Using a simple model to systematically examine the influence of force-velocity profile and power on vertical jump performance with different constraints

ABSTRACT

Power, and recently force-velocity (F-V) profiling, are well-researched and oft cited critical components for sports performance but both are still debated; some would say misused. A neat, applied formulation of power and linear F-V in the literature is practically useful but there is a dearth of fundamental explanations of how power and F-V interact with human and environmental constraints. To systematically explore the interactions of a linear F-V profile, peak power, gravity, mass, range of motion (ROM), and initial activation conditions, a forward dynamics point mass model of vertical jumping was parameterised from an athlete. With no constraints and for a given peak power, F-V favouring higher velocity performed better, but were impacted more under real-world conditions of gravity and finite ROM meaning the better F-V was dependent on constraints. Increasing peak power invariably increased jump height but improvement was dependent on the initial F-V and if it was altered by changing maximal force or velocity. When mass was changed along with power and F-V there was a non-linear interaction and jump improvement could be almost as large for a F-V change as an increase in power. An ideal F-V profile cannot be identified without knowledge of mass and ROM.

KEYWORDS:

• Power
• force-velocity relationship
• F-V relationship
• F-V profile
• Morin

Introduction

Over the past 10–15 years there has been a great advancement in understanding the mechanics that underly vertical jump performance. While vertical jump performance is mechanistically determined by the vertical impulse produced, this quantity is dependent on the interaction of other factors including the fundamental ‘motor or engine’ properties, the environment, and time. A series of works by Samozino, Morin and various co-authors (Jimenez-Reyes et al., 2016; Morin & Samozino, 2016; Samozino et al., 2010, 2012, 2014) around whole body force-velocity profiling have been instrumental in enhancing understanding of these biomechanical factors underlying vertical jump performance. In humans, as with any motor, peak power and the force-velocity profile (think horsepower and torque-revolutions per minute profile for an automobile engine) are key intrinsic variables (Jaric & Markovic, 2009; Jimenez-Reyes et al., 2017; Samozino et al., 2012). This group of authors (Jimenez-Reyes et al., 2016; Morin & Samozino, 2016; Samozino et al., 2010, 2012, 2014) has demonstrated that jump performance is underpinned by the mass the subject must move, the capacity of the system to produce power (measured as the highest value of average power across trials), the fixed range of motion over which the subject can produce vertical force (ROM), and the propulsive force-velocity (F-V) profile. The F-V profile, which is typically a linear estimate of a subject’s ability to produce propulsive vertical force as velocity of movement changes can exhibit different slopes for a given peak average power (Jimenez-Reyes et al., 2018; Samozino et al., 2012, 2014). These findings are of practical importance as they provide useful tools for talent identification and describe mechanical variables that subjects may focus on to improve jump performance. These works (Jimenez-Reyes et al., 2016; Morin & Samozino, 2016; Samozino et al., 2010, 2012, 2014) identify that performance is primarily optimised through two methods: (1) a maximal increase in the capacity to produce power; and (2) optimisation of the F-V profile in accordance with the ROM and mass of the primary movement of interest (i.e., the competition movement). The research from these authors (Jimenez-Reyes et al., 2016; Morin & Samozino, 2016; Samozino et al., 2010, 2012, 2014) has greatly aided understanding of the factors relevant to jump performance, and has demonstrated training programmes can alter F-V profiles that lead to increased jump heights through these profile specific training programmes better than generic training programmes (Jimenez-Reyes et al., 2019). However, it has recently been shown (Bobbert et al., 2023) that these experimentally determined whole body F-V profiles do not represent the intrinsic neuromuscular F-V profile of the athlete, so care is warranted in how they are interpreted and what general fundamental understanding they provide. There is still scope for further conceptual understanding of the biomechanics of jumping, as well as of other ballistic performances even when limited to whole body F-V profiles. Specifically, the experimental and analytical approach previously utilised has not explicitly teased out the role of muscular activation and the resulting rate of force development on jumping performance, which is important as these biomechanical factors are likely central to the execution of some jump types (Bobbert, 2014; Bobbert & Casius, 2005; Bobbert et al., 1996) and to the production of other ballistic performances (e.g., sprinting (Weyand et al., 2010). Additionally, many of the useful and practical relationships provided by these authors are implicit within their mathematical formulation and thus, a more explicit step-by-step investigation could produce a more thorough understanding of the interactions of the underlying mechanics; and consequently, a greater intuitive comprehension.

The aim of this investigation was to build upon the previous F-V profiling literature by providing fundamental insight and intuition into the factors that underpin the mechanics of vertical jump performance. To accomplish this goal, we constructed a simple model that approximated a trained weightlifting subject as a point mass, and in which force production was estimated as functions of velocity and rate of rise to maximal activation. We employed a step-by-step progression whereby the number of constraints and interactions between parameters increased with successive simulations to accommodate the complexity of the mechanical principles underlying real-world vertical jumping (see Table 1 below for roadmap). It is hypothesised that the optimal F-V profile cannot be identified without knowledge of the mass, ROM, gravity, and rate of rise to maximal activation.

Table 1. Roadmap of the groups of simulations performed in this investigation; g, ROM, and act identify if the simulations of the given group were performed under gravity, a limited ROM, or commenced from submaximal activation; F-VP abbreviates F-V profile.

Group	Primary goal of simulation group	Manipulated variable(s)	Constrained variable(s)		g	ROM (y/n)	Act	Figure(s)
1	F-VP manipulation under unlimited ROM, without g	F-VP	PMax, Mass		n	n	n	1
2	F-VP manipulation under defined ROM, without g	F-VP, ROM	PMax, Mass		n	y	n	2
3	F-VP manipulation under unlimited ROM, with g	F-VP	PMax, Mass		y	n	n	3
4	F-VP manipulation under defined ROM, with g	F-VP	PMax, Mass, ROM		y	y	n	4,5
5	PMax manipulation	PMax	Mass, ROM		y	y	n	6
6	Mass manipulation (A: ROM/no g; B: g/unlimited ROM; C: ROM and g)	Mass	F-VP, PMax	A	n	y	n
				B	y	n	n	7
				C	y	y	n
7	PMax and Mass interaction	PMax, Mass			y	y	n	8
8	Mass manipulation starting from submaximal activation	Mass	F-VP, PMax, ROM		y	y	y	9
9	Mass and ROM interaction starting from submaximal activation	Mass, ROM	F-VP, PMax		y	y	y	10

Materials and methods

Identifying the model

To address the aims of this paper a simple forward dynamics simulation model was used to represent the propulsive phase of a vertical countermovement and squat jump. Simple models are useful for looking at principles as there are few variables, so cause and effect can be easily linked (Alexander, 2003; Pandy, 2003). Alexander (1990) used a simple model to demonstrate the principle that long jumpers should run up as quickly as possible but high jumpers have an optimal approach that is typically below their maximal speed. A limitation of simple models is that they may not have sufficient variables to reflect aspects of the real world they are being used as tools to investigate. In our case a simple point mass model, the centre of mass (COM), with a force actuator acting on that point represents a very common experimental method used to investigate vertical jumping. The use of force plate data to calculate the COM motion of the subject, and the applied forces causing the motion, is the most used experimental method of investigating force-jump height relationships. As such, the very simple model employed here is suitable for determining general principles that match the experimental world of interest.

