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1 Introduction

The coordination pattern of speed skating appears to be completely different from all other types of human propulsion. In most patterns of human locomotion, humans generate forces by pushing against the environment in the opposite desired direction of motion. In speed skating humans generate forces by pushing in sideward direction. When we take a closer look at speed skating the straights we observe that a skating stroke can be divided in three phases: the glide, push-off and reposition phase, see Fig. 1. In the push-off phase the skate moves sidewards with respect to the center of mass (COM) of the body till near full leg extension. In the reposition phase the leg is retracted in the direction of the center of mass of the body. During the glide phase the body is supported over one leg that remains at nearly constant height (ankle to hip distance). Double support, where both skates are on the ice, only exists in the first part of the glide phase of one leg and in the second part of push-off phase of the other leg. This coordination pattern with sideward push-off results in a sinus-wave like trajectory of the upper body on the ice [4].

Fig. 1
figure 1

Phases of a stroke: push-off phase, glide phase and reposition phase [1]

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From these observations a number of questions arise. Of the many possible coordination patterns, that is position and orientation of the skates with respect to the upper body, why do skaters use this particular one? What is the optimal coordination pattern for an individual speed skater to reach their optimal performance? How do speed skaters create forward power on ice? Why are speed skaters steering back to their body at the end of the push-off? What is the effect of anthropometric differences on the coordination pattern of a speed skater (like the difference between a tall Dutch skater and a small Japanese skater)? All these questions are highly dependent on the coordination pattern of the speed skater and could well be addressed by simulation and optimization of a biomechanical model of speed skating. But before getting to this optimization approach one needs a model that reasonable matches observed behavior.

Currently, there exist three speed skating models [1, 6, 10]. The first models of speed skating were initiated by Gerrit Jan van Ingen Schenau [12] and further developed by researchers at the VU University Amsterdam [6]. By using power balances of the human and the environment useful information about the posture, athlete physiology and environmental parameters on the performance is obtained. Disadvantages of these models are that the validation is difficult and it is impossible to investigate differences in coordination pattern.

A more recent model was developed by Otten [10], in which forward and inverse dynamics are combined. The model is complex and includes up to 19 rigid bodies and 160 muscles. The model is able to simulate skating and can give insight in the forces/moments in the joints. Limitations of the model are that the kinematics in the model are manually tuned and that the model is not driven and validated with measurements of speed skaters. Unfortunately, no information about this model is available in the open literature, which makes it hard to review.

The most recent speed skater model is developed by Allinger and van den Bogert [1]. they developed a simple, one point mass, inverse dynamics model of a speed skater which is driven by individual strokes. The main limitations of the model are that the model is driven by a presumed leg function in time and that the model is not validated with force measurements. Furthermore, the effect of the assumptions on the model (e.g. constant height) are not investigated. On the other hand the model is possibly accurate and very useful for optimization the coordination pattern of speed skating.

Although three biomechanical models exist, none of these models is shown to accurately predict observed forces and motions. Which is partly due to the lack of experimental kinematic data and force data on stroke level.

In this chapter, we present a 2-dimensional inverse dynamics model on the straights which has minimal complexity. The model is based on three lumped masses and is validated with observed in-plane (horizontal) kinematics and forces at the skates. In the future, this model can be used to provide individual advice to elite speed skaters about their coordination pattern to reach their optimal performance.

2 Methods

We measured in time the 2-dimensional in-plane (horizontal) positions (x,y) of the two skates and the upper body, the normal forces and lateral forces at the two skates and lean angle of the skates. We developed a 2-dimensional inverse dynamic model of a skater. The model is driven by the measured leg extensions, that is relative motions of the skates with respect to the upper body and absolute orientation of the skates with respect to the ice. The upper body motions together with the forces exerted on the ice by the skates are calculated from the model.

A schematic of our 2-dimensional model is shown in Fig. 2. The model consists of three point masses: lumped masses at the body and the two skates. The total mass of the system is distributed over the three bodies by a constant mass distribution coefficient. The motions of the arms are neglected. We do not consider the vertical motion of the upper body, since experiments show that the upper body is at nearly constant height [3]. Air friction and ice friction are taken into account. Idealized contact of the skate on the ice is modeled by a holonomic constraint in the vertical direction and a non-holonomic constraint in the lateral direction.

