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TECHNICAL PAPERS: Joint/Whole Body

# Differences Between Local and Orbital Dynamic Stability During Human Walking

[+] Author and Article Information
Jonathan B. Dingwell1

Department of Kinesiology & Health Education, University of Texas, 1 University Station, D3700 Austin, TX 78712jdingwell@mail.utexas.edu

Hyun Gu Kang

Department of Kinesiology & Health Education, University of Texas, 1 University Station, D3700 Austin, TX 78712

1

Corresponding author.

J Biomech Eng 129(4), 586-593 (Dec 06, 2006) (8 pages) doi:10.1115/1.2746383 History: Received January 28, 2006; Revised December 06, 2006

## Abstract

Currently there is no commonly accepted way to define, much less quantify, locomotor stability. In engineering, “orbital stability” is defined using Floquet multipliers that quantify how purely periodic systems respond to perturbations discretely from one cycle to the next. For aperiodic systems, “local stability” is defined by local divergence exponents that quantify how the system responds to very small perturbations continuously in real time. Triaxial trunk accelerations and lower extremity sagittal plane joint angles were recorded from ten young healthy subjects as they walked for $10min$ over level ground and on a motorized treadmill at the same speed. Maximum Floquet multipliers (Max FM) were computed at each percent of the gait cycle (from 0% to 100%) for each time series to quantify the orbital stability of these movements. Analyses of variance comparing Max FM values between walking conditions and correlations between Max FM values and previously published local divergence exponent results were computed. All subjects exhibited orbitally stable walking kinematics (i.e., magnitudes of Max $FM<1.0$), even though these same kinematics were previously found to be locally unstable. Variations in orbital stability across the gait cycle were generally small and exhibited no systematic patterns. Walking on the treadmill led to small, but statistically significant improvements in the orbital stability of mediolateral $(p=0.040)$ and vertical $(p=0.038)$ trunk accelerations and ankle joint kinematics $(p=0.002)$. However, these improvements were not exhibited by all subjects ($p⩽0.012$ for subject $×$ condition interaction effects). Correlations between Max FM values and previously published local divergence exponents were inconsistent and 11 of the 12 comparisons made were not statistically significant ($r2⩽19.8%$; $p⩾0.049$). Thus, the variability inherent in human walking, which manifests itself as local instability, does not substantially adversely affect the orbital stability of walking. The results of this study will allow future efforts to gain a better understanding of where the boundaries lie between locally unstable movements that remain orbitally stable and those that lead to global instability (i.e., falling).

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## Figures

Figure 5

Orbital stability results for overground (OG) versus treadmill (TM) walking. (A) Average Max FM values for both conditions for all six time series measures examined. Error bars indicate between-subject pooled standard deviations. Smaller Max FM values indicate greater orbital stability. Max FM values tended to be slightly lower (i.e., more orbitally stable) when walking on the treadmill. Differences were not statistically significant for anterior-posterior trunk accelerations (AAP; p=0.434), hip (p=0.286), or knee joint movements (p=0.447). Differences were statistically significant for mediolateral (AML; p=0.040) and vertical (AVT; p=0.038) trunk accelerations and for ankle joint movements (p=0.002). However, the condition × subject interaction effects for all three of these comparisons were also significant (p⩽0.012). (B) Condition × subject interaction plots for the three variables where the ANOVA revealed significant differences between OG and TM walking. Each line type represents the average results for one subject. The differences between OG and TM walking were clearly different for different subjects, with some subjects even showing slight increases in Max FM when walking on the TM.

Figure 6

Correlations between local (λS* and λL*) and orbital (Max FM) stability. (A) Correlations between short-term (λS*) local divergence exponents and Max FM for all six time series examined. (B) Correlations between long-term (λL*) local divergence exponents and Max FM. Only the comparison between λS* and Max FM at the hip was statistically significant after Bonferroni correction (p<0.004) and this relationship was negative: i.e., lesser local stability (larger λS*) predicted greater orbital stability (smaller Max FM). None of the remaining 11 comparisons were statistically significant (0.0%⩽r2⩽19.8%; 0.997⩾p⩾0.049). In general, no consistent relationships between local and orbital stability were found.

Figure 4

Variations in the magnitudes of maximum Floquet multipliers (FM) across the gait cycle for (A) trunk acceleration data and (B) lower extremity sagittal plane joint angles. All FM remained inside the unit circle (magnitude <1) indicating that all subjects exhibited orbitally stable walking patterns. In general, although the FM did vary somewhat across the gait cycle, there were no obvious or consistent patterns to these variations.

Figure 3

(A) Example of reconstructed 3D state spaces from the ankle angle data recorded during overground (OG) and treadmill (TM) walking for a typical subject. While only three dimensions are shown; the data were analyzed in dE=5D state spaces. While the two plots have different shapes, as expected because different time lags (τ) were used in the reconstructions, they both qualitatively appear as noisy closed orbits (i.e., limit cycles). (B) First return maps for the ankle angle data shown in (A), plotting deviations away from the fixed point (θ*) for each stride k+1, relative to the previous stride k. Each small dot represents the value recorded at the same phase (heel strike) of the gait cycle. Thus, the lines between the points serve only to indicate which strides occurred in what order and do not indicate trajectories between points. Fixed points are shown as large red (or gray) dots at [0,0]. These plots provide only a partial snapshot of only one of the five state variables used to define the full state space. Therefore, Floquet multipliers must be computed from the full state space data to determine if consecutive strides return, on average, to points closer to, or farther away from, the fixed point.

Figure 2

Schematic representation of local and orbital dynamic stability analyses. (A) Original time series data, x(t), plotted as a function of time (arbitrary units). (B) Reconstruction of a three-dimensional attractor for x(t) such that S(t)=[(t),x(t+τ),x(t+2τ)]. The two triplets of points indicated in (A) and separated by time lags τ and 2τ each map onto a single point in the 3D state space. (C) Expanded view of a local section of the attractor shown in (B). An initial naturally occurring local perturbation, dj(0), diverges across i time steps as measured by dj(i). Local divergence exponents (λ*) are calculated from the linear slopes of the average logarithmic divergence, ⟨ln[dj(i)]⟩ (Eq. 2), of all pairs of initially neighboring trajectories versus time. (D) Representation of a Poincaré section transecting the state space perpendicular to the system trajectory. The system state, Sk, at stride k evolves to Sk+1 one stride later. The Floquet multipliers quantify whether the distances between these states and the system fixed point, S*, grow or decay on average across many cycles (Eq. (7)).

Figure 1

(A) Schematic representation of an orbitally stable limit cycle that is also locally stable everywhere along the limit cycle. Being locally stable everywhere guarantees that the limit cycle must also be orbitally stable. (B) Representation of an orbitally stable limit cycle composed of both locally stable (solid line) and locally unstable (dashed line) regions. Trajectories that veer away from the limit cycle in the locally unstable regions are then “drawn back” toward it again in the locally stable region (adapted from Ref. 36).

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