**Relationships between Body Build and Knee Joint Flexor and Extensor Torque of Polish First-Division Soccer Players**

#### **Jadwiga Pietraszewska 1, Artur Struzik 2,\*, Anna Burdukiewicz 1, Aleksandra Stacho ´n <sup>1</sup> and Bogdan Pietraszewski <sup>3</sup>**


Received: 18 December 2019; Accepted: 20 January 2020; Published: 22 January 2020

**Featured Application: Body height, body mass and thigh and calf girths are the anthropometric variables that best describe the values of the torque of the knee flexors and extensors that can be achieved by soccer players. Based on these anthropometric variables, the coach can initially assess players' strength abilities, which can help plan individual training programmes.**

**Abstract:** The aim of the study is to identify the relationships between anthropometric variables and knee joint extensor peak torque, knee joint flexor peak torque, and conventional hamstring-to-quadriceps ratio in Polish first-division soccer players. The study examined 37 soccer players aged 19 to 30 years (body mass: 76.8 ± 7.2 kg, body height 1.82 ± 0.06 m). Muscle torques of the knee joint were measured under isometric conditions using a Biodex 4 Pro dynamometer. The anthropometric variables such as body part lengths, breadths, and girths and skinfold thickness were measured. The strongest relationships of knee joint extensors were observed with body mass and variables describing skeleton size and lower-limb muscles. Regarding knee flexor torque, a significant relationship was found only with body mass. However, no significant relationships were observed between the conventional hamstring-to-quadriceps ratio and the anthropometric variables studied. The regression analysis results identified body height, body mass, and thigh and calf girth as the features most associated with knee joint torque. However, anthropometric measurements do not provide full information about the torque proportions of antagonist muscle groups, which is very important for injury prevention. Therefore, measurements using special biomechanical equipment are also necessary for the comprehensive analyses and control of the effects of sports training.

**Keywords:** anthropometric variables; hamstring-to-quadriceps ratio; H/Q ratio; injury prevention; isometric; lower limb; measurement acquisitions and techniques; strength and conditioning; sports training; strength abilities

#### **1. Introduction**

Measurements of muscle torque taken in isometric and isokinetic conditions have often been used for sports training [1–5]. Coaches use biomechanical measurement tools to analyse the sports skill level of athletes, improve movement techniques, or adjust tactics against the opponent [6,7].

Biomechanical analysis provides a fast and reliable evaluation of torque values, helps detect muscle strength imbalance, and screens for lower extremity injury [8,9]. Numerous studies have focused on the evaluation of the value of muscle torque in relation to lower limbs, which are particularly involved during sports movements. Appropriate force and proportion between antagonistic muscle groups are essential for optimal or maximum performance of various movements, which is also true in team sport games [10–14]. The dependence of force on the transverse cross-sectional area of muscles is known, but it should be noted that force is also affected by other factors related to the efficiency of the nervous system (that controls muscle function) and the properties of the muscles themselves [15].

Soccer is a sport in which the requirements for effective play are multi-factorial. The intervals and repetitive nature of the efforts during the match mean that the competitors should demonstrate adequate speed, endurance, strength, and coordination levels. In general, the players do not reach the highest levels of all these motor abilities. However, they must demonstrate a sufficiently high level of preparation in all aspects of motor abilities, with the dominance of some of them (depending on their playing position). Interindividual differences can also be observed in the anthropometric characteristics of soccer players [16]. In soccer, the lower limbs are particularly involved when kicking the ball, striking at a goal, and running at different velocities and with directional changes [17–20]. Soccer training is aimed, among other things, at increasing muscle mass and, consequently, muscle force. The adequate level of force of the knee extensors and flexors allows the player to achieve stronger strikes, jump higher, and sprint faster over short distances [21–23].

In addition to the absolute values of the generated torque, the correct ratio of the torque of extensors to that of flexors is also important. The commonly used conventional hamstring-to-quadriceps ratio (H/Q ratio) represents the ratio of concentric hamstring peak torque during lower limb flexion to concentric quadriceps peak torque during lower limb extension [24]. The functional hamstring-to-quadriceps ratio, in contrast, is a measure of the isokinetic eccentric peak torque of the hamstrings relative to the isokinetic concentric peak torque of the quadriceps during lower limb extension at equivalent angular velocities [25]. An adequately high value of these indexes is likely to effectively prevent injuries of the lower limbs (including those that occur without contact) due to effective eccentric activity to slow down or stop the movement [26].

Measurements of muscle torque under isometric conditions allow for the determination of the maximum values of force generated by a given muscle group to evaluate the potential of the player, the training programme, and muscle imbalance. It should also be noted that the amount of force generated depends on many factors, including physiological and biochemical factors. The aspect of morphological determinants is also important. Previous studies have investigated the relationships of muscle torque with body mass, body mass composition, and somatotype [27–31]. However, the results of such studies have been inconclusive. Norsuriani and Ooi [31] demonstrated positive relationships between muscle torque in the knee joint and body mass, fat-free mass, and body fat. Similar results were presented by Pietraszewski et al. [27]. Lewandowska et al. [28] emphasized positive relationships of muscle torque in the lower limb joints with characteristics of muscularity and skeletal mass (mesomorphy), negative relationships with body slenderness (ectomorphy) and lack of relationships with the fat component (endomorphy). Kim et al. [29] found that men with higher skeletal muscle mass are characterized by higher isokinetic muscle torque values in the lower limb joints. Furthermore, to our knowledge, no previous studies have examined the relationships between the conventional H/Q ratio and anthropometric variables. It should be pointed out that anthropometric measurements are the basic method used to assess the level of development of many morphological features. They can be performed anytime and anywhere, using mobile equipment. Measurements of body girths and skinfolds provide additional indirect information about changes in tissue components, which are very important in the context of functional capabilities of a player. Understanding the relationships between muscle torque and the anthropometric variables would allow coaches to easily make an initial assessment of the players' strength abilities and, at the same time, control the training effects.

