22 KINESIOLOGIA SLOVENICA 4 (1998) 1 : 22-26 
Otmar Kugovnik* 
Bojan Nemec** 
Tomai Pog_ačar 
Milan Coh* 
MEASUREMENT OF TWECTORIES AND 
GROUND REACTION FORCES IN ALPINE 
SKIING 
MERJENJE TRAJEKTORIJ IN REAKCIJSKIH 
Sil PODLAGE PRI ALPSKEM SMUČANJU 
Abstract 
The paper describes measuring system and proce-
dure for measuring of ground reaction forces and 
spatial trajectories of skis in alpine skiing. Duringthe 
measu rement we cap tu red the position and orienta-
tion of the skis and corresponding ground reaction 
forces and force application point. Namely, the main 
purpose of the measurement was to obtain minimal 
set of necessary data for building and verification of 
the mathematical model of the skiing. In our re-
search we concentrate on influence of the ski geo-
metry on skiing, therefore, we measured ground 
reaction force instead of bui lding an inverse dyna-
mic model. 
Keywords: measurement, skiing, mathematical mo-
del 
*University of Ljubljana - Faculty of Sport, Gortanova 22, Sl-1000 
Ljubljana, Slovenia 
phone: ++386 61140-10-77 
fax: ++38661448-148 
e-mail: Otmar.Kugovnik@uni-lj.si 
**Jožef Stefan Institute, Jamova 39, S1-1000 Ljubljana, Slovenia 
Phone: ++38661177-35-65 
Fax: ++38661219-385 
e-mail: Bojan.Nernec@i js.si 
I zvleček 
Prispevek opisuje merilni sistem in merilne postop-
ke, ki smo jih uporabili pri merjenju trajektorij giba-
nja smuči in merjenju reakcijskih sil podlage pri alp-
skem smuča nju. Meri tev trajektorij je obsegala zaje-
manje lege in orientacije smuči v prostoru, pri mer-
jenju sil pa smo zajemali celotno reakcijsko silo pod-
lage ter prijemališče sil na smuč i. Glavni namen me-
ritev je pridobi tev vhodnih podatkov za sintezo in 
verifikacijo takega matematičnega modela smuča­
nja, ki opisuje predvsem vpliv geometri je smuči na 
izvedbo stori tev. 
Ključne besede: merjenje, smučanje, matematični 
model 
Otmar Kugovnik, Bojan Nemec, Tomaž Pogačar, Milan Čoh 
MEASUREMENT OF TRAJECTORIES ANO GROUND REACTION FORCES IN ALPINE SKIING 23 
INTRODUCTION 
Mathematical modell ing is essential also in design 
and production of ski ing equipment. Moreover, an 
appropriate mathematical model can help in analy-
ses and better understand ing of the skiing techni-
ques and for proposing new, more efficient techni-
ques [4, 1,2]. W hile build ing the mathematical mo-
del, the init ial presumptions and defin ition are of 
great importance. We have to define exactly what 
we expect of the model and, based on this observa-
tion, bu ild the minimal set of elements, which des-
cribe the model. Namely, it is well known that too 
complex models can be useless. O n the other hand, 
the model has to be sophisticated enough to descri-
be the behaviour, wh ich is under the investigation. In 
our research, we concentrate mainly on the influen-
ce of the shape of the ski, ski boots andski bindings 
on alpine skiing. Therefore, we included the model 
of the ski, ski bindings andski boots, while the skier 
is modelled on ly asa mass point w ith known mo-
ment of inertia acting asa force vector on the skis. 
Therefore, the measured data includes a minimal set 
of data necessary to veri fy the model. 
METHODS 
M easurement of kinematic parameters 
Measurement of kinematic parameters incl udes 
capturing of positions, velocit ies and accelerations 
of selected points on an object. These points are cal-
led markers. In past, various optica l measuring sys-
tems were developed for kinematic measurement. 
