 Original Research Article
 Open access
 Published:
Exploring the Low ForceHigh Velocity Domain of the Force–Velocity Relationship in Acyclic LowerLimb Extensions
Sports Medicine  Open volume 9, Article number: 55 (2023)
Abstract
Purpose
To compare linear and curvilinear models describing the force–velocity relationship obtained in lowerlimb acyclic extensions, considering experimental data on an unprecedented range of velocity conditions.
Methods
Nine athletes performed lowerlimb extensions on a legpress ergometer, designed to provide a very broad range of force and velocity conditions. Previously inaccessible low inertial and resistive conditions were achieved by performing extensions horizontally and with assistance. Force and velocity were continuously measured over the pushoff in six resistive conditions to assess individual force–velocity relationships. Goodness of fit of linear and curvilinear models (secondorder polynomial function, Fenn and Marsh’s, and Hill’s equations) on force and velocity data were compared via the Akaike Information Criterion.
Results
Expressed relative to the theoretical maximal force and velocity obtained from the linear model, force and velocity data ranged from 26.6 ± 6.6 to 96.0 ± 3.6% (16–99%) and from 8.3 ± 1.9 to 76.6 ± 7.0% (5–86%), respectively. Curvilinear and linear models showed very high fit (adjusted r^{2} = 0.951–0.999; SEE = 17159N). Despite curvilinear models better fitting the data, there was a ~ 99–100% chance the linear model best described the data.
Conclusion
A combination between goodness of fit, degrees of freedom and common sense (e.g., rational physiologically values) indicated linear modelling is preferable for describing the force–velocity relationship during acyclic lowerlimb extensions, compared to curvilinear models. Notably, linearity appears maintained in conditions approaching theoretical maximal velocity. Using horizontal and assisted lowerlimb extension to more broadly explore resistive/assistive conditions could improve reliability and accuracy of the force–velocity relationship and associated parameters.
Key Points

Lowerlimb extensions performed horizontally with assistance resulted in very low inertial and resistive conditions which provides access to assessment conditions approaching neuromuscular limits (i.e., near theoretical maximal velocity) in acyclic lowerlimb extensions.

Compared to curvilinear models, linear modelling of the force–velocity relationship in acyclic lowerlimb extensions displayed the best combination of fitting the underlying data, complexity of the modelling approach, and physiologically rational output parameters.

