More is not always better: modeling the effects of elastic exoskeleton compliance on underlying ankle muscle–tendon dynamics

We chose to model hopping in this study because it is a primarily ankle driven mode of bouncing gait that is simple both mechanically, and from a neural control perspective [21, 30]. Despite its inherent simplicity, however, hopping captures one of the critical features of all bouncing gait: the ability to store and return energy in series tendon at distal joints [22, 23]. Using hopping as a model of functional ankle joint mechanics allowed us to ask fundamental questions concerning the impact of passive exoskeleton assistance on the mechanics and energetics of biological muscle–tendon unit in a simple, straightforward way. Finally, hopping comes with a wealth of experimental whole limb [26, 27], joint-level [21, 25–27], muscle level [22], electromyographic [21, 22, 25], and metabolic [21, 22, 26] data to compare model predictions against.

Our modeled muscle–tendon unit (MTU) consisted of a single Hill-type CE subject to nonlinear force–length (F–L) and –velocity (F–V) actuation dynamics (figure 1); as well as first order activation dynamics. It also had a parallel elastic element capable of generating passive force at lengths greater than ${{l}_{0}}$, the length at which active muscle produces its peak isometric force [31] (figure 1). Parameterization of normalized F–L characteristics was based on experimental observation in Rubenson et al [32]. Normalized F–V and first order activation dynamics were based on parameterized equations from Haeufle et al [33]. Tendon/series elastic element (SEE) dynamics were modeled based on observations from Lichtwark et al [18], and exhibited a nonlinear 'toe' region, after which dynamics can be approximated as linear. All model equations and parameter values used here are detailed in supplementary text.

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System inertial mechanics were modeled by assuming a single rigid attachment point of the biological MTU and exoskeleton (Exo) system to lever arm acting with a fixed mechanical advantage (i.e. pulley) on a point mass under constant gravitational load. This was meant to reflect loading experienced by the triceps surae–Achilles tendon complex at the ankle joint during human hopping (figure 1). Rationale and justification for using this simplified inertial environment are discussed further in Robertson et al [34].

Several key simplifications were made to facilitate implementation of this modeling framework. First, the gastrocnemii and soleus muscle of the triceps surae–Achilles tendon complex were lumped into a single, mono-articular, fusiform muscle. This facilitated use of a simplified limb geometry, and disregards complications that arise from having biarticular and bipennate (i.e. non-fusiform) gastrocnemii muscles (figure 1). While, based on physiological cross-sectional area (PCSA), soleus is the primary force producer within the triceps surae muscle group, these simplifications likely diminished the potential impact that gastrocnemii operating point and architecture played in governing adaptation at the ankle and knee in humans during spring assisted hopping [21, 22]. Secondly, our system operates with a fixed mechanical advantage, disregarding shifts in biological moment arm length that can occur in vivo [35, 36]. Finally, all muscle fibers were assumed to have identical activation/deactivation time constants (${{\tau }_{{\rm act}}}$ and ${{\tau }_{{\rm deact}}}$ respectively), and values used in first order activation dynamics are a weighted average of observed values based on muscle PCSA in cadavers [37–39]. Further justification of model simplifications can be found in Robertson et al [34].

Our model of human hopping with exoskeleton assistance had a linear tension spring in parallel with the biological MTU (figure 1). The slack length of our modeled exoskeleton was the combined length at which the inactive CE developed passive force $\left( {{l}_{0}} \right)$, and slack length of SEE complex $\left( {{l}_{{\rm slack}}} \right)$. The exoskeleton moment arm length is the same as that of the biological MTU $({{l}_{{\rm in}}}=0.04\;{\rm m})$ [37]. This was done to simplify scaling of relative Exo/MTU stiffness relationships explored in this study. We note that stiffness comparisons can be extended to systems where exoskeleton and biological moment arms are not equal by converting linear stiffness $\left( {\rm N}\;{{{\rm m}}^{-1}} \right)$ to rotational stiffness $\left( {\rm N}\;{\rm m}\;{\rm ra}{{{\rm d}}^{-1}} \right)$ using the following relationship

$$\begin{eqnarray*}&&{{k}_{\Theta }}={{k}_{x}}\cdot {{\left( {{l}_{{\rm in}}} \right)}^{2}},\end{eqnarray*}$$

where ${{k}_{\Theta }}$ is rotational stiffness, ${{k}_{x}}$ is linear stiffness, and ${{l}_{{\rm in}}}$ is the distance from the point where force is applied to the center of joint rotation (i.e. moment arm) (figure 1). Finally, transmission of assistive forces was presumed to be ideal, and no attempt was made to model interaction of rigid exoskeleton with soft tissue.