Muscle torque is the product of the force *F* and the length of the lever arm *r*. The lever arm *r* is understood as the distance from the axis of rotation of the lever (joint) along a line perpendicular to the direction of the force *F*. Therefore, it can be expected that the torque of the flexors and extensors of the knee joint are related to the characteristics of muscle development (girths of the body parts) and the length of the individual lower limb parts. These features are related to the overall body size but may also show interpersonal differentiation due to the modifying effect of various environmental factors, including sports training. Due to the specificity of the training, the knowledge gained from the research of players practising various sports cannot be used uncritically. Substantial values of anthropometric variables (e.g., girths of body parts) may result from both muscularity and body fat. However, due to the characteristics of the research group (soccer players) resulting from the specificity of the motor activities performed by such athletes and the usually low body fat content in lower limbs, the relationships should be expected between muscle torque and variables indirectly describing muscularity of the lower limbs. It is important to examine morphofunctional relations in a specific sport.

The aim of the study is to identify the relationships between anthropometric variables and the values of knee joint extensor peak torque, knee joint flexor peak torque, and conventional hamstring-to-quadriceps ratio in a group of Polish first-division soccer players. Furthermore, the authors intended to identify the anthropometric variables that allow for the best possible estimation of the abovementioned biomechanical variables, which would allow coaches to monitor more frequently the effect of training loads used at different stages of the annual training cycle. Knowledge of the abovementioned biomechanical variables is also important during the choice of training loads to reduce the injury risk of athletes.

#### **2. Materials and Methods**

The study examined 37 soccer players aged 19 to 30 years. They were players playing in the Polish Ekstraklasa league. The study included only the players who had actively participated in training and league matches since the beginning of the season and had not been injured. They were previously qualified for the examinations by a sports physician. Their mean training experience was 14 years (8 to 20 years). The mean body mass (±SD) of the subjects was 76.8 ± 7.2 kg, and the body height was 1.82 ± 0.06 m. The tests were conducted in the morning, before the training sessions. On the day before the tests, the players did not play any league matches.

The study was carried out in the Biomechanical Analysis Laboratory and Scientific Research Laboratory of the Faculty of Sport (both with PN-EN ISO 9001:2009 certification) at the University School of Physical Education in Wrocław, Poland. All subjects gave their informed consent for inclusion before they participated in the study. The study was conducted in accordance with the Declaration of Helsinki, and the protocol was approved by the Senate's Research Bioethics Commission of the University School of Physical Education in Wrocław, Poland. Measurements were performed according to International Standards for Anthropometric Assessment (ISAK) [32].

The following anthropometric variables were measured: sitting height (B-vs), lower extremity length (B-tro), shank length (B-ti), biacromial diameter (a-a), biiliocristal diameter (ic-ic), humerus breadth (cl-cm), femur breath (epl-epm), chest girth, waist girth, hip girth, relaxed and flexed biceps girth, thigh girth, and calf girth. Skinfold thickness was measured at the subscapular, abdominal, suprailiac, triceps, and calf locations to assess body fat. Body height, lengths, and breadths were measured to the nearest 0.1 cm with the use of the GPM Anthropological Instruments (Siber Hegner Machinery Ltd., Zürich, Switzerland). Skinfold thickness was measured with a Tanner/Whitehouse skinfold calliper (Holtain LTD, Crosswell, Crymych, Pembs, UK) with a 0.2 mm graduation. Body mass was measured with an electronic weighing scale with an accuracy of 0.1 kg. Body mass index (BMI) was also calculated.

The peak muscle torque under isometric conditions was measured to assess the strength of the lower limbs. Peak muscle torque for extensors and flexors of the knee joint was measured at 75◦ and 30◦, respectively; 0◦ at the knee joint was considered to be a full extension. The measurements were performed separately for the right and left lower limbs. The results of measurements of the dominant limb were used for analyses. Biodex System 4 Pro (Biodex Medical Systems, Inc., Shirley, NY, USA) was used for the torque measurements. The conventional hamstring-to-quadriceps ratio (H/Q ratio) was calculated using the following equation:

$$\text{H/Q ratio} = (T\_f / T\_e) \cdot 100\% \,\tag{1}$$

where *Tf* denotes the concentric peak torque value of knee joint flexors of the dominant lower limb and *Te* is the concentric peak torque value of extensors of the same joint and limb. The conventional H/Q ratio is possible to calculate during the measurement of peak torque only under isometric conditions [33]. Furthermore, the muscle torques of extensors and flexors were added together, and the muscle torque for the knee joint relative to body mass was calculated:

$$(T\_{\mathcal{C}} + T\_{\mathcal{f}}) / \text{BM}.\tag{2}$$

To supplement the information about the general strength abilities of the soccer players studied, back and handgrip strengths were measured using TAKEI dynamometers (Takei Scientific Instruments Co., Ltd., Niigata City, Japan).

Cluster analysis (*k*-means) was used in the statistical analysis to distinguish two groups significantly differing in the magnitude of muscle torque. The intergroup differentiation of anthropometric variables in these clusters was evaluated by means of Student's *t*-test for independent samples. Pearson's *r* correlation coefficient was used to evaluate the relationships of anthropometric variables with muscle torque.

#### **3. Results**

The soccer players studied were characterized by small and moderate intragroup variability in the anthropometric variables (Table 1). The mass-to-height ratios were correct and were confirmed by the BMI values in the entire group (23.3 <sup>±</sup> 1.5 kg/m2).

The obtained values of muscle torque showed a clear interindividual differentiation. To create relatively homogeneous groups in terms of the values of muscle torque generated in the knee extensors and flexors, the *k*-mean cluster analysis was used to identify two subgroups significantly differing in the magnitude of these variables: Group 1—weaker group and Group 2—stronger group. With the adopted criterion of division, the groups differed significantly in terms of the generated muscle torque and in terms of the sum of muscle torque relative to body mass (Table 2). All of the above values were significantly higher in Group 2 (stronger). Higher values were also found in most of the anthropometric variables in this group of players. However, significant intergroup differences were recorded only for body mass, BMI; chest, thigh and calf girths; and humerus and femur breadths, which are a measure of skeleton size. The soccer players from Group 2 also showed significantly higher values in the remaining strength tests, although these variables were not included in the grouping variables. Both back strength and handgrip strength were greater in the athletes from the second cluster. The mean conventional H/Q ratios were 47.0% (Group 1) and 45.6% (Group 2).

