Defining the task and relevant variables

We consider the common task of some biped (ostensibly a human, but could more generally be any robot, walking model, etc.) walking on any clearly defined path with identifiable lateral boundaries (e.g., Fig 1A). We treat walking control as hierarchical. We assume some within-step processes (be they active, passive, or both) generate each individual step [16, 18, 25, 32, 33]. We presume those processes will be different for different bipeds and our models are purposefully agnostic to such details. Indeed, a strength of our approach is precisely that it can apply to any reasonable biped (human, robot, etc.), regardless of how that biped may be actuated and/or controlled at a mechanical (and/or possibly neuromuscular) level to generate each step [9].

What we study here is the between-step regulation of the output of that process. That is, conceptually, we consider the result of each stepping motion as the input to our models [9, 10, 40] of the between-step regulation needed to achieve goal-directed walking [26–28, 38, 39]. We focus on regulating lateral stepping movements in the frontal plane because lateral balance is a significant challenge in bipedal walking [16, 21, 23, 60] and because sideways falls are especially dangerous for elderly adults [19, 20, 61]. We further consider lateral stepping as decoupled from sagittal plane movement (i.e., forward progression). While there is some coupling between frontal and sagittal planes, it is minimal [17, 25, 46, 60] and our prior work has already addressed how humans regulate sagittal plane stepping [9, 10].

The simplest, mechanically sufficient description of the frontal plane configuration of the lateral components of a stepping biped (Fig 1B) includes locations of the body (ostensibly the center-of-mass) and of each of the two feet [33, 46, 62]. Extending our prior sagittal plane work [9, 10], the primary task requirement for any such biped is simply to not step off the path, in this case to either side, i.e.: (1) where we take the z-coordinate to define the lateral direction (Fig 1B and 1C), W_P(x) defines the net width of the walking path, and {z_Ln, z_Rn} define left and right lateral foot placements for the n^th step in a sequence of N consecutive steps. Walking on a treadmill, as studied here, merely implies W_P(x) = W_TM ≡ the constant width of the treadmill belt, but Eq (1) applies generally to any walking path with finite width (e.g., Fig 1A, etc.).

Our objective is to identify what strategies people choose to accomplish this very general walking task. A key observation is that any sequence of {z_Ln, z_Rn} that satisfies the inequality of Eq (1) will achieve this. Thus, an infinite number of choices for each z_Ln and z_Rn exists and many strategies could generate feasible sequences of {z_Ln, z_Rn} that satisfy Eq (1). To identify these strategies, we need to quantify the lateral motion of the walker relative to the path, and the lateral motions of the walker’s feet relative to its body.

While {z_Ln, z_Rn} define the task goal via Eq (1) and ultimately enact any stepping control strategy, we presume {z_Ln, z_Rn} are coordinated to achieve some more general goal related to whole body movement. In particular, {z_Ln, z_Rn} establish where the walker’s body is relative to the path [25, 32, 33, 35]. Here, we take this body position on any given step, z_Bn, as the midpoint between the two feet (Fig 1B and 1C): (2)

This choice of z_Bn approximates the center-of-mass location for steady-state upright walking on level ground [21, 62]. Eq (2) assumes only that stepping movements are symmetrical relative to the center-of-mass, a finding broadly validated in multiple studies of human walking [42, 63–65]. Eq (2) is also consistent with multiple well-established models of pedestrian navigation [52, 54, 55].

In addition to the walker’s location on the path, we must also consider how it moves relative to its path. Goal-directed walking and pedestrian navigation require maintaining direction [52, 53] or “heading” [66, 67]. Likewise, the lateral (z) component of CoM velocity contributes to subsequent foot placement within each step [18, 33, 46] and in the sagittal plane, humans regulate speed and not position [9, 10]. Here, we thus take Δz_Bn as a proxy for lateral speed, or equivalently the lateral component of “heading” (Fig 1C): (3)

Lastly, regulating step width to maintain lateral balance is also important to walking [16, 21, 23, 68]. Here, we define step width as the difference between the lateral locations of the right and left foot (Fig 1C): (4)

Thus, from the most basic definition of walking, any biped walker (human, robot, model, etc.) must therefore exhibit some sequence of {z_Ln, z_Rn, z_Bn, Δz_Bn, w_n} over consecutive steps. However, by definition these variables are not all independent, as can be readily seen by writing Eqs (2) and (4) in matrix form as: (5)

Thus, specifying either [z_Ln, z_Rn] or [z_Bn, w_n] fully specifies the other. We therefore take {z_Bn, Δz_Bn, w_n} to be the biomechanically relevant variables to be regulated and treat {z_Ln, z_Rn} as effectors that enact this regulation. Like our prior sagittal-plane work [10], strategies that regulate some combination of {z_Bn, Δz_Bn, w_n} thus define “control templates” to describe step-to-step regulation of lateral stepping movements that can then compliment mechanical templates of within-step locomotor biomechanics [57, 58]. We then seek the most parsimonious such strategy that captures the variability and step-to-step dynamical structure we observe in experiments.