The model stimulation frequency $\left( {{\omega }_{{\rm stim}}} \right)$ was 2.5 Hz, a frequency at which metabolic cost was consistently reduced in human experiments examining exoskeleton assisted hopping [21, 22, 26]. ${{k}_{{\rm Exo}}}$ was varied in $4.5\;\;{\rm kN}\;{{{\rm m}}^{-1}}$ $\left( 2.5\%\;{{k}_{{\rm SEE}}} \right)$ increments from unassisted $\left( {{k}_{{\rm Exo}}}=\;0\;\;{\rm N}\;{{{\rm m}}^{-1}} \right)$ to that of biological tendon $\left( {{k}_{{\rm SEE}}}=180\;\;{\rm kN}\;{{{\rm m}}^{-1}} \right)$, corresponding to joint level stiffness during perfectly isometric unassisted hopping (i.e. ${{k}_{{\rm MTU}}}={{k}_{{\rm SEE}}}$). Stimulation was provided as a square wave pulse with 10% duty relative to ${{\omega }_{{\rm stim}}}$ (i.e. ${{T}_{{\rm pulse}}}=0.1\times \omega _{{\rm stim}}^{-1}$). This pulse was subject to first order activation dynamics exhibited by biological muscle [31]. Stimulation amplitude $\left( {{A}_{{\rm stim}}} \right)$ ranged from 10% to 100% of maximum possible. All modeled conditions were run for 15 s (∼37 hops) to allow time for the system to become stable and cyclic. For each ${{A}_{{\rm stim}}}/{{k}_{{\rm Exo}}}$ combination, the final 4 complete hopping cycles were used in all subsequent analysis. Parameter values and equations used in model implementation are detailed in supplementary material.

The major criterion for a simulation to successfully 'hop' included having a flight phase where the MTU + Exo system went slack, and the only force applied to our virtual mass was gravity (i.e. ${{F}_{{\rm net}}}=\;{{F}_{{\rm g}}}$). Potential risk of injury was also taken into account as part of determining a successful hop, and normalized peak CE strains $\left( {{\epsilon }_{{\rm peak}}} \right)$ > 0.3 were assumed to pose significant risk of injury based on reports from literature [40–42]. ${{\epsilon }_{{\rm peak}}}$ was computed as follows

$$\begin{eqnarray*}&&{{\epsilon }_{{\rm peak}}}=\;\frac{\left( \mathop{{\rm max} }\limits_{t=0\to {{T}_{{\rm stim}}}}\left( {{l}_{{\rm CE}}}\left( t \right) \right)-{{l}_{0}} \right)}{{{l}_{0}}},\end{eqnarray*}$$

where the cycle period ${{T}_{{\rm stim}}}=\omega _{{\rm stim}}^{-1}$ and ${{l}_{{\rm CE}}}\left( t \right)$ was CE length as a function of time. Finally, successful hopping was required to be cyclic with stimulation, and show no variance when subjected to the integrated return map method detailed in [34].

Once it had been determined that a given condition meets the criterion for hopping in a safe and effective manner, system and component level analysis of mechanics and energetics was performed. First, we determined average positive power over a cycle of hopping for the system (i.e. MTU + Exo) and its respective mechanical components (i.e. MTU, active/passive CE, SEE, Exo) as follows

$$\begin{eqnarray*}&&\bar{P}_{{\rm mech}}^{+}=\;\frac{1}{{{T}_{{\rm stim}}}}\times \int \limits_{t=0}^{{{T}_{{\rm stim}}}} P_{{\rm mech}}^{+}\left( t \right){\rm d}t,\end{eqnarray*}$$

where

$$\begin{eqnarray*}&&P_{{\rm mech}}^{+}\left( t \right)=\;\left\{ \begin{array}{ccccccccccccccc} 0\;{\rm when}\;v\left( t \right)\gt 0, \\ {{P}_{{\rm mech}}}\left( t \right)\;{\rm when}\;v\left( t \right)\lt 0. \\ \end{array} \right.\end{eqnarray*}$$

In other words, power output was positive for shortening (i.e. work done against gravity).