The values of correlation coefficients between muscle torque of the knee extensors and flexors and anthropometric variables were quite low, yet statistically significant in some cases (Table 3). The strongest relationships were observed for knee joint extensors with body mass, BMI, and variables describing skeleton size and lower limb muscles. In the case of the torque of knee flexors, a significant relationship was found only with body mass and BMI. However, no significant relationships were observed between the conventional H/Q ratio and the anthropometric variables studied.


**Table 1.** Statistical characterization of the variables analysed for the entire group of soccer players studied (*n* = 37).

**Table 2.** Statistical characteristics of the variables studied in separate clusters.


<sup>\*—</sup>significant differences between Groups I and II group at the level of *p* < 0.05; *p*—probability of type I error; *t*—*t*-test value; BMI—body mass index; b-vs—sitting height; b-tro—lower extremity length; b-ti—shank length; a-a—biacromial diameter; ic-ic—biiliocristal diameter; cl-cm—humerus breadth; epl-epm—femur breath; *Te*—peak torque value of knee joint extensors; *Tf*—peak torque value of knee joint flexors; H/Q ratio—conventional hamstrings-to-quadriceps ratio; (*Te* + *Tf*) / BM—sum of knee muscle torques relative to body mass.

SD—standard deviation; BMI—body mass index; b-vs—sitting height; b-tro—lower extremity length; b-ti—shank length; a-a—biacromial diameter; ic-ic—biiliocristal diameter; cl-cm—humerus breadth; epl-epm—femur breath; *Te*—peak torque value of knee joint extensors; *Tf*—peak torque value of knee joint flexors; H/Q ratio—conventional hamstrings-to-quadriceps ratio; (*Te* + *Tf*) / BM—sum of knee muscle torque relative to body mass.


**Table 3.** The correlation coefficients between the muscle torque of extensors (*Te*) and flexors (*Tf*) of the knee joint and anthropometric variables for the entire group of soccer players (*n* = 37).

\*—statistically significant at *p* < 0.05; BMI—body mass index; b-vs—sitting height; b-tro—lower extremity length; b-ti—shank length; a-a—biacromial diameter; ic-ic—biiliocristal diameter; cl-cm—humerus breadth; epl-epm—femur breath.

Multiple regression analysis was performed, taking into account body height and anthropometric variables that were most significantly correlated with the examined muscle torque. A significant effect of anthropometric variables on the muscle torque measured in the knee joint was demonstrated. A regression model with four anthropometric variables estimating the biomechanical values was statistically significant, and all predictors explained 38% of the torque variation in the knee extensors and 25% of the torque variation in the knee flexors. Three of the four predictors in the model had a significant effect on the torque of the knee extensors and flexors. The results of the regression analysis are presented in Tables 4 and 5. The estimation error for the extensor Equation (3) was 72.3, whereas for the flexor Equation (4), this value was 42.5.

*Te* = 1943.845 − (10.052 · body height) − (25.424 · thigh girth) + (13.383 · calf girth) + (16.652 · body mass). (3)

*Tf* = 1019.404 − (5204 · body height) − (11,878 · thigh girth) + (5715 · calf girth) + (7756 · body mass). (4)

**Table 4.** Results of regression analysis for knee joint extensor torque.


\*—statistically significant at *p* < 0.05.

**Table 5.** Results of regression analysis for knee joint flexor torque.


\*—statistically significant at *p* < 0.05.

#### **4. Discussion**

The body build of the soccer players studied showed a typical profile of basic anthropometric variables typical for athletes participating in this sport. Interpersonal differences resulted from the different playing positions [34,35]. The mean body height of the soccer players studied (1.82 ± 0.06 m) was similar to the mean body height of the players of most of the best teams in the world and slightly exceeds the mean body height of young men in Poland (1.79 ± 0.07 m) [36]. Therefore, the results obtained in the study are consistent with the information provided in the literature on the body morphology of athletes. Norton and Olds [32] indicated that the mean body height of soccer players (1.79 ± 0.06 m) and its variations are very close to those of the general population. This finding was also confirmed by the research presented by Reilly and Doran [17]. However, these authors pointed to differences depending on sports skill level and nationality. The extreme values in their ranking were obtained from the players from Hong Kong (1.73 ± 0.06 m) and Norway (1.81 ± 0.05 m). The BMI of soccer players in the present study also showed values typical of soccer players [17]. The athletes tested were characterized by good musculature of all body parts, as evidenced by the values of chest and lower and upper limb girths. These features are similar to those of players on other soccer teams [34,35]. The results of our research indicated particularly muscular thighs, which enable players to generate high muscle torque. This ability is also very important when jumping, kicking, rotating, and changing the velocity of movement [14,17,19]. The harmonious development of limb and torso muscles contributes to maintaining player balance on a slippery field and allows for easier control of the ball [17]. Particularly valuable research, whose results can help athletes achieve top-level performance, was carried out among the players of the top teams. Therefore, it seems very valuable that we had an opportunity to conduct research on the Polish first-division soccer players.

Analysis of the muscle torque of knee extensors and flexors shows that human muscles are shaped to a large extent by the gravitational field. This finding is reflected in the significant muscle torque of the knee joint extensors (416.8 ± 86.4 Nm), which act against the gravitational force. The torque values of the knee flexors are approximately two times lower (191.7 ± 46.3 Nm). The mean values of isometric muscle torque in the knee joint obtained for the players in our study are much higher than the mean values obtained in Polish striker soccer players (315.3 ± 53.3 Nm for extensors and 167.7 ± 26.0 Nm for flexors) [37] and higher than those in lower-league soccer players (328.0 ± 69.7 Nm for extensors) [38]. The recorded large values of isometric muscle torque measured in our study may result from substantial training loads to which players were exposed. They trained four times a week in two sessions. In the morning, the athletes had a soccer training session (about 2.5 hours). In the afternoon, they exercised in the gym for about 1.5 hours. The exercises were selected individually. The high sports skill level of the players studied is also of great importance to the results obtained. Previous studies [1,39] indicated differences in the magnitude of muscle torque depending on the sports skill level. Measurements of muscle torque under isokinetic conditions are now also popular. However, these measurements have limitations [33,40]. Furthermore, the aim of the study was to determine the maximum strength abilities of the players, rather than the strength at a certain angular velocity of movement in the knee joint.

