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The Relationship between Pedal Force and Crank Angular Velocity in Sprint Cycling

BOBBERT, MAARTEN FRANK; CASIUS, L. J. RICHARD; VAN SOEST, ARTHUR J.

Medicine & Science in Sports & Exercise: May 2016 - Volume 48 - Issue 5 - p 869–878
doi: 10.1249/MSS.0000000000000845
APPLIED SCIENCES
Free

Purpose Relationships between tangential pedal force and crank angular velocity in sprint cycling tend to be linear. We set out to understand why they are not hyperbolic, like the intrinsic force–velocity relationship of muscles.

Methods We simulated isokinetic sprint cycling at crank angular velocities ranging from 30 to 150 rpm with a forward dynamic model of the human musculoskeletal system actuated by eight lower extremity muscle groups. The input of the model was muscle stimulation over time, which we optimized to maximize average power output over a cycle.

Results Peak tangential pedal force was found to drop more with crank angular velocity than expected based on intrinsic muscle properties. This linearizing effect was not due to segmental dynamics but rather due to active state dynamics. Maximizing average power in cycling requires muscles to bring their active state from as high as possible during shortening to as low as possible during lengthening. Reducing the active state is a relatively slow process, and hence must be initiated a certain amount of time before lengthening starts. As crank angular velocity goes up, this amount of time corresponds to a greater angular displacement, so the instant of switching off extensor muscle stimulation must occur earlier relative to the angle at which pedal force was extracted for the force–velocity relationship.

Conclusion Relationships between pedal force and crank angular velocity in sprint cycling do not reflect solely the intrinsic force–velocity relationship of muscles but also the consequences of activation dynamics.

MOVE Research Institute Amsterdam, Faculty of Human Movement Sciences, VU University Amsterdam, Amsterdam, THE NETHERLANDS

Address for correspondence: Maarten F. Bobbert, Ph.D., MOVE Research Institute Amsterdam, Faculty of Human Movement Sciences, VU University Amsterdam, Van der Boechorstraat 9 1081 BT Amsterdam, The Netherlands. E-mail: M_F_Bobbert@fbw.vu.nl.

Submitted for publication February 2015.

Accepted for publication December 2015.

The relationship between shortening velocity and force of a maximally stimulated isolated muscle is described well by Hill’s hyperbolic equation (20). However, when humans perform maximum effort functional tasks that involve rotations in several joints simultaneously, they generate relations between velocity and force that tend to be almost linear. This has been observed in “explosive” tasks in which the purpose is to maximize the power output during a single contraction. For example, almost linear relationships have been found for average force as a function of average velocity in half-squat exercises performed with a range of added masses (35), force and velocity in leg press against a dynamometer in “isotonic mode” (25), peak force as a function of peak velocity in explosive leg extension against a range of added isotonic forces on a sledge dynamometer (36), and force and velocity in combined knee and hip extension on a servo-controlled dynamometer (48). Why would force drop more with velocity than expected on the basis of Hill’s hyperbolic relationship during these explosive leg press tasks? To answer this question, researchers tend to appeal implicitly or explicitly to “neural mechanisms” (e.g., ref. 48). Essentially, the idea is that at high velocities, subjects are unable to coordinate the muscles at the various joints in such a way that power output is truly maximal. It has recently been shown, however, that there is no need for such an appeal: in simulations of a leg press task against a dynamometer in “isotonic mode,” i.e., with the dynamometer producing a constant force during a single leg press movement, it was found that the force on the dynamometer dropped almost linearly with leg extension velocity even though the force of muscles dropped with muscle shortening velocity according to Hill’s relationship (3). The discrepancy was explained by segmental dynamics, which canceled more and more of the contribution of muscle forces to the force on the dynamometer as leg extension velocity increased.

Force–velocity relationships have also been studied for sprint-type cyclic movement tasks. Here too, almost linear relationships have been found for peak pedal force as a function of crank angular velocity in isokinetic sprint cycling (e.g., refs. 2, 27, 39), average crank moment or tangential pedal force (proportional to crank moment) over a cycle as a function of crank angular velocity during the acceleration phase of sprint cycling (e.g., [7,12,40]), instantaneous crank moment as a function of crank angular velocity at peak power output in cycling against a range of inertial loads (e.g., [32]), crank moment as a function of peak angular velocity reached in all-out sprints against a range of braking forces (e.g., [46]), and crank torque as a function of peak angular velocity reached in all-out arm cranking against a range of inertial loads (45). In one study of the acceleration phase of sprint cycling (12), the force–velocity relationship even tended toward being convex rather than linear ([12], middle panel of their Fig. 3), let alone hyperbolic like Hill’s relationship.

