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Adaptive multi-objective control explains how humans make lateral maneuvers while walking [1]

['David M. Desmet', 'Department Of Kinesiology', 'Pennsylvania State University', 'University Park', 'Pennsylvania', 'United States Of America', 'Joseph P. Cusumano', 'Department Of Engineering Science', 'Mechanics', 'Jonathan B. Dingwell']

Date: 2022-11 

To successfully traverse their environment, humans 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. We identified 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 consistent with these goal functions. Twenty young healthy adults each performed four lateral lane-change maneuvers in a virtual reality environment. Extending 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 allowing the parameters of our previous lateral stepping models to adapt from step to step. To our knowledge, our results are the first to demonstrate humans might 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.

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 maintain balance 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 might accomplish such maneuvers. We show that humans execute a simple lateral lane-change maneuver consistent with our theoretical predictions. Importantly, our simulations show they can do so by adapting at each step the same step-to-step regulation strategies they use to walk straight ahead. 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.

However, here we extend our previous theoretical framework to demonstrate how humans might adapt how they regulate lateral stepping to execute non-steady-state maneuvers without having to resort to some entirely different scheme. We first apply concepts from the general theory in a manner that satisfies the requirements of a non-steady-state lane change maneuver. Specifically, we allow the task goals, task-level costs, and the weighting between competing costs to adapt from each step to the next. We thus derive a model that yields explicit, empirically-testable predictions about how the variability observed during such lateral maneuver tasks will be structured from step to step. We then test these predictions against human experimental data. Lastly, we used the models with these additional adaptive hierarchical elements to simulate how such lateral maneuvers are achieved over a sequence of steps. In doing so, we demonstrate how humans modulate their stepping to trade-off lateral stability for maneuverability to accomplish the lane change task. Thus, our hierarchical motor regulation framework can be applied to a far wider range of walking tasks beyond just straight-ahead, steady-state walking.

It is not clear a priori that our previous lateral stepping regulation framework can also emulate human stepping during lateral maneuvers. This framework was originally developed to model straight-ahead steady-state walking, under assumptions that step-to-step adjustments can be made without changing the fundamental structure of within-step control, and that deviations from perfect performance are small. These assumptions motivated us to select low-dimensional, single-step, linear regulators to model these step-to-step error-correcting processes [ 1 , 30 , 31 ]. However, walking tasks requiring substantial non-steady-state lateral maneuvers would seem to violate these assumptions. Lateral maneuvers that introduce large deviations from steady-state might, for example, require changes to the within-step control structure, or induce substantial nonlinearity. Lateral maneuvers might also require substantially greater impulses to execute, which may necessitate additional compensatory motor and/or kinematic contributions and could result in the need for higher model dimensionality. Furthermore, humans plan maneuvers more than one step in advance in some contexts [ 4 , 5 ], which suggests the possible need for models that depend on more than one prior step. Thus, for any of these reasons, one might reasonably posit that humans regulate stepping using entirely different control schemes when executing lateral maneuvers far from steady state.

However, humans rarely perform long bouts of straight-ahead, continuous walking [ 47 , 48 ]. Instead, humans must frequently perform various kinds of locomotor maneuvers when walking in the real world (e.g., Fig 1A ) [ 7 – 9 ]. Indeed, older adults often fall during such maneuvers because they incorrectly transfer their body weight or they trip [ 49 ]. Because both such causes can be prevented with appropriate foot placement [ 18 , 20 , 22 , 50 ], it is necessary to better understand how humans regulate foot placement as they execute typical lateral maneuvers. Here, we aimed to determine how humans regulate their stepping during a simple lateral maneuver: namely, a single lateral lane-change transition between periods of straight-ahead walking.

