The purpose of this study was to investigate the coordination strategy of maximal-effort horizontal jumping motion in comparison with vertical jumping motion. A horizontal jumping and a vertical jumping with a countermovement were generated using the technique of computer simulation and numerical optimization (Figure 2, Table 2). It was found that the motion of the hip joint was greater in the horizontal jump than in the vertical jump (Figure 2, 3). This observation is consistent with the findings reported in  as well as in . In vertical jumping motions, the orientation of the trunk segment has to be near straight and its angular momentum has to be reduced to near zero at the instant of take-off. This condition is required for the human body to jump up vertically with a straight posture [33, 34]. Therefore a smaller action of the hip joint is allowed in a vertical jump than in a horizontal jump.
The optimal movement time for the horizontal jump was greater than that for the vertical jump (0.92 s and 0.65 s, respectively). This result is consistent with the finding reported in , in which longer movement duration of the trunk segment was observed in horizontal jump than in vertical jump. These movement times were similar to the ones observed in the experimental study , which also suggest that the simulation model and the optimization method employed in this study capture the fundamental nature of human jumping motions.
The magnitude of hip joint flexion during the countermovement was greater in the horizontal jumping than in the vertical jumping (Figure 3). The ankle joint assumed a dorsiflexed posture earlier in the horizontal jumping than in the vertical jumping. Combining these two conditions, the whole body was tilted more in the forward direction in the horizontal jump than in the vertical jump (Figure 2). This result is completely consistent with the experimental observation reported in . This is reasonable considering that it is required to generate a momentum in both forward and upward directions by the instant of take-off in a horizontal jumping. On the other hand, it is required to generate only an upward momentum in a vertical jump. In order to jump upwards with a straight posture, the position of the body's center of mass has to be kept over the feet in a vertical jump. The motions of the hip, knee and ankle joints were coordinated in the vertical jumping to meet this requirement (Figure 2).
When examining the muscle activation (Figure 4) and force development (Figure 5) profiles, it can be observed that the flexor muscles of the leg were recruited to generate greater joint flexion motions during the countermovement phase in the horizontal jumping. This phenomenon was pronounced in the action of the m. iliopsoas, m. rectus femoris and m. tibialis anterior. This was evident from the moment of initiation of the motion. This action had an effect of moving the body's center of mass in the forward direction during the countermovement. This configuration of body segments helped enhance the horizontal momentum delivered to the body's center of mass through the countermovement. The duration of activation of the hip joint extensor muscles (the gluteal muscles and hamstrings) was longer in the horizontal jump than in the vertical jump. This observation is consistent with the finding that the hip joint was utilized more vigorously in the horizontal jump than in the vertical jump. There was a drop in the activation of the mm. vasti during the push-off phase, whereas there was a great activation of the m. biceps femoris during the push-off phase in the horizontal jump. This coordination was needed in order to maintain the forward inclined posture during the push-off phase.
The work outputs of the individual muscles (Table 3) were generally similar between the horizontal jump and the vertical jump. The m. iliopsoas, hip external rotator muscles and hamstrings were the only exceptions in which more than 10 J of difference was observed in the work output. The work output of the m. iliopsoas was greater in the horizontal jumping than in the vertical jumping because this muscle was activated to a greater level during the countermovement in the horizontal jumping (Figures 4 and 5). On the other hand, the work output of the hip external rotator muscles and hamstrings was smaller in the horizontal jumping than in the vertical jumping. Especially, the work output of the hamstrings was negative in the horizontal jumping (-4.8 J, Table 3). This suggests that the hamstrings experienced an eccentric action in which this muscle was stretched at the same time as exerting a muscle force. This is because a great momentum was given to the trunk segment throughout the countermovement in the direction of hip joint flexion, and the hamstrings was utilized to counteract the momentum. Therefore, the force output of this muscle was great (Figure 5) and the work output was negative (Table 3). A similar mechanism seems to have caused the smaller work output of the hip external rotator muscles in the horizontal jump (5.3 J) than in the vertical jump (19.3 J).
