There is ample evidence that making a countermovement enhances performance in fast discrete movements (1,2,10,16,20 2,20,21 1,5
In the literature, several mechanisms have been proposed to explain the enhancement of maximum jump height by a countermovement. These can best be discussed with the help of Figure 1 , which shows for a maximum height CMJ and SJ of one subject the vertical ground reaction force Fz as a function of the height of the center of gravity (CG). The surface under such a plot reflects the change in effective energy, that is, energy contributing to jump height (the sum of potential energy and kinetic energy due to the vertical velocity of CG). If we integrate for the propulsion phase, it is clear that the surface is greater for CMJ than for SJ (the surplus is indicated with the shaded area in Fig. 1 ), because over most of the range of upward motion of CG a greater Fz can be produced in CMJ.
FIGURE 1— Vertical ground reaction force (Fz) plotted against height of center of gravity (CG) for a countermovement jump (CMJ) and a squat jump (SJ) of a single subject (data from (: 5 )). Arrows indicate the direction of time; dots on curve of CMJ have been plotted at 10-ms intervals. The shaded area represents the surplus of effective energy gained in CMJ, which ultimately causes jump height in CMJ ( closed circle ) to be greater than that in SJ ( open circle ). If the body configuration at each height of CG is the same, a greater Fz implies that muscle forces accelerating the body upwards are greater. Why would they be greater in CMJ than in SJ? A first possible explanation is that muscle stretch occurring during the countermovement in CMJ triggers neural responses (13,23 11,14,15,18,25 15 elastic energy , which can be reutilized during the propulsion phase. However, although having series elastic elements does contribute to maximum jump height by the positive effect on the rate at which energy can be released (4,24 1,6,27 3 Fig. 1 ) to be submaximal. This possible explanation has been introduced by Asmussen and Sørensen in 1971 (3 5,12,19,22,26,27 7
To systematically investigate why jump height is greater in CMJ than in SJ, experiments on human subjects are unsuitable, if for no other reason that crucial variables such as individual muscle forces and contractile element lengths cannot be measured. Therefore, researchers have turned to simulation models. Anderson and Pandy (1
Considering the abundance of studies on human subjects reporting greater jump height in CMJ than in SJ, and the plausible explanations offered for this finding, we felt that a greater jump height in CMJ than in SJ should also be borne out in a forward simulation study like the one conducted by Anderson and Pandy (1
METHODS
For simulations of CMJ and SJ we used a two-dimensional forward dynamic model of the human musculoskeletal system (Fig. 2 ). The model, which had muscle stimulation as its only independent input, has been described in detail elsewhere (30 28 17
At the start of each simulation, the model was put in a starting position, and the initial STIM levels were set in such a way that static equilibrium was achieved. In the starting position of CMJ, small hip extension, knee flexion, and plantar flexion moments were needed (Fig. 4 ). To find a unique solution for the initial STIM-levels, we first demanded that hamstrings, rectus femoris, and vasti each produced a small force of 100 N (causing them to take up slack in SEE) and calculated the required STIM levels. Subsequently, we calculated the STIM levels for the other muscles that ensured equilibrium of the system as a whole (the corresponding forces automatically tensed SEE). In the starting position of SJ, hip extension, knee extension, and plantar flexion moments were needed for equilibrium, and we calculated the STIM levels that produced these moments under the constraint that each of the biarticular muscles produced a small force of 100 N. During propulsion, STIM of each muscle was allowed to switch several times. Each switch time initiated a change of STIM from its current level toward either its minimum of 0 or its maximum of 1. Any decrease of STIM toward 0 occurred instantaneously, any increase of STIM toward 1 occurred at a fixed rate (slope of the ramp in Fig. 3 ). Under these restrictions, the motion of the body segments, and therewith performance of the model, depended on a set of switch times. Thus, an optimization problem could be formulated: finding the combination of switch times that produced the maximum value of the height achieved by CG.
FIGURE 4— Body configurations at the start of CMJ and SJ as well as at the start of upward movement of the center of gravity (CG), for subjects (data from (: 5 )) and for the simulation model. Note the difference between CMJ and SJ in ground reaction force magnitude ( arrows pointing upward ) at the start of upward movement of CG. Arrows pointing downward represent the force of gravity, plotted with the origin in CG. FIGURE 3— Stimulation (STIM) and active state (q) as a function of time for gluteus maximus during simulated maximum height CMJ, for the condition in which STIM increased instantaneously (: dashed line ) and the condition in which it increased at a rate of 2.2 s−1 . Note that q depends not only on STIM but also on length of contractile elements, which explains the slight initial increase of q in the latter condition during the first 300 ms. Note also that in this condition the propulsion phase lasted longer because force developed more slowly.