Next, we explored system peak force $\left( {{F}_{{\rm peak}}} \right)$ over a cycle of hopping

$$\begin{eqnarray*}&&{{F}_{{\rm peak}}}=\;\mathop{{\rm max} }\limits_{t=0\to {{T}_{{\rm stim}}}}\left( {{F}_{{\rm MTU}+{\rm Exo}}}\left( t \right) \right)\;=\;{{F}_{{\rm MTU}+{\rm Exo}}}\left( {{t}_{{\rm peak}}} \right),\end{eqnarray*}$$

where ${{F}_{{\rm MTU}+{\rm Exo}}}\left( t \right)$ is MTU + Exo force as a function of time, and ${{t}_{{\rm peak}}}$ is the time point within a cycle where peak system (MTU + Exo) force occurs. We were also interested in how the MTU, Exo, and active/passive CE components shared loads at ${{t}_{{\rm peak}}}$ and computed their respective forces as well.

We were also concerned with how CE F–L and F–V dynamics influenced system mechanics, so we computed the range and average operating strains/velocities from stimulation onset (muscle activation, $\alpha \left( t \right)\gt 0$) to the initiation of flight $\left( {{F}_{{\rm MTU}+{\rm Exo}}}\left( t \right)={{F}_{{\rm g}}} \right)$. Note that this does not include any passive stretch prior to stimulation onset, and only encompassed portions of ground contact where active muscle was likely to make significant contributions.

We were also interested in system energetics, and calculated average metabolic rate $\left( {{{\bar{P}}}_{{\rm met}}} \right)$ using a dimensionless model of metabolic cost [43, 44] as a function of muscle velocity scaled by physiological constants ${{F}_{{\rm max} }}$ (the maximum active isometric CE force), ${{v}_{{\rm max} }}$ (maximum CE shortening velocity), (supplemental text table 1), and normalized muscle active state $\alpha \left( t \right)$ [45]. ${{\bar{P}}_{{\rm met}}}$ was computed as follows

$$\begin{eqnarray*}&&{{\bar{P}}_{{\rm met}}}=\frac{1}{{{T}_{{\rm stim}}}}\times {{F}_{{\rm max} }}\times {{v}_{{\rm max} }}\times \int \limits_{t=0}^{{{T}_{{\rm stim}}}} \left[ \alpha \left( t \right)\times {{p}_{{\rm met}}}\left( {{v}_{{\rm CE}}}\left( t \right)/{{v}_{{\rm max} }} \right) \right]{\rm d}t,\end{eqnarray*}$$

where ${{p}_{{\rm met}}}\left( {{v}_{{\rm CE}}}\left( t \right)/{{v}_{{\rm max} }} \right)$ is dimensionless metabolic cost at time $t$ from [1] (supplemental text table 2). MTU + Exo, MTU, and CE apparent efficiency $\left( {{e}_{{\rm app}}} \right)$ were also a focus of this study, and were computed as follows based on their respective $\bar{P}_{{\rm mech}}^{+\,}$ output

$$\begin{eqnarray*}&&{{e}_{{\rm app}}}=\;\bar{P}_{{\rm mech}}^{+}/{{\bar{P}}_{{\rm met}}}.\end{eqnarray*}$$

Further details on equations, metrics, and constants used as part of model implementation and subsequent analysis can be found in supplementary text.

Experimental data used in all quantitative comparisons made here were taken from Farris et al [21]. Exoskeleton rotational stiffness used in experimental conditions was $91\;{\rm N}\;{\rm m}\;{\rm ra}{{{\rm d}}^{-1}}$ which, when scaled by our modeled biological 'in' moment arm $\left( 0.04\;{\rm m} \right)$ [37] and converted to ${\rm kN}\;{{{\rm m}}^{-1}}$ yields $\sim 56\;{\rm kN}\;{{{\rm m}}^{-1}}$. We constrained our selection of points based on experimental conditions (i.e. unassisted/${{k}_{{\rm Exo}}}=0\;kN\;{{m}^{-1}}$ and assisted/${{k}_{{\rm Exo}}}\approx 54\;000\;kN\;{{m}^{-1}}$), and matched MTU + Exo and MTU average positive power to experimental data for the whole ankle joint with and without an exoskeleton [21]. We selected these two metrics as a basis for comparison because our simplified model does not have the necessary geometric/physiological complexity for quantitative comparisons to muscle level data observed in [22].

