# Exploring the Mechanism of Skilled Overarm Throwing

Although the kinematics and dynamics of overarm throws, such as baseball pitching, have been studied extensively, the relations between these measures remain largely unknown. This review uses a three-dimensional analysis to characterize the mechanical basis of skilled overarm throws by focusing on how each joint angular acceleration is produced by the muscle torques, gravity torques, and velocity-dependent torques.

A 3D analysis identifies the mechanical basis of the large joint angular accelerations in overarm throwing by skilled performers.

^{1}Graduate School of Education, the University of Tokyo, Japan; ^{2}Japan Society for the Promotion of Science, Japan; ^{3}Department of Life Sciences, Graduate School of Arts and Sciences, the University of Tokyo, Japan

*Address for correspondence:* Masaya Hirashima, Ph.D., Graduate School of Education, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan 113-0033 (E-mail: hira@p.u-tokyo.ac.jp; pingdao@tkf.att.ne.jp).

*Accepted for publication*: January 31, 2008.

Associate Editor: Roger M. Enoka, Ph.D.

## INTRODUCTION

Many coaches and scientists have explored the methods and mechanisms of overarm throws, such as a fastball in baseball pitching. Throws in baseball have been characterized in terms of muscle activity (^{13,15}), joint torque (^{10}), joint angle and angular velocity (^{2,7,8,21,22,26}), and translational velocity of the fingertip (^{23}) (Fig. 1). However, knowledge of the cause-and-effect relations between these variables is severely lacking, although it is a fundamental knowledge for improving and maintaining athletes' performance. In this review, we examine previous studies of baseball pitching and introduce our mathematical-based approach to determine the mechanical cause of a skilled overarm throw by focusing on the following: 1) the relation between joint angular velocities and translational ball velocity, and 2) the relation between torques and joint angular accelerations.

## RELATION BETWEEN JOINT ANGULAR VELOCITIES AND TRANSLATIONAL VELOCITY OF THE BALL

The first step in understanding the factors that influence ball velocity is to determine the joint angular velocities that are mostly responsible for the translational velocity of the ball. Here, we consider a serially linked four-segment model of the trunk and the upper limb that has 13 degrees of freedom (DOF). (There are three translational DOF and three rotational DOF for the trunk, three rotational DOF for the shoulder, two rotational DOF for the elbow, and two rotational DOF for the wrist.) Suppose the joint rotates only about the *i*-th DOF axis at the angular velocity of _{i}(Fig. 2). The translational velocity of the fingertip (ṙ_{i}) produced by this rotation is expressed as

where _{i}(=_{i} *ui*) is the angular velocity vector (u_{i} is the unit vector of the joint axis) and p_{i} is the vector from the joint center to the fingertip. Because joints actually rotate about all DOF, the actual translational velocity of the fingertip (*ṙ*)is expressed as the sum of their effects.

After some calculations, we obtain

where _{G} = (_{1} _{2}…_{13})^{T} is the velocities of the generalized coordinates and **J** is a 3 × 13 matrix whose *i*-th column vector **J** _{i} equals *u**i* × *p**i*. (As for the three translational DOF at the trunk (*i* = 1,2,3), *J**i* = *u**i*.) The matrix **J** is called Jacobian matrix that represents the linear relation between joint angular velocities and translational velocity of the end point (^{4}). Note that **J** is not constant during motion but that it changes with the configuration of the limb. Therefore, the critical variables that influence fingertip velocity at ball release are the following: 1) all of the angular velocities (_{G}) and 2) the configuration of the limb (**J**) at the instant of ball release.

We can determine the contribution of each angular velocity to the magnitude of the fingertip velocity by projecting each vector (ṙ_{i}) onto the unit vector (ṙ_{u}) of the translational velocity of the fingertip (^{9,23,28}).

Figure 3 shows the contributions of the 13 DOF velocities to the fingertip velocity at ball release time in skilled baseball players (^{18}). This analysis revealed that it was mainly produced by the leftward rotation of the trunk, internal rotation of the shoulder, elbow extension, and wrist flexion. These results indicate that the skilled throwers produced large angular velocities at only 4 DOF and configured the limb to maximize the contributions by these four angular velocities. From these kinematics, we next examine how the large angular velocities are generated by various torques.

