S ports that involve skating such as speed skating, hockey, and cross-country skiing require a combination of metabolic power, muscular strength, and technique to obtain top speeds. If skating speed is limited by metabolic power or muscle strength, training may improve these constraints to allow an athlete to skate faster. However, skating speed also depends on skating technique (i.e., stroke time, glide time, push-off velocity, push-off direction, preextension knee angle). A change in these variables might improve the performance of an athlete immediately.

The motion of one leg during skating can be divided into three phases: glide, push-off, and recovery (Fig. 1) . During the glide phase the body is supported over one leg that remains at nearly a constant length (ankle to hip distance). The push-off phase begins with the initiation of leg extension and ends near full leg extension when the skate blade lifts off the ice. The skate is returned to the initial position under the body during the recovery phase which begins at the end of the push-off phase and ends when the blade contacts the ice. The gliding phase then begins and the cycle is complete. During the skating cycle of two legs, there is a period of double support when one leg is in the push-off phase and the other leg is in the gliding phase. During this period, body weight is transferred from the push-off skate to the gliding skate. In the context of this paper, a skating stroke will include only the glide and push-off phases of one leg.

Previous publications have stated that the “ideal” technique for speed skating is to maintain a small trunk angle to reduce air friction^{(5)} , glide with a knee angle of between 90° and 110° ^{(1,6)} , fully extend the leg during the push-off phase ^{(4)} , push-off perpendicular to the gliding direction of the skate ^{(7)} , and glide for about 80% of the stroke ^{(7)} . Experimental studies have identified that the work produced by a skater during each stroke is the most important technical factor in speed skating and is associated with faster skaters. It was found that work per stroke is governed by the extension velocity in the hip and knee joints, the preextension knee angle, and the angle between the ice and the push-off leg at the end of the push-off phase^{(1,2,6)} .

This ideal skating technique has been developed over many years using trial-and-error methods and by slower skaters mimicking faster skaters. Although empirical data can demonstrate the techniques used by the fastest skaters, these skaters might not be using the optimal skating technique. In addition, there may be more than one optimal skating technique for an athlete. Speed skating coaches have noted that skaters from one country often demonstrate a different skating technique than skaters from another country^{(2)} . Since speed skaters with different techniques have won international medals, the fastest skating technique is arguable.

The optimal skating technique (e.g., stroke time, glide time, push-off velocity, push-off direction, preextension knee angle) has not been determined analytically. Such an optimal skating technique may vary from one athlete to another because of differences in body size, muscle properties, and fitness level. The purpose of this study was to determine the skating technique that results in the fastest steady-state speed on a straight-away using optimization of a simulation model.

METHODS
The kinetic behavior of a model of a skater was determined using simulation techniques. The model used leg length as a function of time as input to the simulations and the steady-state skating speed as the output. Parameters in the leg length function were adjusted using an optimization routine to determine the technique(s) that resulted in the highest steady-state speed of skating.

Skating model. A skater was modeled as a point mass with a“piston like” leg extending from the center of mass to a mass less skate (Fig. 2A) . The equation of motion for the center of mass (m ) of the skater with forces from the ice on the skate(F _{s} ), from air friction(F _{air} ), and from gravity (m g ) was equation 1

where bold letters represent three-dimensional vectors, r is the position of the center of mass with respect to a reference frame fixed in the ice, and the dots are derivatives with respect to time. All forces are assumed to intersect the center of mass. Air friction is assumed to be proportional to the velocity squared ^{(5)} , equation 2

where k is a constant and ν is the velocity of the center of mass. Ice friction is ignored since it is small compared with other forces on the body. As a result, the push-off force (F _{s} ) is always perpendicular to the skate blade ^{(7)} .