For all modeled conditions, stable and cyclic mechanics were easily achieved within the time allotted, and the final 4 cycles used in analysis are all representative of steady-state behavior. Significant risk of injury was observed for low ${{A}_{{\rm stim}}}$, low ${{k}_{{\rm Exo}}}$ combinations. These regions exhibited CE peak eccentric strains > 0.3 at some point in their hopping cycle (figure 2(A)) [40–42].

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A flight phase was achieved for the majority of the parameter space, with notable exceptions for low ${{A}_{{\rm stim}}}$ (< 20% of max) low ${{k}_{{\rm Exo}}}$ (< 20%${{k}_{{\rm SEE}}}$) combinations, as well as low-to-medium ${{A}_{{\rm stim}}}$ (< ∼ 50% of max) and high ${{k}_{{\rm Exo}}}$ $\left( \gt \tilde{\ }50\%{{k}_{{\rm SEE}}} \right)$ conditions. The greatest flight amplitude was observed at 70% ${{A}_{{\rm stim}}}$ and high ${{k}_{{\rm Exo}}}$ (> 70%${{k}_{{\rm SEE}}}$) conditions (figure 2(B)).

Regions where elevated injury risk or no flight phase were observed are indicated by opaque regions labeled 'Injury Risk' and 'No Flight' in figures 2(A) and (B) respectively and all subsequent contour plots (figures 3(A), (C) and 6(A)–(C)).

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Maximal MTU + Exo $\bar{P}_{{\rm mech}}^{+}$ was generated for high ${{A}_{{\rm stim}}}$, high ${{k}_{{\rm Exo}}}$ conditions (figure 3(A)). Minimums in MTU + Exo $\bar{P}_{{\rm mech}}^{+}$ were generally observed for low stimulation amplitudes regardless of ${{k}_{{\rm Exo}}}$ (figure 3(A)). Data points from a contour of constant Exo + MTU $\bar{P}_{{\rm mech}}^{+}$ spanning 100% ${{A}_{{\rm stim}}}$ with no exoskeleton (i.e. ${{k}_{{\rm Exo}}}=0\;{\rm kN}\;{{{\rm m}}^{-1}}$) through 50% ${{A}_{{\rm stim}}}$ and an ${{k}_{{\rm Exo}}}=90\;\;{\rm kN}\;{{{\rm m}}^{-1}}$ (i.e. ${{k}_{{\rm Exo}}}=50\%{{k}_{{\rm SEE}}}$) were selected for further analysis, and are indicated by white dots in all subsequent contours (figure 3(A)). MTU + Exo workloops from data during ground contact were generated for these selected points and are visualized in figure 3(B).

In general, CE $\bar{P}_{{\rm mech}}^{+}$ was highest for low ${{A}_{{\rm stim}}}$/high ${{k}_{{\rm Exo}}}$ combinations along the border of the 'no flight' region (figure 3(C)), and lowest for high ${{A}_{{\rm stim}}}/{{k}_{{\rm Exo}}}$ conditions where a flight phase was safely achieved (figure 3(C)). Ground contact CE workloops for selected points from figure 3(A) are visualized in figure 3(D).

Selected points of constant MTU + Exo $\bar{P}_{{\rm mech}}^{+}$ (figure 3(A)) traded energy cycled in biological tendon for energy cycled in the modeled exoskeleton, dramatically reducing ${{F}_{{\rm MTU}}}\left( {{t}_{{\rm peak}}} \right)$ and $\bar{P}_{{\rm mech}}^{+}$ from the biological MTU (figure 4). Maintaining constant MTU + Exo $\bar{P}_{{\rm mech}}^{+}$ as in figure 3(A) required a slight increase in MTU + Exo ${{F}_{{\rm peak}}}$ while simultaneously decreasing the force burden on the biological MTU (figure 4(A)). Despite reduced ${{F}_{{\rm MTU}}}\left( {{t}_{{\rm peak}}} \right)$ with increased ${{k}_{{\rm Exo}}}$ for selected data points, there was a small increase in contributions to ${{F}_{{\rm peak}}}$ from passive CE components (figure 4(A)). This had little effect on CE $\bar{P}_{{\rm mech}}^{+}$, which remained nearly constant for these same points in parameter space. CE power actually increased slightly from baseline (unassisted, ${{k}_{{\rm Exo}}}=0\;\;{\rm kN}\;{{{\rm m}}^{-1}}$) to high ${{k}_{{\rm Exo}}}$, low ${{A}_{{\rm stim}}}$ conditions for selected points of constant MTU + Exo $\bar{P}_{{\rm mech}}^{+}$ (figure 4(B)).

