Calculation of mechanical
energy in cross country skiing
Frida Bakkman
Master Thesis
Advanced level, 30 hp
KTH Mechanics, Royal Institute of Technology
Stockholm, August 2012
Abstract
Cross country skiing is considered to be one of the most demanding sports in
terms of endurance. Therefore the skiers are attractive subjects for physiological and biomechanical research whose interest has increased a lot during
the 21st century. The results are used to improve the mechanical knowledge
about the body and to improve the capacity and technique for the competitors.
The aim with this study is to implement a method for mechanical energy
calculation in cross country skiing. This is based on data from 15 skiers
using the double poling technique, where the potential, rotational and translational energies are calculated.
The measurements are made in a lab using a treadmill with stepwise increased velocity. The system used is Vicon MX where the skiers wear reflective markers, whose positions is calculated from data from infra-red light
cameras. The positions of the joint centres are calculated used as input data
to the program. Joint centres and marker data divide the body into segments
where the energies of each segment are calculated and possible to sum up for
the whole body.
The results are examples of obtainable data from the model. It is possible
to compare chosen subjects’ total mechanical energy but also the energies
and segments separately. The results can be used to analyse the different
techniques to improve the capacity of the competitors.
1
Sammanfattning
Längdskidåkning är klassad som en av de tuffaste uthållighetsidrotterna.
Detta är en av anledningarna till att skidåkare är attraktiva test personer
inom forskningen kring fysiologi och biomekanik, vars intresse har ökat under
det senasteårhundradet. Forskningsresultaten används för att öka förståelsen
kring kroppens mekanik men ocksåför att utveckla åkarnas kapacitet.
Målet med denna studie är att implementera en metod för beräkning av
den mekaniska energin vid längdskidåkning. Den är baserad från data från
dubbelstakningsanalys av 15 skidåkare, där både potentiell och kinetisk energi är beräknad.
Mätningarna är gjorda i ett lab där åkarna kör påett rullband vars hastighet
ökar stegvis. Systemet som har använts är Vicon MX där åkarna bär reflektiva markörer, vars positioner är beräknade från datan, som genererats från
infraröda kameror. Utifrån detta beräknas positionerna för alla ledcentrum,
vilka är indata till programmet. Ledcentrum och markörer delar kroppen i
olika segment där energierna är beräknade segmentsvis och möjliga att summera för hela kroppen.
Resultaten från studien är exempel påvad man kan analysera. Det är möjligt
att jämföra valda personers totala mekaniska energi men ocksåenergierna och
segmenten separat. Resultaten kan användas för att analysera olika tekniker
och öka åkarnas kapacitet.
2
Preface
This master thesis is my last step towards my M.Sc in Engineering Physics at
KTH. The thesis is written at KTH Mechanics in cooperation with Swedish
Winter Sport Research Center in Östersund during spring and summer 2012.
I am so pleased that I got the possibility to write about subject where I
could combine two of my biggest interests, sports and mathematics. The interest motivated me to hard work and good results and therefore the barriers
I faced along the road was a bit frustrating. I had to change my focus and
limit the aim. Even so I am satisfied with what I achieved and in particular
what I have learnt about research while writing.
I would like to thank some people who have helped me along the work.
Anders Eriksson-My supervisor who made the cooperation with Östersund
possible. Thanks for the teaching, explanations and help to handle the barriers.
Thomas Stöggl-The one who came up with the subject and helped me
with data and problems along the road. Thanks for your patience.
HC Holmberg-”The spider in the web”. Thanks for you commitment and
a lot of inspiration.
Frida Bakkman, Stockholm 2012
3
Contents
1 Introduction
1.1 Theory . . . . . . . . . . . .
1.1.1 Cross country skiing
1.1.2 Research . . . . . . .
1.2 Objective and goal . . . . .
1.2.1 Limitations . . . . .
1.3 Background . . . . . . . . .
1.4 Biomechanics . . . . . . . .
1.4.1 Anatomical terms . .
1.4.2 Anatomy . . . . . . .
1.4.3 Anthropometric Data
2 Method
2.1 Data Collection . . . . . .
2.1.1 Subjects . . . . . .
2.1.2 Experimental Setup
2.2 Data Processing . . . . . .
2.2.1 Joint centres . . . .
2.2.2 Angles . . . . . . .
2.3 Calculations . . . . . . . .
2.3.1 Approximations . .
2.3.2 Centre of Mass . .
2.3.3 Energy . . . . . . .
2.3.4 MATLAB . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5
5
5
5
6
6
6
7
7
8
8
.
