Humans Adapt Multi-Objective Control of Stepping To Perform Lateral Maneuvers While Walking

To successfully traverse their environment, humans must often perform maneuvers to achieve desired task goals while simultaneously maintaining balance. Humans accomplish these tasks primarily by modulating their foot placements. As humans are more unstable laterally, we must better understand how humans modulate lateral foot placement. We previously developed a theoretical framework and corresponding computational models to describe how humans regulate lateral stepping during straight-ahead continuous walking. This framework yields goal functions for step width and lateral body position that define the walking task and determine the set of all possible task solutions as Goal Equivalent Manifolds (GEMs). Here, we used this framework to determine if humans can regulate lateral stepping during non-steady-state lateral maneuvers by minimizing errors in accordance with these goal functions. Twenty young healthy adults each performed four lateral lane-change maneuvers in a virtual reality environment. Within our general lateral stepping regulation framework, we first re-examined the requirements of such transient walking tasks. Doing so yielded new theoretical predictions regarding how steps during any such maneuver should be regulated to minimize error costs, consistent with the goals required at each step and with how these costs are adapted at each step during the maneuver. Humans performed the experimental lateral maneuvers in a manner consistent with our theoretical predictions. Furthermore, their stepping behavior was well modeled by merely adapting the parameters of our previous lateral stepping models from step to step. To our knowledge, our results are the first to demonstrate humans can use evolving cost landscapes in real time to perform such an adaptive motor task and, furthermore, that such adaptation can occur quickly – over only one step. Thus, the predictive capabilities of our general stepping regulation framework extend to a much greater range of walking tasks beyond just normal, straight-ahead walking. AUTHOR SUMMARY When we walk in the real world, we rarely walk continuously in a straight line. Indeed, we regularly have to perform other tasks like stepping aside to avoid an obstacle in our path (either fixed or moving, like another person coming towards us). While we have to be highly maneuverable to accomplish such tasks, we must also remain stable to avoid falling while doing so. This is challenging because walking humans are inherently more unstable side-to-side. Sideways falls are particularly dangerous for older adults as they can lead to hip fractures. Here, we establish a theoretical basis for how people can accomplish such maneuvers. We show that humans execute a simple lateral lane-change maneuver consistent with our theoretical predictions. Importantly, we show that they can do so using simple adaptations of the same step-to-step regulation strategies they use to walk in a straight line. Moreover, these same control processes also explain how humans trade-off side-to-side stability to gain the maneuverability they need to perform such lateral maneuvers.

person coming towards us). While we have to be highly maneuverable to accomplish such tasks, we must also 28 remain stable to avoid falling while doing so. This is challenging because walking humans are inherently 29 more unstable side-to-side. Sideways falls are particularly dangerous for older adults as they can lead to hip 30 fractures. Here, we establish a theoretical basis for how people can accomplish such maneuvers. We show that 31 humans execute a simple lateral lane-change maneuver consistent with our theoretical predictions. 32 Importantly, we show that they can do so using simple adaptations of the same step-to-step regulation 33 strategies they use to walk in a straight line. Moreover, these same control processes also explain how humans 34 trade-off side-to-side stability to gain the maneuverability they need to perform such lateral maneuvers. 38 To successfully traverse our environment, we humans must adapt to a wide variety of environment contexts 39 and changing task goals (e.g., Fig. 1A), all while maintaining balance. Indeed, humans readily avoid obstacles 40 [2] and/or other humans [3], step to targets [4][5][6], move laterally [7][8][9], navigate complex terrain [10][11][12], and 41 respond to destabilizing perturbations [13,14]. To do so requires a high degree of maneuverability [7] and, determine how humans adjust successive stepping movements [1,29,34,36]. For a given walking task (e.g., 66 Fig. 1A), this framework proposes goal functions that theoretically define the task and, accounting for 67 equifinality, determine the set of all possible task solutions as Goal Equivalent Manifolds (GEMs) [37][38][39][40]. 