Correction to 'Natural Frequency Method: estimating the preferred walking speed of Tyrannosaurus rex based on tail natural frequency'.

[This corrects the article DOI: 10.1098/rsos.201441.][This corrects the article DOI: 10.1098/rsos.201441.].

Natural Frequency Method: estimating the preferred walking speed of Tyrannosaurus rex based on tail natural frequency.

Locomotor energetics are an important determinant of an animal's ecological niche. It is commonly assumed that animals minimize locomotor energy expenditure by selecting gait kinematics tuned to the natural frequencies of relevant body parts. We demonstrate that this allows estimation of the preferred step frequency and walking speed of Tyrannosaurus rex, using an approach we introduce as the Natural Frequency Method. Although the tail of bipedal dinosaurs was actively involved in walking, it was suspended passively by the caudal interspinous ligaments. These allowed for elastic energy storage, thereby reducing the metabolic cost of transport. In order for elastic energy storage to be high, step and natural frequencies would have to be matched. Using a 3D morphological reconstruction and a spring-suspended biomechanical model, we determined the tail natural frequency of T. rex (0.66 s-1, range 0.41-0.84), and the corresponding walking speed (1.28 m s-1, range 0.80-1.64), which we argue to be a good indicator of preferred walking speed (PWS). The walking speeds found here are lower than earlier estimations for large theropods, but agree quite closely with PWS of a diverse group of extant animals. The results are most sensitive to uncertainties regarding ligament moment arms, vertebral kinematics and ligament composition. However, our model formulation and method for estimation of walking speed are unaffected by assumptions regarding muscularity, and therefore offer an independent line of evidence within the field of dinosaur locomotion.

Predicting dive start performance from kinematic variables at water entry in (sub-)elite swimmers.

The dive start is an important component of competitive swimming, especially at shorter race distances. Previous research has suggested that start performance depends on kinematic variables pertaining to the swimmer at water entry, notably the distance from the block, the horizontal velocity of the centre of mass and the angle between body and water surface. However, the combined and relative contributions of these variables to start performance remain to be determined. The aim of the present study was therefore to develop a model to predict start performance (time from take-off to reaching the 15-m line) from a set of kinematic variables that collectively define the swimmer's entry state. To obtain an appropriate database for this purpose, fifteen well-trained, (sub-)elite swimmers performed dive starts under different instructions intended to induce substantial variation in entry state. Kinematic data were extracted from video recordings of these starts, optimised and analysed statistically. A mixed effects analysis of the relation between entry state and start performance was conducted, which revealed a significant and robust dependence of start performance on entry state (χ2(3) = 88, p < .001), explaining 86.1% of the variance. Start time was reduced by 0.6 s (p < .001) when the horizontal displacement at water entry was 1 m further, by 0.3 s (p < .001) when the horizontal velocity of the centre of mass was 1 m/s higher, and by 0.5 s (p < .01) when the entry angle was 1 radian flatter. The robustness of the analysis was confirmed by a similar mixed effects analysis of the relation between entry state and time to the 5-m line. In conclusion, dive start performance can be predicted to a considerable extent from the swimmer's state at water entry. The implications of those findings for studying and improving block phase kinetics are discussed.

Metabolic Cost of Activation and Mechanical Efficiency of Mouse Soleus Muscle Fiber Bundles During Repetitive Concentric and Eccentric Contractions.

Currently available data on the energetics of isolated muscle preparations are based on bouts of less than 10 muscle contractions, whereas metabolic energy consumption is mostly relevant during steady state tasks such as locomotion. In this study we quantified the energetics of small fiber bundles of mouse soleus muscle during prolonged (2 min) series of contractions. Bundles (N = 9) were subjected to sinusoidal length changes, while measuring force and oxygen consumption. Stimulation (five pulses at 100 Hz) occurred either during shortening or during lengthening. Movement frequency (2-3 Hz) and amplitude (0.25-0.50 mm; corresponding to ± 4-8% muscle fiber strain) were close to that reported for mouse soleus muscle during locomotion. The experiments were performed at 32°C. The contributions of cross-bridge cycling and muscle activation to total metabolic energy expenditure were separated using blebbistatin. The mechanical work per contraction cycle decreased sharply during the first 10 cycles, emphasizing the importance of prolonged series of contractions. The mean ± SD fraction of metabolic energy required for activation was 0.37 ± 0.07 and 0.56 ± 0.17 for concentric and eccentric contractions, respectively (both 0.25 mm, 2 Hz). The mechanical efficiency during concentric contractions increased with contraction velocity from 0.12 ± 0.03 (0.25 mm 2 Hz) to 0.15 ± 0.03 (0.25 mm, 3 Hz) and 0.16 ± 0.02 (0.50 mm, 2 Hz) and was -0.22 ± 0.08 during eccentric contractions (0.25 mm, 2 Hz). The percentage of type I fibers correlated positively with mechanical efficiency during concentric contractions, but did not correlate with the fraction of metabolic energy required for activation.