Lateral stepping dynamics in humans

To test different hypothesized control strategies, we re-analyzed data from a prior study in which young healthy human participants walked on a 1.77 m wide motorized treadmill at a pre-determined comfortable walking speed [69, 70]. We measured time series of left (z_L) and right (z_R) lateral foot placements. We used Eqs (2)–(4) to calculate corresponding time series of body positions (z_B), heading (Δz_B), and step widths (w), which we then analyzed.

To quantify variability, we computed standard deviations for each time series. To quantify temporal correlations across consecutive steps, we computed scaling exponents, α, using Detrended Fluctuation Analysis (DFA) [9, 71] (see Methods). An α > ½ indicates statistical persistence (deviations in either direction are more likely to be followed by deviations in the same direction). An α < ½ implies anti-persistence (subsequent deviations are more likely to be in the opposite direction). An α = ½ indicates uncorrelated fluctuations (subsequent deviations are equally likely to be in either direction). A value of α = 1½ indicates brown noise (i.e., integrated white noise), equivalent to Brownian motion. In the context of control, variables not tightly regulated typically exhibit strong persistence (α >> ½). Conversely, variables that are more tightly regulated typically exhibit approximately uncorrelated fluctuations (α ≈ ½) [9, 10, 12].

All of the candidate stepping variables, {z_Bn, Δz_Bn, w_n}, and also foot placements, {z_Ln, z_Rn}, fluctuated (e.g., Fig 2A), with magnitudes of variability well above the level of measurement noise (Fig 2B). Fluctuations in lateral body position (z_B) exhibited strong statistical persistence (α >> ½). Fluctuations in heading (Δz_B) exhibited mostly statistical anti-persistence (α < ½), consistent with heading being the first-difference of position (Fig 2C). Fluctuations in step width (w) exhibited weak statistical persistence (½ < α << 1½).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Experimental values of primary stepping variables.

A) Example time series data for a representative trial from a typical participant. Each plot shows 290 consecutive steps of left (z_L; red) and right (z_R; blue) foot placements, body position (z_B), heading (Δz_B), and step width (w). For the stepping plot (z_L; z_R), black vertical dashed lines indicate the lateral edges of the treadmill (±0.885 m). For the other time series plots (z_B, Δz_B, and w), red vertical dashed lines indicate ±5 standard deviations, as determined from the average of the standard deviations of all participants. B) Standard deviation (σ) values for all trials for all participants for each variable (z_B, Δz_B, and w). C) DFA scaling exponent (α) values for all trials for all participants for each variable (z_B, Δz_B, and w). For (B) and (C), each subplot shows a summary boxplot (blue–left) indicating the median, 1^st and 3^rd quartiles, and whiskers extending to 1.5× the inter-quartile range, with values beyond that range shown as individual data points. Each subplot also shows individual data points (red dots–right) indicating all individual trials for all participants. These experimental data were aggregated across 65 total trials of 290 steps each, as obtained from 13 participants (5 trials each).

https://doi.org/10.1371/journal.pcbi.1006850.g002

Given that all variables exhibited both substantial variability and varying degrees of statistical persistence, one cannot conclude from these data alone which variable (or combination of variables) were or were not regulated. Indeed, we might be tempted to conclude that Δz_B and w were more tightly regulated than z_B because they both exhibited α ≈ ½ [9, 10, 12]. However, any such inference would be incorrect. As we demonstrate, such a conclusion would succumb to the formal logical fallacy of “affirming the consequent” [72]: i.e., while processes that are tightly regulated typically yield α ≈ ½, not all processes that exhibit α ≈ ½ are tightly regulated. What is needed are appropriate computational models that can accurately predict these variability and statistical persistence relationships and indicate what gives rise to them.

Uni-objective stepping regulation

We consider first the simplest such stepping control strategies, namely those that regulate only one of the variables (Eqs (2)–(4)) to maintain some constant value of that variable. The strategy “stay in the middle of the path” implies one would adjust consecutive foot placements to maintain z_Bn = z_B* ≡ Constant. We note that other reasonable choices for z_Bn different from Eq (2) exist. Each would yield a different corresponding z_B*, but would not change the step-to-step dynamics quantified here. Likewise, because we can define the origin of our coordinate system (Fig 1) at any point we choose, we could always take z_B* = 0, but we retain the more general form for now. The strategy “maintain heading” (regardless of lateral position) implies that one would try to maintain Δz_Bn = Δz_B* ≡ Constant. Here, Δz_B* = 0 would represent the goal to walk “straight ahead” (i.e., with no lateral deviation), but again we retain the more general form for now. Lastly, the strategy “maintain step width” (e.g., as in [16, 22, 23]) implies that one would try to maintain w_n = w* ≡ Constant (regardless of absolute position or heading). Here, we would take w* = 〈w〉 ≡ some average (or typical) step width (e.g., for an individual or group, etc.) as the desired step width to maintain.