The point mass model with its linear vertical force actuator was constrained to provide realistic performances. The maximal force (FMax) and maximal velocity (VMax) values, as well as the range of motion over which force production was possible (ROM), were modelled from the propulsive phase of a countermovement jump (CMJ), without arm swing of a trained Olympic weightlifter. As often seen in the whole body F-V profile literature, and specifically in vertical jumping, as in Samozino et al. (2010, 2012, 2014) a linear F-V relationship was used, parameterised as the line defined by the FMax and VMax variables (F-V profile). It is important to distinguish that while these previous topical works (Morin et al., 2019; Samozino et al., 2010, 2012) chose to utilise experimentally derived average values (average force, velocity, power), it has recently been shown (Bobbert et al., 2023) that this method does not provide a true estimate of the underlying force-velocity capabilities of the relevant lower body musculature. Therefore, the present work utilises instantaneous peak values (e.g., FMax is defined here within as the maximum force the model is able to generate at 0 velocity) as these fundamental properties of mechanical systems are better suited for an iterative topical exploration from first principles.

A linear F-V profile results in a quadratic power curve with peak power being achieved at ½ maximal force, which is also ½ maximal velocity, and can be presented as:

(1) $\mathit{System}\mathit{peak}\mathit{power}\mathit{capability}\left(\mathit{PMax}\right)=\frac{\mathrm{Fmax}\ast \mathrm{Vmax}}{4}$(1)

Changes in F-V profiles occurring at constant Pmax were operationally defined as F-V profile rotations; and, during these manipulations an increase in Fmax necessarily corresponded with a decrease in Vmax (and vice versa). As informed by the subject, the baseline model parameters were: Fmax = 2667 N, Vmax = 4.09 m∙s⁻¹, Pmax = 2726 W (collectively known as the initial model conditions (IC)); and a mass of 83.7 kg. Various aspects of this model including F-V profile, ROM (0.54 m as informed by the subject), mass, PMax, and rate of rise to maximal activation; or the environment (i.e., gravitational acceleration (set at 9.8 m∙s⁻²)), were then manipulated to tease out their influence on simulated jumping performance (see Table 1 below). Simulations performed in groups 1–7 (Table 1) all commenced from a maximal activation state (i.e., force production at t₀=Fmax); groups 8 and 9 (Table 1) were performed under modelled CMJ and squat jump (SJ) propulsive conditions which were distinguished by their initial activation states (Bobbert & Casius, 2005; Bobbert et al., 1996). For the CMJ, initial activation for the bodymass or higher than bodymass simulations was set at maximal due to the high initial propulsive forces demonstrated by the subject, and supported by the previous modelling work of Bobbert (Bobbert, 2014). For masses lighter than bodymass, initial activation was linearly decreased with mass based on two points: maximal activation at bodymass, and 85% activation at 70% of bodymass as informed through both experiment and simulation (Bobbert, 2014; Jaric & Markovic, 2009). For the squat jump (SJ), initial activation was set to the level that produced the isometric force necessary to oppose gravitational force and maintain a stationary squat position. Rate of rise to maximal activation (simulation groups 8 and 9 only) was modelled as a function of time as per Infantolino et al. (2019):

(2) $\mathit{a}\left(\mathit{t}\right)=\frac{(a_f-a_0)}{1+e^{\left(-\frac{t-t_0}{t_r}\right)}}$(2)

Where a_f is the final activation, a₀ is the initial activation, t₀ is the time at midpoint of activation rise, and t_r governs the rate of activation rise. a_f was set at 1 and a_o was set at 0; time to maximal activation was modelled as 350 ms (i.e., t₀ = 0.175) and t_r was set at 0.075.

How six factors interact to determine the modelled jump performance

The aim of this study was to utilise a simplified model of an elite jumping performance to evidence how the six factors of: instantaneous peak power (PMax), F-V profile as determined by peak variables (FMax and VMax), ROM, gravity, mass, and rate of rise to maximal activation interact to produce the jump performance, which is represented by final velocity for the remainder of this work. To achieve this, simulations proceeded as laid out in Table 1.

Results

The influence of F-V profile on performance in the absence of ROM and gravity constraints (simulation group 1)

In this section it is shown that without the real-world constraints of gravity and ROM, F-V profiles favouring lower FMax take longer to achieve PMax and VMax but always perform more work and achieve better performances as compared to profiles favouring Higher FMax.

Along with PMax, the F-V profile is a fundamental property of motors (whether biological or mechanical), and is of incredible importance for this investigation as F-V rotations under constant PMax are theoretically capable of producing infinitely large and infinitely small performances when simulated in isolation. Without the effects of ROM and gravity, simulations were conducted at fixed mass and a constant PMax for the initial conditions (IC), +40% FMax, and −40% FMax F-V profiles (Figure 1). As the F-V profile was manipulated from favouring higher FMax to lower FMax (therefore higher Vmax), the simulations performed more work (Figures 1(b,c)) and thus, produced better performances (Figure 1(d)). Although performance increased with decreasing FMax, so did the required ROM and time to both achieve PMax and complete the simulation (Figure 1). Although all profiles achieved PMax, it occurred at greater displacements as the forces and accelerations were lower up until PMax was reached. While this decreased acceleration also increased the displacement and time required to complete a jumping simulation, the complementary factor was force production continued at velocities higher than those achieved under profiles favouring higher FMax. Because these simulations were allowed unlimited displacement and were not under the influence of gravity, they completed only when a velocity was reached whereby no further force could be produced, that is, VMax. Crucially, because the performance can be defined by the final velocity across all simulations, in this simplified scenario, at a given mass, a profile with a higher VMax always demonstrates an ability to perform more work and therefore always produces a better performance. Even though in most real-world scenarios profiles demonstrating relatively higher or lower FMax may be superior based on additional factors, when run in these unconstrained conditions with the simulation running to completion (and under constant mass), profiles possessing lower FMax always produce more power over the duration of the simulation, always produce more force over the distance of the simulation, and always yield a higher final velocity. The ability of a simulation to perform work and achieve a resulting final velocity in isolation is defined operationally here as its work capacity.