Towards determination of power loss at a rowing blade: Validation of a new method to estimate blade force characteristics.

To analyze on-water rowing performance, a valid determination of the power loss due to the generation of propulsion is required. This power los can be calculated as the dot product of the net water force vector ([Formula: see text]) and the time derivative of the position vector of the point at the blade where [Formula: see text] is applied ([Formula: see text]). In this article we presented a method that allows for accurate determination of both parameters using a closed system of three rotational equations of motion for three different locations at the oar. Additionally, the output of the method has been validated. An oar was instrumented with three pairs of strain gauges measuring local strain. Force was applied at different locations of the blade, while the oar was fixed at the oarlock and the end of the handle. Using a force transducer and kinematic registration, the force vector at the blade and the deflection of the oar were measured. These data were considered to be accurate and used to calibrate the measured strain for bending moments, the deflection of the oar and the angle of the blade relative to its unloaded position. Additionally, those data were used to validate the output values of the presented method plus the associated instantaneous power output. Good correspondence was found between the estimated perpendicular blade force and its reference (ICC = .999), while the parallel blade force could not be obtained (ICC = .000). The position of the PoA relative to the blade could be accurately obtained when the perpendicular force was ≥ 5.3 N (ICC = .927). Instantaneous power output values associated with the perpendicular force could be obtained with reasonable accuracy (ICC = .747). These results suggest that the power loss due to the perpendicular water force component can be accurately obtained, while an additional method is required to obtain the power losses due to the parallel force.

Real-Time Feedback on Mechanical Power Output: Facilitating Crew Rowers' Compliance With Prescribed Training Intensity.

PURPOSE: Athletes require feedback in order to comply with prescribed training programs designed to optimize their performance. In rowing, current feedback parameters on intensity are inaccurate. Mechanical power output is a suitable objective measure for training intensity, but due to movement restrictions related to crew rowing, it is uncertain whether crew rowers are able to adjust their intensity based on power-output feedback. The authors examined whether rowers improve compliance with prescribed power-output targets when visual real-time feedback on power output is provided in addition to commonly used feedback. METHODS: A total of 16 crew rowers rowed in 3 training sessions. During the first 2 sessions, they received commonly used feedback, followed by a session with additional power-output feedback. Targets were set by their coaches before the experiment. Compliance was operationalized as accuracy (absolute difference between target and delivered power output) and consistency (high- and low-frequency variations in delivered power output). RESULTS: Multilevel analyses indicated that accuracy and low-frequency variations improved by, respectively, 65% (P > .001) and 32% (P = .024) when additional feedback was provided. CONCLUSION: Compliance with power-output targets improved when crew rowers received additional feedback on power output. Two additional observations were made during the study that highlighted the relevance of power-output feedback for practice: There was a marked discrepancy between the prescribed targets and the actually delivered power output by the rowers, and coaches had difficulties perceiving improvements in rowers' compliance with power-output targets.

Huxley-type cross-bridge models in largeish-scale musculoskeletal models; an evaluation of computational cost.

A Huxley-type cross-bridge model is attractive because it is inspired by our current understanding of the processes underlying muscle contraction, and because it provides a unified description of muscle's mechanical behavior and metabolic energy expenditure. In this study, we determined the computational cost for task optimization of a largeish-scale musculoskeletal model in which muscles are represented by a 2-state Huxley-type cross-bridge model. Parameter values defining the rate functions of the Huxley-type cross-bridge model could be chosen such that the steady-state force-velocity relation resembled that of a Hill-type model. Using these parameter values, maximum-height squat jumping was used as the example task to evaluate the computational cost of task optimization for a skeletal model driven by a Huxley-type cross-bridge model. The optimal solutions for the Huxley- and Hill-type muscle models were similar for all mechanical variables considered. Computational cost of the Huxley-type cross-bridge model was much higher than that of the Hill-type model. Compared to the Hill-type model, the number of state variables per muscle was large (2 vs about 18,000), the integration step size had to be about 100 times smaller, and the computational cost per integration step was about 100 times higher.