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Force–length (F–L) and –velocity (F–V) operating points during active force production for data points selected in figure 3(A) can be seen in figure 5. As ${{A}_{{\rm stim}}}$ decreased and ${{k}_{{\rm Exo}}}$ increased, the modeled CE increased its range of operating lengths as well as average values of both operating length and velocity (bars, figure 5). Average operating strain increased from approximately 0.9-1.15${{l}_{0}}$ (∼4.9–6.3 cm) (markers, figure 5(A)), while average operating velocity more than doubled from ∼0.07 to 0.16 ${{v}_{{\rm max} }}$ (∼3−7 cm s⁻¹) (markers, figure 5(B)).

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Average metabolic rate scaled almost directly with ${{A}_{{\rm stim}}}$/integrated muscle activation (figure 6(A)), but there was a slight increase in ${{\bar{P}}_{{\rm met}}}$ when ${{A}_{{\rm stim}}}$ was held constant and ${{k}_{{\rm Exo}}}$ was increased (figure 6(A)). For the data points of constant MTU + Exo $\bar{P}_{{\rm mech}}^{+}$ selected in figure 3(A), ${{\bar{P}}_{{\rm met}}}$ decreased with decreasing ${{A}_{{\rm stim}}}$ and increasing ${{k}_{{\rm Exo}}}$ (figure 6(A)).

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MTU ${{e}_{{\rm app}}}$ decreased with decreasing ${{A}_{{\rm stim}}}$ except in low-to-no ${{k}_{{\rm Exo}}}$ conditions (figure 6(B)). MTU ${{e}_{{\rm app}}}$ also decreased as ${{k}_{{\rm Exo}}}$ increased unless ${{A}_{{\rm stim}}}$ was ≥90% of maximum, at which point it increased slightly from unassisted (figure 6(B)). For the data points selected in figure 3(A), MTU ${{e}_{{\rm app}}}$ declined rapidly with decreasing ${{A}_{{\rm stim}}}$ and increasing ${{k}_{{\rm Exo}}}$ (figure 6(B)).

CE ${{e}_{{\rm app}}}$ (active + passive components) generally decreased with increasing ${{k}_{{\rm Exo}}}$, and increased with decreasing ${{A}_{{\rm stim}}}$ (figure 6(C)). For the data points selected in figure 3(A), CE ${{e}_{{\rm app}}}$ increased with decreased ${{A}_{{\rm stim}}}$ and increased ${{k}_{{\rm Exo}}}$ (figure 6(C)).

An MTU $\bar{P}_{{\rm mech}}^{+}$ of ∼2.4 W kg⁻¹ was observed for the 60% ${{A}_{{\rm stim}}}$, ${{k}_{{\rm Exo}}}=0$ condition, and output ∼1.7 W kg⁻¹ was observed for the 50%${{A}_{{\rm stim}}}$, 30%${{k}_{{\rm SEE}}}$ (i.e. ${{k}_{{\rm Exo}}}=54\;{\rm kN}\;{{{\rm m}}^{-1}}$) condition, matching findings for whole ankle $\bar{P}_{{\rm mech}}^{+}$ in a single limb from previous human studies [21] (figure 7(A)). These conditions are highlighted on most contour plots by a white star (figures 3(A), (C) and 6(A)–(C)), and compared directly in figure 7. Similarities between experimental and modeled conditions include little to no change in $P_{{\rm mech}}^{+}$ for the CE, large reductions in $\bar{P}_{{\rm mech}}^{+}$ for both the SEE and whole MTU, with a considerably greater fraction of MTU $\bar{P}_{{\rm mech}}^{+}$ coming from the CE (figure 7(A)), and a reduction in ${{\bar{P}}_{{\rm met}}}$ (figure 6(A)) when springy assistance was provided. There were also strong qualitiative similarities in the time-course of MTU/MTU + Exo force production and MTU/CE excursion dynamics during ground contact (figures 7(B), (C)). The major difference between modeled/experimental data had to do with rate of force onset/offset (figure 7(B)) and CE excursion (figure 7(C)) in late/early stance.

