A model-based analysis of the mechanical cost of walking on uneven terrain

Human walking on uneven terrain is energetically more expensive than on flat, even ground. This is in part due to increases in, and redistribution of positive work among lower limb joints. To improve understanding of the mechanical adaptations, we performed analytical and computational analyses of simple mechanical models walking over uneven terrain comprised of alternating up and down steps of equal height. We simulated dynamic walking models using trailing leg push-off and/or hip torque to power gait, and quantified the compensatory work costs vs. terrain height. We also examined the effect of swing leg dynamics by including and excluding them from the model. We found that greater work, increasing approximately quadratically with uneven terrain height variations, was necessary to maintain a prescribed average forward speed. Greatest economy was achieved by modulating precisely-timed push-offs for each step height. Least economy was achieved with hip power, which did not require as precise timing. This compares well with observations of humans on uneven terrain, showing similar near-normal push-off but with more variable step timing, and considerably more hip work. These analyses suggest how mechanical work and timing could be adjusted to compensate for real world environments.

It is energetically more expensive for humans to walk on natural, complex terrain than 2 on hard, flat surfaces [1][2][3][4][5]. A multitude of factors, such as terrain unevenness, damping, 3 stiffness, friction, and other surface characteristics can affect gait biomechanics and 4 energy expenditure. Such potential causes for increases in energy consumption can be 5 roughly classified into two categories: gait adaptations that facilitate control or preempt 6 falls when walking under more challenging conditions and gait adaptations that are 7 mechanically necessary to account for terrain variation. The first category includes, for 8 example, increased muscle co-activation for greater joint stiffness during walking on 9 slippery surfaces [6][7][8], or a more crouched posture for improved stability during 10 running [9]. Such adaptations might lead to sub-optimal gait through a voluntarily 11 made trade-off between gait stability and gait economy. In contrast, increases in 12 June 8, 2020 1/23 energetic cost associated with mechanically necessary adaptations on uneven terrain 13 cannot be offset by alternative gait adjustments. Even when walking at the theoretical 14 optimum, surface properties can require more mechanical work to be performed against 15 the ground. For example, walking on energy-dissipating surfaces, such as sand, 16 necessitates more mechanical work than walking on hard surfaces [10]. Similarly, surface 17 height variations impact center of mass trajectories [11], which may lead to greater 18 energy expenditure [12]. It is often difficult to experimentally distinguish the specific 19 contribution of a gait adaptation to an increase in energetic cost, since humans likely 20 make multiple adaptations at the same time. An alternative way to investigate how 21 uneven terrain affects gait energetics is through a model-based analysis that focuses on 22 one single gait characteristic. 23 To illustrate the effects of center of mass motion on energy expenditure, consider a 24 simple example of moving along an upward step of height d followed by an equal 25 downward step of height d. When the center of mass moves upwards, potential energy 26 changes by E = mgd (with m being the total body mass and g being the gravitational 27 acceleration). Without any other adaptations, this change in potential energy must be 28 generated by mechanical work, W , performed over the course of the step, such that 29 W = E = mgd. As the downwards step is associated with an equal amount of 30 negative work W , the net energetic effect of the two steps is zero. There may, 31 however, be metabolic energy expended for negative work, if actively performed by 32 muscle. It has been estimated that when humans walk up and down steep inclines they 33 perform positive work with an efficiency of ⌘ + = 25 % and negative work with an 34 efficiency of ⌘ = 120 % [13]. That is, generating positive or negative mechanical work 35 can require positive net metabolic effort. The amount of additional metabolic energy 36 E VCOM to move up and down vertically would be: and has a net positive magnitude. However, this does not account for the possibility 38 of gait adaptations that might occur in response to the irregular surface, and an optimal 39 gait that mitigates some of the effects of Equation (1) likely exists. For example, 40 strategies other than direct work production could account for fluctuations in potential 41 energy. By slowing down when going up a step and speeding up when going down, 42 potential energy could be gained from kinetic energy, in turn reducing the amount of 43 necessary active work [14]. Similarly, taking advantage of collisions to passively perform 44 negative work could further reduce the necessary metabolic effort. In addition to optimal walking strategy over irregular terrain requires a detailed model-based analysis. 48 The amount of work necessary for steady-state locomotion of a model on uneven 49 terrain strongly depends on how the gait is powered. Past research has shown that 50 energy expenditure of simulated locomotion is sensitive to the relative timing of active 51 push-off done by the trailing limb and collision during heel-strike of the leading 52 limb [15]. Work production is most effective when done through push-off immediately 53 before collision of the leading leg, otherwise additional work must be done at the 54 hip [16]. On uneven terrain, unperceived irregularities may disrupt the relative timing 55 of push-off and heel-strike and demand work production at the hip. In addition, and 56 especially on highly irregular terrain, theoretically optimal push-off magnitudes may 57 actually be biologically unfeasible; although push-off and collision work must always be 58 positive and negative, respectively, in biological gait, theoretically optimal gaits may 59 dictate otherwise. This means that there exists a limit to the maximum amount of 60 positive work that can be done through push-off, after which additional work must be 61 done through the hip.

