Vertical

displacement

T1 T2

Recording Triggering Pre-

buffer

Post- buffer

Figure 3.6: Barbell vertical position signal with triggering, buffers and actual recording

The triggering is based on the barbell’s vertical position and two thresholds are used, T1 for at which height the recording should start and T2 for where it should stop. These triggers have directions assigned to them, so that they only trigger when the barbell passes the threshold in the chosen direction of the test leader. An example of this is shown in Figure 3.6 on page 30 where the direction for T1 is upwards and for T2downwards. This could typically be e.g. a JS.

To avoid unintentional triggering, the thresholds are set higher/lower than the start and stop position. This means that the motion starts a bit before and ends a bit after the actual triggering. This is compensated by having the pre- and post-recording buffers along with the data trimming.

3.2.2 Filtering

The vertical displacement value used for trigger detection is calculated using the LPT signals filtered using a non-weighted moving average with a width of 150 samples.

When the whole movements raw data is captured (stage called “filter and cut out data” in Figure 3.5 on page 29), all signals are filtered using a third-degree Butterworth zero phase filter with a cut-of frequency of 8 Hz. This is to remove the impact of the high frequency noise from the sensors in the final calculations.

The cut-of frequency was decided by use of trial. This cut-off frequency was found to capture the essence of the motions, whilst eliminating as much of the

noise as possible. Because the signal is first recorded in its full length and then filtered, the zero-phase filtering process offers the possibility to remove phase shift introduced by real time filters.

3.2.3 Phase intervals

In most strength and power testing, the whole motion of an exercise is not of interest. More often it is only certain aspects of the movement that are of interest, such as the concentric/eccentric phase etc.

To be able to measure different parts of the movement, the software features two selectable intervals for which both a set of key values are calculated. The software itself finds and suggests intervals of interest based on where the velocity is zero. These points are then filtered in such a way that they are not to closely lumped both in the time aspect and in the vertical displacement aspect, making it easier for the user to quickly choose and change intervals. For most tests however, often only one phase is of interest.

The result of this for a JS is shown in the second part of Figure 3.8 on page 33.

3.2.4 Saving

All filtered time data from the sensors are saved in a measurements file of the type TDMS (Technical Data Management System) from NI. The TDMS is searchable and allows for saving of multiple tests in the same file along with time data and test parameters [28]. The structure of a measurements file is shown in Figure 3.7 on page 32.

To allow corrections of parameters after the test is done, all time data needs to be saved so that all calculations can be redone.

NAME MEASUREMENT

FILE

2011-06-13 09:45:32

LOAD

LPT 1

LPT 1

GFR Y

GFR Z

2011-06-13 09:45:54

Figure 3.7: File structure of the measurements file

3.3 PC Interface

To operate the test apparatus, a graphical user interface (GUI) was made using LabVIEW.

Figure 3.8: Screenshot of the user interface

The GUI for the test apparatus is shown in Figure 3.8 on page 33 with the numbered features;

1. Button for creating or opening test data files.

2. Test data inputs such as personal information and parameters for the test and exercise. This is saved as parameters in the test file (Figure 3.7 on page 32).

3. Display showing the current barbell height during the recording.

4. Test history. Showing tests that have been made with calculated key values and parameters. Possibility to edit parameters (and thereby redoing the calculations), export data and present data exists.

5. Data presentation plot. To show time data such as speed, power etc. Also possible to plot data series against each other, e.g. the relation between power-speed, force-position etc.

6. Choice of x- and y-axis data series for the data presentation plot. Possible to choose multiple y-axis data series so that e.g. both speed and power is plotted simultaneously against time.

7. Plot of phase intervals for calculation of key values (3.3.1).

8. Choice of phase intervals for calculation of key values.

3.3.1 Key values

To give the athletes and researchers easily used indicators, certain key values calculated from the measurements are made. These calculated key values, used in the application, for assessing the athletes’ physical status are;

Peak Power, Force and Acceleration calculated for each method, phase and (except for power) dimension.

Average Power, Force and Acceleration calculated for each method, phase and (except for power) dimension.

Duration for the whole movement and for each phase.

Phase Intervals and directions start and stop points and their respective passing direction.

Chapter 4

Trial

A small trial test was carried out consisting of 9 male, division 1 junior ice hockey player, participants (BW = 84 ± 1, 5 kg), who performed JS at six different intensities; 30,50,75, 85, 100 and 125 % of their BW:s. Four different methods were used to calculate peak and average power during the concentric part of the motion; Ia, II, III and IV. The result is presented in Figure 4.1 on page 36.

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Figure 4.1: The peak and average power with standard deviation within the trial group, measured during the concentric phase of the JS

Chapter 5

Results and Discussion

The objectives and goals for this thesis, set up in Section 1.2, were to make an application able to replicate the methods and models used in Cormie et. al.

2007 [9], and also explore the possibility of other, previously not mentioned, methods for muscle power output calculation. Both these goals were met where a test apparatuses was constructed and set up at CSD and one new model for muscle power output was derived.

This test apparatus has the ability to completely determine the barbell’s position in 3D and its orientation around two angles (not radially around the bar). It also measures the GRF:s in 3D. This is more than any other previously used test setup.

The newfound method of power estimation, method IV, accounts for the barbell and athlete as two separate masses, whereas previous methods uses a single mass model. This reflects more of the reality than previos methods, with the only idealization that the athlete’s COM is at rest at the beginning of the motion.

This is something, however, that for most exercises is true.

5.1 Commercial potential

The commercial potential of this product is somewhat limited.

Firstly, it requires much more equipment and space to operate than other com- ercial systems. The trilateration process requires quite a large space to insure that the angles between the wires are sufficiently large. Also this test system uses five LPT’s, whereas all the commercial off-the-shelf systems only use one.

Secondly, the advantage of this systems regarding precision probably does not motivate the much higher cost. Eventough the system is more accurate in terms of measuring the barbell position in 3D and using a high precision FP,