The relationship between the muscle torque of extensors and flexors of the knee joint is important from the standpoint of maintaining knee stability during some movement activities. For example, damage to the hamstrings can occur during the kicking of the ball with the lower limb. This damage may occur when, during knee extension, a group of antagonistic muscles does not ensure the effective eccentric deceleration of the movement [8,41,42]. It is assumed that conventional H/Q ratio values exceeding 60% can effectively prevent injuries and damage to the anterior cruciate ligament (ACL) and hamstring strains [43–48]. According to Kim and Hong [44], soccer players with a conventional H/Q ratio of more than 60% are less likely to suffer non-contact injuries to their lower limbs. In this study, the conventional H/Q ratio was 46.1% for all the soccer players studied, which is lower than the desired value and indicates unfavourable proportions of muscle torque in these muscle groups. This result suggests that coaches should pay more attention to the improvement of the strength level of the knee

flexors during training. No injuries were reported the last time among the players studied. However, it can be presumed that this was influenced by motor control, which means mutual adjustment of the musculoskeletal and nervous systems [49]. In order to control movement, the central nervous system must integrate multiple sensory information (both external and proprioceptive) and trigger the signals necessary for muscle recruitment to achieve the intended goal. The performance of the movement consists in the planning of the motor strategy and its economic and flawless execution. Disturbed motor control has a negative effect on the quality of movement and increases the risk of injury. It should be expected that in elite players, sport training had led to the improved motor control. Disproportionate torque of agonist and antagonist muscle groups has been observed in male soccer players [27,37,48]. However, there are groups of soccer players for which the value of the conventional H/Q ratio remains within the recommended standard [45]. Cometti et al. [39] demonstrated that sports skill level in soccer players increases with higher H/Q ratios.

The lack of significant relationships between the conventional H/Q ratio and anthropometric variables indicates that low values of the conventional H/Q ratio in the group of soccer players (i.e., not entirely correct proportion of torque of antagonistic muscle groups) cannot be observed based on the variables describing body structure. Therefore, muscle imbalance cannot be diagnosed without special biomechanical measurements because the body build will not suggest such an abnormality. It should also be noted that high H/Q ratios may exist in the presence of relative weakness in both the hamstring and quadriceps muscle groups [50], which can also explain the lack of the above relationships. Another limitation may be that the peak values of extensors and flexors occur at different knee flexion angles [26,33].

Dividing the soccer players studied into two clusters allowed for the assessment of body build differences between the players, who significantly differed in terms of muscle torque generated within the knee joint. The athletes from both clusters did not show any differences in body height or its components, which is attributable to the genetic determinants of this feature [51]. Furthermore, the differences observed in body mass and girths that describe the size of the skeleton and muscles indicate relationships between these features and strength abilities. This is also confirmed by significant correlation coefficients for some anthropometric features (body mass, humerus breadth, femur breadth, calf girth, subscapular skinfold, suprailiac skinfold, and calf skinfold) and isometric muscle torque of the knee joint extensors and flexors. As in other studies, positive relationships were obtained between muscle torque and body mass, breadths and girths that describe the size of the skeleton and muscles [27–31]. However, contrary to previous reports, positive relationships between muscle torque and skinfolds were also found [28].

The regression analysis was used to identify the variables that can be indirectly used to estimate torque values for knee joint extensors and flexors generated by the player. The results of this analysis identify the body height, body mass, and thigh and calf girth as the features that best describe these variables. Therefore, based on these anthropometric variables, the coach may initially assess the strength abilities of soccer players, and these assessments can lead to the adjustment of training programmes to the individual predispositions of the players.

The limitations of our study are incomplete information about the applied training loads (resulting, e.g., from club changes of the players studied), the sex of the players (slightly different results can be expected in female groups, which are characterized by higher body fat), and the type of sport being practised (it is not advisable to directly refer to the results of people practising other sports, e.g., individual sports). On the other hand, all of the players studied belonged to the national soccer elite, which allows for the assumption that their training was carried out professionally, translating into a high sports skill level.

#### **5. Conclusions**

The results allow for the conclusion that the muscle torque of the knee joint extensors and flexors in soccer players is significantly related to some anthropometric variables. Due to the less complicated nature of anthropometric apparatuses and the high mobility of this type of equipment, the prediction of muscle torque in the knee joint based on anthropometry seems to be useful for screening soccer players for preliminary estimation of the abovementioned biomechanical variables. However, anthropometric methods do not provide full information about the mutual relationships between the muscle torque of the knee joint extensors and flexors, which is very important from the standpoint of injury prevention. For this reason, measurements using special biomechanical equipment are also necessary for the comprehensive analyses and control of the effects of sports training.

**Author Contributions:** Conceptualization, J.P. and B.P.; methodology, J.P. and B.P.; software, J.P. and B.P.; validation, J.P., A.S. (Artur Struzik), A.B., A.S (Aleksandra Stacho ´n). and B.P.; formal analysis, J.P., A.S. (Artur Struzik) and B.P.; investigation, J.P., A.B., A.S. (Aleksandra Stacho ´n) and B.P.; resources, J.P., A.B., A.S. (Aleksandra Stacho ´n) and B.P.; data curation, J.P. and A.S. (Artur Struzik); writing—original draft preparation, J.P., A.S. (Artur Struzik) and B.P.; writing—review and editing, J.P., A.S. (Artur Struzik) and B.P.; visualization, J.P. and A.S. (Artur Struzik); supervision, J.P.; project administration, J.P.; and funding acquisition, B.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors thank the study participants for their effort, time devotion, and collaboration during the study.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **E**ff**ect of Landing Posture on Jump Height Calculated from Flight Time**

**Daichi Yamashita 1,\*, Munenori Murata 1,2 and Yuki Inaba <sup>1</sup>**


Received: 31 December 2019; Accepted: 20 January 2020; Published: 22 January 2020