They can be classified into 
• Systems with automatic recognition of the mar-
kers, known as systems with automatic digitalisa-
t ion 
• Systems that requi re manual digital isation of the 
markers on each image frame of the recorded mo-
tion. 
Systems with automatic d igital isation can be further 
classified into systems w ith active and passive mar-
kers. For measurements in sports only systems with 
passive markers are suitable, because active sensors 
require wi ring. For automatic digitalisation cameras 
with infrared orwide range (visible) spectrum can be 
used [S]. Although systems based on infrared-came-
ras are usually more efficient and precise, they are 
not suitable for outdoor measurements. Optical 
measurementsystems can be used for measurement 
system dynamics through inverse dynamic model-
ling. However, inverse dynamic modelling require 
exact measurement of body accelerations. Accele-
rations are obta ined by a second derivative of the 
measured position t rajectories. However, second 
derivative amplifies measuri ng noise and results are 
appl icable on very limited cases in spite of very sop-
histicated fi ltering in data acquisition. 
In our measurements we used ARIEL and CON-
SPORT measuringsystems. The position of the mar-
kers was calculated based on the video-image of two 
calibrated cameras. More cameras can be applied in 
order to enlarge the measurement space, w hich is 
necessary in order to study sports like alpine skiing. 
The cameras captu re 50 images per second. 
Synchron isation between cameras is accomplished 
using optical signal, which can cause d istortion due 
to the imprecise synchron isation of the cameras. 
Measuri ngsystem ARIEL includes module for auto-
matic digitalisation, but of l imited functionality; the-
refore ali trajectories were obtained w ith manual di-
gitalisation. 
M easurement of ground reaction forces 
Cround reaction force measuringsystem consists of 
two subsystems - subsystem for outdoor measure-
mentand data analysis system . Ground reaction for-
ces are measured using our own developed strain 
gauge based block sensors, inserted into ski-boot 
sole. Our system d iffers from the similar systems for 
ground reaction force measurement, because itdoes 
not requ ire virtually any change inski equipment [3]. 
The only modification is a mi nor one inski boot sole 
protectors, w hich does not change the ski boots 
fu nctional ity. 
The system incorporates also a camcorder. The vi-
deo-image is synchronised with the force measure-
ment using radio-data modem or photographic 
flash, which is activated at the start of the measure-
ment. Measured forces are saved on data-logger 
with a sampling rate up to 200 Hz. System for 
ground reaction force data analysis then calculates 
force magnitude and force vector for each leg and 
d isplays it in synchronisation w ith the digitised vi-
deo-image. The function block d iagram of the 
ground reaction measuring system is presented in 
Fig 1. 
Synchronisation of measurement systems 
As previously mentioned, we applied two indepen-
dent systems, one for kinematic measurement and 
one for ground reaction force measurement. As both 
systems used their own data storage, exact synchro-
nisation between two systems had to be obtained. 
As present, video cameras of ARI EL and CONSPORT 
measuring system can be synchronised only opti-
cally, using LED array, wh ich has to be visible to all 
24 
Otmar Kugovnik, Bojan Nemec, Tomaž Pogačar, Milan Čoh 
MEASUREMENT OF TRAJECTORIESAND GROUND REACTION FORCES IN ALPINE SKIING 
i))))) 
~ eameorder 
((((i ''~ ..,.l,on~alioo 
• micf'o-controtter 
eX11emal synchronlZ81ion 
MEASUREMENT SVSTEM 
OATAANALYSES SYSTEM 
video monitor 
carncorder 
Figure 1. Ground reaction force measuring system 
the cameras. Ground reaction measuringsystem has 
two modes of synchronisation: first, by sending ra-
dio signal from synchronisation micro-controller to 
the data logger and second, when data logger lights 
a flash at the initiation of the measurement. Flash as 
well as the data logger are carried by the skier and 
are therefore motion during measurement. Thus, 
optical synchronisation of both systems is not feasib-
le, because both LED array and flash would have to 
be visible to ali cameras. Therefore, we decided to 
use another way. The synchronisation was accom-
plished using photocell, which triggers both LED ar-
ray and synchronisation micro-controller, which is 
illustrated in Fig. 2. 
