A Model of Postural Control in Quiet Standing: Robust Compensation of Delay-Induced Instability Using Intermittent Activation of Feedback Control

Abstract

The main purpose of this study is to compare two different feedback controllers for the stabilization of quiet standing in humans, taking into account that the intrinsic ankle stiffness is insufficient and that there is a large delay inducing instability in the feedback loop: 1) a standard linear, continuous-time PD controller and 2) an intermittent PD controller characterized by a switching function defined in the phase plane, with or without a dead zone around the nominal equilibrium state. The stability analysis of the first controller is carried out by using the standard tools of linear control systems, whereas the analysis of the intermittent controllers is based on the use of Poincaré maps defined in the phase plane. When the PD-control is off, the dynamics of the system is characterized by a saddle-like equilibrium, with a stable and an unstable manifold. The switching function of the intermittent controller is implemented in such a way that PD-control is ‘off’ when the state vector is near the stable manifold of the saddle and is ‘on’ otherwise. A theoretical analysis and a related simulation study show that the intermittent control model is much more robust than the standard model because the size of the region in the parameter space of the feedback control gains (P vs. D) that characterizes stable behavior is much larger in the latter case than in the former one. Moreover, the intermittent controller can use feedback parameters that are much smaller than the standard model. Typical sway patterns generated by the intermittent controller are the result of an alternation between slow motion along the stable manifold of the saddle, when the PD-control is off, and spiral motion away from the upright equilibrium determined by the activation of the PD-control with low feedback gains. Remarkably, overall dynamic stability can be achieved by combining in a smart way two unstable regimes: a saddle and an unstable spiral. The intermittent controller exploits the stabilizing effect of one part of the saddle, letting the system evolve by alone when it slides on or near the stable manifold; when the state vector enters the strongly unstable part of the saddle it switches on a mild feedback which is not supposed to impose a strict stable regime but rather to mitigate the impending fall. The presence of a dead zone in the intermittent controller does not alter the stability properties but improves the similarity with biological sway patterns. The two types of controllers are also compared in the frequency domain by considering the power spectral density (PSD) of the sway sequences generated by the models with additive noise. Different from the standard continuous model, whose PSD function is similar to an over-damped second order system without a resonance, the intermittent control model is capable to exhibit the two power law scaling regimes that are typical of physiological sway movements in humans.

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

“…The second relies on the observation that the interplay between time delays and noisy perturbations can transiently stabilize an unstable fixed-point [12,14,15,68,71]. The third strategy is a nonlinear-type of control mechanism which relies on the properties of a saddle point [10,4]. The analytical explanation for transient stabilizations shown in Figure 12 is not presently known.…”

“…Two general types of explanations have been advanced. First, it is possible that the intermittency may be event-driven, namely corrective forces are generated whenever the controlled variable crosses a threshold [4,10,12,33,34,39,68,69,71]. A well known example of threshold crossing in human balance control is the "safety-net" character of the ankle-hip-step control strategy used by humans to maintain balance in the face of increasingly large perturbations [18,86].…”

Ageing of internal models: from a continuous to an intermittent proprioceptive control of movement

“…Intermittency is characterised by sub-movements which are triggered by an error crossing a threshold (Asai et al 2009;Hanneton et al 1997;Miall et al 1993;Wolpert et al 1992). In both self-selected and fast speed conditions, the weak correlations observed between the peak velocity and the constant error suggested that the young adults used adjustments to reach the endpoint position (Messier and Kalaska 1999).…”

Modelling human balance using switched systems with linear feedback control

“…The model we propose is similar to the one studied in Asai et al [25]-the PD control is switched on or off depending on the value of state variables. The model studied in Asai et al [25] is an extension of the studies conducted in Bottaro et al [26,27] where the authors introduce intermittently switched on/off controller that allows for a bounded motion, and it is this type of control which Asai et al [25] propose as the possible explanation for the sway motion during quiet standing. In the study of Asai et al [25], it is also shown that the bounded dynamics is linked with a motion along the stable manifold of a saddle point.…”