The time-delayed inverted pendulum: Implications for human balance control

Abstract

The inverted pendulum is frequently used as a starting point for discussions of how human balance is maintained during standing and locomotion. Here we examine three experimental paradigms of time-delayed balance control: (1) mechanical inverted time-delayed pendulum, (2) stick balancing at the fingertip, and (3) human postural sway during quiet standing. Measurements of the transfer function (mechanical stick balancing) and the two-point correlation function (Hurst exponent) for the movements of the fingertip (real stick balancing) and the fluctuations in the center of pressure (postural sway) demonstrate that the upright fixed point is unstable in all three paradigms. These observations imply that the balanced state represents a more complex and bounded time-dependent state than a fixed-point attractor. Although mathematical models indicate that a sufficient condition for instability is for the time delay to make a corrective movement, tau(n), be greater than a critical delay tau(c) that is proportional to the length of the pendulum, this condition is satisfied only in the case of human stick balancing at the fingertip. Thus it is suggested that a common cause of instability in all three paradigms stems from the difficulty of controlling both the angle of the inverted pendulum and the position of the controller simultaneously using time-delayed feedback. Considerations of the problematic nature of control in the presence of delay and random perturbations ("noise") suggest that neural control for the upright position likely resembles an adaptive-type controller in which the displacement angle is allowed to drift for small displacements with active corrections made only when theta exceeds a threshold. This mechanism draws attention to an overlooked type of passive control that arises from the interplay between retarded variables and noise.

Human Balance Control: Dead Zones, Intermittency, and Micro-chaos

“…Our focus is on the effects of a sensory dead zone on the control of postural sway when there is state-dependent feedback [39,40,70,90]. The phrase 'statedependent feedback' means that the feedback only contains terms related to…”

“…The observation that it is easier to balance a longer stick at the fingertip than a shorter one demonstrates the importance of time-delayed feedback: once the stick becomes sufficiently long its rate of movements become slow relative to the time required by the nervous system to make a corrective movement. Consequently mathematical models of these Newtonian systems take the form of second-order delay differential equations [39,40,50,70,85,88,89,90] θ(t) − ω 2 n sin θ(t) = f (θ(t − τ ),θ(t − τ ),θ(t − τ )) ,…”

Modelling human balance using switched systems with linear feedback control

“…In the study of Milton et al [21,22], the switch-like balance controller was considered as a type of intermittent control. The idea of switched control was further explored in Milton et al [23,24]. The authors considered a model with switched control, random perturbations that model noise, and time delay.…”

Balancing on tightropes and slacklines

“…In figure 5b, we show the stability region as a function of the feedback delay t and the normalized sag R. We note that increasing the value of the delay makes the performance worse and can lead the system to instability, as expected [8,27]. Since the natural frequency of the system decreases as ffiffiffiffiffiffiffiffi ffi 1=R p as R ) 1, we expect the critical value of t to increase as R becomes larger.…”