It was assumed that the center of mass remained at a constant height(h ) above the ice as has been shown to occur experimentally^{(2)} . This assumption prevented the skater from falling to the ice as a result of the instability of this inverted pendulum type of model. Using the vertical (z ) component ofequation 1 , the magnitude of the push-off force(F _{s} ) is dependent on the instantaneous leg length(L ), equation 3

In the plane of the ice (xy-plane), the equations of motion for the center of mass from equations 1 to 3 reduce toequations 4a,4b

where L _{h} is the projection of leg length L into the xy-plane, γ describes the direction of the skate blade on the ice, and x and y are coordinates for the center of mass relative to the reference frame fixed in the ice. The projection of leg length (L ) in the horizontal plane (L _{h} ) was used to describe leg length for convenience.

Since leg extension occurs perpendicular to the skate blade, the instantaneous direction of the skate blade (γ) is calculated from geometrical constraints (Fig. 2B) as,[5a] [5b]

where L _{h} is the velocity of leg extension in the plane of the ice and ν and β are the speed and direction of the center of mass. The orientation of the skate blade also describes the direction of the push-off force since they are perpendicular to each other.

The model was constrained by three relationships: leg length, instantaneous power, and average power.

Leg length was constrained to be within physiological limits(-L _{hmax} ≤ L _{h} ≤L _{hmax} ).
The instantaneous power (P _{inst} ) during a stroke must not exceed the maximum possible power (P _{fv} ) from a leg extension motion (P _{inst} ≤ P _{fv} ). Instantaneous power could be determined from the push-off force and velocity of leg extension in the horizontal plane, since no work is done in the vertical direction,Equation 6 The maximum possible power from a single leg extension as a function of leg extension velocity (ankle relative to the hip) was based on force-velocity data extracted from the leg press results of Vandervoort et al. ^{(12)} and fit to a fifth order polynominal function( Fig. 3 , solid line) and then multiplied by u , Equation 7 where u is the leg extension velocity in m·s ^{-1} and P _{fv} is power in W. The average power per stroke (P _{avg} ) must be less than the available aerobic power for skating (P _{avg} ≤P _{aer} ). Average power per stroke was calculated asEquation 8 During skating, the anaerobic system can contribute significantly to increase the average power that is available above that of the aerobic system ^{(11)} . Assuming that aerobic power represents the power that can be exerted over a long period of time, these results are limited to steady-state skating speeds.
Optimization procedure. The behavior of the skating model is completely determined by the input function, the horizontal leg length which is a function of time. Once this function is chosen, the push-off angle(θ) is calculated (equations 5a and 5b) and the equations of motion (equations 4a and 4b) are solved using second and third order Runge-Kutta methods (Matlab, The Math Works Inc., Natick, MA) for one stroke. This procedure is repeated alternating strokes with a left push-off and a right push-off until the skater reaches a steady-state speed. The model contains no double support phase, so the push-off forces (F _{s} ) are transferred instantaneously from the end of the push-off phase for one leg to the beginning of the glide phase for the other leg. Parameters used in the model were based on average findings for elite male speed skaters from the literature (Table 1) .

Instead of choosing the horizontal leg length as a function of time and solving for the push-off angle during each interation, it is possible to select the push-off angle as a function of time and solve for the leg length using equations 5 . This technique may be more appropriate when the skate glides in a straight line (e.g., the skating technique used in cross-country skiing ).

A function that has the basic characteristics of a skating stroke^{(2,9)} was chosen to express the horizontal leg length (L _{h} ) as a function of time. This function represents one stroke and consists of a glide phase and a linear push-off phase(Fig. 4) . Three parameters describe the function: stroke time, glide time, and push-off velocity. Stroke time(t _{stroke} ) is the amount of time required to complete one stroke. Glide time is the proportion of the stroke time spent in the gliding phase (0 ≤ t _{glide} < 1). During the glide phase, the horizontal leg length is zero so the center of mass is directly above the skate. Leg extension velocity (˙L _{h} ) describes the rate of leg extension in the plane of the ice. The duration of the push-off phase can be determined from the other two parameters (t _{stroke} ×(1 - t _{glide} )). The constraint functions limited leg length to less than L _{hmax} .

