## Introduction

In human aquatic locomotion, low hydrodynamic resistance from the water (active drag; Da) is often considered to be a key variable. However, due to the complex unsteady fluid phenomena, it is currently impossible to directly measure Da. Therefore, researchers have established indirect methods to estimate Da, which often require special devices. For example, di Prampero et al.1 measured swimmers’ oxygen consumption while swimming under assisted and resisted conditions in a circular swimming channel. They plotted the oxygen consumption against the external load on a two-dimensional plot, established a linear regression line on the plot, and mathematically estimated Da by extrapolating the regression line to zero oxygen consumption. A similar mathematical method has also been developed in the last decades, such as the use of the residual thrust during swimming trials with different flow velocities while swimmers maintain their stroke frequency2,3,4. However, these methods require a swimming channel or flume, which is not accessible for many practitioners.

Another device that has been frequently used to assess Da is the Measuring Active Drag (MAD) system, which requires the swimmer to propel by pushing off submerged pads equipped with force transducers5. Although the MAD-system has been widely used6,7,8, this method also requires sets of large pushing pads that are often not accessible to practitioners. Furthermore, the MAD-system enables researchers to estimate Da only in the arm-only front crawl stroke, and investigating Da in the four whole-body competitive swimming strokes (butterfly, backstroke, breaststroke and front crawl) with this system is not possible.

Currently, one of the simplest ways to assess Da is the velocity perturbation method (VPM) proposed by Kolmogorov and Duplishcheva9, which only requires athletes to swim with their maximal effort with and without a known external resisted force via a non-elastic cord and an external load or object. Da can then be mathematically computed using the equation below.

$${D}_{a}=\frac{{F}_{add}\cdot {v}_{2}\cdot {v}_{1}^{2}}{{v}_{1}^{3}-{v}_{2}^{3}},$$
(1)

where Fadd is the additional resistive force due to the external load or object, and v1 and v2 are the velocities measured without and with the external load or object, respectively. Despite the simplicity, this method also has limitations, such as the assumption that the swimmer produces an equal amount of power during the free-swimming condition and the resisted swimming condition, which is questionable10. Furthermore, VPM assumes that Da increases with the square of the swimming velocity; however, when the swimmer actively propels forward, this is not always the case3,11. Given these simplified assumptions, the accuracy of the data obtained by VPM might be questionable. However, despite the accuracy being not guaranteed, the method would be practically useful if it has strong reliability as the method can then be used to monitor the short-term and long-term changes in Da.

Researchers have applied this method using a wide range of additional resistances12,13,14. However, it is currently unknown how much force should be assigned to swimmers to ensure reliable outcomes. As violating the bespoken equal-power assumption systematically affects the outcome10, assigning a load or object that causes a small additional resisted force is probably preferable to make the two conditions as similar as possible. However, assigning too small resistance might cause a large random error because when the assigned resisted force is too small, a slight swimming motion (such as kicking and consequent splashes) might cause random movements of the cord (and the pulled object). This likely affects the swimmer’s velocity or the external force to which the swimmer is exposed. However, the effect of choosing different resisted forces in Da calculation using VPM has not been assessed in the literature.

In summary, VPM is one of the most practical methods to quantify Da15,16 among many methods, but it is unknown how much additional force should be used for Da calculation with this method. Furthermore, the reliability of the method has not been reported in the literature. Therefore, the purposes of the present study were to assess the reliability of VPM with different resisted forces and investigate the difference in the calculated outcomes between distinct conditions. It was hypothesised that small resistance conditions would result in low reliability in Da calculation using VPM.

## Material and methods

### Participants

Eight males (17.0 ± 1.8 years age, 1.85 ± 0.05 m height, 73.0 ± 6.4 kg mass, 690.4 ± 67.8 FINA Points) and eight females (17.6 ± 1.2 years age, 1.71 ± 0.06 m height, 64.8 ± 7.2 kg mass, 689.6 ± 91.4 FINA Points) who specialised in front crawl were recruited.