Experimental estimation of energy absorption during heel strike in human barefoot walking.

Metabolic energy expenditure during human gait is poorly understood. Mechanical energy loss during heel strike contributes to this energy expenditure. Previous work has estimated the energy absorption during heel strike as 0.8 J using an effective foot mass model. The aim of our study is to investigate the possibility of determining the energy absorption by more directly estimating the work done by the ground reaction force, the force-integral method. Concurrently another aim is to compare this method of direct determination of work to the method of an effective foot mass model. Participants of our experimental study were asked to walk barefoot at preferred speed. Ground reaction force and lower leg kinematics were collected at high sampling frequency (3000 Hz; 1295 Hz), with tight synchronization. The work done by the ground reaction force is 3.8 J, estimated by integrating this force over the foot-ankle deformation. The effective mass model is improved by dropping the assumption that foot-ankle deformation is maximal at the instant of the impact force peak. On theoretical grounds it is clear that in the presence of substantial damping that peak force and peak deformation do not occur simultaneously. The energy absorption results, due the vertical force only, corresponding to the force-integral method is similar to the results of the improved application of the effective mass model (2.7 J; 2.5 J). However the total work done by the ground reaction force calculated by the force-integral method is significantly higher than that of the vertical component alone. We conclude that direct estimation of the work done by the ground reaction force is possible and preferable over the use of the effective foot mass model. Assuming that energy absorbed is lost, the mechanical energy loss of heel strike is around 3.8 J for preferred walking speeds (≈ 1.3 m/s), which contributes to about 15-20% of the overall metabolic cost of transport.

Improved determination of mechanical power output in rowing: Experimental results.

In rowing, mechanical power output is a key parameter for biophysical analyses and performance monitoring and should therefore be measured accurately. It is common practice to estimate on-water power output as the time average of the dot product of the moment of the handle force relative to the oar pin and the oar angular velocity. In a theoretical analysis we have recently shown that this measure differs from the true power output by an amount that equals the mean of the rower's mass multiplied by the rower's center of mass acceleration and the velocity of the boat. In this study we investigated the difference between a rower's power output calculated using the common proxy and the true power output under different rowing conditions. Nine rowers participated in an on-water experiment consisting of 7 trials in a single scull. Stroke rate, technique and forces applied to the oar were varied. On average, rowers' power output was underestimated with 12.3% when determined using the common proxy. Variations between rowers and rowing conditions were small (SD = 1.1%) and mostly due to differences in stroke rate. To analyze and monitor rowing performance accurately, a correction of the determination of rowers' on-water power output is therefore required.

Mechanical power output in rowing should not be determined from oar forces and oar motion alone.

Mechanical power output is a key performance-determining variable in many cyclic sports. In rowing, instantaneous power output is commonly determined as the dot product of handle force moment and oar angular velocity. The aim of this study was to show that this commonly used proxy is theoretically flawed and to provide an indication of the magnitude of the error. To obtain a consistent dataset, simulations were performed using a previously proposed forward dynamical model. Inputs were previously recorded rower kinematics and horizontal oar angle, at 20 and 32 strokes∙min-1. From simulation outputs, true power output and power output according to the common proxy were calculated. The error when using the common proxy was quantified as the difference between the average power output according to the proxy and the true average power output (P̅residual), and as the ratio of this difference to the true average power output (ratiores./rower). At stroke rate 20, P̅residual was 27.4 W and ratiores./rower was 0.143; at stroke rate 32, P̅residual was 44.3 W and ratiores./rower was 0.142. Power output in rowing appears to be underestimated when calculated according to the common proxy. Simulations suggest this error to be at least 10% of the true power output.

An accurate estimation of the horizontal acceleration of a rower's centre of mass using inertial sensors: a validation.