Figure 1. The effects of rotating the F-V profile at constant PMax (2726 W) and mass (83.7 kg) over an unlimited ROM in the absence of gravity on simulated jumping performance. (a) The F-V profiles of the modelled initial conditions (IC) as well as +40% FMax and −40% FMax rotations in F-V profile under constant PMax; the work performed by simulations under each of these profiles as shown by the areas under the (b) force-distance and (c) power-time curves; and (d) the performance of simulations under each profile. The colour coding of black, red, and green delineate the IC, +40% FMax, and −40% FMax profile rotations; the same colour coding is used in all subsequent figures. Horizontal line in panel C delineates PMax of 2726 W.

The influence of ROM on performance (simulation group 2)

In this section it is shown that under real-world ROM constraints (but without gravitational influence), F-V profiles favouring lower FMax do not always produce superior performances as compared to F-V profiles favouring higher Fmax. This is because the former require greater distances to realise their full work capacities, and thus are typically impacted more by limited ROMs (see Figure 2 below).

Figure 2. The effects of rotating the F-V profile at constant PMax (2726 W) and mass (83.7 kg) over a defined limited ROM in the absence of gravity on simulated jumping performance. (a) Simulations for the same F-V profiles as shown in Figure 1 over a 0.05 m ROM; and (b) over a 0.54 m ROM as informed by the subject; the work performed for each of these simulations as shown by the areas under the (i) force-distance and (ii) power-time curves; (iii) the peak power achieved for each simulation; and (iv) the performance achieved by each simulation. Horizontal line in rows ii and iii delineate PMax of 2726 W.

While the greater distance required to perform a given quantity of work by F-V profiles favouring relatively lower FMax has no influence on performance given an unlimited displacement, it can have a major effect when a finite ROM is introduced. The ROMs typical to real-world jumping are capable of demonstrating profound influence in practice (Jaric & Markovic, 2009; Morin et al., 2019; Samozino et al., 2012) and are primarily determined by both intrinsic and extrinsic factors. Intrinsic factors can be subject-specific segment lengths, ranges of force production of the muscle-tendon units, and attachment sites of tendons (moment arms), while extrinsic factors can be appropriate instruction, coached technique, and possible technological equipment interventions. These factors interact to determine the ROM over which propulsive force may be produced, which constrains performance. Although the −40% FMax profile has the largest work capacity, for a 0.05 m ROM it produced the least amount of work (Figure 2(a(i,ii))), lowest instantaneous peak power (Figure 2(a(iii))) and the poorest performance (Figure 2(a(iv))). This is because the relatively low FMax dictated a low acceleration, thereby permitting only a small fraction of work capacity to be realised before the ROM limit was reached.

Given the larger ROM of 0.54 m the −40% FMax simulation achieved its PMax of 2726 W. While the −40% FMax profile was the least successful over the 0.05 m ROM it was the most successful for the 0.54 m ROM condition as the ROM was sufficient to allow the simulation to achieve PMax (Figure 2(b(ii,iii))) and to realise much of its work capacity (Figure 2(b(i,ii))). Although F-V profiles with lower Fmax have the advantage of larger work capacities, the price paid under constant PMax conditions is lower acceleration capabilities (Figures 1 and 2); and thus, a higher likelihood that a given ROM will prevent the simulation from achieving a large percentage of its work capacity. Therefore, under constant mass neither a profile favouring higher FMax or VMax can be considered superior as it pertains to performance without knowledge of the ROM. As the ROM is made shorter and shorter, eventually a crossover ROM is reached where the profile that favours higher FMax can deliver more work within the confined displacement, and thus produce a superior performance.

The influence of gravity on performance (simulation group 3)

In this section it is shown that under the real-world gravity constraint (but infinite ROM), F-V profiles favouring lower FMax do not always produce superior performances as compared to profiles favouring higher FMax, as the percentage of force production capability (i.e., FMax) allocated to opposing gravity is larger for lower FMax F-V profiles.

While jump testing, training and performance can be executed in non- and reduced-gravity conditions (e.g., jumping sleds that move purely horizontally (Kramer et al., 2010; Samozino et al., 2012) the swimming start, and turns), many competition sporting movements and associated training exercises occur in a direction with a substantial component that must oppose gravity. Gravity produces a force, weight, equal to the mass times gravitational acceleration, which is −820 N for bodymass only simulations of the subject in this study, and weight opposes propulsion for the duration of the vertical jump ROM.

The effect of weight is to decrease accelerations compared to the non-gravity case, and also, to limit the velocity at which further propulsive force may be produced. Under gravity, simulations end when no further propulsive force can be produced, or in other words, when the entirety of the force production capabilities must be allocated to oppose weight thereby leaving no force available for propulsion. While the performance capability of a given F-V profile is equal to VMax, the realisable performance under gravity is the X coordinate for a given F-V profile where Y = |weight|. As demonstrated in Figure 3(a) the maximal attainable velocity was equal to the intersection point of the given F-V profile with the horizontal line at F = 820 N, which was highest for the −40% FMax profile and lowest for the + 40% FMax profile. In fact, even though simulation under −40% FMax used more than 50% of its force producing capabilities to oppose weight, it still performed more work and therefore produced a better performance as compared to both IC and + 40% FMax; simulations that used only 31 and 22% of their force capabilities to counteract weight.

Figure 3. The effects of rotating the F-V profile at constant PMax (2726 W) and mass (83.7 kg) over an unlimited ROM in the presence of gravity on simulated jumping performance. (a) F-V profiles used for this group of simulations (same profiles as Figure 1 plus an additional −60% FMax profile); the work performed for each of these simulations as shown by the areas under the (b) force-distance and (c) power-time curves; (d) the peak power achieved for each simulation; and (e) the performance achieved by each simulation. Horizontal line in panels C and D delineates PMax of 2726 W; horizontal line in panel a delineates the gravitational force of 820 N representative of subject bodyweight; the point of intersection between a F-V profile and this horizontal line shows the highest velocity (and thus performance) the subject can achieve under gravity.