.
.
.
.
.
.
.
.
.
.
11
11
11
11
12
12
14
15
15
15
15
18
3 Results
20
3.1 Centre of mass . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Discussion
4.1 Centre of mass . . . . . . . . .
4.2 Energy . . . . . . . . . . . . . .
4.2.1 Total Energy . . . . . .
4.2.2 Potential and Rotational
4.3 Translational Energy . . . . . .
4.3.1 Segment analysis . . . .
. . . . .
. . . . .
. . . . .
Energy
. . . . .
. . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
26
26
26
26
26
27
27
5 Conclusions
28
5.1 Future Perspectives . . . . . . . . . . . . . . . . . . . . . . . . 28
4
1
1.1
1.1.1
Introduction
Theory
Cross country skiing
Cross country skiing was first invented as a way of transport during the
winter, but is now developed to a way of train and compete [5]. The original
style of cross country skiing is the classic technique, which is characterized
by the parallel movement of poles and skis. This technique can be divided
into four different gears which are used dependent of terrain and capacity
[1].
In the 1980s the skating technique was introduced[8]. This way of skiing
requires higher strength in the upper body, which has further been stimulated
with the introduction of the new racing forms, sprint and mass start, during
the last decades. In these type of races the skiers need a fast finish and
thereby the competitors put more emphasis on upper body strength training
and endurance training[9][8]. These changes have also increased the ability
to reach higher velocities in certain parts of the classic races too[9].
1.1.2
Research
Cross country skiing is considered to be one of the most demanding sports
in terms of endurance. This is one of the reasons to the increased interest
of research in this field during the 21st century. Since the skiers have such
a high physiological level and use the whole body during the work, they are
very attractive as test subjects[5].
From the beginning of physiological test, dated to the early 20th century,
until now the knowledge and possibilities has increased a lot. The technique
development has made it possible to measure, for example, the uptake of
oxygen, blood pressure, cardiac output and even the metabolic response and
blood flow in specific muscles[10]. Most of the research today is done in a lab
where the skiers uses roller skis on a treadmill[5]. It is possible to capture
the movements in 3D and measure the forces in the shoes and poles.
The results of the research are used to increase the understanding of this
complex sport in a mechanical and physiological way. It is also important
for the development of the technique and for the performers to improve their
capacity[10][5].
5
1.2
Objective and goal
The aim with this study is to develop a method for mechanical energy calculations. This can be used to analyse and compare for better understanding
around the cross country skiing technique.
1.2.1
Limitations
• Double poling is the technique used in this study
• The results are in 2D, the sagittal plane
• The results are examples of what can be obtained
1.3
Background
In 1987 Robert W. Norman and Paavo V. Komi presented their study: Mechanical Energetics of World Class Cross-Country Skiing. Their purpose was
to determine eventual differences in mechanical output and energy transfers
in different inclines[6].
This differs from my study where the aim is to present a model that can
be used for this type of calculations, but doesn’t put any emphasis on the
numbers in the results.
The measurements in the Norman study were maid at a 15km race at the
World Championships in 1978. The segment model of the bodies contained
15 segments, including skis and poles, where the Cartesian coordinates of the
”joints” where obtained from the capture. The data was sampled from the
diagonal stride technique from 11 subjects at 1.6◦ and 9◦ with a frequency of
60 Hz respectively 30 Hz. The data was filtered at an upper cut off frequency
of 4.5 Hz before the displacements and velocities were determined[6].
The measurements in this study are made in a lab, using roller skis on a
treadmill. The segment model consists of 19 segments all together, where
the energies of each segment are determined separately. The technique that
is used is the double poling and the incline is constant on 1◦ . The velocity is
24.5 km/h from the beginning and will increase stepwise during the sampling.
The data is captured by eight cameras emitting infra red light reflected by
markers attached on the subjects, and is sampled at a frequency of 250 Hz.
The data output is not filtered.