68 The goal functions are incorporated into task-level costs, which we then use in an optimal control formalism 69 [41,42] to generate relatively simple, phenomenological models of step-to-step motor regulation (i.e., "motor 70 regulation templates" [1,29,30]). For any proposed goal function, these templates predict how humans 71 would attempt to drive that goal function to zero at the next step, thus achieving perfect task performance on 72 average. For an appropriate choice of goal functions, these models successfully replicate human step-to-step dynamics in both the fore-aft [29] and lateral [1] directions. In the lateral direction in particular, multi- 74 objective regulation of primarily step width (w) and secondarily lateral position (z B ) captures human step-to- 75 step dynamics during continuous straight-ahead walking ( Fig. 1B-C) [1,13,43]. Whether explicit models are 76 constructed or not, this overall theory and its hierarchical control/regulation schema provides a powerful 77 framework from which to interpret experimental results [13,31,[43][44][45][46][47][48][49][50]. 78 79 However, humans rarely perform long bouts of straight-ahead, continuous walking [51,52]. Instead, humans 80 must frequently perform adaptive locomotor behaviors (i.e., maneuvers) when walking in the real world (e.g., 81 Fig. 1A) [7][8][9]. Indeed, older adults often fall during such maneuvers because they incorrectly transfer their 82 body weight or they trip [53]. Because both such causes can be prevented with appropriate foot placement 83 [18][19][20][21], it is necessary to better understand how humans regulate foot placement as they execute typical 84 lateral maneuvers. Here, we aimed to determine how humans regulate their stepping during a simple lateral 85 maneuver: namely, a single lateral lane-change transition between periods of straight-ahead walking. 86 87 It is not clear a priori that our previous lateral stepping regulation framework can also emulate human 88 stepping during lateral maneuvers. This framework was originally developed to model straight-ahead steady- 89 state walking, under assumptions that step-to-step adjustments can be made without changing the fundamental 90 structure of within-step control, and that deviations from perfect performance are small. These assumptions 91 motivated us to select low-dimensional, single-step, linear regulators to model these step-to-step error- Participants completed nearly all lateral maneuvers in 4 non-steady-state steps. Participants first took a ; blue) and right (z R ; red) foot placements for (i) 38 transitions when the cue was given on a contralateral step relative to the direction of transition, (ii) 41 transitions when the cue was given on an ipsilateral step relative to the direction of transition, and (iii) all 79 transitions with respect to the initiation of the transition, defined as the last step taken on the original path. In (i)-(ii), the black dotted lines at step 0 indicate the onset of the audible cue. All transitions are plotted to appear from left to right. B) Time series of lateral position (z B ) and step width (w) for all transitions with respect to the initiation of the transition, with all transitions plotted to appear from left to right. C) Errors with respect to the stepping goals, [z B * , w * ]. For steps in the interval [-3, 0], z B * was defined as the experimental mean z B during steady state walking before the transition. For steps in the interval [1,7], z B * was defined as the experimental mean z B during steady state walking after the transition. For all steps analyzed, w * was defined as the experimental mean w during steady state walking. Error bars indicate experimental standard deviations at each step. Gray shaded regions indicate the mean standard deviation (± 1) from all steady state walking steps. 