Comparison between CMJ and SJ, which was carried out for different conditions, always started with simulation of CMJ from the average equilibrium starting position of human subjects (5 Fig. 4 ). To find the optimal solution for CMJ, two constraints were introduced by adding a penalty value to the optimization criterion. First, a penalty was set on deviation of the minimum height reached by CG from a prespecified value. By specifying a value derived from experimental data, we obtained simulated CMJ that could be compared with CMJ of subjects collected in a previous study (5
CMJ and SJ were compared for different conditions, each of which involved a combination of a target minimum height of CG (71, 74, or 77 cm) and a prespecified rate of change of STIM during the ramp (ranging from 1.1 s−1 to infinite). For each condition, STIM for CMJ was optimized with the help of a parallel genetic algorithm (29 29
Once the solutions were found for CMJ and SJ in each condition, we used them to determine the cause for differences in muscle work performed. Muscle work is the integral of muscle force with respect to MTC length, with muscle force depending on active state, CE length, and CE velocity. We estimated the effect of differences in active state between CMJ and SJ by substituting at each MTC length in SJ the active state at the same MTC length in CMJ, and subsequently recalculating muscle force and work. The same procedure was followed to estimate the effect of differences in CE velocity.
RESULTS
The simulation model behaved very well. First, in all conditions, regardless of the number of STIM switch times allowed per muscle, the optimization ended with a solution for CMJ in which each muscle only switched “off” and then “on,” and in a solution for SJ in which each muscle only switched “on.”
Second, optimal solutions for CMJ and SJ reasonably well reproduced the average configurations reported elsewhere for subjects (5 Fig. 4 ). This was true regardless of the rate of increase of STIM. The only difference was that in the optimal solutions for CMJ the model tended to have the HAT segment rotated less forward at the minimum height of CG (with a compensatory further backward rotation of the leg segments) than the subjects. When the rate of increase of STIM was set to 2.2 s−1 , the average rate of increase of ground reaction force was satisfactorily reproduced with the model (Fig. 5 ), and over time, kinematics of simulated jumps satisfactorily resembled those of jumps by human subjects (Fig. 6 ).
FIGURE 5— Time histories of the vertical ground reaction force in CMJ (left) and SJ (right) for individual subjects (data from (: 5 )), and for the simulation model with the rate of increase of stimulation set to 2.2 s−1 . Time is expressed relative to the instant of take-off ( t = 0). FIGURE 6— Stick diagrams at various time instants for CMJ (left) and SJ (right) for one subject (data from (: 5 )) and for the simulation model with the rate of increase of stimulation set to 2.2 s−1 . Arrows pointing upward represent ground reaction forces; arrows pointing downward represent the force of gravity, plotted with the origin in the center of gravity. Time is expressed relative to the instant of take-off ( t = 0). Third, maximum jump heights of CMJ and SJ changed systematically with variations in conditions, and so did the difference in maximum height between CMJ and SJ. At a given target minimum height of CG, when the rate of increase of STIM was greater, maximum jump height was greater; the increase in height was smaller in CMJ than in SJ, so the difference in maximum height between CMJ and SJ became smaller (Fig. 7 ). At a given rate of increase of STIM, when the target minimum height of CG was lower, maximum jump height was greater (Fig. 7 ), which supports the contention that for a “fair” comparison of CMJ and SJ it must be ensured that the starting height of CG in SJ matches the lowest height of CG reached in CMJ (1,5
Regardless of condition, maximum height in CMJ was greater than that in SJ. Even when stimulation was switched instantaneously to its new value, the difference in jump height was still on the order of 0.4 cm. The latter is not surprising, because active state follows STIM via a first-order process and hence does not become maximal instantaneously (Fig. 3 ).
