Introduction
Muscular power is widely considered to be a key determinant in athletic performance, particularly in sports that require large amounts of force generation in a short period of time (6,13,22 2,12,17 4,15,17,19
The criterion (or reference) method of measuring LBPP, used to validate estimates or as a “gold standard” measure, is based on performance in a vertical jump, where the subject is required to jump off a force platform (FP) (4,9,12,14 P ) is then calculated from the product of the force (F ) and velocity (v ): P = F × v (7,21
In order for the criterion method to achieve maximum validity and reliability, it is necessary to control and specify several key variables: vertical force range and resolution, force sampling frequency and resultant force integration frequency, method of integration, determination of body weight (BW), and determination of the initiation of the jump. Whereas some investigators have considered the effect of some of these key variables, for example, the effect of varying sampling frequency on several jump performance measurements including peak instantaneous power (11 16
Methods
Experimental Approach to the Problem
Force–time histories were collected for 15 professional rugby players, each performing 1 CMJ with arms akimbo (hands on hips). The force–time histories were then used to determine the influence of vertical force range, sampling frequency, determination of BW, identification of jump initiation, and method of numerical integration on LBPP of each subject's CG by systematically varying each 1 and monitoring the effect on LBPP. Body mass (BM) was determined from BW and taken to be BW·g −1 (kg) with g = acceleration due to gravity.
Subjects
The participants were 15 male international rugby union players (age = 25.8 ± 3.4 years, stature = 1.89 ± 0.09 m, mass = 102.5 ± 13.9 kg) who were familiar with CMJs, as they formed an element of their training and testing regime. The University Ethics Committee approved all experimental procedures, and all participants were volunteers and gave informed written consent.
Experimental Procedure
The participants undertook the tests at the beginning of a preseason training session. All participants underwent the same warm-up consisting of 5 minutes of light cycling followed by dynamic movements with emphasis placed on the musculature associated with jumping. After warming up, participants completed 1 maximal CMJ. All participants were given standardized instructions to stand on the FP (model number 92866AA, Kistler Instruments Ltd., Farnborough, United Kingdom) and jump when a signal lamp illuminated. The platform's vertical range was set to its maximum, 0–20 kN (i.e., 0–5 kN per corner transducer). The signal to jump and data collection were triggered when the participant remained stationary for more than 1 second, as observed by the operator. This ensuring that the first second of data collection (1 second of pretrigger) could be used to accurately determine BW, and as a reference to from which to determine a jump initiation time. The analog signal from the FP was sampled at a frequency of 1,000 Hz through a 16-bit analog to digital converter (ADC) using Kistler's Bioware (version 3.2.7.0; Kistler Instruments Ltd., Farnborough, United Kingdom). An initial sampling frequency of 1,000 Hz was chosen because it is the highest sampling frequency normally used to measure LBPP in a CMJ (2,8,14
Measurements
Calculation of Power
Power was determined from the unfiltered force–time history using the impulse momentum principle. The resultant VGRF was numerically integrated at the sample frequency of the force–time history and divided by BM to determine instantaneous velocity for time points that corresponded with the original force–time history. Instantaneous power was then taken as the product of the instantaneous velocity and VGRF at corresponding time points.
Analysis of Vertical Force Range and Body Weight
The VGRF measured by a FP consists of the arithmetic sum of 4 individual vertical force signals originating from the 4 transducers of the platform (1 transducer mounted in each corner) (Figure 1 ). Consequently, it was necessary to consider the force transmitted through each corner transducer as well as the combined, gross vertical force.
Figure 1: Vertical force–time history of a countermovement jump showing the 4 corner vertical force components. Fz = resultant vertical force Fz1–Fz4 are corner vertical force components or generally Fzc.
The vertical force–time histories for each participant's CMJ were recorded and the maximum unfiltered values of the gross force and the corner transducers' components of the gross force were determined by inspection for each subject. Body weight was taken to be the mean value of the VGRF during the first second of data collection. Body weights for the resampled data (500 and 100 Hz) were determined using the same procedure.
Analysis of the Use of Different Sampling Frequencies
To investigate the effect of sampling frequency on the determination of LBPP from performance in a CMJ, force–time histories recorded for the vertical force range analysis were used. These 15 force–time histories were then resampled using Bioware's resampling function at 500 and 100 Hz. The LBPP was determined for each jump at the 3 unfiltered sampling frequencies using Simpson's rule at the corresponding frequency to determine the velocity–time data. Body weight was defined as mean VGRF during 1 second of stance. The integration start time, t i SD . The instant t i
Analysis of Method of Numerical Integration
To investigate the effect of the method of integration on LBPP from a CMJ, the 15 unfiltered force–time histories sampled at 1,000 Hz were used. The start point for the integration was taken as t s
Analysis of Identification of the Initiation of a Countermovement Jump
During the stance phase of a CMJ, the VGRF will vary constantly because of slight movement of the subject and noise in the instrumentation, both internal and external. Therefore, it was necessary to define a threshold value of VGRF during the stance phase, beyond which the jump was defined as having been initiated. If the threshold was set too low, a mistriggering could occur, if set too high the defined initiation of the jump would be too late. Thus the initiation time, t s , was defined as the instant, after the signal to jump has been given, that the VGRF exceeded the mean plus or minus 5 SD of the BW as measured in the stance phase. This threshold would reduce the probability of mistriggering in the stance phase to p < 0.0000006 (i.e., 1 mistrigger every 1,744 jumps on average, for a stance phase of 1-second sampled at 1,000 Hz).