**Abstract:** Flight time is widely used to calculate jump height because of its simple and inexpensive application. However, this method is known to give different results than the calculation from vertical velocity at takeoff. The purpose of this study is to quantify the effect of postural changes between takeoff and landing on the jump height from flight time. Twenty-seven participants performed three vertical jumps with arm swing. Three-dimensional coordinates of anatomical landmarks and the ground reaction force were analyzed. Two methods of calculating jump height were used: (1) the vertical velocity of the whole-body center of mass (COMwb) at takeoff and (2) flight time. The jump height from flight time was overestimated by 0.025 m compared to the jump height from the takeoff velocity (*p* < 0.05) due to the lower COMwb height at landing by −0.053 m (*p* < 0.05). The postural changes in foot, shank, and arm segments mainly contributed to decreasing the COMwb height (−0.025, −0.014, and −0.017 m, respectively). The flight time method is reliable and had low intra-participant variability, but it cannot be recommended for a vertical jump when comparing with others (such as at tryouts) because of the potential "cheating" effect of differences in landing posture.

**Keywords:** flight time; vertical jump; center of mass; landing

#### **1. Introduction**

Jumping ability is regarded as one of the most important aspects of many sports. Vertical jump measurement is a method to assess lower limb power [1], strength [2], and neuromuscular status [3]. Therefore, the vertical jump test has been used to assess the impact of training [2] and to select high-level players at tryouts in many sports such as American football [4] and basketball [5].

The force platform is one of the most widely used methods of vertical jump measurement and is considered the gold standard for determining the mechanical outputs of jumping [6]. Force platforms are used to measure the ground reaction force (GRF) and derive the velocity of the whole-body center of mass (COMwb) using the impulse–momentum relationship. However, they are costly for sports teams and strength coaches, so their use is limited mainly to university laboratories and research institutes.

Recently, the calculation of jump height from flight time using a contact mat, a photoelectric cell, and a smartphone that utilizes a high-speed camera application has become increasingly popular due to its low cost and straightforward assessment methods. In this method, jump height is calculated using a uniform acceleration equation. The equation justifies the method only if the height of the COMwb is the same at takeoff and landing. It has been reported, however, that the method overestimates the countermovement jump without arm swing (CMJ) height by 2% [7], 3–4% [8], 8% [8], and 11% [9] compared to the method using vertical velocity at takeoff from a force platform. These results suggest that the height of the COMwb at landing is lower than that at takeoff, making the flight time longer. Consequently, the jump height from flight time is overestimated.

One potential determining factor for this difference, suggested by the previous studies, is that participants landed with their lower limbs partially bent, resulting in an inflated flight time [8,9]. Kibele [7] showed that knee and ankle joints were more flexed, and the COMwb height was lower at landing than at takeoff. Also, a different arm posture at takeoff and landing seems to affect the difference in the COMwb height when arm swing is permitted [10]. Previous studies reported that the COMwb height at takeoff in the vertical jump with arm swing (VJ) was 0.024 m [11] and 0.034 m [12] higher than that in CMJ.

The flight time method of calculating jump height is widely used by laboratories and sports teams, even though many researchers have acknowledged the postural differences at takeoff and landing. However, there have been few studies which have aimed to understand the sources of error in jump height from flight time. In order to fully understand the sources of error, it is helpful to quantify the relationship between the postural difference and the difference in the COMwb height. This is because the height of the center of mass of a system is given by a mathematical formulation: the mass-weighted average of the heights of the segments. Therefore, the purpose of this study is to quantify the effect of postural changes in each segment on the COMwb height difference between takeoff and landing. We hypothesized that lower limb bending and arm movement are the primary factors that affect the overestimation of jump height from flight time. Understanding the sources of error in jump height from flight time would be useful for better instruction to reduce systematic bias and interpersonal variability when using the simple and low-cost method of vertical jump measurement.

#### **2. Materials and Methods**

#### *2.1. Participants*

Twenty-four males and three females (age: 19 to 42 years; height: 1.77 ± 0.11 m; mass: 75.3 ± 11.9 kg) participated in this experiment. They provided written informed consent to undergo the experimental procedures, which were conducted in accordance with the Declaration of Helsinki and were approved by the ethics committee of the Japan Institute of Sports Sciences (H29-0065).

#### *2.2. Instrumentation*

Three-dimensional coordinates of the anatomical landmarks were acquired using a 3D optical motion capture system with ten cameras (500 Hz; Vicon, Oxford, UK). Forty-seven reflective markers were placed on each participant's body—the same as in the previous study [13]. All kinematic data was filtered and interpolated using a Woltring quintic spline [14]. To choose the optimal cut-off frequency of 4.6–7 Hz, a residual analysis was performed [15]. Participants wore their athletic shoes. GRF data was obtained at 1000 Hz using two force platforms (0.9 m × 0.6 m, type 9287B; Kistler, Winterthur, Switzerland).

#### *2.3. Procedures*

Participants performed three maximal VJs after warm-up and familiarization. They were instructed to stand upright and motionless for 1 s then began the movement of the jump. They were required not to bend their lower limbs before landing. Two or more experimenters watched each trial, and if they noted that the requirement was not met (i.e., leg tucking), the trial was repeated.

#### *2.4. Data Reduction*

Two methods of calculating jump height were used: (1) the vertical velocity of the COMwb at takeoff, and (2) flight time. The vertical GRFs (*F*ver) were integrated by trapezoid rule integration to estimate the vertical velocity [6]. The vertical velocity at takeoff (*V*to) was calculated using the following equation:

$$V\_{\rm to} = \frac{1}{m\_{\rm wb}} \int\_{t\_{\rm st}}^{t\_{\rm bw}} (F\_{\rm ver} - m\_{\rm wb}g) \mathrm{d}t \tag{1}$$

where *m*wb, *F*ver, *g*, *t*st, and *t*to represent the body mass, vertical GRF, gravitational acceleration (9.806 m/s2 [16]), the time of the start of the initial jumping motion, and the time of its termination at takeoff, respectively. The body mass was calculated by averaging *F*ver over the 0.3 s quiet stance [7] and dividing by gravitational acceleration. We confirmed that the coefficient of variance (CV) of *F*ver during the quiet phase in each trial was low (less than 1%). The start of the motion was identified as the first *F*ver detected to deviate above or below body weight by 1%. To eliminate the influence of inter-participant variance in body weight, takeoff and landing times were defined as the first intersection of *F*ver with 1% of body weight (7.4 ± 1.2 N, range 5.0 to 9.9 N). The jump height from *V*to (*H*v) was calculated using the following equation:

$$H\_{\rm V} = \begin{array}{c} \frac{1}{2\xi} V\_{\rm to} \,^2. \tag{2}$$

*H*v was used in this study as the criterion for comparison. Jump height from flight time (*H*t) was calculated using the following equation:

$$H\_{\rm t} = \frac{1}{8}gt\_{\rm flight}^{\prime}\tag{3}$$

where *t*flight represents the flight time (see Appendix A).