The energy is calculated by summing the energies of the segment. The energies are divided in potential, rotational and translational energies and calcu6
lated separately for each segment. Norman also investigates the mechanical
work output and mechanical energy transfers between segments. The results
are presented in tables of numbers to compare different skiers and slopes[6].
The method for energy calculations in this study is similar. The three types
of energies are calculated for each segment and then summarized. The translational energy is calculated in all three directions. The results of this study
are only examples of possible results that can be generated and no further
discussion or analyse around the numbers is presented.
1.4
Biomechanics
1.4.1
Anatomical terms
The anatomy of the body is often explained in specific terms and uses shortenings of words. To increase the understanding for the reader there is a table
of the most used ones in this study shown below.
Figure 1: The three different planes of the body. Reproduced from[13].
7
Sagittal plane
The 2D plane of the body seen
from the side, divides the body in
lateral and medial.
The 2D plane of the body seen
from above, divides the body in
upper and lower.
The 2D plane of the body seen
from the front, divides the body
in frontal and posterior.
The end of a segment nearest
midpoint of the body
Opposite from Proximal
Centre of mass
Bending and stretching in a joint
(In the sagittal plane)
Movement in joint from and towards the body (In the coronial
plane)
Markers placed on the frontal
pelvis
The markers placed on the posterior pelvis
Transverse plane
Coronal plane
Proximal
Distal
COM
Flexion/Extension
Abduction/Adduction
LASI/RASI
LPSI/RPSI
Table 1: Anatomical Terms
1.4.2
Anatomy
To make these calculations possible it is necessary to do some simplifications
and approximations of the anatomy of the body. The body is divided into
smaller parts, in this case 15 different segments, where the soft tissue is
neglected and the segments are assumed to be stiff rods. This is a good
approximation for the bones of the limbs, whose names are used for their
corresponding segments.(See figure 3) The names of all the segments and
their endpoints are presented in the table in the next section.
1.4.3
Anthropometric Data
The segments in the model have their specific COM, mass and radius of
gyration, determined by Dempster, according to the tables below[2].
8
Figure 2: Bones and endpoints used for segment determination. Based on[14].
Figure 3: The segment model. Based on[15].
Dempster used eight cadavers to define the segment parameters. First he
determined the joint centres by Reulaux’s (1876) method, which he used as
endpoints for the segment. Then he used a balancing technique to determine
the center of gravity and a pendulum technique for the moment of inertia
determination[2].
9
Segment length parameters
Segment Proximal to Distal
Head
C7 to Midhead
Thorax
L5 to C7
Pelvis
Midhip to L5
Humerus Shoulder joint center to Elbow joint center
Radius
Elbow joint center to Wrist joint center
Hand
Wrist joint center to Finger knuckle
Femur
Hip joint center to Knee joint center
Tibia
Knee joint center to Ankle joint center
Foot
Ankle joint center to Toetip
Dempster
1.0
0.37
0.895
0.43
0.45
0.429
0.47
0.43
0.5
Table 2: Length parameters, the distance from the proximal end as a fraction of
the segment length [11].
Segment mass and radius of gyration according to Dempster
Segment m
Kcg
Head
1.0
1.0
Thorax
0.37 0.37
Pelvis
0.895 0.895
Humerus 0.43 0.43
Radius
0.45 0.45
Hand
0.429 0.21
Femur
0.47 0.45
Tibia
0.43 0.44
Foot
0.5
0.47
Table 3: Segment mass parameters, the fractional weight of the body. Radius of
gyration according to COM of each segment[11].
The radius of gyration is one of the parameters used to calculate the
moment of inertia, which is needed for the rotational energy. The relation
between these parameters is given by,
2 2
Icg = mKcg
l
(1)
Where Icg is the moment of inertia at COM of the segment, m is the mass
and l is the length of the segment.
10
2
2.1
Method
Data Collection
The data collection was made in the ski lab at Swedish Winter Sport Research
Center in Östersund.
2.1.1
Subjects
The subjects are 15 volunteering skiers in the range of level from elite amateur
to world champions. The range of weight is between 64.9 kg and 87kg with an
average of 76.1 ± 5.1kg (mean ± st.d), the height varies from 171.5 cm to 187
cm with an average of 180 ± 4.8cm. The pole length differs between 141cm
and 152.5 cm, the average length is 148.8 ± 3.8cm. All skiers were familiar
with roller skiing on treadmill from earlier tests and training sessions.