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In this study, we isolated the effects of surface unevenness on energy expenditure by 63 studying walking of simulated legged models over terrain of increasing surface 64 variability. Specifically, we evaluated the energetic consequences of the mechanical 65 adaptations required to move up and down over irregular terrain, and the mechanisms 66 by which mechanical work can be performed to do so. To this end, we investigated two 67 simple models of legged locomotion walking over uneven surfaces: the powered rimless 68 wheel [17] and the powered simplest walking model [18][19][20]. For simplicity, we 69 represented uneven terrain as a surface consisting of alternating up and down steps of 70 equal height. Both models were powered by impulsive push-off and hip work. Similar 71 models have been able to closely approximate, among other parameters, center of mass 72 and joint dynamics, ground reaction forces, and swing leg motion during locomotion on 73 level ground [17,21,22]. In addition, such models have made suggestions for economical 74 methods to power locomotion [14,15]. For the two models, we analytically calculated 75 the effects of terrain irregularity on step timing and average forward speed, and 76 numerically simulated a range of adaptation strategies. We then identified optimal 77 locomotion patterns and calculated the effective costs of transport assuming different We adapted two simple inverted pendulum models of walking to investigate how cost of 85 transport is affected by varying terrain variability, gait types, and methods of energy 86 input. The powered rimless wheel [17] takes steps of fixed length and has no swing leg 87 dynamics. It is modeled as rigid spokes fixed at an inter-leg angle of 2↵ (Fig. 1A). In 88 contrast, the powered simplest walking model [19,20] includes passive swing leg 89 dynamics, and is modeled as two rigid legs connected at the hip by a frictionless hinge 90 joint (Fig. 1A). Both models have legs of length l, a point mass m located at the hip (or 91 wheel center) and representing the pelvis and torso, and feet with negligible mass m f .

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Stance leg dynamics for both models are simply that of an inverted pendulum: where ✓(t) is the stance leg angle relative to vertical and g is gravity.

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For the simplest walking model, (t) is the swing leg angle relative to vertical. A 95 torsional hip spring (with stiffness k hip ) applies a torque between the stance and swing 96 legs at a desired swing leg frequency, ! n = q k hip m f l 2 . The torsional spring is analogous to 97 hip flexor and extensor muscles activity observed during steady human walking.