The COMwb position was calculated as the weighted sum of a 15-segment model (i.e., head, upper trunk, lower trunk, upper arms, forearms, hands, thighs, shanks, and feet) based on body-segment parameters [17]. To compare the difference in the whole-body posture between takeoff and landing, we used a seven-segment model of the head, arm, upper trunk, lower trunk, thigh, shank, and foot (Figure 1). The positions of the arm, thigh, shank, and foot segments were the average of the right and left side.

**Figure 1.** The definition of (**a**) vertical component of the segment center of mass length (COM*k*) and (**b**) vertical component of the segment length (SEG*k*).

Once an object is projected into the air, the COM of the system must follow a parabolic trajectory, and the trajectory cannot be altered in the air until landing. When the position of a segment moves relative to the COMwb, it affects the other segments' positions relative to the COMwb to keep the COMwb trajectory constant. As a result, the difference in a segment posture influences the COMwb height at landing. To understand the effects of the postural difference between takeoff and landing on the COMwb height, we quantified the contributions of the changes in the vertical component of each segment on the COMwb height. When one segment changes its posture, it affects (1) the segment COM

height and (2) the COM height of all segments above it. We defined the vertical component of the segment COM length (COM*k*) by the following equations:

$$\text{COM}\_{k} = h\_{\mathbb{C}\_{k}/l\_{k}}\left(k = 1 \text{ to } 6\right) \tag{4a}$$

$$\text{COMP}\_k = h\_{\mathbb{C}\_k/l\_\delta}(k=7) \tag{4b}$$

where *k* represents the segment number (see Figure 1) and *hCk*/*Jk* represents the height from the lower edge point (joint) of the segment to the segment COM (Figure 1a). COM7 (i.e., the arm segment) was defined relative to the proximal joint (the suprasternal notch). In the same way, we defined the vertical component of the segment length (SEG*k*) by the following equation:

$$\text{SEG}\_k = h\_{\mathbb{I}k + 1/\mathbb{J}\_k} \tag{5}$$

where *hJk*+1/*Jk* represents the height from the lower edge point (joint) of the segment to the proximal joint (Figure 1b). Then, we calculated the contributions for all seven segments (CONT*k*) using the following equations:

$$\text{CONT}\_{k} = \frac{m\_{k}}{m\_{\text{wb}}} \Delta h\_{\text{C}\_{k}/l\_{k}} + \sum\_{i=k+1}^{7} \frac{m\_{i}}{m\_{\text{wb}}} \Delta h\_{l\_{k+1}/l\_{k}} \left(k = 1 \text{ to } 5\right) \tag{6a}$$

$$\text{CONT}\_k = \frac{m\_k}{m\_{\text{wb}}} \Delta h\_{\text{C}\_k/l\_6} \tag{6} \\ \tag{6b}$$

where *m* and Δ represent the segment mass and the difference in a variable between takeoff and landing, respectively.

When a lower COMwb at landing is observed, the difference makes the flight time longer, meaning that the jump height from flight time is overestimated. To understand the influence of the difference in the COMwb height on jump height overestimation (Δ*H*), we created a contour color map using the following equations:

$$
\Delta H = H\_\text{t} - H\_\text{v} \tag{7}
$$

$$
\Delta H = \frac{1}{8} \mathcal{g} \left( \sqrt{\frac{2H\_\mathrm{v}}{\mathcal{g}}} + \sqrt{\frac{2(H\_\mathrm{v} + \Delta \mathrm{COM}\_\mathrm{wb})}{\mathcal{g}}} \right)^2 - H\_\mathrm{v} \tag{8}
$$

where ΔCOMwb represents the difference in the COMwb height. The term in brackets on the right side of Equation (8) is the flight time (see Appendix B). All numerical calculations were performed using MATLAB 2018b (The MathWorks, Inc., Natick, MA, USA).

#### *2.5. Statistical Analysis*

The three jumps performed with each device were averaged to provide a representative value for each variable. Means and standard deviations (SDs) were calculated after verifying the normality of distributions using Kolmogorov–Smirnov statistics. Paired-sample *t*-tests were used to compare the mean differences between methods and between time phases (takeoff and landing). One-sample *t*-tests were used to examine CONT*<sup>k</sup>* against zero. The magnitude of the difference was also assessed using Cohen's *d*, where *d* > 0.8 is a large effect, 0.5 ≤ *d* ≤ 0.8 is a moderate effect, 0.2 ≤ *d* ≤ 0.5 is a small effect, and *d* < 0.2 is a trivial effect [18]. The intra-participant reliability of the variables of the three jumps was examined by the intraclass correlation coefficient, one-way random-effects model (ICC1,1). Acceptable reliability was defined as an ICC > 0.70 [19]. The analysis of the fixed bias with its upper and lower limits of agreement (LOA) between the jump heights for all 81 trials obtained from the two calculations was performed by using a Bland–Altman plot [20]. Heteroscedasticity of error (proportional bias) was defined as a coefficient of determination (*r*2) > 0.1 [21]. Statistical significance

was determined by a probability level of *p* < 0.05. All calculations were performed using IBM SPSS Statistics version 19 (IBM Co., Chicago, IL, USA).