Researchers and practitioners can confidently use linear modelling to describe the force–velocity relationship in acyclic lower limb extensions up to 75% of maximum lowerlimb extension velocity in average, and up to 85% for some individuals.
Background
Ballistic movements are common in daily life and crucial in many sports. Success during such maximal efforts relies on high force and power production over the entire movement. Human dynamic maximal force and power generation capabilities depend on movement velocity and are well described by the force–velocity (F–v) and powervelocity (P–v) relationships [e.g., 1]. These two relationships have four main output variables of interest: (i) P_{max}, the apex of the P–v relationship representing the maximal power that can be reached at a specific velocity, called optimal velocity (v_{opt}); (ii) F_{0}, the forceintercept of the F–v relationship, corresponding to the theoretical maximal force produced at zero velocity; (iii) v_{0}, the velocityintercept of the F–v relationship, corresponding to the theoretical maximal velocity until which force can be produced, and (iv) the slope (or curvature) of the F–v relationship representing the rate at which force production capabilities decrease when velocity increases. F_{0} and v_{0} represent strength indexes of force production capabilities at low and high velocities, respectively, i.e., in the high forcelow velocity and low forcehigh velocity domains of the F–v relationship. F–v and P–v relationships have seen wide adoption in testing and training of ballistic performance. For instance, biomechanical modelling [2,3,4] and experimental results [5] indicate that ballistic performance depends on both P_{max} and the slope of the F–v relationship. Several studies have provided a basis for training guidelines [6], which revolve around individualization and subsequently improved training efficiency [e.g., 7–9]. Consequently, the F–v relationship interests both practitioners and coaches.
In lowerlimb ballistic extensions, F–v and P–v relationships can be evaluated via i) cyclic extensions, such as during running [e.g., 10] or cycling [e.g., 11], and with ii) acyclic extensions, like during vertical [12] and horizontal jumping [e.g., 13] or on inclined/horizontal legpress devices [e.g., 3, 14]. While cyclic extensions involve force orientation technique, and thus their transferability is limited, acyclic extensions rather consider the quasitotal external force developed by lower limbs, assessing less exercisespecific strength indexes. Once collected, mathematical modelling is used to determine the F–v relationship, from which the variables of interest are extracted (i.e., F_{0}, v_{0}, P_{max} and the slope). In the case of lowerlimb acyclic extensions, the F–v relationship has been mostly described using linear modelling [e.g., 13, 15, 16]. The linear model is based on the basic firstorder polynomial function and typically exhibits very high goodness of fit (GoF; i.e., high coefficient of determination [r^{2}] and low standard error of estimate [SEE]) on force and velocity data. Nevertheless, the use of linear models has been questioned [e.g., 17–19] since the F–v relationship’s evaluation typically includes force measurements across a restricted range of velocity conditions (~ 20 to ~ 50–60%v_{0}) [e.g., 13, 16, 17]; accordingly, because more than half of the F–v relationship is typically undescribed by experimental data, any linearity observed might instead represent a partial range of an overall curvilinear shaped relationship. Two empirical arguments have been proposed to support the use of curvilinear models in acyclic lowerlimb extensions. Firstly, in monoarticular human movements or singlemuscle in vitro conditions [20, 21], curvilinear models fit a wide range of velocity conditions (from ~ 0 to ~ 75–99%v_{0}). Under these conditions, curvilinear models were based on i) an exponential function (F&M's_{Eq}; [20]); ii) a reciprocal function (rectangular hyperbola, Hill’s_{Eq}; [21,22,23]); or iii) a combination of the two [24], which showed very high GoF and SEE on force and velocity data. The second argument was that the basic secondorder polynomial (Poly_{2}) or exponential functions typically exhibit higher GoF compared to the one of the linear model, when fitted to data situated within the typical restricted range of velocity [e.g., 17, 18, 25]. Of note, before studying F–v relationship on isolated muscles, Hill and colleagues studied it on single (elbow flexion; [26]) and multijoint (pedalling; [27]) movements, using linear model. Several experimental studies have explored the F–v relationship beyond the typical 20–60%v_{0} range and assessed the GoF of functions of the linear and curvilinear models (n.b., from this point on, fitting quality [i.e., r^{2} and SEE] of the function of a model will be discussed directly as fitting quality of a model).
In the high forcelow velocity domain (i.e., from 0 to ~ 20%v_{0}), researchers typically agree that linear models simply and accurately fit force and velocity experimental data and estimate F_{0} [e.g., 25,26,27,28]. In the low forcehigh velocity domain (i.e., from ~ 60 to 100%v_{0}), only three studies have explored lowerlimb force production in conditions nearing maximal velocity [14, 29, 30]. Yamauchi et al. [14] reported higher GoF of the linear model than curvilinear models (using the basic exponential function) up to ~ 97%v_{0}. However, force and velocity were collected at specific joint angles as peak values, which limits the transferability of the results to other experimental conditions [14, 31]. Lindberg et al. [30] reported very high GoF of the linear model until ~ 85%v_{0}, without considering curvilinear models in their analyses. Alcazar et al. [29] observed the F–v relationships of some participants were better described by the linear model and others by a curvilinear model (using Hill’s_{Eq}), although the underlying force and velocity data in the low forcehigh velocity and the high forcelow velocity domains were obtained via different exercise conditions and analyses. In addition to these three studies, two empirical arguments support adopting a ‘simpler’ linear model: firstly, Bobbert [32] showed that the F–v relationship displayed a “quasilinear” shape from ~ 5 to 90%v_{0} in a simulation of lowerlimb extensions performing a legpress task (each muscle’s F–v relationship was described by Hill’s_{Eq}), and; secondly, the F–v relationship has been described by the linear model in other multijoint movements, such as during cycling and running, with high GoF on experimental data covering the wide range of ~ 20–90%v_{0} [11, 33]. In any case, while ultimately there is no consensus (likely owing to a dearth of research exploring low forcehigh velocity domains), the current evidence indicates linear modelling likely best describes the F–v relationship.