Fig. 2
figure 2

Free body diagrams of the three point mass model (horizontal plane, top view). The masses are located at the COM of the body and at the COM of the skates. F ls and F rs are perpendicular with the skate blades, θ ls and θ rs are the steer angles of the skates with respect to the x-axis. The x- and y-axes are the inertial reference frame fixed to the ice rink

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Values for the mass distribution and air friction are found experimentally. The best agreement between the measurements and model can be achieved if we use accurate values for these parameters. Therefore we constructed an objective function J min and minimized the error between the measurements and model. Details on the objective function can be found in Appendix 8.3.

3 Model Analysis

In the model analysis for speed skating, three stages can be distinguished. First, the unconstrained equations of motion of the speed skater of a single stroke are derived. Secondly, the constraints are formulated and incorporated into the unconstrained equations of motion. Finally all equations are derived in terms of generalized coordinates and solved by numerical integration of these constrained equations of motion.

3.1 Equations of Motion

The equations of motion for each separate body (upper body, right skate and left skate) can be derived in x and y direction. Friction forces (air and ice friction) as well as the constraint forces are acting on the bodies. All constraints acting on the bodies will be explained in the next paragraph. The unconstrained equations of motions for all bodies are,

$$ \begin{array}{l} m\ddot{x}_i = -F_{\mathit{friction}X_i}+F_{\mathit{constraints}X_i} \\ [3pt] m\ddot{y}_i = -F_{\mathit{friction}Y_{i,}}+F_{\mathit{constraints}Y_i} \end{array} \quad i = B, LS, RS $$
(1)

where \(F_{\mathit{friction}X_{i}}\) is the component of the friction force in x direction and \(F_{\mathit{friction}Y_{i}}\) the component of the friction force in y direction. F constraintsX are the constraint forces in x direction and F constraintsY the constraint forces in y direction.

3.2 Constraints

The first set of constraints are the leg extension constraints, they connect the skates to the upper body. The positions of the skates are prescribed by the position of the upper body and the leg extension coordinates. The second set of constraints are at the skates. A holonomic constraint is applied in the vertical direction which establish that the skate is on the ice and a non-holonomic constraint in the lateral direction of the skate to express that there is no lateral slip of the skate on the ice.

3.3 Generalized Coordinates

The generalized coordinates of the skater model are chosen such that we can express the coordination of the motion of the skater in terms of the leg extensions and the skate orientations (steer angles). Therefore the configuration of the skater is expressed by the motion of the upper body and the leg extensions (relative motions of the skates with respect to the upper body, see Fig. 3) and can be described by the generalized coordinates,

$$ \mathbf{q} = (xb,yb,u_{LS} ,v_{LS} , \theta_{LS} ,u_{RS} ,v_{RS}, \theta_{RS})^T, $$
(2)

in which θ LS and θ RS are the steer angles of the skates with respect to the global x-axis. These steer angles, which are prescribed coordinates, are needed to apply the non-holonomic skate constraints. The equations of motion will be written in terms of the generalized coordinates. Detailed information on the transformation of the equations of motions in terms of the generalized coordinates can be found in Appendices 8.1, 8.7, and 8.8.

Fig. 3
figure 3

Definition of generalized coordinates

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3.3.1 Leg Extension Constraints

The position of the right and left skate can be expressed as function of the generalized coordinates and will be incorporated into the equations of motion means by holonomic constraints. The left skate leg extension constraints are,

$$ c_1 = x_{LS} - x_B + \cos ( {\theta_{LS} } )u_{LS} + \sin ( {\theta_{LS} } )v_{LS} = 0, $$
(3)
$$ c_2 = y_{LS} - y_B - \sin ( {\theta_{LS} } )u_{LS} + \cos ( {\theta_{LS} } )v_{LS} = 0, $$
(4)

and the right skate leg extension constraints are,

$$ c_3 = x_{RS} - x_B + \cos ( {\theta_{RS} } )u_{RS} + \sin ( {\theta_{RS} } )v_{RS} = 0, $$
(5)
$$ c_4 = -y_{RS} + y_B - \sin ( {\theta_{RS} } )u_{RS} + \cos ( {\theta_{RS} } )v_{RS} = 0. $$
(6)

3.3.2 Skate Constraints

When the skate is on the ice we assume no lateral slip between the ice and skate, that is the lateral velocity of the skate is zero. This can be expressed by a non-holonomic constraint which are for the left and right skate respectively,

$$ c_5 = -\sin ( {\theta_{LS} } )\dot{x}_{LS} - \cos ( {\theta_{LS} } )\dot{y}_{LS} = 0, $$
(7)
$$ c_6 = -\sin ( {\theta_{RS} } )\dot{x}_{RS} + \cos ( {\theta_{RS} } )\dot{y}_{RS} = 0. $$
(8)

Since we do not consider vertical motions no constraints in the vertical direction are needed. Contact or no contact is described by on/off switching of the corresponding non-holonomic constraint.