FIGURE 3

FIGURE 3

A few researchers have explicitly wondered why force–velocity relationships for sprint-type cyclic movements are not hyperbolic (e.g., [7,8,27,39]), but to the best of our knowledge, this issue is still unresolved. Just like in explosive leg press tasks, segmental dynamics may affect pedal forces (8,23,24,28) and hence the relationship between peak pedal force and crank angular velocity. Then there is a second factor of importance in sprint cycling: the purpose is not to maximize the pedal force but to maximize the average pedal power over a cycle, henceforth referred to as average power output. For the latter purpose, muscles need to be switched “on” and “off,” and because the active state has relatively slow dynamics, muscles need to switch “off” a certain amount of time before they start to lengthen, which affects average power output (28,29,42), and presumably also affects the shape of the relationship between peak pedal force and crank angular velocity. Finally, it has been suggested on the basis of EMG analyses that coordination becomes suboptimal at high crank angular velocities during sprint cycling (38).

The purpose of the present study was to understand why relationships between pedal force and crank angular velocity in sprint cycling are less curved than the intrinsic force–velocity relationship of muscles. For this purpose, we simulated isokinetic sprint cycling at different crank angular velocities with a forward dynamic model of the human musculoskeletal system. The only input of the model was muscle stimulation over time, which we optimized to maximize average power output. We manipulated inertial properties and active state dynamics of the model and studied how this affected the relationship between pedal force and crank angular velocity.

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MATERIALS AND METHODS

The simulation model (Fig. 1) and the optimization approach used for the current study have been described previously (9,42). Briefly, the model consisted of a planar linkage of rigid segments of which the top one, representing the upper body, was fixed in space (cf. [33]). Because crank angular velocity was also imposed, the legs were mechanically decoupled, and it was sufficient to simulate the movement of only one lower extremity consisting of a thigh, a shank, and a foot. The foot was connected by a frictionless hinge joint to the crank, enabling the exertion of both pushing and pulling forces (e.g., [30]), and the crank rotated with a constant angular velocity about a point fixed in space. The only input of the model was neural drive (STIM, ranging from 0 to its maximum of 1) to eight muscle–tendon complexes (Fig. 1), each modeled as a Hill-type unit consisting of a contractile element, a series elastic element, and a parallel elastic element. Force of the contractile element depended on length, velocity, and active state, defined as the relative amount of Ca2+ bound to troponin (15). Following Hatze (18), the active state depended on free Ca2+ concentration (γ) according to an algebraic saturation relationship influenced by contractile element length, and γ was related to STIM by first-order dynamics:

where m and c are constants. It is important to note that although the dynamics of γ are linear, deactivation is a slower process than activation because of the algebraic saturation relationship relating γ to the active state. If we start from equilibrium at STIM = 0 and increase STIM to 1, the active state will rapidly increase and then saturate. If we start from equilibrium at STIM = 1, the active state will linger at a high value before it starts to drop rapidly. In models lumping the whole process of transforming STIM into the active state into one first-order differential equation, the same behavior is obtained by using a much longer time constant for deactivation than for activation (e.g., [31,34,47]).

FIGURE 1

FIGURE 1

None of the parameter values of the model was tuned to subjects participating in any cycling study. As detailed elsewhere (42), values for anatomy and inertial properties of the musculoskeletal system were based on anthropometrics of a group of volleyball players. Parameter values that may be relevant for comparison between simulation results and experimental results in the literature are as follows: mass of the upper body 55 kg, mass of each of the legs 13.2 kg, thigh length 0.49 m, shank length 0.46 m, and crank length 0.17 m; the seat height was set to 0.96 times the trochanteric height, and the line from the crank axis to the hip joint was 100° relative to the right horizontal. For more inertial parameter values of the model, see Ref. (42). Values for parameters describing the transformation of STIM to the active state in fast muscle were adopted from the literature (18), the force–length relationship of contractile elements was based on the sliding filament theory and numbers of sarcomeres in series in human muscle fibers, and Hill parameters for the force–velocity relationship of the contractile element (a/F0 = 0.4, where F0 is the maximal force, and b = 5.2LCE,opt/s, where LCE,opt is the optimum muscle fiber length) were based on the force–velocity relationship of cat medial gastrocnemius ([41], see Ref. [5] for derivation of values). The latter causes the muscles in our model to be relatively fast: cat gastrocnemius consists for about 75% of fast twitch muscle fibers (1), which is at the high end of the range of percentages of fast twitch fibers reported for human muscles (e.g., [19]).

