# Identifying the Inertial Properties of a Padel Racket: An Experimental Maneuverability Proposal

^{*}

## Abstract

**:**

## 1. Introduction

^{2}) with enclosing walls made of glass and fences and solid stringless rackets. Padel is played two against two. The ball has the exact dimensions as the one used for tennis matches. The game was invented in Mexico in 1969, spread during the 1980s in South America, mainly in Argentina, and has been since the 1990s a common and popular sport in Spain with corresponding tournaments, associations, sports clubs with padel courts, and racket manufacturers. Although Spain is the major player country, with over 4 million players [1], padel has also spread outside these countries in recent years [2,3]. Current estimations show that there are over 18 million players and 300,000 federated players around the world. The International Padel Federation (FIP), founded in 1991, is the leading association to regulate and promote padel and is formed by more than 50 national federations representing almost 100 countries [4]. FIP recently launched the Premier Padel

^{TM}, a global padel tour backed by the Professional Players Association (PPA), with 10 tournaments scheduled for 2022 and 2023 and 25 tournaments for 2024. Another professional circuit, World Padel Tour, was created by a private firm (Setpoint Events, S.A.) that organizes tournaments in cities from more than 15 countries considering Open, Master, and Challenger Tournaments and Exhibitions. The repercussions of padel, the number of players worldwide, and the research interest is increasing [5].

## 2. Measurement Process

#### 2.1. Preliminary Measurements

#### 2.2. MOI Calculation Methods

_{p}and the mass of the racket m

_{r}(Equation (4)), and knowing the pendulum base plate MOIs, I

_{p}, it is possible to obtain the racket MOI I

_{r}.

#### 2.3. Comparison of the Approaches

#### 2.4. Test Stand Design and Implementation

^{®}(Dassault Systèmes, France, 2022) of the test stand is 2.16181 kg cm

^{2}, higher than the MOI that passes through the COM 2.157 kg cm

^{2}. That does not significantly influence the results since the vertical MOI of padel rackets is higher.

_{1}+ d

_{2}+ d

_{3}(Figure 7) when the holder is not displaced because the thread knots are on the holders’ outer sides. After displacing the holders with the racket, the effective thread length is d

_{2}. To consider the observation, the mean value L = d

_{2}+ (d

_{1}+ d

_{3})/2 is used in the follow-up experiments and calculations.

#### 2.5. Sensor Data Processing

^{®}function process the data and determine the period time of the oscillation. It acquires the raw data from the I/O device for a determined time, optimizes the signal in several steps, and performs a fast Fourier transformation (FFT) to obtain the power spectral density. The first and higher harmonic is the vertical oscillation frequency. The different steps of the processing signal are:

- Raw sensor data are noisy and coarse, so signals are filtered first with low pass at 1500 Hz and second with a simple moving average. It replaces every data point by the mean of its initial value and the value of the preceding and the following two data points. Filters reduce the noise, and the leaps in the signal are smoothed out, as shown in Figure 10.
- The signal is trimmed (Figure 10b) to consider only a whole multiple of the repetitive period. That means that the dataset starts at the same point of the period where it ends. An offset is applied to the data so that the mean of the values is just zero. After that, complete cycles can be detected by searching for the points where the sign changes from positive to negative or vice versa.
- It shows the frequency spectra (Figure 11). After obtaining the dominating frequency, the period determines the time of an entire cycle.

#### 2.6. Interface Design Software

^{®}interface facilitates the experiments with the test stand. Apart from the primary function of data acquisition and calculating the MOI from the sensor data and racket parameters, it visualizes the raw (Figure 10a), optimized signal (Figure 10b), and the computed period (Figure 11). Figure 12 shows a screenshot of the interface. Every pendulum needs different parameters (Equations (2) and (3)). This selection enables and disables the corresponding edit fields for the parameters. Before measuring, the I/O device is connected by specifying the device’s name. The measurement starts with an adjustable sample frequency and total sample time. After measuring, the program displays the original sensor signal, the optimized signal, and the resulting period duration from the FFT. The frequency spectrum can be plotted additionally.

## 3. Calibration, Testing, and Error Estimation

#### 3.1. Propagation of Error and Repeatability for Horizontal Rotation

_{r}around the horizontal rotation axes x and y.

_{0}is the mass of the test stand without the padel racket, m

_{r}is the mass of the padel racket, I

_{0}is the MOI with no load, T is the period, and d is the distance between the rotation pivot and the COM. The cradle, holder, screws, and nuts are set to have an MOI I

_{0}that must be subtracted from the result. The COM of the test stand without a padel racket lies on the rotation axis. Hence, it cannot oscillate without a racket and must be estimated using CAD software SolidWorks

^{®}. The approximation cannot be very accurate because the software assumes uniform density for every part, which is not the case for 3D-printing parts. MOI of the empty test stand is 24.998 kg cm

^{2}. Nevertheless, the approximation shows that I

_{0}is a mere fraction of the MOI of a padel racket (I

_{0}/I

_{r}≈ 0.1%), so the influence of the errors in I

_{0}is insignificant.

