Presented at NACOB 98: North American Congress on Biomechanics Canadian Society for Biomechanics - American Society of Biomechanics University of Waterloo Waterloo, Ontario, Canada August 14-18, 1998

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SELF-ORGANIZED TRANSITION OF THE LEG MOTION
IN THE LEARNING PROCESS OF A LOCOMOTION

Takashi Yokoi ¹ , Akihiko Takahashi ¹ , Makoto Muraoka ² , Chieko Fujita ^1
1 National Inst. of Bioscience and Human-Engineering, 1-1 Higashi, Tsukuba, Ibaraki 305-8355, Japan
² University of Tsukuba, 1-1-1 Ten-oudai, Tsukuba, Ibaraki 305-8574, Japan

INTRODUCTION

Dynamical system approaches is applied to behavioral analysis, and self-organization and chaos were observed in our behavior (Woolacott et al., 1989; Kelso,1995; Borghese, 1996; Yokoi et al.,1996). This present study, using the dynamical system approaches, is intended as an investigation of the learning process of human movements for acquiring skilled performance. The main purpose hear is to kinematically describe the learning process of coordinated movements while focusing our attention on the aspect of self-organization and chaos of the leg motion in a human locomotion.

METHODS

For the above purpose, we employed constant-speed, closed-eye, backward walking on a treadmill as an imposed exercise yet to experience. Eight adult male subjects (age: 25.9 ± 3.6 yr, stature: 1.720 ± .426 m, weight: 63.9 ± 5.9 kg) were asked to perform this walking on a treadmill (at a belt speed of 50 m/min.) for five minutes four times. The two-dimensional coordinate values of the landmarks on the right leg of each subject were continuously measured for five minutes at 60 Hz by Quick-MAG System II (OKK, Inc.). The angles of the hip, knee and ankle joints were calculated from the measured coordinates (Fig.1). The transition of the lower limb motion with the progress of learning was perceived as the angular relation pattern among the three lower limb joints. The attractor of a dynamical system creating the flexion- and extension-movement of each joint was reconstructed from the time-series data of the joint angle by the method of Takens (1981).

In order to obtain the magnitude of chaos and the degree of freedom of the attractor, fractal dimension and maximum Lyapunov exponent were calculated for each attractor by the methods of Grassberger & Procaccia(1983) and Sano & Sawada (1985) . Lyapunov exponent of an attractor measure the rate of divergence of points on arbitrarily close trajectory, providing an estimate of the chaotic phenomenon. Fractal dimension is a parameter to describe geometrical structure (and/or degree of freedom) of the attractor. The attractor has chaotic property when the dimension is fractional.

Fig.1 Definitions of lower limb joint angles.

RESULTS AND DISCUSSION

Fig.2 shows the angular relation pattern of the ankle, knee and hip joints of a subject in motion in the first to fourth trials (Trial 1 to Trial 4) . The data for the 50 seconds immediately after the beginning of measurement are plotted in each trial of the figure. As the trial number advanced, a gradual change was seen in the angular relation pattern of the three joints. The trajectory appeared to converge into a particular pattern. These changes in trajectory patterns, typical examples of self-organized transition of the leg motion, may indicate that the motion is improved in stability and accuracy with the progress of learning.

Fig.2 Transition of angular relation patterns of the ankle, knee and hip joints in motion in Trials 1 through 4 of the subject O.

Fig.3 shows attractor trajectories of a dynamical system governing the flexion and extension of each joint. These attractors were reconstructed into a three-dimensional phase space from the time-series of lower limb joint angle. The attractor pattern of the ankle joint changes from chaotic in Trial 1 to distinctive one in Trial 4 while gradually changing its geometric shape. The attractor patterns of the hip and knee joints are more distinctive than that of the ankle joint, and there is little change accompanying the advancement of trials. A resembling tendency was found for the other 8 subjects.

Fig.3 Attractor trajectory of a dynamical system, reconstructed into a three-dimensional phase space according to the time-series data of angle of each lower limb joint for the subject O.

The Lyapunov exponent value was positive (i.e., chaos) for any joint and was largest for the ankle joint (Fig.4 A). The difference in the exponent among the leg joints became small with the repetition of trials mainly due to the decrease in the exponent value for the ankle joint. The fractal dimension was largest for the attractors of the knee joint (Fig.4 B), meaning that the degree of freedom for the dynamical system controlling knee joint motion is largest in the three joints. The fractal dimension was fractional for the attractors of the three joints.

Fig.4 Lyapunov exponents (A) and fractal dimensions(B) of the attractors for the leg joints in Trial 1 through Trial 4.

These results may indicate 1) the transition pattern of the joint motion with the progress of learning is different among the three leg joints, 2) chaotic dynamics exists in leg joint motion of the present locomotion, and 3) the chaos observed in the leg joint motion is related to the motor learning of the locomotion.

For the future, we shall perform both experimental and theoretical approaches more to identify the role and meaning of chaos in motor learning of human locomotion.

REFERENCES

Borghese,N.A. et al, J. Physiology, 494-3:863-879, 1996.

Grasberger,P. et al. Physica D, 9:189-208, 1983.

Sano,M. et al., Physical Review Letters, 55-10:1082-1085, 1985.

Takens,F., Dynamical Systems and Turbulence, Lecture Notes in Mathematics 898:366-381, Springer-Verlag, 1981.

Yokoi,T., et al., J. Robotics and Mechatronics, 8-4:364-371, 1996.

Woolacott, M.H. et al. (Eds.),Development of posture and gait, University of Southern California Press, 1989.