Other functions (polynomials of orders one to six) describing the horizontal leg length were analyzed before choosing the above function. Changing the type of function describing the horizontal leg length and little effect on the resulting skating speed. The reason is that the optimal path for the center of mass in this model is a straight line trajectory. The center of mass has a large velocity component in the desired direction of travel and a small velocity component perpendicular to the desired direction of travel that is caused by the push-off leg. The use of higher order polynomial functions or even functions fit to actual leg extension data^{(2,9)} result in little or no improvement in skating speed and complicate the interpretation of results.

Since no solution for an optimal skating technique was unique, effects of changes in the three technique parameters on the resulting steady-state speed was determined. Stroke time and glide time were held constant while the maximal skating speed was determined by optimizing the push-off velocity, subject to the constraints, using a sequential quadratic programming method(Matlab, The Math Works, Inc.). Stroke time and glide time were systematically changed in repeated computations to determine the effects of the three parameters on maximal skating speed. The constraint equations of average power and instantaneous power were also altered to determine their effects on maximal skating speed and skating technique.

RESULTS
Simulation results. Results from the optimizations reveal a surface that represents the maximum speed this hypothetical skater can go for a chosen glide and stroke time at the optimal leg extension velocity(Fig. 5) . At a chosen glide and stroke time if the leg extension velocity is not optimal, the skating speed will be below the surface.

When no constraints are applied to the model, the maximum skating speed is described by a surface that increases asymptotically as stroke time decreases towards zero seconds (Fig. 5A) . Maximal skating speed varies little with changes in the glide time. The maximum skating speed for this system, with a mass less leg and skate, approaches infinity as stroke time approaches zero. Because of inertial properties of the leg and skate and because the force produced by muscle is dependent on the velocity of shortening ^{(3)} , the stroke time will be limited.

The instantaneous power constraint creates two limiting surfaces on the maximal skating speed. One of these surfaces is related to the limited leg extension velocity (Fig. 5B) . The leg extension velocity for all points on this new constraint surface are close to the maximal unloaded velocities (L = 3.24 m·s^{-1} orL _{h} = 4.58 m·s^{-1} ). The other limiting surface caused by the instantaneous power constraint creates a third boundary at the higher skating speeds (Fig. 5C) . The power-velocity relation for leg extension (Fig. 3) creates this broad constraint region on skating speed that varies with stroke and glide time. For a given leg extension speed, the instantaneous power during the stroke must fall below the power-velocity curve (Fig. 3) . This constraint essentially cuts off the ridge created between the surface of the model and the surface of the maximum leg extension velocity inFig. 5B .

The average power constraint creates an additional constraint surface that is nearly flat with respect to maximal skating speed (Fig. 5D) . Maximal skating speed for this constraint surface increases only slightly as stroke time decreases because the trajectory of the center of mass becomes more straight.

A range of skating techniques can be used to obtain the same skating speed. The range of skating techniques (i.e., the area enclosed by a given skating speed contour) decreases as skating speed increases (Fig. 6) . For example, a skating speed of 12 m·s^{-1} can be achieved with different combinations of stroke and glide times at points I, II, and III(Fig. 6 and Table 2) . Technique I could be described as a slow leg extension; technique II is a glide followed by a more explosive leg extension; and technique III uses quick explosive strokes. The leg extension velocity must be optimal to achieve the 12 m·s^{-1} skating speed (Fig. 6 , dotted lines andTable 2 ). The optimal leg extension velocity must increase as stroke time decreases and as glide time increases. Stroke and glide times within the 12 m·s^{-1} contour line can be used to achieve a skating speed less than or equal to 12 m·s^{-1} ; however, the velocity of leg extension will not be optimal.

The paths of the skate blade and the center of mass in the plane of the ice will be different depending on the technique chosen, even for the same skating speed. The path of the center of mass deviates more from the desired direction of travel for technique I followed by technique II and then technique III(Fig. 7) . Technique III requires about twice the number of strokes as the other two techniques for the same skating speed. Note that the angle of the push-off leg relative to the desired direction of travel is directed slightly rearward at the beginning of the push (techniques I, II, and III), sideward as the push continues (techniques I and II), and then even slightly forward at the end of the push (technique I). Thus, the optimal push-off direction varies during the stroke but remains perpendicular to the direction of the gliding skate.