#### **3. Results**

*H*<sup>t</sup> was significantly higher than *H*<sup>v</sup> (0.421 ± 0.081 and 0.396 ± 0.074 m, respectively, *p* < 0.001, *d* = 1.046). The mean fixed bias (with 95% LOA) between *H*<sup>t</sup> and *H*<sup>v</sup> was 0.025 m (with range −0.028 to 0.079 m) (Figure 2a). The further analysis of the Bland–Altman plot (Figure 3) revealed very low *r*<sup>2</sup> values (*r*<sup>2</sup> = 0.068), meaning outcomes estimated from *H*<sup>t</sup> had no proportional bias to overestimate or underestimate jump performance. Acceptable intra-participant reliabilities were observed for both *H*t and *H*<sup>v</sup> (ICC1,1 = 0.964 and 0.979, respectively).

**Figure 2.** (**a**) The difference in jump height (Δ*H*) and (**b**) the difference in whole-body COM height (ΔCOMwb) for each participant. Each bar represents a participant. They are arranged in descending (**a**) and ascending (**b**) order.

**Figure 3.** Bland–Altman plot for the jump height from the vertical velocity at takeoff (*H*v) and the jump height from flight time *(H*t). The central line represents the absolute average difference between the methods, and the upper and the lower lines represent ± 1.96 standard deviation (SD).

The COMwb was significantly lower at landing than at takeoff (1.087 ± 0.100 m and 1.140 ± 0.071, respectively, *p* < 0.001, *d* = 1.17) (Table 1). Acceptable intra-participant reliabilities were observed for the COMwb at both takeoff and landing (ICC1,1 = 0.991 and 0.967, respectively). Inter-participant variability ranged from −0.182 to 0.008 m (Figure 2b). COMarm, COMshank, and COMfoot showed lower values at landing compared to at takeoff (*p* < 0.05, large effect size) (Table 1). SEGshank and SEGfoot showed lower values at landing (*p* < 0.05, large effect size) compared to at takeoff (Table 2). The intra-participant

reliabilities of those variables were acceptable (ICC1,1 > 0.7). CONTarm, CONTshank, and CONTfoot showed lower values compared to zero (*p* < 0.05, moderate to large effect size) (Table 3 and Figure 4).


**Table 1.** The vertical component of the segment center of mass (COM*k*) at takeoff and landing.

**Table 2.** The vertical component of the segment length (SEG*k*) at takeoff and landing.


\* *p* < 0.05.

**Table 3.** The contribution of the difference in the vertical component of the segment length (SEG*k*) to the difference in the whole-body center of mass (COMwb) height (CONT*k*).


**Figure 4.** The contribution of the difference in the vertical component of the segment length (SEG*k*) to the difference in the whole-body center of mass (COMwb) height (contributions for all seven segments, CONT*k*). The sum of the differences in all of the segments is the difference in the COMwb height. Each bar shows the result for a participant; the values are arranged in ascending order.

From Equation (8), the difference in jump height (Δ*H*) was influenced by ΔCOMwb and jump height (*H*v). The contour map (Figure 5) showed that the jump height did not greatly affect the overestimation of jump height.

**Figure 5.** An illustration of the influence of the difference in the COMwb height (ΔCOMwb) and jump height (*H*v) on jump height overestimation (Δ*H*) (Equation (8)). The top-left triangular area in white shows there are no real roots because ΔCOMwb cannot be greater than *H*v. The red circles represent each experimental data point.

#### **4. Discussion**

The purpose of the present study was to quantify the effect of postural changes between takeoff and landing on jump height overestimation. The jump height from flight time was 0.025 m (6.4%) higher than the jump height calculated from velocity. We confirmed that the current result was reasonable compared to the previous studies, which showed 2–11% overestimation of CMJ height [7–9].

The difference in the vertical components of the foot and shank segment lengths were the main contributions to the difference in the COMwb height. Also, the inter-participant differences were large (range −0.092 to 0.018 m for foot and −0.040 to 0.003 m for the shank, see Figure 4). Therefore, the observed lower COMwb height at landing was mostly due to lower ankle dorsiflexion. In some previous studies, experimenters instructed each participant not to flex their knees [22] and hips [23] at landing. In other studies, experimenters instructed participants to land in a similarly extended position at takeoff [24,25]. From these previous studies, it is suggested that ankle dorsiflexion at landing was considered a less serious effect on the jump height, whereas our results indicate that it is the most critical motion. In these studies, each trial was also watched and judged by the experimenter subjectively to ensure that the instructions had been followed. To reduce the difference in the COMwb height, it is recommended that an experimenter instruct participants about the landing technique for the jump tests, primarily focusing on foot and shank segments, such as "landing with toes pointing downwards" [26]. However, such instruction seems to be inappropriate. One reason for this is that the posture at landing is essential because high impacts cause lower joint injuries [27]. Preparatory flexing at the hip, knee, and ankle is an effective strategy to reduce the impact of landing [28]. Moreover, a previous study revealed that extra attention increased the impact of landing [29]. It may be difficult to control the posture at landing in detail without increasing the risk of injury and excess stress.

It is unlikely—but not impossible—that the control of the upper body affects the lower limb bending. Because the total momentum and total angular momentum of a system both remain constant unless acted upon by an external influence, when a segment moves relative to the COMwb, the other segments have to move to compensate. Therefore, there is a possibility that the upper trunk movement affected the shank and foot postures. It is notable that the causal relationships between these postural effects are unknown.

In this study, the difference in the arm COM height relative to the suprasternal notch also affected the difference in the COMwb height between takeoff and landing, though there was a large inter-participant variability (from −0.046 m to 0.008 m). Although no previous study has reported the effect of arm posture on flight time overestimation, some studies reported that an arm swing contributes to increased the COMwb height [11]. The height of the arm COM in 23 out of 27 participants was above the proximal joint at takeoff, and in 14 of the 23 participants, it was below the proximal joint at landing. When VJ is performed, experimenters do not instruct the participants regarding the height of arm movement before takeoff, because they want to evaluate jump performance using arm swing as much as possible, comparing it to jumps without arm swing. On the other hand, it might be possible to control the height of the arm movement at landing through instruction, such as "arms above the shoulder at landing." No studies justify that the height of the COMwb should be the same at takeoff and landing. At least, the current instruction focusing on lower limb posture cannot prevent the potential "cheating" that can be accomplished by lowering arms as much as possible at landing. Previous studies have shown that arm swing improved jump height [30], but the improvement might be somewhat overestimated when the jump height was calculated from flight time. Therefore, the flight time method cannot be recommended for a vertical jump with arm swing, especially when compared with others, such as at tryouts.