While all profiles further skewed towards lower FMax have greater work capacities, their performances are hindered by the disproportionately large percentage of force allocated to opposing weight. This is evidenced by the −60% FMax F-V profile included in Figure 3, which under simulation in a gravitational environment produced a relatively small amount of work (Figure 3(b,c)), failed to achieve PMax (Figure 3(d)) and produced a poor performance (Figure 3(e)). If FMax is lower or equal to weight then no jump will be possible as all the vertical force production is allocated towards the offsetting of weight; and although not shown, simulations with bodymass performed under profiles where FMax ≤ 820 N failed for this reason. This is the extreme case of work capacity failing to be utilised. Of the large number of F-V profiles where FMax>|weight|, all produced jump performances, with the best performance resulting from simulation under the F-V profile where ½FMax = |weight| (FMax = 1641 N for the 83.7 kg subject in the current investigation; simulations not shown).

With only gravity as an extra constraint some analytical outcomes can still be more easily seen than in later cases where multiple variables and constraints are altered simultaneously. Given that the different F-V profiles are linear and with the same peak power, then the area under F-V line is the same for all cases and is the area of a triangle. This allows a geometrical solution to be found for which F-V profile will give the largest take-off velocity, as it is the one that has its intersection with the horizontal weight line the furthest to the right (Figure 3(a)). In a simplified form this can be seen as the weight line being used to describe a rectangle with a right-hand side defined by the intersection with the F-V line, and where we want to make the area of the rectangle as large as possible. This occurs when the F-V line intersects the weight line at half the height of the triangle, and this means the F-V profile that will give the best performance is one where FMax = 2 × weight.

The influence of ROM and gravity on performance (simulation group 4)

This section quantifies the combined influence of real-world ROM and gravity constraints on simulated jumping performance under different F-V profiles. Critically, although F-V profiles favouring lower FMax have greater work capacities, they don’t always produce better performances in the real-world as they are more negatively impacted by the combination of the ROM and gravity constraints as compared to F-V profiles favouring higher FMax.

Real-world jumping performance typically occur under both the ROM and gravity constraints. When both these factors were included in fixed-mass jumping simulations, IC produced the greatest amount of work (Figures 4(a,b)) and the best performance (Figure 4(d)), which is in line with previous works suggesting that humans are designed to maximise power output during jumping at bodymass (Jaric & Markovic, 2009, 2007). This outcome differed from simulations performed under only the ROM constraint, or gravity constraint, as these sets of simulations performed best under the −40% FMax profile (Figures 2(b(iv)) and 3(e)). As mentioned previously, one consequence of adding gravity is a decrease in acceleration as subject’s force production formerly used for propulsion is instead used to oppose gravity. By adding gravity to a 0.54 m ROM constraint, the −40% FMax simulation no longer possessed the required distance to achieve PMax (Figures 4(b,c)). While the addition of gravity also lowered the acceleration of the IC and + 40% FMax profiles, their performances were less affected as evidenced by the ability of these simulations to still achieve PMax.

Figure 4. The effects of rotating the F-V profile at constant PMax (2726 W) and mass (83.7 kg) over a 0.54 m ROM in the presence of gravity on simulated jumping performance. The work performed for each of these simulations (simulations for the same F-V profiles as displayed in Figure 1) as shown by the areas under the (a) force-distance and (b) power-time curves; (c) the peak power achieved for each simulation; and (d) the performance achieved by each simulation. Horizontal line in panel C delineates PMax of 2726 W.

While the optimal F-V profile for the maximisation of jump performance for the simulations performed under only gravity is ½FMax = |weight| (FMax = 1641 N in the current investigation), under both gravity and ROM the optimal profile is less obvious. To provide more detailed insight, we started with IC (FMax = 2667 N) and performed additional simulations with ± 100 N changes in FMax. As reported in Figure 5(a), we found that the best performance of 2.58 m∙s⁻¹ resulted from simulation under FMax = 2467 N; however, all simulations within FMax = ±400 N of this performance produced results that were 2.53 m∙s⁻¹ or better. Thus, a ‘flat spot’ was reached where substantial changes in profile led to similar performance outcomes. It is intuitive that this performance ‘flat spot’ exists around the global maximum as profiles that favour a relatively higher FMax perform more work from simulation commencement up until the achievement of PMax, but then experience an earlier drop in work rate, and reach lower final levels of power production as compared to lower FMax profiles (Figure 5(b)). Even though PMax was achieved at different time points, or not achieved for the FMax = 2067 N profile, and simulations performed different percentages of work in relation to their work capacities, the total amount of work tended to balance out leading to similar performances. In fact, the F-V profile-performance relationship can be modelled as a quadratic function (Figure 5(a)), which by definition demonstrates an initially slow, and then increasing rate of change around a global maxima or minima. These findings are of practical importance as they suggest that as subjects approach their optimal F-V profile for a constant mass and ROM, attempting to further fine tune the profile may be of less benefit as compared to focusing training resources into increasing Pmax.

Figure 5. The effects of 100 N F-V profile rotations in FMax around the initial modelled conditions (IC; FMax=2667 N) at constant PMax (2726 W) and mass (83.7 kg) over a 0.54 m ROM in the presence of gravity on simulated jumping performance. (a) The performance achieved by each simulation; and (b) the work performed for each of these simulations as shown by the areas under the power-time curves. The black trace and squares show initial conditions (IC); all other colour coding is specific to this figure; purple open square shows the profile that optimised performance (FMax = 2467 N). Horizontal line in panel B delineates PMax of 2726 W.

The influence of increasing PMax on performance (simulation group 5)

In this section it is shown that increases in PMax result in increases in performance for the simulated subject; however, the magnitude of increase depends on the initial F-V profile in combination with the mechanism by which PMax is increased (i.e., whether PMax increased via a bump in FMax, VMax, or a combination of the two).