For a valid determination of a rower's mechanical power output, the anterior-posterior (AP) acceleration of a rower's centre of mass (CoM) is required. The current study was designed to evaluate the accuracy of the determination of this acceleration using a full-body inertial measurement units (IMUs) suit in combination with a mass distribution model. Three methods were evaluated. In the first two methods, IMU data were combined with either a subject-specific mass distribution or a standard mass distribution model for athletes. In the third method, a rower's AP CoM acceleration was estimated using a single IMU placed at the pelvis. Experienced rowers rowed on an ergometer that was placed on two force plates, while wearing a full-body IMUs suit. Correspondence values between AP CoM acceleration based on IMU data (the three methods) and AP CoM acceleration obtained from force plate data (reference) were calculated. Good correspondence was found between the reference AP CoM acceleration and the AP CoM accelerations determined using IMU data in combination with the subject-specific mass model and the standard mass model (intraclass correlation coefficients [ICC] > 0.988 and normalized root mean square errors [nRMSE] 3.81%). Correspondence was lower for the AP CoM accelerations determined using a single pelvis IMU (0.877 < ICC < 0.960 and 6.11% < nRMSE < 13.61%). Based on these results, we recommend determining a rower's AP CoM acceleration using IMUs in combination with the standard mass model. Finally, we conclude that accurate determination of a rower's AP CoM acceleration is not possible on the basis of the pelvis acceleration only.

Comparison of the validity of Hill and Huxley muscle-tendon complex models using experimental data obtained from rat m. soleus in situ.

The relationship between mechanical and metabolic behaviour in the widely used Hill muscle-tendon complex (MTC) model is not straightforward, whereas this is an integral part of the Huxley model. In this study, we assessed to what extent Huxley- and Hill-type MTC models yield adequate predictions of mechanical muscle behaviour during stretch-shortening cycles (SSCs). In fully anaesthetized male Wistar rats (N=3), m. soleus was dissected completely free, except for the insertion. Cuff electrodes were placed over the n. ischiadicus. The distal end of the tendon was connected to a servo motor, via a force transducer. The setup allowed for full control over muscle stimulation and length, while force was measured. Quick-release and isovelocity contractions (part 1), and SSCs (part 2) were imposed. Simulations of part 2 were made with both a Hill and a Huxley MTC model, using parameter values determined from part 1. Modifications to the classic two-state Huxley model were made to incorporate series elasticity, activation dynamics, and active and passive force-length relationships. Results were similar for all rats. Fitting of the free parameters to the data of part 1 was near perfect (R(2)>0.97). During SSCs, predicted peak force and force during relaxation deviated from the experimental data for both models. Overall, both models yielded similarly adequate predictions of the experimental data. We conclude that Huxley and Hill MTC models are equally valid with respect to mechanical behaviour.

Strapping rowers to their sliding seat improves performance during the start of single-scull rowing.

In this study, the effect of strapping rowers to their sliding seat on performance during 75 m on-water starting trials was investigated. Well-trained rowers performed 75 m maximum-effort starts using an instrumented single scull equipped with a redesigned sliding seat system, both under normal conditions and while strapped to the sliding seat. Strapping rowers to their sliding seat resulted in a 0.45 s lead after 75 m, corresponding to an increase in average boat velocity of about 2.5%. Corresponding effect sizes were large. No significant changes were observed in general stroke cycle characteristics. No indications of additional boat heaving and pitching under strapped conditions were found. The increase in boat velocity is estimated to correspond to an increase in average mechanical power output during the start of on-water rowing between 5% and 10%, which is substantial but smaller than the 12% increase found in a previous study on ergometer starting. We conclude that, after a very short period of adaptation to the strapped condition, single-scull starting performance is substantially improved when the rower is strapped to the sliding seat.

From bicycle chain ring shape to gear ratio: algorithm and examples.

A simple model of the bicycle drive system with a non-circular front chain ring is proposed and an algorithm is devised for calculation of the corresponding Gear Ratio As a Function Of Crank Angle (GRAFOCA). It is shown that the true effective radius of the chain ring is always the perpendicular distance between the crank axis and the line through the chain segment between the chain ring and the cog. It is illustrated that the true effective radius of the chain ring at any crank angle may differ substantially from the maximum vertical distance between the crank axis and the chain ring circumference that is used as a proxy for the effective chain ring radius in several studies; in particular, the crank angle at which the effective chain ring radius is maximal as predicted from the latter approach may deviate by as much as 0.30 rad from the true value. The algorithm proposed may help in designing chain rings that achieve the desired GRAFOCA.

