Upper extremity reaching movements are characterized by a stereotypical, bell-shaped 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 bell-shaped 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.

Figure 1

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The kinematic goal for accuracy may be expressed quantitatively as minimization of the final endpoint position variance (Harris and Wolpert, 1998). Non-smooth motions introduce inaccuracy because motor noise increases with motor command amplitude, a phenomenon termed signal-dependent 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 (‘bang-bang 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 minimum-variance 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 ‘torque-change’ (integral of squared joint torque derivatives; Uno et al., 1989). But many such costs produce non-smooth 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 well-understood. Energy is expended in proportion to force and time (‘tension-time 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 less-appreciated 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 steady-state cyclic reaching motions. The expectation was that the proposed force-rate cost would cost metabolic energy in excess of what could be explained by mechanical work. We next applied the empirically derived cost for both force-rate and work to an optimal control model of discrete, point-to-point reaching, and tested whether it could predict the smooth, bell-shaped velocities normally attributed to minimum-variance. If the proposed cost is observed as expected and predicts bell-shaped profiles, it could potentially provide a re-interpretation of existing theories based on kinematics alone, and integrate energy expenditure into a general framework for planning reaching movements.

Figure 2

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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 force-rate cost, also expected to increase with ${\displaystyle f^{5/2}}$ . 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 force-rate were as follows: movement amplitude decreasing according to $f^{-3/2}$ (Figure 3B), joint torque increasing with ${\displaystyle f^{1/2}}$ (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 force-rate cost can be tested for an increasing contribution to overall metabolic cost.

Figure 3

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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 R² = 0.50; p = 1e-8; Figure 4a; Table 1).

Figure 4

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Figure 5

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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 R² = 0.52; p = 4e-9) $f^{-1/2}$ (Figure 5D; R² = 0.93; p = 7e-29). Consistent with the fixed-power condition, average positive mechanical power did not change significantly with frequency $f$ (Figure 5E; slope = 0.081 ± 0.13 W.s^–1; mixed-effects 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 force-rate (Figure 6). This is demonstrated with metabolic power as a function of movement frequency $f$, and as a function of force-rate 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 force-rate metabolic power, increasing approximately with $f^{5/2}$ (Figure 6A). This same force-rate cost could also be expressed as a linear function of the empirical torque rate per time, with an estimated coefficient of $c_t$ = 8.5e-2 (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, force-rate accounted for about 6% and 74% of net metabolic rate at the two respective frequencies.

Figure 6

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Muscle EMG amplitudes increased with movement frequency (Figure 7). Deltoid and pectoralis both increased approximately with $f^{3/2}$ (pectoralis: R² = 0.65; p = 1.1e-6; deltoid: R² = 0.56; p = 1.5e-5), as did the co-contraction index (R² = 0.58; p = 0.0009). This was consistent with expectations of muscle activation increasing faster than torque for increasing movement frequencies.

Figure 7

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Cross-validation of metabolic cost during cyclic reaching

Separate cross-validation trials agreed well with force-rate 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 cross-validation costs reasonably well (R² = 0.42; p = 2.7e-6).

Figure 8

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Force-rate-dependent cost predicts point-to-point reaching motions and durations

We applied the energy cost from cyclic reaching to predict discrete, point-to-point 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 bell-shaped velocities (Figure 9) similar to the minimum variance model and to empirical data (Harris and Wolpert, 1998). Also compared were minimum torque-rate (Uno et al., 1989), and minimum activation squared using a Hill-type muscle model (Kistemaker et al., 2014). Each objective approximately reproduces the empirical bell-shaped profile, with metabolic cost (Equation 1), torque-rate-squared, and activation-squared all having correlation coefficients above 0.8 (0.99, 0.98, and 0.82, respectively). The metabolic energy cost including empirically tested work and force-rate terms therefore predicts trajectories similar to other, non-energetic costs proposed previously, and to human data.

Figure 9

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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 bell-shaped 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.

Figure 10

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We tested whether the metabolic cost of reaching movements is predicted by the hypothesized energetic cost including force-rate. Our experimental data showed a cost increasing with movement frequency as predicted with force-rate, more so than did the mechanical work performed. The same cost model was also cross-validated 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 force-rate hypothesis as an energetic basis for reaching movements.

The force-rate 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 force-velocity 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 force-rate hypothesis thus explains these data better than previous quantitative models or relationships.

The force-rate hypothesis was also consistent with three other observations: (1) electromyography, (2) cross-validation, and (3) point-to-point 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 cross-validated 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 cross-validation data and the force-rate prediction (Figure 8). The overall energy cost (${\displaystyle \dot{E}}$ 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 force-rate hypothesis also explains discrete, point-to-point reaching. The characteristic bell-shaped 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 force-rate hypothesis, independently predicted by a single model.

The force-rate 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 minimum-variance from competing hypotheses such as minimum-jerk and -torque-change (Kawato, 1999). Similarly, the present study does not disambiguate force-rate from minimum-variance, since both predict similar point-to-point movements. In fact, minimum-variance also has some dependency on effort, albeit implicitly, due to the mechanism of signal-dependent noise (Harris and Wolpert, 1998). It also explains well the trade-off between movement speed and endpoint accuracy, where energy expenditure is unlikely to be important. However, the ambiguity also means that both variance and force-rate 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 torque-change 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 bell-shaped velocities through inverse optimization. However, multiple objectives can reproduce such velocities non-uniquely (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 non-trivial 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 half-cycle reaching action, analogous to a point-to-point 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. Force-rate 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 (${\displaystyle {\dot{E}}_{FR}}$) 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 force-rate 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 force-rate 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 maximal-effort 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 minimum-variance hypothesis alone, reaching seems unusually dominated by kinematics. But our results suggest that metabolic energy expenditure may have been over-shadowed by the minimum-variance hypothesis, because it makes similar predictions for point-to-point movements. There is need to both quantify and test the force-rate hypothesis more specifically, perhaps in combination with minimum-variance. Nonetheless, there is a meaningful energetic cost to reaching that can also explain the smoothness of reaching motions.

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 force-rate measure, particularly for faster frequencies of movement. Key to the experiment (Figure 2) was to isolate the hypothesized force-rate 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, cross-validation test, with different combinations of movement amplitude and frequency. Finally, we applied this same force-rate 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.

Data has been deposited to Dryad Digital Repository, accessible here: doi:http://doi.org/10.5061/dryad.qfttdz0gn.

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Decision letter after peer review:

Thank you for submitting your article "The energetic basis for smooth human arm movements" for consideration by eLife. Your article has been reviewed by 2 peer reviewers, one of whom is a member of our Board of Reviewing Editors, and the evaluation has been overseen by Richard Ivry as the Senior Editor. The following individual involved in review of your submission has agreed to reveal their identity: Mazen Al Borno (Reviewer #2).

The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this to help you prepare a revised submission.

Essential revisions:

1) The authors claim to present a theory of movement control that unifies smoothness and duration. They currently only show that their metabolic cost model can explain smoothness. However, their metabolic cost model on its own cannot explain the speed-accuracy tradeoff and minimum variance on its own cannot explain duration. Thus to support a unifying theory, they need to show that their metabolic cost model and minimum variance combined can explain all three movement properties: smoothness, the speed-accuracy tradeoff and duration. They also need to show that this new unifying model performs better than models using a combination of minimum variance and other effort models (such as mech work, hill muscle models, and torque-change). Alternatively, instead of this additional analysis, the authors could temper their claims of a unifying theory and instead focus on their result that smoothness can also be explained by energy minimization. Although a weaker finding, this is also a an important contribution to the field.

2) A comparison of the predicted velocity profiles by their metabolic model and the predictions of models based on squared muscle activations, torque rate, and Hill-type muscle model is needed.

3) To support their claim that energy explains duration, the authors should demonstrate the ability of their metabolic model to explain movement duration, and compare its predictions to the predictions of other effort models (mechanical work, squared muscle activations, torque rate, Hill-type muscle models). This analysis should also include whether their metabolic model predicts a non-zero metabolically optimal movement duration and if so, how the metabolically optimal duration scales with movement distance.

Reviewer #1 (Recommendations for the authors):

1. For the predictions of the bell-shaped velocity profile, it is not clear how the effort cost was represented in the cost function. Equation (1) represents the effort cost per movement, and even Equation (9) seems to represent the average energy per time associated with force-rate, since it is a function of the frequency of the contraction and the amplitude of the torque rate of the contraction. Thus, there is no expression for effort cost as a function of instantaneous time (i.e. similar to effort models based on torque rate squared or force squared or activation squared), rather it is calculated based on the total mechanical work and the frequency (or duration) of the movement. It would help to have the authors clarify how effort was represented in their optimization approach.

2. Additional figures should be provided plotting individual and average subject trajectory data and emg data across the different conditions. A figure of the experimental setup and visual feedback should also be provided. It is unclear how a single cursor was used to show the average joint angles.

3. As written, it seems that Equations 7 and 9 are dependent upon the fixed power assumption. This would mean that the fitted coefficient in Equation 9 as a whole could only apply when comparing conditions with identical power. Specifically, Equation (2) states the hypothesis that the force-rate per contraction increases proportionally with the amplitude of force rate (or amplitude of torque rate). For the model in Equation (3) and the sinusoidal motion shown, the amplitude of torque rate increases with Af3 (the product of the amplitude of the motion A, and the frequency, f, cubed). In the fixed power condition, the amplitude of the motion is a function of the frequency, so torque rate increases proportionally with bf3/2, where b is a fitted coefficient. Thus, Equation 7 (and Equation 9) which related amplitude of torque rate to cost per contraction, and cost per contraction per unit time, appear specific to the fixed power constraint. The authors should more clearly explain how this derivation, and the fitted coefficient, can generalize to non-fixed power conditions. I assume that any differences in mechanical power are absorbed in the first term in Equation (1), thus the coefficient would generalize, but this should be clarified.

4. Related to the point above, more effort should be made to explain these frequency-based experiments and findings to a single movement with varying distance and duration.

5. The authors should also plot metabolic cost vs movement duration for the different distances, as this is more of a standard form. This would allow for an easier understanding of the data in the context of a single reach.

6. Related to the point above, the authors do not address how their metabolic model predictions compare to published data on the metabolic cost of a single reach as a function of distance and duration. It would strengthen the impact of their findings to demonstrate whether the current dataset exhibits similar relationships between metabolic cost, movement duration and distance.

7. The authors seem to be underselling the cross-validation. If we understand correctly, the coefficient fitted in one experiment with one group of subjects, was able to explain data in a different experiment with different subjects with different frequencies and amplitudes, and the conditions had different power requirements. This is a lovely finding and should be highlighted more strongly. To help highlight the strengths of the cross-validation, there should be a more detailed explanation of the purpose of the cross-validation: to check whether the coefficient generalizes to different subjects as well as different frequencies/amplitudes/mech.power conditions.

8. A more quantitative comparison is needed (in addition to the figure presented in figure 8), when comparing the metabolic model velocity profile predictions to empirical data and the minimum variance predictions.

9. Why was the experiment a bilateral movement? Did the authors divide the measured power by 2? I couldn't find mention of this in the methods.

10. It seems that the elbow is externally fixed during the movement. Otherwise the subjects would need to stabilize the elbow and the model used of a single joint/rigid body would be inappropriate. However this is not clear from the text. Please confirm.

11. Sometimes the authors refer to 'energy cost per contraction' and sometimes 'energy cost per movement'. Please clarify the distinction, if any.

Reviewer #2 (Recommendations for the authors):

The paper states that minimizing squared muscle activations does not predict smooth movements. My understanding is that optimal feedback theory does predict smooth movements (Todorov 2002; Al Borno 2020). For instance, in your model, if you included only the squared muscle activations term (or just used the Hill-type model), would you obtain a non-smooth bell-shaped velocity profile?

As pointed out in the paper, in minimum-variance theory, an effort term is implicitly there. This could indicate that minimum-variance theory is sufficient to explain all the data (in addition to the speed-accuracy tradeoff, which this work does not address). If this is incorrect, it would be useful for the authors to point out where minimum-variance theory falls short. For instance, recent work by Nagamori et al., 2021 questions the main assumption behind the theory.

Essential revisions:

1) The authors claim to present a theory of movement control that unifies smoothness and duration. They currently only show that their metabolic cost model can explain smoothness. However, their metabolic cost model on its own cannot explain the speed-accuracy tradeoff and minimum variance on its own cannot explain duration. Thus to support a unifying theory, they need to show that their metabolic cost model and minimum variance combined can explain all three movement properties: smoothness, the speed-accuracy tradeoff and duration. They also need to show that this new unifying model performs better than models using a combination of minimum variance and other effort models (such as mech work, hill muscle models, and torque-change). Alternatively, instead of this additional analysis, the authors could temper their claims of a unifying theory and instead focus on their result that smoothness can also be explained by energy minimization. Although a weaker finding, this is also a an important contribution to the field.

We were unclear about the unified objective, and have amended the discussion to make it clearer that energy economy is not treated as the sole objective for reaching movements. For example:

“…both variance and force-rate 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.”

We have also added more explanation for how economy can predict duration (Figure 10), and have added discussion about how “metabolic energy expenditure is a physiological, independently testable measure of effort” (in contrast to more abstract objectives such as minimum activation or torque change).

2) A comparison of the predicted velocity profiles by their metabolic model and the predictions of models based on squared muscle activations, torque rate, and Hill-type muscle model is needed.

As suggested, we have included these predictions in an updated figure (now Figure 9). We also expanded discussion regarding them: “…multiple objectives can reproduce such velocities non-uniquely, making additional tests important for disambiguating them. The advantage of metabolic energy expenditure is that it is a physiological, independently testable measure of effort.

3) To support their claim that energy explains duration, the authors should demonstrate the ability of their metabolic model to explain movement duration, and compare its predictions to the predictions of other effort models (mechanical work, squared muscle activations, torque rate, Hill-type muscle models). This analysis should also include whether their metabolic model predicts a non-zero metabolically optimal movement duration and if so, how the metabolically optimal duration scales with movement distance.

We have added simulations using force-rate cost to predict preferred movement duration as a function of distance (new Figure 10), along with comparison to recent experimental literature (Reppert et al., 2018). The figure shows how predicted duration is non-zero and increases with distance. We did not test the other objectives, because to our knowledge they have not been proposed to predict duration.

Reviewer #1 (Recommendations for the authors):

1. For the predictions of the bell-shaped velocity profile, it is not clear how the effort cost was represented in the cost function. Equation (1) represents the effort cost per movement, and even Equation (9) seems to represent the average energy per time associated with force-rate, since it is a function of the frequency of the contraction and the amplitude of the torque rate of the contraction. Thus, there is no expression for effort cost as a function of instantaneous time (i.e. similar to effort models based on torque rate squared or force squared or activation squared), rather it is calculated based on the total mechanical work and the frequency (or duration) of the movement. It would help to have the authors clarify how effort was represented in their optimization approach.

We agree, and have added the explicit energy equation that is minimized during the fixedtime optimization, on page 16.

2. Additional figures should be provided plotting individual and average subject trajectory data and emg data across the different conditions. A figure of the experimental setup and visual feedback should also be provided. It is unclear how a single cursor was used to show the average joint angles.

We have added text to explain the motion of the cursor located in the methods section Page 13. We neglected to note clearly that the second panel includes individual subject data, and we have amended the caption appropriately. Finally, we have added an additional figure (new figure 4) of average reaches.

3. As written, it seems that Equations 7 and 9 are dependent upon the fixed power assumption. This would mean that the fitted coefficient in Equation 9 as a whole could only apply when comparing conditions with identical power. Specifically, Equation (2) states the hypothesis that the force-rate per contraction increases proportionally with the amplitude of force rate (or amplitude of torque rate). For the model in Equation (3) and the sinusoidal motion shown, the amplitude of torque rate increases with Af3 (the product of the amplitude of the motion A, and the frequency, f, cubed). In the fixed power condition, the amplitude of the motion is a function of the frequency, so torque rate increases proportionally with bf3/2, where b is a fitted coefficient. Thus, Equation 7 (and Equation 9) which related amplitude of torque rate to cost per contraction, and cost per contraction per unit time, appear specific to the fixed power constraint. The authors should more clearly explain how this derivation, and the fitted coefficient, can generalize to non-fixed power conditions. I assume that any differences in mechanical power are absorbed in the first term in Equation (1), thus the coefficient would generalize, but this should be clarified.

We have added specific equations to show how Equation (2) applies to point-to-point movements (new Equations 10 and 11), along with clarification how the coefficient to force-rate is independent of the fixed power constraint (new subsection Model of point-to-point reaching). In Methods:

“This force-rate coefficient is not specific to cyclic movements alone… The model may therefore make testable predictions of energetic cost even for movements that are acyclic and not constrained to fixed power.”

4. Related to the point above, more effort should be made to explain these frequency-based experiments and findings to a single movement with varying distance and duration.

The new subsection Model of point-to-point reaching (page 16) provides greater detail on single movements. Briefly, we applied the coefficient found from cyclic movements to predict discrete movements, which use a state-dependent form of the objective, plus different boundary constraints at movement start and end.

5. The authors should also plot metabolic cost vs movement duration for the different distances, as this is more of a standard form. This would allow for an easier understanding of the data in the context of a single reach.

We have added a second axis to our (new Figure 5)A to include movement amplitude.

6. Related to the point above, the authors do not address how their metabolic model predictions compare to published data on the metabolic cost of a single reach as a function of distance and duration. It would strengthen the impact of their findings to demonstrate whether the current dataset exhibits similar relationships between metabolic cost, movement duration and distance.

The new point-to-point subsection adds detail about duration. We have included predictions of increasing movement time with distance (new Figure 10), which re-uses the force-rate coefficient found in experiment.

7. The authors seem to be underselling the cross-validation. If we understand correctly, the coefficient fitted in one experiment with one group of subjects, was able to explain data in a different experiment with different subjects with different frequencies and amplitudes, and the conditions had different power requirements. This is a lovely finding and should be highlighted more strongly. To help highlight the strengths of the cross-validation, there should be a more detailed explanation of the purpose of the cross-validation: to check whether the coefficient generalizes to different subjects as well as different frequencies/amplitudes/mech.power conditions.

Yes, that understanding is correct. We have modified the Methods topic sentence:

“As a cross-validation test of the force-rate cost, we tested the generalizability of coefficient ;;_! against a second set of conditions with a separate set of subjects”.

Later in same paragraph:

“we expected the model to roughly predict trends regarding mechanical and metabolic rates for the cross-validation conditions.”

8. A more quantitative comparison is needed (in addition to the figure presented in figure 8), when comparing the metabolic model velocity profile predictions to empirical data and the minimum variance predictions.

We have added a quantitative comparison of the different models to capture bell-shapes, page 14. We report correlation coefficients for the models. Even though our model yielded the highest correlation (0.99), we state in Discussion:

“However, multiple objectives can reproduce such [bell-shaped] velocities non-uniquely (page 10)… metabolic energy expenditure is a physiological, independently testable measure of effort.”

9. Why was the experiment a bilateral movement? Did the authors divide the measured power by 2? I couldn't find mention of this in the methods.

In Methods, we have added:

“The movements were cyclic and bimanually symmetrical to induce steady energy expenditure sufficient to be distinguished easily by expired gas respirometry.”

Also in Methods:

“We report net metabolic rate ;;; for bimanual movement…” We did not divide by 2, except in Discussion, where we have added “And per reaching movement, the metabolic cost (at two movements per cycle) was about 3.5 J per arm.”

10. It seems that the elbow is externally fixed during the movement. Otherwise the subjects would need to stabilize the elbow and the model used of a single joint/rigid body would be inappropriate. However this is not clear from the text. Please confirm.

We have amended the text to clarify that both shoulder and elbow torques were computed from inverse dynamics, and the same force-rate coefficient identified for both. That same coefficient was then used to predict point-to-point movements.

11. Sometimes the authors refer to 'energy cost per contraction' and sometimes 'energy cost per movement'. Please clarify the distinction, if any.

We have simplified to “cost per movement,” and in Discussion consider the cost for each half-cycle movement.

Reviewer #2 (Recommendations for the authors):

The paper states that minimizing squared muscle activations does not predict smooth movements. My understanding is that optimal feedback theory does predict smooth movements (Todorov 2002; Al Borno 2020). For instance, in your model, if you included only the squared muscle activations term (or just used the Hill-type model), would you obtain a non-smooth bell-shaped velocity profile?

We added now Figure 9 includes a comparison to hill-model activation-squared and other objectives. We agree that some specific models can produce bell-shaped velocities, and have amended the text accordingly. We also note:

“However, multiple objectives can reproduce such [bell-shaped] velocities non-uniquely (page 10)…Accordingly metabolic energy expenditure is a physiological, independently testable measure of effort.”

As pointed out in the paper, in minimum-variance theory, an effort term is implicitly there. This could indicate that minimum-variance theory is sufficient to explain all the data (in addition to the speed-accuracy tradeoff, which this work does not address). If this is incorrect, it would be useful for the authors to point out where minimum-variance theory falls short. For instance, recent work by Nagamori et al., 2021 questions the main assumption behind the theory.

We did not intend to dismiss the minimum-variance theory, and have amended the text to be clearer. In Discussion:

“However, the ambiguity also means that both variance and force-rate 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.”

We agree with the reviewer that minimum-variance remains a viable theory, but it does not explain the metabolic cost predicted and measured here.

Author details

1. Jeremy D Wong

   University of Calgary, Faculty of Kinesiology, Department of Biomedical Engineering, Calgary, Canada

   Contribution

   Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Project administration, Resources, Software, Visualization, Writing - original draft, Writing – review and editing

   For correspondence

   jeremy.wong2@ucalgary.ca

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-5564-1794

2. Tyler Cluff

   1. University of Calgary, Faculty of Kinesiology, Department of Biomedical Engineering, Calgary, Canada
   2. Hotchkiss Brain Institute, University of Calgary, Calgary, Canada

   Contribution

   Formal analysis, Funding acquisition, Methodology, Writing – review and editing

   Competing interests

   No competing interests declared

3. Arthur D Kuo

   University of Calgary, Faculty of Kinesiology, Department of Biomedical Engineering, Calgary, Canada

   Contribution

   Conceptualization, Funding acquisition, Project administration, Supervision, Visualization, Writing - original draft, Writing – review and editing

   Competing interests

   No competing interests declared

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