In our previous study23, we adopted and revised a seven-segment bipedal model of a walking human introduced by Yang et al.16 and obtained the FSR at the toe-off instant of the swing foot (Foot2) where the BOS was the standing foot (Foot1) in contact with the ground (Fig. 2). The lower and upper limits of the FSR as a function of the BOS perturbation amplitude and frequency are reported in Tables Tables11 and and22 of this reference23 and are used in the present work. The lower limit of FSR defined the backward loss-of-balance as a need for stepping backward to prevent falling. The upper limit of FSR defined the forward loss-of-balance as the inability to maintain balance by terminating gait where the anterior foot is located and without taking further steps forward. During continuous walking, the body COM state voluntarily leaves this FSR (FSR1) during the swing phase period, which would not necessarily result in loss-of-balance. This is because the BOS expands to the area under and between both feet as soon as Foot2 touches the ground in front of Foot1 at the beginning of the double-support phase (i.e., the heel-strike instant of Foot2).

During the double-support phase, until the succeeding toe-off instant of Foot1, forward loss-of-balance does not occur before the COM state passes the upper limit of the succeeding FSR (FSR2) in which the anterior foot (Foot2) determines the BOS. In the present paper, to expand the definition of FSR to assess the stability of consecutive steps, we define the ExFSR (Extended FSR) as the region including the FSRs for all instants within an entire step (from the toe-off instant of Foot2 to the toe-off instant of Foot1). Given that each step is composed of a swing phase and a succeeding double-support phase, the ExFSR is the COM state-space between the lower limit of FSR1 and the upper limit of FSR2 (Fig. 2). Note that during the double-support phase, FSR1 and FSR2 are separated by the distance between the toe’s tip of Foot1 and the heel of Foot2. We assumed that FSR1 and FSR2 are identical and can be obtained based on the frequency and amplitude of the external perturbations according to 23. During walking, the COM, BOS and, thus, ExFSR progress step by step.

We introduce the ‘Index of Stability’, i.e., ISL(n) and ISU(n), to characterize the risk of backward and forward loss-of-balance, respectively, during one isolated step during gait (n is the step index). ISL(n) (or ISU(n)) are defined as follows (Fig. 2):

The shortest distance, a positive value, from the trajectory of the COM state to the lower (or upper) limit of the ExFSR, if the COM state lies inside the ExFSR near its lower (or upper) limit.

The longest distance, a negative value, from the trajectory of the COM state to the lower (or upper) limit of the ExFSR, if the COM state lies outside the lower (or upper) limit of the ExFSR.

IS(n) (i.e., ISL(n) or ISU(n)) depends on both the COM position and velocity, and illustrates how close the individual can be to loss-of-balance for the duration of a step. When IS(n) is a positive value, the smaller the IS(n), the more probable the loss-of-balance. A negative IS(n) is indicative of temporary loss-of-balance. Based on the physiological condition of the walker and the value of the negative IS(n), the temporary loss-of-balance can either be recovered or lead to an incidence of falling.

We also defined the ‘Index of Balance Challenge’, BCL and BCU, as the percentage of steps (out of all steps) during a walking trial, in which ISL(n) and ISU(n), respectively, was negative. As such, BCL and BCU are indicators of challenge in maintaining backward and forward balance, respectively, during a perturbed walking trial.

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