3.4 Mass Distribution

The number of bodies in the model is based on an investigation of the shift in position of the center of mass on a complete anthropometric model of a speed skater during the gliding and the push-off phase of a stroke. A minimum of three bodies was shown to be necessary for describing the shift of the center of mass [8].

The total mass m of the skater is now distributed over the three point masses (body, left skate, right skate) by using a mass distribution coefficient α (Fig. 4). The distribution of the masses are given by m B =(1−α)m, m LS =(α/2)m, and m RS =(α/2)m.

Fig. 4
figure 4

Positions of the COM of the bodies during the push-off together with the mass distribution

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3.5 Friction Forces

The total friction forces can be roughly divided in 80 % air friction and 20 % ice friction [5]. The ice friction in the model, following de Koning [7], is described by Coulomb’s friction law,

$$ F_{\mathrm{ice}} = \mu F_N $$
(9)

where μ is the friction coefficient and F N the normal force of the skate on the ice. Here we assume that the height of the skater is constant and that there is no double stance phase. Therefore, the ice friction can be written as F ice=μmg, in which m the total mass of the skater and g the earth gravity. The air friction can be described by,

$$ F_{\mathrm{air}} = \frac{1}{2} \rho C_d A v^2 = k_1 v^2 $$
(10)

where ρ represents the air density, C d the drag coefficient, A the frontal projected area of the skater, and v the velocity of the air with respect to the skater. The air drag forces at each individual mass are calculated by multiplying the mass distribution coefficient of that mass by the total air drag. The drag coefficient k 1 can only be estimated experimentally. With an experimental method (see Appendix 8.5) both the drag coefficients μ and k 1 for every individual subject are estimated.

3.6 Model Summary

The equations of motion together with the constraint equations are completely defined by the state of skater. Combining the equations of motion for the individual masses (1) and including of the constraint forces and the constraints (3)–(6) on the acceleration level results in the constraint equations of motion for the system, Au=b, with

(11)
$$ \mathbf{u} = \left [ { \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c @{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} {\ddot{x}_{B} } & {\ddot{y}_{B} } & {\ddot{x}_{LS} } & {\ddot{y}_{LS} } & {\ddot{x}_{RS} } & {\ddot{y}_{RS} } & {\lambda_1 } & {\lambda_2 } & {\lambda_3 } & {\lambda_4 } \end{array} } \right ]^T $$
(12)
(13)

where h c1…h c4 are the convective acceleration terms of the constraints (Appendix 8.8) and λ 1…λ 4 are the constraint forces (Lagrange multipliers). Here λ 1 and λ 2 are the constraint forces in the left leg, and λ 3 and λ 4 the constraint forces in the right leg. The non-holonomic skate constraints are not yet included in this system, but will be in a later stage.

The model consists of 3 bodies with each 2 degrees of freedom, thus the unconstrained system has 6 degrees of freedom. However, there are 4 coordination constraints and 1 non-holonomic constraint of the skate on the ice (no double stance); therefore 1 degree of freedom remains. If there is a double stance phase then both skates are on the ice, the system is over-constrained and no degree freedom is left. Therefore for the model we will assume only single stance phases, and the model will alternatively switch between the right skate en left skate constraint. This assumption is validated by the experimental force data, where we see only a short period of double stance with load transfer.

We rewrite the equations of motion (11)–(13) (still without the non-holonomic skate constraints) in terms of the generalized coordinates (2), where the prescribed coordinates (leg extension coordinates (u LS ,v LS ,θ LS ,u RS ,v RS ,θ RS )) are pushed to the right-hand side (Appendix 8.1). Next, the constraint of the skate on the ice (left or right) is added to the equations. Finally the reduced constrained equations of motion are given by, for when the left skate is on the ice,