Previous studies have typically evaluated models by detecting significant differences between GoF, only comparing nonadjusted r^{2} and SEE. This is problematic as these two indexes naturally inflate with models’ complexity, but without penalizing the use of higher degrees of freedom, favoring thus more complex models. Moreover, should a better fit be detected, it is impossible to clarify whether adding degrees of freedom describes the experimental data well enough to justify their higher complexity over simpler models. Indeed, more degrees of freedom increases variance, which can lead to noise in the model fit and biased or physiologically illogical estimations of outputs (e.g., F_{0} or v_{0}; [25]). Hence, previous works did not consider the principle of parsimony, which dictates that “Numquam ponenda est pluralitas sine necessitate”, as stated by William of Ockham (transl. plurality must never be posited without necessity; [34]). Applied here, models with higher degrees of freedom should not be preferred when simpler models are equally experimentally and statistically evidenced [35], as recommended in sport and exercise science [36].
In this study we aimed to compare the accuracy and relevance of linear and curvilinear models to describe the force–velocity relationship in acyclic lowerlimb extensions across a broad range of velocity conditions. We hypothesized that, despite higher GoF of curvilinear models, their greater complexity would not improve the description of force and velocity data to an extent that would warrant their use instead of the simpler linear model.
Methods
Participants
Nine healthy participants (8 males and 1 female, age = 21.3 ± 0.5 years, mass = 70.6 ± 9.1 kg and stature = 1.78 ± 0.07 m) gave their written informed consent to take part in this study, which was approved by the local ethics committee and complied with the standards of the declaration of Helsinki. All participants practiced regular physical activities (strength and endurance training) with no common training program between them (in terms of volume and intensity), and were free of musculoskeletal pain or injury during the study.
Design of the Study
This study comprised three sessions separated by 24 to 48 h of rest. The first session familiarized participants with performing ballistic lowerlimb extensions on the ergometer at high forcelow velocity settings and viceversa. The final two sessions were dedicated to assessing individual F–v and P–v relationships and each involved performing ballistic lowerlimb extensions in 6 resistive conditions. Two sessions were planned to ensure that participants could maximize force production at very high velocities [e.g., 35].
Ergometer
A shared limitation of previous works characterizing lowerlimb acyclic force production is an inability to access extremely high velocities due mainly to the mechanical constraints imposed by the body weight and inertia. We addressed this issue by building an innovative instrumented legpress ergometer (vide infra). The ergometer was a custombuild horizontal legpress equipped with a flywheel surrounded by a friction belt (Fig. 1). It comprised of a metal frame supporting a fixed seat to which each participant was harnessed, with adjustable pads above their shoulders. Participants were positioned with their lower limbs flexed, and feet placed upon a chariot, in a position that approximated the bottom of a squat jump. The chariot was set on lowfriction rails along which it was free to slide. In this manner, the ergometer allowed assessment of extensions without moving the entire body mass, where the user instead drove the chariot with the lower limbs. A friction belt and lateral traction springs provided control over resistance and assistance applied to the chariot motion, respectively, and enabled access to a broad range of mechanical conditions (notably, very high movement velocities). For each trial, the chariot was held in its starting position via electromagnets that were released by the participant via a handheld button, allowing the chariot to move under ballistic intent (i.e., feet losing contact with the chariot at the end of the extension). For each lowerlimb extension, participants were asked to apply force onto the chariot, which resulted in its acceleration and the concomitant acceleration of the flywheel linked by a chain. Instantaneous linear and angular displacements of the chariot and the flywheel were measured with linear (Kübler Group, VillingenSchwenningen, Allemagne, 250 Hz) and angular (Baumer, Fillinges, France, 250 Hz) encoders, respectively. The friction forces applied by the belt on the flywheel (\(F_{fb}\)) was measured with a strain gauge (Futek, Irvine, USA, 250 Hz).
Protocol
Each session began with a warmup consisting of ~ 15 min of dynamic movements including submaximal and maximal unloaded squats, squat jumps and lowerlimb ballistic extensions on the ergometer at high forcelow velocity and viceversa.
The first session focused on familiarizing athletes with the testing protocol. This included placing the participant in a position and adjusting the ergometer until they felt able to express maximum force. This placement was recorded for latter sessions and all subsequent trials. Participants then performed twenty to thirty ballistic lowerlimb extensions on the ergometer in six different resistive and inertial conditions, interspersed with a minimum of 10 s passive rest periods, to habituate the participants with maximal effort (i.e., maximal neuromuscular activation) in each extension condition performed on the ergometer. To conclude, participants performed two maximum ~ 3s maximum isometric contractions separated by 5 min of rest. Athletes were instructed to “push as hard and as fast as possible” for each trial, and verbally encouraged during the trial. The ergometer chariot was set in the previous selfselected preferred starting position (with knee and hip angle ranging from 72 to 114° and from 98 to 125°, respectively) with friction force set at maximum to prevent the chariot from any displacements on the rail.