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Assuming that m f ⌧ m, the swing leg does not affect stance leg dynamics [15], and 99 swing leg dynamics for the simplest walking model reduce to: When the leading leg contacts the ground, the roles of the stance and swing switch, 101 and the models undergo an instantaneous and perfectly inelastic collision that redirects 102 the center of mass velocity to be tangential to the new stance leg. At this point, the leg 103 angles update such that ✓ + = and + = ✓ for the simplest walking model, and 104 ✓ + = ✓ + 2↵ for the rimless wheel. In these and the following equations, the 105 superscripts '-' and '+' refer to states immediately before and after collision, 106 respectively. Energy is lost due to the negative work done by the collision and must be 107 replaced through positive work to maintain steady state walking. 108 Two strategies have been proposed as analogs of human lower limbs doing positive 109 work on the center of mass: ankle push-off initiated by the trailing leg shortly before 110 heel-strike of the leading leg, and the application of hip torques throughout the stance 111 phase [15,17]. In our models, we approximated both strategies using impulses (Fig. 1B). 112 The ankle push-off impulse,P , is directed along the trailing leg towards the COM and 113 applied immediately before heel-strike of the leading leg. That is, push-off is 114 perpendicular to the direction of motion of the COM before collision. In contrast, the 115 hip impulseH is applied along the COM direction of motion after collision. The 116 push-off impulse acts analogously to work done by the lower leg muscles before toe-off, 117 while the hip impulse is analogous to the hip torque produced by hip extensor muscles 118 in the human, applied between the torso (not included in our simplified models) and 119 stance leg [15,23,24].

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Considering a passive collision, as well as the magnitudes of active push-off and hip 121 impulses, post-collision velocities of the stance and swing legs can be derived with 122 impulse-momentum. For the simplest walking model they are given by: and for the rimless wheel by: The above equations are a modification of the derivations presented in [15] and [19], and 125 include the effect ofH on stance leg angular velocity, also referred to as "late 126 push-off" [14,25].

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The post-collision stance leg angular velocity,✓ + k , of each step k is thus a direct 128 function of the pre-collision velocity,✓ k , and the magnitudes of the impulses H k and P k 129 at this step. The pre-collision velocity,✓ k+1 , at the subsequent heel-strike is then 130 obtained by integrating the continuous dynamics given by Equations (2) and (3), 131 starting from✓ + k . For the rimless wheel, the fixed leg angle and repeating structure of 132 the terrain result in the same COM height at the start and end of each step, independent 133 of whether the model is moving up or down. Since stance phase of the models is 134 energetically conservative, the integration leads to a trivial result for the rimless wheel: 135 In contrast, the inter-leg angle of the simplest walking model can vary from step to step 136 and the above relationship no longer holds. The velocity✓ k+1 and leg angles ✓ k+1 and 137 k+1 of the simplest walking model can thus only be computed via numerical 138 integration of the equations of motion.

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Given the periodic nature of the simulated surface, we restricted our analysis to 140 two-step periodic motion and assumed that leg angles and velocities for each second •✓ + 1 and✓ + 2 : velocities immediately after collision, leading into an up or down step, 147 respectively.

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Through numerical integration, we can determine the step time, t k , and the step 149 distance, x k , of each step by taken by the simplest walking and rimless wheel models.

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From this, the average speedv over two steps is computed as: Step durations and distances are dependent on step height d and on the initial angular, 152 or step, velocities✓ + 1 and✓ + 2 . Each initial step velocity needs to be sufficiently large in 153 order to overcome the apex of the COM trajectory and to complete the step in a finite 154 amount of time. Without adaptions in the initial step velocities, walking over uneven 155 terrain leads to larger average step durations,t = 1 2 (t 1 + t 2 ), and shorter effective step 156 lengths,x = 1 2 (x 1 + x 2 ). This, in turn, influences the average forward speedv (for 157 details, see Appendix S1). In order to understand the mechanical adaptations necessary 158 for walking on uneven terrain, we inverted this dependency; we constrainedv to a fixed 159 value, analogous to walking with a constant speed on a treadmill. This means that if 160 step height d is increased, initial step velocities must be adapted to maintain a desiredv. 161 This essentially introduces an implicit relationship between the initial step velocities of 162 the up and down steps. If the desired walking speed and one initial step velocity is 163 prescribed, the second initial step velocity can be uniquely determined. Once we 164 determined the relationship between✓ + 1 and✓ + 2 given a desired forward speed, we 165 computed the magnitudes of P and H that achieved this relationship.

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To compute the required mechanical work and associated metabolic effort to maintain a 168 desired forward speedv, we need to quantify the work done in Equations (4) and (6).

