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Edited by: Ana Filipa Silva, Polytechnic Institute of Maia, Portugal

Reviewed by: Uday C. H. Hasan, Al-Kitab University, Iraq; Chiang Liu, University of Taipei, Taiwan; Tomoyuki Nagami, Kitasato University, Japan

This article was submitted to Biomechanics and Control of Human Movement, a section of the journal Frontiers in Sports and Active Living

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

This study explored the mechanical factors that determine accuracy of a baseball pitching. In particular, we focused on the mechanical parameters at ball release, referred to as release parameters. The aim was to understand which parameter has the most deterministic influence on pitch location by measuring the release parameters during actual pitching and developing a simulation that predicts the pitch location from given release parameters. By comparing the fluctuation of the simulated pitch location when varying each release parameter, it was found that the elevation pitching angle and speed significantly influenced the vertical pitch location, and the azimuth pitching angle significantly influenced the horizontal pitch location. Moreover, a regression model was obtained to predict the pitch location, and it became clear that the significant predictors for the vertical pitch location were the elevation pitching angle, the speed, and spin axis, and those for the horizontal pitch location were the azimuth pitching angle, the spin axis, and horizontal release point. Therefore, it was suggested that the parameter most affecting pitch location weas pitching angle. On the other hand, multiple regression analyses revealed that the relation between release parameters varied between pitchers. The result is expected to contribute to an understanding of the mechanisms underlying accurate ball control skill in baseball pitching.

In various sport-related motor skills, such as throwing, kicking, and hitting, accurately controlling an object (typically a ball) to a target position is one of the most important skills. In this skill, there is a difference in performance, i.e., reproducibility of the ball arrival position, even among experts (Kawamura et al.,

It is also known that the choice of values of release parameters varies widely between different pitchers, even if the same position is targeted (Jinji and Sakurai,

The relationship between parameters influencing the results of movements have been evaluated in previous studies. The relation in which each parameter does not vary independently each time, but fluctuates while maintaining a relationship that compensates for the variability of other parameters has been reported. This is considered to be the best method, which is used by some skilled players, for improving accuracy of the arrival position without increasing reproducibility. The existence of such “compensatory coordination” which contributes to the stability of performance has been reported using many constrained virtual tasks (Müller and Loosch,

Some sports skills, such as baseball pitching, involve more parameters for determining the arrival position than the number used in previous studies. In baseball pitching, it has been shown that the spin rate, which is the number of times the ball rotates per unit time, and the spin axis, which is the orientation of the ball rotation, differs between pitchers, and contribute to the ball trajectory in addition to the release position and release velocity vector (Jinji and Sakurai,

The purpose of this study was to investigate the degrees of influence of each baseball pitching release parameter on pitch location. In addition to the release point and the velocity vector used in the previous studies that performed the throwing task (Kudo et al.,

Seven skilled baseball pitchers participated (sex: male; age: 28.1 ± 9.9 years; height: 175.9 ± 5.1 cm; body mass: 76.5 ± 3.5 kg; 6 right-handed and 1 left-handed), including one former professional pitcher from the NPB (Nippon Professional Baseball; Japan's top baseball league). They pitched 30 fastballs on the mound in an indoor stadium and were instructed to aim at the catcher's mitt, which was 90 cm above the ground, 40 cm outside from the center of home base (outside is defined based on the same side batters, e.g., a right-handed batter in case of a right-handed pitcher), and 50 cm behind home base. The data of 187 pitches were obtained, excluding data in which measurement errors occurred. The release parameters were measured using TrackMan Baseball (TRACKMAN). TrackMan Baseball detected release timing and the release parameters were taken. For the measurement of pitch location, a DV camera (Panasonic HC-V 100 M, Japan) was installed 7–8 m in front of home base, and the moment when the catcher caught the ball was photographed from the front side of the catcher (30 Hz). To obtain the position coordinates, we calibrated 3 points in the horizontal direction (1.5 m intervals) and 5 points in the vertical direction (0.5 m intervals) on the plane, giving a total of 15 calibration points for the catching position. The calibration points were digitized using numerical analysis software (MATLAB, Mathworks, Japan), and the average standard error was set to 1.0 cm or less. The position coordinates of the pitch location were calculated by digitizing the center point of the ball at the moment of catching and by using direct linear transformation. All participants provided informed consent, and the study was performed in accordance with the Declaration of Helsinki and with the approval of the ethics committee of the University of Tokyo.