Plots of the leg length, force, and power as a function of time illustrate the differences in the three techniques (Fig. 8) . Full leg extension occurs for techniques I and II but not for technique III (70% of leg extension used). The push-off force is directly related to the leg length(1 m = 954.5 N) because of the assumption that the height of the center of mass is constant (Fig. 8) . The peak push-off force is 30% lower for technique III compared with techniques I and II. Although the average power is similar for all three techniques, the instantaneous power was very different. Techniques II and III required instantaneous powers that were 33% and 61%, respectively, larger than technique I.

Sensitivity of model to the constraints. Changes in the constraints of this skating model will effect the maximum possible skating speed. According to this model, the average power constraint limits the maximal skating speed. An increase in the average power will raise this constraint vertically (surface D, Fig. 5D ). The maximum skating speed corresponding to average power can be estimated using ν =(P _{avg} /k )^{1/3} . A 10% increase in average power(from 300 to 330 W) results in an increase in the maximum skating speed of 3%(from 12.54 to 12.95 m·s^{-1} ).

Increasing or decreasing the power-velocity curve (isometric force) by 20% results in an upward or downward shift, respectively, of the instantaneous power constraint surface (Fig. 3) . If skating speed is not limited by average power, increasing the instantaneous power can result in faster skating speeds. Increasing the instantaneous power by 20% results in a 10% increase in skating speed (15.4 m·s^{-1} ), while decreasing the instantaneous power by 20% results in a 12% decrease (12.2 m·s^{-1} ) in skating speed, based on the skating speed (14.0 m·s^{-1} ) at a stroke time of 0.4 s and a glide time of 0.0 using the original instantaneous power curves (Fig. 9) .

Increasing the instantaneous power constraint also increases the range of techniques that can produce the same skating speed (Fig. 9D) . Using the contour lines for a skating speed of 12 m·s^{-1} , the reduced instantaneous power (curve C) results in a smaller range of techniques than with the larger instantaneous power (curve B), as evidenced by the area enclosed by the curves (Fig. 9D) . In general, the weaker skater must use shorter stroke times to achieve the same skating speed as the stronger skaters for a given glide time.

Changes in the instantaneous power constraint can also be made by increasing the maximum velocity of leg extension. Increasing the maximum velocity of leg extension shifts that constraint surface in the direction of zero stroke time. Because the instantaneous power constraint is almost flat in the direction of reduced stroke time, such a shift in the maximum velocity of leg extension will result in little or no possibility to increase skating speed.

Increasing the height of the center of mass shifts the surface created by the model (surface A, Fig. 5 ) downwards to slower skating speeds (Fig. 10A-C) . This decrease in skating speed is not related to an increase in air friction caused by an increase in frontal area. A larger height (i.e., a larger knee angle during gliding) requires the skater to decrease the stroke time to skate the same speed(Fig. 10D) . Increasing the height of the center of mass decreases the range of skating techniques available as illustrated by the decrease in area of the constant speed contours (Fig. 10) . The height of the center of mass is inversely related to the distance the push-off leg can be extended.

DISCUSSION
This study was performed to determine the skating technique that produces the fastest steady-state speed on a straight-away. A simulation model was developed to achieve this goal since empirical data from athletes can only show what technique the best skaters use. Empirical data does not necessarily show what the optimal technique for a given speed skater is, what parameters must be changed to increase speed, or what is the most important parameter to improve or change (e.g., aerobic power, strength, or technique).

Model validity. The model from this study produces results that compare favorably with studies that have measured the skating technique of world class speed skaters. This model predicts stroke and glide times of 0.98 s and 0.25, respectively, for a skating speed of 12 m·s^{-1} . This is comparable to 0.88 s and 0.74 for the stroke and glide times of speed skaters during the 1988 Winter Olympics at the Olympic Oval, Calgary^{(2)} . The experimental data was based on the average results of 29 males during the straight-away portions of the 5000-m races(skating speed 12.12 ± 0.2 m·s^{-1} ).