Each of these strategies can be defined in terms of a scalar “goal function” [6, 7, 12] in each variable separately, each having the same general form: (6) where q ∈ {z_B, Δz_B, w} and the goal is to drive F_q → 0 (i.e., minimize errors with respect to the goal). For each candidate control variable, we write a corresponding state-update equation governing the regulated step-to-step dynamics as a 1D map in q: (7) where q is the controller “state” being regulated, u_q(q_n) is the relevant control input, σ_mν_m and σ_aν_a are multiplicative and additive noise terms, respectively, and g is an additional gain (introduced in our previous work: [9, 10]) that could be used to tune the regulator away from the derived optimal control. Here, no such tuning was necessary, so we set g = 1 for all subsequent analyses. Single variable regulators such as in Eq (7) thus allow the remaining unregulated variables to “drift” under the action of motor noise.

Following prior work [9, 10, 12], these controllers were derived analytically (see Methods) as stochastic optimal single-step controllers with direct error feedback, following the Minimum Intervention Principle [3, 4]. This yielded the following stochastically optimal control input, u_q, as a function of the current state, q_n: (8)

We used these models to simulate walking trials that we then compared directly to experimental data from humans. Each such model depends on three parameters (σ_a, σ_m, and γ/α). We conducted parameter sensitivity analyses (see S1 Appendix) across wide ranges of each of these parameters to determine if any of these candidate uni-objective control models could adequately replicate the experimental findings of Fig 2.

None of the uni-objective control models came close to capturing the behavior observed experimentally, regardless of parameter choices (see S1 Appendix for details). Nearly all simulations either regularly stepped off the prescribed path, or took biomechanically unrealistic steps, or both. We therefore also imposed additional biomechanically realistic constraints to each control model (see S1 Appendix for details) to ensure they did not either walk off the path (here, the treadmill: z_Max = ±0.885 m), or take steps that were unrealistically too wide or too narrow (defined here as no more than ±5 standard deviations, based on experimental means; Fig 2). All of these control models fully satisfied (by construction) the requirements of the walking task (Eq 1) and did so by taking steps that were biomechanically feasible [34, 62, 73]. Nevertheless, all of the model configurations tested (e.g., Fig 3) exhibited step-to-step dynamics that were both qualitatively and quantitatively substantially different from humans (see S1 Appendix for details) and thus failed to replicate the lateral stepping behavior observed in experiment (Fig 2).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Typical model simulation results for biomechanically constrained position control.

A) Example time series data for a single representative trial. The time series plotted, axes, and axis limits are all the same as in Fig 2. B) Standard deviation (σ) values and (C) DFA scaling exponent (α) values for all trials for each variable (z_B, Δz_B, and w). Each subplot in (B) and (C) shows boxplots (blue–left) and for 30 representative simulated trials from the model (red–right). All boxplots were constructed in the same manner as described in Fig 2. This model failed to replicate experimental findings from humans, as did all other uni-objective control models across all parameter ranges tested (see S1 Appendix for complete details).

https://doi.org/10.1371/journal.pcbi.1006850.g003

Multi-objective stepping regulation

Because no uni-objective controller model (with or without added biomechanical constraints) adequately replicated experimental findings, we next considered multi-objective control of 2 candidate parameters. The general 2-state vector-valued goal function [6, 7, 12] for any of these becomes: (9) where q₁ and q₂ are any two distinct variables selected from q_1,2 ∈ {z_B, Δz_B, w} and the goal is again to drive F → 0. The corresponding state-update equation becomes: (10) where the q_1,2 are the controller states being regulated. We then followed the same analytical derivation process [9, 10, 12] (see Methods), which yielded the following stochastically optimal control input: (11)

Both the state update equation (Eq (10)) and control inputs (Eq (11)) were chosen to have no coupling between their q₁ and q₂ components (see Methods). Thus, at each new step, each individual optimal controller (for q₁ and q₂ separately) predicts a different foot placement for the next foot. As one cannot place one’s foot in two places at once, the net result (i.e., final predicted foot placement) becomes a weighted average of the foot placements predicted by each of the uni-variate models for q₁ and q₂ separately.

We used these multi-objective models to simulate walking trials that we then compared directly to experimental data from humans (Fig 2). We conducted parameter sensitivity analyses (see S2 Appendix) across wide ranges of each of the governing parameters to determine if any of these candidate multi-objective control models could adequately replicate the experimental findings of Fig 2. Here, we emphasized varying the simulation parameter (ρ) that specified the weighted average between q₁ control and q₂ control (see S2 Appendix for details).

Simulations that regulated combinations of position and heading (i.e., [q₁, q₂] = [z_B, Δz_B]) failed to replicate human stepping dynamics, mainly by regularly violating reasonable step width limits (Fig. S2-2 in S2 Appendix). Across a substantial range of the parameter space tested, simulations that regulated combinations of heading and step width (i.e., [q₁, q₂] = [Δz_B, w]) did successfully achieve the task goal (Eq (1)) and did so using biomechanically reasonable step widths (Fig. S2-4 in S2 Appendix). Thus, many of these simulations predicted stepping strategies humans could have used to accomplish the task. However, all of these simulations yielded variability and statistical persistence of lateral position (σ(z_B) and α(z_B)) that far exceeded experimental values (Fig. S2-4 in S2 Appendix). Thus, all simulations regulating heading and step width failed to replicate human stepping dynamics, even though this strategy was physiologically feasible.

Simulations that regulated combinations of position and step width (i.e., [q₁, q₂] = [z_B, w]) also failed to replicate human stepping dynamics over a wide range of ρ (0% ≤ ρ < ~89% and ρ > ~97%; see Supplement S2; Fig. S2-6). However, for baseline parameter values, when the distribution of control was weighted to favor ~93% step width control, vs. ~7% position control, these simulations did replicate the basic statistical properties of human stepping dynamics (namely, both the standard deviations (σ) and statistical persistence (α) for all of the specified variables (z_B, Δz_B, and z_B) (Fig 4; same as Fig. S2-5 in S2 Appendix).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Typical model simulation results for multi-objective position-step width control.

A) Example time series data for a single representative trial simulated for baseline parameter values (see S2 Appendix) and a control proportion that was weighted at 93% step width / 7% position control. The time series plotted, axes, and axis limits are all the same as in Fig 2. B) Standard deviation (σ) and (C) DFA scaling exponent (α) values for all trials for each variable (z_B, Δz_B, and w). Each subplot in (B) and (C) shows boxplots for the experimental data (blue–left; from Fig 2) and for 30 representative simulated trials from the model (red–right) using the same parameter values as in (A). All boxplots were constructed in the same manner as described in Fig 2.

https://doi.org/10.1371/journal.pcbi.1006850.g004

Indeed, more refined parameter sensitivity analyses across a narrower range of 89% ≤ ρ ≤ 97% (Fig 5) yielded a range of solutions weighted strongly in favor of step width control that successfully replicated human stepping dynamics (Fig 2) and were also robust to variations in additive noise (σ_a) and cost function weights (γ/α) (Fig 5A). For all parameter variations tested across the range 89% ≤ ρ ≤ 97%, no simulation ever took any steps that exceeded either the lateral boundary limits (i.e., stepped off the treadmill; Fig 5C) or the step width limits (i.e., took excessively wide or narrow steps; Fig 5D). Thus, across all of these cases, this model successfully achieved the walking task (Eq (1)), did so by taking biomechanically feasible steps, and also successfully replicated all of the same stepping dynamics observed experimentally in humans.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Parameter sensitivity results for multi-objective position-step width control.

A) Standard deviations for each primary output variable (z_B, Δz_B, and z_B). B) DFA scaling exponents (α) for each primary output variable. In both (A) and (B), horizontal gray bands indicate the mean ± 1SD band exhibited by humans for that variable to indicate the range of values observed experimentally. For display purposes, all vertical axes are scaled to the mean ± 2SD band exhibited by humans. C) Percentage of steps (z_L and z_R) taken in each simulated trial that exceeded the Lateral Boundary Limits (±0.885 m): i.e., stepped off the treadmill. D) Percentage of steps that exceeded Step Width Limits (±5σ as determined from experimental data; equivalent step width range: −0.15 cm to +25.54 cm), reflecting biomechanically unrealistically wide or narrow steps (see S1 Appendix for details). Stepping data shown here were simulated at multi-objective proportions from 89% step width (11% position) control, every 2% up to 97% step width (3% position) control. Black symbols indicate ‘baseline’ parameter values (see S2 Appendix): σ'_a based on experimental values, σ'_m = 0.1∙σ'_a, and (γ/α)' = 0.1. Red and Green symbols indicate these same baseline parameter values, except γ/α = 0.0 and 0.2, respectively. Blue and Cyan symbols indicate these same baseline parameter values, except σ_a = 0.9∙σ'_a and 1.1∙σ'_a, respectively.

https://doi.org/10.1371/journal.pcbi.1006850.g005

Measures of variability (e.g., Fig 5A, etc.) quantify average magnitudes of differences across all steps, but do not indicate how stepping is executed dynamically over time. DFA analyses (e.g., Fig 5B, etc.) quantify statistical persistence (average step-to-step linear dependence) across time [9, 71], but independent of the magnitude of the variance. Thus, neither of these calculations (σ or α) quantifies how the proportionality of subsequent corrections (in time) of step-to-step fluctuations might differ with the magnitude of those initial fluctuations. Conversely, the Minimum Intervention Principle [3] and N-step capturability concept [44] both suggest that corrections to errors could well vary nonlinearly with the magnitudes of those errors: i.e., one could readily ‘ignore’ small errors and correct only larger ones. Neither σ nor α would capture any such dependence. Therefore, we conducted a third independent set of analyses to verify these findings [10].

If steps are adjusted at each step based on the previous step (as they are designed to be in our models; Eqs (7) and (10)), we would hypothesize that a stepping regulator trying to maintain some constant average value of any (Eqs (6) & (9)), when observing any deviation on a given step (i.e., ), should correct that deviation on the subsequent step by making a corresponding change, Δq_n+1 = q_n+1 − q_n, in the opposite direction [10]. For each simulated and experimental (human) data set, we constructed plots of Δq_n+1 vs. q'_n for each q ∈ {z_B, Δz_B, w}. We computed the linear slopes (using least-squares regression) and strength of correlation (r²) for each corresponding relationship [10]. We assumed our simulations, which enacted precisely this form of control, would exhibit evidence of strong linear control of step width (w), reflected by slopes close to −1 with high correlation, but weak linear control of position (z_B), reflected by slopes close to 0 with very low correlation. We hypothesized that if humans exerted step-to-step control in a similar manner, they would exhibit results consistent with our simulations. Indeed, this was precisely the case (Fig 6). All of the same simulations that precisely matched human stepping dynamics in Fig 5 (above) also just as precisely matched the strength of direct step-to-step corrections of all three time series variables (Fig 6).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. Direct control analysis results for multi-objective position-step width control.

A-B) Example plots of how errors in relative position (z'_Bn) and relative step width (w') were corrected on subsequent strides (Δz_B(n+1) and Δw_(n+1)). Data are shown for (A) one typical experimental trial from typical human participant and for (B) one typical trial from the multi-objective position-step width controller adopting baseline parameter values (see Fig 6). C) Linear regression slopes for each corresponding relationship for each primary output variable (z_B, Δz_B, and z_B). D) Corresponding linear correlation (r²) values for each linear regression for each primary output variable. In both (C) and (D), horizontal gray bands indicate the mean ± 1SD band exhibited by humans for that variable to indicate the range of values observed experimentally. In both (C) and (D), horizontal axis limits and symbol / color designations are the same as shown in Fig 5.

https://doi.org/10.1371/journal.pcbi.1006850.g006

Together, the results of Figs 5 and 6 make clear that the multi-objective position–step width control model accurately replicates the basic step-to-step dynamics of the fundamental variables {z_Bn, Δz_Bn, w_n}. Of course, these models not only need to regulate these variables appropriately, but should also generate appropriate stepping movements of the feet themselves {z_Ln, z_Rn}, as these are the effectors that enact this regulation. Therefore, as a final confirmation of the preceding results, we assessed how the above dynamics then mapped onto the stepping dynamics of {z_Ln, z_Rn}, which is easily accomplished using Eq (5).

The goals to maintain constant z_Bn = z_B* and w_n = w*, when considered individually, each yield linear GEMs that are orthogonal to each other in the [z_L, z_R] plane (Fig 7A). For the multi-objective position–step width control models, the [z_Bn, w_n] dynamics (i.e., including both variability and temporal correlation structure) maps into [z_Ln, z_Rn] dynamics precisely as defined by Eq (5). We thus predicted that if humans also adopted control consistent with this policy, their experimental [z_Bn, w_n] dynamics would also map into [z_Ln, z_Rn] dynamics in the same way as in the models. Indeed, this was precisely the case (Fig 7B and 7C), further supporting the validity of these multi-objective position–step width control models.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 7. Projection of [z_B, w] control variables onto the [z_L, z_R] stepping plane.

A) Example plots of stepping data for simulations of uni-objective controllers projected onto the [z_L, z_R] plane (see Eq (5)). One typical simulation trial each is shown for controllers regulating either position (z_B) only (left) or step width (w) only (right), each simulated using baseline parameter values (see S1 Appendix). In each plot, diagonal lines indicate the z_B* = constant (green, left) or w* = constant (red, right) GEM’s, as determined from the average position or step width (respectively) exhibited on that trial. All combinations of [z_L, z_R] that lie along either GEM equally satisfy the respective goal. B) Example plots of stepping data for one typical simulated trial from the multi-objective position–step width (z_B-w) controller, simulated using baseline parameter values (see S2 Appendix) and for one typical experimental trial. For each plot, the z_B* and w* GEM’s were determined from the average position and step width exhibited on that trial. C) Standard deviation (σ) and (D) DFA scaling exponent (α) values for all trials for left (z_L) and right (z_R) steps, body position (z_B), and step width (w) for humans (blue) and for the multi-objective position–step width controller, simulated using baseline parameter values (red). All boxplots were constructed in the same manner as described in Fig 2. Note that as all variables are in units of meters, variability measures can be compared directly, as coordinate dependence is not an issue here. All simulation values are well within the range of experimental results. Thus, the multi-objective position–step width control model captured the observed left/right stepping dynamics in [z_L, z_R] as it regulated [z_B, w].

https://doi.org/10.1371/journal.pcbi.1006850.g007

Ethics statement

Prior to participating, all participants signed informed consent statements approved by the Institutional Review Boards of both Brooke Army Medical Center and The University of Texas at Austin.