While a primary purpose of this investigation is to elaborate on how rotations in F-V profile under constant PMax influence performance, it is equally important to demonstrate how changes in PMax starting from a single F-V profile affect performance, as PMax is arguably the most important variable determining the performance of the modelled subject. To demonstrate the influence of PMax manipulation on performance, we started with three modelled F-V profiles from the previous simulations; IC and ± 40% FMax, and increased PMax from 2726 W to 3816 W through a 40% bump in either FMax (BUMPFMax) or VMax (BUMPVMax), or an 18.3% bump in both (BUMPFMaxVMax; Figure 6(i)), holding all other constraints constant (83.7 kg mass, 0.54 m ROM, gravity).

Figure 6. The effects of increasing PMax at constant mass (83.7 kg) over a 0.54 m ROM in the presence of gravity on simulated jumping performance. Starting from the same F-V profiles as shown in Figure 1 ((a) IC, (b) +40% FMax, and (c) −40% FMax) PMax was increased through either a pure 40% bump in FMax (BUMPFMax; dashed purple), a pure 40% bump in VMax (BUMPVMax; dashed light blue) or a balanced 18.3% bump in both (BUMPFMaxVmax; dashed olive) (all bumps achieve PMax of 3816 W). (i) The F-V profiles of the simulations used in this simulation group organised by column; (ii) the work performed by simulations under each of these profiles as shown by the areas under the power-time curves; (iii) the peak power achieved during simulations under each profile; and (iv) the performance of simulations under each profile. Horizontal dotted lines show baseline instantaneous peak power of 2726 W; horizontal broken lines show increased instantaneous peak power of 3816 W.

As shown in Figures 4 and 6(b), although the + 40% FMax profile delivered work relatively quickly, it possessed a relatively low work capacity and was sub-optimal in combination with the simulated mass and ROM. While increasing PMax through a further bump in FMax (BUMPFMax) increased the work performed and improved performance, this improvement was relatively small as the profile was rotated further away from optimal thereby failing to address the profile’s main deficiency of work capacity (Figure 6(b)). In contrast, elevating PMax through a bump in VMax (BUMPVMax) rotated the slope of the modelled force-velocity relationship towards optimal and enhanced performance to a much larger extent, as the increase in PMax resulted primarily from the needed increase in work capacity (Figure 6(b)). Although PMax was now achieved at a further displacement and longer time (Figure 6(b(ii))), it was still achieved well within the ROM; and thus, a large percentage of work was realised relative to capacity. In fact, (when starting from the + 40% FMax profile) BUMPVMax, although superior to the BUMPFMaxVMax and BUMPFMax simulations, remained suboptimal as PMax was still achieved relatively early (i.e., only 84 ms into the 248 ms simulation; Figure 6(b(ii))).

The starting profile reported in Figure 6(c) demonstrated the opposite deficiency, the profile of −40% FMax possessed a relatively large work capacity; however, it was only able to realise low quantities of work relative to this capacity as is evidenced by the sub-maximal peak power of 2481 W achieved during simulation (Figures 4(c) and 6(c)). While increasing PMax through a further bump in VMax (BUMPVMax) improved performance, this improvement was relatively small as the profile was rotated further away from optimal thereby failing to address the profile’s main deficiency, the realisation of work capacity under the gravitational and ROM constraints. In contrast, starting with the −40% FMax profile and elevating PMax through a bump in FMax (BUMPFMax) rotated the slope of the modelled force-velocity relationship towards optimal and enhanced performance to a much larger extent (Figure 6(c)), as the increase in FMax allowed a larger percentage of force to be allocated to propulsion and allowed work to be delivered quicker.

Finally, the baseline IC F-V profile, which was derived from our modelled subject, was already close to optimal in relation to bodymass and ROM (i.e., highest performance of all three pre-PMax bump conditions; Figures 4 and 6(a)). It is therefore not surprising that a balanced bump in FMax and VMax yielded the greatest performance benefit with an increase in VMax also yielding a relatively large performance gain (Figure 6(a)). This is because a bump in PMax resulting from a balanced increase in both FMax and VMax does not produce a rotation in F-V profile; and therefore, the approximately optimal profile during IC remained approximately optimal under the increase in PMax. This can be seen in Figure 6(a(ii)), where although the magnitudes of PMax are different, both the IC and BUMPFMaxVMax simulations achieved PMax at the same time.

PMax is the capacity of the system to instantaneously deliver work and is defined by the energy resource and the energy conversion rate. PMax is determined by the combination of FMax and VMax (see Equation 1(1) $\mathit{System}\mathit{peak}\mathit{power}\mathit{capability}\left(\mathit{PMax}\right)=\frac{\mathrm{Fmax}\ast \mathrm{Vmax}}{4}$(1) ); therefore, under gravity and ROM, an increase in either FMax or VMax without a concurrent decrease in the other variable always results in an increase in PMax, and almost always results in an increase in performance. However, depending on the modelled F-V profile, mass, and ROM, the mechanism by which PMax is increased plays a crucial role in determining the magnitude of the performance benefit (Morin & Samozino, 2016; Samozino et al., 2012). Thus, although we have operationally defined a F-V profile rotation to be a change in either FMax or VMax at the expense of the other (i.e., at constant PMax), changes in PMax can also rotate the slope of the force-velocity model towards or away from optimal depending on the parameters of the individual and the mechanism by which PMax is increased.

The influence of changing mass on performance (simulation group 6)

This section illustrates the influence of changing mass during simulations under constant F-V profile (and PMax). Under a limited ROM and no gravity, performances decrease with increasing mass as greater distances are required to realise work capacity; under gravity (and unlimited ROM), performances decrease with increasing mass as a higher percentage of force production must be allocated to opposing weight; these effects are greatest when both ROM and gravity constraints are present as during real-world jumping.

During jumping, there is a mass that is required to be moved, which is usually the bodymass of the subject, although manipulations both increasing (e.g., weighted vest) and decreasing (e.g., assistance bands) ‘mass’ are often encountered, especially during training for athletic performances. The mass encountered during jumping at bodymass can vary dramatically between subjects (e.g., think basketball subjects of 70 kg and 140 kg), which can alter the acceleration and velocity during propulsion and thus, the optimal F-V profile for a given ROM.

All simulations for this sub-section were performed with the same PMax and F-V profile (IC). Under mass manipulation without ROM and gravity constraints, all simulations achieved the same performance of VMax; however, as the mass increased, the acceleration decreased and both greater distances and longer times were required (not reported). Under the ROM constraint and no gravity (Figure 7(a)), performance of simulations under all masses decreased as none of the simulations achieved VMax within the allowed 0.54 m ROM; with the performance of simulations under heavier masses realising less work in relation to capacity within the allotted ROM due to the lower accelerations (Figure 7(a)).