Muscles were assumed to be maximally stimulated during part of the crank cycle and to receive no stimulation during the remainder of the crank cycle. Thus, the leg motion depended on the STIM-onset and STIM-offset times of the muscles. To find a unique solution of these time objective function:

where Pext is the average power output (note that minimization of −Pext leads to maximization of +Pext), xi are the state variable of the system, c1,i and c2 are positive constants, and Δ is the difference between the value of a variable at the start of a cycle and the value of that variable at the start of the next cycle. Thus, the second and third terms of H penalized nonperiodic behavior.

For constant angular velocities of the crank, a parallel genetic algorithm (43) was used to find the set of STIM-onset and STIM-offset times that minimized H for two different models. The first model, model REF, was the reference model as described above. From the solutions obtained with model REF, we also isolated the muscular contribution to the pedal force (REF-mc) as proposed by Fregly and Zajac (17), which was straightforward and unambiguous to do because we had the full system of dynamic equations of motion (9). [Kautz and Hull (23) were actually the first to propose a method to calculate the muscular contribution to the pedal force, and this method was later adopted by others (e.g., Ref. [28]). Kautz and Hull calculated “… the force that would be measured at the pedal if the leg was fixed in the observed configuration and the net intersegmental moments were applied in the absence of gravity” (23). If we inserted our simulation results in the equations of Kautz and Hull, we ended up with muscular contributions to the pedal force different from ours, which implies that those equations do not yield the contribution that we were interested in here.] The second model was obtained from the reference model by setting segmental masses and moments of inertia to negligible values and by setting γ = c STIM, so that activation dynamics was bypassed; this model will be referred to as model NINAD (negligible inertia, no activation dynamics). Optimal solutions obtained for model NINAD are interesting because they yield a good approximation of the maximal tangential pedal force that all muscles together can produce at each crank angle at the given crank angular velocity. After all, the tangential force is the only force that produces power on the pedal, and average power output, which was the optimization criterion, is maximal when the tangential force at each crank angle is maximal. Thus, relationships between pedal force and crank angular velocity of model NINAD will closely reflect the intrinsic force–velocity relationship of muscle.

We optimized control for isokinetic crank angular velocities ranging from 30 to 200 rpm. For up to 150 rpm, the optimizations yielded physiological cycling motions, but for higher crank angular velocities, they yielded motions that were qualitatively different, in that the foot was oriented almost vertically at the top dead center and rotated to an almost horizontal position during the downward stroke. Apparently, in our model with two kinematic degrees of freedom without constraints on joint motion, maximum average cycle power at crank angular velocities above 150 rpm was generated with unphysiological motion. Instead of reducing the kinematic degrees of freedom to one (e.g., [49]) or constraining the range of motion of the ankle, we accepted this limitation of our model because we felt that this approach lent more credibility to the results obtained at crank angular velocities from 30 to 150 rpm. Because the purpose of our study was to explain a phenomenon observed in cyclists performing physiological motions, we will not present and analyze any results obtained for crank angular velocities above 150 rpm.

The reader may wonder why we allowed STIM to switch only between 0 and 1 rather than between 0 and a muscle-specific submaximal value to be included in the parameters to be optimized. We performed one optimization in this way for cycling at 120 rpm. This optimization converged to a solution in which the STIM bursts of all muscles had an amplitude of exactly 1.0 and STIM onsets and STIM offsets were the same as the ones obtained with the restricted formulation of the optimization criterion, suggesting that this does in fact lead to maximal power output over a cycle. To save on computer time, we used the restricted formulation of the optimization criterion, allowing STIM to switch only between 0 and 1, in all other optimizations.