Experimental participants

We compared model predictions to human experimental data taken from the same data set as analyzed in [69, 70]. Thirteen young healthy adults participated (Table 1). All participants were screened to exclude anyone who reported any history of orthopedic problems, recent lower extremity injuries, any visible gait anomalies, or were taking medications that may have influenced their gait.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Participant characteristics.

All values except Sex are given as Mean ± Standard Deviation. Walking speeds shown are the participants’ actual speeds [m/s]. These corresponded to the same pre-designated non-dimensional walking speed (Froude speed ≡ Fn = 0.16) [69, 70] for all participants.

https://doi.org/10.1371/journal.pcbi.1006850.t001

Experimental protocol and data processing

Experimental protocols were described in detail previously [69, 70]. In brief, all participants walked on a 1.77m wide treadmill in a Computer Assisted Rehabilitation Environment (CAREN) virtual reality system (Motek, Amsterdam, Netherlands). Each participant walked at a constant speed (Table 1), non-dimensionally scaled to their leg length [69]. Following a 6-minute acclimation period, each participant completed five 3-minute walking trials with normal optic flow and visual scene movement matched to the treadmill belt speed.

Kinematic data were collected at 60 Hz using a 24-camera motion capture system (Vicon Motion Systems, Oxford, UK) and 57 reflective markers (Wilken et al., 2012). These kinematic data were post-processed using Vicon Nexus and Visual3D (C-Motion Inc., Germantown, MD). For the analyses presented here, only foot and pelvis marker data were used.

Heel strikes were determined using a velocity-based detection algorithm [82]. To increase precision of these calculations, raw marker data were first interpolated from 60 Hz to 600 Hz using a piecewise cubic spline interpolation [40, 47]. For each step, lateral foot placements (z_L and z_R) were taken as the lateral location of the heel marker at that step (Fig 1C). From these foot placements, body position (z_B), heading (Δz_B), and step width (w) were computed using Eqs (2)–(4) (Fig 1C). For consistency, only the first 290 consecutive steps from each trial (e.g., Fig 2A) were analyzed.

Uni-objective stochastic optimal regulator models

We consider three potential stepping control strategies (Fig 1C): while walking, try to maintain approximately constant absolute position (z_B) of the body on the path, constant lateral speed or heading (Δz_B), or constant step width (w). Each of these strategies yields a goal function [6, 7, 12] in each variable separately, each having the same form (see also Eq (6)): (12) where q ∈ {z_B, Δz_B, w}. The relevant step-to-step dynamics of q_n are then modeled as a 1-D map (see also Eq (7)): (13) where q is the controller state being regulated and u_q(q_n) is the control input to be derived. Here, ν_m and ν_a are each independent random variables with zero mean and unit variance that represent multiplicative (ν_m) and additive (ν_a) noise, respectively. The σ_m and σ_a terms then give the standard deviations of each noise term.

Similar to our prior work [9, 10], this state update equation (Eq (13)) models the discrete-time inter-step dynamics in q. As such, it represents a simple “control template” model that regulates noise-induced fluctuations away from perfect performance by adjusting q_n at each consecutive step. The choice of possible states to control (q ∈ {z_B, Δz_B, w}; Fig 1) was motivated by the fact that any biped (human, robot, model, etc.) must generate some sequence of these variables to be “walking”. We note that in the absence of noise and control input, successive strides simply repeat (i.e., q_n+1 = q_n), which is consistent with well-established notions of limit cycle behavior (e.g., [16, 46, 75]) of walking. Thus, many reasonable models of continuous-time within-step walking dynamics, whether relatively very simple (e.g., [75, 76]), or highly complex (e.g., [78]) could be used to generate all of the relevant step-to-step time series (z_Bn, Δz_Bn, w_n, z_Ln, z_Rn) analyzed here.

Given Eqs (12) and (13), we then follow the same process as described in [9] to derive the control input, u_q(q_n). As in prior work [9, 10, 12], these controllers were modeled as stochastic optimal single-step controllers with direct error feedback, based on the Minimum Intervention Principle (MIP) [3, 4]. These controllers are optimal with respect to the expected value of the following cost function: (14) in which the first term penalized the error with respect to the goal function (Eq (12)) at the next step (i.e., the deviation of F_q from 0 at step n + 1) and the second term penalized “effort” in terms of the magnitude of the control input (u_q) used to drive the state from step n to step n+1 [9]. Here, α and γ were positive constants that weighted the different components in the total cost, C. The form of Eq (14) is sufficiently general to include all cases studied in this work.