We posit that locomotion is functionally hierarchical, consisting of both within-step control and step-to-step regulation. To compliment within-step control models, we developed a motor regulation framework to determine how humans adjust successive stepping movements [ 1 , 30 , 34 , 35 ]. For a given walking task (e.g., Fig 1A ), we use this framework to propose goal functions that theoretically define the task and, accounting for equifinality, determine the set of all possible task solutions as Goal Equivalent Manifolds (GEMs) [ 37 – 40 ]. The goal functions are incorporated into task-level costs, which we then use in a stochastic optimal control formalism [ 41 ] to generate relatively simple, phenomenological models of step-to-step motor regulation (i.e., “motor regulation templates” [ 1 , 30 , 31 ]). For any proposed goal function, these templates predict how humans would attempt to drive that goal function to zero at the next step, thus achieving perfect task performance on average. For an appropriate choice of goal functions, these models successfully replicate human step-to-step dynamics in both the fore-aft [ 30 ] and lateral [ 1 ] directions. In the lateral direction in particular, multi-objective regulation of primarily step width (w) and secondarily lateral position (z B ) captures human step-to-step dynamics during continuous straight-ahead walking ( Fig 1B and 1C ) [ 1 , 13 , 42 ]. Whether we construct explicit models or not, this overall theory and its hierarchical control/regulation schema provides a powerful framework from which to interpret experimental results [ 13 , 42 – 46 ].

Human walking exhibits considerable variability [ 27 , 28 ], redundancy (i.e., the body has more mechanical degrees of freedom than necessary to perform any movement) [ 29 ], and equifinality (i.e., humans can perform most tasks an infinite number of ways) [ 30 , 31 ]. Computational models are necessary to fully understand how humans perform accurate, goal-directed walking movements in the context of these challenges. Most prior models of human walking (e.g., [ 21 , 32 , 33 ]) address within-step “control,” defined here as the processes that drive the dynamics of each step to remain viable (i.e., to prevent falling) [ 34 ]. While ensuring viability is indeed necessary to walk [ 35 , 36 ], it is not sufficient: humans engage in purposeful walking with a destination to reach and/or other tasks to achieve. Control models that “just walk” (i.e., remain viable) do not address how humans achieve such goal-directed walking.

Humans accomplish these sorts of walking maneuvers primarily by modulating their foot placements. Appropriate foot placement at each step can redirect center-of-mass accelerations, which enables humans to maintain balance while walking straight ahead [ 18 – 20 ]. This is especially important in the lateral direction where humans are thought to be more unstable [ 21 – 23 ]. Indeed, laterally-directed falls are particularly injurious in older adults [ 24 , 25 ]. However, the express purpose of any lateral maneuver is to specifically interrupt ongoing straight-ahead walking to achieve some particular walking task goal that requires a walker to alter their foot placements [ 8 , 26 ]. Thus, it is important to better understand how humans adapt their lateral foot placements to maintain balance while simultaneously executing such lateral maneuvers.

A) Examples of people walking in common contexts that require adaptability, including walking on a winding path and avoiding an obstacle. B-C) Configuration of bipedal walking in both the frontal (B) and horizontal (C) planes. Coordinates are defined in a Cartesian system with {x,y,z} axes shown in (B) and (C) and the origin defined as the geometric center of the available walking surface. The lateral positions of the left and right feet {z L , z R } are used to derive the primary lateral stepping variables that are regulated from step-to-step: step width (w = z R −z L ) and lateral body position (z B = ½(z L + z R )), which reflects a once-per-step proxy for the lateral position of the center-of-mass (CoM) ([ 1 ]; see S1 Text ).

To successfully traverse our environment, we humans must adapt to a wide variety of environment contexts and changing task goals [ 1 ] (e.g., Fig 1A ), all while maintaining balance. Indeed, humans readily avoid obstacles [ 2 ] and/or other humans [ 3 ], step to targets [ 4 – 6 ], move laterally [ 7 – 9 ], navigate complex terrain [ 10 – 12 ], and respond to destabilizing perturbations [ 13 , 14 ]. To do so requires a high degree of maneuverability [ 7 ] and, potentially, the ability to trade-off stability for maneuverability [ 8 , 9 ]. While maneuverability has been widely studied in animal locomotion [ 15 , 16 ] and robotics [ 16 , 17 ], far less is known about how humans accomplish such tasks.