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RESULTS

Figure 2 shows for the individual muscles as a function of crank angle the optimal STIM solutions as well as the corresponding active state and muscle–tendon complex length (LMTC) obtained for the reference model at crank angular velocities of 50 and 140 rpm. Slight differences between the motions at these two crank angular velocities caused some differences between the LMTC curves of the muscles crossing the ankle joint but hardly affected the LMTC curves of muscles crossing the other joints. Figure 2 clearly shows that deactivation is slower than activation, for reasons explained in the Methods section, and that deactivation takes a substantial part of the cycle at 140 rpm. The results shown in Figure 2 also exemplify systematic changes that were observed over the whole range of crank angular velocities investigated: when isokinetic crank angular velocity was higher, the STIM bursts of iliopsoas, hamstrings, glutei, rectus femoris, and vasti occurred earlier in the cycle and covered a smaller range of crank angles. Figure 3 shows Ftan, Frad, pedal power, and the active state of the vasti as a function of crank angle for crank angular velocities of 50, 80, 110, and 140 rpm. As expected on the basis of the intrinsic force–velocity relationship, the magnitude of the excursions of Ftan and Frad decreased as crank angular velocity increased. Ftan reached its peak value at a crank angle of about 100° (dotted vertical lines in Figs. 2 and 3), to which we will henceforth refer as the test angle. We constructed the force–velocity relationship by plotting Ftan at the test angle as a function of crank angular velocity (Fig. 4A), and the power–velocity relationship by plotting pedal power at the test angle as a function of crank angular velocity (Fig. 4B). We chose to select Ftan at the test angle instead of the maximal value of Ftan because the latter could occur at different crank angles depending on the crank angular velocity, causing the force–velocity and power–velocity relationship to be contaminated by the effects of variations in transfer of muscle forces to pedal forces. However, the results would have been almost the same if we had plotted the maximal value of Ftan rather than Ftan at the test angle, as will be clear from Fig. 3D. Obviously, even when selecting Ftan at the test angle, muscle fiber lengths may slightly differ over speeds, if only because at a given LMTC muscle force affects the length of series elastic elements. Although the effect of differences in muscle fiber lengths on Ftan is difficult to quantify, the effect on individual muscle forces was small; for example, variations in muscle fiber length of vasti at the test angle over speeds had an isolated effect of only 2.8% on force of these muscles.

FIGURE 2

FIGURE 2

FIGURE 4

FIGURE 4

In Figures 3 and 4, it can be seen that results for muscle contributions only (REF-mc) were nearly the same as results for REF, which means that in isokinetic cycling at the crank angular velocities investigated here, segmental dynamics can safely be neglected when it comes to evaluating the force–velocity relationship. At the lowest crank angular velocities, results of REF were close not only to results of REF-mc but also to results of NINAD. At high crank angular velocities, however, peak forces and pedal power of REF and REF-mc were considerably lower than those of NINAD. For example, at 140 rpm, the actual tangential pedal force and power of REF were less than 70% of the values of NINAD (cf. Fig. 3C with K and Fig. 3D with L). Clearly, in our model, the relationships between pedal force and crank angular velocity and pedal power and crank angular velocity do not reflect the intrinsic muscle properties. The reason for this underestimation is that as crank angular velocity goes up, the active state at the test angle becomes lower and lower. This can be seen in Figure 3A for the vasti muscle group, the largest single contributor to pedal power at the test angle. It also occurs to the same extent in rectus femoris (cf. Fig. 2L and M) and to a lesser extent in glutei (Fig. 2N). For maximizing average power output at higher crank angular velocities, it is optimal that muscles are already partly deactivated when the test angle is reached (42), but for producing as much force and power as possible at the test angle, it is obviously not optimal. In the NINAD model, where activation dynamics is bypassed, the active state was maximal at the test angle regardless of crank angular velocity (Fig. 3I), the shortening velocities of the muscles fibers of all muscles were proportional to crank angular velocity, and there were only minimal variations in muscle force caused by muscle length variations with crank angular velocity (Fig. 2Q–X). Thus, the force–velocity relationship (Fig. 4A) and power–velocity relationship (Fig. 4B) closely reflected the intrinsic relationships. It is important to note that although the relationship between pedal force and crank angular velocity of NINAD (Fig. 4A) is hyperbolic, it has a linear appearance over the range of crank angular velocities investigated. The reason is simple if we look at the relationship of NINAD: we are studying only the low-velocity part! In fact, the highest shortening velocity of muscle fibers at the test angle at 150 rpm, which occurred in the vasti, was only 3.5 muscle fiber optimum lengths per second.

