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6
Getting in shape: reconstructing threedimensional long
track speed skating kinematics by comparing several body
pose reconstruction techniques.
E. van der Kruk, A.L. Schwab , F.C.T. van der Helm & H.E.J. Veeger (2017),
Getting in shape:
reconstructing threedimensional longtrack speed skating kinematics by comparing several
body pose reconstruction techniques.
Accepted with revisions at Journal of Biomechanics
The definition of power became clear in the previous chapter (CH 5); now we can move on
to the mechanical power determination in speed skating. Two proceedings are inevitable in
determining the mechanical power in speed skating: inverse kinematics and inverse dynamics.
The methods for each of them have been widely explored in other human movement studies,
e.g. gait studies, but now need to be addressed for the speed skating motion. This chapter
discusses the inverse kinematics handlings. The aim of this chapter is to determine to what
extend the choice for a body pose reconstruction technique influences the estimation of joint
power. We compare four global optimization methods in terms of marker residual reduction
and model fidelity.
‘Besides, it is a disgrace to grow old through sheer
carelessness before seeing what manner of man you may
become by developing your bodily strength and beauty
to their highest limit. But you cannot see that, if you are
careless; for it will not come of its own accord
.
’ Socrates
PART III CHAPTER 6
Abstract
In gait studies body pose reconstruction (BPR) techniques have been widely explored, but
no previous protocols have been developed for speed skating, while the peculiarities of the
skating posture and technique do not automatically allow for the transfer of the results of
those explorations to kinematic skating data. The aim of this paper is to determine the best
procedure for body pose reconstruction and inverse dynamics of speed skating, and to what
extend this choice in BPR influences the estimation of joint power. The results show that an
eight body segment model together with a global optimization method with revolute joint
in the knee and in the lumbosacral joint, while keeping the other joints spherical, would be
the most realistic model to use for the inverse kinematics in speed skating. To determine joint
power, this method should be combined with a leastsquare error method for the inverse
dynamics. Reporting on the BPR technique and the inverse dynamic method is crucial to enable
comparison between studies. Our data showed an underestimation of up to 74% in mean joint
power when no optimization procedure was applied for BPR and an underestimation of up to
31% in mean joint power when a bottomup inverse dynamics method was chosen instead of
a least square error approach. Although these results are aimed at speed skating, reporting on
the BPR procedure and the inverse dynamics method, together with setting a standard should
be common practice in all human movement research to allow comparison between studies.
1. Introduction
Speed skating is, except for cycling, the fastest way for humans to propel themselves over
flat land. Humans seem to have developed several skating techniques, each subjected to the
one constraint that, due to the construction of the skate, there can only be a pushoff lateral
to the gliding direction of the blade. What the optimal technique is, has yet to be discovered.
Kinetic data for biomechanical analysis are essential in this search.
A complicating factor in the biomechanical research of speed skating is the complexity of
performing threedimensional kinetic measurements on an ice rink. One skating stroke can
cover a distance of 18m, which results in a huge volume (18m x 4m x 2m) in terms of motion
capture. However, with the recently developed wireless instrumented klapskates (van der Kruk,
den Braver, Schwab, van der Helm, & Veeger, 2016) and the rapidly improving techniques for
3D motion capture, we managed to capture 3D kinetic data of elite speed skaters for 50m
of the straight part, which implies about three to four speed skating strokes, for this project.
For a full biomechanical analysis, recorded marker positions need to be transformed into
segment position and orientation. The general assumption is that the body segments are
rigid. The actual marker data will however never exactly describe actual rigid bodies, due to
instrumental errors and soft tissue artefacts, a wellknown phenomenon (Cappozzo, Cappello,
Croce, & Pensalfini, 1997; Cappozzo, Catani, Leardini, Benedetti, & Della Croce, 1996).
Therefore, body pose reconstruction techniques (BPR) play an important role. Stateofthe
art BPR technique is the global optimization method (GOM) (Lu & O’connor, 1999), which
searches for the optimal pose of the multibody system, such that the measured data points
and the estimated data points from the biomechanical model are minimized in a leastsquare
error sense. The biomechanical model can vary in model complexity e.g. number of segments
and joint constraints (Andersen, Benoit, Damsgaard, Ramsey, & Rasmussen, 2010; Charlton,
Tate, Smyth, & Roren, 2004; Duprey, Cheze, & Dumas, 2010; Reinbolt et al., 2005).
In gait studies these techniques have been widely explored (Ojeda, MartínezReina, & Mayo,
2016), but no previous protocols have been developed for BPR in speed skating, while the
peculiarities of the skating posture and technique do not automatically allow for the transfer
of the results of those explorations to kinematic skating data. Moreover, previous studies on
speed skating do not report on any of the methods used for the inverse kinematics or the
inverse dynamics to determine joint power (van der Kruk, van der Helm, Veeger, & Schwab,
2017). It is also unclear to what extent the choice for these methods influences the joint power
estimations.
Inverse kinematics: comparing BPR techniques
The aim of this paper is to determine the best procedure for body pose reconstruction and
inverse dynamics of speed skating, and to what extend this choice influences the estimation
of joint power. We present an eight segment rigid body model and compare two inverse
dynamics methods  bottomup and least square error, and four global optimization methods
in terms of marker residual reduction and model fidelity  such that the joint angles obtained
from the inverse kinematics meet the biomechanical restrictions of the human joints.
This paper is organized as follows; first the data collection, the body pose reconstruction
techniques and the evaluation criteria are presented in the method section. Second we present
the results on the marker residuals reduction and the model fidelity together with the effect of
the choice of a BPR technique on the joint power estimation. Finally the results are discussed
to determine the best BPR procedure for speed skating analysis.
2. Method
2.1 Experimental setup
Data for this study were drawn from a larger study on eight Dutch elite speed skaters. Here
we use the data of three strokes for one participant, since the objective of this paper is to show
the influence of the different data manipulation procedures on the inverse kinematics and
kinetics on the same set of data.