2.1.2
Experimental Setup
Vicon The system used in this study is VICON MX for motion capture
with the VICON Nexus software. This system is designed for Life Science
applications and is used as well in clinical and research laboratories as in
sports performance centres and universities[4].
This system uses infra-red light from cameras to register the motion from
reflective markers worn by the subject. In this case there are eight of these
cameras attached in the sealing at computed positions. To make 3D measurements it is necessary that each marker on the subject is registered by at
least three cameras.
Preparations Before using the Vicon system it is necessary to calibrate
the cameras. This is done by swinging a T attached with four markers around
the room to define the capture volume and then place this T at a specifics
point to set the origin of the room and define the global coordinate system.
In this case the X-axis points from the right to the left side of the subject,
the Y-axis points backwards and the Z-axis points upwards.
The subject is attached with approximately 30 markers at very precise positions. Since they are attached straight to the skin the clothing is just a
pair of shorts. The positions of the markers is specific to fit the script of the
Vicon program. The program will scale the generic model to the anatomy
of each person and calculate the different joint centres for the subject. Together with the marker data it is necessary to measure the length, leg length
11
Figure 4: Emil Jönsson during measurements with Vicon MX. Reproduced
from[12].
and the width of the joints to make the calculations more accurate[17]. The
subject will then stand in a ”T-pose”, with the arms in a horizontal position,
to make sure that all markers are visible for the cameras.
Measurements During the measurement the cameras stream marker data
to a real-time engine. This engine calculates 3D positions of the markers and
underlying biomechanics specific for the model of the subject[3].
The treadmill starts at a velocity of 24.5km/h and increases with 1.5km/h
every 10th second. The skier goes double poling until he falls off the treadmill,
and get caught by the attached lift. The cameras sample the marker position
at a frequency of 250Hz.
2.2
Data Processing
The marker data is then exported to the Plug In Gate model(PIG) that
together with the anatomical measurements calculates joint positions and
segment angles for the subject[11].
2.2.1
Joint centres
Pelvis and Hip joints In order to calculate the data for the lower limbs,
the PIG model starts to define the coordinate system of pelvis. The origin
of this system is determined from the ASIS markers and the axes defined
from the directions between and within the ASIS and PSIS markers. The
12
centres of the hip joints is then determined from this system, together with
a defined offset. The offset is calculated in all three directions dependent of
the different leg lengths. The results of these centres is then used for further
calculations of the knee and ankle centres.
Knee and Ankle
Offset The offset of the ankle and knee joints are the distance from the
joint centre to the centre of the marker, which means adding the radius of
the marker to half the measured width of the joint.
Centres The positions of the joint centres are calculated by using a
modification of a defined chord function. The general version of this function
is used in static trials, but in the dynamic calculation the segment might be
rotated and therefore an extra marker is needed.
The function uses a previous calculated joint centre, the joint marker and, in
the modified version, a segment marker to define a plane. The joint centre
is located at the offset distance from the marker in the plane in a direction
perpendicular to the line between the known joint centre and the calculated
one. See figure 5
Upper body The calculations of the joint centres of the upper body begins
with defining a coordinate system in thorax. This is done from the surrounding markers and then using claviclula to find the centre of the shoulder joint.
Offset The offset of the upper body joints is measured and calculated
in the same way as for knee and ankle.
Centres The centre of the shoulder is determined from the chord function by using thorax origin, shoulder marker and a defined ’wand’ shoulder
as points. Furthermore the elbow is found in the same way but by using
the wrist markers a new vector is constructed and creates the plane together
with the shoulder joint centre and the elbow marker.
The wrist is just defined from the elbow and the wrist joint markers, without
using the chord function.
2.2.2
Angles
The angles calculated by PIG are cardan angles derived from the orientations of the segments attached to the joints. The angles are either relative, a
13
Figure 5: Relationship between hip joint centre, knee joint centre, knee marker
and thigh marker. Parametres for the chord function used for knee joint centre
determination. Re-drawn from[16].
comparison between the segments, or absolute, the position of one segment
according to an axis. The description of the angles can be in two, mathematically equivalent, different sets, ordered and goniometric.