$$ \left [ { \begin{array}{c@{\quad}c@{\quad}c} m & 0 & { - s_{LS} } \\ 0 & m & { - c_{LS} } \\ { - s_{LS} } & { - c_{LS} } & 0 \end{array} } \right ] \left [ \begin{array}{c} {\ddot{x}_B } \\ {\ddot{y}_B } \\ {\lambda_5 } \end{array} \right ] = \left [ \begin{array}{c} {\mathbf{T}^T_{,\mathbf{q}}(\mathbf{f} - {\mathbf{M}}\mathbf{h})} \\ { h_{c5}} \end{array} \right ], $$
(14)

and for when the right skate is on the ice,

$$ \left [ \begin{array}{c@{\quad}c@{\quad}c} m & 0 & { - s_{RS} } \\ 0 & m & { c_{RS} } \\ { - s_{RS} } & { c_{RS} } & 0 \end{array} \right ] \left [ \begin{array}{c} {\ddot{x}_B } \\ {\ddot{y}_B } \\ {\lambda_6 } \end{array} \right ] = \left [ \begin{array}{c} {\mathbf{T}^T_{,\mathbf{q}}(\mathbf{f} - {\mathbf{M}}\mathbf{h})} \\ {h_{c6}} \end{array} \right ], $$
(15)

where λ 5 and λ 6 are the lateral constraint forces on the skate and h c5 and h c6 are the convective acceleration terms of the skate constraints, the latter are presented in Appendix 8.8. Clearly both systems have one degree of freedom left, one can think of it as being the forward motion.

3.7 Model Constants

Experimental data was obtained from four different riders. Listed in Table 1 are the values of the model parameters used in the simulations for these four riders. The total mass of the skater and gravity are a measured quantities. The other parameters are found by an optimization process as described in Appendix 8.3.

Table 1 Parameter values for the four riders
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4 Model Analysis

4.1 Parametrization of the Coordination Body Functions

Input to the model are the measured motion coordinations, the leg extensions and the skate steer angles, and their velocities and accelerations. To determine these all measured positions have to be differentiated with respect to time. To get rid of model errors due to numerical differential and filtering errors (spikes), all positions are first parameterized by smooth functions. The required parametrization functions have to be twice differentiable. The combination of a linear and periodic functions satisfies this requirement. The used parametrization function is,

$$ f = c_0 + c_1 t + \sum_{k = 1}^5 {a_k \sin \biggl( {2k \pi }\frac{t}{T} \biggr) + b_k \cos} \biggl( {2 k \pi \frac{t}{T}} \biggr). $$
(16)

The fit is not accurate at the beginning and end of the stroke, which results in a mismatch of the initial conditions on the velocities and accelerations. Therefore the coordinates are fit at a somewhat longer time period and then cut off afterwards. We tried also other parametrization functions, like polynomial and cubic splines. The differentials of polynomial functions became unstable with increasing order, while piecewise cubic splines have no filtering which results in high frequent components in the positions. The measured positions of the body, left and right skate in x and y direction of a single stroke are parameterized according to (16) and by differentiating the equations of the fitted function the velocities and acceleration are calculated.

4.2 Integration of the Differential Equations

The differential algebraic equations (14), (15) describing the motion of the system cannot be solved analytically. Therefore, the equations will be numerically integrated, using the classic Runge-Kutta 4th order method (RK4). The stepsize h is taken constant during the whole simulation, and chosen identical to the sample time of the measurements T s =1/100 [sec]. After each numerical integration step the constraints are fulfilled by a projection method (Appendix 8.2).

4.3 Data Collection

The data collection of the skater includes the 2-dimensional in-plane positions (x,y) of the two skates and the upper body, the normal and lateral forces at the two skates and lean angle of the skates. The global positions are measured by a radio frequency based so-called local position measurement system (LPM) from Inmotio.Footnote 1 This system is installed at the Thialf speed skate rink in Heerenveen, The Netherlands. The LPM system has been used for analysis of soccer matches, and can handle up to 22 active transponders at 1000/22 Hz. The transponders are approximately placed at the positions of the point masses.

We have developed two instrumented clap skates to measure the normal and lateral forces (N i ,L i ) at the blades of the skates, see Fig. 5. To be able to compare these with the model output, which are the global lateral forces F Tls and F Trs , the lean angles of the skates, ϕ i , has be measured too. These angels are measured using an inertial measurement unit from Xsens,Footnote 2 where only the lean angle is used.

