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x(t+2r)

State Space

Figure 1: Reconstruction of a attractor trajectory in the state space (in case of the embedding
dimension d = 3).

log. Embedding dimension d was determined by using the global false nearest neighbors algorithm
(Cao 1997). In our case, the embedding dimension was four, to form a valid state space.

Lyapunov exponent quantifies the average exponential rate of divergence of neighboring trajecto-
ries in a reconstructed state space. The estimation of the largest Lyapunov exponent performed
with the method proposed by Kantz (Kantz 1994). The Lyapunov exponent A was defined as the
following.
XAt
D(At) = D(0)e . (2)
The notation D(At) denotes the displacement between neighboring trajectories after The nota-
tion At interval. -D(O) is the initial distance between neighboring point. Lyapunov exponent A
quantifies the average exponential rate of divergence of neighboring trajectories in a reconstructed
state space. A Higher value of Lyapunov exponent indicates a larger divergence of the attractor
in the state space, suggesting less stability of the dynamical system. We calculated the exponent
from ten steps acceleration data in steady state of the walking trial. Data were analyzed without
filtering to avoid complications associated with filtering nonlinear signals.

EXPERIMENT
We developed a portable device consisted of monolithic IC accelerometers (±2 G, ADXL202E;
Analog Devices Inc.. MA, USA) with 16-bit duty cycle converter, Li-Ionic batteries, micro proces-
sor units and CompactFlash card. This equipment is small (100x55x18.5 mm) and lightweight
enough to carry without any restriction. The equipment was attached to the center of lower
back representing the center of gravity of the body using a back supporter, as shown in Figure 2.
Three-dimensional acceleration as lateral, vertical, and anteroposterior direction were measured
by the portable equipment with sampling frequency of 100 Hz.

Seven healthy young adults (25.0 ± 1.6 yr.) and fifty-four elderly adults aged (76.7 ± 4.6 yr.)
participated in the experiment. All subjects gave signed informed consent. Prior to the experi-
ment, physical conditions and exercise habit were examined by questionnaires. The subjects were
instructed to walk at their self-selected speed on a, 16 m straight track without any restriction.
The beginning and the end of the strait track 3 m were considered as transition phases of the
walking. Constant walking phase in middle 10 m of the track was applied to the calculation.