## RELATIONS BETWEEN TORQUES AND JOINT ANGULAR ACCELERATIONS

In this section, we calculate the muscle torques with the inverse dynamics technique and further examine the production of each joint angular acceleration by torques from various sources. In multi-DOF movements, joint rotations are produced not only by gravity and muscle forces but also by interjoint interactions (^{19,29}). Many researchers in the field of biomechanics and neuroscience have been interested in the ability of humans to coordinate multiple muscles during the performance of accurate movements in the presence of multijoint dynamics (^{3,5,12,14,16,18,19,25}). After Zajac and Gordon (^{29}) gave a seminal review on the cause-and-effect relation between muscle torques and joint angular accelerations in the two-joint movements in two-dimensional (2D) space (As for the three-dimensional (3D) movements, Zajac and Gordon (^{29}) only briefly described the single-joint movements of the ball-and-socket joint (3-DOF) without mathematical equations.), researchers have examined the influence of each muscle on the acceleration of the body during walking and pedaling (^{1,11,20,30}).

This approach, however, has not been extended to 3D movements, such as baseball pitching and tennis stroke. This lack of progress is likely attributable to the absence of a method to examine the relations between various torques and joint angular acceleration for multijoint and multi-DOF movements in 3D space. Therefore, we next develop a method that extends a single-joint and 1-DOF system analysis to the multijoint and multi-DOF system in 3D space. One consequence of this development is that motor control studies on the multijoint dynamics (^{5,12,16,18,25}) often include erroneous interpretations about the influence of torques on joint angular acceleration.

### Methods to Identify the Cause-and-Effect Relation

#### One-Joint and 1-DOF system

In a single-joint and 1-DOF system, the effect of a torque on the joint rotation is very simple because there is only the one equation of motion. An angular acceleration produced by a torque is determined by the magnitude of that torque and the moment of inertia (*I*) about the joint axis. For example, consider a 1-DOF elbow joint movement (θ, elbow joint angle) in a vertical plane. Because there is a gravity torque (*g*(θ)) and a net muscle torque (τ) due to the activity of all muscles crossing the elbow joint, the equation of motion is

and the angular acceleration (θ¨) produced by the two torques is simply expressed as:

#### Two-Joint and 2-DOF system in 2D space

The relations become more complex when multiple joints are involved in the movement. Even in 2D space, for example, there are more than two equations of motion, and furthermore, these equations interact with each other. For example, consider a 2-DOF (*e.g*., θ_{1}, shoulder joint angle; θ_{2}, elbow joint angle) movement in a vertical plane. Two equations of motion can be written in the following form:

where θ = (θ_{1}θ_{2})^{T} are the generalized coordinates, **JOURNAL/essrv/04.02/00003677-200810000-00007/ENTITY_OV0496/v/2017-07-29T043525Z/r/image-png** = (**JOURNAL/essrv/04.02/00003677-200810000-00007/ENTITY_OV0496/v/2017-07-29T043525Z/r/image-png** _{1} **JOURNAL/essrv/04.02/00003677-200810000-00007/ENTITY_OV0496/v/2017-07-29T043525Z/r/image-png** _{1})^{T} are the first derivatives, and **V**(θ, ) is the angular-velocity-dependent torque. The complete mathematical derivation is described in the appendix of reference (^{16}). Consider the forward dynamics case of how much angular accelerations (θ¨_{1} and θ¨_{2}) occur if muscle torques (τ_{1} and τ_{2}) are applied at two joints when the system is in the state of θ and **JOURNAL/essrv/04.02/00003677-200810000-00007/ENTITY_OV0496/v/2017-07-29T043525Z/r/image-png**. We cannot solve this problem by dealing with each equation separately because unknown variables (θ¨_{1} and θ¨_{2}) are present in both equations. Therefore, we must solve them as simultaneous equations in the following way:

The first point that should be emphasized is that each angular acceleration is affected by both muscle torques. Thus, a muscle torque at one joint induces angular accelerations at both joints; this is an interjoint interaction. Note that the muscle-induced accelerations (** I**(θ)

^{−1}τ ) instantaneously emerge with the production of muscle torques.