To derive u_q(q_n), we first substitute for e using F_q from Eq (12) into Eq (14): (15)

We then replace q_n+1 with the right hand side of the state update map, Eq (13) to give: (16)

We next expand Eq (16) and gather like terms in and u_q to obtain: (17)

We then take the expected value of C (i.e., ), noting that the noise processes have zero mean, unit variance, and are uncorrelated. That is, we set:

This then gives: (18)

In the general case, defines the Lagrangian, Λ, for this system. The optimal controller then extremizes . We thus differentiate Eq (18) with respect to u_q, set the resulting expression equal to zero, and solve the resulting algebraic equation for our final controller, u_q (see also Eq (8)): (19) (20) where the term in brackets, [•], represents the effective gain of the controller that defines the amount of the error, (q_n−q*), to be corrected at each step (see S1 Appendix for additional details). For each q ∈ {z_B, Δz_B, w}, these models (Eqs (13) and (20)) were used to simulate the respective q_n time series. For each such simulated q_n time series, the relationships between each q_n and the other stepping variables (i.e., Eqs (2)–(4); Fig 1C) were then used to generate all of the relevant step-to-step time series (z_Bn, Δz_Bn, w_n, z_Ln, z_Rn) analyzed for each simulation.

Unlike our previous models for sagittal plane stepping regulation [9, 10], the controllers derived here (Eq (20)) were not “unbiased”, meaning they were not required to exhibit perfect performance on average. However, it remains relevant to derive the corresponding optimal control input for the more restrictive unbiased case. Requiring u_q(q_n) to be unbiased would mean additionally requiring that the expected value of the goal function, F_q, at step n+1 be zero [9]: (21)

Substituting for q_n+1 in F and taking expected values, this additional constraint becomes: (22)

Adding this additional constraint to our controller yields the augmented Lagrangian, Λ, as: (23) where μ is a Lagrange multiplier. The optimal controllers again extremize Λ. We thus make the appropriate substitutions in Eq (23), differentiate Λ, and set the resulting expression equal to zero to obtain: (24)

Typically, Eqs (22) and (24) would be used together to solve for both u_q and μ [9, 12]. Here, as we are interested only in u_q, we obtain the unbiased stochastic optimal u_q directly from Eq (22) as: (25)

Thus, this unbiased controller reduces to simple direct proportional feedback control. Alternatively, we could also obtain Eq (25) directly from Eq (20) in the special limiting case of assuming both no multiplicative noise and no cost for control effort (σ_m = γ = 0).

In particular, we note that for the case of q = w, Eq (25) is the same direct step width controller postulated by Kuo [16]. The present work thus extends that work in at least 3 important ways. First, all controllers considered here were derived from a stochastic optimal control framework, not proposed by conjecture. Second, these derived more general control policies (Eqs (8) and (11)) allow us to consider stochastically optimal controllers in cases less restrictive than the unbiased case. Third, they also allow us to consider stochastic control of other potentially relevant stepping variables beyond just step width.

Multi-objective stochastic optimal regulator models

We consider strategies that control two controller states (q₁ & q₂) simultaneously, with the goal to drive each variable to its own desired value. Thus, the generic dual-objective goal function has the form of Eq (9), with the expanded two-state update map of Eq (10). We then take the cost function for regulating two variables to be the sum of those used for each single variable: (26)

The subsequent steps in the derivation are then the same as those described above for the single variable case, as outlined in Eqs (15)–(20). Here, this yields the two-variable control law of Eq (11). We considered three different specific versions of this general model, corresponding to all the possible combinations of any two of the three potential control state variables: i.e., [q₁, q₂] ∈ {[z_B, Δz_B], [Δz_B, w], [z_B, w]}.

We note that the goal function of Eq (9) indicates that each error, e_i, in Eq (26) will depend only on the goal level error for its own corresponding control state variable, q_i. Likewise, Eq (26) contains no cross terms in either the goal-level errors (e₁, e₂) or controller inputs (u₁, u₂). Hence, the final derived control law of Eq (11) and corresponding final state update equation of Eq (10) are both fully diagonal, such that the fluctuation dynamics of q₁ and q₂ are uncoupled. While it is certainly possible to imagine controller templates that include interactions between q₁ and q₂, we found in practice that the more parsimonious forms of Eqs (9) and (26) were adequate for our analyses.

Simulations of lateral stepping

In total, we generated simulations for three sets of each of three models. We first assessed the original (unconstrained) uni-objective models for each of q ∈ {z_B, Δz_B, w}, for which we imposed no additional task constraints. We assessed each control strategy across a range of parameter values of each model. We then compared the dynamical behavior predicted by each set of simulations directly to experimental values (i.e., Fig 2) to determine the likelihood humans might have adopted any of these uni-objective control strategies. Full details of these simulations and the parameter sensitivity analyses conducted on these models and the results of those analyses are in S1 Appendix.

All of the unconstrained uni-objective models failed to capture human stepping dynamics (S1 Appendix). They did so mainly by either exceeding lateral boundary limits (i.e., they walked off the path) and/or by taking steps that were unrealistically too wide or too narrow.

Because none of the unconstrained uni-objective models captured human stepping dynamics, we then assessed corresponding “constrained” uni-objective models for each of q ∈ {z_B, Δz_B, w}. In those simulations we additionally imposed physiologically relevant constraints that the simulations could not take steps that would walk off the path or would yield step widths unrealistically too wide or too narrow. We assessed each of these constrained control strategies across a range of parameter values of each model. We then compared the dynamical behavior predicted by each set of these simulations directly to human experimental values (i.e., Fig 2). Full details of these simulations and the parameter sensitivity analyses conducted on these models and the results of those analyses are in S1 Appendix.