Figure 7. The effects of changing mass (23.7 kg to 183.7 kg) under constant F-V profile (IC) on simulated jumping performance. (a) The effects of changing mass over a 0.54 m ROM and in the absence of gravity on simulated jumping performance; (b) the effects of changing mass over an unlimited ROM and in the presence of gravity on simulated jumping performance; and (c) the effects of changing mass over a 0.54 m ROM and in the presence of gravity on simulated jumping performance. The work performed for each of these simulations as shown by the areas under the (i) force-distance and (ii) power-time curves; (iii) the peak power achieved for each simulation; and (iv) the performance achieved by each simulation. Horizontal line in panel D delineates PMax of 2726 W. IC is coded in black, all other colour coding is specific to this figure; bodymass is abbreviated as BM.

By removing ROM and adding the gravity constraint, performances decreased with increasing mass as the greater masses dictated larger percentages of force production were allocated to opposing gravity (Figure 7(b)). Contrary to simulations under the ROM constraint where all simulations achieved PMax (Figure 7(a(iii))), simulations ≥+60 kg failed to achieve PMax under gravity (Figure 7(b(iii))). Gravity also had a larger influence on performance than did ROM, as the final velocity was lower for simulations under all masses for the former as compared to the latter (Figures 7(a(iv),(b(iv))). When simulations were performed including both gravity and ROM constraints (Figure 7(c)), the resulting performance reflected a combination of both factors, which was characterised by fewer simulations achieving PMax (Figure 7(iii)) and lower performances across all simulated masses (Figure 7(iv)).

The interaction between PMax and mass on performance (simulation group 7)

This section illustrates how bumps in PMax interact with different mass conditions in the production of simulated jumping performance. While a bump in PMax almost always results in a performance improvement, the amount of increase depends on the interaction between the mass and the mechanism by which PMax is bumped (i.e., whether PMax increased via a bump in FMax, VMax, or a combination of the two). Intuitively, as mass increases, the optimal bump changes towards those bumps that occur via increases in Fmax.

Understanding how increases in PMax differentially influence performance based on mass is important as the primary opportunity for increasing jump performance is through concurrent increases in PMax and F-V relationship optimisation as: (1) the mass of the subject may be malleable, and (2) many ballistic sporting movements differ in the mass that must be moved, with comparisons between some tasks deviating to a large degree (e.g., martial arts striking vs maximal weightlifting clean). The simulations in this section examined the interaction between bumps in PMax (Figure 8(i)) and changes in mass utilising the same modelling parameters as the ‘The influence of increasing PMax on performance’ and ‘The influence of changing mass on performance’ subsections.

Figure 8. The effects of increasing PMax and changing mass over a 0.54 m ROM in the presence of gravity on simulated jumping performance. This simulation group utilised the same starting F-V profiles and bumps in PMax as shown in Figure 6 ((a) IC, (b) +40% FMax, and (c) −40% FMax)). (i) The F-V profiles of the simulations used in this simulation group (organised by column); (ii) the peak power achieved during simulations under each profile; and (iii) the performance of simulations under each profile. Figure is not colour coded for mass; bodymass is abbreviated as BM.

The mechanism of PMax bump influenced the mass-peak power relationship. Simulations under BumpFMax tended to achieve PMax across a greater range of masses, whilst also achieving higher instantaneous peak powers for the simulations under masses that did not achieve PMax, whereas BumpVMax produced the opposite pattern (Figure 8(ii)). Performance decreased as the mass increased for all simulations, however, both the performance of the simulation under the lightest mass and the rate of performance decrement with increasing mass were sensitive to the modelled F-V relationship (Figure 8(iii)). For example, of the starting F-V profiles, −40% Fmax produced the best performance for simulations under −60 kg mass but had the greatest rate of decrease in performance with increasing mass. In fact, simulations at +100 kg mass were unable to produce propulsion for this profile-mass combination as |FMax| < |gravitational force| (Figure 8(c(iii))). While performance increased for all simulations under the lightest mass regardless of mechanism of PMax bump, both this performance and the rate of performance decrement with increasing mass were sensitive to the mechanism by which PMax was bumped. Increasing PMax by BumpFMax resulted in the smallest increase in performance under −60 kg mass but decreased the rate of performance decrement with increasing mass, with BumpVMax having the opposite effect. Increasing PMax through BumpFMaxVMax produced an iso-slope shift in the performance-mass relationship.

Viewing the performance-mass relationship across a spectrum of masses and multiple F-V profiles provides insight into F-V profile optimisation at constant PMax and ROM. Profiles favouring relatively lower FMax produce better performances and are closer to optimal for simulations under lighter masses, however, their performance drops off with increasing mass at a higher rate, dictating the existence of a crossover mass in which they become less optimal compared to simulations under profiles favouring higher FMax. This can be seen visually by comparing the increased (i.e., bumped) PMax F-V profiles reported in Figure 8(iii). For example, although BumpFMaxVMax from the IC profile produced a poorer performance as compared to the BumpVmax profile for simulations at −60 kg to −20 kg masses, a crossover was reached for simulation under BM where BumpFMaxVMax performed better, which was due to the relatively flatter drop-off of this profile’s performance-mass relationship (Figure 8(a(iii))). It is important to note that once the crossover mass is reached, the profile favouring higher FMax remains comparatively better for simulations under all masses that are heavier (provided |FMax| > |gravitational force|).

The influence of activation and rate of rise to maximal activation on performance (simulation groups 8 and 9)

This section quantifies the influence of commencing propulsion from submaximal activation, as may occur during real-world SJ and CMJ. Critically, performances are influenced to a greater extent under lighter masses as the rate of rise to maximal activation is a function of time and lighter masses traverse their allotted ROM quicker as compared to heavier masses. For SJ, this impact is magnified as lighter mass simulations begin propulsion with lower initial activation levels as less activation is required to oppose gravity in the starting position.

Figure 9. Differences in performance between simulations commencing from maximal activation and (a) CMJ and (b) SJ for different masses under constant F-V profile at constant PMax (IC) over a 0.54 m ROM in the presence of gravity. The work performed for each of these simulations as shown by the areas under the (i) force-distance and (ii) power-time curves; (iii) the peak power achieved for each simulation; and (iv) the performance achieved by each simulation; horizontal lines in panels (ii) and (iii) delineate PMax of 2726 W. Solid curves and squares delineate simulations commencing from maximal activation; these simulations are colour matched by mass with dashed curves and open squares that delineate simulations commencing from submaximal activation. IC is coded in black, all other colour coding is specific to Figures 9 and 10; bodymass is abbreviated as BM.