This way of defining joint angles works well unless the rotations approaches
90 degrees. This type of rotations may cause perfectly aligned axes. Then
it is impossible to distinguish the next rotation around the other axes and
there by no solution of the angle calculation is obtained.
Another issue occurs when a set of rotation can correspond to another set
of rotations with the same result. This problem can cause switches between
the solutions and thereby discontinuities in the result.
These problems particularly occur in the shoulder.
14
2.3
2.3.1
Calculations
Approximations
Segments The segments are handled as stiff rods between each proximal
and distal point. This approximation may cause small errors according to the
trunk, pelvis and head. The possible bending motions within these segments
makes the stiff line assumption inaccurate.
The endpoints of these segments are approximated from the trunk and pelvis
markers according to the PIG model (See next section).
Angles The method assumes a 2D motion in the sagittal plane. The abduction and rotation angles is therefore neglected and the model uses only
the flexion/extension angles for rotational energy calculations.
Foot The endpoint for the foot should be the heel and the toe, according to
Dempster, taken from the positions of the markers. In this study the ankle
joint centre is used instead of the heel. Since the angular velocity of the foot
segment is the time derivative of the ankle angle these two points have to
be in the same segment, and this assumption is necessary for the rotational
energy. This will cause a small error in the position of the COM of the foot,
but since the foot is 1.45% of the body weight this will not cause a significant
error in these results.
2.3.2
Centre of Mass
The Centre of Mass is calculated for each segment according to the anthropometric data. These results are then used for the energy calculations of
each segment. The COM, at time frame i, of the whole body is the weighted
sum of all COMs of the body segments according to the formula below.
P15
COMi =
15
X
mk rk,i
⇔
fk rk,i
M
k=1
k=1
(2)
Where rk is an three dimensional vector to the centre of mass point of segment
k, mk is the segment weight, M is the total weight of the body and fk is the
percent fractional part of body segment k .
2.3.3
Energy
The energy calculations are divided into the three different types of energy;
potential, translational and rotational. These are calculated for each segment
in each time frame and then summarized for the whole body.
15
Potential Energy The potential energy at time frame i for the whole body
is shown in the formula below.
P Ei =
19
X
mk ghk,i
(3)
k=1
Where mk is the mass of the segment k, g is the gravity constant and h is the
z coordinate of the centre of mass point of the segment k at time i, according
to the ground.
Translation Energy The translational energy is calculated by a modification of the usual translation energy formula given by;
T Ei =
19
X
1
k=1
2
2
mk vk,i
(4)
TE is the translational energy in one direction at time frame i, mk is the
mass of segment k and vk,i is the velocity of segment k at time frame i in
the specific direction. This energy is calculated for the body, skis and poles
in all three directions.
Since the subjects ski on a treadmill, it seems to be in almost the same
position according to the global coordinate system during the measurement.
This means that the skiers move in the same velocity as the treadmill and
this must be taken into account in the calculation of the translation energy
in the Y and Z direction. The incline of the treadmill of 1 degree determines
the small amount in the Z direction.
T Eiy =
19
X
1
k=1
T Eiz =
2
19
X
1
k=1
2
mk (vk,i,y + V el ∗ cos(Θ))2
(5)
mk (vk,i,z + V el ∗ sin(Θ))2
(6)
V el is the velocity of the treadmill and Θ is the incline.
The velocity vector for each segment is approximated as the difference
between the points in the contiguous time frames for each time i divided by
the difference in time according to:
vk,i =
rk,i+1 − rk,i−1
2∆t
16
(7)
v is defined in all three directions but since this study is limited to the
sagittal plane it is only the velocity in y and z direction that is used.
The sampling frequency is 250 Hz and the movement is not that fast, this
approximation gives a good estimation of the velocity.
Rotational Energy The rotational energy at time frame i is calculated
for the body segments and the poles according to the formula:
REi =
17
X
1
k=1
2
2
IkC ωk,i
(8)
Here IkC is the moment of inertia of segment k around its center of mass
and ω is the angular velocity, calculated by the differences in angles.
ωk,i =
θk,i+1 − θk,i−1
2∆t
(9)
Skis and Poles The COM of the poles and skis are given by the PIG
model and used in this model too. The potential and translational energies
are calculated in the same way as the other segments with a weight of 0.3kg
for each pole and 1.2kg for each ski.