The second point is that each angular acceleration is affected by a new term **V**(θ, **JOURNAL/essrv/04.02/00003677-200810000-00007/ENTITY_OV0496/v/2017-07-29T043525Z/r/image-png**) that depends on the angular velocities of both joints. Thus, rotation at one joint induces angular accelerations at both joints, which is another interjoint interaction. Because angular velocity is an integrated value of the angular acceleration, the velocity-induced acceleration at a certain instant (** I**(θ)

^{−1}

**V**(θ, )) reflects the history of effects from muscle and gravity torques until that instant and does not correspond to the instantaneous effects from muscle and gravity torques at that instant.

In summary, each angular acceleration is produced by the following: 1) muscle torques at all joints, 2) gravity torques applied to all joints, and 3) velocity-dependent torques at all joints.

#### Multijoint and multi-DOF system in 3D space

We finally examine the multi-DOF system in 3D space by considering the 7-DOF system of the human upper extremity. Although 3D movements seem to be difficult, the basic approach is the same as the 2-DOF system in 2D space. This is accomplished by writing the Newton-Euler equation for each segment and then to express them by using the acceleration of generalized coordinate as in the equations 7 and 8. In the 2-DOF system in 2D space, we selected the joint angles as generalized coordinates, although the segment angles relative to the ground could also be used as generalized coordinates. The joint angle expression seems easier to interpret and, hence, is widely used by clinicians and sports coaches. Accordingly, we defined the three orthogonal joint coordinate axes for each joint as follows: 1) shoulder - internal-external rotation, elevation-depression, and third-axis rotation; 2) elbow - extension-flexion, pronation-supination, and varus-valgus; and 3) wrist - longitudinal rotation, flexion-extension, and ulnar-radial deviation (for detailed definition, see Fig. 3B in (^{17})).

The Newton-Euler equation of a segment *i* (*i =* 1: upper arm, 2: forearm, 3: hand) is

where *mi* is the mass, **I****i** is the inertia tensor, ɑ*gi* is the translational acceleration of the center of gravity, ω*i* and ω˙*i* are the angular velocity and acceleration relative to the ground, *f**i* is the sum of all forces, *n**i* is the sum of all moments applied to the segment. Please note that these vectors and tensors are expressed in terms of the global coordinate system.

To use the joint coordinate expression, we expand ω˙*i* by using joint angular acceleration (*i*θ¨*i*) in the following way:

where *i* *i* is the joint angular velocity vector of i-th joint (i = 1: shoulder, 2: elbow, 3: wrist) that is expressed in the *i*-th joint coordinate system. (Upper-left subscript indicates the joint coordinate system in terms of which the vector is expressed. When a vector is expressed in terms of the global coordinate system, the upper-left subscript is omitted.) For example, ^{1} _{1} = (_{1−IE} _{1−ED} _{1−k})^{T} consists of three angular velocities about the internal-external axis, elevation-depression axis, and third axis at the shoulder. *i*θ¨*i* is the joint angular acceleration vector expressed in the *i*-th joint coordinate system. For example,

*i* ** R** is the coordinate transformation matrix from the

*i*-th joint coordinate system to the global coordinate system. Note that Ω

_{i}is not the angular velocity vector of the

*i*-th segment but the angular velocity vector of the

*i*-th joint coordinate system expressed in the global coordinate system.

By substituting equation 11 into equation 10 for all three segments and reorganizing, we obtain the desired form of the equation as follows:

where

where *i*τ*i* is the muscle torque vector at the *i*-th joint that is expressed in the *i*-th joint coordinate system. For example, *1*τ*1* = (τ*1−IE* τ*1−ED* τ*1−k*)*T* consists of three muscle torques about the internal-external axis, elevation-depression axis, and third axis at the shoulder. The complete mathematical expression of equation 12 can be seen in equations 15, 19, and 20 of (^{17}) (see also Erratum). Note that the system inertia matrix (**I**(**θ**) is now a 9 × 9 matrix. Because the human upper extremity has only the 7 DOF, we eliminated 2 DOF (varus-valgus at the elbow and the longitudinal rotation at the wrist) and obtained the 7-DOF equation of motion that can be expressed in the same form as equation 12.

To understand the effect of the torques on the joint angular acceleration, we must multiply the inverse of the ** I**(θ)∈

*R*^{7−7}as follows:

Because this equation is the same form as the equation 9 in the 2-DOF system, we can discuss the cause-effect relations between torques and accelerations in a similar way as the 2-DOF system.