Data were collected on an indoor ice rink in Thialf Heerenveen, the Netherlands. Twenty
Qualisys cameras (300 Hz) were placed on both sides of the straight part of the rink, covering
an area of 50m (Qualisys, 2015) (Figure 6.1). Subjects were equipped with a full body passive
marker set consisting of 22 markers (Van Sint Jan, 2007) (Figure 6.2). Equipped with a LPM
motion tracking sensor, the skaters were tracked by four dome cameras to gather video
footage (30 Hz).
The subjects skated on two wireless instrumented klapskates (van der Kruk et al., 2016).
The instrumented skates each consist of two threedimensional force sensors which measure
the force in normal (
N
F
) and lateral direction (
L
F
) between the shoe and the blade (Figure
6.2C) (100Hz). Additionally the position of force application, the center of pressure (COP) was
Figure 6.1 Research set up of the Qualisys system. Twenty cameras were located along the straight part
of the rink. 50 meters of the straight part were covered by the calibrated volume. The participants were
equipped with a full body marker set consisting of 29 markers of which six markers were used only
in the static trials. The pink areas indicate the calibrated volumes. The field of view of each camera is
shown.
PART III CHAPTER 6
measured. The moment of the environment acting on the blade,
d
B
M
, was not measured
since it was expected to be small. The skates were equipped with a Maple hinge mechanism
and Maple blades (Maplez, 2017). Each skater placed his or her own shoe in the instrumented
bridge. Force data, kinematic data and the IMUs were synchronized via a digital startend
pulse (Shimmer3, 2015). The dome cameras run on a global time stamp (GMT), equal to the
timestamp of the kinematic measurement system.
The longitudinal force (ice friction) was not measured, but estimated using Coulomb’s law
of friction ice N
FF
µ
=
(De Koning, De Groot, & Van Ingen Schenau, 1992), where
µ
is the
friction coefficient and N
F
is the normal force of the skate on the ice. The air frictional forces
were estimated based on the study of van Ingen Schenau (1982) (Appendix 6.B.5).
2.2 Rigid body model
The skater is modelled as a chain of linked rigid bodies (
i
), or segments. After a first analysis
on the number of segments, we divided the skater into eight segments: the skates (s), the legs
Figure 6.2 A) Local coordinate systems (SCS) of each segment with the markers defining them; see
appendix A for further explanation on the SCS. B) Skater equipped with the markerset of 23 markers 
marker names adopted from (Van Sint Jan, 2007). Green indicates the marker on the right side, blue the
markers on the left side, and yellow are the six markers that were removed after the static trial. Markers
indicated with ACC were positioned at an IMU. C) The instrumented klapskate to measure the pushoff
forces of the skaters (normal and lateral direction). The force sensors are integrated in the bridge; data
were logged on the logger on the rear side of the skate (van der Kruk et al., 2016).
Inverse kinematics: comparing BPR techniques
(e), the thighs (t) , pelvis (p) and the HAT (h), which is the head, arms and torso (Figure 6.3). The
arm movements are thus neglected, since we assume HAT is a rigid body. A seven segment
model in which the pelvis was part of the HAT segment was tested in our first analysis, but
proved to be insufficient, since the COM of the HAT then showed unacceptable translations.
Therefore the lumbosacral joint (LJ) was added to obtain an eight segment model. The local
axes of the system are specified in appendix 6.A. The global reference frame xyz is specified,
where y is up, x is in the longitudinal direction of the straight and z is in the lateral direction of
the straight part of the rink (right, facing forward), in agreement with the ISB convention. The
Euler rotations of a segment correspond to the order Y,X and Z, which are referred to as yaw,
roll, pitch. The joint rotation, which is the rotation between two segments, is rotated in the
Euler sequence Z, X, and Y, around the SCS of the proximal segment, further referred to as the
flexionextension (Z’), internalexternal rotation (Y’’) and adductionabduction (X’’’).
2.3 Inverse dynamics and joint power
To apply inverse dynamic techniques, first the NewtonEuler equations of motion for each of
the segments need to be determined in a global reference frame. These equations of motion
are laid out in Appendix 6.B. The center of mass (COM) and the mass of the separate segments
were determined by the specifications as given in Table 4.1 of Winter (2009). The inertial
tensor specifications given in Table 2 of Dumas, Chèze, & Verriest (2007)most of the predictive
equations are ambiguously applicable in the conventional 3D segment coordinate systems
(SCSs are applied to determine the inertial matrix for each segment.
2.3.1 inverse dynamics: LSE and CS
The joint moments and forces can be determined via different inverse dynamics methods.
In this paper two commonly applied methods were used in order to determine if the impact
of the choice for a BPR procedure on the joint moments differs for different inverse dynamics
techniques.
Consecutive Solving (CS) or the bottomup technique for lower extremities (Miller &
Nelson, 1973). The equations of motion are solved from distal to proximal until the upper
joint is reached, in our case the lumbosacral joint. This method leaves residual moments and
forces at the HAT, in previous studies also referred to as
the hand of god
, which are indicative
of the accuracy of the approximation procedure.
LeastSquare Error (LSE) Since the system of equations for the speed skater model is
overdetermined (in our system we have 39 variables and 48 equations, see Appendix 6.B),
the solution can be found by solving the system of linear equations with a leastsquare
error fit (Kuo, 1998). The method minimizes the moment and forces residuals and spreads
the remaining residuals out over the seven joints. The system equations and minimization
problem are explained in Appendix 6.C.