_{0}and d

_{r}are the distances between the rotation pivot and the COM of the non-loaded stand and the racket respectively. As for this case, d

_{0}= 0 applies because of the test stand symmetry, thus Equation (6) simplifies to:

_{r}, and the distance d

_{r}cause deviations in the final results for the MOI I

_{r}. According to error propagation, the standard deviations ΔT, Δm

_{r}, Δd

_{r}, and the partial derivatives to the parameters, must be determined to calculate ΔI

_{r}.

_{r}, m

_{r}, and T has a similar influence.

_{0}from estimated with the CAD, the standard deviation can be estimated ΔI

_{0}could have a maximum value of 0.025 kg cm

^{2}. Concerning the calculated average value of 137 padel racket, MOI in axes x and y I

_{r}= 1.963 × 10

^{−4}kg m

^{2}, the error propagation amounts to 0.0127%.

_{xy}) of the calculation of I

_{r}from horizontal pendulum rotation is shown in Equation (12) and is 0.820 kg cm

^{2}, where standard deviations were taken to be error estimation. The error propagation amount is 0.418%, considering average values of MOI x-y 196.313 kg cm

^{2}.

#### 3.2. Propagation of Error and Repeatability for Vertical Rotation

_{0}must be subtracted from the result to obtain the pure inertia of the padel racket. It is determined by measuring the time of the period without racket T

_{0}. The equation is:

_{z}) of the calculation of I

_{z}from vertical trifilar pendulum rotation is shown in Equation (21) and is 0.114 kg cm

^{2}, where standard deviations were taken to be error estimation. The error propagation amount is 0.658%, considering the average values of MOI axis z 17.282 kg cm

^{2}. This result is consistent with the findings [39] in the error analysis with tennis rackets, where period T and plate radius R are the most influential parameters in the relative error. According to Equations (15)–(20), if the length of the threads increases, the relative errors decrease.

^{2}) which have been measured 10 times and average values are: length (0.247, 0.236, and 0.225 m) and mass (0.381, 0.367, and 0.347 kg) (Figure 14).

#### 3.3. Summary on Error Estimation and Considerations

- Some rackets are manipulated with extra grips or protectors on the side. These items change the weight and balance point.
- The variance of the parameters between rackets of the same model is significant. Some manufacturers specify the weight of the rackets with a range of 20 g, which causes a variance that makes it impossible to generalize the results for all rackets of the same model accurately.
- The handle end of some models is oddly shaped or has a strap attached, making it difficult to accurately measure the distance to the COM and position the racket in the cradle.
- Even though ball bearings used in the horizontal pendulum have low friction, this friction damping the swing, and, after approximately ten cycles, the padel rackets stop due to the friction force.
- In the case of the vertical rotation axis, the rotation goes not precisely through the COM; since the thickness of the padel rackets varies, the axes do not always coincide.
- The trifilar vertical pendulum achieves worse results than the horizontal pendulum. However, it allows calculating the MOI when the pendulum’s axis passes through the COM of the padel racket, as occurs with the z axis. The trifilar pendulum requires a more complex and heavier cradle to fix the padel racket, the slight deviations between the pendulum axis and the distance to the COM influence the result. The initial manual impulse given to the trifilar pendulum makes it not only oscillate around its vertical axis, creating an overlapping of harmonic movements with different frequencies that add error to the measurement. The horizontal pendulum is an effortless and reliable design, but it only can be used when the axis of rotation of the pendulum and the COM are significantly separated.

## 4. Experiments and Results

#### 4.1. Correlation between Values

_{com}between handle end and COM (i.e., balance distance), and the moments of inertia I

_{x}, I

_{y}, I

_{z}through the COM are measured. Figure 1 shows the corresponding axes. Table 7 summarizes the data measured.

_{com}and I

_{x}is 0.82, and for d

_{com}and I

_{y}is 0.80. In addition, the two considered moments of inertia correlate strongly with each other. As shown in Figure 15, a padel racket with a high result for I

_{x}also has a high result for d

_{com}(balance) and also a high result for I

_{y}(Figure 16). Here the determination correlation coefficient (R

^{2}) amounts to 0.88 and 0.97, removing two outsiders.

_{z}, no correlation can be found. As expected, the location of the COM does not influence the inertia around the z axis. Moreover, neither the mass nor other axis MOI shows any correlation. The highest correlation coefficient amounts to only 0.41 between I

_{y}and I

_{z}.

#### 4.2. Maneuverability Parameter

^{2}) are not comfortable outside of technical language.

_{p}).

_{x}, I

_{y}, and I

_{z}are the MOIs of the racket on each axis and I

_{xR}, I

_{yR}, and I

_{zR}are the MOIs of the reference racket on each axis.

