The energetic basis for smooth human arm movements
Abstract
The central nervous system plans human reaching movements with stereotypically smooth kinematic trajectories and fairly consistent durations. Smoothness seems to be explained by accuracy as a primary movement objective, whereas duration seems to economize energy expenditure. But the current understanding of energy expenditure does not explain smoothness, so that two aspects of the same movement are governed by seemingly incompatible objectives. Here, we show that smoothness is actually economical, because humans expend more metabolic energy for jerkier motions. The proposed mechanism is an underappreciated cost proportional to the rate of muscle force production, for calcium transport to activate muscle. We experimentally tested that energy cost in humans (N = 10) performing bimanual reaches cyclically. The empirical cost was then demonstrated to predict smooth, discrete reaches, previously attributed to accuracy alone. A mechanistic, physiologically measurable, energy cost may therefore explain both smoothness and duration in terms of economy, and help resolve motor redundancy in reaching movements.
Editor's evaluation
This paper will be of interest to researchers in the fields of biomechanics, movement control, and decision making. A novel, mechanistic model of metabolic cost is presented to account for a phenomenon not explained by current models of metabolic energy. This is followed by a demonstration of how this metabolic model can improve our understanding of movement control by revealing an energetic basis for smooth movements.
https://doi.org/10.7554/eLife.68013.sa0Introduction
Upper extremity reaching movements are characterized by a stereotypical, bellshaped speed profile for the hand’s motion to its target (Figure 1A). The profile’s smoothness seems to preserve kinematic accuracy (Harris and Wolpert, 1998) and have little to do with the effort needed to produce the motion. But effort or energy expenditure appear to affect other aspects of reaching (Huang et al., 2012; Shadmehr et al., 2019), and influence a vast array of other animal behaviors and actions (Alexander, 1996). It seems possible that effort or energy do influence the bellshaped profile, but have gone unrecognized because of incomplete quantification of such costs. If so, then dynamic goals including effort could play a key role in movement planning.
The kinematic goal for accuracy may be expressed quantitatively as minimization of the final endpoint position variance (Harris and Wolpert, 1998). Nonsmooth motions introduce inaccuracy because motor noise increases with motor command amplitude, a phenomenon termed signaldependent noise (Matthews, 1996; Sutton and Sykes, 1967). It predicts well the speed profiles for not only the hand but also the eye. It explains why more curved or more accurate motions need to be slower, and also subsumes an older theory for minimizing kinematic jerk (Flash and Hogan, 1985). The single objective of movement variance explains multiple aspects of smooth movements, and makes better predictions than competing theories (Diedrichsen et al., 2010; Haith et al., 2012; Todorov, 2004).
There are nonetheless reasons to consider effort. Many optimal control tasks must include an explicit objective for effort, without which movements would be expected to occur at maximal effort (‘bangbang control’, Harris and Wolpert, 1998; Bryson and Ho, 1975). In addition, metabolic energy expenditure is substantial during novel reaching tasks and decreases as adaptation progresses (Huang et al., 2012). Such a cost also helps to determine movement duration and vigor (Shadmehr et al., 2016), not addressed by the minimumvariance hypothesis. Indeed, optimal control studies have long examined effort costs such as for muscle force (Kolossiatis et al., 2016), mechanical work (Alexander, 1997), squared force or activation (Nelson, 1983; Ma et al., 1994), or ‘torquechange’ (integral of squared joint torque derivatives; Uno et al., 1989). But many such costs produce nonsmooth velocity profiles (Figure 1B), or lack physiological justification, or both. Some studies have included explicit models of muscle energy expenditure, but without testing such costs physiologically (Kistemaker et al., 2010). There is good evidence that energy expenditure is relevant to reaching (Shadmehr et al., 2016), but no physiologically tested cost function predicts the velocity profiles of reaching as well as the minimum variance hypothesis.
The issue could be that metabolic energy expenditure for muscle is not quantitatively wellunderstood. Energy is expended in proportion to force and time (‘tensiontime integral’) in isometric conditions (Crow and Kushmerick, 1982), and in proportion to mechanical work in steady work conditions (Barclay, 2015; Margaria, 1976), neither of which apply well to reaching. There is, however, a lessappreciated cost for muscles that increases with brief bursts of intermittent or cyclic action. It is due to active calcium transport to activate/deactivate muscle, observed in both isolated muscle preparations (Hogan et al., 1998) and whole organisms (Bergström and Hultman, 1988). It has also been hypothesized quantitatively (Doke and Kuo, 2007), as a cost per contraction roughly proportional to the rate of change of muscle force. Such a cost has indeed been observed in a variety of lower extremity tasks (Dean and Kuo, 2011; Doke et al., 2005; van der Zee and Kuo, 2020). It has a mechanistic and physiological basis, is supported by experimental evidence, and would appear to penalize jerky motions due to their energetic cost. What is not known is whether this energetic cost can explain reaching.