During the glide phase, experimental studies have measured a small leg extension velocity during the glide phase^{(2,9,10)} . Reducing the leg extension to zero during the gliding phase of the model simplifies the model without removing the major leg movement. As a result, the leg length-time curves and power-time curves (Fig. 4 and 8) do not exactly match the behavior of speed skaters during the glide phase^{(9,10)} . Also leg extension-time curves from the model do not match the experimental results at full leg extension where the leg extension velocity decreases to zero^{(2,9,10)} ; however, during this deceleration period, the skate is not in contact with the ice ^{(7)} .

Results from the model resemble those of speed skaters even though the leg length function was a simplification of what speed skaters actually do. The paths of the skate blade and the center of mass for this model qualitatively resemble what is observed in speed skating ^{(7)} . The leg extension speeds found in the model are within the range of peak leg extension speeds found for elite (1.59 m·s^{-1} ) and trained (1.10 m·s^{-1} ) speed skaters ^{(9)} .

Results from this model are acceptable considering that the force-velocity curves for leg extensions that were used in the model are not from the same athletes used in the experimental study. Speed skaters may be able to produce more force at a given velocity of leg extension than was used in the model. An increase in the instantaneous power constraint would bring the results from the model closer to the results of other studies^{(2,9)} .

Model limitations and considerations. In this skating model, the center of mass was assumed to remain at a constant height during skating. Although this assumption compares well with what many skaters do^{(2)} , it may be advantageous to allow the center of mass to rise and fall during the stroke.

Power calculations in this model are purely mechanical; therefore, the model does not fully describe the metabolic cost of skating with different techniques. Increases in metabolic energy may occur for a skater gliding with a small knee angle (a lower center of mass height) compared with a large knee angle due to the increased force of an isometric contraction. For similar reasons, metabolic energy may increase as glide time increases and as leg extension velocity increases. Based on this hypothesis, skating techniques with no glide time would be preferable.

In reality, more metabolic energy will be required to use a technique that has a small stroke time because more strokes will be required to cover the same distance than a technique that uses a longer stroke time. The mass of the skate and leg, which was neglected in this model, would have to be repeatedly accelerated and decelerated, requiring more power. For this reason, when the model predicts the same performance for a range of techniques, the techniques with longer stroke times are probably more optimal for actual skaters.

The double support phase was not included in this model under the assumption that double support should be reduced to zero to get optimal performance, as many coaches will agree. However, recent investigations have proposed that a gradual shift of weight from one leg to the other during the double support phase results in less energy expenditure than if the weight was shifted more rapidly ^{(13)} . This model assumes an instantaneous transfer of weight from the push-off skate to the gliding skate.

In the model's present form, it applies to skating techniques where the foot remains parallel to the ice surface. This is because of the method in which the instantaneous power curves were measured. Some skating techniques incorporate larger ankle plantarflexion angles at push-off such as speed skating with the slap skate ^{(8)} or cross-country skiing skating techniques. This model could incorporate these techniques if the instantaneous power curve were measured using a similar lower limb motion^{(11)} .

Coaching applications. The skating model presented here predicts that there are many different techniques a skater may use to get the same resulting skating speed. This result is not surprising since different skating techniques have been used by World and Olympic Champions over the past 24 years. Although it appears that skating technique is irrelevant to performance, this is not true at the high skating speeds. As the speed to the athlete increases, a smaller and smaller set of possible techniques exists.

This skating model may be customized to individual skaters. The leg extension force-velocity relationship and aerobic power may be measured for a skater. These established tests can give the average and instantaneous power surfaces for this skater. It can then be determined if he/she needs to develop average power, instantaneous power, or alter technique to improve his/her skating speed. As stated earlier, it is probably better to increase instantaneous power through increases in strength rather than increases in the speed of leg extension.