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Based on observations from previous experimental studies we hypothesized that by increasing exoskeleton stiffness and reducing stimulation amplitude, our model system (MTU + Exo) would be able to maintain a constant $\bar{P}_{{\rm mech}}^{+}$ output and 'stiffness' from an unassisted condition [21, 25–27]. Model findings agreed with this hypothesis, and it was possible to start at any stimulation level ≥60% without exoskeleton assistance, and maintain $\bar{P}_{{\rm mech}}^{+}$ production by decreasing ${{A}_{{\rm stim}}}$ and increasing ${{k}_{{\rm Exo}}}$ (figure 3(A)). By following any one of these contours, it was observed that there were nearly identical force–displacement (e.g. 'stiffness') dynamics during stance as demonstrated by MTU + Exo workloops from selected points in figure 3(A) (figure 3(B)).

Our second hypothesis was that, by following a contour of constant MTU + Exo $\bar{P}_{{\rm mech}}^{+}$, there should be reduced MTU forces at ${{t}_{{\rm peak}}}$ as well as a decrease in SEE and MTU contribution to MTU + Exo $\bar{P}_{{\rm mech}}^{+}$ [21, 22, 25–27]. By examining figure 4, it can be seen that this is indeed the case. As ${{k}_{{\rm Exo}}}$ was increased and ${{A}_{{\rm stim}}}$ decreased, contributions to peak MTU + Exo force from MTU were reduced, while those from Exo increased (figure 4(A)). A similar trend emerges for power sharing, where reductions in MTU/SEE $\bar{P}_{{\rm mech}}^{+}$ were offset by increased output from the Exo (figure 4(B)). Given that the ankle plantar flexor muscle groups depend primarily on inertial dynamics and elastic energy cycling as a means for joint level $\bar{P}_{{\rm mech}}^{+}$ generation [16, 22, 23], it follows that reduced loading results in reduced $\bar{P}_{{\rm mech}}^{+}$ production in the SEE and MTU as a whole.

Our third hypothesis predicted that, in spite of reduced biological loading, average positive power in the CE would remain constant. The predicted mechanism by which this would occur was increased excursions in conjunction with reduced CE forces, which would result in nearly constant CE average positive power production [22]. By examining figure 4(B), it can be seen that, as ${{A}_{{\rm stim}}}$ was decreased and ${{k}_{{\rm Exo}}}$ increased, CE $\bar{P}_{{\rm mech}}^{+}$ remained nearly constant, and even increased slightly for ${{k}_{{\rm Exo}}}\gt \tilde{\ }20\%{{k}_{{\rm SEE}}}\;\left( \tilde{\ }36\;{\rm kN}\;{{{\rm m}}^{-1}} \right)$ conditions (figures 3(C) and 4(B)). Our model also predicted that this occurred in conjunction with reduced biological loading (figure 4(A)) as well as increased CE excursion (figures 3(D) and 5(A)) and average operating velocity (figure 5(B)) in full agreement with hypothesis (3). Increased excursions also occurred in conjunction with greater values of ${{\epsilon }_{{\rm peak}}}$, many of which were $\gt \;0$ (i.e. ${{l}_{{\rm CE}}}\gt {{l}_{0}}$) (figures 2(A) and 5(A)), resulting in passive CE force generation (figure 4(A)). This coupling of force generation to CE length also played a role in keeping CE $\bar{P}_{{\rm mech}}^{+}$ at near constant levels (figure 4(B)).

For our fourth and final hypothesis, we predicted that by following a contour of constant MTU + Exo $\bar{P}_{{\rm mech}}^{+}$ it would be possible to reduce ${{\bar{P}}_{{\rm met}}}$ from the unassisted condition [21, 26]. Following the contour of white dots from figure 3(A), increasing ${{k}_{{\rm Exo}}}$ required a reduction in ${{A}_{{\rm stim}}}$, which was the primary determinant of ${{\bar{P}}_{{\rm met}}}$ for our modeled data (figure 6(A)). This was in spite of greater average operating velocities (figure 5(B)), and no reduction in $\bar{P}_{{\rm mech}}^{+}$ output from the CE (figures 3(B) and 4(B)).