The rotational energy for the skis is not calculated. The endpoints and angles
of these are not known, and since they have contact with the treadmill during
the whole motion there will be no rotational energy produced.
The rotational energy of the poles is determined, and therefore the moment
of inertia is needed. This parameter is calculated from the formula of a stiff
rod where the center of gyration is at the midpoint.
Im id =
mL2
12
(10)
The poles used in the study is not homogeneous and the COM is not
at the midpoint. The moment of inertia around the COM is determined by
Steiner’s theorem.
IC = Im id + md2
(11)
Where m is the mass and d is the distance between the midpoint and COM,
whose position is given in the input data.
This is the parameter used for the rotational energy calculation for the poles.
17
2.3.4
MATLAB
The program that is used for the calculations is M AT LAB(2011a). The
imported data are text files with sampling information in the top of the
document and then columns with data for each time frame. The columns
represent the X,Y,Z coordinates for each joint center and chosen markers.
The first text rows are deleted and the data from each marker/joint center
is stored in matrices of three columns, for the different coordinates, named
by the column header. The MATLAB commands are easy to use to operate
on these large types of data matrices.
Parameters
Length The segments are approximated to stiff rods between their endpoints and the length of each segment is calculated by normalizing the vector
between these points. The endpoints are given from the PIG model and by
using the norm()-function in MATLAB these numbers are calculated.
Mass The mass of each segment is given by the product of the fraction,
given by the anthropometric data, and the mass of the subject.
Endpoints The endpoints are in most cases given by the joint centres
or markers, but for pelvis and thorax, L5 and midhip is needed and thereby
calculated.
The midhip is the midpoint between the two hip joint centres and thereby
defined as the average value of these two coordinates.
The L5 value is calculated by the formula used in the PIG model. PIG
defines the coordinate system of pelvis with origin at the mid point between
the ASIS markers. The X-axis points forward in the direction from the mean
of the PSIS markers to the mean of the ASIS markers. The Y-axis points
from RASI to LASI, and the Z-axis is pointing upwards perpendicular to the
X and Y axes.
L5 is defined by the following equation:
˙ − LJC)
L5 = midhip + [0, 0, 0.828]norm(HJC
(12)
Where the Z-term added corresponds to the direction of the local coordinate
system of pelvis.
Functions The following functions are written separately for the different
types of calculations.
18
COM Calculates the COM for each segment separately. The input is
the endpoints and the anthropometric length parameter, and the output is
the 3D positions of the COM of the segment in each time frame.
Translational The translational energy function is called for each segment and calculated in all three directions. The input is the positions of the
COM, the speed vector, the incline, the mass parameter and the weight of
the subject. The output is a three dimensional matrix with the translational
energy in the different directions in each time frame. Since the velocity is
calculated from the differences of the contiguous positions the output matrix
will be 2 elements shorter than the input.
Rotational The input to this function is the endpoints of the segment,
the radius of gyration, the mass parameter, the weight of the subject and
the sagittal plane angle of the proximal joint of the segment. The output
is a column vector with the rotational energy in the sagittal plane for each
segment at each time. Since the angular velocity is used, this vector will be
2 elements shorter than the input.
The program The program is built up in a straight forward way. All data
files are stored in the same folder and imported to the program. The program
loops through the files and calculates the energies for one subject at the time.
The program stores the input data in proper named matrices. Then the
COM of each segment is calculated by numbers and position data from the
anthropometric tables and the endpoints. The total COM of the body is then
calculated. The energies are calculated from their separate function and the
data is stored both separately and summarized in different cell arrays.
Since the results are stored in matrices the results are plotted after all files
are calculated. It is possible to choose segment or type of energy to compare
in the different plots.
19
3
3.1
Results
Centre of mass
The COM of each segment that is determined by this method is an important part of the results since these are used for the energy calculations. The
figure below illustrates a subject at an example instance with the segments
represented by blue lines and the COM of each segment is represented by
dots in different sizes.
Figure 6: Centre of mass illustration in 3D
20
3.2
Energy
The results are examples that illustrate possible comparisons between subjects with just a caption underneath. More detailed comments will follow in
the discussion section.