Results

Lateral maneuvers in humans We analyzed data from a prior study conducted in our laboratory [51]. Young healthy participants walked in a virtual reality environment. They walked on one of two parallel paths, the centers of which were 0.6m apart, projected onto a 1.2m wide motorized treadmill (Fig 2A). Following an audible cue, they executed a lateral maneuver from the path they were walking on over to the adjacent path. Importantly, participants were given no explicit instructions about how to execute this maneuver. We analyzed seventy-nine total maneuvers from twenty participants. PPT PowerPoint slide

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TIFF original image Download: Fig 2. Experimental Stepping Time Series and Errors. A) Time series of left (z L ; 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. https://doi.org/10.1371/journal.pcbi.1010035.g002 Participants variably took 0-to-3 steps between the cue and initiating the maneuver (Fig 2A[i-ii]). However, once initiated, they performed each maneuver consistently (Fig 2A[iii]-B). Participants completed nearly all maneuvers with an ipsilateral transition step that involved a single, large step to reach the new path. One participant, however, completed one of their maneuvers by taking a large cross-over step. Additionally, one maneuver from another participant required three steps to reach the new path. These two non-conforming maneuvers demonstrate that participants had many options: the task itself did not require them to complete the maneuver in any specific way. Participants completed nearly all lateral maneuvers in 4 non-steady-state steps. Participants first took a “preparatory” step (step 0) prior to the transition, during which they slightly narrowed their step width and incrementally moved towards the new path. They then took a large “transition” step (step 1) to cover most of the transition distance. Participants took a subsequent “recovery” step (step 2) that again exhibited a slightly narrowed step width. Participants then reached their final new goal position (step 3) and returned to steady-state walking on their new path.

Steady-state stepping regulation cannot replicate lateral transitions We first used our previous multi-objective model of lateral stepping regulation to test whether humans could regulate stepping during lateral maneuvers using a constant regulation strategy like that used for continuous, straight-ahead walking [1]. This model selects each new foot placement (z L or z R ) as a weighted average of independent predictions that minimize errors with respect to either a constant step width (w*) or lateral position (z B *) goal, consistent with multi-objective stochastic optimization of error costs with respect to these two quantities (see Methods). For this model, the relative proportion of step width to lateral position regulation was defined by ρ, where ρ = 0 indicates 100% z B control (and hence, 100% weight on the z B cost) and ρ = 1 indicates 100% w control (100% weight on the w cost) [1]. This stepping regulation model reproduced the key features of human stepping dynamics during continuous, straight-ahead walking for constant values in the approximate range 0.89 ≤ ρ ≤ 0.97 [1]. We assessed whether this model, with any constant value of ρ, could emulate the non-steady-state stepping dynamics experimentally observed during the lateral maneuver task. We found that it was not capable of doing so. The model did emulate steady-state stepping dynamics (Fig 3C; Step -3) for constant values in the range of approximately 0.83 ≤ ρ ≤ 0.92, consistent with previous findings [1]. However, no simulations over this range of ρ completed the lateral maneuver as observed in the experiment (Fig 3A and 3B; red). Furthermore, while model simulations that weighted position and step width regulation similarly (i.e., ρ ≈ 0.5) approximately emulated average experimentally observed stepping time series and errors (Fig 3A and 3B; blue), they failed to emulate the stepping variability humans exhibited during either steady-state (Step -3) or transition (Step 1) steps (Fig 3C). Indeed, no single constant value of ρ emulated both steady-state and transition stepping dynamics (Fig 3C). PPT PowerPoint slide

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TIFF original image Download: Fig 3. Constant Parameter Model Results. A) Simulated stepping time series (mean ± SD) of 1000 lateral transitions using the original, constant parameter model with ρ = 0.90 (red) and ρ = 0.55 (blue). For steps in the interval [–3, 0], we set z B * as the experimental mean z B during steady state walking before the transition. For steps in the interval [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). https://doi.org/10.1371/journal.pcbi.1010035.g003

Re-thinking stepping regulation for non-steady-state tasks Any biped (human, animal, robot, etc.) must enact step width and/or lateral position regulation via left and right foot placement (Fig 1C). The stepping goals, [z B *, w*] that guide steady-state walking individually form diagonal and orthogonal Goal Equivalent Manifolds (GEMs) when plotted in the [z L , z R ] plane (Fig 4A) [1]. The intersection of these GEMs represents the multi-objective goal to maintain both z B * and w* and therefore defines the foot placement goal, [z L *, z R *], for the task. Along any GEM, deviations tangent to the GEM are “goal equivalent” because they do not introduce errors with respect to the goal. Conversely, deviations perpendicular to the GEM are “goal relevant” because they do introduce such errors [30]. Humans typically exhibit greater variability along GEMs they exploit [37,41,52]. Here, because the z B * and w* GEMs are orthogonal, goal equivalent deviations with respect to either GEM are goal relevant with respect to the other [1]. Hence, the ratio of the δ zB and δ w deviations with respect to both the z B * and w* GEMs theoretically reflects the relative weighting of z B and w regulation. During steady-state walking, the distribution of human steps is strongly anisotropic (Fig 4A): steps are strongly aligned along the w* GEM (such that δ zB /δ w >> 1) because humans heavily weight regulating step width over position [1]. PPT PowerPoint slide