We decided to further analyze the difference in jump height between CMJ and SJ for the condition in which the target minimum CG height was 0.74 m and STIM increased at a rate of 2.2 s−1 . In this condition, the maximum CG height reached was 1.530 m in CMJ and 1.510 m in SJ (Fig. 8 ), corresponding to an increase in effective energy during the propulsive phase of 637 J and 621 J, respectively. The total work output of the muscles during the propulsion phase was 729 J in CMJ and 710 J in SJ (Table 1 ), which means that the efficacy ratio (the fraction of muscle work converted into effective energy) was 0.86 in CMJ and 0.87 in SJ. The greater work in CMJ was due to a greater work output of hamstrings and gluteus maximus in CMJ (Table 1 ). Figure 9 presents for gluteus maximus the force, active state, and CE-shortening velocity as a function of MTC length. The work done during the propulsion phase is equal to the integral of force with respect to MTC length during this phase. The surplus of work for the propulsion phase in CMJ was attributed to a greater force over the first 30% of MTC shortening in the hip extensors. To estimate the effect of active state on the surplus work, we substituted at each MTC length in SJ the active state at the same MTC length in CMJ, and subsequently recalculated muscle force and work. In this thought experiment, the work output of the muscles in SJ increased by 33 J (Table 1 ), now causing a virtual work surplus in SJ of 14 J. However, in SJ the CE-shortening velocity remained lower than in CMJ (not surprisingly, because the take-off velocity was lower), and this could fully explain the virtual work surplus (Table 1 ).
DISCUSSION
In the literature, it has been described abundantly that people jump higher in CMJ than in SJ, and various possible explanations have been offered. To systematically investigate why jump height is greater in CMJ than in SJ, experiments on human subjects are unsuitable, and one needs to resort to simulation models. In the present study, a simulation model was used to investigate whether the difference in jump height between CMJ and SJ could be explained by a difference in active state during the propulsion phase.
If effects are to be evaluated quantitatively, a realistic model is needed. In previous studies it has been shown that the model produces SJ that closely resemble jumps of human subjects (4 Figs. 4 and 6 ). Although different subjects converge toward a common kinematic pattern at the end of the propulsion phase, there is substantial interindividual variation in body configuration in the first part of the propulsion phase (7 Fig. 5 ). In SJ, the simulated maximum height jumps fit well within the confidence intervals corresponding to the interindividual variation (Fig. 5 ), although admittedly in CMJ the ground reaction force increased faster in the model than in the subjects. Jump height of the model (45.6 cm in CMJ, 43.7 cm in SJ) was only slightly less than that of the human subjects from which the initial conditions were derived (48.1 ± 3.6 cm for CMJ and 44.7 ± 4.4 cm for SJ (5
In the simulations performed in this study, maximum jump height in CMJ was always greater than that in SJ, regardless of condition. This was in line with findings in studies on human subjects but contrary to the result of Anderson and Pandy (1
In our simulation model, which did jump higher in CMJ than in SJ, no reflexes or potentiation were incorporated and all simulated jumps corresponded to optimal solutions. This means that in the model the greater jump height in CMJ was completely due to the fact that in CMJ active state could develop during the preparatory countermovement, whereas in SJ it inevitably developed during the propulsion phase (Figs. 8 and 9 ). This allowed the hip extensor muscles to produce more force and more work during shortening in CMJ (Fig. 9 ), which was reflected in a higher Fz over the first 5 cm of upward motion of CG (Fig. 8 ) and in a greater jump height. In fact, in our thought experiment, in which we introduced in SJ the advantage of a higher active state during shortening as found in CMJ, we even ended up with a virtual muscle work surplus in SJ as compared with CMJ (Table 1 ). This virtual work surplus, however, could be explained by the fact that the CE-shortening velocities remained lower in SJ than in CMJ (Table 1 ).
The greater force and work that the hip extensor muscles could produce over the first part of shortening in CMJ could easily cause jump height to be about 2 cm greater in CMJ than in SJ (Fig. 7 ). Obviously, the difference in jump height between CMJ and SJ decreased when STIM increased faster (Fig. 7 ), because the latter caused a reduction of the distance covered at submaximal active state in SJ. Considering the variation among subjects in the rate at which they developed force (Fig. 5 ), it might be expected that the difference in jump height between CMJ and SJ also varies among subjects; a large difference is expected in subjects that build up force slowly, whereas a small difference is expected in subjects that build up force quickly. However, the resolution of experimental studies might be too low to detect such subtle differences. More importantly, according to the simulations, jump height may increase by several centimeters if active state is developed more quickly (Fig. 7 ). This raises the question why some subjects develop active state and force so slowly (Fig. 5 ). It has been speculated elsewhere (8,9
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