/ 
video-<ecorder 7 1)))) 
.,~. ; \ i E_l-'.'."'" 
v<leo-ca,.,ras / \ 
1 
~ . '•, 
'"-{ 
synchromzatK>n 
!Jasi! 
~ 
Figure 2. Synchronisation scheme of kinematic measuring system 
and ground reaction measuring system 
M easurement procedure and data calcula-
tion 
For our measurements we have chosen a fiat terrain 
with constant slope along the measurement area. 
The me asu rement area was large enough to cap ture 
one parallel ski turn. As is well known, the measure-
ment accuracy is inversely proportional to the mea-
surement area. 
In order to determine t he position and orientation 
of a rigid object like skis, at least three non collinear 
markers are required. In order to increase the accu-
racy of the measurement and overcome the prob-
lem of hiding markers, more markers are used forthe 
si ngle body. On the other hand, t he distance bet-
ween markers affects measurement accuracy. Best 
results are obtained if vectors connecting markers 
are orthogonal and the distance between markers is 
as big as possible. 
Because skis are narrow and direction of the skis of-
ten coincides with the camera viewpoint, it is favou-
rable to place third marker on the support, as shown 
in Fig. 3. 
Figure 3. Marker pfacement and notation 
Kinematic measurementsystem returns the position 
of each marker asa vector m = [x y z] with regard to 
the base co-ordinate system, defined at camera ca-
libration phase. Any point of interest on the rigid 
body can be calculated using the following formu-
las. 
b=[
11
::=::
11
] č = iixb (1) 
[
a
1 
b
1 
C
1
] [dl T = a, b, c, p = T O 
a, b, c, O 
(2) 
Here ,it denotes marker position and Tis transfor-
mation matrix between local co-ordinate of the 
body (skis) and basic co-ordinate system. Orien-
tation of the body with respect to the base co-ordi-
nate system axesx y z is described with set of angles 
a f3 y. Orientation o f the body can be calculated 
from transformation matrix T usingfollowingformu-
las 
a = - ArcTa11(~) 
a1 
p = ArcSin(b,) 
b 
y = - ArcTan (-1.) 
b, . 
(3) 
• 
Otmar Kugovnik, Bojan Nemec, Tomaž Pogačar, Milan Čoh 
MEASUREMENT OF TRAJECTORIESANO GROUND REACTION FORCES IN ALPINE SKIING 25 
At the ski tu rn analyses, the skidd ing angle and 
edging angle are of interest. We define skidding 
angle </J as an angle between tangent on the ski turn 
trajectory and vector aligned the skis, as shown in Fig 
4. 
Figure 4. Skidding angle and edging angle 
First, we have to define the surface, where the mo-
tion trajectory lies. In general this surface is not fiat. 
For our analyses we suppose that this surface can be 
decomposed into a number offlatsurfaces, connec-
ting two adjacent marker points, represented by vec-
tor m2 and correspondingvectoras,ii 1 as long as the-
se points are not co-linear. Thus, one tirne segment 
of the trajectory is described with the vectors mlk) 
mlk-t) and the fiat surface is defined with this seg-
ment and the vector m 1(k), where k de notes k-th 
tirne instance. Next, we have to define zero skidding 
angle, which is not, as could be expected, a tangent 
to the ski trajectory. More appropriate definition of 
zero skidding angle can be obtained by defining a 
vector of length L, that connects current marker po-
sition mlk) with previous marker position mlk-t). 
Here, L denotes the distance between markers. It is 
notstraightforward to determine the vector mi(k-t). 