Figure 7: Total mechanical energy
21
Figure 8: Total mechanical energy per kg
22
Figure 9: Potential Energy
Figure 10: Rotational Energy
23
Figure 11: Translational energy Y direction
Figure 12: Translational Energy X direction
24
Figure 13: Total mechanical energy right arm
Figure 14: Angular Velocity Right Elbow
25
4
4.1
Discussion
Centre of mass
In figure 4 there is a picture of the COM of the body where the purple star
correspond to the result from my method. The different sizes of the dots is
proportional to the fraction of body weight for each segment. The purple
diamond is the COM parameter from the PIG model, which is calculated in
the same way by adding the segments positions and masses[11]. The lines
in the picture are drawn between the endpoints from raw data and the dots
are plotted separately from the calculations in this model. The placements
of these dots according to the lines correspond to the expected positions for
the COM of the segments.
The results for the two total COM’s shows a remarkable difference in this
picture. The difference varies through the cycles but is mostly in this magnitude when the upper body is bended.
Further results in this study is based on the result calculated by this method.
4.2
Energy
The data plotted in the energy graphs is not filtered. The pattern in these
figures are quite noisy except for the potential energy. These spikes occurs
due to the calculation of the velocities in the different directions. The rounding of the differences between the positions will cause too high or too low
velocities and the squaring of these numbers enlarges the errors and results
in this spiky pattern.
4.2.1
Total Energy
Figure 5 shows the total energy for three chosen subjects. The total energy
includes the potential, rotational and translational in the Y and Z directions.
These are plotted after the running of the program which makes it possible
to analyse and compare specific subject to each other.
To show the difference in results and the affection of the weight of the subject,
figure 6 shows the same subjects where the energy is divided with the mass
of each subjects.
4.2.2
Potential and Rotational Energy
The different energies can be analysed separately, which is shown in figure
7-10. Figure 7 illustrates the potential energy of the whole body. The difference between the curves is mainly affected by the difference in height of the
26
subjects.
These oscillations correspond mostly to the bending of the trunk in the push
off phase of the cycle. The trunk is almost 40% of the body weight which
means that its movements of the energy level becomes quite obvious. The
increased velocity during the measurements results in shorter cycles which
causes the difference in frequency between the beginning and end of the
curves.
The rotational energy is shown in figure 8. This is only available in the
sagittal plane since the moment of inertia is only determined around the Xaxis in this case. The amount of this is less than 0.1% of the total energy of
the body according to this method.
4.3
Translational Energy
Figure 9 shows the translational energy in the forward direction. The step
pattern is caused by the increase in velocity of the treadmill every 10th second.
The maximum value of the translational energy cycle is almost at the time
of the minimum value in the potential energy oscillations. This is explained
by the energy generated at the end of the push off phase when the trunk is
most bended.
Figure 10 illustrates the translational energy in the x-direction. This is useful data to see how much energy is lost sideways, especially in the skating
technique.
As one can see, in this case, these numbers are negligible compared to the energy produced in the forward direction. In double poling this small amount
of energy correspond to the small abduction of the arms and poles when
pulling the poles forward for push off.
4.3.1
Segment analysis
The method also generates energy results for the different segments. Figure
11 exemplifies this for the right arm, where the energies of humerus and radius are added. This picture shows the total energy for these two segments,
but it is also possible to analyse the different energies separately for the segments. This can be used to adjust small energy consuming movements to
improve the technique.
For further analyse of the different segments one can also compare the an27
gular velocity. Figure 12 shows that data for the elbow joint, which is the
angular velocity for radius.
The angular velocity in the elbow, the hip and the knee joints are necessary
parameters in speed adaptation. High velocities in double poling is reached
by higher poling frequency and longer cycles achieved by increased pole force,
assisted by higher angular velocities[9].
5
Conclusions
This method makes it possible to compare the mechanical energy produced
by different subjects. By generating the results of the energies separately
and also by segment makes it possible to analyse even the small differences
between the skiers that can improve their technique.
The program can generate translational energy in three directions, even
though only two of them is used in the examples of results of this study.
This makes it possible to do analyses in three dimensions for the translational energy which is necessary for the techniques with more lateral movements. The rotational energy three dimensions is dependent of the moment
of inertia around all three axes. This data is not available in any literature
today. The amount of this energy will probably not affect the total energy
results remarkably according to the low angular velocities of the segments,
and therefore this method, with the translational energy, is sufficient for 3D
analysis.