The simulation results may be summarized as follows. First, when simulating isokinetic cycling at 30–150 rpm, we are only studying the effects of the low-velocity part of the muscle force–velocity relationship on the pedal force, and this part has a low curvature to begin with. Second, activation dynamics cause pedal force and power to drop more with crank angular velocity than expected on the basis of intrinsic muscle properties, not only when we extract force and power at the test angle (Fig. 4A–B), but also if we take the average force and power over a cycle (Fig. 4C–D). As a result, the relationship between average tangential force (

) over a cycle and crank angular velocity (Fig. 4C) is almost linear and can be approximated by the regression line:

which yields an optimal pedaling rate of 116 rpm, at which

is 264 N and the average power output is 547 W per leg, or 1094 W = 13.5 W·kg−1 for two legs together.

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DISCUSSION

The purpose of the present study was to understand why relationships between pedal force and crank angular velocity in cycling are less curved than the intrinsic force–velocity relationship of muscles. For this purpose, we simulated isokinetic cycling at crank angular velocities ranging from 30 to 150 rpm with a model of the human musculoskeletal system driven by muscle stimulation over time, which we optimized to maximize average power output. Just like relationships described for cyclists in the literature cited in the introduction, the simulated relationship between pedal force and crank angular velocity was almost linear (Fig. 4A, REF). It turned out that over the range of speeds investigated, i.e., 30–150 rpm, one is studying only the effects of the low-velocity part of the force–velocity relationship of the muscles on the pedal force, and this part has a low curvature to begin with. Furthermore, it was found that pedal force (Fig. 4A, REF) and pedal power at the test angle (Fig. 4B, REF) dropped more with crank angular velocity than expected on the basis of intrinsic muscle properties (Fig. 4A–B, NINAD). In contrast to what was found in leg press tasks (3), this was not explained by segmental dynamics in cycling; there were only small differences between the total tangential pedal force (REF) and the muscular contribution to this force (REF-mc) (Figs. 3 and 4). Rather, it was explained by the active state at the test angle becoming lower and lower as crank angular velocity went up. Below, we will discuss the validity of the simulation study and the explanation, interpretation, and relevance of the simulation results.