In general this vector can be calculated as crossing 
of the sphere with radiusL and centre in mlk) with 
a trajectory, formed by vectors m2 . More practical 
solution is obtained with the numerical procedure, 
where we start at the pointn1lk) and search back 
the vector ,ii1 (k-t), until the distance between 1nlk-t) and mi(k) is close enough to L. 
Based on this definition skidd ingangle can be calcu-
lated using following equations 
cos(<j,)= arJJ~IJ a=m,(k)-m,(k-1) E = m,(k)-m,(k) (4) IJaJlb , , . 
Please note that for calculation of the skidding angle 
only two markers rn1 and ,ii2 are necessary. The 
equation (4) always returns positive angle. Negative 
skidding angles are not feasible, but can be obtained 
due to the inaccurate digitalisation. 
Similarly, edging angle š is defined as the angle bet-
ween normal vector to the_plane defined by 1nJ(k) 
mlk) mlk-t) and vector W =m2(k) - mJ(k). Thus, 
edging angle can be calculated using equations 
-r-
cos(i;) = ll~lll;II , ii = a x b . (5) 
RESULTS 
The proposed measu rement procedure was used to 
measure a series of parallel ski turns and side skid-
ding. Here, the results of the parallel ski turn are gi-
ven. Ski turn was performed by an experienced ski 
instructor using skis with emphasised side cut (car-
ving skis). Side cut radius of the skis was 12 m. The 
snow temperature was -2° C and the air temperatu-
re was + 3°C. The captured video images are shown 
in Fig 5. Fig 6 shows sequences of ski positions du-
ringthe ski turn. The correspondingskiddingangle is 
shown in Fig 7. The ground reaction forces and for-
ce application point are presented in Fig 8. 
Fig 5. lmages of the measured para/le/ ski turn 
 
26 Otmar Kugovnik, Bojan Nemec, Tomaž Pogačar, Milan Čoh MEASUREMENT OFTRAJECTORIESAND GROUND REACTION FORCES IN ALPINE SKIING 
CONCLUSION 
In the paperwe presented a new measurement met-
hod for evaluation of the ski turn. We measured tra-
jectory and orientation of the skis and ground reac-
tion forces and the correspond ing force application 
point. In orderto accomplish this measurement, we 
synchronised two measurementsystems. We deve-
loped a mathematica l procedure for calculation of 
the skidding angle and edging angle. The presented 
measurementprocedure wi ll serve to obtain the da-
ta for verification of the mathematical model of the 
skiing, which is under the investigation. 
Parallel sk.i tum 
-10 
Figure 6. Measured ski turn (top view). Plotted trajectories con-
nect marker position. Skis position is out/ined for each third mea-
sured position. 
20 
l " 
f,o E;B . 
00 0 .5 o.• 0.7 
Fig 7. Ca/cu/ated skidding angle 
RESUL T ANT FORCES • SORS 1.MES 
r:l : j 
-O.S O 0.5 1 1.5 2 
tirne (sec) 
i~tJ =::~ .. ~ 1 
og 5 O 0.5 1 1.5 2 
Ume (sec) 
REFERENCES 
1. Hirano Y. Numerical simulation of a turning alpine ski during 
recreational skiing. lnt. J. Medicine and Science in Sports and 
exercise 1996; 28(9): 1209-121 3 
2. Hirano Y., Tada N. Mechanics of a turn ing snow ski, lnt. J. 
Mech. Sci. 1994; 5( 36): 421-429 
3. Nemec B. A system for measuring ground reaction forces in 
alpine skiing. Coachingand Sport ScienceJournal 1997; 2(3): 
46-55 
4. Renshaw AA., M ote CD. A model for turning snow ski. lnt. J. 
Mech. Sci. 1989; 31 (1 O): 721-736 
S. Wilson OJ., Smith B.K., Gibson J.K., Choe BK., Gaba BC., 
Voeltz JT. Accuracy of digital ization using automated and ma-
nual methods. Proc. of I5B5'98. Konstanz: UKV - Univesi-
tatsverlag Konstanz, 1998: 590-592