Contours of constant MTU + Exo $\bar{P}_{{\rm mech}}^{+}$ in figure 3(A) all ran into the same problem as ${{k}_{{\rm Exo}}}$ increased and ${{A}_{{\rm stim}}}$ decreased: there came a point where the system was no longer capable of hopping. In these conditions, the mechanical response of biological and exoskeleton model components to cyclic stimulation was such that they counteracted each other, resulting in force and power production dynamics that did not facilitate a flight phase (figure 2(B)). In order to achieve hopping behavior at higher values of ${{k}_{{\rm Exo}}}$ it became necessary to raise ${{A}_{{\rm stim}}}$, and ultimately increase ${{\bar{P}}_{{\rm met}}}$ (figures 3(A) and 6(A)). This outcome was in line with observations from Grabowski et al (2009), which demonstrated that a more compliant exoskeleton achieved greater reductions in ${{\bar{P}}_{{\rm met}}}$ than a stiffer one [26]. Unfortunately, Grabowski et al did not record EMG as part of their study, so there is no way of knowing whether or not reduced metabolic benefit in stiffer exoskeletons was accompanied by increased muscle activation as predicted here [26].

Even though there were significant reductions in metabolic rate when ${{A}_{{\rm stim}}}$ was reduced in conjunction with increasing ${{k}_{{\rm Exo}}}$, it did not make the biological system as a whole more efficient. In fact, it had the exact opposite effect (figure 6(B)). By applying mechanical assistance and reducing the force burden on the MTU, the ability for an SEE to effectively cycle energy was reduced considerably (figure 4). The ultimate result was a large decrease in MTU apparent efficiency (figure 6(B)) despite concomitant reductions in metabolic rate (figure 6(A)).

This system level trend did not hold at the muscle level, and CE $\bar{P}_{{\rm mech}}^{+}$ output remained constant as ${{\bar{P}}_{{\rm met}}}$ declined with increasing ${{k}_{{\rm Exo}}}$ (figure 4(A)). This resulted in muscle ${{e}_{{\rm app}}}$ rising from ∼25% in the unassisted condition to ∼40% at the highest level of assistance for the data point in figure 3(A) $\left( 50\%{{A}_{{\rm stim}}},{{k}_{{\rm Exo}}}=90\;{\rm kN}\;{{{\rm m}}^{-1}} \right)$ (figure 6(C)). This increased CE apparent efficiency, however, came with greater ${{\epsilon }_{{\rm peak}}}$ values/risk of injury (figure 2(A)) and increased energy cycling in passive CE elements (figure 4(B)).

Despite the simplicity of the model implemented here, it could be compared rather nicely to human joint level data observed in Farris et al [21, 22]. By constraining our ${{k}_{{\rm Exo}}}$ values based on those used in experimental studies, and sweeping our ${{A}_{{\rm stim}}}$ for values of MTU $\bar{P}_{{\rm mech}}^{+}$ observed experimentally, we identified a point of best fit for the Exo with no spring (${{A}_{{\rm stim}}}=70\%$ of max, ${{k}_{{\rm Exo}}}=0\%{{k}_{{\rm SEE}}}$) and Exo with spring $\left( {{A}_{{\rm stim}}}=50\%\;{\rm ofmax},{{k}_{{\rm Exo}}}=30\%{{k}_{{\rm SEE}}} \right)$ conditions from [21, 22] for hopping at 2.5 Hz (figure 7(A), stars in figures 3(A),(C) and 6). These points maintained constant MTU + Exo $\bar{P}_{{\rm mech}}^{+}$ (figure 3(A)), reduced ${{A}_{{\rm stim}}}$ and ${{\bar{P}}_{{\rm met}}}$ requirements from unassisted (figure 6(A)), and traded reduced CE force (figure 7(B)) for increased excursion (figure 7(C)) to keep $\bar{P}_{{\rm mech}}^{+}$ constant (figure 3(C)). Direct comparisons of muscle/tendon mechanics was difficult, since experimental data were only for the soleus MTU, and our model has all three plantarflexors lumped into a single monoarticular MTU. We did, however, directly compare % of total $\bar{P}_{{\rm mech}}^{+}$ from CE/SEE in both studies, and found a quantitative match easily within ±1 Standard Error of the Mean (SEM) of experimental data (figure 7(A)).