Finally, we assessed each of the three 2-variable multi-objective models (i.e., [q₁, q₂] ∈ {[z_B, Δz_B], [Δz_B, w], [z_B, w]}). We assessed each of these control strategies across a range of parameter values of each model. We then compared the dynamical behavior predicted by each set of these simulations directly to experimental values (i.e., Fig 2) to determine the likelihood humans might have adopted any of these multi-objective control strategies. Full details of these simulations and the parameter sensitivity analyses conducted on these models and the results of those analyses are in S2 Appendix.

Calculation of dependent measures

The primary gait variables obtained from each simulated or experimental walking trial consisted of time series of left and right lateral foot placements (z_L & z_R), lateral body position and heading (z_B & Δz_B), and step width (w) (Figs 1C and 2). We then analyzed these times series from both experimental and simulated walking trials in the same ways for each to make direct comparisons between simulation predictions and experimental findings.

First, we computed standard deviations of each relevant time series to quantify the magnitudes of the variance in them. For the computational models, variance in the respective variable(s) being regulated reflected the “goal relevant” variance [6, 7] because fluctuations in each respective controlled q_n directly reflected deviations away from the corresponding desired q*. Conversely, any variance in those variables not being directly controlled was “goal equivalent” [6, 7] because fluctuations in those variables did not directly affect the designated task goal(s).

Standard deviations, however, quantify only the average magnitude of differences across all steps, regardless of temporal order. They thus yield no information about how each step affects subsequent steps. Therefore, here we used Detrended Fluctuation Analysis (DFA) to define a convenient lag-independent measure of statistical persistence across successive steps in each time series [9, 12, 71]. In brief, DFA calculates a scaling exponent, α, that quantifies the degree of statistical persistence or anti-persistence in a time series. To compute these exponents, each time series x(n), where n ∈ {1, …, N } steps [3–5], was first integrated to form a cumulative sum: (27) where x(n) is the value of x for the n^th step and is the mean value of x across all N strides. Each integrated series was divided into equal, non-overlapping segments of length j. Each segment was detrended by subtracting a least squares linear fit to that segment. The squares of the residuals were then averaged over the entire data set and the square root of the mean residual, F(j), was calculated: (28)

This process was repeated for different segment lengths, j. Here, fifty values of j evenly distributed between 4 and N/4 were used. Typically, F(j) increases with j and a graph of log[F(j)] versus log(j) will exhibit a power-law relationship indicating the presence of scaling, such that F(j) ≈ j^α. These log[F(j)] versus log(j) plots were then fit with a linear function using least squares regression. The slope of this line defined the scaling exponent α [9, 12, 71].

Standard deviation calculations quantify only average magnitudes of fluctuations and DFA α calculations quantify only the degree to which, on average, these deviations are corrected in time (from step-to-step). Neither analysis quantifies how step-to-step corrections of fluctuations might depend nonlinearly on the initial magnitudes of those fluctuations. Thus here, we directly quantified the degree to which deviations, , from the mean value,, of a given time series were corrected on the subsequent step by corresponding changes, Δq_n+1 = q_n+1 − q_n, in the opposite direction [10]. We performed these analyses on each relevant time series obtained from both our human subjects (Fig 2) and from each of our final multi-objective position-step width control models (Fig 5). For each relevant time series, q ∈ {z_B, Δz_B, w}, we constructed plots of Δq_n+1 vs. q'_n (Fig 6A). We then computed the linear slopes (using least-squares regression) and strength of correlation (r²) [10] for each corresponding relationship (Fig 6B and 6C).

Statistical comparisons

The primary comparisons of interest were between the model predictions and experimental observations (Fig 2). Here however, because the simulated and experimental data were structured in very different ways, standard inferential statistical tests (e.g., t-test, ANOVA, etc.) would not be appropriate. Additionally, because for the models, we could simulate as many trials as we wanted and of any arbitrary length, we could thus ensure p-values for most any comparisons that could be arbitrarily small and thus would not be meaningful. Therefore, to present descriptive data (Figs 2, 3, 4 & 7), for each relevant measure, data were plotted as boxplots indicating the median, 1^st and 3^rd quartiles, with whiskers extending to 1.5× the inter-quartile range and any values beyond that range indicated by individual data points. To directly compare model predictions to human experimental results (Figs 5 and 6), for each relevant measure, the expected range of values exhibited by humans was taken as the mean ± 1 standard deviation for the experimental data set. Model predictions are then shown as the mean ± 95% confidence interval for that mean for each set of corresponding simulations. We then infer that whenever the mean simulation prediction for a given measure lies within the experimental mean ± 1 SD band, that computational prediction is statistically consistent with experimental findings [10].

Data

All relevant experimental data are available from Dryad (https://doi.org/10.5061/dryad.p254480) [83].