Quaternions and joint angles in an analysis of local stability of gait for different variants of walking speed and treadmill slope

The authors describe an example of application of a nonlinear time series analysis directed at identifying the presence of deterministic chaos in human gait kinematic data by means of the largest Lyapunov exponent (LLE). A positive LLE value is interpreted as an indicator of local instability. The research was aimed at assessment of the influence of both walking speed and ground slope on the resilience of gait control to infinitesimally small perturbations that occur naturally during walking. The analysis of treadmill gait data was carried out twofold: 1) for the time series representing the following joint angles: hip flexion/extension, knee flexion/extension and dorsiflexion/plantarflexion of the ankle, and 2) for the time series representing rotations of foot, tibia and femur segments through Euler angles converted to a quaternion representation. A comparison between both approaches as well as a dependency between treadmill inclination and LLE values constitute the original aspects of this study. The LLE value was estimated threefold for every time series: as the short-term LLE for both the first step and the first stride and as the long-term LLE over a fixed interval between the fourth and the tenth stride. It was confirmed that all considered movements are characterized by positive LLE values which quantify a local instability. Moreover, a tendency to attenuate the perturbation consequences is evident in all variants of walking speed and treadmill slope.