It is interesting that the amount of translational mechanical energy gain of the body's center of mass throughout the jumping motion was greater in the horizontal jump (258.9 J) than in the vertical jump (180.3 J; 70% as compared to in the horizontal jump). This result is consistent with what has been reported in , in which the translational energy gain of the body's center of mass during a vertical jump was as much as 64% of the translational energy gain of the body's center of mass during a horizontal jump. As there was only a minor difference in the total muscle work output (the total value was 357.3 J for the horizontal jumping and 375.8 J for the vertical jumping; the difference was only 5%), it was suggested that muscular work was transferred to the mechanical energy of the body's center of mass more effectively through the horizontal jumping than through the vertical jumping. An explanation for this finding is in the difference of transfer of mechanical energy during the countermovement. In the horizontal jumping, a reduction of potential energy as the body segments were moved to a lower position was coupled with an increase of kinetic energy of those segments moving in the forward direction (Figure 2). Therefore, there was a smaller loss of mechanical energy during the countermovement in the horizontal jumping. However, in the vertical jumping, all the downward momentum generated during the countermovement had to be cancelled by muscular efforts before the body started moving upward (Figure 2). Therefore, there was a greater energy loss during the countermovement. The contribution of the angular component of the mechanical energy (0.5*(moment of inertia)*(angular velocity)^2) was rather small in both types of jumping at the instant of take off. The magnitude was less than 6% for horizontal jumping, and less than 3% for vertical jumping. In other words, the translational components had much greater contributions. This might be because the value of moment of inertia of human body segments is generally small.
In this study, the resultant velocity of the body's center of mass at the instant of take-off was greater in the horizontal jumping than in the vertical jumping (Table 2). This result seems to be inconsistent with the results reported in , in which the resultant velocity of the body's center of mass at the instant of take-off was almost identical between a horizontal jump and a vertical jump. This difference can be explained by the existence/absence of a countermovement. As the jumping motion simulated in this study employed a countermovement, and as an effective transfer of mechanical energy was observed in the horizontal jumping, the body's center of mass experienced a greater gain of mechanical energy by the instant of take-off. It is suggested that this mechanism of energy transfer was less evident in the horizontal jumping motion studied in , as that motion was a squat jumping instead of a countermovement jumping.
The optimal angle of projection obtained in this study was 34 deg, whereas this parameter obtained in a preceding experimental study was 48 deg . This discrepancy can be explained by the difference in the musculoskeletal properties of the model and the subjects, i.e., the subjects that participated in  were 'stronger' than the model utilized in this study. This is evident by comparing the vertical jumping height obtained in this study (38.5 cm) with the vertical jumping height of the subjects (41.0 cm), although both of these figures are within the range of experimental observations reported in numerous preceding studies on vertical jumping. This difference seems reasonable considering that the subjects were trained athletes (Australian Football) in . As discussed previously, the horizontal distance traveled in the horizontal jump was calculated as
By analyzing the right-hand side of this formula, it can be derived that the second term is maximized when the projection angle is 45 deg. However, the first term (Xt.o.) also made a substantial contribution in this study. The computer simulation model and the numerical optimization chose the strategy of increasing Xt.o. with a smaller angle of projection. Although it is assumed that the optimal projection angle will become closer to 45 deg when the muscular parameters of the model are strengthened, customizing a computer simulation model to a specific subject population requires very complex and sophisticated treatments. This issue needs to be addressed in future studies.
In this study, the optimal pattern of muscle activation, including the initial muscle activation level, was searched using numerical optimization. This resulted in reasonable movements both for horizontal and vertical jumping motions. It is observed that the initial level of muscle activation and muscle forces were not identical between these motions (Figure 4 and 5). This result, i.e., the discrepancies in the initial conditions of the simulation, might seem controversial at a first glance. However, these initial conditions were not given to the model: instead, the numerical optimization procedure found these initial conditions to be the most suitable for the model to perform jumping starting from the identical upright posture. We believe the general similarity between the simulated body dynamics and that of the human subjects suggest the validity of the approach taken in this study. As the objective function utilized in this study considered only the translational motions of the body's center of mass, rotational component of the mechanical energy was not explicitly analyzed. Techniques of foot placement at the time of landing were not discussed either. However, these might become more important when performing more precise comparisons between the simulated and experimentally captured motions, or when applying the methods and findings of this study to sports scenes. More sophisticated modeling and simulation that include the landing phase will be valuable in future studies with a goal of reducing the risk of injuries in athletes.