However, there is similarly a high probability that t s , as described above, while identifying an instant that has a very low probability of being part of the stance phase, will be in the jump phase of the CMJ. Therefore, to investigate the effect of varying t s , and consequently its suitability as a start point, t s was identified for the 15 CMJ's unfiltered force–time histories. The LBPP was determined using an integration starting point equal to, t s = 100 ms, for each subject. The point, t s = 100 ms was chosen because it was clearly in the stationary phase of the jump. Values of LBPP were then determined using integration starting points of t s = 90 ms through t s , to, t s + 30 ms at intervals of 10 ms for each participant. Consequently any within-participant variation in LBPP could be attributed to the integration starting point.
Statistical Analyses
All statistical analysis was performed using SPSS 19 (SPSS Inc., Chicago, IL, USA).
Analysis of a Vertical Force Range and Body Weight
Mean and SD were determined for the vertical force–time histories' maximum values of the gross force and the corner transducers' components of the gross force. Mean and SD were determined for values of BW at sampling frequencies of 100, 500, and 1,000 Hz.
Analysis of the Use of Different Sampling Frequencies
To investigate the difference in LBPP for different sampling frequencies, limits of agreement and mean systematic bias of power output produced by the 100 and 500 Hz sampling frequencies, in relation to the power outputs of the1,000 Hz sampling frequency were assessed using Bland and Altman plots (Bland and Altman (3
Analysis of Method of Numerical Integration
Limits of agreement and mean bias of LBPP produced by the 2 methods of numerical integration were assessed using Bland and Altman plots (3 3
Analysis of Identification of the Initiation of a Countermovement Jump
The LBPP determined with an integration start point of t s t s t s t s t s SD were determined for the NPDs. Thus, the mean ± 3 SD was taken to represent the difference value of LBPP, plus associated uncertainty, for any integration start point after t s p ≤ 0.003) as compared with PRV. To investigate the rate of change in the uncertainty of the difference value of LBPP, the first derivative of the SD of time series NPD (dSD/dt) was numerically determined giving the rate of change of uncertainty in the NPD. If an integration start point was taken after the jump had started, there would clearly be an increase in the NPD because the integration process would not have included part of the jump. Equally, as the integration start point moves further into the jump, the rate of change of uncertainty would increase rapidly. Hence, it is reasonable to expect the first derivative of the time series of SD of NPD to identify when the jump had started. It was necessary to analyze the SD of the NPD because the mean value NPD has the potential to show very little change at the beginning of a CMJ, as approximately half of the participants executing a CMJ start by first moving up that is, a countermovement before the CMJ, and half start by immediately dipping. Thus at the start of a CMJ, it is likely that an increase in positive lower-body power would be mirrored by a corresponding increase in negative lower-body power in approximately half of participants. However dSD/dt is not sign-dependent and hence will identify a positive or a negative change in NPD.
Results
Selection of Vertical Force Range
The maximum and minimum total vertical force, maximum and minimum vertical component forces, and mean and SD of all CMJs are displayed in Table 1 . There was no difference in BW for the 3 different sample frequencies (to 1 d.p. ± 1 digit) (Table 2 ).
Table 1: Vertical forces produced during a countermovement jump.*
Table 2: Mean and SD of body weights determined at different sampling frequencies during 1 second of quiet standing (n = 15).
Selection of Sampling Frequency
The results of the comparison of sampling frequencies can be seen in Figures 2 and 3 . The sampling frequency of 100 Hz, when compared with 1,000 Hz, produced a mean difference of 87 W and limits of agreement (mean ± 1.96 × SD ) of 144 and 31 W. The sampling frequency of 500 Hz, when compared with 1,000 Hz, produced a mean difference of 8 W with limits of agreement of 24 and −11 W.
Figure 2: Bland and Altman plot comparing peak vertical mechanical power outputs of countermovement jumps using sampling frequencies of 100 and 1,000 Hz.
Figure 3: Bland and Altman plot comparing peak vertical mechanical power outputs of countermovement jumps using sampling frequencies of 500 and 1,000 Hz.
Method of Numerical Integration
Figure 5 shows the results of the integration methods comparison, the differences between LBPPs calculated using Simpson's rule, and the trapezoidal rule are plotted on the y axis, and the mean (mean between Simpson's rule value and trapezoidal rule value) is plotted on the x axis. The analysis resulted in a mean of the difference of 13 W and limits of agreement (mean ± 1.96 × SD ) of 6 and 19 W.
Identification of the Initiation of a Countermovement Jump
Figure 4 shows the rate of change of NPD. The graph shows a slight negative gradient between t s t s t s t s
Figure 4: Graph of the rate of change of the SD of NPD.
