Keywords

1 Introduction

Tennis forehand is one of the basic strokes used in every game. In this shot two phases may be specified: the preparation and the moment of impact [1]. Each step is an important aspect of a properly performed stroke. The aspect of the repeatability of tennis shots plays an important role in archiving better results, both for professional and amateur players. Each time the players should be able to perform strikes with the same or similar parameters, including both the body and the racket.

The first purpose of this paper is to create the algorithms for calculating the motion parameters for the preparation phase of the forehand. They are: the position of the head of the tennis racket relative to the player’s body and the placement of the player’s feet towards the front of the pelvis. The elbow angle (i.e. whether the arm is bent or straight) is read from the biomechanical data. The second aim of the study is to calculate the repeatability of the selected motion parameters (both for the body and the racket) of the preparation phase of the tennis forehand using numerical evaluation. In this research the preparation phase begins when the racket’s position reaches the maximum along XY plane and ends before the moment of impact.

The shots were performed in the motion capture laboratory using the Plug-in Gait biomechanical model and a racket model [2]. The described parameters are computed from three dimensional data, stored in C3D file. The above-described parameters were computed for ten consecutive forehand strikes of a professional tennis coach.

2 Research Overview

Much research concerns analysis of the serve as the main stroke making an advance during the game. The motion repeatability of the tennis serve of three experienced and three novice players was studied in [3]. The participants were asked to serve a ball into another side of the tennis court 10 times using an inertial motion capturing system consisting of 17 sensors. The sampling rate was 40 fps. Three-dimensional position fluctuations of the lumbar vertebra were measured as the centre of the body. Three-dimensional angular velocities of 23 segments and 22 joints were captured. The tennis player’s serve analysis (during the backswing preparation as well as the forward swing motions) using accelerometers is presented in [4]. The kinematic chain pattern and correct angular and translational movements of the knee, waist and wrist of the player are discussed.

The created model for the tennis serve is presented in [5]. The authors created a tool for evaluating the individual technique of a tennis player based on the kinetic energy transmission pattern using a complete mechanical body model. The model, created for the purpose of the research, consisted of 28 markers with five solid-rigid ones. The tool treated the individual technique of a player as a mathematical function. The significant discriminant function evaluated the tennis player’s serve as “good” or “bad”. The kinetic energy is composed of a linear velocity of a body segment and a rotational component which considers the angular velocity of the segment.

The sport technique in male tennis was studied in [6], for three shots. The analysis was done in four key points: when the time of maximum acceleration of upper tennis racket occurred, when the maximum speed of upper tennis racket was recorded, when the time of contact between the tennis racket and the ball occurred, and for the highest and lowest values of kinematic parameters of all segments of the dominant upper extremity (right). The kinematic parameters chosen for the study were: velocity, acceleration of the dominant upper extremity, centre of gravity and upper racket, height, lateral and forward-backward distance of the centre of gravity, angles in both knees, elbows and body bending.

Three aspects of the racket are analysed: its velocity, swing and orientation. Both motion capture systems (optical and markerless) and video cameras are used for the research. The motion and pose in 3 dimensions were computed from a video recording [7]. This method also allows for computing racket performance. The swing of the racket was investigated in [8]. The speed of a tennis racket (as well as that of a golf club and a baseball bat) was analysed using 3-dimensional images obtained from two cameras. Reflective tape was placed on the elbow, around the wrist and around the racket’s handle in two places. The maximum effort (both in standing and sitting positions) was calculated for each swing. The swing speed was approximately constant during the test.

Tracking a tennis racket using a markerless system (based on multiple cameras) and a visual hull is presented in [9]. Views of a tennis racket were recorded and segmented into binary images. The racket’s shape was obtained. The visual hull of the racket was created using the intersection of the volume of space formed by back-projecting the silhouettes from all input views. In [10] the velocity of the tennis racket was calculated based on a monochrome recording made using a high-speed camera.

The research containing a tennis racket tracking and its analysis using Vicon system is presented in [2]. Seven retro-reflective markers were attached to it, so the racket’s movement could be calculated. The racket’s velocity in the consecutive frames, the orientation of the racket in both sagittal and axial planes of the racket were indicated. The racket motion was studied in [11].