139 We first used our previous multi-objective model of lateral stepping regulation to test whether humans could 140 regulate stepping during lateral maneuvers using a constant regulation strategy like that used for continuous, 141 straight-ahead walking [1]. This model selects each new foot placement (z L or z R ) as a weighted average of 142 independent predictions that minimize errors with respect to either a constant step width (w * ) or lateral 143 position (z B * ) goal, consistent with multi-objective stochastic optimization of error costs with respect to these 144 two quantities (see Methods). For this model, the relative proportion of step width to lateral position 145 regulation was defined by ρ, where ρ = 0 indicates 100% z B control (and hence, 100% weight on the z B cost) 146 and ρ = 1 indicates 100% w control (100% weight on the w cost) [1]. This stepping regulation model 147 reproduced the key features of human stepping dynamics during continuous, straight-ahead walking for 148 constant values in the approximate range 0.89 ≤ ρ ≤ 0.97 [1]. 149 150 We assessed whether this model, with any constant value of ρ, could emulate the non-steady-state stepping 151 dynamics experimentally observed during the lateral maneuver task. We found that it was not capable of  Tasks   162 Any biped (human, animal, robot, etc.) must enact step width and/or lateral position regulation via left and 163 right foot placement (Fig. 1C). The stepping goals, [z B * , w * ], which guide steady-state walking, individually 164 form diagonal, orthogonal Goal Equivalent Manifolds (GEMs) when plotted in the [z L , z R ] plane (Fig. 4A) [1]. 165 The intersection of these GEMs represents the multi-objective goal to maintain both z B * and w * and therefore 166 defines the foot placement goal, [z L * , z R * ], for the task. When viewed in this manner, it is evident that nearly 167 any substantive change in either stepping goal (i.e., Δz B * and/or Δw * ) will induce a diagonal shift of the corresponding GEM(s) in the [z L , z R ] plane (Fig. 4A). This will necessitate corresponding changes in both left 169 and right foot placement, which therefore cannot be accomplished in a single step (Fig. 4A). At minimum, two consecutive steps are required to execute nearly any maneuver involving some Δz B * and/or  [1,7], we set z B * as the experimental mean z B during steady state walking after the transition. For all steps, w * was defined as the experimental mean w during steady state walking. B) Stepping errors (mean ± SD) at each step relative to the stepping goals, [z B * ,w * ], for the same data as in (A). For both (A) and (B), gray bands indicate the middle 90% range from experimental data. C) Means and standard deviations of both regulated variables (z B and w) during a steady state step (Step -3; left) and during the transition step (Step 1; right) for all values of 0 ≤ ρ ≤ 1. Gray bands indicate 95% confidence intervals from the experimental data computed using bootstrapping. Green bands indicate ±1 standard deviation from 1000 model simulations at each value of ρ. Model simulations over the approximate range of 0.83 ≤ ρ ≤ 0.92 fell within the experimental ranges for all variables for steady-state walking (Step -3; Left), as indicated by the region highlighted in yellow. However, no such range captured the experimental data during the transition step (Step 1; Right).

Re-Thinking Stepping Regulation for Non-Steady-State
Δw * (Fig. 4B). The first step must be taken by either the left or right foot to either of two possible placements of steps that can be used to accomplish any Δz B * and/or Δw * maneuver. For example, we 176 experimentally observed a four-step strategy to be most typical (Fig. 2). To represent such a strategy, we  plane, goals to maintain constant position (z B * ) or step width (w * ) each form linear Goal Equivalent Manifolds (GEMs) that are diagonal to the z L and z R axes and orthogonal to each other. Any change from some initial (green) to final (blue) stepping goals, such as the theoretical rightward shift in z B * (Δz B * ) and increase in w * (Δw * ) depicted here, displaces these GEMs diagonally in the [z L , z R ] plane. This requires changes in both z L and z R to accomplish, and so cannot be achieved in any single step. B) At least two consecutive steps (z R →z L , or z L →z R ) are required to accomplish a change in stepping goals (Δz B * and/or Δw * ). Both possible intermediate steps (labeled 'a' or 'b') have their own distinct stepping goals, [z B * , w * ]. C) An idealized four-step strategy, like that most commonly observed experimentally (Fig. 2), includes 3 intermediate steps, each with distinct stepping goals. D) Goal relevant deviations (δ zB and δ w ) with respect to both the z B * and w * GEMs. These deviations characterize the stepping distribution at a given step, reflecting the relative weighting of z B and w