There were, however, some shortcomings of model findings as well. Most notably, our model predicted significant passive CE stretch in early stance (figure 7(C)). This was not observed in human studies, where the ability to actively modulate MTU position with antagonist muscles (i.e. tibialis anterior) and knee flexion likely work together to limit CE lengthening in early stance (figure 7(C)). Secondly, while qualitative trends in metabolic demand agreed (i.e. ${{\bar{P}}_{{\rm met}}}$ reduced with Exo, figure 6(A)), the model underestimated values of reductions in whole body metabolic cost attributed to reduced plantar-flexor loads from [22]. Again, this was likely attributable to model simplifications, which do not allow for bi-articular plantar flexors (i.e. medial and lateral gastrocnemii) that contribute to knee joint mechanical demands in humans [21].

With all of these model predictions and simplifications in mind, the question now becomes one of optimization. How can metabolic benefits, MTU and CE efficiency, and risk of injury all be balanced to best assist a compliant MTU? Was there a point or region in parameter space where detrimental and beneficial aspects can be balanced to achieve an idealized outcome? We are confident that humans would follow a path through our parameter space that kept Exo + MTU stiffness and average positive power nearly constant [27](figures 3(A)–(B) and 4(A)), maintained near constant CE positive power [22] (figures 3(C)–(D) and 4(A)), reduces muscle activation [21, 22, 25], and allowed for reduced metabolic energy consumption [21, 22, 26] (figure 6(A)). A significant region of parameter space remained accessible even with these restrictive criteria in place, and all of these regions terminated at a point where flight was no longer achieved or injury risk was too great beyond some terminal increase in ${{k}_{{\rm Exo}}}$/reduction in ${{A}_{{\rm stim}}}$ (figure 3(A)). In every case, the only way to sustain hopping beyond this point was to increase ${{A}_{{\rm stim}}}$ in conjunction with increasing ${{k}_{{\rm Exo}}}$, ultimately resulting in increased metabolic cost (figure 6(A)) and average positive power from the MTU + Exo system (i.e. enhanced performance) (figure 3(A)). This, again, agrees with observed trends in human data [26], and leads us to the conclusion that springy assistance will be optimized for metabolic benefit when muscle activation is minimized and MTU + Exo power/stiffness remains consistent with the baseline/unassisted condition.

This model also made a second interesting prediction: by increasing assistance beyond the point of optimal metabolic benefit, it was possible to obtain better mechanical performance (i.e. hop height, Exo + MTU positive power) than the biological system alone (figures 2(B) and 3(A)). These regions of enhanced performance were also coincident with reduced CE positive power (figure 3(C)) and injury risk (figure 2(A)), as well as an increased metabolic cost relative to conditions where expenditure is minimized (figure 6(A)). This agrees with observations from Grabowski et al, where increased performance/hop height was also accompanied by greater variability in touchdown position, indicating that greater performance may come at the cost of reduced stability [26]. In conclusion, our model does predict that metabolic and mechanical performance benefits can be achieved simultaneously, but that neither is likely to be optimal while the other persists in a compliant MTU.

By examining the interplay of reflex feedback, feed-forward control, and a wearable device, we hope to better understand how design of assistive devices may be optimized to achieve a desired outcome, be it reduced metabolic cost or enhanced performance. Towards this end, future modeling work will integrate simulated spinal cord reflex pathways in conjunction with feed-forward control to further explore the possibilities and limitations of spring-based assistive exoskeletons. Given the possible variation in muscle fascicle length, velocity and force production dynamics observed in this study, it seems likely that feedback through spindle and/or golgi tendon organs could play a role in shaping adaption to steady state dynamics in the context of external assistance. Finally, future experimental work will include studies of human hopping for a range of exoskeleton stiffness' with instrumentation similar to Farris et al [21, 22] to better understand neural and metabolic adaptation to springy exoskeletons, and either verify or refute predictions made here regarding tradeoffs between reductions in metabolic cost and enhanced mechanical performance.