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In general, mechanical work done by an impulse,J, on a particle with mass m moving 170 at an initial velocityṽ t0 , can be computed as W =ṽ t0 ·J + 1 2m |J| 2 . Based on the 171 direction of push-offP (perpendicular to the direction of motion of the COM) and hip 172 impulseH (along the direction of motion of the COM) the work of these impulses can 173 be computed from their magnitudes according to: where W P and W H are the work done by the push-off and hip impulses, respectively 175 (similar to [15]). The work performed by the impulsive push-off is strictly positive, while 176 hip work can be positive or negative. If positive and negative work efficiency is defined 177 as ⌘ + and ⌘ , respectively, the metabolic cost of transport (COT, non-dimensional) 178 over two steps is given by: where W + H and W H are the positive and negative hip work, respectively.   preemptive push-off is impossible [14]. units [26]. Hence, all gaits had an average speed ofv=0.342 p lg, with swing leg natural 219 frequency of the simplest walking model equal to ! n = 1.02 rad/s.

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Based on the desired gait characteristics, we numerically identified repetitive, 221 steady-state gaits for each model that resulted in identical system states every two steps. 222 To this end, we performed simulations over two steps, consisting of one up and one 223 down step of equal height d. Numerical integration began immediately after heel-strike 224 leading into the up step and ended immediately after heel-strike following the down step. 225 Integration was interrupted at each heel-strike in order to define a new stance leg. At velocity˙ + according to Equation (5). Simulations were conducted over a range of 228 initial step velocities of the first step✓ + 1 , with the initial step velocity of the second step 229 ✓ + 2 calculated such that the average forward speed over the two steps was equal tov.

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This relationship between the two initial step velocities was determined for step heights 231 ranging from d=0 l to d=0.1 l. The simulated two-step gaits relied on a combination of 232 push-off and hip work for power, given that any desired post-collision stance leg velocity 233 can be achieved with an appropriate choice of impulse magnitudes P and H (per 234 Equations (4) and (6)). Impulses P and H were restricted to follow the energy input 235 strategies described previously. We identified all two-step periodic gaits numerically, 236 using a nonlinear root search (similar to [27]).

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We show that, given a desired average forward speed and step height d, the initial up 239 and down step velocities,✓ + 1 and✓ + 2 , were interdependent for the rimless wheel and 240 simplest walking models (Fig. 2). This dependency defined a range of possible gaits, 241 which was particularly limited for the simplest walking model due to step timing 242 constraints introduced by swing leg dynamics. For both models, cost of transport was 243 directly related to step height and the initial up-step velocity. The relationship between 244 these three variables was further influenced by the work input strategy (Fig. 3). In response to larger step heights was larger for the simplest walking model than for the 250 rimless wheel (Fig. 6). Finally, we compare these model results to observations of 251 humans walking on uneven terrain [5].

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All possible gait solutions for the rimless wheel model lay on an extended surface in the 254 space defined by✓ + 1 ,✓ + 2 , and step height d (Fig. 2). To maintain the desired average 255 speed ofv=0.342 p lg, smaller values in✓ + 1 required larger values in✓ + 2 and vice versa. 256 On level ground, the relationship between the initial step velocities was symmetric 257 about the point where the two velocities were equal ( Fig. 2A, insert). Increasing d shifts 258 the symmetry point in the direction of a larger initial up-step velocities and a smaller 259 down-step velocity.

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The most energetically economical choice of initial step velocities✓ + 1 and✓ + 2 261 depended on the work input strategy. When powered by positive variable (VP) or 262 constant (CP) push-off impulses, the most energetically optimal solution was practically 263 indistinguishable from gaits with identical initial step velocities, independent of step 264 height (Fig. 3A,B). In contrast, for gait powered only through hip work (HO), the most 265 energetically optimal gaits were those with equal apex velocities at mid-step (Fig. 3C). 266 These gaits generally required a larger initial velocity at the up-step than on the down 267 step. Gaits with with equal step durations were most energetically costly, independent 268 of work input strategy. The VP work input strategy tended to be the most economical 269 overall, but only allowed for a limited region of gaits to be powered purely by push-off 270 (light grey shaded part of the surface in Fig. 3A). As initial step velocities✓ + 1 and✓ + 2 271 diverged, one of the push-off impulses grew, while the other got smaller. Since push-off 272 impulses can only be positive, the smaller impulse had to be replaced by negative work 273 done at the hip as soon as it reached a value of zero (unshaded part of the surface).