regulation. E) During steady-state walking, humans exhibit stepping distributions strongly aligned to the w * GEM, reflecting strong prioritization of step width over position regulation. For a two-step maneuver strategy (as in B), stepping distributions at the intermediate step are theoretically predicted to be approximately isotropic (i.e., nearly circular). F) For a four-step maneuver strategy with a large primary transition step and smaller preparatory and recovery steps (as in C), stepping distributions are theoretically predicted to be most isotropic at the primary transition step and intermediately isotropic at the preparatory and recovery steps. preparatory, transition, and recovery steps (Fig. 4C). However, specifying these foot placements alone does not capture how any given biped might perform these 182 steps. Along any GEM, deviations tangent to the GEM are "goal equivalent" because they do not introduce 183 errors with respect to the goal. Conversely, deviations perpendicular to the GEM are "goal relevant" because 184 they do introduce such errors [29]. Humans typically exhibit greater variability along GEMs they exploit [37, 185 41, 42]. For multi-objective lateral stepping regulation, the z B * and w * GEMs are orthogonal (Fig. 4A). Thus, 186 goal equivalent deviations with respect to either GEM are goal relevant with respect to the other (Fig. 4D) [1]. 187 Hence, the ratio of the δ zB and δ w deviations with respect to both the z B * and w * GEMs theoretically reflects the 188 relative weighting of z B and w regulation. For example, during steady-state walking, the distribution of human 189 steps is anisotropic: that is, the steps are strongly aligned along the w * GEM (such that δ zB /δ w >> 1) because 190 humans heavily weight step width over position regulation [1].  regulator's relative importance, ρ, and the additive noise, σ a , that represents the strength of physiological 232 perceptual/motor noise. Here, we conducted three sequential numerical experiments to assess the effects of adapting each of these model parameters on key stepping dynamics: time series, errors, and variance 234 distributions. We first incorporated adaptive stepping goals that we derived theoretically to approximate an 235 idealized four-step transition strategy (Fig. 4C). Next, we adapted the proportionality parameter, ρ, to reflect 236 the predicted stepping distributions (Fig. 4F). Finally, we increased the additive noise to reflect the observed 237 increase in variability during the lateral maneuver. Importantly, we did not attempt to precisely estimate the 238 model parameters, but rather selected parameters based on the idealized theoretical considerations described 239 above. This approach is analogous to using "templates" of legged locomotion to reveal the basic principles of 240 human walking by testing fundamental hypotheses about the underlying regulation strategies [59,60]. the stepping goals to adapt from step to step appears necessary, but is not sufficient to elicit experimentally 256 plausible stepping dynamics during lateral maneuvers.
Next, we added the ability to adaptively modulate ρ at each step ( heavily weights regulating step width over lateral position (i.e., ρ ≈ 0.9) [1], presumably to maintain lateral 261 stability. Conversely, we expect people to trade off stability to gain maneuverability [8,9] while executing 262 this maneuver. We therefore set ρ = 0.50 at the transition step (Fig. 7A) to specify equal weighting of step Here, stepping goals, [z B * ,w * ] were updated at each step to reflect an idealized four-step transition strategy. The control proportion (ρ) and additive noise (σ a ) were held constant across all steps. B) Stepping time series (mean ± SD) of 1000 simulated lateral transitions using the parameters in (A). C) Stepping errors (mean ± SD) for the simulations in (B). In both (B) and (C), gray bands indicate the middle 90% range from experimental data. D) Simulated stepping data from the preparatory, transition, and recovery steps projected onto the [z L ,z R ] plane. Gray ellipses represent 95% prediction ellipses at each step from the experimental data. Blue ellipses represent 95% prediction ellipses from the 1000 simulated lateral transitions. The diagonal dotted lines indicate the predicted constant-z B * and constant-w * GEMs at the preparatory, transition, and recovery steps. E) Ellipse characteristics (mean ± SD) at each step (as defined in Fig. 5): aspect ratio (top), area (center), and orientation (bottom). Gray bands indicate ±95% confidence intervals from the experimental data derived using bootstrapping. Adapting the stepping goals alone yielded experimentally plausible stepping time series (B), errors (C), and locations (D), but not stepping distributions (D). Thus, adaptive stepping goals are necessary but not sufficient to replicate human stepping during lateral maneuvers. width and position regulation (Fig. 4F), thereby maximizing maneuverability. We then set ρ = 0.7 at the 264 preparatory and recovery steps (Fig. 7A) to reflect an intermediate multi-objective cost weighting (Fig. 4F). 