2.3.2 Joint Power
Joint power is part of the mechanical power balance in speed skating and stands for the
mechanical power generated in the joints. A precise definition of mechanical power in speed
skating is presented in van der Kruk et al. (2017). The joint power is calculated with the
moments in the joint and the rotations around the joint, as in
( )
77
, ,1 1
11
j tot o o o o j j
ij
P++
= =
= −=
∑∑
MM
ωω ω
(6.1)
In which
o
ω
is the segment angular velocity,
j
M
are the joint moments and j
ω
are the
PART III CHAPTER 6
Figure 6.3 The skater is divided into eight segments; the skates (s), the legs (l), the thighs (t) , the pelvis
(p) and a HAT (h). The forces acting on the skater are the ground reaction forces, ice frictional force and
the air frictional forces. There are joint forces and moments acting in the Ankle (A), Knee (K), Hip (H) and
Lumbosecral (L) joints. Indicated are the Center of Mass (COM) of each segment, the Center of Pressure
of the air friction (CP), where the air frictional force acts upon, and the Center of pressure of the ground
reaction force (COP). Although indicated at different positions in the figure, in this paper it was assumed
that the CP is positioned at the COM of each segment. The NewtonEuler equations for this FBD are
presented in Appendix B.
Inverse kinematics: comparing BPR techniques
joint angular velocities. If we now write the complete equation for all joints, we obtain:
( ) ( ) ( ) ( )
,
ddd ddd dd
j tot A s e K e t H t p L p h
ddd
P= −+ −+ −+ −
∑∑∑
M M MM
ωω ωω ωω ωω
(6.2)
In which the subscripts denote the segments (
, ,, ,set ph
) and the joints (ankle (A), knee
(K), hip (H) and lumbosacral joint (L)). d denotes either left of right, so both sides (legs) are
incorporated in the joint power.
2.4 Body pose reconstruction procedures
In this paper we compare four different body pose reconstruction procedures based on
global optimization (GOM) with different joint modelling and an unoptimized technique
(UNO). The names are adopted from (Ojeda et al., 2016).
UnOptimized (UNO) The unoptimized technique defines a segment by the origin and
three orthogonal axes which are defined by single markers measured at each frame, with a
minimum of three markers per segment. UNO method constructs the local reference frame at
each point time point in the movement. The vector running from the origin to the markers is
measured in the static trial and used in every frame, without a leastsquare error estimation.
UNO does not correct for skin tissue artefacts and has no kinematic constraint, which entails
that the separate segments can detach during movement and the length of the segments
may vary. The UNO method and the local coordinate systems are expounded in appendix 6.A.
Global Optimization Method (GOM) This technique searches for the optimal pose of the
multibody system, such that the measured data points and the estimated data points from
the biomechanical model are minimized in a leastsquare error sense (Lu & O’connor, 1999).
The position of a marker at any moment in time
0,
m
i
r
can be described by
0
0, , ,
'
m
i oi g oi JC m
R
→→
=+⋅
r JC v
(6.3)
In which
o
JC
is the joint center as well as the origin of the segment
o
,
go
R→
is the rotation
matrix from the global system to the segment system and
0
'
JC m→
v
is the vector running from
the joint center to marker
m
, expressed in the segment coordinate system, measured in the
static trial. When the linked segment system is indeed completely rigid, the estimated marker
position
0,
m
i
r
should be consistent with the measured marker position
,
0,
m meas
i
r
. The difference
Procedure
Joints
Kinematic constraints
STA reduction
UNO

NO
NO
GOMs
All spherical
no translations in joints
YES
GOMt
Ankle, Lumbosacral and
Hip spherical, Knee joint
twoaxes.
 no translations in joints
 twodegree knee joint
(flexionextension,
internalexternal)
YES
GOMr
Ankle, Lumbosacral, and
Hip spherical, Knee joint
revolute
 no translations in joints
 onedegree knee joint
(flexionextension)
YES
GOMrr
Ankle and Hip spherical,
Lumbosacral and Knee
joint revolute
 no translations in joints
 onedegree knee joint
(flexionextension)
 onedegree
lumbosacral joint
(flexionextension)
YES
Table 6.1 Overview of the Body Pose Reconstruction Procedures.
PART III CHAPTER 6
between the two defines the marker residual
,
, ,,
m m m meas
oi oi oi
= −res r r
(6.4)
, ,,
m mT m
oi oi oi
=res res res
(6.5)
In which
,
m
oi
res
is the residual vector (xyz) at time
i
for marker
m
, which in total makes 22
residuals. Input to the optimization function are the measured marker positions at each point
in time and
0
'
JC m
v
→
measured in the static condition. The system is a nonlinear multivariable
function and was solved in a nonlinear optimization using the function
fmincon
, in Matlab.
The model uses the UNO technique to find a start position of the model. Output of the
optimization are the rotation matrices for the skates, legs, thighs, pelvis and torso and the
positions of AJC, KJC and HJC and LJC.
GOM reduces the residuals and adds kinematic constraints, restricting any translation within
joints (between segments) and guarantees a constant segment length. Within this technique
there are several ways to model the joint constraints. In this study we applied four (Table 6.1):
1. a model with only spherical joints (GOMs) (27 DOF);
2. a model with a twoaxes knee joint allowing for only flexionextension and internal
external rotation (GOMt); The lumbosacral, hip and ankle joint are spherical (25 DOF).
3. a model with a revolute (oneaxis) knee joint, only allowing for flexionextension (GOMr);
The lumbosacral, hip and ankle joint are spherical (23 DOF).
4. a method in which in addition to the revolute joint in the knee, also the lumbosacral
joint is modelled as a hinge joint, only allowing flexionextension (GOMrr). Since the LS
joint was only added to improve the CoM translation of the HAT in the inverse model, a
revolute joint was deemed to be sufficient (21 DOF).
2.5 Criteria for evaluating the BPR procedures
To evaluate the various BPR procedures, we determined kinematic and kinetic criteria. For the
kinematic criteria there are two evaluation measures. First, marker residuals, which are often
used in literature to quantify the fit of the model on the experimental data (Lu & O’connor,
1999; Ojeda, MartínezReina, & Mayo, 2014). The residuals depend on STA and instrumental
errors as well as the global optimization fit. To evaluate the residuals for the procedures, the
sum of the marker residuals over time for each marker
m
RES
is determined
,
1
1
T
mm
oi
i
RES T
=
=
∑
res
(6.6)
In which
T
is the total time of the three consecutive strokes. Additionally, the average total
marker residual of all markers (
22
m
N=
) was determined by:
22
,
11
11
T
tot m
oi
im
m
RES NT
= =
=∑∑ res
(6.7)
For the second kinematic evaluation criteria, the procedures were evaluated on the obtained
joint kinematics which are the joint angles. The results were tested on their model fidelity.