Motor commands for fast point-to-point arm movements are customized for small changes in inertial load.

For repeated point-to-point arm movements it is often assumed that motor commands are customized in a trial-to-trial manner, based on previous endpoint error. To test this assumption, we perturbed movement execution without affecting the endpoint error by using a modest manipulation of inertia. Participants made point-to-point elbow flexion and extension movements in the horizontal plane, under the instruction to move as fast as possible from one target area to another. In selected trials the moment of inertia of the lower arm was increased or decreased by 25%. First, we found that an unexpected increase or decrease of inertia did not affect the open loop controlled part of the movement path (and thus endpoint error was not affected). Second, we found that when the increased or decreased inertia was presented repeatedly, after 5-11 trials motor commands were customized: the first 100ms of agonistic muscle activity in the smoothed and rectified electromyographic signal of agonistic muscles was higher for the high inertia compared to the low inertia. We conclude that endpoint error is not the only parameter that is used to evaluate if motor commands lead to movements as planned.

Are fast interceptive actions continuously guided by vision? Revisiting Bootsma and van Wieringen (1990).

In an influential study, R. J. Bootsma and P. C. W. van Wieringen (1990) argued that 2 of their 5 participants used visual information continuously during the attacking forehand drive in table tennis, its brief duration vis-à-vis the visuomotor delay notwithstanding. The authors repeated Bootsma and van Wieringen's experiment and included a condition in which vision was obscured after drive initiation. The authors replicated most of Bootsma and van Wieringen's findings but found no significant differences between the full-vision and no-vision conditions, which goes against the interpretation of these findings as evidence for continuous visual guidance. A subsequent simulation study found that a single preprogrammed muscle stimulation pattern resulted in spatiotemporal convergence similar to that observed experimentally but not in other important behavioral characteristics. The results contain no indications that visual information that becomes available after drive initiation affects arm motion and suggest that a form of model-based predictive control is operative rather than continuous visual guidance.

Estimation of the energy loss at the blades in rowing: common assumptions revisited.

In rowing, power is inevitably lost as kinetic energy is imparted to the water during push-off with the blades. Power loss is estimated from reconstructed blade kinetics and kinematics. Traditionally, it is assumed that the oar is completely rigid and that force acts strictly perpendicular to the blade. The aim of the present study was to evaluate how reconstructed blade kinematics, kinetics, and average power loss are affected by these assumptions. A calibration experiment with instrumented oars and oarlocks was performed to establish relations between measured signals and oar deformation and blade force. Next, an on-water experiment was performed with a single female world-class rower rowing at constant racing pace in an instrumented scull. Blade kinematics, kinetics, and power loss under different assumptions (rigid versus deformable oars; absence or presence of a blade force component parallel to the oar) were reconstructed. Estimated power losses at the blades are 18% higher when parallel blade force is incorporated. Incorporating oar deformation affects reconstructed blade kinematics and instantaneous power loss, but has no effect on estimation of power losses at the blades. Assumptions on oar deformation and blade force direction have implications for the reconstructed blade kinetics and kinematics. Neglecting parallel blade forces leads to a substantial underestimation of power losses at the blades.

Isometric torque-angle relationships of the elbow flexors and extensors in the transverse plane.

Maximal voluntary isometric torque-angle relationships of elbow extensors and flexors in the transverse plane (humerus elevation angle of 90 degrees ) were measured at two different horizontal adduction angles of the humerus compared to thorax: 20 degrees and 45 degrees . For both elbow flexors and extensors, the torque-angle relationship was insensitive to this 25 degrees horizontal adduction of the humerus. The peak in torque-angle relationship of elbow extensors was found at 55 degrees (0 degrees is full extension). This is closer to full elbow extension than reported by researchers who investigated this relationship in the sagittal plane. Using actual elbow angles during contraction, as we did in this study, instead of angles set by the dynamometer, as others have done, can partly explain this difference. We also measured electromyographic activity of the biceps and triceps muscles with pairs of surface electrodes and found that electromyographic activity level of the agonistic muscles was correlated to measured net torque (elbow flexion torque: Pearson's r=0.21 and extension torque: Pearson's r=0.53). We conclude that the isometric torque-angle relationship of the elbow extensors found in this study provides a good representation of the force-length relationship and the moment arm-angle relationship of the elbow extensors, but angle dependency of neural input gives an overestimation of the steepness.