where θ = (θ1θ2)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 (τ1 and τ2) 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)−1τ ) 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, Ii is the inertia tensor, ɑgi is the translational acceleration of the center of gravity, ωi 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 (iθ¨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. iθ¨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 Ωi 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 iτi is the muscle torque vector at the i-th joint that is expressed in the i-th joint coordinate system. For example, 1τ1 = (τ1−IE τ1−ED τ1−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 A1i 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 (τ1−IE) but also by the muscle torques about the other two axes at the shoulder joint (τ1−ID, τ1−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 (ωi × I iωi) 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 (τ1 and τ2) 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 (τ1 and τ2) 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 NET1 is the torque required to produce θ¨1 and that it is produced by MUS1, INT1, and GRA1. To quantify the contributions of MUS1, INT1, and GRA1 to the angular acceleration of shoulder (θ¨1), they compared the magnitude of MUS1, INT1, and GRA1 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.
The authors thank Drs. Daichi Nozaki and Katsu Yamane for valuable discussions about the multijoint dynamics.
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Keywords:©2008 The American College of Sports Medicine
baseball pitching; multijoint movements; three-dimensional movements; muscle torque; velocity-dependent torque; forward dynamics; inverse dynamics