We therefore tested whether there is an energetic basis for reaching movements (Figure 2). We did so by measuring oxygen consumption during steadystate cyclic reaching motions. The expectation was that the proposed forcerate cost would cost metabolic energy in excess of what could be explained by mechanical work. We next applied the empirically derived cost for both forcerate and work to an optimal control model of discrete, pointtopoint reaching, and tested whether it could predict the smooth, bellshaped velocities normally attributed to minimumvariance. If the proposed cost is observed as expected and predicts bellshaped profiles, it could potentially provide a reinterpretation of existing theories based on kinematics alone, and integrate energy expenditure into a general framework for planning reaching movements.
Results
Model optimization resulted in a prediction of the metabolic cost of cyclic reaching movements. With movement amplitude decreasing with movement frequency, metabolic cost was predicted to increase with movement frequency $f$ to the 5/2 power (Figure 3A). This is in proportion to the forcerate cost, also expected to increase with $f}^{5/2$ . We also expected a fixed metabolic cost for mechanical work, because these conditions result in fixed mechanical power across frequencies. The specific movement conditions needed to separate the costs of work and forcerate were as follows: movement amplitude decreasing according to ${f}^{3/2}$ (Figure 3B), joint torque increasing with $f}^{1/2$ (Figure 3C), and hand speed decreasing with ${f}^{1/2}$ (Figure 3D). Thus, even though mechanical power is expected to contribute substantially to metabolic cost, the forcerate cost can be tested for an increasing contribution to overall metabolic cost.
We found that the rate of metabolic energy expenditure increased substantially with movement frequency, even as the rate of mechanical work was nearly constant. We first confirmed that cyclic reaching was performed largely by sinusoidal motions at the shoulder, across all conditions (Figure 4). These were accompanied by approximately sinusoidal torque and power, and fairly consistent EMG profiles. Under such conditions, subjects expended more than triple (a factor of 3.56) the net metabolic power for about twice the frequency (a factor of 2.33), with 5.32 ± 2.73 W at the lowest frequency of 0.58 Hz, compared to 18.95 ± 6.02 W at the highest frequency of 1.36 Hz (Figure 5A). As predicted, metabolic rate increased approximately with ${f}^{5/2}$ (Equation 9; adjusted R^{2} = 0.50; p = 1e8; Figure 4a; Table 1).
Other aspects of the cyclic reaching task were as prescribed and intended (Figure 5B–E; Table 1). Reach amplitudes decreased according to the targets, approximately with ${f}^{3/2}$ (Figure 5B). Shoulder torque amplitude and endpoint speed also changed with respectively ${f}^{1/2}$ (Figure 5C; adjusted R^{2} = 0.52; p = 4e9) ${f}^{1/2}$ (Figure 5D; R^{2} = 0.93; p = 7e29). Consistent with the fixedpower condition, average positive mechanical power did not change significantly with frequency $f$ (Figure 5E; slope = 0.081 ± 0.13 W.s^{–1}; mixedeffects linear model with a fixed effect proportional to ${f}^{1}$ , and individual subject offsets as random effects; p = 0.16). Amplitude of torque rate per time increased more sharply, approximately with ${f}^{5/2}$ (Figure 5E), with coefficient $b$ of 78.93 ± 6.55 CI, 95% confidence interval.
The net metabolic cost was also consistent with the hypothesized sum of separate terms for positive mechanical work and forcerate (Figure 6). This is demonstrated with metabolic power as a function of movement frequency $f$, and as a function of forcerate per time. With positive mechanical work at a fixed rate of about 1.2 W, the metabolic cost of work was expected to be constant at approximately 5 W. The difference between net metabolic rate and the constant work cost yielded the remaining forcerate metabolic power, increasing approximately with ${f}^{5/2}$ (Figure 6A). This same forcerate cost could also be expressed as a linear function of the empirical torque rate per time, with an estimated coefficient of ${c}_{t}$ = 8.5e2 (Figure 6B; see Equation 11); joint torque is treated as proportional to muscle force, assuming constant shoulder moment arm. In terms of proportions, mechanical power accounted for about 94% of the net metabolic cost at 0.58 Hz, and 26% at 1.36 Hz. Correspondingly, forcerate accounted for about 6% and 74% of net metabolic rate at the two respective frequencies.
Muscle EMG amplitudes increased with movement frequency (Figure 7). Deltoid and pectoralis both increased approximately with ${f}^{3/2}$ (pectoralis: R^{2} = 0.65; p = 1.1e6; deltoid: R^{2} = 0.56; p = 1.5e5), as did the cocontraction index (R^{2} = 0.58; p = 0.0009). This was consistent with expectations of muscle activation increasing faster than torque for increasing movement frequencies.
Crossvalidation of metabolic cost during cyclic reaching
Separate crossvalidation trials agreed well with forcerate coefficients. The second group of subjects moved with slightly increasing mechanical power, and slightly higher metabolic cost (Figure 8). But applying the cost coefficient ${c}_{t}$ derived from the primary experiment, the model (Equations 1; 10) was nevertheless able to predict crossvalidation costs reasonably well (R^{2} = 0.42; p = 2.7e6).
Passive elastic energy storage during cyclic reaching