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Furthermore, for highly unbalanced steps (✓ + 1 >>✓ + 2 ) and large values of d, the 275 push-off prior to the up-step became so large that it required a negative collision 276 impulse for the rimless wheel to remain on the ground. In this region (dark grey shaded 277 part of the surface) push-off values had to be constrained and, in order to maintain the 278 required average forward speed, supplemented with positive hip work.

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Differences in energy expenditure between the three work input strategies may be 280 examined in terms of the positive and negative work done at every step (Fig. 4). For the 281 energetically optimal VP-powered gait (green line, Fig. 3), larger step heights d required 282 increasing push-off before the up-step and decreasing push-off before the down-step.

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Since a push-off impulse helps reduce impact velocity, larger push-offs were accompanied 284 by smaller subsequent collision losses. Conversely, smaller push-offs resulted in larger terrain.

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In contrast to CP-and VP-powered gaits, HO-powered gaits can only provide work 296 input after collision, leading to much larger energy losses during collision. As d 297 increased, more hip-work was done in the up-step and less in the down-step, with no hip 298 work done in the down-step at very high values of step height. As a result, HO-powered 299 gaits required more positive work for walking on level (0.116 gml at d=0 l) and uneven 300 (0.124 gml at d=0.1 l) ground when compared to gaits powered by other work input 301 strategies. This is consistent with past analyses of the effects of using hip work on total 302 energy expenditure [20]. On level ground, powering gait only through hip work led to 303 an almost three-fold increase in COT compared to other work input strategies. However, 304 the rate of growth in total positive work as a function of d was smaller for HO-powered 305 gaits compared to the VP-and CP-powered gaits. None of the energetically optimal 306 rimless wheel gaits required negative hip-work, independent of work input strategy.

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Thus, optimal COT results are equivalent to the positive work performed.

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All of the strategies exhibited increasing COT with step height, although varying 309 considerably in magnitude (Fig. 6). The rimless wheel with variable push-off had least 310 COT, but even constant push-off was more economical than the simplest walking model. 311 In addition, the COT increased approximately quadratically with step height 312 (R 2 > 0.99) for all rimless wheel models, including with costlier HO-power.

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In contrast to the rimless wheel, the simplest walking model did not allow for gaits with 315 significant differences in up and down step duration. This is because step timing was 316 dependent on time required for the passive swing leg to travel forward. Consequently, 317 simplest walking model solutions had a smaller range of possible initial step velocities 318 ✓ + 1 and✓ + 2 . In fact, the range of possible velocities formed a closed loop in the✓ + 1 and 319 ✓ + 2 space ( Fig. 2A, insert). Over increasing step height d, the loop widened to allow for 320 gaits with more variable step durations (Fig. 2B).

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Similarly, simplest walking model COT surfaces also formed a tubular manifold, 322 which lay in proximity to rimless wheel gaits with equal step durations (Fig. 3). As a 323 result, the COT of the energetically optimal gaits of simplest walking model was 324 significantly higher compared to optimal gaits of the rimless wheel, independent of work 325 input strategy. Simplest walking model gaits with the largest difference in step time 326 tended to be the most energetically optimal on surfaces with larger step heights d.

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Similar to the rimless wheel, optimal VP-powered gaits were most economical for the 328 simplest walking model at all step heights when compared to gaits using other work 329 input strategies (Fig. 6, lower bound of shaded regions). However, in contrast to the 330 rimless wheel, optimal HO-powered gaits had lower COT values than CP-powered gaits 331 at higher values of step height d (all d greater than ⇡ 0.05l).

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On level ground, the most energetically optimal simplest walking model gaits for 333 each work input strategy required the same amount of positive work as their respective 334 rimless wheel counterparts (Fig. 5, first row). As step height increased, the simplest 335 walking model required more positive and negative work during the up and down steps, 336 respectively, when compared to the rimless wheel. This was caused by the need to 337 maintain similar step durations, which consequently led to larger total negative work 338 requirements. At lower step heights, and for VP-and HO-powered gaits, the increase in 339 negative work could largely be attributed to greater, passive collision costs. At higher 340 step heights, much of the negative was also actively generated by the hip. This is in 341 contrast to the rimless wheel, for which negative work was always passive and did not 342 contribute to the total COT. For CP-powered gaits, negative work was done actively 343 through the hip even at lower step heights. This led to CP-powered gaits being more 344 energetically expensive than HO-powered gaits at larger step heights (Fig. 5, second 345 row), since additional negative hip work required by CP-powered gaits was more costly 346 than the additional positive work required by HO-powered gaits.

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We explored the mechanical effects of terrain unevenness on the cost of transport for the 349 rimless wheel and simplest walking models. We simulated an uneven surface of equal up 350 and down steps, and found periodic gaits for both models over a range of step heights. 351 Uneven terrain could entail greater mechanical work for three main reasons. First, 352 collision losses and cost of transport increase approximately quadratically with step 353 height variations, for fixed step lengths. Those losses must then be restored through 354 positive work, most economically with push-off and less so with hip work. Second, 355 uneven surfaces limit the ability to predict heel-strike events, because push-off is most 356 economical if timed pre-emptively before heel-strike [15,16]. Otherwise, less economical 357 hip work (or late push-off, [14,25]) would be needed to maintain nominal speed. Third, 358 swing leg dynamics also influence collision losses. The passive swing leg of the simplest 359 walking model yields steps of varying length and duration, in contrast with the fixed 360 step lengths of the rimless wheel. More variable steps have higher collision losses on 361 average, and therefore require more positive work. It might be less costly overall to 362 actively control the swing leg, depending on the trade-off its energetic cost [28] against 363 a passive swing leg.

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Although uneven terrain is more costly, it is fundamentally similar to level ground. 365 In an inverted pendulum model, kinetic and potential energy are exchanged 366 conservatively, and positive work adds to that energy. The only dissipation is in the 367 heel-strike collision. This remains the case on uneven terrain, except for greater collision 368 losses. Down-steps dissipate more energy, and up-steps less energy than nominal gait.

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Up-steps lose more time and speed, and the opposite for down-steps. But the differences 370 are asymmetric, so that the losses are greater, and require more positive work with step 371 height, even if there is no net height change over the two steps.

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It is instructive to compare our model results with observations of humans walking 373 on uneven terrain. The most relevant human experiments include mechanical and 374 metabolic human data on level ground and on uneven terrain, with a 2.54 cm maximum 375 step height [5], roughly analogous to the present model with d = 0.013 l. The net 376 metabolic rate of walking humans increased by approximately 28% (from 2.65 W/kg to 377 3.38 W/kg, respectively) [5]. This corresponds to a COT increase from 0.093 to 0.118 378 (Fig. 6).

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Such comparison allows for quick elimination of two models: vertical center of mass 380 motion and hip-powered gait. For comparable height variation, humans walk with 381 considerably less cost than the vertical center of mass model ( E VCOM in Equation (1) 382 and Fig. 6), and the HO-powered strategy. The E VCOM cost ignores all joint motion or 383 forward motion of the body, and yet still predicts higher costs than human. Other 384 models show uneven terrain may be negotiated for less cost than the vertical motion 385 would imply ( [14,25]), and previous studies show that E VCOM is a poor predictor for 386 even level walking [12]. The HO-powered gait is far more costly than humans, even on 387 level ground (Fig. 6). This is supported by both models and empirical data, which show 388 that it is disadvantageous to input work via hip alone [15,16,29,30]. We therefore do 389 not consider either model explain the metabolic cost of human walking on level or 390 uneven terrain.