To create a simulation for pitch location prediction, it was necessary to consider the mechanical elements acting on the ball at release. The three-dimensional orthogonal coordinates were defined as follows; the origin was taken as the center of the pitcher plate on the mound, the orientation of the x axis was in the direction of home base, the y axis was oriented vertically upwards, and the z axis was oriented in the third base direction (

Three-dimensional orthogonal coordinate setup in this study. _{1} (ranging from −90° to 90°) and azimuth angle θ_{2} (ranging from −90° to 90°), and the spin axis is shown by the elevation angle θ_{3} (ranging from −90° to 90°) and the azimuth angle θ_{4} (ranging from −90° to 90°).

Next, an equation of motion was set based on the coordinate axes, as follows. Generally, drag (F_{d}) and lift (F_{l}) are expressed as follows.

where ρ (= 1.2 kg/m^{3}) represents air density, S (= 4.3 × 10^{−3} m^{2}) is the sectional area of the ball, and V is the velocity magnitude. C_{d} (= 0.35) and C_{l} (= (πS) 0.5 × spin rate/2) are the drag coefficient and lift coefficient, respectively, which are the same as those reported in (Kray et al.,

When the spin axis of the ball is perpendicular to the traveling direction, the drag increases as the speed of the ball increases, and the lift increases as the rotation speed increases. When the spin axis is not perpendicular to the traveling direction, the lift force is perpendicular to both the traveling direction and spin axis, which means the lift force can be represented by the cross product of the traveling direction and spin axis of the ball. From the above discussion, the equation of motion of the ball considering drag and lift can be expressed as follows.

where m (= 0.145 kg) is the mass of the ball, H is the magnitude of the cross product of speed and spin axis, and g (= 9.81 m/s^{2}) is gravitational acceleration.

a_{x}, a_{y}, and a_{z} are the rotational shaft angles of the ball as follows:

At this time, the equation of motion can be viewed as a second order differential equation with time t as a variable. The pitch location of the ball can be calculated by solving this equation of motion as an initial value problem of a differential equation. However, it is not possible to solve the equation algebraically if specific functions, such as the pitching trajectory, are not determined. Therefore, it is necessary to approximate the pitching trajectory through a numerical analysis. In this study, the Dormand–Prince method (Dormand and Prince, _{2}), spin rate (n), and spin axis (θ3, θ4). The pitch location (y, z), which was 50 cm behind home base in this study, is calculated based on these parameters and numerical analysis. Due to the limited functionality of TrackMan Baseball, it was not possible to measure the horizontal spin axis (θ4); thus, its simulated value was set to a constant of 30° because several studies (Jinji and Sakurai,

The variation in pitch location was simulated while varying each parameter individually. Each parameter was varied from its minimum to the maximum value for each pitcher, and the other parameters were fixed to the average for each pitcher. The results indicated that the larger the variation in the pitch location is, the higher the possibility that the pitch location is changed by the parameter.

In addition to simulation analysis, a regression model was obtained to predict the pitch location. By backward-forward stepwise multiple regression analysis, the explanatory rate of the pitch location of each release parameter was calculated. This method finds the optimal combination of explanatory variables by reducing the number of explanatory variables from the most complex model (using all explanatory variables). If there is parameter whose p value is larger than 0.05, the parameter was reduced from the model. The regression was run separately for each pitcher. We used MATLAB to find a regression model with a coefficient of determination as close to 1 as possible.

The mean speed of the ball in this study was 32.6 ± 2.2 m/s, whereas it was 33.8 ± 1.7 m/s in Jinji and Sakurai (

Mean release parameters and pitch location for each pitcher.

