Fig 1.
Regulating Stride Length (Ln) and Stride Time (Tn) When Walking on a Treadmill.
(A) Schematic figure of a person walking on a treadmill of total length LTM with the center position defined as zero. The only strict requirement of the task is that the person not walk off either the front (+LTM/2) or back (−LTM /2) end of the treadmill (Eq 1). (B) Example data for stride times (Tn) and stride lengths (Ln) for a typical trial for a typical subject. Each dot represents the particular [Tn, Ln] combination for one individual stride, n. The solid diagonal line indicates the set of all combinations of Ln and Tn that achieve the exact same speed, v. This line defines one possible Goal Equivalent Manifold (GEM) for walking (Eq 2): i.e., for walking while trying to achieve the goal of maintaining constant speed. Unit vectors then define directions perpendicular to (êP) and tangent to (êT) the GEM. Deviations δP and δT define the deviations of each data point in the êP and êT directions, respectively (see Ref. [16]).
Table 1.
Participant Characteristics.
Fig 2.
Means of Basic Gait Variables.
(A) Stride Lengths (Ln), (B) Stride Times (Tn), and (C) Stride Speeds (Sn = Ln/Tn). In each sub-plot, data shown are for healthy human subjects (HU), the Position Control model with MIP control (PMIP), the Position Control model with “over-correcting” control (POVC), the Speed Control model with MIP control (SMIP), and the Speed Control model with “over-correcting” control (SOVC). For the HU data, error bars indicate between-subject ± standard deviations. For the model data, error bars indicate between-trial ± standard deviations. All models yielded mean stride parameters well within the experimental range. However, both MIP controllers (PMIP and SMIP) yielded much greater between-trial variance for both Ln (A) and Tn (B) than observed experimentally.
Fig 3.
Variability (σ) and Statistical Persistence (α) of Stride Lengths (Ln) and Stride Times (Tn).
(A) Variability (within-trial standard deviations (σ) of humans (HU) and of each of the four model configurations (PMIP, POVC, SMIP, and SOVC). (B) DFA scaling exponents (α) of humans (HU) and of each of the four model configurations (PMIP, POVC, SMIP, and SOVC). For the HU data, error bars indicate between-subject ± standard deviations. For the model data, error bars indicate between-trial ± standard deviations. Both models that implemented pure MIP-type control (i.e., PMIP and SMIP), regardless of whether controlling for speed or position, yielded much higher levels of variance (A) and much greater degrees of statistical persistence (indeed, approaching random walk behavior: α ≈ 1.5) (B) in the typical walking parameters that are not directly controlled for (i.e., Ln and Tn) than did humans (HU).
Fig 4.
Example Time Series of Stride Speeds (Sn) and Absolute Position (Pn).
Data are shown for 240 consecutive strides from a typical trial for a typical human subject (HU) and for one typical trial each of each of the four model configurations (PMIP, POVC, SMIP, and SOVC). All four model configurations yielded Sn time series that initially appeared qualitatively similar both to each other and also to humans (HU). Conversely, both of the Position Control models (PMIP and POVC) qualitatively exhibited far less variance and far less “drift” in their absolute positions on the treadmill (Pn) than did humans (HU).
Fig 5.
Variability (σ) and Statistical Persistence (α) of Stride Speeds (Sn) and Absolute Positions (Pn).
(A) Variability (within-trial standard deviations (σ) of humans (HU) and of each of the four model configurations (PMIP, POVC, SMIP, and SOVC). (B) DFA scaling exponents (α) of humans (HU) and of each of the four model configurations (PMIP, POVC, SMIP, and SOVC). For the HU data, error bars indicate between-subject ± standard deviations. For the model data, error bars indicate between-trial ± standard deviations. Position control yielded uncorrelated fluctuations (α ≈ 0.5) of absolute positions (Pn) under optimal MIP control conditions (PMIP) and slightly anti-correlated fluctuations (α < 0.5) of Pn when over-correcting for position (POVC). This in turn yielded highly anti-correlated fluctuations (α < 0.1) of stride speeds (Sn). Speed control yielded uncorrelated fluctuations (α ≈ 0.5) of stride speeds (Sn) under optimal MIP control conditions (SMIP) and slightly anti-correlated fluctuations (α < 0.5) of Sn when over-correcting for position (SOVC). This in turn yielded strongly correlated fluctuations for absolute position (Pn), which approached random walk behavior: α ≈ 1.5), consistent with position reflecting integrated speed (Eq 1, 3, and 13). Both of the position control models (PMIP and POVC) exhibited fluctuation dynamics very different from those of humans (HU). The over-correcting speed control model (SOVC) exhibited fluctuation dynamics most consistent with humans.
Fig 6.
Direct Correction of Errors in Position and Speed.
(A) Exemplary plots of how errors in relative position were corrected on each subsequent stride (ΔPn+1 = Pn+1 − Pn). Data are shown from 1 trial each for each of the two sub-optimal control model configurations (POVC and SOVC) and for 1 typical human subject. Diagonal lines indicate least-squares fits to each data set. “Perfect” error correction would yield a relationship with a slope of −1 and a strong correlation. (B) Summary results for the slopes of these relationships for all trials for both humans (HU) and for each of the four model configurations (PMIP, POVC, SMIP, and SOVC). For the HU data, error bars indicate between-subject ± standard deviations. For the model data, error bars indicate between-trial ± standard deviations. (C) Summary results for the strength of correlation (r2) of these relationships [error bars are defined in the same manner as in part (B)]. While both position control models (PMIP and POVC) exhibited steep slopes (B) with strong correlations (C), both humans and both speed control models (SMIP and SOVC) exhibited nearly zero slopes with extremely weak correlations, consistent with not correcting deviations in position. (D) Exemplary plots of how errors in relative speed
were corrected on each subsequent stride (ΔSn+1 = Sn+1 − Sn). Data are shown in the same manner as in (A). As in (A), “perfect” error correction would yield a relationship with a slope of −1 and a strong correlation. (E) Summary results for the slopes of these relationships for all trials for both humans (HU) and for each of the four model configurations (PMIP, POVC, SMIP, and SOVC) [error bars are defined in the same manner as in part (B)]. (F) Summary results for the strength of correlation (r2) of these relationships [error bars are defined in the same manner as in part (B)]. Human subjects exhibited nearly perfect corrections (i.e., slope ≈ −1: (E)) for errors in speed with strong correlations (F), as did both of the speed control models (SMIP and SOVC).