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The remaining models all suggest that humans should use push-off to power gait.

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The models gain better economy by applying pre-emptive push-off, for both level and 393 uneven terrain. And if push-off cannot be timed precisely and varied appropriately, it is 394 still more economical to apply constant push-off with added hip work (CP-powered 395 gait), rather than hip work alone. On comparison terrain (d = 0.013 l), humans produce 396 ankle push-off power comparable to level ground, albeit with more variable timing 397 (about 25% more step period variability, no significant differences in ankle work [5]). It 398 is unclear whether human push-off is limited by ankle power or poorer timing, but the 399 additional work needed for uneven terrain appears to be supplied by the hip (about 60% 400 more positive hip work [5]). None of the models make good numerical predictions for 401 human costs, but they do provide a mechanistic basis for using push-off, and they 402 approximately predict how costs should increase with step height (Fig. 6).

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The models also suggest that it would be uneconomical for humans to allow the 404 swing leg to move passively. The resulting collision losses are considerably higher than 405 with fixed step lengths (VP simplest walking model vs. rimless wheel, Fig. 6). The 406 human swing leg does have dynamics, unlike the rimless wheel, but can also be actively 407 controlled, unlike the simplest walking model. Humans appear to actively control the 408 swing leg and step length [31], with a metabolic cost [28]. That cost might be preferable 409 to the collision losses predicted for a passive swing leg.

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A limitation of this study was the use of a regular pattern of equal up and down 411 steps for uneven terrain. It would be more realistic to model stochastic terrain structure. 412 This would be expected to have little effect on the examination of work input strategies, 413 since the strategies only assumed either full or zero ability to adapt to the terrain. In 414 addition, simulations with stochastic terrains show very similar trends to the patterned 415 terrain, albeit with greater effects on speed and other variables such as step duration or 416 COT (see Appendix S1). Another limitation is that we also modeled relatively small 417 terrain variations, which would be expected to induce less timing variability than than 418 larger variations. For greater step heights, we would expect push-off timing to be more 419 critical, and more likely to require hip work. As an extreme case, a single very high step 420 or sequence of up-steps could bring the model to a complete halt, and require additional 421 work input. We suspect larger step heights might cause hip work to become more 422 dominant.

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As a further limitation, we excluded the question of control from our study, focusing 424 only on periodicity and not stability. In particular, our models relied on simple impulse 425 control. However, simple model analysis shows an increased risk of falling on stochastic 426 terrain [32]. Control optimization methods for walking models have been proposed to 427 improve gait stability on uneven terrain [33,34], and push-off could be optimized to 428 simultaneously maintain stable gait and minimum work input [14,25]. More 429 sophisticated control methods could be used to study trade-offs between energy 430 economy and perturbation rejection.

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Our models are also drastically simplified compared to human. Notably, humans 432 perform more positive and especially negative work at the knee on uneven terrain [5], 433 and may dissipate collision losses more actively than modeled here. It would also be 434 helpful to model the torso, which can be useful for stabilizing walking in response to 435 perturbations [35]. We also assumed a fixed relationship between work and energy 436 expenditure, which is quite complex in human. Not only are there non-work 437 contributions to energetic cost such as force production ( [28]) and muscle co-activation, 438 but other factors such as body posture, altered gait kinematics (e.g. to increase swing 439 foot clearance), and soft tissue deformation [36] can all affect energy expenditure. It is 440 possible that performing an energetic analysis with a more complex model and the 441 addition of gait stability analysis and control, would allow more insight into how 442 humans balance the trade-off between stability and energetic cost.

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Uneven terrain poses increasing mechanical demands on bipedal walking. There 444 appear to be multiple ways to meet those demands by performing work to maintain 445 forward speed despite unevenness. In our models, trailing leg push-off appears to be an 446 economical means to add work, but only if it can be timed appropriately, and not 447 limited by saturation or other constraints. Hip work is less economical, but also far 448 more forgiving of timing. Humans appear to retain similar push-off on uneven terrain, 449 but more of the modulation appears to take place with hip. Simple models cannot