_{2} |
||||||||||
---|---|---|---|---|---|---|---|---|---|---|

A | 32.2 ± 1.0 | 1.38 ± 1.1 | −3.66 ± 1.09 | 26.9 ± 2.3 | −53.0 ± 3.8 | 1.72 ± 0.039 | 1.49 ± 0.025 | 0.25 ± 0.041 | 0.63 ± 0.30 | −0.23 ± 0.20 |

B | 33.0 ± 0.50 | −1.27 ± 0.75 | −3.81 ± 0.47 | 31.3 ± 0.87 | −31.3 ± 3.7 | 1.73 ± 0.047 | 1.79 ± 0.029 | 0.50 ± 0.024 | 0.50 ± 0.25 | −0.16 ± 0.11 |

C | 33.4 ± 0.34 | −0.72 ± 0.69 | −2.70 ± 0.56 | 31.1 ± 0.64 | −21.0 ± 2.9 | 1.69 ± 0.024 | 1.62 ± 0.022 | 0.38 ± 0.020 | 0.51 ± 0.21 | −0.16 ± 0.14 |

D | 33.2 ± 0.72 | −0.26 ± 0.83 | −4.00 ± 0.55 | 29.9 ± 1.0 | −32.4 ± 5.3 | 1.73 ± 0.029 | 1.58 ± 0.012 | 0.48 ± 0.027 | 0.62 ± 0.26 | −0.27 ± 0.13 |

E | 35.9 ± 0.38 | −0.49 ± 1.51 | −3.85 ± 0.91 | 33.4 ± 1.1 | −41.3 ± 3.9 | 1.63 ± 0.030 | 1.60 ± 0.019 | 0.58 ± 0.033 | 0.74 ± 0.43 | 0.048 ± 0.21 |

F | 32.5 ± 0.49 | 1.50 ± 0.99 | −3.38 ± 1.07 | 27.3 ± 4.7 | −9.47 ± 5.9 | 1.78 ± 0.049 | 1.38 ± 0.022 | 0.81 ± 0.034 | 0.99 ± 0.26 | −0.085 ± 0.35 |

G | 28.0 ± 0.27 | 1.36 ± 1.64 | −1.52 ± 0.61 | 24.6 ± 0.80 | 2.39 ± 6.1 | 1.81 ± 0.021 | 1.59 ± 0.001 | 0.15 ± 0.019 | 0.66 ± 0.44 | −0.078 ± 0.21 |

Mean | 32.6 ± 2.2 | 0.28 ± 0.99 | −3.11 ± 0.80 | 29.0 ± 2.76 | −25.2 ± 17.4 | 1.72 ± 0.06 | 1.55 ± 0.08 | 0.43 ± 0.20 | 0.66 ± 0.15 | −0.17 ± 0.11 |

Comparing the variation in the pitch location caused by changing each release parameter, when the elevation pitching angle (θ1) was changed, the fluctuation of the vertical pitch location was the largest.

When the azimuth pitching angle (θ2) was changed, the fluctuation of the horizontal pitch location was the largest.

The fluctuation of the vertical and horizontal pitch location is summarized in

Fluctuation of the vertical and horizontal pitch locations when each release parameter was changed. The elevation pitching angle (θ_{1}) and speed significantly influenced the vertical pitch location, and the azimuth pitching angle (θ_{2}) significantly influenced the horizontal pitch location. **0.01≤

Multiple regression was used with the release parameters to independently establish regression equations for each pitcher. The average ^{2} values of regression models to predict the vertical pitch and horizontal pitch location for each pitcher were 0.97 ± 0.02 and 0.96 ± 0.04, respectively.

Coefficient of regression equation for each patient.