## 5. Study with Padel Users

#### 5.1. The Procedure of the Study

#### 5.2. Results of the Study

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 6.**Setup for the horizontal rotation. (

**a**) Computer-aided design (CAD) model. (

**b**) The limiter positions the racket with a defined distance between the rotation axis and grip end.

**Figure 7.**Setup for the vertical rotation. (

**a**) CAD model, (

**b**) final assembly, and (

**c**) detail of strings.

**Figure 8.**The location where the accelerometers are attached. (

**a**) For horizontal pendulum and (

**b**) for the vertical pendulum.

**Figure 10.**Plots of the signal in the different steps of the processing. (

**a**) Raw signal. (

**b**) Processed signal (filtered and trimmed).

**Figure 11.**Frequency spectrum of the signal after the FFT. The dominating frequency can be found at 0.920 Hz, which corresponds to a period duration of T = 1.086 s.

**Figure 15.**Moments of inertia I

_{x}(

**a**) and I

_{y}(

**b**) and median values as a function of the balance distance.

Device | Advantages | Disadvantages |
---|---|---|

Atwood machine | Easy calculation | Long time for preparation Controllable motor recommended |

Pendulum | Simple design | Different designs depending on the axis |

Springs | Easy calculation Simple design | Spring characteristics must be determined |

**Table 2.**Average values, partial derivation, standard deviation, and the influence of every parameter in error for the case of horizontal rotation.

Parameter | x_{i} | ∂I/∂x_{i} | ∆xi | ∂I/∂x_{i}∆xi10^{−4} |
---|---|---|---|---|

T | 1.103 s | 0.0485 | 0.00121 s | 0.587 |

m_{r} | 0.363 kg | 0.0736 | 0.000516 kg | 0.380 |

d_{r} | 0.265 m | 0.1007 | 0.000425 m | 0.428 |

Model | Theoretical MOI X (kg cm ^{2}) | Measured MOI X (kg cm ^{2}) | MOI Standard Deviation | Accuracy Error (%) | Standard Deviation/Theoretical MOI (%) |
---|---|---|---|---|---|

a | 202.936 | 203.69 | 0.704 | −0.37 | 0.35 |

b | 186.116 | 184.53 | 0.262 | 0.85 | 0.14 |

c | 168.074 | 168.49 | 0.265 | −0.25 | 0.16 |

**Table 4.**Average values, partial derivation, standard deviation, and the influence of every parameter in error for the case of vertical rotation.

Parameter | x_{i} | ∂I/∂x_{i} | ∆xi | ∂I/∂x_{i}∆xi10^{−4} |
---|---|---|---|---|

T | 0.4829 s | 0.0084 | 0.00108 s | 0.0903 |

T_{0} | 0.31008 s | −0.0016 | 0.0016 s | −0.0258 |

m_{r} | 0.3630 kg | 0.0039 | 0.000516 kg | 0.0201 |

m_{0} | 0.1568 kg | 0.0023 | 0.000168 kg | 0.0039 |

L | 0.1498 m | −0.0018 | 0.000087 m | −0.0015 |

R | 0.10008 m | 0.0354 | 0.000174 m | 0.0607 |

Model | Theoretical MOI X (kg cm ^{2}) | Measured MOI X (kg cm ^{2}) | MOI Standard Deviation | Accuracy Error (%) | Standard Deviation/Measured MOI (%) |
---|---|---|---|---|---|

A | 20.000 | 19.750 | 0.0777 | −1.14 | 0.39 |

B | 17.506 | 17.527 | 0.0412 | −0.12 | 0.24 |

C | 15.231 | 14.971 | 0.0412 | 1.71 | 0.28 |

Test Stand | Error Propagation (%) | Accuracy Error (%) | Repeatability (%) |

Horizontal | 0.418 | 0.491 | 0.215 |

Vertical | 0.658 | 0.990 | 0.303 |

Parameter | m (g) | d_{com} (mm) | I_{x} (kg cm^{2}) | I_{y} (kg cm^{2}) | I_{z} (kg cm^{2}) |
---|---|---|---|---|---|

Avg | 363.07 | 265.52 | 190.599 | 202.027 | 17.282 |

Min | 316 | 246 | 161.405 | 169.631 | 14.633 |

Max | 387 | 283 | 209.869 | 224.166 | 19.812 |

**Table 8.**Results of the test with users. Each value represents the percentage of users classifying the racket in the category.

Padel Racket Index | High | Medium | Low |
---|---|---|---|

0.90 | 91% | 9% | 0% |

1.02 | 9% | 59% | 32% |

1.12 | 0 | 32% | 68% |

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**MDPI and ACS Style**

Blanes, C.; Correcher, A.; Beltrán, P.; Mellado, M.
Identifying the Inertial Properties of a Padel Racket: An Experimental Maneuverability Proposal. *Sensors* **2022**, *22*, 9266.
https://doi.org/10.3390/s22239266

**AMA Style**

Blanes C, Correcher A, Beltrán P, Mellado M.
Identifying the Inertial Properties of a Padel Racket: An Experimental Maneuverability Proposal. *Sensors*. 2022; 22(23):9266.
https://doi.org/10.3390/s22239266

**Chicago/Turabian Style**

Blanes, Carlos, Antonio Correcher, Pablo Beltrán, and Martin Mellado.
2022. "Identifying the Inertial Properties of a Padel Racket: An Experimental Maneuverability Proposal" *Sensors* 22, no. 23: 9266.
https://doi.org/10.3390/s22239266