From the contour curve (Figure 5), jump height did not greatly affect the overestimation. For example, if the jump height is 0.20 m and the COMwb height is 0.04 m lower at landing compared to takeoff, then the jump height from flight time is overestimated by 0.0195 m. if the jump height changes to 0.60 m, the overestimation from the same difference in the COMwb height is 0.0198 m. The relationship between the difference in the COMwb height and the overestimation of jump height is in a ratio of almost two to one.

Many studies have considered force platforms as the "gold standard" to evaluate jump height [22, 31,32], but this confuses the instrumentation with the calculation method. We can calculate jump height by two methods using force platforms: (1) vertical velocity at takeoff, and (2) the time in the air [8,9]. To clarify the validity and reliability of the simple methodology to calculate jump height from flight time, force platforms are considered the gold standard to calculate flight time because the method contained is valid under certain conditions, as described above. The jump height from vertical velocity at takeoff is the true gold standard for jump height measurement.

Calculating jump height from flight time is still useful for coaches who want to measure changes in an individual resulting from their training program because of its low cost, simplicity, and ease of implementation. Recently, many commercial devices are have been developed to measure jump height from flight time, such as an iPhone app [33] and inertial measurement unit [24]. Other methods have also been in development, such as linear position transducers, but these showed overestimation by 7.0 cm compared to the jump height from flight time [34]. In this study, we confirmed that the flight time method has high intra-participant reliability and no proportional bias, though there is a fixed bias. Researchers and coaches are usually interested in comparing jump height before and after training. If the same device is used for both pre- and post-tests, it is useful.

#### **5. Conclusions**

In conclusion, we found that jump height from flight time is overestimated compared to the jump height from takeoff velocity as a result of the lower limb and arm postures at landing. Understanding the sources of error in jump height from flight time can be used to develop better instruction to reduce the systematic error.

**Author Contributions:** Conceptualization, D.Y., M.M. and Y.I.; methodology, D.Y., M.M. and Y.I.; software, D.Y., M.M. and Y.I.; validation, D.Y., M.M. and Y.I.; formal analysis, D.Y., M.M. and Y.I.; investigation, D.Y., M.M. and Y.I.; writing—original draft preparation, D.Y., M.M. and Y.I.; visualization, D.Y.; funding acquisition, D.Y., M.M. and Y.I. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the research grant of the Japanese Society of Biomechanics.

**Acknowledgments:** The authors would like to thank the executive committee members of "KEIHIROBA", a conference of the Japanese Society of Biomechanics, for their helpful comments.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Calculation of Jump Height from Flight Time**

It is noted that the assumption for this calculation is that the height of the COMwb is the same at takeoff and landing of the jump (*H*<sup>v</sup> is equal to *H*t). Once an object is projected into the air, the COMwb must follow a parabolic trajectory, and the trajectory cannot be altered in the air until landing because only the gravitational acceleration is applied to it. Therefore, the vertical velocity of the COMwb is calculated as

$$V(t) = V\_0 - \mathfrak{g}t \tag{A1}$$

where *V(t)* represents the vertical velocity, *V*<sup>0</sup> represents the initial velocity, and *t* represents the time of travel. As *V(t)* becomes zero at the highest point during flight phase the time from the takeoff to the highest point (*t*top) is expressed as

$$V\_{\text{to}} - \lg t\_{\text{up}} = 0 \tag{A2a}$$

$$V\_{\text{to }} = \text{gt}\_{\text{up}}\tag{A2b}$$

*t*up should be half of the flight time, with the peak of the jump happening at exactly the midpoint of the flight time, expressed as follows:

$$V\_{\text{to }} = \frac{1}{2}gt\_{\text{flight}}\tag{A3}$$

Substituting Equation (A3) for Equation (2), we obtain Equation (3) as follows:

$$H\_{\rm lt} = H\_{\rm V} = \frac{1}{2g} \Big(\frac{1}{2}gt\_{\rm fllight}\Big)^2\tag{A4}$$

$$H\_{\rm t} = \frac{1}{8}gt\_{\rm flight}^{2}\tag{A5}$$

#### **Appendix B. Calculation of the Flight Time from the Vertical Displacement of the COMwb**

The vertical displacements of the COMwb travelling from takeoff to the highest point (vertical velocity becomes zero) and from the highest point (vertical velocity is zero) to landing are both expressed as

$$h(t) = \frac{1}{2}gt^2\tag{A6}$$

where *h(t)* represents the vertical displacement. During the time from the takeoff to the highest point (*t*up), *h(t)* is equal to *H*v, and Equation (A6) gives

$$H\_{\rm V} = \frac{1}{2}gt\_{\rm up}^{-2} \tag{A7}$$

$$t\_{\rm up} = \sqrt{\frac{2H\_{\rm V}}{g}} \tag{A8}$$

On the other hand, during the time from the highest point to landing (*t*down), *h(t)* is the sum of *H*<sup>v</sup> and Δ*COMwb*, and Equation (A6) gives

$$H\_{\rm V} + \Delta \text{COM}\_{\rm wb} = \frac{1}{2} \text{g} t\_{\rm down} \,\,^2 \tag{A9}$$

$$t\_{\text{down}} = \sqrt{\frac{2(H\_{\text{v}} + \Delta \text{COM}\_{\text{wb}})}{\mathcal{g}}} \tag{A10}$$

The flight time (*t*flight) is the sum of *t*up and *t*down. Therefore, *t*flight is expressed as

$$t\_{\text{flight}} = \sqrt{\frac{2H\_{\text{v}}}{\mathcal{S}}} + \sqrt{\frac{2(H\_{\text{V}} + \Delta \text{COM}\_{\text{wb}})}{\mathcal{S}}} \tag{A11}$$

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© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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