_{2} |
^{∧}2 |
||||||||
---|---|---|---|---|---|---|---|---|---|

A | – | 0.195 | – | – | – | – | – | – | 0.945 |

B | 0.054 | 0.318 | – | – | 0.009 | – | – | – | 0.988 |

C | 0.060 | 0.315 | – | – | 0.008 | – | – | – | 0.978 |

D | 0.100 | 0.322 | – | – | 0.014 | – | – | – | 0.986 |

E | 0.061 | 0.300 | – | – | −0.009 | – | – | – | 0.991 |

F | 0.143 | 0.298 | −0.024 | – | – | – | 0.988 | – | 0.984 |

G | – | 0.235 | – | 0.054 | – | – | – | – | 0.931 |

Mean | 0.97 ± 0.02 | ||||||||

_{2} |
^{∧} |
||||||||

A | – | – | 0.173 | – | −0.009 | – | – | – | 0.911 |

B | – | – | 0.215 | – | −0.007 | – | – | 0.624 | 0.988 |

C | – | – | 0.225 | – | −0.004 | – | – | 1.164 | 0.992 |

D | – | −0.015 | 0.223 | – | −0.005 | – | – | 0.626 | 0.992 |

E | – | 0.015 | 0.242 | – | −0.006 | – | – | 0.592 | 0.990 |

F | – | – | 0.310 | – | −0.014 | – | – | – | 0.983 |

G | – | – | 0.318 | – | – | – | – | – | 0.885 |

Mean | 0.96 ± 0.04 |

Predictive parameters of vertical (top row) and horizontal (bottom row) pitch locations for each pitcher. The values indicate the

The present study investigated the extent to which pitch location changes when the release parameters vary within a realistic range. By comparing the fluctuation of the simulated pitch location when varying each release parameter, it was found that the elevation pitching angle and speed significantly influenced the vertical pitch location, and the azimuth pitching angle significantly influenced the horizontal pitch location. Moreover, a regression model was obtained to predict the pitch location, and it became clear that the significant predictors for the vertical pitch location were the elevation pitching angle, speed, and spin axis, and those for the horizontal pitch location were the azimuth pitching angle, spin axis, and horizontal release point. Therefore, it was suggested that the parameter most affecting pitch location was pitching angle.

It can be considered that the method used in this study identifies the variability of the pitch location when each release parameter fluctuates from the average value of each pitcher. However, how easily the release parameters themselves vary may be different. Therefore, here, by interpreting the fluctuation range of the measured value of each release parameter as the easiness of variation of each release parameter, the influence of each release parameter on the pitch location was compared as fairly as possible.

Based on the above, it can be considered that the variation in the pitch location is mainly caused by the variation in the pitching angle, suggesting that adjustment of the pitching angle is a crucial factor for accuracy of baseball pitching. By comparing the pitch location when each parameter was varied independently, it was found that the pitching angle and speed, i.e., the velocity vector, significantly affected the pitch location for all pitchers. In particularly, with respect to pitching angle, it was found that a deviation of several degrees produces a deviation of several tens of centimeters. This result was further supported by multiple regression analysis. The reason why such a result was obtained may be that pitching is a task for which the speed of the projectile at the time of release is relatively large compared to the other throwing tasks, and the distance to the target is sufficiently long. It is considered that the variation in the pitching angle, which is the direction of the velocity vector, greatly affected the variation in the pitch location, the distance to which is long. Moreover, how release parameters are defined also the one of the reasons. However, the release parameters used in this study are common in expressing the ball movement. In previous studies, the same way to define the release parameters as this study (ex. Nagami et al.,

Although the fluctuation of the spin axis was very small in the simulation, its explanatory rate for pitch location was high for many pitchers. This result showed that spin axis had little influence when it was fluctuated independently but had a significant explanatory rate when combined with other parameters. Therefore, it was suggested that the spin axis covariated with other parameters to affect pitch location. The release point was conversely shown to cause fluctuation of pitch location in the simulation, but it had no explanatory rate for pitch location. This result suggested that the influence of release point was canceled by changes of other parameters. Thus, the release point may have a cooperative relationship with other parameters. Some parameters, such as spin rate, showed little influence on the variability of pitch location in both the simulation and regression analysis. However, in the previous study, the ratio of the spin rate to ball speed considering the direction of spin axis (i.e., effective spin parameter) would affect the lift coefficient more strongly than the spin rate and the spin axis separately (Nagami et al.,