265 The same adaptive stepping goals (Fig. 6) were again incorporated here. All other model parameters were 266 assigned constant values across all steps (Fig. 7A). 267 268 Therefore, in addition to adapting both the stepping goals (Fig. 6) and ρ (Fig. 7) from step to step, we then 269 doubled the additive noise (σ a ) in the model at the preparatory and transition steps (Fig. 8A). Additive noise is  Fig. 6), the control proportion (ρ) was also varied during the preparatory, transition, and recovery steps to reflect the hypothesized maneuverability and error correction at each step. Additive noise (σ a ) remained unchanged from the original, constant parameter model. B-E) Results obtained from 1000 simulated lateral transitions using the parameters in (A), with data plotted in an identical manner to Fig. 6B-E. Adding step-to-step modulation of ρ again captured experimental stepping time series and errors at each step (B-C). Here however, allowing ρ to adapt at each step also induced changes in prediction ellipse aspect ratios during the transition steps that were qualitatively similar to those observed experimentally. Modulating ρ however, did not induce corresponding changes in ellipse area (D-E). Thus, adapting ρ at each step is also necessary but not sufficient to emulate human stepping during lateral maneuvers. thought to reflect physiologic noise from a variety of sources, including sensory, perceptual, and/or motor 271 processes [29]. Here, we assessed whether increasing this additive noise could emulate the increases in the 272 stepping distribution areas observed experimentally (Fig. 5B-C).  (Fig. 8B-D) and qualitatively captured the shapes of the stepping distributions (Fig. 8D-E). Here, in addition to adapting the stepping goals (Fig. 6) and ρ (Fig. 7), additive noise (σ a ) was also doubled at the preparatory and transition steps. B-E) Results obtained from 1000 simulated lateral transitions using the parameters in (A), with data plotted in an identical manner to Figs. 6-7. Adding σ a modulation again emulated experimental stepping time series and errors (B-C), as well as the qualitative changes in the aspect ratio of the fitted ellipses during the maneuver (D-E). Furthermore, modulating σ a emulated the experimentally observed increases in ellipse areas. Modulating σ a also affected orientations of the fitted ellipses, although not entirely in the same ways as the experimental data (D-E). Therefore, adaptively modulating the stepping goals ([z B * , w * ]), control proportion (ρ), and additive noise (σ a ) in our existing model can elicit changes in stepping dynamics qualitatively similar to those observed in humans during this lateral maneuver.
Increasing σ a also increased the areas of the stepping distributions at the preparatory, transition, and recovery 278 steps ( Fig. 8D-E). Interestingly, increasing σ a also influenced the orientations of these distributions, inducing 279 a clockwise rotation at the preparatory step and a counterclockwise rotation at the transition step (Fig. 8D-E). Understanding how humans perform accurate, goal-directed walking movements in the face of inherent 293 variability, redundancy, and equifinality remains a fundamental question in human motor neuroscience. In 294 pursuit of this aim, we previously developed a theoretical framework to describe how humans regulate 295 stepping to achieve continuous, straight-ahead walking [1]. Models developed from this framework provide 296 goal-directed "stepping regulation templates" that are both analogous and complimentary to mechanical 297 templates that describe the within-step mechanics and dynamics of walking (e.g., [22,23,32,33]). Here, 298 because humans rarely perform long bouts of steady-state walking [51,52], we therefore examined how our 299 basic regulation template scheme could be used to model human stepping dynamics during a prescribed 300 lateral maneuver that, by necessity, strongly deviated from steady state motion.