The estimated natural frequency of cyclic arm motions was 2.83 ± 0.56 Hz. This suggests a rotational stiffness about the shoulder joint of about 250 N·m·rad^{–1}, if series elasticity were assumed for shoulder muscles. With passive elastic energy storage, the average positive mechanical power of muscle fascicles would decrease slightly, from about 0.5 W per arm to 0.33 W. Thus, series elasticity would cause active mechanical power to decrease with movement frequency, as energy expenditure increased.
Hilltype model does not predict experimentally observed energy cost
The Hilltype model’s predicted net energy cost increased approximately linearly with movement frequency, from 33 W to 47 W. The model dramatically overpredicted the net metabolic cost for all movements (by up to a factor of 6.2), and metabolic cost rose across frequency by less than half as found experimentally (a factor of 1.42 vs. empirical 3.56). Current musculoskeletal models do not accurately predict the cost of cyclic reaching.
Forceratedependent cost predicts pointtopoint reaching motions and durations
We applied the energy cost from cyclic reaching to predict discrete, pointtopoint reaching (Figure 9) of fixed and free durations. The prediction from trajectory optimization (Equation 11) was for a standard movement of fixed duration and distance (0.65 s and 30 cm, respectively; Harris and Wolpert, 1998), using the energy cost coefficients ${c}_{W}$ and ${c}_{t}$ derived from the primary experiment. This yielded bellshaped velocities (Figure 9) similar to the minimum variance model and to empirical data (Harris and Wolpert, 1998). Also compared were minimum torquerate (Uno et al., 1989), and minimum activation squared using a Hilltype muscle model (Kistemaker et al., 2014). Each objective approximately reproduces the empirical bellshaped profile, with metabolic cost (Equation 1), torqueratesquared, and activationsquared all having correlation coefficients above 0.8 (0.99, 0.98, and 0.82, respectively). The metabolic energy cost including empirically tested work and forcerate terms therefore predicts trajectories similar to other, nonenergetic costs proposed previously, and to human data.
Similar predictions were made for different distances, leaving duration unconstrained (Figure 10). The predicted, optimal durations increased with movement distance, roughly similar to human preferred durations (Reppert et al., 2018). The associated trajectories also retained the bellshaped velocities across all distances. The proposed metabolic energy cost, plus a penalty for long durations, therefore predicts both trajectories and durations roughly similar to human data.
Discussion
We tested whether the metabolic cost of reaching movements is predicted by the hypothesized energetic cost including forcerate. Our experimental data showed a cost increasing with movement frequency as predicted with forcerate, more so than did the mechanical work performed. The same cost model was also crossvalidated with a separate set of reaching movements, and predicts smooth reaching movements, similar to the minimum variance model. We interpret these findings as suggesting the forcerate hypothesis as an energetic basis for reaching movements.
The forcerate hypothesis explains the observed metabolic energy cost increases better than more conventionally recognized costs. For example, the cost of mechanical work alone cannot explain the higher cost at higher movement frequencies, because the rate of work remained fixed (Figure 5). A possible explanation is that the energetic cost per unit of work (${c}_{W}$ in Equation 1) could increase with faster movements, due to the muscle forcevelocity relationship (Barclay, 2015). But the conditions here actually yielded slower hand speeds with higher frequencies (Figure 5D), and thus cannot explain the higher cost. Nor were our results explained by a current musculoskeletal model (Umberger, 2010), which drastically overestimated the overall cost and underestimated the increases with movement frequency. The proposed forcerate hypothesis thus explains these data better than previous quantitative models or relationships.
The forcerate hypothesis was also consistent with three other observations: (1) electromyography, (2) crossvalidation, and (3) pointtopoint reaching. First, muscle EMGs increased more rapidly (approximately with ${f}^{3/2}$ ; Figure 7) with movement frequency than did joint torques (approximately with ${f}^{1/2}$ ; Figure 5C). The proposed mechanism is that brief bursts of activation require greater active calcium transport (and thus greater energy cost), because muscle force production has slower dynamics than muscle activation (van der Zee and Kuo, 2020). Second, we crossvalidated the primary experiment, by applying its cost coefficients (${c}_{t}$ and ${c}_{W}$ , Figure 6) to predict an independent set of conditions. We found good agreement between crossvalidation data and the forcerate prediction (Figure 8). The overall energy cost ($\dot{E}$ from Equation 1) depends on a particular combination of work, force, and movement frequency, yet only has one degree of freedom (${c}_{t}$). Third, the forcerate hypothesis also explains discrete, pointtopoint reaching. The characteristic bellshaped velocity profile is predicted by optimal control, using the cost coefficients derived from cyclic movements (Figure 9). Moreover, movement duration is predicted to increase with distance, approximately similar to human reaches (Reppert et al., 2018). These observations serve as tests of the forcerate hypothesis, independently predicted by a single model.
The forcerate cost is surely not the sole explanation for reaching. The optimal control approach has been used to propose a variety of abstract mathematical objective functions that can predict movement. But there may be multiple objectives that predict similar behavior. As such, careful experimentation (Harris and Wolpert, 1998; Kawato, 1999) was required to disambiguate minimumvariance from competing hypotheses such as minimumjerk and torquechange (Kawato, 1999). Similarly, the present study does not disambiguate forcerate from minimumvariance, since both predict similar pointtopoint movements. In fact, minimumvariance also has some dependency on effort, albeit implicitly, due to the mechanism of signaldependent noise (Harris and Wolpert, 1998). It also explains well the tradeoff between movement speed and endpoint accuracy, where energy expenditure is unlikely to be important. However, the ambiguity also means that both variance and forcerate could potentially contribute to movement. It is quite possible that minimum variance dominates for fast and accurate movements, and energy cost for the trajectory and duration of slower ones, with both contributing to a unified objective for reaching.
Effort objectives have long been considered potential counterparts to the kinematic performance objective. For example, the integrated squared muscle force or activation or torquechange all emphasize effort and arm dynamics as explicit features for reaching (Uno et al., 1989). Effort is also important for selection of feedback control gains (Kuo, 1995; Todorov and Jordan, 2002), adaptation of coordination (Emken et al., 2007), identification of control objectives from data (Vu et al., 2016), and determination of movement duration (Shadmehr et al., 2016). The problem is that these manifestations of effort are abstract constructs with limited physiological basis, justified mainly by their ability to reproduce bellshaped velocities through inverse optimization. However, multiple objectives can reproduce such velocities nonuniquely (Figure 9), making additional and independent tests important for disambiguating them. Accordingly, metabolic energy expenditure is a physiological, independently testable measure of effort.The change in metabolic cost during adaptation (Huang et al., 2012) and the effect of metabolic state on reaching patterns (Taylor and Faisal, 2018) strongly suggest a role for energy in reaching. The present study offers a means to incorporate a truly physiological effort cost into optimal control predictions for smooth and economical movements.
There is a measurable and nontrivial energetic cost for cyclic reaching. Even though the arms were supported by a planar manipulandum, at a movement frequency of 1.5 Hz, we observed a net metabolic rate of about 19 W. For comparison, the difference in cost between continuous standing and sitting is about 24 W (Mansoubi et al., 2015), making the reaching task nearly as costly as standing up. For each halfcycle reaching action, analogous to a pointtopoint movement, the metabolic cost was about 3.5 J per arm. The cost may not be particularly high, but the nervous system may nonetheless prefer economical ways to accomplish a reaching task.
There are several limitations to this study. One is that energetic cost was experimentally measured for the whole body, and not distinguished at the level of the muscle. Forcerate was also estimated from joint torque and not from actual muscle forces. We therefore cannot eliminate other physiological processes as possible contributions to the observed energy cost. In addition, the hypothesized cost ($\dot{E}}_{FR$) is thus far a highly simplified, conceptual model for a muscle activation cost. More precise mechanistic predictions of this cost would be facilitated with specific models for muscle activation, myoplasmic calcium transport, and force delivery are needed (e.g. Baylor and Hollingworth, 1998; Ma and Zahalak, 1991). We also tested the forcerate cost in continuous reaches of fixed work, predominantly by the shoulder, and with the arms supported against gravity. Further studies are needed to test more ecological movements such as discrete reaching in arbitrary directions, and while interacting with objects.
The forcerate hypothesis suggests a substantial role for effort or energy expenditure in upper extremity reaching movements. Some form of effort cost is often employed to examine selection of feedback gains or muscle forces, and even generally expected for optimal control problems where maximaleffort actions are to be avoided (Bryson and Ho, 1975). And in the experimental realm, energy expenditure is regarded as a major factor in animal life and behavior (Alexander, 1996), even to the small scale of a single neural action potential (Sterling and Laughlin, 2017). Under the minimumvariance hypothesis alone, reaching seems unusually dominated by kinematics. But our results suggest that metabolic energy expenditure may have been overshadowed by the minimumvariance hypothesis, because it makes similar predictions for pointtopoint movements. There is need to both quantify and test the forcerate hypothesis more specifically, perhaps in combination with minimumvariance. Nonetheless, there is a meaningful energetic cost to reaching that can also explain the smoothness of reaching motions.
Materials and methods
There were three main components to this study: (1) a simple cost model, (2) a set of human subjects experiments with cyclic reaching, and (3) an application of the model to predict discrete reaching trajectories. The model predicts that metabolic cost should increase with the hypothesized forcerate measure, particularly for faster frequencies of movement. Key to the experiment (Figure 2) was to isolate the hypothesized forcerate cost, by applying combinations of movement amplitude and frequency that control for the cost of mechanical work. This primary test was accompanied by a secondary, crossvalidation test, with different combinations of movement amplitude and frequency. Finally, we applied this same forcerate cost to the prediction of discrete reaching movement trajectories. This was to test whether the energetic cost, derived from continuous, cyclic reaching movements, could also predict the smooth, discrete motions often found in the literature.
Model predictions for forcerate hypothesis
Request a detailed protocolWe hypothesized that the energetic cost for reaching includes a cost for performing mechanical work, and another for the rate of force production. These costs are implemented on a simple, twosegment model of arm dynamics, actuated with joint torques. These torques perform work on the arm, at an approximately proportional energetic cost (Margaria, 1976) attributed to actinmyosin crossbridge action (Barclay, 2015). The forcerate cost is hypothesized to result from rapid activation and deactivation of muscle, increasing with the amount of force and inversely with the time duration. It is attributable to active transport of myoplasmic calcium (Bergström and Hultman, 1988; Hogan et al., 1998), where more calcium is required for higher forces and/or shorter time durations, hence forcerate (Doke and Kuo, 2007).
For the simple motion employed here, the prediction of the total metabolic energy $E$ consumed per movement is the sum of costs for work and forcerate,
where W is the positive mechanical work per movement, ${c}_{W}$ the metabolic cost per unit of work, and ${E}_{FR}$ is the hypothesized forcerate cost
where $\dot{F}$ denotes the amplitude of forcerate (timederivative of muscle force) per movement, and ${c}_{f}$ is the energetic cost for forcerate. This cost is to be distinguished from the earlier torquechange hypothesis (Uno et al., 1989), which integrates a sumsquared forcerate over time, and which had no hypothesized relationship to metabolic energetic cost. During cyclic reaching, the peak forcerate $\dot{F}$ increases with both force amplitude and the frequency of cyclic movement. Here, positive and negative work are performed in equal magnitudes, and so their respective costs are lumped together into a single proportionality ${c}_{W}$ . We assigned ${c}_{W}$ a value of 4.2, from empirical mechanical work efficiencies of 25% for positive work and –120% for negative work (Margaria et al., 1963).
The work and force of the cyclic reaching movements about the shoulder are predicted by a simple model of arm dynamics. In the horizontal plane of a manipulandum supporting the arm,
with shoulder angle $\theta \left(t\right)$ , shoulder torque ${\rm T}\left(t\right)$ (treated as proportional to muscle force), and rotational inertia $I$. Applying sinusoidal motion at amplitude $A$ and movement frequency $f$ (in cycle/s),
The torque is therefore
and amplitude of mechanical power $\dot{W}$
We apply a particular movement condition, termed the fixed power constraint (Figure 2A), where the average positive mechanical power is kept fixed across movement frequencies, so that the hypothesized forcerate cost will dominate energetic cost (Figure 3A). This is achieved by constraining amplitude to decrease with movement frequency (Figure 3B),
This fixed power condition also means that hand (endpoint) speed, proportional to $\dot{\theta}$ , should have amplitude varying with ${f}^{1/2}$ , and torque amplitude with ${f}^{1/2}$ (Figure 3C and D).
Applying fixed power to the forcerate cost yields the energetic cost prediction. Torquerate amplitude $\dot{T}$ with Equation 2 and Equation 7 yields
where $b$ is a constant coefficient. The proportional cost per movement is therefore (Equation 2)
where ${c}_{f}$ is a constant coefficient across conditions. Experimentally, it is most practical to measure metabolic power $\dot{E}$ (Figure 3a) in steady state. Multiplying $E$ (cost per movement, Equation 2) by $f$ (movement cycles per time) yields the predicted proportionality for average metabolic power,
The net metabolic rate $\dot{E}$ is expected to increase similarly, but with an additional offset for the constant work cost $\dot{E}}_{W$ under the fixedpower constraint (Figure 3A). Finally, the metabolic energy per time associated with forcerate would be expected to increase directly with torquerate per time $f\cdot \dot{T}$ ,
where movement frequency $f$ represents cycles per time, and coefficient ${c}_{t}$ is equal to ${c}_{f}$ divided by $b$.
This forcerate coefficient is not specific to cyclic movements alone. The general metabolic cost model (Equations 1; 2) is potentially applicable to pointtopoint and other motions, with different amounts of work and forcerate. The forcerate cost $\dot{E}}_{FR$ is independent of mechanical work, and may be predicted using the cost coefficient derived from cyclic experiments. The model may therefore make testable predictions of energetic cost even for movements that are acyclic and not constrained to fixed power.
Experiments
Request a detailed protocolWe measured the metabolic power expended by healthy adults ($N$ = 10) performing cyclic movements at a range of speeds but fixed power (Equation 7). We tested whether metabolic power would increase with the hypothesized forcerate cost $\dot{E}}_{FR$, in amount not explained by mechanical work. We also characterized the mechanics of the task in terms of kinematics, shoulder torque amplitude, and forcerate for shoulder muscles. These were used to test whether the mechanics were consistent with the model of arm dynamics, and whether forcerate increased as predicted (Equations 7–10). We first describe a primary experiment with fixed power conditions, followed by an additional crossvalidation experiment. All subjects provided written informed consent, as approved by University of Calgary Ethics board.
Subjects performed cyclic bimanual reaching movements in the horizontal plane, with the arms supported by a robotic exoskeleton (KINARM, BKIN Technologies, Inc). The movements were cyclic and bimanually symmetrical to induce steady energy expenditure sufficient to be distinguished easily by expired gas respirometry. The exoskeleton was used to counteract gravity in a lowfriction environment (with no actuator loads), and to measure kinematics, from which shoulder and elbow joint torques were estimated using inverse dynamics. Subjects were asked to move each arm between a pair of targets, reachable mostly by mediolateral shoulder motion, with relatively little elbow motion (less than 1 deg average excursion across all conditions). A single visual cursor (5 mm in diameter) was displayed for the right hand, along with one pair of visual targets (circles 2.5 cm in diameter), all optically projected onto the movement plane. To encourage equal bimanual motion, the cursor’s position was not for one hand alone, but rather computed as an average of right and left arm joint angles, making it insufficient to move one arm alone.
Timing was set with a metronome beat for reaching each of the two targets, and amplitude by adjusting the distance between the targets. Prior to data collection, subjects completed a 20min familiarization session (up to 48 hr before the experiment) where they received task instructions and briefly practiced each of the tasks.
The primary experiment was to test for the predicted energetic cost for reaching, in five conditions of cyclic reaching at increasing frequency and decreasing amplitude. The frequencies were 0.58, 0.78, 0.97, 1.17, 1.36 Hz, and amplitudes were 12.5, 8, 5.8, 4.4, 3.5°, respectively. These cyclic movements were chosen to be of moderate hand speed, with peak speeds between 0.4 and 0.6 m/s.
We estimated metabolic rate using expired gas respirometry. Subjects performed each condition for 6 min, analyzing only the final 3 min of data for steadystate aerobic conditions, with standard equations used to convert O2 and CO2 rates into metabolic power (Brockway, 1987).We report net metabolic rate $\dot{\mathrm{E}}$ for bimanual movement, defined as gross rate minus the cost of quiet sitting (obtained in a separate trial, 98.6 ± 11.5 W, mean ± s.d.).
We also recorded arm segment positions and electromyographs simultaneously at 1000 Hz. These included kinematics from the robot, and electromyographs (EMGs) from four muscles (pectoralis lateral, posterior deltoid, biceps, triceps) in a subset of our subjects (five subjects in primary experiment, five in crossvalidation). The EMGs were used to characterize muscle activation and coactivation.
The metabolic cost hypothesis was tested using a linear mixedeffects model of net metabolic power. This included the hypothesized forcerate cost (Equation 10) as a fixed effect, yielding coefficient ${c}_{f}$ for the forcerate term proportional to ${f}^{5/2}$ . A constant offset was included for each subject as a random effect. In addition, the forcerate cost $\dot{E}}_{FR$ was estimated by subtracting the fixed mechanical work cost $\dot{E}}_{W$ from net metabolic power $\dot{E}$ , and then compared against torque rate amplitude per time (Equation 11). Sample size was appropriate to yield a statistical power of 0.99 based on statistical characteristics of previous reaching studies of metabolic cost (Wong et al., 2018). Both the main experiment and cross validation experiment were performed a single time.
We tested expectations for movement amplitude and other quantities from kinematic data. Hand velocity was filtered using a bidirectional lowpass Butterworth filter (first order, 12 Hz cutoff). Shoulder and elbow torques were computed using inverse dynamics, based on KINARM dynamics (BKIN Technologies, Kingston), and subjectspecific inertial parameters (Winter, 1990). The approximate rotational inertia of a single human arm and exoskeleton about the shoulder was estimated as 0.9 kg⋅m^{2}. The positive portion of bimanual mechanical power was integrated over total movement duration and divided by cycle time, yielding average positive mechanical power. Linear mixedeffects models were used to characterize the powerlaw relations for mechanical power, movement amplitude, movement speed, torque amplitude, and torque rate amplitude (Figure 3). The latter was estimated by integrating the torque rate amplitude per time (Equation 11) for each joint, and then summing the two. The forcerate hypothesis was also tested by comparing $\dot{E}}_{FR$ with torque rate per time (Figure 3A), assuming torque is proportional to muscle force.
Electromyographs were used to test for changes in muscle activation and coactivation. Data were meancentred, lowpass filtered (bidirectional, second order, 30 Hz cutoff), rectified, and lowpass filtered again (Roberts and Gabaldón, 2008), from which the EMG amplitude was measured at peak and then normalized to each subject’s maximum EMG across the five conditions. We expected EMG amplitude to increase with muscle activation, with simplified firstorder dynamics between activation (EMG) and muscle force production (van der Zee and Kuo, 2020). This treats the ratelimiting step of force production as a lowpass filter, so that greater activation or EMG amplitudes would be needed to produce a given force at higher waveform frequencies. The firstorder dynamics mean that EMG would be expected to increase with torque rate ${f}^{3/2}$ rather than torque, as tested with a linear mixedeffects model. We also computed a cocontraction index for EMG, in which the smallest value of antagonist muscle pairs was computed over time, and then integrated for comparison across conditions (Gribble et al., 2003). All statistical tests were performed with threshold for significance of p < 0.05.
As a crossvalidation test of the forcerate cost, we tested the generalizability of coefficient ${c}_{t}$ against a second set of conditions with a separate set of subjects (also $N$ = 10; two subjects participated in both sets). The conditions were slightly different: frequencies ranging 0.67–1.3 Hz and amplitudes 12.5–4.42°, which resulted in higher mechanical work and forcerate. We applied the model (Equation 1, Equation 11) and ${c}_{t}$ coefficient identified from the primary experiment to the crossvalidation conditions. As a further test of the central hypothesis, we expected the model to roughly predict trends regarding mechanical and metabolic rates for the crossvalidation conditions.
Estimation of elastic energy storage in shoulder muscles
Request a detailed protocolWe estimated the resonant frequency of cyclic reaching, to account for possible series elasticity in shoulder actuation. Series elasticity could potentially store and return energy and thus require less mechanical work from muscle fascicles. We estimated this contribution from resonant frequency, obtained by asking subjects to swing their arms back and forth rapidly at large amplitudes (at least 15°) for 20 s, and determining the frequency of peak power (PWelch in Matlab). We then used this to estimate torsional series elasticity, and the passive contribution to mechanical power.
Musculoskeletal model to simulate experimental conditions
Request a detailed protocolWe tested whether a Hilltype musculoskeletal model could explain the metabolic cost of cyclic reaching. The hypothesized forcerate is not explicitly included in current models of energy expenditure, and would not be expected to explain the experimental metabolic cost. We therefore tested an energetics model available in the OpenSim modeling system (Seth et al., 2018; Uchida et al., 2016; Umberger, 2010), applied to a model of arm dynamics with six muscles (Kistemaker et al., 2014). We used trajectory optimization to determine muscle states and stimulations, with torques from inverse dynamics as a tracking reference. Optimization was performed using TOMLAB/SNOPT (Tomlab Software AB, Sweden; Gill et al., 2002), to minimize meansquared torque error, squared stimulation level, and squared stimulation rate. The optimized muscle states were then fed into the metabolic cost model (Umberger, 2010).
Model of pointtopoint reaching movements
Request a detailed protocolThe forcerate cost hypothesis was also used to predict pointtopoint reaching movements and their durations. Here, we form an overall objective function $J$ that includes the energetic cost per movement $E$ (Equation 1) as a physiological effort term, and a simple penalty proportional to movement duration ${t}_{f}$ . This may be regarded as an adaptation of Shamehr et al.’s (2016) hypothesis that duration is a tradeoff between effort and (the inverse of) a temporally discounted reward. The overall objective is thus:
where the energy expenditure is expressed a function of joint angle $\theta \left(t\right)$ and torque $\tau \left(t\right)$ trajectories, and duration is penalized with proportionality $k$. Minimization of this objective can predict pointtopoint reaching trajectories both of fixed duration (by constraining ${t}_{f}$) and of free duration. We show that this objective predicts smooth, bellshaped velocity profiles similar to minimumvariance, as well as durations increasing with movement distance.
We used this objective in trajectory optimization of planar, twojoint reaching movements. For fixed duration ${t}_{f}$ , the objective $J$ depends only on energetic cost $E$, with the mechanical work and forcerate terms expressed as a time integral for both joints:
with joint torques ${\tau}_{i}$ ($i=\mathrm{1,2}$ for elbow and shoulder, respectively). To compare with the minimum variance model of Harris and Wolpert, 1998, we used a similar straight reaching movement of amplitude 30 cm and duration ${t}_{f}$ of 650 ms. We also used the empirically estimated ${c}_{t}$ from the cyclic reaching experiment (assuming the same coefficient for both shoulder and elbow), along with a pointtopoint constraint to have zero initial and final acceleration of the hand, again using TOMLAB. The predicted hand velocity trajectory was qualitatively compared with the empirical bellshaped velocity from minimum variance (Harris and Wolpert, 1998).
We also examined movements of unconstrained duration, which have been shown to take longer with greater movement distance (Reppert et al., 2018). We selected $k$ so that the average movement speed was approximately equal to the average preferred movement duration across the empirically measured reach speeds ($k$ = 25 J/s). We qualitatively compared the trajectories and durations from model against data for movements ranging 8–40 cm (Reppert et al., 2018).
Data availability
Data has been deposited to Dryad Digital Repository, accessible here: doi:http://doi.org/10.5061/dryad.qfttdz0gn.

Dryad Digital RepositoryThe energetic basis for smooth human arm movements.https://doi.org/10.5061/dryad.qfttdz0gn
References

A minimum energy cost hypothesis for human arm trajectoriesBiological Cybernetics 76:97–105.https://doi.org/10.1007/s004220050324

Model of sarcomeric Ca2+ movements, including ATP Ca2+ binding and diffusion, during activation of frog skeletal muscleThe Journal of General Physiology 112:297–316.https://doi.org/10.1085/jgp.112.3.297

Energy cost and fatigue during intermittent electrical stimulation of human skeletal muscleJournal of Applied Physiology 65:1500–1505.https://doi.org/10.1152/jappl.1988.65.4.1500

Derivation of formulae used to calculate energy expenditure in manHuman Nutrition. Clinical Nutrition 41:463–471.

Chemical energetics of slow and fasttwitch muscles of the mouseThe Journal of General Physiology 79:147–166.https://doi.org/10.1085/jgp.79.1.147

Energetic costs of producing muscle work and force in a cyclical human bouncing taskJournal of Applied Physiology 110:873–880.https://doi.org/10.1152/japplphysiol.00505.2010

The coordination of movement: optimal feedback control and beyondTrends in Cognitive Sciences 14:31–39.https://doi.org/10.1016/j.tics.2009.11.004

Mechanics and energetics of swinging the human legThe Journal of Experimental Biology 208:439–445.https://doi.org/10.1242/jeb.01408

Energetic cost of producing cyclic muscle force, rather than work, to swing the human legThe Journal of Experimental Biology 210:2390–2398.https://doi.org/10.1242/jeb.02782

Motor adaptation as a greedy optimization of error and effortJournal of Neurophysiology 97:3997–4006.https://doi.org/10.1152/jn.01095.2006

The coordination of arm movements: an experimentally confirmed mathematical modelThe Journal of Neuroscience 5:1688–1703.https://doi.org/10.1523/JNEUROSCI.050701688.1985

Snopt  an SQP algorithm for nonlinear optimizationSiam Journal of Optimization 12:979–1006.

Role of cocontraction in arm movement accuracyJournal of Neurophysiology 89:2396–2405.https://doi.org/10.1152/jn.01020.2002

Evidence for hyperbolic temporal discounting of reward in control of movementsThe Journal of Neuroscience 32:11727–11736.https://doi.org/10.1523/JNEUROSCI.042412.2012

Contraction duration affects metabolic energy cost and fatigue in skeletal muscleThe American Journal of Physiology 274:E397–E402.https://doi.org/10.1152/ajpendo.1998.274.3.E397

Reduction of metabolic cost during motor learning of arm reaching dynamicsThe Journal of Neuroscience 32:2182–2190.https://doi.org/10.1523/JNEUROSCI.400311.2012

Internal models for motor control and trajectory planningCurrent Opinion in Neurobiology 9:718–727.https://doi.org/10.1016/s09594388(99)000288

The central nervous system does not minimize energy cost in arm movementsJournal of Neurophysiology 104:2985–2994.https://doi.org/10.1152/jn.00483.2010

The cost of moving optimally: kinematic path selectionJournal of Neurophysiology 112:1815–1824.https://doi.org/10.1152/jn.00291.2014

An optimal control model for analyzing human postural balanceIEEE Transactions on BioMedical Engineering 42:87–101.https://doi.org/10.1109/10.362914

A distributionmoment model of energetics in skeletal muscleJournal of Biomechanics 24:21–35.https://doi.org/10.1016/00219290(91)90323f

The Optimal Control of a Movement of the Human Upper Extremity 1IFAC Proceedings Volumes 27:455–460.https://doi.org/10.1016/S14746670(17)463026

Energy cost of runningJournal of Applied Physiology 18:367–370.https://doi.org/10.1152/jappl.1963.18.2.367

Physical principles for economies of skilled movementsBiological Cybernetics 46:135–147.https://doi.org/10.1007/BF00339982

Movement vigor as a traitlike attribute of individualityJournal of Neurophysiology 120:741–757.https://doi.org/10.1152/jn.00033.2018

Interpreting muscle function from EMG: lessons learned from direct measurements of muscle forceIntegrative and Comparative Biology 48:312–320.https://doi.org/10.1093/icb/icn056

OpenSim: Simulating musculoskeletal dynamics and neuromuscular control to study human and animal movementPLOS Computational Biology 14:e1006223.https://doi.org/10.1371/journal.pcbi.1006223

A Representation of Effort in DecisionMaking and Motor ControlCurrent Biology 26:1929–1934.https://doi.org/10.1016/j.cub.2016.05.065

Movement Vigor as a Reflection of Subjective Economic UtilityTrends in Neurosciences 42:323–336.https://doi.org/10.1016/j.tins.2019.02.003

The variation of hand tremor with force in healthy subjectsThe Journal of Physiology 191:699–711.https://doi.org/10.1113/jphysiol.1967.sp008276

Optimal feedback control as a theory of motor coordinationNature Neuroscience 5:1226–1235.https://doi.org/10.1038/nn963

Optimality principles in sensorimotor controlNature Neuroscience 7:907–915.https://doi.org/10.1038/nn1309

Stance and swing phase costs in human walkingJournal of the Royal Society, Interface 7:1329–1340.https://doi.org/10.1098/rsif.2010.0084

The high energetic cost of rapid force development in cyclic muscle contractionJournal of Experimental Biology 10:266965.https://doi.org/10.1101/2020.08.25.266965

BookThere Is an Energetic Cost to Movement Jerk in Human ReachingSociety for Neuroscience.
Article and author information
Author details
Funding
University of Calgary (Benno Nigg Chair)
 Arthur D Kuo
Natural Sciences and Engineering Research Council of Canada
 Arthur D Kuo
Alberta Health Services
 Arthur D Kuo
Natural Sciences and Engineering Research Council of Canada
 Tyler Cluff
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
This work was funded by NSERC (Discovery and CRC Tier 1), Dr. Benno Nigg Research Chair, and Alberta Health Trust. We acknowledge Dinant Kistemaker for sharing simulation code for Hilltype muscle model energetics.
Ethics
Human subjects: Informed consent was obtained from all subjects and the Health Research Ethics Board approved of all procedures (REB181521).
Copyright
© 2021, Wong et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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