### Procedures

A cross-sectional study design was used. The testing was performed in a 25 m indoor swimming pool (27 and 28 °C water and air temperature, respectively), where participants performed their individual warm-up procedure on land and in water as they usually do in competitions. Thereafter, swimmers were instructed to perform 5 × 25 m sprints with their maximum effort with five isotonic external loads (1, 2, 3, 4 and 5 kg for females and 1, 3, 5, 7 and 9 kg for males). Swimmers had at least 4 min of rest between each trial. The external load was assigned to the swimmer via a non-elastic cord using a portable robotic resistance device, 1080 Sprint (1080 Motion AB, Lidingö, Sweden), which also measured the swimming velocity and the tethered force (333 Hz sampling frequency). The device was positioned on the starting block resulting in the location of the origin of the cord exactly 1 m above the water surface. The cord was connected to the swimmer’s waist with an S11875BLTa swim belt (NZ Manufacturing, OH, United States), meaning that the measured velocity was the velocity of the abdomen region of the swimmer rather than their centre of mass. To investigate the reliability of Da assessment, the same procedure was repeated twice at the same time on different days with a 1–5 days interval.

Three stroke cycles around the mid-pool were extracted using the time-velocity curve from the obtained data. The mean swimming velocity (vadd) and tethered force (Fadd) during the three-cycle period at each condition were calculated. The horizontal component of the measured velocity and force were obtained using the equation below for the analysis using the trigonometric ratios17,18,19

$${var}_{H}=var\cdot cos\left[{sin}^{-1}\left(\frac{1.00}{{L}_{cord}}\right)\right],$$
(2)

where varH and var are respectively the horizontal component and measured value of the variable, 1.00 is the height of the origin of the cord from the water surface, and Lcord is the length of the cord at the time. The maximum velocity (vmax) at a free-swimming condition was estimated using the load-velocity profiling17,18, and Da at vmax was computed using Eq. (1) with vmax, vadd and Fadd as inputs for each external load condition.

### Statistical analyses

The day-to-day reliability was assessed using intra-class correlation (ICC) with a two-way random single-measure model, and the absolute and percentage (relative to the mean) typical errors were quantified20. ICC was interpreted as showing a meaningful agreement when the 95% confidence interval (95% CI) lower bound was larger than 0.7521. A two-way repeated-measures analysis of variance (ANOVA) with Bonferonni correction for multiple post hoc comparisons was employed to investigate systematic day and external load effects. The normality of data was assessed using the Shapiro–Wilk test and confirmed for all Da outcomes. ICC analysis and ANOVA were conducted using the Statistical Package for Social Sciences (SPSS) version 24.0 (IBM Corp, Armonk, NY, United States) with significance level of p = 0.05.

### Ethics approval and consent to participate

The study was approved by the local Ethical Committee and the National Data Protection Agency for Research in accordance with the Declaration of Helsinki. All participants (or a legal guardian for minors) were provided detailed verbal and written explanations of the purpose, procedure and risks related to the study and provided written informed consent.

## Results and discussion

Descriptive statistics and results from the reliability analyses are presented in Tables 1 and 2, respectively. In both sexes, trials with an external load smaller than 3 kg showed < 0.75 of the 95% CI lower bound. In male swimmers, this was also the case for 7 and 9 kg trials. The absolute and relative typical errors were similarly small at 3 and 5 kg trials in males and smallest at the 5 kg trial in female swimmers. In both sexes, a significant external load effect on Da outcome was observed (p < 0.001), while neither a significant day effect nor the interaction between the effects was found (Fig. 1). In male swimmers, Da measured with 7 and 9 kg external load were smaller than Da obtained from 1, 3 and 5 kg trials. In females, all Da values differed from those obtained in other trials, except for the comparison between 3 and 4 kg (p = 0.07).

A non-significant day effect for both sexes showed that there was no systematic bias in the day-to-day reliability assessment. The low ICC observed in 1 kg (both males and females) and 2 kg (females), corresponding to Fadd of 16–27 N (Table 1), suggests that low resistance should not be used to assess Da with VPM and supports the initial hypothesis. Nevertheless, using too large resistance is also not advisable as male swimmers showed lower reliability when Da was assessed with 7 and 9 kg (Fadd = 80–102 N) compared with 3 and 5 kg load trials (Fadd = 37–60 N). From the reliability perspective, researchers and practitioners should assess Da with Fadd of 37–60 N in both male and female swimmers.

The between-participants mean of Da varied from 46 to 84 N in males and from 39 to 60 N in females (at the velocity of about 1.82 m/s and 1.56 m/s, respectively), depending on the external load assigned to the swimmer. Furthermore, the larger the external load used in VPM, the lower Da, as illustrated in Fig. 1. Even though it is not possible to discuss the accuracy of the method as there is currently no method that directly measures Da, comparing the results from the current study with the literature is helpful to examine whether the obtained Da in the present study is reasonably aligned with previous studies. Kolmogorov and Duplishcheva9 analysed Da for whole-body front crawl swimming in both males and females and reported that, on average, males showed about 83 N at 1.78 m/s and females exhibited about 53 N at 1.60 m/s. These values were similar to the results of the present study (with 1 kg external load). Another study22 investigated Da in arms-only front crawl swimming using MAD-system and established Da equation for males (Da = 28.9·v2.12) and females (Da = 20.4·v2.28). These equations, in combination with the mean vmax in the present study, generate Da = 100.5 N for males and Da = 56.2 N for females. This Da for females is comparable to the result of the present study (when the resisted load was 1 kg). However, the Da calculation for males produced a slightly larger value than the present study, which might be due to the previous study having included water polo players in their samples22. However, a recent study23 assessed Da for both males and females using the assisted towing method and reported considerably larger Da than other studies, including the present one (mean Da = 89.0 N for females at v = 1.60 m/s, and mean Da = 140.5 N for males at v = 1.87 m/s).

The comparisons between the previous studies and the present study showed that the results of the present study were the closest to the literature when the external load for VPM was 1 kg. Furthermore, Da calculated in heavy load conditions (such as 5–9 kg loads for males and 3–5 kg loads for females) were close to, or even smaller than, passive drag results reported in the literature. For example, Zamparo et al.24 showed the passive drag of 70 N at 1.80 m/s and 47–60 N at 1.42–1.62 m/s for male and female competitive swimmers, respectively. These passive drag values are larger than Da found in the present study with 5–9 kg (males) and 3–5 kg loads (females), which indirectly suggests that the Da obtained at heavy load conditions were probably underestimated.

These examples imply that there is likely a trade-off between the accuracy and the reliability of VPM, i.e. the lighter the external load, the more accurate but less reliable the Da outcome. As indicated in the introduction, VPM is very sensitive to the violation of its assumption that the swimmer’s power output is equal between without and with external force/load conditions10. As measuring the power output during swimming is currently a very challenging task, it is unclear how much the external force/load affected the power output of swimmers. However, assuming that the power output is more similar when the two conditions (with and without external resistance) are closer, it is reasonable to consider that the power output in a semi-tethered condition is closer to free-swimming when assigning a smaller force/load.

Therefore, it is necessary to choose the external load which can produce reliable results while avoiding assigning a heavy load to the swimmer. For male swimmers, considering that 3 kg and 5 kg trials showed high reliability and there were no statistical differences in Da between these trials, Da assessment with Fadd of 37–60 N can be equally recommended. In females, among the three trials that exhibited high reliability (3–4–5 kg), 5 kg load produced a significantly lower Da than 3 kg and 4 kg trials, meaning that the underestimation of Da was probably more severe in the 5 kg trial than in the other two trials. Therefore, even though assessing Da with Fadd of 37–60 N could produce reliable results, limiting Fadd to 37–47 N might be preferable to minimise the underestimation of Da for females.

In conclusion, VPM can produce reliable results when assigning swimmers with a 3–5 kg load (37–60 N Fadd) for both male and female competitive swimmers, and assigning smaller or larger Fadd than the suggested range to swimmers can cause low measurement reliability. The calculated Da outcomes with this range of Fadd are likely underestimated. Nevertheless, due to strong reliability, VPM with Fadd of 37–60 N can be used to assess differences in Da between groups or to assess a long-term change in Da, as long as the same setting is utilised. However, it is advisable to limit Fadd to 37–47 N for females due to the underestimation of Da being more severe when assigning a larger Fadd, such as 60 N. The present study only focused on post-puberty age swimmers, but VPM has also often been used to assess Da in young swimmers16. Therefore, the reliability of this method for age group swimmers should be further investigated.