Since there are no results on 3D speed skating kinematics in literature to compare the results
to, we can only evaluate these results based on general biomechanical knowledge.
For the kinetic evaluation criterion, the methods were evaluated on their dynamic consistency.
Inverse kinematics: comparing BPR techniques
The joint forces and moments, obtained with the two inverse dynamics techniques, were
evaluated based on the NewtonEuler residuals, i.e. the residuals left in each Newton (
,
d
Fo
e
)
and Euler (
,
d
Mo
e
) equation of motion (Appendix 6.B):
,
d d dd
Fi o o o
em= −
∑
Fa
(6.8)
( )
,
d d dd
Mi o o o
d
eI
dt
= −
∑
M
ω
(6.9)
The NewtonEuler residuals were summed over all segments and averaged over time. We
normalized to the estimated joint forces and moments (also summed over the joints and
averaged over time):
8
,,
11
7
11
100%
T
dT d
Fo Fo
Newton io
T
dT d
jj
ij
RES
= =
= =
= ⋅
∑∑
∑∑
ee
FF
(6.10)
8
,,
11
7
11
100%
T
d Td
Mo Mo
Euler io
T
dT d
jj
ij
RES
= =
= =
= ⋅
∑∑
∑∑
ee
MM
(6.11)
Note that for CS there are only NewtonEuler residuals at the HAT segment, while for
LSE there are residuals for every segment. Finally, the mean (
,j tot
P
) and peak (
,
max
j tot
P
) joint
power (for two consecutive strokes) were estimated for each combination of BPR and inverse
dynamics technique, to quantify the influence of a choice in terms of joint power estimation.
3. Results
For easier interpretation of the results and clarification on the terminology on the speed
skating for this paper, an infographic was constructed from the measured 3D kinetic data
(Figure 6.4). The caption provides a description of the phases and terms.
3.1 Marker residuals and joint gaps
The average values of the marker residuals for the three analysed strokes are shown in
Figure 6.5. Among the GOM methods we only see small differences in marker residuals,
none of the methods sticks out from these results. The largest residuals are found
in the upper body, since the rigid body assumption will hold least for the HAT segment.
UNO is the only method that allows joint gaps. The mean gaps for the Ankle, Knee, Hip and
Lumbosacral joint were 2.1 cm, 7.3 cm, 0 cm and 3.9 cm respectively for both left and right.
Joint kinematics
The joint angles obtained using the five procedures (UNO, GOMs, GOMr, GOMt, GOMrr) for
two consecutive strokes are given in Figure 6.6 . Additionally a 3D visualisation whereby all
five procedures are sketched together with the measured marker positions is given in Figure
6.7.The joint angles are clearly changed by the optimization method compared to the un
optimized method UNO, which shows unrealistic rotations (too large knee and hip extensions
(Figure 6.6)). The GOMs method, with spherical joints only, shows unrealistic large knee joint
adduction angles and hip endo rotations. The GOMt, GOMr and GOMrr methods all solve the
PART III CHAPTER 6
Figure 6.4 Overview of the speed skating motion, reconstructed from the data of one participant. A) skating
motion front view, divided into the four phases: glide phase, push off phase, repositioning phase and the double
stance, where both skates are on the ice. The pushoff angle of the leg is the angle the leg makes with the hori
zontal during the pushoff motion in the frontal plane. The arrows indicate the pushoff force in global space,
the scale is indicate in the topright corner. The grey line indicates the CoM motion of the HAT. B) Top view of
the skating motion. The red, blue and black lines indicate the trajectories of respectively the right skate, left skate
and CoM of the HAT on the ice. The steer angle is the angle the skate makes with the global xaxis while on the
ice. C) measured (local) normal and lateral push off force on the skate (see E) for the right (red) and left (blue)
skate during the right stroke. D) lean angle (roll), steer angle (yaw) and foot angle (opening of the skate, pitch)
in the global system (see E). F) Center of Pressure (COP) measured on the skate together with the upward global
force (Fy). Instances 1 and 7 have too little force on the skate, to determine the COP. F) joint flexion angles of the
angle knee and hip.
A right stroke in speed skating can be described by instances 1 to 7: 1) The skater places the right skate on the
ice, while the normal force on the left skate almost reaches its peak value. 2) The weight of the skater is evenly
distributed over the left and right skate. 3) All the weight is shifted to the right skate, the left skate is retracted
from the ice, which ends the double stance phase. 4) The skater lowers his upper body by decreasing the knee
angle. Lowering the upper body causes a dip in the normal force curve of the skate. In this phase, the gliding
phase, the lean angle transforms from negative to positive, so the skate shifts from the lateral to the medial side
of the blade. The steering angle of the skate is at maximum when the lean angle is zero. 5) The skater moves his
upper body away from his skate, thereby increasing the force on his skate. Since the lean angle is now positive
and the steering angle still has a positive angle, the skater has a force component in both the forward and the
sideways direction of the rink. 6) The skater keeps increasing his force, by stretching his knee (pushoff phase),
until the peak force. Just before the peak, the left skate reentered the ice. 7) The skater shifts his weight to the
left skate, until all weight is shifted and the skater retracts his skate from the ice. The skater then repositions his
right skate for the next stroke. During the stroke the upper body of the skater has an upanddown movement
of about 0.15 m. The distance covered in the visualized stroke was 12.6 m.
Inverse kinematics: comparing BPR techniques
unrealistic knee adduction by restricting it and thereby also improve the hip endo rotation
into more realistic values. Comparing the latter three, the GOMt method shows quite arbitrary
endoexo rotation in the knee joint, not related to the skating pattern. In the GOMr and GOMrr
, where the knee endoexo rotation is constrained, an increased ankle endoexo rotation is
visible up to about 30 degrees compared to GOMt, however these only occur when the skate
is of the ice and are therefore feasible.