302
Our key theoretical contribution is that we show how our previously developed theoretical stochastic optimal 303 control framework can model how humans adapt their lateral stepping to enact non-steady-state lateral 304 maneuvers (Fig. 4). Specifically, we show this can be accomplished by adding an additional layer to our 305 previous control hierarchy: namely, the processes responsible for adapting, on a step-by-step basis, the cost 306 landscapes governing stepping regulation. Our key empirical contribution is we then demonstrate (Fig. 5) that 307 humans do indeed execute lateral maneuvers in a manner consistent with our theoretical predictions. Our key 308 computational contribution is that we show that allowing the parameters in our lateral stepping regulation 309 model to adapt from each step to the next can emulate the changes in lateral stepping dynamics exhibited by 4), only loosely related to general trends observed in our experiment. While our resulting simulations did not 316 precisely "fit" the experimental data, they were not intended to. On the contrary, our aim was to demonstrate 317 that reasonable values of the parameters could yield the same basic dynamical and statistical stepping features 318 that humans exhibited during this lateral maneuver task (Fig. 8). This approach is directly analogous to using 319 mechanical "templates" of within-step dynamics of legged locomotion to reveal basic principles of walking 320 and to propose fundamental hypotheses about what high-level control strategies might be acting [59,60]. 321 Here, we used our theoretical framework to derive such hypotheses (Fig. 4). We then used both experiments Many studies have used optimal control theory to model a myriad of individual motor tasks (e.g., [61][62][63][64][65]). 326 Such efforts, however, did not consider tasks where the task goals and/or movement objectives change as the 327 task is being performed. Likewise, many studies have addressed motor adaptation (e.g., [66,67]) and/or 328 motor learning (e.g., [68,69]), including for walking tasks (e.g., [70][71][72]). These paradigms, however, track 329 how task performance changes slowly over many repetitions (typically at least 10's to 100's), and not from 330 one repetition to the next. Furthermore, while multiple studies have demonstrated that such longer-term 331 adaptation can be replicated by "lag-1" type computational models that explicitly correct only for errors 332 experienced on one previous iteration of the task [73][74][75][76][77][78][79], those models were not themselves "adaptive". On 333 the contrary, they presumed some constant process (with constant model parameters) that achieved adaptation 334 over multiple repetitions of the task under consideration. In sum, none of these extremely well-trodden 335 paradigms quite capture the nature of the task we studied here. 336 337 Conversely, rapid adaptation in response to changing environmental contexts has been demonstrated in both 338 birdsong [80][81][82] and human speech [83]. This apparent plasticity of a well learned, crystalized behavior 339 suggests that vocalization may be controlled by a "malleable template," in which trial-by-trial variability is 340 used to adapt learned behaviors [84]. Such rapid adaptations have also been observed in both animal [15,16,341 [85][86][87][88] and human [7][8][9] locomotion (i.e., maneuvers). As the associated motor planning processes occur 342 nearly instantaneously [89][90][91], this prior work supports our findings that humans can and do make rapid 343 adaptations to their stepping regulation to enact lateral maneuvers. Thus, in demonstrating that our lateral 344 stepping regulation framework successfully predicts how humans perform lateral maneuvers, our findings 345 support the notion that humans use a malleable template [84] to make "embodied decisions" [6] about how to 346 regulate their stepping movements in real time during ongoing locomotion. 347 8]. However, in the absence of a coherent, predictive, theoretical framework, this stability-maneuverability 350 trade-off has not been adequately defined, much less confirmed. We propose that our lateral stepping 351 regulation framework, and the models derived from it, provides the necessary theoretical and computation 352 foundation needed to describe how humans trade-off stability for maneuverability during lateral movements. 