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TIFF original image Download: Fig 4. Conceptual Depiction of Task Performance. A) When viewed in the [z L , z R ] 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. Deviations (δz B and δw) with respect to both the z B * and w* GEMs characterize the stepping distribution at a given step and reflect the relative weighting of z B and w regulation. In steady-state walking, humans strongly prioritize regulating w over z B , producing stepping distributions strongly aligned to the w* GEM. B) Any maneuver would then involve a substantial change from some initial (green) to some new final (blue) stepping goals that will displace these GEMs diagonally in the [z L , z R ] plane, such as the theoretical rightward shift in z B * (Δz B *) and increase in w* (Δw*) depicted here. To accomplish such a maneuver requires changing both z L and z R . This cannot be achieved in any single step. C) At least two consecutive steps (either ‘a’: z R →z L , or ‘b’: z L →z R ) at minimum are required to execute a lateral maneuver (Δz B * and/or Δw*). Each possible intermediate step (‘a’ or ‘b’) has its own distinct stepping goals. D-F) For any given intermediate step, numerous feasible strategies to execute the maneuver are theoretically possible. D) One such strategy might be to maintain strong prioritization of w over z B regulation (i.e., as in A) at the intermediate step. Enacting this strategy would produce a stepping distribution at the intermediate step that would remain strongly aligned to the new constant step width GEM (w*a) at that intermediate step. E) Another feasible strategy might be to simply put the first (here, right) foot at its new desired location (z R ). This would produce a stepping distribution at the intermediate step that would be strongly aligned to that new desired location (here, z R ) for that step. F) A third feasible strategy might be to maximize maneuverability. Here, foot placement at the intermediate step should be as accurate as possible. This would produce an approximately isotropic (i.e., circular) stepping distribution at the intermediate step. https://doi.org/10.1371/journal.pcbi.1010035.g004 When viewed in this manner, it then becomes evident that nearly 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 4B). This will necessitate corresponding changes in both left and right foot placement, which therefore cannot be accomplished in a single step (Fig 4B). At minimum, two consecutive steps are required to execute nearly any maneuver involving some Δz B * and/or Δw*. The first step must be taken by either the left or right foot to either of two possible intermediate foot placement goals in the [z L , z R ] plane (Fig 4C). Indeed, we can derive exact stepping goals for this intermediate step algebraically (see Methods). However, equifinality exists in both the number and placements of steps that can be used to accomplish any Δz B * and/or Δw* maneuver. People could, if they so choose, adopt any number of strategies involving nearly any number of steps (e.g., two such examples are shown in Fig 3A and 3B). However, specifying these foot placements alone does not capture how any given biped might perform these steps. We must also consider the distribution of steps (i.e., δ zB /δ w ) at each intermediate step as the stepping goals change (i.e., Δz B * and/or Δw*). Here, our theoretical framework thus allows us to posit different empirically-testable hypotheses (e.g., Fig 4D–4F) about how these δ zB /δ w ratios should also change at each step. For example, a walker that strongly prioritized regulating step width (e.g., Fig 4A), as humans do in straight-ahead walking [1], could maintain that strategy but then simply take first one step and then a second, each with an appropriately larger step width (Fig 4D). Alternatively, the walker could instead seek to simply take the first step to the new stepping goal for that foot, followed by an appropriate second step (Fig 4E). If on the other hand, a walker wanted to trade-off stability to maximize maneuverability [8, 9], foot placement at the intermediate step should be as accurate as possible to minimize stepping errors at the final step. Theoretically then, stepping distributions at this ideal intermediate step should be approximately isotropic (i.e., circular), reflecting no preference for either step width or position regulation (Fig 4F). One could imagine similarly proposing other alternative competing hypotheses.

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