While the F-V profile represents the capacity of the system to perform work, during propulsion the realisation of this capacity is sensitive to both the initial activation level, hence force level, and rate of rise to maximal activation, hence the rate of force development. During jumping, performances are maximised when propulsive motion begins with the maximal activation of the relevant musculature (Pandy & Zajac, 1991); however, physiological and mechanical constraints may dictate that propulsion begins at submaximal levels of activation, as occurs during SJ and may occur during CMJ under some conditions. When beginning propulsion from submaximal activation, performances are optimised by increasing the activation level of the appropriate musculature to maximal as quickly as possible (see Pandy and Zajac (1991) for more detailed information). During this initial period in which activation and force production are increasing, the range of motion is traversed under levels of force that are less than predicted by the F-V profile of the subject, and the subject performs relatively less work, tends to produce less total work and thus a poorer performance as compared to the equivalent simulation commencing from maximal activation.

In the present investigation, all simulations that commenced under submaximal activation performed less work as compared to equivalent conditions starting from maximal activation (Figures 9(i,ii) and 10(ii)). For these simulations, the velocity equal to ½VMax (i.e., 2.05 m∙s⁻²) was achieved prior to maximal activation being reached, which dictated the force production was also submaximal at this time and resultantly, that peak power was less than PMax (see Equation 1(1) $\mathit{System}\mathit{peak}\mathit{power}\mathit{capability}\left(\mathit{PMax}\right)=\frac{\mathrm{Fmax}\ast \mathrm{Vmax}}{4}$(1) ; Figure 9(iii)). The simulations that achieved the lowest force production at ½VMax also realised the largest decrease in peak power production. Although none of the simulations of the current investigation achieved PMax when commencing from submaximal activation, some simulations under heavier masses (e.g., +50 kg) would be capable of achieving PMax given sufficient ROM.

CMJ and SJ differ in that the activation state can be built up during the preceding eccentric phase during CMJ (Bobbert & Casius, 2005; Bobbert et al., 1996). For the modelled subject this ‘head start’ allowed for maximal activation and force production at the beginning of propulsion across all simulations of bodymass and heavier (Figure 9(a(i))), as well as CMJ having higher initial levels of activation and force production for all lighter simulations as compared to SJ. Because of this, CMJ always performed more work over the early ROM and the total ROM, achieved higher peak power and produced a better performance as compared to SJ, under equivalent conditions. It is worth mentioning that although the difference between SJ and CMJ is substantial under the lightest mass (i.e., 3.53 m∙s⁻¹ vs 3.67 m∙s⁻¹ for simulations under −60 kg), this discrepancy is less substantial under heavier masses with simulations at bodymass producing performances of 2.48 m∙s⁻¹ and 2.57 m∙s⁻¹ for SJ and CMJ. This demonstrates that under the real-world conditions of vertical jumping under bodymass, that the ability to activate the musculature quickly may not be a major performance-determining factor for some subjects (see Kozinc et al. (2022)) for experimental agreement). It is also worth mentioning that compared to simulations commencing from maximal activation, CMJ performance was only modestly decreased, even during simulations under the lightest modelled masses (e.g., 3.71 m∙s⁻¹ vs 3.67 m∙s⁻¹ for simulations under −60 kg).

Figure 10. The effects of commencing propulsion from submaximal activation under the SJ condition for different masses under different ROMs (0.135–0.810 m) under constant F-V profile at constant PMax (IC) in the presence of gravity on simulated jumping performance. Simulations comparing maximal activation with SJ activation conditions for the (a) 23.7 kg, (b) BM; and (c) 123.7 kg mass conditions. (i) Modelled activation across the ROM for all SJ simulations. The work performed for each of these simulations is shown by the areas under the (ii) force-distance and (iii) power-time curves; and (iv) the performance achieved at each ROM for each simulation; solid curves and squares delineate simulations commencing from maximal activation; these simulations are colour matched by mass with dashed curves and open squares that delineate SJ simulations. IC is coded in black, all other colour coding is specific to Figures 9 and 10.

Simulated jumping performances were not only sensitive to the initial level of activation, but also the mass. SJ simulations did not commence from zero activation, as they would under conditions without gravity, as the subject was required to sub-maximally activate the musculature to produce the level of force required to support their weight in the pre-propulsive squat position. Because heavier weights required higher levels of force to maintain the squat position, SJ simulations under heavier masses began propulsion at higher levels of activation and required less time to achieve maximal activation as compared to simulations under lighter masses (Figure 10(i)). This resulted in SJ performance with higher masses being closer to the CMJ equivalent than SJ performance with lower masses.

In addition to benefiting from higher initial activation and less time to maximal activation, SJ simulations under heavier masses also benefitted from the lower accelerations of the subject (see Figures 1 and 2) as compared to simulations under lighter masses. This is because the ROM is fixed, and activation increases as a function of time. Therefore, simulations under heavier masses achieved maximal activation earlier in the ROM (Figure 10(i)), not only because they commenced from higher initial activation, but also because the lower acceleration produced a smaller change in distance per unit time and thus, a smaller percentage of the ROM traversed per unit increase in activation (Figure 10(i)). The higher initial activation and lower acceleration worked together to produce more work (Figures 9(b(i,ii); Figure 10(ii)) and better performances in relation to maximal activation simulations for SJ simulations under heavier masses as compared to lighter masses (Figure 9(b(iv)).