353 By extension, our findings suggest that stability and maneuverability in the context of locomotion are not 354 distinct and independent concepts, but rather different manifestations of the same underlying stepping 355 regulation process. Indeed, if stability and maneuverability were independent, as often assumed, humans 356 should be able to remain stable and maneuverable simultaneously. However, our theoretical framework (Fig.   357 4) demonstrates precisely how and why humans must trade-off some degree of stability (i.e., w-regulation) to 358 gain the maneuverability needed to perform lateral maneuvers. Walking in the real world often requires maneuverability to adapt to changing environmental conditions or 361 goals. We suggest that such maneuvers are governed by a hierarchical control/regulation schema with at least 362 three distinct layers: low-level processes that govern within-step dynamics to ensure viability [34,59,60], 363 step-to-step regulation to achieve goal-directed walking [1,29,34,36], and the presently demonstrated 364 modulation of stepping regulation to achieve adaptability. The ability of our lateral stepping regulation 365 framework to emulate human stepping during the lane change maneuver studied here demonstrates that its 366 predictive capabilities extend to a much greater range of walking tasks than initially thought, encompassing 367 not just steady state walking, but transient behaviors as well.  The experimental protocols were described in detail previously [58]. Briefly, participants walked in an "M-385 Gait" system, comprised of a 1.2m wide motorized treadmill in a virtual reality environment (Motek, 386 Amsterdam, Netherlands). Each participant walked at a constant speed of 0.75 m/s. Following a 4-minute 387 acclimation trial, participants completed several different walking trials involving path navigation. The data 388 analyzed here were generated from one such trial, during which participants were instructed to switch 389 between two parallel paths, centered 0.6m apart, following an audible cue (Fig. 2). Participants completed 6 390 maneuvers during one 4-minute walking trial, and were instructed to walk normally on their current path 391 between maneuvers. The first and last maneuvers occurred too close to the beginning and end of the walking 392 trial, respectively, to ensure participants were walking normally both before and after the maneuver. strikes were determined using a velocity-based detection algorithm [92]. Lateral foot placements (z L and z R ) 401 were defined as the lateral location of the heel marker at each step.
Step width (w) and lateral position (z B ) 402 were then determined at each step using Eq (1) (Fig. 1C). determined at each step of each maneuver as the differences in each from these steady-state stepping goals 415 (Fig. 2C).

418
To model lateral stepping, the simplest, mechanically sufficient biped includes the lateral locations of the 419 center-of-mass and each of the two feet (z Ln and z Rn ; Fig. 1B) [32,93,94]. We presume that left and right foot 420 placement are coordinated to achieve some more general walking task goal or goals. One such goal is to 421 maintain lateral balance by regulating step width (w n ; Fig. 1C) [23,[95][96][97]. Humans also regulate the lateral 422 position of the body's mass center with respect to their path [18,94,98,99], approximated by the midpoint 423 between the two feet (z Bn ; Fig. 1C) during upright walking [93,95]. By selecting appropriate foot placements, The control inputs, u zB (z Bn ) and u w (w n ), were derived analytically as stochastic optimal single-step regulators 442 with direct error feedback [1,29,30,40], following the Minimum Intervention Principle [42,64]. Such 443 controllers are optimal with respect to the following quadratic cost functions: The first term of each cost function penalizes errors with respect to the goal function (Eq (2)) at the next step. 448 The second term penalizes "effort", quantified as the magnitude of the control input. Here, α and γ were 449 positive constants that weighted the two terms of each cost function [1]. The subsequent analytical derivation 450 [1] yielded control inputs as: where ρ = 0 indicates 100% position control and ρ = 1 indicates 100% step width control [1]. Hence, stepping 459 regulation is conceived here as arising from a "mixture of experts" [63,100]. In this way, the value of ρ Stepping Regulation for Non-Steady-State Tasks 483 When the z B * and w * GEMs are viewed in the [z L , z R ] plane, it is clear that at least one intermediate step is 484 necessary to accomplish any Δz B * and/or Δw * maneuver (Fig. 4A). For the minimum two-step maneuver 485 strategy (Fig. 4B), the stepping goals for either transition step can be determined algebraically from the 486 steady-state initial, [z Bi * , w i * ], and final, [z Bf * , w f * ], stepping goals. We first determined the initial and final foot 487 placement goals from the initial and final stepping goals using Eq (1). The foot used to take the intermediate 488 step must be placed at that foot's final foot placement goal, while the