Equation 13 tells us that the angular acceleration of shoulder internal rotation (θ¨_{1−IE}) is produced by the following: 1) all seven muscle torques, 2) the velocity-dependent torque, and 3) the gravity torque (see equation 14).

where A*1i* is the (1, *i*) component of the matrix ** I**(

**θ**)

^{−1}. It is important to note that θ¨

_{1−IE}is affected not only by the muscle torque about its own axis (τ

_{1−IE}) but also by the muscle torques about the other two axes at the shoulder joint (τ

_{1−ID}, τ

_{1−k}) and the muscle torques about all the DOF at the other joints. Thus, a muscle torque at 1 DOF induces angular accelerations on all the DOF. Therefore, this phenomenon could be referred to as an inter-DOF interaction instead of an interjoint interaction.

There is a difference between 2D and 3D space for the velocity-dependent torque (**V**(θ,)). It consists of a centripetal torque and Coriolis torque in 2D space but includes a gyroscopic torque (ω_{i} × **I** _{i}ω_{i}) in 3D space.

### Comments on Motor Control Studies of Multijoint Dynamics

Many researchers have been interested in how humans control complex multijoint dynamics because a joint rotation is affected not only by the muscle activation and gravity but also by interjoint interaction. They have studied this problem by analyzing various types of movements such as reaching (^{3,25}), drawing (^{5}), throwing (^{16,18}), kicking, and walking (^{24,27}). However, there have been limitations in the methods used to analyze these actions. First, the definition of the interaction torque varies across studies, which makes it difficult to compare the results from different studies. Second, some studies have incorrectly interpreted the influence of torques on joint angular accelerations largely because of a lack of attention on whether they are solving an inverse or forward dynamics problem. To clarify the situation, we demonstrate a representative approach and make clear what is right and what is wrong.

For simplicity, consider a 2-DOF system (shoulder and elbow) in 2D space. Previous studies have partitioned torques in equations 7 and 8 into the four types (NET, net torque; MUS, muscle torque; INT, interaction torque; GRA, gravity torque) in the following way:

Hollerbach and Flash (^{19}) mentioned that a multijoint movement requires the controller to generate the muscle torque to include the interaction torque that is not present in a single-joint movement. They actually calculated the interaction torque during reaching movements and demonstrated that it is large and must be compensated for by the controller. Gribble and Ostry (^{14}) compared the onset-time of muscle activities and the interaction torque calculated from the observed motion and found that muscle activities preceded the emergence of the interaction torque. They concluded that the central nervous system compensates for the interaction torque in a feed-forward manner. These seminal studies solved only the inverse dynamics problem, that is, how much muscle torque (τ_{1} and τ_{2}) is required to generate the angular acceleration (θ¨_{1} and θ¨_{2}) when the system is in the state of θ and . Because unknown variables (τ_{1} and τ_{2}) are separately present in equations 15 and 16, they could solve the problem by dealing with each equation separately.

In contrast, recent studies have both calculated the MUS and INT torques by inverse dynamics and quantified the contributions of the MUS and INT torques to each joint acceleration (^{5,12,16,18,25}). Thus, these studies intended to solve the forward dynamics problem. This required dealing with the two equations as simultaneous equations as shown in equation 9 and solving the unknown variables (θ¨_{1} and θ¨_{2}) that are present in both equations.

Instead, these previous studies dealt with each equation separately. For the shoulder equation 15, they considered that NET_{1} is the torque required to produce θ¨_{1} and that it is produced by MUS_{1}, INT_{1}, and GRA_{1}. To quantify the contributions of MUS_{1}, INT_{1}, and GRA_{1} to the angular acceleration of shoulder (θ¨_{1}), they compared the magnitude of MUS_{1}, INT_{1}, and GRA_{1} without considering the elbow equation 16. Similarly for the elbow, they conducted the same analysis without considering the shoulder equation 15. This approach is not correct because the solution for one unknown variable (*e.g*., θ¨_{1}) includes the other unknown variable (*e.g*., θ¨_{1}). Although one may argue that both θ¨_{1} and θ¨_{2} are known variables because they are obtained from the observed motion, these studies intended to solve the forward dynamics problem after the inverse dynamics calculation. Rather, it is necessary to treat the angular accelerations as unknown variables because the purpose of the analysis is to solve the forward dynamics problem of how the angular accelerations are generated.