Discussion
Body weight was determined by taking the mean GRF value, as measured by the FP, for 1 second of the stance phase immediately before the signal to jump being given. In this study, sampling frequency (100, 500, and 1,000 Hz) had no effect on the determination of BW (to 1 d.p. ± 1 digit) (Table 2 ).
The maximum resultant VGRF recorded in all trials was 2,950 N in a jump by a subject with a BW of 1,166 N (Table 1 ). This is consistent with Kibele (14 14 SD , 988 N + (145 N × 3) = 1,423 N. The corresponding resultant maximum vertical force range for the FP would then be 1,423 N × 4 = 5,692 N, that is 5.7 times BW. Setting a FP's range to this value, or higher, would reduce the probability of it being exceeded to p ≤ 0.003.
Generally, when sampling a signal to represent it elsewhere, the higher the sampling frequency the greater the fidelity of the representation of the original signal. Specifically Nyquist's sampling theorem (18 Figure 3 ). It is also highly likely that 500 Hz would also achieve this precision. The worst case scenario for the precision of a sampling frequency of 500 Hz would be that the mean difference and limits of agreement of 1,000 Hz sampling frequency compared with 2,000 Hz sampling frequency were the same as for 500 Hz compared with 1,000 Hz giving a mean difference between 500 and 2,000 Hz of +0.2%, with an upper limit of agreement of +1.0% and a lower limit of agreement of −0.4%. However, as a sampling frequency of 1,000 Hz produces more accurate results than a sampling frequency of 500 Hz, and given the convenience of sampling in time intervals of milliseconds, a sampling frequency of 1,000 Hz was chosen as the preferred frequency for the determination of power output by the criterion FP method in this study.
There appears to be no agreed method in the literature of determining the instant when a CMJ has been initiated. Some researchers use a relative threshold such as 5% BW during the stance phase and define the jump initiation as the instant that the VGRF falls below that threshold (5,11,20 8 14,9 1,10 SD of the NPD shows a slight negative gradient between times t s t s Figure 4 ), indicating that there is a small constant variation in the SD of the NPD corresponding to instrument drift and small unbalanced impulses. Between t s t s SD of the NPD starts to increase rapidly, indicating that starting integration after this point would miss some of the beginning of the CMJ. Therefore, the initiation of the CMJ must be before t s t s
Figure 5 shows that the maximum error, ΔP , in the determination of peak power between Simpson's rule and the trapezoidal rule would be, ΔP ≤ 0.13% (confidence interval = 95%). However, because it is unclear which of these methods of integration gives the more correct result, the best estimate of the correct value of peak power can be taken as the mean of the 2 results. It can therefore be concluded that if a maximum error of 0.13% in the determination of peak power is acceptable, then the 2 methods of numerical integration, Simpson's rule, and the trapezoidal rule can be used interchangeably. Based on the above conclusions, Table 3 , the criterion method for determining peak vertical mechanical power output in a CMJ was defined.
Figure 5: Bland and Altman plot comparing peak vertical mechanical power outputs of countermovement jumps using Simpson's rule and the trapezoidal rule.
Table 3: Criterion method specification for the measurement of peak mechanical power in a CMJ by the criterion force platform method.* †
This study did not investigate the effect of ADC resolution on LBPP because generally it is not adjustable in commercial systems. However, it will clearly affect the determination of LBPP. The 2 most common resolutions currently in use are 12 bit (e.g., AMTI) and 16 bit (e.g., Kistler and Bertec). A 12-bit ADC is capable of representing an analog signal as 4,096 that is 212 , discrete steps, thus theoretically representing an analog force signal with a range of 0–20 kN in discrete steps of 4.9 N, whereas a 16-bit ADC would theoretically represent the same signal with a resolution of 0.3 N. However, in practice, the resolution of a system is dependent on other factors in addition to ADC bits including system noise and actual force range as opposed to stated maximum range. For example, the Kistler 9281E FP has a maximum VGRF range of 20 kN, however the absolute range of the platform is 30 kN because it is also capable of measuring a vertical “pull” of 10 kN. Clearly, this would not be the case with a portable system. It is reasonable to expect a 16-bit ADC to better represent an analog signal than a 12-bit ADC, therefore, whenever possible, a 16-bit ADC should be used in preference to 12-bit ADC, particularly when high force ranges are used. The absolute range of the platform should also be clearly stated.
Practical Applications
Testing peak mechanical power in a CMJ using a FP is widely used as a method of monitoring and assessing elite athletes. The criterion method described in this study, Table 3 , is valid and reliable and could be adopted as the basis of a standard method of testing LBPP in a CMJ when using a FP.
Further Research
Peak vertical mechanical power output in a CMJ as measured by a FP is one of a number of performance variables (probably the most common) that are used in the assessment of elite athletes. Future work should focus on the influence of ADC resolution on LBPP and also developing criterion methods for the determination of other commonly used variables derived from force–time histories of a CMJ, for example, rate of force development (both eccentric and concentric), velocity, and jump spring stiffness.
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