Fig. 5
figure 5

Forces in local reference frame (N ls , T ls , N rs , T rs ) and global reference frame (F Nls , F Tls , F Nrs , F Trs ): (a) Left skate, (b) right skate

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For data acquisition a DAQ unit of National instrumentsFootnote 3 is used. All the force and orientation data is collected from the DAQ via a USB connection on a mini laptop which is carried by the skater in a backpack. The different measurement systems are synchronized by means of images from a high speed camera. See Appendix 8.4 for detailed description of the synchronization method.

Data sets of four trained speed skaters are used to validate the model. The data collection is performed with a standard measurement protocol which includes: skating two laps at an estimated 80 % of maximal performance level. The tests are repeated at least three times.

4.4 Fitting the Model to the Observed Data

The model is validated by showing how closely it can simulate the observed forces and motions. Quantification of the model errors are analyzed similar to that of McLean [9]. The measured data has different scales and units and therefore we constructed a measurement of error, J min, between the model and the measured data which includes the error of the upper body position, velocities and local normal forces (N ls and N rs ). The measurement of error is dimensionless, reasonably scaled and independent of the number of time samples. See Appendix 8.3 for a detailed description of the measurement error function J min.

5 Results

Plots of the measured and simulated forces and motions (output of model) as a function of time for a sequencing left and right stroke are shown in Fig. 7 (the parameters are according to the first rider from Table 1). The corresponding measured and parameterized leg extensions (input of model) of the left and right leg are shown respectively in Fig. 6(a) and Fig. 6(b). At the beginning of the left stroke (t=0) the skate is placed in front of the upper body, resulting in a negative u ls . During the stroke the skate is moving sidewards and backwards, u ls and v ls increase. At the end of the stroke the skate is retracting to the upper body, u ls and v ls decrease. At the beginning of the right stroke (t=1.25), the skate is again moving sidewards, v rs increase. However the motion pattern of the u rs is somewhat different in comparison with u ls . The u rs remains approximately constant during the stroke, which eventually will results in a different output motion of the upper body in y direction.

Fig. 6
figure 6

Measured and parameterized leg extension coordinates u i ,v i and θ i as a function of time for a sequencing left and right stroke for rider 1 from Table 1. Gray filled area means that the skate is not active. (a) Left skate, (b) right skate

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The skater has an average forward speed of ≈32 km/h. The upper body describes a sine-wave like trajectory with respect to the ice during speed skating the straights (Fig. 7(a), y b ), which has also been observed by de Boer [4]. The velocity pattern sidewards, \(\dot{y}_{b}\), are alike for left and right stroke. However, the forward acceleration/deceleration pattern differ per stroke. This was observed for every rider.

Fig. 7
figure 7

Simulated (black lines) and measured (gray lines) upper body positions, velocities, accelerations and local normal forces on the skates (N i ), as a function of time for a sequencing left and right stroke, for rider 1 from Table 1 (mg=647 N)

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The local normal forces N LS and N RS of the active skate are shown in Fig. 7(b), where the height of the body is assumed constant. At the large force drop in the measured force data a switch is made in the model from the left skate to the right skate. Note that the sum of the measured left and right force corresponds well to the calculated value. At the beginning of the stroke the normal force is rising above the body weight of the skater. Then a small force drop appears and at the end of stroke the normal forces rises again well above the body weight. The maximal normal force during push-off is approximately 150 % of the body weight.

Agreement exists between the measured and simulated positions and velocities. The largest error is in the force data, which mainly appears at the beginning and end of the stroke.

For all skaters the net error J min (24) of all straight left strokes is calculated. This net error is divided by the number of optimization parameters being the upper body positions, upper body velocities and the local normal forces and presented in Table 2.

Table 2 Net error J min per subject (average of all left straight strokes of all tests) divided by the number of optimization parameters
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Averages of the magnitudes of the residuals are calculated similar to that of Cabrera [2] by \(R_{j} = \sum_{i = 1}^{N} {\vert {\tilde{y}_{ij} - y_{ij} } \vert } /N\). In which N the number of collected data points, y i the measured value of the variable and \(\tilde{y}_{i}\) the simulated value of the variable from the model. For all variables j the R j is shown in Table 3. The residuals of the upper body are less than 0.10 m for the forward position, 0.031 m sidewards, 0.20 m/s in the forward velocity, 0.06 m/s sidewards, and 53 N for the local normal forces in the skate.