The first question to be asked is whether sprint cycling in humans is represented in a sufficiently valid way by our simulation and optimization approach. After all, the main conclusion to be drawn on the basis of simulation results should not depend on the specific values assigned to parameters in the model. The musculoskeletal model used here is by definition a simplification of the human musculoskeletal system, but it did include the major muscle–tendon complexes of the lower extremity and the fundamental behavior of their elements. It was originally designed for jumping and, after optimization of STIM(t) for jump height, has successfully reproduced various types of jumps in terms of jump height, kinematics, kinetics, and even muscle activation patterns (e.g., [4,44]). The reference model was then used without any further tuning of parameter values for simulation of sprint cycling (42) where, after optimization of STIM(t) for maximum average power output in cycling at 120 rpm, a remarkably good correspondence was obtained in both shape and magnitude between the tangential pedal force profile of the model and the profile measured in cyclists during isokinetic sprint cycling (2). Radial force profiles pertaining to the optimal solution for the model were also similar to profiles measured in cyclists during isokinetic sprint cycling, except for crank angles between 150° and 200° where they were larger than measured radial forces (34,42). Because the force profiles result from joint moment profiles, which in turn are caused by the forces generated by the muscles, it should not come as a surprise that individual muscles of the simulation model were stimulated (Fig. 2) in the phases of the cycle where EMG bursts were found in the muscles in cyclists performing isokinetic sprints (e.g., [13]). Interestingly, Dorel et al. (13) found that in isokinetic sprint cycling, the EMG of hamstrings, gluteus maximus, and tibialis anterior reached peak values of only 70%–80% of the values during maximal voluntary contractions. We expected that if we limited STIM to submaximal values and conducted new optimizations, STIM bursts would become wider and performance would be. By performing one additional optimization at 120 rpm in which we limited the maximal STIM of the muscles mentioned, we confirmed that it was indeed the case (results not shown). Hence, if cyclists limit the maximum activation of muscles as reported by Dorel et al. (13), they probably do not do so for reasons of maximizing performance. Quite a number of studies in the literature on cycling also report values for the “index of mechanical effectiveness” (11,12) or “force effectiveness ratio” (24), i.e., the ratio of tangential force integrated over a section of a cycle to total force integrated over that same section of a cycle. We consider this variable irrelevant when it comes to maximizing performance in sprint cycling. After all, if increasing the radial force is necessary to increase the tangential force and hence power output, then so be it. For evaluation of the simulation results, however, the “index of mechanical effectiveness” (IE) can still be used. For human sprint cycling, values for IE over the full cycle ranged from a maximum of about 0.8 on average around 50 rpm to about 0.5 on average around 150 rpm (ref. 12, their Fig. 3). In our simulation study, the corresponding values were 0.71 and 0.48, respectively. During the downstroke (from 30° to 150°), IE values in our simulation study were about 0.81, which was within the range of values measured in subjects (ref. 12, their Fig. 4). The optimal pedaling rate of the model, i.e., the rate at which maximum average cycle power was produced, was 116 rpm. This is close to the values reported for healthy active male subjects (see, for example, Ref. [14] and data of Ref. [2] presented in Ref. [42]) but somewhat lower than the average value of 123 ± 11 rpm reported for athletes trained in sports other than cycling (21) and substantially lower than the value of 130 ± 5 rpm reported for world-class sprint cyclists (14). It has been shown that optimal pedaling rate is correlated with the percentage of fast twitch muscle fibers (19). Because the model had relatively fast muscles compared with human subjects, one would actually expect the model to have a higher optimal pedaling rate than human subjects solely based on muscle contractile properties. But Van Soest and Casius (42) clearly showed that activation dynamics, and especially the rate of deactivation, also has a major effect on the optimal pedaling rate. Activation dynamics of real muscles have many intricacies. For example, the active state decreases more rapidly during shortening than during isometric contraction (e.g., [16]), and this shortening-induced deactivation is to some extent velocity dependent (e.g., [10]). This phenomenon was not included in our simulation model. Hence, the rate of deactivation might well be underestimated in the model, which would then tend to cause the model’s optimal pedaling rate to be lower than that of human subjects. Inevitably, some uncertainty will remain as to the exact properties of the human musculoskeletal system, and therefore, one should not expect the model to predict the exact values of variables such as the optimal pedaling rate of human subjects. The same is true for the average power output at the optimal pedaling rate. In the model, we found a value of 13.5 W·kg−1, which is within the range of experimental values reported in the literature: higher than the value of 12.6 ± 1.9 W·kg−1 reported for athletes trained in “explosive sports” other than cycling (21), but somewhat lower than the value of 15.4 ± 1.5 W·kg−1 reported for the third French league volleyball players (37), and substantially lower than the value of 19.3 ± 1.3 W·kg−1 reported for world-class sprint cyclists (14). Admittedly, the actual power output of the athletes will have been somewhat higher than the reported values because it was measured at the crank, whereas 8%–10% power loss may occur because of friction in the crank set, chain, and flywheel bearings. At the same time, a 10% increase in power of the model could be easily realized by increasing all muscle forces by 10%, and this would not change the behavior of the model. All things considered, in our opinion, the simulation model satisfactorily reproduced the important features of human isokinetic sprint cycling, especially during the downstroke and at the test angle, and certainly the mechanisms that explain the general behavior of the simulation model will also explain the general behavior of human subjects. We see no reason to suspect, therefore, that the main conclusion of the study depends on specific values assigned to parameters in the model.

