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

Abstract

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.

Fig 1. Defining Relevant Lateral Stepping Variables.

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).

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 ±1 standard deviation from all steady state walking steps.

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).

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.

Fig 5. Experimental Stepping Distributions.

A) Predicted stepping goals and distributions for the experimentally-imposed lane-change maneuver during both the steady-state walking periods (top) and transition steps (bottom). During steady-state walking before and after the maneuver, we predicted stepping distributions would be strongly aligned to the w* GEM, reflecting strong prioritization of step width over position regulation, as we have previously observed [1]. Participants most commonly performed the maneuver using a four-step maneuver strategy (Fig 2), which included 3 intermediate steps with distinct stepping goals. Stepping distributions were predicted to be most isotropic at the large primary transition step, and intermediately isotropic at the smaller preparatory and recovery steps. B) Experimental stepping data during the steady state periods before and after the transition (top) and during the preparatory, transition, and recovery steps from 78 analyzed maneuvers, plotted to appear from left to right (bottom), projected onto the [z_L, z_R] plane. The steady-state data were pooled across 16 steady-state steps before and after the transition from 20 participants performing 4 transitions each (1280 steps total). The diagonal dotted lines indicate the initial and final constant-z_B* and constant-w* GEMs. The gray dots indicate individual steps, and the red ellipses depict 95% prediction ellipses derived using a χ² distribution. C) Experimental 95% prediction ellipse characteristics at each step: aspect ratio (top), calculated as the ratio of the eigenvalues of the covariance matrix of the ellipse, area (center), and orientation (bottom), calculated as the angular deviation (positive angles indicate counter-clockwise) from the orientation of the constant-w* GEM. Error bars indicate ±95% confidence intervals at each step derived using bootstrapping. Gray dashed lines indicate the mean of each characteristic from the two steady state ellipses in (B; top).

Fig 6. Adaptive Stepping Goals.

A) Model parameters. 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-E). Thus, adaptive stepping goals are necessary but not sufficient to replicate human stepping during lateral maneuvers.

Fig 7. Adaptive Control Proportion.

A) Model parameters. Here, in addition to adapting the stepping goals ([z_B*, w*]; 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–6E. 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.

Fig 8. Adaptive Additive Noise.

A) Model parameters. 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 and 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.