Jumps commencing from submaximal activation, such as SJ, were also sensitive to the allotted ROM. As shown in Figure 10, SJ simulations over shorter ROMs produced poorer performances in relation to maximal activation simulations as compared to those over longer ROMs (Figure 10(iv)). This can be attributed to two factors: the discrepancy in initial activation between SJ and maximal activation simulations, and the modelled F-V relationship (Figures 10(i,iii))). The lower force production resulting from submaximal activation during SJ (Figure 10(ii)) caused a smaller change in velocity as compared to the equivalent maximal activation simulation early in the ROM (Figure 10(iii)), which in turn, allowed the subject to produce more force as dictated by the F-V profile. The discrepancy in velocity between simulation conditions widened over the early ROM (Figure 10(iii)) while simultaneously, the activation continued to increase during SJ (Figure 10(i)). Eventually, a crossover point was reached where the SJ simulation realised a large enough percentage of its force producing capability (as determined by activation), and a slow enough velocity in relation to the maximal activation stimulation, that it began and then continued to produce greater force and perform more work (Figure 10(ii)). For all the simulated distances, this greater work produced during the latter phase of the SJ did not offset the lesser work produced over the early ROM, however, it did serve to partially decrease the discrepancy in work (Figure 10(ii)) and performance (Figure 10(iv)) between conditions. The larger the allowed ROM, the greater the distance travelled under slower velocity for SJ (Figure 10(iii)), and thus, the greater the opportunity for SJ simulations to perform work approaching that of the equivalent maximal activation simulation (Figure 10(ii)). Theoretically, if the ROM was unlimited, the subject would have continued to produce greater force and more work during SJ until force production, velocity, and performance equalised between conditions.

Discussion and implications

The current work reported that for a given PMax, a F-V profile favouring lower FMax always possesses a larger work capacity and produces a better performance as compared to a profile favouring higher FMax under constant mass, an unlimited ROM, and no gravity. However, when a defined ROM, and gravity are inserted into the simulation, as occurs during real-world jumping, lower FMax profiles suffer larger performance decrements for a given ROM and/or gravitational force as compared to profiles possessing higher FMax. Therefore, when comparing the performance of two profiles, there is a trade-off between higher work capacity (profiles favouring lower FMax), and the ability to realise this capacity (profiles favouring higher FMax) within the allowed constraints. The current work also found that while increasing the capability of the system to produce power almost always improves the jump performance, the magnitude of improvement is dependent on the mechanism by which PMax is increased in combination with the fit of the starting F-V profile with the mass, ROM, and gravitational conditions. A bump in PMax may result from an increase in FMax, VMax, or a combination of the two; and thus, even though a F-V profile rotation is defined as a change in relationship slope at constant PMax, meaningful changes in the F-V relationship may occur via bumps in PMax. The current work also reported that all simulated jumping performances based on the modelled subject are negatively influenced by commencement from submaximal activation, however, performances under shorter ROMs and lighter masses (and the interaction of the two factors) are influenced to a greater extent.

Given this is based on a simple modelling paradigm, some broader practical considerations are warranted. It is easy to change parameters in a model, but equivalent changes may not be possible in the real world. Consideration should be given as to whether ROM can be altered meaningfully, and can maximum velocity be altered with training commensurately to how maximum force can be as we have done in the model.

Although ROM is easy to alter in a model, ROM can often be thought of as more of a fixed parameter in humans; however, in many activities there is room for altering ROM without lengthening bones! In a simple muscle driven model of a squat jump, Domire and Challis (2007) demonstrated that the lower the starting position for the squat the greater the jump height. This very deep squat start position and improved performance has not been seen in human studies but Scholz et al. (2006) compared the squat jumping ability of a bonobo to a human and found the bonobo was superior and started from a full butt to floor squat position. Most humans do not go as deep into a squat position as they could when performing a vertical jump. This could be due to real joint geometry limitations not seen in the simple models, or limitations in flexibility that do not allow full muscle activation whilst directing the COM vertically from a deep squat. For many, getting into a full squat can lead to ankle plantar flexion and thus remove some of the ROM needed during the latter high velocity extension phase just before take-off.

During sprinting propulsion is only possible during foot contact and the short duration of contact limits the propulsive ROM, however, sprint speed is not limited by a single cycle of action but it is a culmination of multiple foot contacts (Labonte et al., 2024) that vary in characteristics from start to maximum velocity. This has led to F-V profiling of sprinting over strides (Samozino et al., 2022) but this is outside the scope of this study. Compared to normal sprinting, speed skating has much higher velocities due to the pushing leg being separate from the support leg during most of the cycle, with the support leg also moving forward, and thus a ‘stride’ having a greater effective ROM over which it can function (there are also much smaller braking forces due to the continuously moving support leg). ROM being altered by changes in equipment and improving power output was elegantly demonstrated with the introduction of the klapskate into long track speed skating, where it extended the effective range of motion, in the direction required, over which power could be applied without the blade catching on the ice (Houdijk et al., 2000).

Jiménez-Reyes et al. (2019) demonstrated that those with an initial force deficit in their F-V profile took about 50% longer to get to a no deficit F-V profile than those with an initial velocity deficit when undergoing training tailored to their specific deficit profile (12.6 vs 8.7 weeks). Changing intrinsic velocity properties of the muscle tendon system may be difficult to achieve, requiring increased sarcomeres in series, pennation angle changes alongside changes in the muscle belly gearing, or tendon changes to enhance fibre range of operation and shortening velocity. However, there are also many technique/coordination, neural factors at the whole body level that can be more easily and rapidly adapted to allow changes in the experimentally derived F-V curve. Depressed activation during single joint movements has been well reported during eccentrics but it has also been seen in high velocity concentric actions and when near full extension (Forrester & Pain, 2010; Pain & Forrester, 2009a; Pain et al., 2013; Voukelatos & Pain, 2015; Voukelatos et al., 2022). Activation dynamics varies with contraction type and speed meaning the total available torque for a given joint angle and angular velocity is not always achieved (Tillin et al., 2012a, 2018). Dynamic training, and training over only a few weeks, can alter the motor unit recruitment patterns and reduce neural inhibition to improve performance (Tillin et al., 2012b; Van Cutsem et al., 1998; Voukelatos et al., 2018). Between elite athletes, and between joints for a given athlete, there can be very large differences in the joint level F-V profile, especially with regard to power production at higher velocities, in a manner that is specifically beneficial for the athlete’s sport, indicating this a likely a very trainable aspect, although genetic predisposition cannot be ruled out (Pain & Forrester, 2009b).

In conclusion, the current findings support the hypothesis that the optimal F-V profile cannot be identified without knowledge of the mass, ROM, gravity (or more likely the direction of propulsion relative to gravity), and rate of rise to maximal activation and these parameters may differ between subjects and jump types; and, in the wider context of ballistic performances may exhibit dramatic differences (e.g., a boxing jab as compared to a maximal weightlifting clean). Thus, when considering across activities it is important to be aware that these different activities may require large differences in F-V capabilities which can inform both athlete selection and optimal training principles.

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The author(s) reported there is no funding associated with the work featured in this article.