In addition to the similar trend among all pitchers in the simulation analysis, the multiple regression analysis showed that the explanatory rates of each parameter were different for each pitcher. This indicates that the elevation pitching angle and speed are common factors in determining pitch location, but other parameters, such as the spin axis and release point, likely have different relations among the release parameters for each pitcher. One of the reasons for this difference might be that there can be specific combinations of release parameters in individual pitchers, even when targeting the same location (Jinji and Sakurai,

Some limitations of this study should be noted. The experimental environment in this study was different from that in actual baseball games. The data was measured indoors so as to reduce the effect of wind and air currents on the aerodynamic characteristics of the ball that affect the ball trajectory. The participants always threw at the same target position, and there was no batter. Some release parameters might be influenced by these factors. However, the main results of study are considered not to be different largely by these factors because the release parameters used in this study did not have great difference from the studies have referred before. Moreover, sample size of this study was small. Only seven skilled baseball pitchers participated in the study. Different results may be obtained with more participants with various skill levels. It may need to investigate more participants in more practical settings in the future study.

It should be noted that the result may be specific to the fastball used in this study. The participants threw only 4 seam fastballs in this study, but pitchers throw various types of pitches, such as braking balls, in actual baseball games. Previous studies investigating release parameters for various ball types have shown that, depending on the type of ball, the spin rate and the spin axis can significantly influence on the trajectory of the ball (Jinji and Sakurai,

This is the first study to investigate the influence of release parameters including spin parameters, on the pitch location. The fluctuation of the pitch location was simulated for variation of each release parameter, and it was revealed that each parameter's contribution to the pitching accuracy varied. In previous studies, the pitch location was found to be related to variability in joint kinematics and ball release timing (Hore,

With recent advances in science and technology, measurement equipment, and data analysis technology have made remarkable progress. For example, TrackMan Baseball (TrackMan, Denmark), which was developed in 2003, is generally used as a data analysis system in major league baseball. Because it is possible to easily measure various parameters of the ball with high precision and in real time, practice, and teaching can be aimed at measurable numerical values rather than ambiguous feeling. From this research, we were able to show that the contribution to pitching accuracy varies depending on the parameters. As it is difficult to be conscious of multiple aspects during actual movements, extracting important elements may be useful for practice and teaching. Moreover, understanding the differences of individuals may contribute to performance improvements. Pitching involves multiple skills, such as increasing the ball speed, improving control, and learning breaking balls. Therefore, various styles can coexist even among skilled pitchers. When targeting tasks as pitching, it is important for coaches and players not only to know the tendency seen among skilled players and but to pay attention to cases that deviate from it. The advantage of the approach in this study with the incorporation of theoretical knowledge of body and ball movements, is that the knowledge about what is typical and when it may not be valid is acquired.

This study revealed the degree of influence of each release parameter on the pitch location in baseball pitching. The fluctuation of the pitch location was simulated for variation of each release parameter. It was revealed that, the elevation pitching angle and speed significantly influenced the vertical pitch location, and the azimuth pitching angle significantly influenced the horizontal pitch location. Moreover, a regression model was obtained to predict the pitch location, and it became clear that the significant predictors for the vertical pitch location were the elevation pitching angle, speed, and spin axis (θ3), and those for the horizontal pitch location were the azimuth pitching angle, spin axis, and horizontal release point. Therefore, it was suggested that the parameters most affecting pitch location were pitching angle. In future work, we will consider a relationship of parameters more clearly to further elucidate the factors affecting pitch location.

The datasets generated for this study are available on request to the corresponding author.

The studies involving human participants were reviewed and approved by the ethics committee of the University of Tokyo. The patients/participants provided their written informed consent to participate in this study.

AK, KK, KN, and SW contributed conception and design of the study. HK, TM, and MK performed experiments. AK performed the analysis and wrote the first draft of the manuscript. KN revised partially the manuscript. All authors contributed to manuscript revision, and read and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors would like to thank Dr. Nasu and other members of NTT Communication Science Laboratories for technical assistance with the experiments. We also thank Dr. Tanabe of Aoyama Gakuin University for support of research and the members of National Institute of Fitness and Sports in Kanoya for their hospitality during our visit. This work was in part supported by CREST, Japan Science and Technology Agency.

The Supplementary Material for this article can be found online at: