Humans use multi-objective control to regulate lateral foot placement when walking

A fundamental question in human motor neuroscience is to determine how the nervous system generates goal-directed movements despite inherent physiological noise and redundancy. Walking exhibits considerable variability and equifinality of task solutions. Existing models of bipedal walking do not yet achieve both continuous dynamic balance control and the equifinality of foot placement humans exhibit. Appropriate computational models are critical to disambiguate the numerous possibilities of how to regulate stepping movements to achieve different walking goals. Here, we extend a theoretical and computational Goal Equivalent Manifold (GEM) framework to generate predictive models, each posing a different experimentally testable hypothesis. These models regulate stepping movements to achieve any of three hypothesized goals, either alone or in combination: maintain lateral position, maintain lateral speed or “heading”, and/or maintain step width. We compared model predictions against human experimental data. Uni-objective control models demonstrated clear redundancy between stepping variables, but could not replicate human stepping dynamics. Most multi-objective control models that balanced maintaining two of the three hypothesized goals also failed to replicate human stepping dynamics. However, multi-objective models that strongly prioritized regulating step width over lateral position did successfully replicate all of the relevant step-to-step dynamics observed in humans. Independent analyses confirmed this control was consistent with linear error correction and replicated step-to-step dynamics of individual foot placements. Thus, the regulation of lateral stepping movements is inherently multi-objective and balances task-specific trade-offs between competing task goals. To determine how people walk in their environment requires understanding both walking biomechanics and how the nervous system regulates movements from step-to-step. Analogous to mechanical “templates” of locomotor biomechanics, our models serve as “control templates” for how humans regulate stepping movements from each step to the next. These control templates are symbiotic with well-established mechanical templates, providing complimentary insights into walking regulation.

We systematically varied each of these parameters ( a ,  m , ) one-at-a-time to determine their effects on the model outputs. For each variable regulated (z B , z B , w), we defined the 'baseline' value for additive noise ( a ) for that variable as the group mean of the within-subject standard deviation for that variable, based on the experimental data ( We took the baseline amplitude of multiplicative noise ( m ) to be 10% of the baseline additive noise (i.e.,  m = 0.10• a ), as multiplicative noise effects are quite small for steady-state walking [1,2]. We also presumed error correction was weighted more heavily than control effort, and thus we took the baseline ratio of the cost function weights to be () = 0.10.
Each parameter to be varied was then varied over a range of values much wider than expected to be observed empirically.
Additive noise amplitudes ( a ) were varied from 0 to 5 times  a . Multiplicative noise amplitudes ( m ) were varied from 0 to 20 times  a (i.e., 200 times 'baseline'). Relative cost function weights () were varied from 0 to 2 (i.e., up to twice as much weight placed on control effort than error correction). These parameter variations are summarized as follows: The "Normalized Parameter Range [a.u]" shown as {0, …, 1} on the horizontal axes in each of the following "Parameter Sensitivity Results" figures (Figs. S1-2, S1-4, S1-6, S1-8, S1-10, and S1-12) corresponds to the ranges listed above. pg. 2 of 9 We took the 'desired' or 'goal' values for each variable to be: Position: z * B = 0.0 m (center of path) Heading: z * B = 0.0 m (straight ahead) Step Width: (w * ) = 0.127 m (from experiment - Fig. 2) Where the desired step width (w * ) was the group mean of the within-subject mean step width from the experiment (Fig. 2).
For each model and each set of parameter values tested, we generated 20 sets of time series data, each simulating 500 consecutive walking steps (analogous to simulating 20 walking trials). Time series of left & right foot positions (z L and z R ), body position (z B ), heading (z B ), and step width (w) were generated for each simulated walking trial and analyzed in the same manner as experimental data: standard deviations () and DFA exponents () were computed for each time series.
Beyond the control inputs as specified by the model, these initial models were otherwise "unconstrained" and thus capable of making stepping movements that could step off the path, or with very wide or very narrow step widths. We therefore also computed the percentage of steps (z L and z R ) taken in each simulated trial that exceeded the Lateral Boundary Limits: i.e., for the experiments analyzed here, the left and right edges of the treadmill belt (±0.885 m, see Fig. 2).
We also computed the percentage of steps that exhibited step widths that we defined as unrealistically either "too wide" or "too narrow". We defined this as exceeding ±5 standard deviations (±5), as determined from the group-average of the within-subject standard deviations, averaged across all participants (Fig. 2). This yielded an allowable step width range of −0.15 cm to +25.54 cm (see Fig. 2). Statistically, ±5 encompasses >>99.99% of the observed data, reflecting an ~1 in 3.5 million chance of stepping outside this range. However, when asked to do so, healthy adults can walk with steps as wide as ~29 to 42 cm [3][4] [5] and can take crossover steps [6][7][8] with step widths well below the −0.15 cm. Thus, while exceeding these ±5 limits would be very unusual for normal walking, doing so would still be very biomechanically feasible.

Imposing Realistic Biomechanical Constraints:
All of the initially "unconstrained" models (Figs. S1-1 thru S1-6) failed to adequately capture the stepping dynamics exhibited by humans (Fig. 2), regardless of parameter choices. In most cases, these models either regularly exceeded either the lateral boundary limits (i.e., they stepped off the treadmill) or the ±5 step width limits, or both.
We therefore ran additional simulations and repeated the same parameter sensitivity analyses (Figs. S1-7 thru S1-12) where this time we retained each of the same three controllers, but imposed constraints on foot placement (z L and z R ) at each step such that the simulations could not step off the treadmill (±0.885 m) and could not take steps that would exceed the ±5 step width limits (i.e., −0.15 to +25.54 cm) described above.
It is important to note that all of these "constrained" uni-objective control models both fully satisfied the requirements of the task (i.e., Eq. 1) and did so by taking steps that were biomechanically feasible [3][4][5][6][7][8]. That is, all of these simulations reflected stepping strategies people could have executed successfully, had they chosen to.
It is also important to note that the constraints imposed were intended to reflect those minimally necessary to achieve the task goal (Eq. 1). Had we imposed constraints on foot placement more (or less) restrictive than those imposed here, neither the overall qualitative findings, nor the final conclusions, would have been any different.

Figures and Plotting Conventions:
For each of the six total control models, we present two figures. In the first figure for each model (odd numbered figures), we display example time series and comparisons to human values for 'baseline' parameter values. For each output variable, we show comparisons of boxplots to demonstrate the range and distribution of values within each data set.
For each second (even numbered) figure, we show the mean ± 1 bands for humans (gray shaded areas) to indicate the range of values observed experimentally for that variable. We plot simulation results as mean ±95% confidence intervals for each mean to identify where the predicted simulation means would lie inside or outside of the experimental range.   Step Width Control:    Step Width Control -Constrained:  Unconstrained Uni-Objective Control Models (Figs. S1-1 to S1-6): Comparisons across models exhibit clear redundancy between the output variables (z B , z B , and w): when any one variable was regulated, the others clearly were not. All model configurations were highly sensitive to additive noise ( a ), highly insensitive to even very large changes in multiplicative noise ( m ), and only modestly sensitive to changes in relative cost function weights ().
All of these models failed to capture the stepping dynamics exhibited by humans (Fig. 2) across all parameter combinations tested. In addition to not matching variability and/or DFA results, most simulations regularly exceeded either lateral boundary or step width limits, or both. In particular, while the DFA results were highly sensitive (between models) to which stepping variable was regulated, they were highly insensitive to any of the imposed variations in the parameters governing each model. Thus, control of z B (Fig. S1-2) always resulted in (w) ≈ 1.5, control of z B always resulted in (z B ) ≈ (w) ≈ 1.5, and control of w (Fig. S1-6) always resulted in (z B ) ≈ 1.5. In each case, the DFA  ≈ 1.5 reflected an uncontrolled random walk process [9,10]. The failure of these models to replicate human stepping dynamics was thus not due to some limitation in the choice of parameters, but rather to fundamental structural differences in the models themselves.
Constrained Uni-Objective Control Models (Figs. S1-7 to S1-12): Adding relevant constraints to choices of stepping locations (z L & z R ) ensured that ALL of these simulations solved the main task requirement (Eq. 1) without ever stepping off of the treadmill or taking biomechanically implausible steps (% of steps exceeding lateral boundary or step width limits all = 0). This mitigated some of the excessive variability observed in some of the unconstrained control models and thus some values of some stepping variables improved for some parameter choices.
However, all of these constrained models still failed to capture the stepping dynamics exhibited by humans (Fig. 2) across all of the parameter choices tested. Thus, in spite of the fact that all of these models produced biomechanically viable strategies, none of them came close to replicating what we had observed in the experimental findings.