Table 3 Table of the residuals between measured and simulated values of the variables. Body position in x direction [m], body position in y direction [m], body velocity in x direction [m/s], body velocity in y direction [m/s], body acceleration in x direction [m/s2], body acceleration in y direction [m/s2], local normal forces [N]
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6 Discussion

6.1 Model Error

All position residuals are within the accuracy of the position measurement system (≈0.15 m). The accuracy of the LPM can be increased if two transponders, instead of one transponder are positioned at the skates and the upper body. The forward velocities \(\dot{x}_{B}\) are less accurate than the sideward velocities \(\dot{y}_{B}\), which is reasonable due to the fact that the forces are mainly in sideward direction instead of forward. Orientation errors have therefore more influence on the \(\dot{x}_{B}\) than on the \(\dot{y}_{B}\).

No total agreement exists between the measured forces and the forces calculated in the model, generally at the beginning and at the end of the stroke. There is no normal force drop in the calculated data which is a result of the simplification that there is no double stance phase, but the sum of the measured left and right force do correspond well with the calculated one. Conversion from global to local forces resulted in a force error, caused by the accuracy of the lean angle sensors. The accuracy of these sensors are < 2 deg root mean square, resulting in a local normal force error between ≈20/−20 N. Besides conversion errors, crosstalk exists of ≈3 % of the lateral forces to the normal forces (max. −9/9 N). The maximal error due to inaccuracy of the measurement equipment is then approximately 29 N.

The net error J min of all measurements are in the same magnitude, which shows that the model is valid for all subjects.

6.2 How Does the Fit Depend on Mechanical Constants

The sensitivity of the mechanical constants is obtained by minimizing the net error J min (24). This net error is calculated by letting the upper body motions variable while fixing all other parameters to their optimal fit value, except for the wanted minimization parameter (mass distribution α, air friction coefficient k 1 or mass of the skater m). In Fig. 8 the normalized net error J min are plotted as function of the minimization parameters. The minimal values in the figures correspond to the values of the parameters at the optimal fit.

Fig. 8
figure 8

Plots of J min versus a single parameter value, mass distribution α, air friction coefficient k 1 and total mass of the skater m, as the parameter is varied about the nominal value for rider 1 from Table 1. The filled circles correspond to the value of J min at the nominal parameter value

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The mass is the most sensitive mechanical parameter, however this parameter is measured accurately and therefore of no concern here. The value of the mass distribution α as well as the friction coefficient k 1 are more uncertain. The figure shows clearly that the fit depends little on these mechanical constant.

6.3 Fitting False Data

If the fits which are obtained are a result of good curve fitting, then it should be able to obtain good fits to false data. To test the model a pure sine function, Acos(2πt/T), with amplitude A, and stroke time T, is added to the measured velocity data of the upper body in either directions. In Fig. 9 the minimal error function versus the amplitude of the sinus wave is plotted. The total error between the model and the measured variables is minimal if the amplitude of the added function is zero. The model shows the best fit if there is not added corrupted data to the velocity data of the upper body. These results shows that the fits are not a result of good curve fitting, but rather the result of a good model.

Fig. 9
figure 9

Plot of error J min versus the amplitude of the sine wave corrupting the velocity data of the upper body of the skater

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6.4 Kinematic Complexity

The double stance phase was not included in the model. However, the sum of the measured left and right force during the short double stance phase do correspond well with the calculated forces (Fig. 7(b)), which demonstrates that there is little need for modeling this double stance phase.

Another major simplification of the model is that it was assumed that the center of mass remains at a constant height during skating, which was based on de Boer [3]. However, in accelerometer data of the upper body it was found that at the end of the stroke the upper body accelerates about 1.5 times gravity, which really influences the forces in the model. Therefore it seems beneficial to include the vertical motion of the body in the model.

7 Conclusions

We have constructed a simple 2-dimensional model of speed skating that does a reasonable job of imitating the forces and kinematics as observed in actual speed skating. The model reproduces these forces and motions reasonably well, even if we do not fit for that. The model is limited in accuracy due to the limited accuracy of the LPM position measurement system. Adding the (small) vertical motion of the upper body can increase the accuracy of the model.

The model seems promising for individual training advice. Coordination patterns of individual skaters can be optimized by using the model if psychological constraints of individual skater are added to the model. In Appendix 8.6 a detailed description of the needed constraints on the model is given. The model can also be used to give insight in the biomechanics of speed skating, like why speed skaters steer back to their body at the end of the stroke. Finally the effect of anthropometric differences between speed skaters can be determined.