Next, we demonstrate the difference between these previous analyses and the more appropriate analysis presented in this review. We analyze a simple 2-DOF (shoulder and elbow) movement in the horizontal plane, which was simulated (Initial angles are 0 degree for the shoulder and 60 degree for the elbow. Initial angular velocities are both 0. The length and weight of each segment are the same as one of the subjects in (^{18}). The equations were integrated using the Runge-Kutta algorithm with a constant time step of 1 ms.) by the constant muscle torques (shoulder MUS = 4.0 Nm; elbow MUS = 1.0 Nm) for 0.4 s. Figure 4C and D shows the result of the appropriate analysis using equation 9 and indicates how each joint angular acceleration (thin line) is produced by the following: 1) shoulder muscle torque (bold line), 2) elbow muscle torque (gray line), and 3) velocity-dependent torque (dotted line). In contrast, Figure 4A and B shows the result of the previous analysis in which MUS and INT are divided by *I* _{11} (equation 17 for Fig. 4A) or *I* _{22} (equation 18 for Fig. 4B).

According to the previous analysis, one may interpret that the initial shoulder angular acceleration (thin line in Fig. 4A) during first 100 ms as mainly being produced by the shoulder MUS (bold line) with a slight counteraction by the INT (dashed line). However, Figure 4C shows that the contribution of the shoulder MUS is much greater, that the elbow MUS strongly counteracts it, and that the velocity-dependent torque slightly assisted this effect. The two analyses, therefore, produced different results about how this relatively simple action is controlled by the nervous system.

In summary, researchers must make clear whether the purpose of a study is to solve the inverse dynamics problem alone or to solve the forward dynamics problem too after the inverse dynamics calculation. When solving the inverse dynamics alone, the classification of the torques as shown in equations 15 and 16 is meaningful because the INT, which is absent in single-joint movements, is considered as an additional torque that the controller must compensate for in generating the MUS in multijoint movements. When solving the forward dynamics, however, it is necessary to solve the equations as simultaneous equations as shown in equation 9 or 13. In this case, there are three torques (muscle torque, gravity torque, and velocity-dependent torque) that exert an inter-DOF interaction on the joint angular acceleration. As a consequence of this interaction, the joint angular acceleration at 1 DOF is affected by muscle torques at all DOF, gravity torques at all DOF, and velocity-dependent torques at all DOF.

### CONCLUSIONS AND IMPLICATIONS FOR BASEBALL PITCHING

This review began by showing the four angular velocities that are the important contributors to the translational ball velocity at the ball-release time. The dynamic analysis we introduced here can further reveal how the angular accelerations of these four important DOF are produced by the muscle torques, gravity torques, and velocity-dependent torques at all DOF. Such knowledge informs us about the control of overarm throws (including baseball pitching and similar actions in tennis, handball, javelin throw, etc.). If an important angular acceleration is mainly produced by a muscle torque at a certain DOF in skilled throwing, coaches can focus on training the muscle torque at that DOF. If an important angular acceleration is mainly produced by a velocity-dependent torque, the situation is more complex. As previously described, the velocity-dependent torque reflects the history of the muscle torques until that time. Therefore, we must first determine the muscle torque that is the original source of the velocity-dependent torque, and because the velocity-dependent torque depends on the 3D spatial relation between angular velocity vectors and limb configuration, we must then determine this spatial relation in skilled performers. Finally, it is also important to consider the configuration of the entire limb because the angular acceleration produced by torques depends on the system mass matrix (** I**(θ)) as well as the magnitude of the muscle and velocity-dependent torques. These relations underscore the significance of "form" in throwing actions.

Some previous studies of 3D sports movements have calculated the muscle torque at each joint with the inverse dynamics technique (^{6,10}), which can identify the characteristics of movement kinematics associated with high joint forces and torques. To improve a throwing performance, however, we propose that it is necessary to use a forward dynamics approach after the inverse dynamics calculations. This new approach enables us to determine the direct mechanical cause of a skilled overarm throw and, as a result, provides cues on how to improve performance.

## Acknowledgments

The authors thank Drs. Daichi Nozaki and Katsu Yamane for valuable discussions about the multijoint dynamics.

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**Keywords:**

baseball pitching; multijoint movements; three-dimensional movements; muscle torque; velocity-dependent torque; forward dynamics; inverse dynamics