3.2 NewtonEuler residuals and Joint Power
The NewtonEuler residuals, relative to the summed joint forces and moments, are given
in Table 6.2. For both inverse dynamics methods, the Newton residuals (RESNewton) are lower
than the Euler residuals (RESEuler). LSE has significant lower RESEuler compared to CS. In absolute
numbers, the maximal average residual of CS, which only results in residuals at the HAT
segment, is 160 N and 133 Nm. The maximal residual of the LSE method is 23 N and 9 Nm, at
the skate segment. Within the LSE method, the influence of the BPR procedure on RESEuler and
RESNewton is only marginal, where GOMs and GOMrr seem most dynamic consistent (Table 6.2).
Estimation of joint power is influenced by both the BPR procedure and the inverse dynamics
method, each of the combinations is presented in Figure 6.8. The figure makes clear that the
differences in joint power between methods is large.
Figure 6.5 Marker residuals of each marker for the five BPR procedures. Marker definitions are given in
figure 6.2; Note the graphs have different scaling, while we want to compare the BPR procedures rather
than the mutual markers.
PART III CHAPTER 6
Figure 6.6 Joint angles in the segment coordination system for three consecutive strokes (rightleft
right) with the five BPR techniques. The solid line is right (r), the dotted line denotes the left (l) side. The
angles are in degrees. The grey areas indicate whether the right or left skate is on the ice and the double
stance phases. The upright conditions for the extension angles are 180
0
for the knee and hip and 90
0
for
the ankle. All other rotations have a 0
0
value in the standing upright position.
Figure 6.7 3D plot of the measured markers (black dots) and the estimated joint centers (coloured cir
cles) for the five BPR procedures. A coloured line is drawn from joint to joint (the joint gaps for UNO are
therefore not shown) and a thin black line from marker to marker.
Inverse kinematics: comparing BPR techniques
4. Discussion
Both the choice in BPR procedure and inverse dynamics method have a large impact on
the estimation of joint power. The results underline the importance for setting a standard for
future studies and reporting on both procedures to allow for comparison of studies  also
when these methods are embedded in a software. This applies not just for speed skating, but
also to other studies, where motion capturing in large volumes is involved (van der Kruk &
Reijne, 2017).
For the inverse dynamics method, the bottomup approach (CS) is dynamically less consistent
than the leastsquare error approach (LSE) (Table 2). In this speed skating study, where the
accuracy of the motion capture data is low due to the large recording volume, the LSE method
has made the results more robust by incorporating full body dynamics (Table 3). Therefore LSE
is here the better choice for inverse dynamics.
An optimization procedure for BPR is essential, as the unoptimized procedure, UNO, results
in unacceptable joint rotations (Figure 6.6). The results show that a model with spherical joints
only (GOMs), results in unrealistic knee adduction and hip endo rotations and therefore is not
sufficient. GOMt, with a twodegree knee joint, shows arbitrary endoexo rotations in the joint
angles, which is expected to be merely a compensation variable caused by the minimization
problem, than an actually existing motion, since the motion is not periodically. Although
GOMr and GOMrr show an increased ankle endoexo rotation (compared to GOMt), these
rotations occur when the skates are in the air, which makes these joint angles acceptable. Both
methods seem sufficient for the inverse kinematics of a speed skater. The GOMrr however has
a slightly better dynamic consistency compared to GOMr (Table 6.2), therefore GOMrr is the
best choice in BPR procedure.
Table 6.3 presents the percentage differences between using the GOMrr for inverse kinematics
together with a LSE inverse dynamics method, and using any of the other combinations. The
table makes the necessity for reporting the applied methods clear; Compared to GOMrr, an
unoptimized method estimates the joint power lower, with a 45 to 74 % difference. This
structural underestimation is partly related to the neglected translational powers in the joint,
which only apply for UNO. Using a GOM procedure, but a CS method instead of LSE, causes
differences of up to 31% in mean joint power and 29% in peak power. As long as a GOM
procedure with a LSE method is applied, the difference stays within the 10%, again ruling to
always use a leastsquare error approach for the inverse dynamics.
Figure 6.8 Joint power obtained with the five BPR procedures with the bottomup approach (CS) (left)
and leastsquare error approach (LSE) (right) for inverse dynamics.
PART III CHAPTER 6
4.1 Soft tissue artefacts in speed skating
The residuals found in the speed skating experiment are sometimes ten times larger than
in gait studies (Ojeda et al., 2014). The main cause of this difference is the size of the volume
in which the data were captured, which influences the instrumental error of the marker data.
The calibration error in this experiment was 4.54.7 mm, which in lab environments usually
is <0.7mm (van der Kruk & Reijne, 2017). Second cause is the skating posture; while in gait
analysis the dynamic posture is similar to the static (upright) trial, in speed skating the skater
bends forward, with an increased knee and hip flexion. Due to this bending, the skating suit of
the skaters with the attached markers will significantly shift relatively to the skin (and skeletal);
this highly undermines the rigid body assumption. In future studies it is advised to use an
additional static calibration, where the participant is positioned in a speed skating posture.
4.2 Inverse Dynamics
Accurate kinematic measurements are essential for inverse dynamics. Measurement data can
benefit from sensor integration or sensor fusion. The leastsquare error approach used in this
study, can be interpreted as a way of sensor fusion: adding the equations of the HAT segment
to the total system of motion equations, forming a closed loop, enables the fusion of kinematic
data and force data to solve the joint moment and forces in a leastsquare manner. Possible
improvement of LSE as applied in the current study, is to use an inverse covariance matrix for
the weighting as done in Van Den Bogert & Su (2008). This might further improve the Euler
residuals for LSE. Also, in addition to the NewtonEuler equation, the power balance equation
could be implemented to the system, thereby further improving the dynamic consistency of
the model. Such a method is introduced and further discussed in a sequencing paper (van der
Kruk, Schwab, & van der Helm, 2017), in which we also discuss the influence of instrumental
errors and body parameters on joint power estimations.
RES
CS
LSE
Newton
Euler
Newton
Euler
UNO
7.3%
34.2%
8.0%
25.3%
GOMs
7.0%
33.7%
7.7%
22.5%
GOMt
7.2%
34.3%
7.8%
23.8%
GOMr
7.2%
34.0%
7.9%
23.2%
GOMrr
7.1%
33.6%
7.8%
22.6%
Table 6.2 Dynamic residuals of each inverse kinematics and inverse dynamics combination. The New
tonEuler residuals were summed over all segments, averaged over time and normalized to joint forces
and moments (eq.7.10 & 7.11).
j
P
CS
LSE
mean
peak
mean
peak
UNO
74%
60%
45%
57%
GOMs
29%
15%
9%
10%
GOMt
31%
29%
6%
10%
GOMr
26%
15%
3%
6%
GOMrr
3%
+9%
**
**
Table 6.3 Difference between the mean and peak joint power of the best method (GOMrr with LSE),
indicated by **, and the mean and peak joint power with any of the other combinations. Data are based
on two consecutive strokes.
Inverse kinematics: comparing BPR techniques
5. Conclusion
An eight body segment model together with a global optimization method with revolute
joint in the knee and in the lumbosacral joint would be the most realistic model to use for the
inverse kinematics in longtrack speed skating. To determine joint power this method should
be combined with a leastsquare error method for the inverse dynamics. Reporting on the
BPR optimization technique and the inverse dynamic method is crucial to enable comparison
between studies. Our data showed an underestimation of up to 74% in mean joint power when
no optimization procedure was applied for BPR and an underestimation of up to 31% when
a bottomup inverse dynamics method was chosen instead of a least square error approach.
Acknowledgements
The authors express their gratitude to Frida Bakkman, Daniel Thompson, Erik Westerström,
and Marcus Johansson of Qualisys, Wouter van der Ploeg of the KNSB, Andre Zschernig of
the company Moticon, and Frédérique Meeuwsen, Niels Lommers, and Jos Koop of the TU
Delft and the Hague university of applied sciences for their help and support during the
measurements. Also we express gratitude to Thialf for giving us the opportunity of overnight
measurements at their ice rink. This study was supported by the NWOSTW under grant 12870.
References
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Inverse kinematics: comparing BPR techniques
Appendix 6.A Segment Coordinate Systems (SCS)
For each segment, a Segment Coordinate System (SCS) was determined (Figure 6.2).
Symmetry is assumed between the left and the right leg.
A.1 Skate
The Xaxis of the skate SCS runs from the AJC (estimated as the midpoint between FAL and
TAM) to the midpoint of the FM2 and FM5 marker. The Yaxis is normal to a plane containing
the AJC, FM2 and FM5 marker. The Zaxis is the cross product between the Y and X axes. The
origin is AJC.
A.2 Leg
The Ankle joint Center (AJC) was estimated as the midpoint between FAL and TAM. The
Yaxis of the leg runs from the AJC to the KJC (midpoint between LFE and MFE). The Zaxis is
normal to a plane containing the AJC, KJC and TTC. The Xaxis is the cross product between
the Y and Z axis. The origin is AJC.
A.3 Thigh
The Knee joint Center (KJC) was estimated as the midpoint between LFE and MFE. The Yaxis
of the thigh runs from the HJC to the KJC. The Xaxis is normal to a plane containing the HJC,
LFE and MFE, pointing anteriorly. The Zaxis is the cross product between the X and Y axis.
A.4 Pelvis
The SCS of the Pelvis has its Zaxis running through the markers R_IAS and L_IAS. The Yaxis
is the vector perpendicular to the plane containing the R_IAS,L_IAS and SACR marker. The
Xaxis is the cross product of the Y and Zaxis. The origin is the midpoint MIAS between the
R_IAS and L_IAS marker. The hip joint center (HJC) is estimated according to (Reed, Manary, &
Schneider, 1999), where the vector from MIAS to HJC is estimated based on the pelvis width (
PW
):
[ ]
0.24 0.30 0.36PW PW kk PW⋅
, in which
kk = −
for the left HJC (LHJC) and
1kk =
for the right HJC (RHJC).
A.5 HAT
The SCS of the HAT has his YAxis running through the markers C7 and SACR. The Xaxis is
the vector perpendicular to the plane containing the C7, RSAE and LSAE markers. The Zaxis is
the cross product of the X and Yaxis. The origin is the C7 marker.
PART III CHAPTER 6
Appendix 6.B NewtonEuler Equations of Motion
In this appendix the NewtonEuler equations for the sevenbodyrigid model are given. The
body consists of eight segments: the skates, the legs, the thighs, the pelvis and the HAT which
is the head, trunk and arms, see Figure 6.3.
B.1 Skate ( )
The NewtonEuler equations at the COM of the skate in the three global directions are,
,,
d dd dd d dd
s B ice A G s air s s s
m=+++ + =
∑
FFFFF F a
(B.1)
Where indicates the left ( ) or right ( ) skate.
dd
ice b
+FF
are the reaction forces acting at
the COP of the blade of the skate, where
d
ice
F
works in the longitudinal direction of the blade
and
d
B
F
in the normal (
d
N
F
) and lateral direction (
d
L
F
) of the blade. We make a distinction
between the two, since we measure
d
B
F
and can only estimate
d
ice
F . The air frictional force
,
d
air s
F
has its own center of pressure (CP) on the segment, where the force acts.
,
d
Gs
F
is the
gravitational force acting in the COM of the skate segment.
d
A
F
is the force acting in the
Ankle joint center ( ). The sum of the forces should add up to the mass of the segment times
the acceleration of the COM of this segment,
dd
ss
ma
. For the rotational part we write the Euler
equation at the COM expressed in the global reference frame in the three global directions:
( )
,,,
BA
d dd dd d d dd
s B F A F s air s ice s s s
d
dt
=+ ++ + + =
∑
ω
MMM MM M M I
(B.2)
Where
B
d
F
M
is the moment caused by the force
d
B
F
and
,
d
ice s
M
is the moment due to the ice
frictional force.
,
A
d
Fs
M
is the moment implied by the ankle joint force
d
A
F
as in
,/
A
d dd
F s As A
= ×M rF
(B.3)
Where
/
d
As
r
is the vector running from the center of mass of the skate to the Ankle joint center.
d
B
M
is the external moment of the environment acting on the blade, which was assumed to
be small and therefore neglected.
,
d
air s
M
is the moment induced by the air frictional force.
This moment only exists when the CP is different from the COM. After all, a force that acts
at the COM, would, like the gravitational force and the acceleration force, not contribute to
the sum of moments around the COM. To complete, there is a moment acting in the ankle
joint
d
A
M
. The sum of moments should add up to the change in angular momentum of the
segment at the COM
( )
dd
ss
d
dt
ω
I
, where
d
s
ω
is the angular velocity and
d
s
I
its inertia tensor
of the skate expressed in a global reference frame. The global parameters
d
o
ω
and
d
o
I
were
determined via the orientation of the segment. The segment orientation is described by the
Euler sequence yaw (
γ
), roll (
α
), and pitch (
β
), with the rotation matrix of the segment o
R
:
d ddd
o ooo
γαβ
=R RRR
(B.4)
Inverse kinematics: comparing BPR techniques
The global angular velocity vector (
d
o
ω
) was then determined by:
00
00
00
d
o
d d d dd
o o o oo
d
o
γ γα
α
γ
β
=++
ω
R RR
(B.5)
In which the Euler rotational velocities were determined by differentiation of the Euler angles,
filtered with a twoway second order Butterworth filter with a 12Hz cutoff frequency. The
global inertial tensor was determined by:
d d d dT
o oo o
′
=I RI R
(B.6)
In which
d
o
I′
is the inertial tensor in the segment frame. Next, the global inertial tensor (
d
o
I
)
was multiplied by the global segment angular velocity (
d
o
ω
) , differentiated once and filtered
with a second order Butterworth filter with a 4Hz cutoff frequency to obtain
( )
dd
oo
d
dt
ω
I
.
B.2 Leg (e) and thigh (t)
The equations of motion for the leg and the thigh are derived in a similar manner to that of
the skate. Each segment introduces a force and a moment in the consecutive joint (Knee (K)
and Hip (H)). The leg has the following equations of motion:
,,
d ddd d dd
e A K G e air e e e
m=−++ + =
∑F FFF F a
(B.7)
( )
,,,
AK
d d d d d d dd
e A K F e F e air e e e
d
dt
=−+ + + =
∑
ω
M M M +M M M I
(B.8)
Where
d
A
F
is the force acting in the ankle joint, but with an opposite sign to the one acting
in the skate segment.
d
K
F
is the force acting in the knee joint. The moments
,
A
d
Fe
M
and
,
K
d
Fe
M
are induced by the forces in the respective joints.
,
A
d
Fe
M
is different from
,
A
d
Fs
M
due to
the different moment arm
/
d
Ae
r
, running from the center of mass of the segment e to the joint
center A. The moment
d
A
M
is equal but opposite to the ankle moment as appearing in B.2.
d
K
M
is the moment in the Knee joint.
,
d
air e
M
is the moment induced by the air frictional force.
( )
dd
ee
dI
dt
ω
is the change in angular momentum of the segment at the COM of the leg, where
d
e
ω
is the angular velocity and
d
e
I
its inertia tensor expressed in a global reference frame.
Moving to the next segment, the equations of motion for the thigh are given by:
,,
d ddd d dd
t K H G t air t t t
m=−++ + =
∑
F FFF F a
(B.9)
( )
,,,
KH
d d d d d d dd
t K H F t F t air t t t
d
dt
=−+ + + =
∑
ω
M M M +M M M I
(B.10)
PART III CHAPTER 6
Where
d
H
F
is the force acting in the hip joint and
d
H
M
is the moment in the Hip joint.
,
K
d
Ft
M
is again different from
,
K
d
Fe
M
due to the different moment arms
/
d
Kt
r
.
( )
dd
tt
d
dt
ω
I
is
the change in angular momentum of the segment at the COM of the thigh, where
d
t
ω
is the
angular velocity and
d
t
I
its inertia tensor expressed in a global reference frame.
B.3 Pelvis (p)
Pelvis has the most forces and moments acting on it. There are three joints at the pelvis: the
left and right HJC and the LJC. The Newton equation of motion for the pelvis is:
,,
rl
p H H L G p air p p p
m=−− ++ + =
∑
F F F FF F a
(B.11)
Here
r
H
F
and
l
H
F
are the forces acting in respectively the right and the left hip and L
F
is
the force acting at the lumbosacral joint.
,Gp
F
is the gravitational force acting in the COM of
the segment and
,air p
F
is the air frictional force acting on the pelvis. The Euler equation of
motion for the pelvis is:
( )
,, ,,
HH L
l rl r d
p H H Fp Fp L Fp airp p p
d
dt
=−− − − ++ + =
∑
ω
M MMM M MM M I
(B.12)
( )
pp
dI
dt
ω
is the change in angular momentum of the segment at the COM, where
p
ω
is
the angular velocity and
p
I
its inertia tensor expressed in a global reference frame.
,
H
d
Fp
M
is
again different from
,
H
d
Ft
M
due to the different moment arms
/
d
Hp
r
.
L
M
is the moment acting
in the lumbosacral joint.
B.4 HAT (h)
We assume the trunk, head and arms to be one rigid body segment. Therefore the lumbosacral
joint is the most superior joint. So in the equation of HAT, no additional joint force or moment
is introduced. The Newton Equations of HAT are therefore:
,,h L G h air h h h
m=−+ + =
∑F FF F a
(B.13)
,Gh
F
is the gravitational force acting in the COM of the segment and
,air h
F
is the air frictional
force acting on HAT. The Euler equation of motion of the HAT at the COM, expressed in a
global reference frame is:
( )
,,
L
d
h L F h air h h h
d
dt
= +=
∑
ω
M M M M I
(B.14)
( )
hh
dI
dt
ω
is the change in angular momentum of the segment at the COM of the HAT,
where
h
ω
is the angular velocity and
h
I
its inertia tensor expressed in a global reference
frame.
Inverse kinematics: comparing BPR techniques
B.5 Air frictional force
The air frictional force (
,air i
F
) acting at a segment
i
was estimated by first determining the
total air frictional force acting on the skater, based on the study of van Ingen Schenau (1982):
22
,1
1
2vv
air tot d xyz xyz
F AC k
ρ
= =
(B.15)
where
d
C
represents the drag coefficient,
A
the frontal projected area of the skater,
ρ
the
air density and
xyz
v
the velocity of the air with respect to the skater. Based on frontal video
analysis of the experiment, the ratio in frontal area between the segments was estimated and
used to determine the air frictional force per segment.
B.6 CS method: the bottomup approach
The solution for the CS approach, was found as following; first, the ankle force (
A
d
F) was
determined (eq. B.1) to find the moment in the ankle joint (
A
d
M
) (eq. B.2). Next, using
A
d
F
and
A
d
M, the knee force (
K
d
F) was determined (eq. B.4), in order to estimate the moment
around the knee (
K
d
M) (eq. B.5). Then, using
K
d
F and
K
d
M, the forces in the hip joints (
H
d
F
) are determined (eq. B.6), to then estimate the moments in the hip joints (
d
H
M
) (eq. B.7).
Finally, with
H
d
F
and
d
H
M
, the force in the lumbosacral joint (
L
F
) is determined (eq. B.8), to
estimate the moment in the lumbosacral joint (
L
M
) (eq. B.9). Note that the equations of the
HAT segment (eq. B.10, B.11) are not used in the CS approach.
PART III CHAPTER 6
Appendix 6.C LSE method: System Equations and the minimization problem
The LSE method is a leastsquare error method, based on a linear equation. First the Newton
Euler equations (appendix B) for all segments have to be written in terms of the unknown
variables, which are the joint forces and joint moments. Therefore we have to rewrite the
moment caused by the joint force in terms of this joint force. This moment is the cross product
between the vector that runs from the COM of segment
o
to the joint center of joint
j
(
/
d
jo
r
)
and the force acting in this joint (
d
j
F
), so that:
,
/, ,
/ /, ,
/, ,
jo
dd
jox j x
d ddd d
F jo j joy j y
dd
joz j z
rF
rF
rF
=×= ×
M rF
(C.1)
In which
,jo
d
F
M
is the moment induced by the forces acting in the joint. To obtain a linear
equation, we can replace the cross product, by introducing the matrix
/
d
jo
r
:
,
/, /, ,
/ /, /, ,
/, /, ,
0
0
0
jo
dd d
joz joy j x
d dd d d d
F jo j joz jox j y
dd d
joy jox j z
rr F
r rF
rr F
−
= = −
−
M rF
(C.2)
We use this matrix in the set of linear equations. The complete system of equations (eq. B.1,
B.2, B.4B.11) is:
/
/
//
//
//
0 0 0 0 0 000000
0 0 0 0 0 000000
0 0 0 0 000000
0 0 0 0 000000
0 0 0 0 000000
0 0 0 0 000000
0 0 0 0 000000
0 0 0 0 0 00000
0 0 0 0 0 0 0000
0 0 0 0 0 000
0 0 000000
00 0 00000
000
rc
As
lc
As
rc rc
Ae K e
lc lc
Ae K e
rc rc
Kt Ht
−
−
−
−
−−
−−
−−
−−
−
I
I
II
II
II
II
II
rI
rI
r r II
r r II
r r II
,,
//
//
0 000 0
0 0 0 0 0000
rrrr r r
s s B G s air s ice
l
ss
r
A
l
A
r
K
l
K
r
H
l
H
r
A
l
A
r
K
l
K
r
H
lc lc l
Kt Ht H
rc lc
Hh Hh
mF
ma
−− − −
=
−
− − −−
aFF F
F
F
F
F
F
F
M
M
M
M
M
r r II
M
r r II
( )
( )
( )
,,
,,
,,
,,
,,
,,
,,
,,
B
B
lll l l
B G s air s ice
rr r r
e e G e air e
ll l l
e e G e air e
rr r r
t t G t air t
ll l l
t t G t air t
h h G h air h
rr r r r r
s s B F ice s air s
ll l l l l
s s B F ice s air s
rr
ee
FF F F
ma F F
ma F F
ma F F
ma F F
ma F F
dI MM M M
dt
dI MM M M
dt
dIM
dt
ω
ω
ω
−− − −
−−
−−
−−
−−
−−
−− − −
−− − −
−
( )
( )
( )
( )
,
,
,
,
,
r
air e
ll l
e e air e
rr r
t t air t
ll l
t t air t
h h air h
dIM
dt
dIM
dt
dIM
dt
dIM
dt
ω
ω
ω
ω
−
−
−
−
(C.3)
Inverse kinematics: comparing BPR techniques
In which is a three by three identity matrix. We can write this equation as
Cf =n
(C.4)
When solving the system, we will obtain an error , where is either a translational (
) or rotation ( ) error
,
(
d
qo
e C )=fn
(C.5)
The system in C.3 is overdetermined and therefore is solved in a leastsquare manner. We
minimize hereby the error while solving for
,,
min
dT d
qo qo =f
ee
(C.6)
Optionally a weighing factor can be added to the minimization problem, adding a weighing
factor . This could be an option when it is known that one of the segment measurements
is less reliable than others.
, ,, min
dT d d
qo qo qo =f
e we
(C.7)
In this paper we choose to keep . The solution is computed by performing a least
squares fit, which minimizes the sum of squares of the deviations of the data from the model,
via the solution:
( )
1
TT
CC C
−
= ⋅fn
(C.8)
PART III CHAPTER 6