Coaches can apply this skating model on a working basis since the technique parameters are easily measured: stroke time, glide time, and skating speed. Stroke time can be measured as the time taken to complete a number of straight-away strokes (t _{stroke} = time/number of strokes). Stroke and glide time can be determined by counting the number of video frames in each phase from a video recording. These values can then be plotted on the graph with the model surface predictions for a specific skater. From this, the constraining factor for skating speed can be determined. If a skater is not limited by average or instantaneous power constraints, an increase in speed may be possible through a change in technique (e.g., a decrease in stroke time or a change in leg extension velocity). As discussed in the limitations section, it is probably best to choose the longest stroke time for a given speed. In this case, an increase in speed can result only with a decrease in the glide time and/or stroke time.

Results from this study show that:

A number of skating techniques can be used to achieve the same steady-state speed.
As skating speed increases the range of techniques decreases.
Either average power or instantaneous power constraints can limit the steady-state skating speed.
Increasing average power raises the top skating speed with an accompanying reduction in the range of skating techniques.
Increasing instantaneous power or decreasing the height of the center of mass increases the range of possible skating techniques.
It is more advantageous to increase instantaneous power through increases in strength than increases in the speed of leg extension.
Full leg extension is not necessarily optimal to reach a top speed.
This model may be applied to all skating sports: speed skating, in-line skating, hockey, and cross-country skiing . In the future this model may be applied to starting techniques, cornering, double support phase, and changes in the height of the center of mass.

Figure 1-Frontal view of a speed skater showing the three phases of a skating stroke.
Figure 2-(A) Free body diagram of the skating model. The ice is the xy-plane with y directed in the desired direction of travel. (B) Geometrical representation of the velocity vectors in the plane of the ice that determine the push-off angle (γ). Figure 3-Instantaneous power (force × velocity) for the model was based on the force-velocity curves for a single leg extension motion(ankle relative to hip). The force-velocity curve was extracted from data of Vandervoort et al. : ^{(9)} (circles) and fit to a fifth-order polynomial function (solid line). Changes in the force-velocity and power-velocity curves are shown as a result in a 20% increase (1.2X) and decrease (0.8X) in the isometric force.

Figure 4-The function representing the horizontal leg length(: L _{h} ) as a function of time is described by three parameters: stroke time ( t _{stroke} ), glide time( t _{glide} ), and leg extension velocity( ˙L _{h} ).

Figure 5-Surface plots describe the maximum skating speed(m·s-1) for the skating model with no constraints (A), the addition of a maximum speed for leg extension constraint (B), an instantaneous power constraint (C), and an average power constraint (D). Surfaces are at the optimal speed of leg extension. Contour lines in the stroke-glide plane are isokinetic lines for 10, 11, and 12 m·s-1 of skating speed. Figure 6-A detail of the stroke-glide time plane of: Fig. 5D . Solid lines are contour lines for a constant skating speed. The optimal horizontal leg extension speeds necessary to reach these skating speeds are shown as dotted lines. The circles identify three different skating techniques that result in a skating speed of 12 m·s^{-1.} Figure 7-Resulting center of mass (circles) and skate blade (dashed line) paths in the plane of the ice for the three different skating techniques identified in : Fig. 6 . The line connecting the circle and the dashed line represents leg length and the direction of the push-off leg. Figure 8-Leg length and power-time curves for the three skating techniques in : Fig. 7 . Force is directly proportional to leg length in the top graph (1m = 945 N). Figure 9-(A) Surface plot of : Fig. 5D along with plots of the effects of a 20% increase (B) and a 20% reduction (C) in the power-velocity constraint as shown in Fig. 3 . Solid contour lines are for 10, 11, and 12 m·s^{-1} skating speed. (D) The 12 m·s^{-1} skating speed contour lines are from figures A, B, and C. Figure 10-Surface plots similar to : Fig. 5D describing of the effects of the height of the center of mass (A) 0.75 m, (B) 0.87 m, and (C) 0.96 m which corresponds to gliding knee angles of 90°, 110°, and 130°, respectively. Solid contour lines are for skating speeds of 10, 11, and 12 m·s^{-1} and dotted contour lines are for horizontal leg extension velocities of 1, 2, and 3 m·s^{-1} . (D) The 12 m·s^{-1} skating speed contour lines are from figures A, B, and C. REFERENCES
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