The finding that pedal force in cycling dropped more with crank angular velocity than expected on the basis of Hill’s hyperbolic relationship (Fig. 4) is similar to what was found in explosive leg press tasks (3). In explosive leg press tasks, the explanation was that segmental dynamics canceled more and more of the contribution of muscle forces to the force on the dynamometer as leg extension velocity increased. In cycling simulations, however, segmental dynamics hardly affect the force–velocity relationship according to our calculations (Fig. 4). How can this discrepancy be explained? The answer is that leg extension velocities, which quadratically affect segmental dynamics, remained lower in cycling than in explosive leg press tasks. To a very rough approximation, the configuration of the leg at the test angle in cycling is comparable with the configuration in which the relation between leg extension velocity and dynamometer force was studied previously (3), and the direction of the tangential pedal force is not too different from that of the dynamometer force. When the test angle was reached in cycling at the highest crank angular velocity of 150 rpm, leg extension velocity was only 2.2 m·s−1. During the simulated explosive leg press task, the forces due to segmental dynamics at 2.2 m·s−1 were less than half of those at 2.8 m·s−1 (3). So the apparent discrepancy in the role of segmental dynamics is primarily due to the fact that we studied the force–velocity relationship up to lower velocities in cycling than in explosive leg extension. Nevertheless, even at crank angular velocities between 100 and 150 rpm, the force exerted on the pedal at the test angle—in solutions obtained by optimizing average power output—is substantially below the maximal force that the muscles could produce (Fig. 4A). At crank angular velocities of 140 rpm and higher, the peak tangential pedal force is less than 70% of its maximal value (compare model REF-mc with model NINAD), and the mean tangential pedal force is only about 60% of its maximal value (Fig. 4C). As a result, the relationship between pedal force and crank angular velocity for isokinetic sprint cycling tends to be linear rather than hyperbolic, just like the relationships measured in experiments on human subjects. However, we should not expect a highly curved relationship to begin with because, as suggested by the results obtained with the NINAD model (Fig. 4A), we are actually studying only the low-velocity part of the force–velocity relationship. In any case, we can now conclude that during explosive leg extension, dynamometer force drops faster with velocity than the intrinsic muscle force because part of the generated muscle force causes accelerations of the segments and is not “seen” by the dynamometer (3), whereas in cycling, pedal force drops faster with velocity than the intrinsic muscle force because part of the intrinsic muscle force is simply unrealized.

The simulation results help us understand why relationships between pedal force and crank angular velocity in cycling are less curved than the intrinsic force–velocity relationship of muscles. Maximizing average power in cycling requires muscles to bring their intrafilamentary Ca2+ concentration from as high as possible during shortening to as low as possible during lengthening through active sequestration of Ca2+ by the sarcoplasmatic reticulum (28,29,42). Deactivation is a relatively slow process (6,26,29), so the instant of switching off muscle stimulation must occur a certain amount of time before lengthening starts. As crank angular velocity goes up, this amount of time corresponds to a greater change in crank angle, so the instant of switching off muscle stimulation must occur earlier in the cycle and, for all muscles contributing to the pedal force, earlier relative to the test angle where the pedal force was extracted for the force–velocity relationship (Fig. 3). Obviously, this cause of affairs is only related to the angular velocity of the crank and the purpose of maximizing average power output and not to the specific sprint cycling conditions. Thus, the effects that we have described not only occur during sprint cycling on an isokinetic dynamometer but will surely also occur during sprint cycling on mechanically or electronically braked flywheel ergometers, and they will also apply to the acceleration phase of sprint cycling on such dynamometers. Hence, the mechanism revealed by the simulations seems sufficient to explain why relationships between peak or average pedal force and crank angular velocity measured in all the studies cited in the introduction tend to be less curved than the intrinsic force–velocity relationship. As a matter of fact, the same holds for work-loop studies on isolated muscles. For example, James et al. (22) imposed oscillatory length changes at different frequencies to rabbit latissimus dorsi muscles and optimized the onset and offset of stimulation to maximize average cycle power. If we divide their power values by the corresponding velocities (Table 3 in [22]), we obtain an average force that drops linearly with velocity, exactly as we found in the current study on cycling (Fig. 4C). Therefore, just like in explosive leg press tasks, we see no need for the assumption that suboptimal coordination is hampering force production at high crank angular velocities (38). In any case, our key finding is that relationships between pedal force and crank angular velocity measured during sprint cycling in vivo do not reflect solely the intrinsic force–velocity relationship of muscles, and hence are deceptive. It follows that changes in the relationship measured in vivo, for example, as a result of training, are not necessarily due to changes in intrinsic muscle characteristics and should be interpreted with caution.

The authors declare that they have no funding for this study and that they have no professional relationships with companies or manufacturers who will benefit from the results of the present study. Results of the present study do not constitute endorsement by the American College of Sports Medicine.

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Keywords:

CYCLIC MOVEMENTS; MUSCULOSKELETAL MODEL; FORWARD DYNAMICS; OPTIMIZATION; MUSCLE POWER OUTPUT

© 2016 American College of Sports Medicine