This paper describes the algorithms of four parameters of the preparation phase of the forehand strike. The position of the head of the tennis racket relative to the player’s body was presented in [6], where the position is specified as below or above the head. In this paper the position of the tennis racket is indicated as: above the forehead, between the forehead and the shoulder, between the shoulder and the pelvis and below the pelvis. The elbow angle was also studied in [6]. In this paper the angle is computed by the Plug-in Gait model. The setting of the feet was studied in [6]. In this paper a new algorithm is presented which indicates the feet’s positions.

3 Method

3.1 Participant

One right-handed 30 year-old tennis coach was the participant in this study. He was 1.74 m tall and weighed 71 kg. He signed the ethical approval form for this research.

3.2 Motion Capture System

A passive optical motion capture system was used to track the participant and the racket while performing tennis strokes at the Laboratory of Motion Analysis and Interface Ergonomics at the Lublin University of Technology in Poland where interdisciplinary tests are performed [12]. The motion capture system consisted of: eight NIR T40S cameras operating in near infrared, two reference video Bonita cameras, a Giganet hub collecting data, a desktop computer and a set of accessories (e.g. markers, a calibration wand, double-sided tape). The system recorded the positions of the markers placed on the subject’s body (each marker must be seen by at least two cameras). The equipment was supplied with Vicon’s Nexus 2.0 software, used for system calibration, data recording and data processing. The movements were registered with 100 Hz.

3.3 Experimental Procedure

The participant was prepared for the experiment according to the Plug-in Gait Model [13]. Thirty-nine retroreflective markers were attached to the participant using hypoallergenic double-sided tape as specified in the model. This model allows for calculating angles, torques and forces in a subject’s joints. Then, the person was measured for the purpose of creating and scaling a new subject in the Nexus software (height, weight, leg length, arm offset, knee, ankle, elbow and the thickness of both hands). The subject’s calibration was performed as the next step in preparation due to the verification that the markers are visible and correctly attached.

After the warm-up, the participant was told to perform ten forehand strikes without the ball while running and avoiding a bollard placed on the floor. Because the participant was running, the strokes were more natural than hitting the ball from a standing pose. The racket was tracked during the research. Seven retro-reflective markers were attached to the racket according to the scheme presented in Fig. 1. They allow one to reconstruct the shape of the racket.

Fig. 1.
figure 1

Tennis racket with seven markers attached [2].

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3.4 Post-processing

Each 3D recording was post-processed using the Vicon Nexus software that consisted of four main steps: marker labelling, gap filling using interpolation methods, data cleaning (e.g. deleting all unlabelled markers) and applying the Plug-in-Gait model (only for the human body). A new subject was created for the racket. It consisted of seven markers [2]. The post-processed recordings were exported as C3D files. Each C3D file contains ten consecutive forehand strikes. For the purpose of this research the preparation phases were designated. These files were used for further analysis. The files were processed by the author’s own piece of software created in C++ using the Biomechanical toolkit (b-tk) and Eigen libraries.

3.5 Coefficient of Variation

The coefficient of variation – CV (see formula 1) [14] was used for computing the repeatability of tennis forehand preparation phase for selected parameters.

$$ CV = \frac{\sigma }{\mu } *100 $$
(1)

where σ denotes the standard deviation and μ – mean.

The parameter was considered as repeatable when CV was lower than ten.

4 Algorithms

Two algorithms calculating the selected motion parameters were implemented for the purpose of this research. They base on three dimensional positions of the markers attached to the body and the racket stored in the C3D file. The first indicates the racket’s positions relative to the player’s body in the consecutive frames of the recording (Fig. 2). The second describes the positions of the feet and the distance from the frontal part of the pelvis in the consecutive frames (Fig. 3). The right elbow angles (RElbowAngle) were read from the biomechanical data given by implementing the Plug-in Gait model.

Fig. 2.
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Algorithm indicating the position of the head of the tennis racket towards the player’s body.

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Fig. 3.
figure 3

The algorithm computing the distance between the frontal pelvis and both toes and indicating which leg is placed further.