stance foot remains at its initial foot 489 placement goal. For example, if the right foot is used to take the intermediate step ( Fig. 4B; a) However, equifinality exists in the number of steps and their placements that can be used to accomplish any 501 Δz B * and/or Δw * maneuver. Experimentally, we most commonly observed a four-step maneuver strategy with 502 distinct preparatory, transition, and recovery steps (Figs. 2, 4C) The stepping goals themselves, however, do not provide insight into how humans regulated their stepping 513 during the lateral maneuver. For a four-step maneuver strategy with a large primary transition step and 514 smaller preparatory and recovery steps, as was observed experimentally (Fig. 2), stepping distributions are 515 theoretically predicted to be most isotropic at the primary transition step and intermediately isotropic at the 516 preparatory and recovery steps. Thus, we expect 95% of the data points to lie inside this ellipse, assuming a bivariate normal distribution. We 530 then characterized each such ellipse by its shape, size, and orientation (Fig. 5C). We defined the shapes of 531 each ellipse by their aspect ratio, computed as the ratio of the major-and minor-axis eigenvalues: λ 1 / λ 2 . We 532 computed the sizes of each ellipse as their area: accordance with our previous steady-state stepping regulation model [1]. We first replaced the stepping goals 549 z B * and w * in the model (Eq (2)) with the estimated adaptive stepping goals (see previous section) (Fig. 6A). 550 We next adapted ρ, the model parameter that specifies the relative weighting of step width and position Finally, in addition to the adaptive stepping goals and ρ modulation, we doubled the additive noise parameter 558 (σ a ) at the preparatory and transition steps (Fig. 8A). The increased area of the stepping distributions at the 559 preparatory, transition, and recovery steps reflects greater overall variability with respect to both the w * and 560 z B * GEMs. Additive noise is thought to reflect physiologic noise from sensory, perceptual, and/or motor 561 processes [29], which is expected to increase during the lateral maneuver task. Here, we increased the additive 562 noise (Fig. 8A) to determine the extent to which this would qualitatively capture the types of increases in 563 stepping distribution areas that we observed experimentally. 1000 times for each model iteration. All maneuvers were oriented from left-to-right, and the transition step 569 was specified to be taken with the ipsilateral (i.e., right) foot relative to the direction of the transition,   consistent with all but one of the experimentally observed maneuvers. Time series (mean±SD) of foot 571 placement (z L and z R ), position (z B ), and step width (w) were calculated at each step of the simulated 572 maneuvers. Errors with respect to both position and step width (mean±SD) were calculated as the difference 573 in the simulated position and step width relative to the stepping goals at each step of the simulated maneuvers. 574 The simulated stepping time series and errors were compared to the middle 90% range of the experimental 575 data at each step. The simulated stepping distributions were characterized by the aspect ratio, area, and 576 orientation of a fitted 95% prediction ellipse (see previous section). Error bars at each step for each variable 577 were calculated as 95% confidence intervals using bootstrapping. The simulated ellipse characteristics were 578 compared to the bootstrapped 95% confidence intervals from the experimental data at each step for each 579 variable. 580 581

582
As the experimental and simulated data were structured in very different ways, standard inferential statistics 583 (e.g., t-test, ANOVA, etc.) would not be appropriate to compare these results. Furthermore, we could generate 584 a sufficiently large number of model simulations to ensure small p-values for almost any comparison, thereby 585 diminishing the comparative power of any such assessments. Instead, we used descriptive statistics (e.g., 586 standard deviations, confidence intervals, etc.) to quantify the experimental observation values. We then 587 compared model predictions to these observations. We inferred that model predictions that fell within 588 experimentally observed standard deviations/confidence intervals were statistically consistent with the 589 experimental results. Additionally, the aim of this analysis was to determine if hierarchical adaption of our 590 goal-directed multi-objective lateral stepping regulation models could qualitatively emulate the same types of 591 changes in stepping dynamics observed during the lateral maneuver task. Descriptive statistics were sufficient 592 to accomplish this aim.