The second and third sessions largely mirrored the first, except the resulting data were recorded to determine individual F–v and P–v relationships over the largest range of resistive conditions possible (in order of decreasing resistance): (1) resistive friction eliciting a movement velocity of ~ 0.3 m s^{−1}, as the typical extension velocity observed during a one repetition maximum squat (C_{1RM}; determined at the end of the familiarization session), (2) resistive friction corresponding to ~ 50% of maximal isometric force (C_{50%Fmax}), only accelerating the chariot and the flywheel, (3) the friction belt being removed, without and (4) with two springs assisting the motion (C_{ØFric0S} and C_{ØFric2S,} respectively) and (5) only accelerating the chariot, the chain between chariot and flywheel being removed without and (6) with two springs in assistance (C_{Char0S} and C_{Char2S}, respectively; Table 1). Two to three trials were performed for each resistive condition. For each trial, participants were asked to trigger the electromagnets and to hold lateral handles for upperbody stabilization, while producing as much force as possible and extending their lower limbs as fast as possible, aiming to push the chariot ballistically.
Data Analysis
During isometric tests, force output was measured with the strain gauge on the friction belt and the maximal isometric force was calculated as the maximum averaged force over one second. During lowerlimb extensions, as hip was fixed and feet were constantly in contact with the chariot, the instantaneous extension velocity (m s^{−1}) and acceleration (\(a_{{{\text{chariot}}}}\), in m s^{−2}) of the lower limbs were determined as first and secondorder derivative of the lowpass filtered (20 Hz, Butterworth, 4th order) position signal obtained via the linear encoder. During each trial of all conditions, instantaneous force (in Newtons) was computed using Eq. 1 (detailed computations for each of the six resistive conditions in Table 1).
where F_{flywheel} is the force to accelerate the flywheel (Eq. 2), F_{chariot} to accelerate the chariot (Eq. 3), F_{friction} the force to overcome the frictional forces applied by the belt on the flywheel (Eq. 4), F_{limbs} the force to accelerate the center of mass of the lower limbs (Eq. 5), which was estimated from 2D biomechanical model (detailed in the next paragraph), F_{roll} the internal resistive force of the flywheel (6.06 N) and F_{spring} the force of the tension springs (Eq. 6).
where I is the moment of inertia of the flywheel (0.131 kg m^{2}), ∝ (rad s^{−2}) the instantaneous angular acceleration of the flywheel determined as the secondorder derivative of the lowpass filtered (20 Hz, Butterworth, 4th order) position signal obtained from the angular encoder, d_{p} (m) the cog radius (0.032 m), m_{chariot} the mass of the chariot (15.15 kg) including the mass of the chain (1.05 kg), d_{flywheel} the radius of the flywheel (0.24 m) m_{limbs} the mass of the lower limbs, α_{limbs} (m s^{−2}) the instantaneous acceleration of the lower limb’s center of mass estimated from a 2D biomechanical model (see below), n the number of springs in assistance during the lowerlimb extension, k the spring’s stiffness (320 N m^{−1}), b the initial spring tension at free length (40 N) and x the instantaneous length of the spring determined from the instantaneous position of the chariot. Note that the rolling friction of the chariot on the rail was counterbalanced by the very low linear encoder traction force, and thus negligible (~ 0.01 N).
As proposed by Rahmani et al. [37] for the bench press exercise, the use of a simplified 2D model with three segments is accurate enough to estimate center of mass displacement of the upper limbs. Thus, the 2D model of the lower limbs used in the present study comprised three segments (thighs, shins and feet), with the length and mass of each estimated as a fraction of body height and mass, respectively [38]. The model allows for the determination of the center of mass instantaneous horizontal position of the three body parts during lowerlimb extensions, as the barycenter of the thighs, shins and feet center of mass. Then, the center of mass instantaneous horizontal position of the lower limbs was estimated.
Force, velocity, and power were averaged over lowerlimb extensions, which started when a_{chariot} became positive and ended when:
for conditions with the frictional forces on the flywheel, or for other conditions, respectively. Here, m_{flywheel} (126 kg) being the linear equivalent mass of the flywheel’s moment of inertia, which was computed as:
For each participant, F–v and P–v relationships were determined from mean force, velocity and power values obtained from the best trial (i.e., highest mean power output) of the six different resistive conditions across all trials performed in the second and third sessions. These values were fitted with the basic firstorder polynomial function to model a linear shaped individual F–v relationships (for the linear model) and with Poly_{2}, F&M’s_{Eq} and Hill’s_{Eq} to model a curvilinear shaped individual F–v relationships (for the curvilinear model). As described by Hill in 1938, Hill’s_{Eq} corresponded to [22]:
where F and v correspond to mean force and velocity over lower limb pushoff, and a and b are constants. Accordingly to the Fenn & Marsh’s work published in 1935, F&M’s_{Eq} corresponded to [20]:
where A and B are constants.
The optimization procedure to fit the function of each model on the experimental force and velocity data consisted of applying of least squares method with polynomial regression for the basic firstorder polynomial function and Poly_{2}, or the Levenberg–Marquardt algorithm for F&M’s_{Eq} and Hill’s_{Eq}. Optimizations were aiming to minimize the sum of squared errors, with the Levenberg–Marquardt algorithm set to 1.10^{7} model evaluations (i.e., number of evaluations of the loss function) and 1.10^{6} iterations (i.e., number of increments of the function’s variables). As the Levenberg–Marquardt algorithm finds only a local minimum of the loss function, which is highly dependent on the function’s starting parameters, the procedure of optimization was repeated one thousand times, considering at each repetition, a random starting value (within the range [0; + ∞] and [− 100; + ∞] for Hill’s_{Eq} and F&M’s_{Eq}, respectively).
Individual F_{0} and v_{0} values were computed as the force and velocityaxis intercept for each model. Individual P–v relationships were determined by integrating over velocity the F–v relationship. Then, P_{max} and v_{opt} were determined as the apex of the P–v relationship and the velocity condition at which P_{max} occurred, respectively. F_{0} and P_{max} were additionally expressed relative to body mass for (_{Rel}F_{0} and _{Rel}P_{max}, respectively).
Statistical Analysis
All data are presented as mean ± standard deviation (SD). To locate the mean force, velocity and power values obtained in the six resistive conditions, these outputs were expressed relative to F_{0}, v_{0} and P_{max} obtained with the linear model (F_{0L,} v_{0L}, and P_{maxL}, respectively). When Hill’s_{Eq} was used in curvilinear models, the magnitude of the curvature of the F–v relationship was quantified by computing the ratio a/F_{0} [22].
The effect of fit function (the basic firstorder polynomial function, Poly_{2}, F&M’s_{Eq} and Hill’s_{Eq}) on F_{0}, _{Rel}F_{0}, v_{0}, P_{max}, _{Rel}P_{max} and v_{opt} was tested with ANOVAs. These ANOVAs were performed after checking for normal distribution and sphericity with Shapiro–Wilk’s and Mauchly’s tests, respectively. If not met, a sphericity correction was applied. If the effect of the main factor was significant, Holm’s post hoc test was used to highlight significant differences. The magnitude of effect for each factor within the model was quantified via η^{2} and ω^{2}, which were both interpreted as trivial, small, moderate and large when matching value of < 0.01, 0.01–0.06, 0.06–0.14 and > 0.14, respectively [39]. The magnitude of the difference (i.e., effect size; d) between outputs of the four functions (posthoc tests) was reported via standardization to the betweensubject standard deviation, as well as their associated confidence intervals. Effect sizes, d, were interpreted using qualitative thresholds, with < 0.2, 0.2 to < 0.6, 0.6 to < 1.2, 1.2 to < 2.0 and > 2.0 representing trivial, small, moderate, large, and very large effect, respectively [40]. For all statistical analyses, an alpha level of 0.05 was set.
To describe the GoF of the four functions, adjusted r^{2}, SEE and distribution of residuals in force across velocity condition were computed. The magnitude of the differences between adjusted r^{2} and SEE from the four functions was assessed using specific scales as proposed by Rudsits et al. [41]. A clear improvement in adjusted r^{2} was identified when its value increased from one magnitude threshold to the next on the scale: 0.99, 0.92, 0.74, 0.50, and 0.20. This scale was also used to describe the magnitude of adjusted r^{2} corresponding to extremely high, very high, high, moderate, and low values, respectively. SEE values were compared using the qualitative thresholds above, but magnitude thresholds for assessing the standardized effect were halved [41].
Since SEE and residuals in force do not represent criterions for model selection with parsimony and statistical inferences, and interpreting change in adjusted r^{2} could be limited, models were compared using and Akaike Information Criterion (AIC) analysis (for details, please see [42]). Briefly, this method was used to detect whether Poly_{2}, F&M’s_{Eq} and Hill’s_{Eq} lead to a great enough improvement of the GoF to justify their higher complexity (i.e., increased number degrees of freedom), in comparison to the firstorder polynomial function (i.e., linear modelling). This analysis was conducted on each individual force and velocity data set. To perform this test, i) the sum of standard error of each model (SSE), ii) the corrected AIC index (AICc; used due to the ratio sample size/degrees of freedom being inferior to 40; Burnham and Anderson 2004), iii) the difference in AICc between the model with the smallest AICc and other models (ΔAICc), iv) the relative likelihood of each model, v) the AICc weight (AICc_{w}) for each model and vi) the relative and vii) absolute AICc evidence ratio (AICc_{wER}) were computed (for detailed definitions of these parameters, please see [43]).
Results
Mean force, velocity and power developed over the pushoff in the six resistive conditions are presented in Table 2, in raw values and expressed relative to F_{0L}, v_{0L} and P_{maxL}, respectively.
Typical examples of F–v relationships drawn using the linear and curvilinear models, associated with their resulting P–v relationship, are presented in Fig. 2. When using Hill’s_{Eq}, a/F_{0} value was 1.06 ± 0.72 (no unit). F_{0}, _{Rel}F_{0}, v_{0}, P_{max}, _{Rel}P_{max} and v_{opt} values are presented in Table 3. When using Poly_{2}, v_{0} could not be calculated for eight participants, due to the absence of an intercept with the velocity axis (i.e., the fit trended towards infinity; see examples of two individuals on Fig. 2, dashed black line on the left middle and bottom panels). There was a significant main effect of fit function on F_{0}, _{Rel}F_{0}, v_{0}, P_{max} and _{Rel}P_{max} (all p < 0.05; η^{2} = 0.720, 0.786, 0.618, 0.785, 0.817, respectively, and ω^{2} = 0.047, 0.105, 0.271, 0.043 and 0.102, respectively), but not on v_{opt} (p = 0.380). Posthoc analyses’ pvalues and effect sizes are presented in the Table 3.
GoF of the linear and curvilinear models, assessed with adjusted r^{2}, SEE and the distribution of force residuals on the velocity conditions spectrum, are presented as individual values on the panels of Figs. 3 and 4. Effect size of change in SEE were large, when comparing the linear model to the curvilinear models, but a clear improvement of adjusted r^{2} was observed for only three participants (Fig. 3, left panel). Comparisons of the linear and curvilinear models using AICc analysis are presented in Table 4.
Discussion
An innovative legpress ergometer allowed lowerlimb external force production measurements over very broad mean extension velocity range (individual values ranged from ~ 0.2 to ~ 3.1 m s^{−1}). Expressed relative to individual force–velocity relationships, the range corresponded to ~ 8 and ~ 77%v_{0L} (individual values ranged from ~ 5 to ~ 86%v_{0L}). Over the extended range of experimental data, following the principle of parsimony, the linear model was very likely the best model to describe the force–velocity relationship, compared with curvilinear alternatives.
The novel ergometer presented here allows ballistic and horizontal (i.e., without the resistance of the body weight) lowerlimb extensions, with assistance to the motion, without upperbody movement and with low external masses (i.e., only lower limb and chariot masses). Where the methodological settings (e.g., technology) of previous studies allowed ‘high’ velocity conditions of only ~ 1.7–2 m s^{−1} [13, 17, 29, 44], the ergometer made it possible to reach mean extension velocities of up to 2.7–3 m s^{−1} (Table 2). Such high velocity conditions were only accessible due to a combination of both very low resistive and inertial conditions and technical assistance, since the best individual in C_{Char0S} showed similar value as the mean of individual in C_{Char2S}. In acyclic lowerlimb extensions, reaching very high movement velocity is challenging since each effort starts at zero velocity and requires the inertia of the moving masses to be overcome in each resistive/loading condition. In contrast, up to 90%v_{0L} is attainable without cumbersome methods and equipment in cyclic movements (e.g., running and cycling), because highvelocity lowerlimb extensions occur when the moving masses have been already accelerated [11, 33].
In the present study, linear and curvilinear models showed equally distributed residuals across velocity conditions (i.e., 5–86%v_{0L} range) and small SDs within 100 N (Fig. 4). These results highlight that all models describe the force and velocity data over the tested experimental range with similar precision. Further, each model provided a very to extremely highquality fit, with low SEE (Fig. 3). Change in SEE from the linear model to curvilinear models were large with a clear improvement of adjusted r^{2} observed for only three participants. This was mainly caused by isolated cases of high errors for the linear model (Fig. 4). Overall, even if curvilinear models fit the data minorly better compared to the linear model [e.g., 17, 25, 27], the GoF of all models was in the high to very high quality range. Nevertheless, when examining the prediction error and relative quality of the different models (i.e., AIC, see Table 4) the linear model had a ~ 99% chance to be the best model and displayed extremely strong evidence in its favor. Consequently, despite higher GoF of curvilinear models, their higher degrees of freedom did not improve accuracy of F–v relationship description to an extent warranting their utilization. This follows the principle of Occam’s razor, which states that among models with similar accuracy, the one with the fewest assumptions and parameters is preferable [34]. These results support using simpler linear model to describe the F–v relationship in acyclic lowerlimb extensions over a broad range of resistive conditions.
The validity of a model describing the F–v relationship relies on the practical and physiological relevance of its output parameters – here corresponding to F_{0}, v_{0}, P_{max} and its slope. In the present study, P_{max} values estimated from the linear model were very close to the experimental power output measured around v_{opt} (e.g., right bottom panel, Fig. 2). These findings are in line with previous studies, which reported similar results on legpress and horizontal and vertical squat jumps [14, 25, 29]. Consequently, P_{max} values estimated from the linear model are likely accurate estimates of true values and are physiologically relevant. Comparatively, curvilinear models exhibited lower P_{max} values than the linear model by ~ 100–120 W (Table 3 and Fig. 2), and these values can be lower than experimental power output measured around v_{opt} (e.g., right bottom panel, Fig. 2). Thus, despite the values being rational, curvilinear models probably underestimate P_{max}. These results contrast reports of higher P_{max} estimated by curvilinear models (using Poly_{2} and Hill’s_{Eq} [25, 45]), which might be explained by these studies lacking experimental data beyond 50%v_{0}—leading to higher estimations of v_{0}, and accordingly P_{max}. Nevertheless, the linear and curvilinear models appear to provide comparable estimates of P_{max} when including additional resistive/loading conditions around v_{opt} [25]. In the present work, F_{0} estimated from the linear model was slightly to moderately lower compared to estimations of curvilinear models (Table 3 and Fig. 2), and both congruent with typical maximum strength values (e.g., ~ 1.5 to 1.8 times body mass for halfsquat 1RM). This is in line with a previous study where a curvilinear model (using Hill’s_{Eq}) estimated higher values of F_{0} in comparison to the linear model [45]. Nevertheless, curvilinear models can provide quasisimilar values of F_{0} when including additional resistive/loading conditions in the high forcelow velocity domain, notably close to F_{0} [25, 45]. Overall, linear and curvilinear models provide physiologically coherent F_{0} and P_{max} values, but the precision of the latter relies on sufficient velocity conditions (i.e., longer testing duration) to avoid over or underestimation.
Where P_{max} and F_{0} values appear similar across models, v_{0} values extrapolated from linear and curvilinear models can diverge strikingly. In the same manner as with F_{0} and P_{max}, one clear means of testing the physiological relevance of v_{0} is measuring lowerlimb force production at extremely high velocities (i.e., close to the graphical intercept) and comparing the values. Nonetheless, despite the very high extension velocities attained in this study (Table 2), substantial differences in v_{0} persisted between the linear model and curvilinear models (from ~ 4 to ~ 50%, Table 3). In the present study, the linear model estimated v_{0} values of ~ 3.2 m s^{−1}. These values are comparable to estimations of v_{0} with a linear model during a simulated legpress task including force collection from ~ 5 to 90%v_{0} [32]. In addition, comparable values can be estimated from acyclic monoarticular knee extensions under very low resistive and inertial conditions [12, 21] by applying the 2D model previously mentioned (~ 650 and 750°/s, corresponding to lowerlimb extension linear velocities of ~ 2.5 and 3 m s^{−1}, respectively). Finally, theoretical maximal pedaling cadences for active individuals ( ~ 230 rotations per minute; Dorel et al. 2010) and experimental maximal pedaling cadences of elite track cyclists (~ 270–300 rotations per minute) would correspond (considering a typical crank length of 0.175 m) to lowerlimb extension velocities of ~ 2.7 m s^{−1} and ~ 3.2–3.5 m s^{−1}, respectively. Overall, these values are in line with v_{0} values estimated here from the linear model (Table 3) and lower than those extrapolated here from curvilinear models. Thus, although the compared lowerlimb extension linear velocities are representative of individuals with different anthropometrics and training histories, and were measured using different movements, their proximity to v_{0} values estimated from the linear model support the physiological relevance of the latter. Furthermore, they highlight the potential overestimation of v_{0} values estimated by curvilinear models in acyclic lower extensions; this overestimation is important to consider when evaluating the F–v relationship with a narrow range of velocity conditions (e.g., ~ 20–60%v_{0}), the likes of which are common in field testing, since v_{0} values are more likely to be overestimated. In this context, the linear model should be preferred to avoid estimations of potential erroneous values. A critical limitation of curvilinear models is the potential for values that are not physiologically plausible [25]. For example, the curvilinear model including Poly_{2} exhibited the lowest SEE of all curvilinear models (Fig. 3, right panel), but did not define v_{0} for 8 out of 9 participants (e.g., left middle and bottom panels, Fig. 2). If taken at face value, the practical interpretation for these athletes is that their force production at very high velocities greatly exceed that at low velocities—trending toward infinite force capabilities. This interpretation is nonsensical, and supports the argument that higher GoF of a model does not systematically lead to more accurate and valid outcomes. Consequently, the most appropriate model should be selected per its ability to describe at best the properties of the system studied (e.g., the human biological features of external force production capabilities during a multijoint movement), rather than solely according to the mathematical function which fits at best the experimental data.