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4.1 Head of the Tennis Racket Towards the Player’s Body

The pseudocode, presented in Fig. 2, is a method that inputs the height from the ground of the specified markers in a form of the vectors so that the data for all frames can be stored. The height is expressed as the Y coordinates of the markers. The markers that were taken into consideration are: RH1, RFHD, RSHO and RASI. Each iteration of the for loop indicates the position of the head of the tennis racket towards parts of the body in the current frame as an integer value, defined in Table 1. The algorithm returns an array (racketPosition) storing the positions of all frames of the recording.

Table 1. The racket positions towards body parts
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4.2 Feet Positions

The pseudocode, presented in Fig. 3, is a method that computes the distance between the line indicated by two markers placed on the frontal pelvis (RASI and LASI) and two markers attached to the toes (RTOE and LTOE) projected on the XY plane. For the consecutive frames, using for loop, the line is defined based on A, B and C coefficients. Then, the distances from the right and the left toe are calculated using Formula 2. The integer value is calculating which represents which leg is placed further from the pelvis. If the value is set to one, the left leg is further from the pelvis. If the value is set to two, the right leg is further from it. Otherwise, two legs equidistant from the pelvis. The algorithm returns these values in the dist vector.

$$ \frac{{\left| {A \cdot x + B \cdot y + C} \right|}}{{\sqrt {A \cdot A + B \cdot B} }} $$
(2)

where A, B, C stand for the line coefficients and x, y are the marker coefficients.

5 Results

5.1 Head of the Tennis Racket Towards the Player’s Body

The participant performed ten strikes (trials). The percentage share of each racket’s position towards the body (defined in Table 1) was computed using algorithm presented in Fig. 2. Two extremes (Nos. 2 and 9) were excluded from the further research due to the average time phase criterion failure (furthest values relative to the average number of grates). The obtained results are gathered in Table 2.

Table 2. The percentage of the racket positions towards the body in the preparation phase
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The mean value, standard deviation (SD) and the coefficient of variation (CV) were computed for the time of the strikes (frame numbers) as well as for the racket’s three positions (for percentages values). The results are gathered in Table 3.

Table 3. The statistical values for time and the percentage of the racket positions towards the body in the preparation phase
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5.2 Feet Positions

The feet positions are calculating using algorithm presented in Fig. 3. At the beginning stage of the preparation phase the participant placed his right foot ahead of the left one. It was calculated after what time (percentage) he reversed the feet order. The results are presented in Table 4. The statistical values are: mean – 46.05, SD – 3.67 and CV – 7.97.

Table 4. The percentage time of the left foot being ahead of the right one
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5.3 Elbow Angle

The average elbow angles in the following positions indicated by the head of the tennis racket towards the player’s body for preparation phase are gathered in Table 5. The statistical values are presented in Table 6.

Table 5. The average elbow’s angle in the following positions of forehand preparation phase
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Table 6. The statistical values for the average elbow angle in the following positions of the forehand preparation phase
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6 Conclusions

The paper presents the idea of calculating the repeatability of selected parameters of the preparation phase of the tennis forehand without the ball. The head of the tennis racket towards the player’s body, the feet positions and the elbow angles were analysed. The first aim of this paper was both to present the algorithms computing the selected parameters of the preparation phase of the tennis forehand and to verify them on the basis of C3D data of one participant (Tables 2, 4, and 5). The second purpose was to calculate the repeatability of the obtained parameters using the coefficient of variation.

The results were grouped according to the first parameter analysed – the head of the tennis racket towards the player’s body. The statistical values, presented in Table 3, clearly show that only position 0 (above forehead) and 1 (between the forehead and shoulder) are repeatable. Other positions of the racket towards the body change differently in the following trials.

The results presented in Table 4 indicate that the participant begins the preparation phase with his right foot (put) forward. After average 46% of the phase time he puts his left leg forward. This parameter is repeatable. This is the correct arrangement of the lower limb in this part of the forehand.

The arrangement of the arm during the preparation phase of the forehand also plays an important role. It should be neither totally bent nor totally straight. The results gathered in Table 6 allow to draw the conclusions that in position 0 the arm is more straight (the elbow angle is higher) than in other positions. It seems that this shot is thoroughly trained.

This study may play a very interesting aspects for both beginners and professional players. The precise values, obtained by the algorithms presented here, may give directions for the future training in order to improve the player’s performance and repeatability of the tennis forehand.