5.1
Future Perspectives
This model for calculations assumes some approximations which might affect
the results. Below there are listed suggestions of improvements o increase
the accuracy and the utility of the program.
Endpoints The ambiguities around the endpoints of the segments of the
upper body and how to define these from the markers can be more precise.
The PIG model is just one way of doing this.
Parameters The parameters used as the anthropometric data are an average of eight men, which are not specialized for the athletic built bodies.
One way of improving this might be to measure the width of the segments
and approximate these as solid cylinders which will give more accurate parameters, unique for each subject. To do more accurate 3D analyses it is
28
also necessary to determine the moment of inertia around all axes to get the
rotational energy in all directions too.
Filtered Data If the noisy data of the energy graphs would be low pass
filtered, the curves would show a smoother pattern. Then it is possible to
do further calculations according to work and power. This can be useful
to compare to the metabolic cost and relate to the results of the Norman
study[6].
29
References
[1] Abrahamsson Joakim, Klassisk Teknik. SISU Idrottsböcker, 2005.
[2] Robertson et al. Research methods in biomechanics Human Kinetics, 2004
[3] Vicon. Sports performance. http://www.vicon.com/applications/sports.html.
Accessed July 20, 2012.
[4] Vicon motion system. Vicon Nexus Product Guide-Foundation Notes.
2008.
[5] Holmberg, Hans-Christer. Physiology of cross country skiing.
http://www.gih.se/FORSKNING/Forskningsgrupper/Fysiologi/Avhandlingar/HC-Holmbergs-disputation/. October 2011. Accessed 28 July, 2012.
[6] Norman, Robert. W; Komi, Paavo. V. Mechanical Energetics of world
class Cross-Country Skiing. International journal of sport biomechanics.
1987, 3, pp 353-369.
[7] Smith, Gerald A; Nilsson, Johnny; Kvamme, Bent; Ure, Jarle; Ingjer,
Frank. Arm, Trunk and Leg Contributions to Double Poling Performance
in Cross-Country Skiing. Medicine & Science in Sports & Exercise. Volume 36(5) Supplement, May 2004, p S14.
[8] Holmberg, Hans-Christer; Lindinger, Stefan; Stöggl, Thomas; Eitzlmar,
Erich; Muller, Erich. Biomechanical Analysis of Double Poling in Elite
Cross-Country Skiers. Medicine & Science in Sports & Exercise. Volume
37(5), May 2005, pp 807-818
[9] Lindinger, Stefan Josef; Stöggl, Thomas;Muller, Erich; Holmberg, HansChrister. Control of Speed during the Double Poling Technique Performed
by Elite Cross-Country Skiers. Medicine & Science in Sports & Exercise.
Volume 41(1), January 2009, pp 210-220
[10] Smith, Gerald A.;
Holmberg, Hans-Christer. Nordic skiing biomechanics and physiology. July 2010. http://w4.ub.unikonstanz.de/cpa/article/viewFile/4380/4071. Accessed 28 July, 2012
[11] Plug-In Gait. http://www.irc-web.co.jp/vicon web/news bn/PIGManualver1.pdf.
Accessed July 10, 2012.
[12] Nationellt Vintersportcentrum. http://www.ostersund.se/vinterstaden/nationelltvintersportc
2012. Accessed August 5, 2012.
30
[13] Barefoot
Fitness.
Movement
http://www.vicon.com/applications/sports.html.
July 2012.
[14] University
of
Maryland
http://www.umm.edu/imagepages/1115.htm.
20, 2012.
not
March
muscle.
2012. 10
medical
center.
2011.Accessed
July
[15] Sawers, Anders; Hahn, Michael.E. Regulation of whole-body frontal
plane balance varies within a step during unperturbed walking.
http://ars.els-cdn.com/content/image/1-s2.0-S0966636212000793gr1.jpg. 2012. Accessed July 20, 2012.
[16] Plug-In Gait. http://www.irc-web.co.jp/vicon web/news bn/PIGManualver1.pdf.
Accessed July 10, 2012.
[17] Vicon. Getting started with Vicon Nexus, Tutorial 2. 2006
31