It is important to note that, even if the true F–v relationship in acyclic lowerlimb extensions were nonlinear beyond ~ 86%v_{0L}, it would not challenge the application of the linear model within the range 0–86%v_{0L}. Indeed, this range represents most of the practical field situations, with the linear extrapolation of F_{0} and v_{0} representing the theoretical limits of the neuromuscular system. Thus, the use of the linear model within this ~ 86% range does not challenge scientific applications in performance, testing and training related to the linear shaped F–v relationship in multijoint movements [e.g., 9, 46]. Consequently, practitioners and coaches should be confident in using field approaches, while acknowledging their accuracy is reliant on various methodological factors and rigorous measurements.
Beyond the GoF of a model and the physiological relevance of its output parameters, the reliability of the latter is also a key point to test the quality of a model. High reliability has been often reported for F_{0} and P_{max} and moderate to high reliability for v_{0}, when estimated from the linear model [17, 47–49]. Only one study has compared the reliability of linear and curvilinear models (using the basic exponential function and Poly_{2}), which showed similar (F_{0}) and lower (v_{0} and P_{max}) reliability for curvilinear models over an assessed range of ~ 10–50%v_{0L} [25]. However, it is important to note that such restricted ranges of velocity conditions will very likely reduce the reliability of the estimated parameters, especially from complex models, since they are more likely to vary with measurement error. Consequently, even if the linear model seems to yield greater reliability, further studies using a wider range of experimental conditions are needed. While determination of models’ outputs reliability in this study wasn’t feasible, intertrial reliability indicate coefficient of variation scores fell within acceptable ranges of 1.6 and 5.8% for mean force and velocity across inertial/resistive/assisted conditions, considering 4—8 participants.
Finally, differences in models used to describe the F–v relationship between acyclic lowerlimb extensions and during singlejoint or in vitro singlemuscle contraction have been supported by the fact that the former refers to external rather than intrinsic muscle force production. Indeed, the former involves specific underlying mechanisms, including neural control of various muscle groups, activation and segmental dynamics, which are not all encompassed in the two latter conditions [1, 14, 32]. In this sense, Bobbert [32] reported a “quasilinear” F–v relationship over a wide range of simulated velocity conditions (~ 5–90%v_{0L}) in acyclic lowerlimb extensions, despite using Hill’s curvilinear equation to characterize intrinsic force production capabilities of individual muscles. Furthermore, the linearity of the F–v relationship in acyclic lowerlimb extensions is in line with the linearity observed in other multijoint movements, such as cycling and running, where lower limb force production was measured over a wide range of velocity conditions, notably on the velocity end (i.e., ~ 20–90%v_{0L} [11, 33]). Consequently, biomechanical simulations and studies on other multijoint movements tend to align with linear modelling on F–v relationship obtained in acyclic lowerlimb extensions.
Perspectives
The unique design of the ergometer used in this study allowed lowerlimb force production measurements from very low (similar to onerepetition maximum) to very high (approaching estimates of physiological maximums) velocity conditions. Similar devices that can generate comparable conditions could provide a means of targeting the development of force produced at very high, and otherwise inaccessible velocities during training; this is particularly interesting for weak population to train their specific deficit in velocity capabilities [50]. This type of design allows the force–velocity relationship to be evaluated i) without carrying external loads, which may be safer notably for frail populations, and ii) on a wide range of velocity conditions, which could increase the accuracy and the reliability of v_{0} and P_{max} [51]. When examining a greatly expanded range of velocity conditions the linear model was the most appropriate to describe the force–velocity relationship in acyclic lowerlimb extension. Since most field situations occur within the explored range, actual testing and training methods applying such a model to multijoint movements are justifiable [e.g., 52, 53].
Conclusion
Very high lower limb extension velocities can be reached using a specialized legpress (assisted horizontal acyclic lower limb extensions without moving the rest of the body). The implementation of such an ergometer allowed a much larger portion of the force–velocity relationship to be examined than previously accessible. Over this wide range, the force–velocity relationship appeared well described by the linear model, since curvilinear models did not improve accuracy to a degree warranting their utilization. Moreover, where curvilinear models can produce irrational outputs (e.g., v_{0}) under typical testing settings, the linear model has provided physiologically appropriate values. With this in mind, practitioners should feel confident in adopting linear modelling when assessing the force production capabilities of the lower limbs at different velocities during acyclic ballistic extensions. Technical and methodological improvements of the ergometer could potentially help further widen the range of accessible velocity conditions and test the linearity of the force–velocity relationship in velocity conditions close to the maximal extension velocity.
Availability of Data and Material
The data set supporting the conclusions of this article will be made available by the authors on reasonable request.
Abbreviations
 95% CI:

95% Confidence intervals
 %v _{0} :

Percentage of the velocityaxis intercept of the F–v relationship
 %F _{0} :

Percentage of the forceaxis intercept of the F–v relationship
 %v _{0L} :

Percentage of the velocityaxis intercept of the linear F–v relationship
 %F _{0L} :

Percentage of the forceaxis intercept of the linear F–v relationship
 %P _{maxL} :

Percentage of the apex of the P–v relationship, derived from the linear F–v relationship
 \(\propto\) :

The instantaneous angular acceleration of the flywheel
 AIC:

Akaike Information Criterion
 \(a_{{{\text{chariot}}}}\) :

The instantaneous acceleration of the feet support
 \(a_{{{\text{limbs}}}}\) :

The instantaneous acceleration of lower limb’ centre of mass
 C _{ØFric2S} :

Condition of lower limb extension, during which only the feet support and the flywheel of the ergometer is accelerated with the assistance of 2 springs
 C _{ØFric0S} :

Condition of lower limb extension, during which only the feet support and the flywheel of the ergometer is accelerated with no spring assistance
 C1RM:

Condition of lower limb extension performed against resistive frictional force close to the maximal isometric force, leading to extension velocity close a onerepetition maximum’s typical performed velocities
 C_{50%Fmax} :

Condition of lower limb extension performed against resistive frictional force corresponding to ~ 50% of maximal isometric force
 C_{Char2S} :

Condition of lower limb extension during, which only the feet support of the ergometer is accelerated with the assistance of 2 springs
 C_{Char0S} :

Condition of lower limb extension during, which only the feet support of the ergometer is accelerated with no spring assistance
 \(d_{{{\text{flywheel}}}}\) :

The radius of the flywheel
 \(d_{p}\) :

The cog radius of the ergometer
 d:

Effect size
 \(F\) :

The instantaneous force developed by lower limbs during extension
 F _{0} :

The theoretical maximal force that lower limbs could produce over one extension at zero velocity
 F&M’s_{Eq} :

Fenn and Marsh’s equation
 \(F_{{{\text{chariot}}}}\) :

The force developed by lower limbs to accelerate the feet support of the ergometer
 \(F_{fb}\) :

The friction force applied by the belt on the flywheel of the ergometer
 \(F_{{{\text{limbs}}}}\) :

The force developed by lower limbs to accelerate its own mass
 \(F_{{{\text{flywheel}}}}\) :

The force developed by lower limbs to accelerate the flywheel
 \(F_{{{\text{friction}}}}\) :

The force developed by lower limbs to overcome the frictional force applied by the belt on the flywheel
 \(F_{{{\text{roll}}}}\) :

The internal resistive force of the flywheel
 \(F_{{{\text{spring}}}}\) :

The force produced by springs in assistance to the movement
 F–v :

Forcevelocity
 GoF:

Goodness of fit
 Hill’s_{Eq} :

Hill’s equation
 \(I\) :

The moment of inertia of the flywheel
 m s−1:

m per second
 \(m_{{{\text{chariot}}}}\) :

The mass of feet support
 \(m_{{{\text{limbs}}}}\) :

The mass of lower limbs
 \(m_{{{\text{flywheel}}}}\) :

The linear equivalent mass of the flywheel’s moment of inertia
 N :

Newtons
 P _{max} :

The maximal power capacity at the optimal extension velocity
 Poly2:

the secondorder polynomial function
 Pv :

Powervelocity
 r ^{2} :

coefficient of determination
 SD:

standard deviation
 SEE:

standard error of estimate
 v _{opt} :

optimal velocity
 v0:

The theoretical maximal velocity until which lower limbs could produce force over one extension
 W :

Watts
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Acknowledgements
We are grateful to Baptiste Denopce and Nicolas Masson (master’s students in engineering and ergonomics of physical activities at Université Savoie Mont Blanc) for their assistance during the experimental phase of this project. No funding was received for this work.
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JRR, PS, and MB conceived and designed the experiments. JRR conducted experiments. JRR, MB, and PS analyzed the data. JRR, LAM, and JBM wrote the manuscript. All authors read and approved the final manuscript.
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Rivière, J.R., Morin, JB., Bowen, M. et al. Exploring the Low ForceHigh Velocity Domain of the Force–Velocity Relationship in Acyclic LowerLimb Extensions. Sports Med  Open 9, 55 (2023). https://doi.org/10.1186/s40798023005980
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DOI: https://doi.org/10.1186/s40798023005980