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.

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Human upright posture, as a mechanical system, is characterized by an instability of saddle type, involving both stable and unstable dynamic modes. The brain stabilizes such system by generating active joint torques, according to a time-delayed neural feedback control. What is still unsolved is a clear understanding of the control strategies and the control mechanisms that are used by the central nervous system in order to stabilize the unstable posture in a robust way while maintaining flexibility. Most studies in this direction have been limited to the single inverted pendulum model, which is useful for formalizing fundamental mechanical aspects but insufficient for addressing more general issues concerning neural control strategies. Here we consider a double inverted pendulum model in the sagittal plane with small passive viscoelasticity at the ankle and hip joints. Despite difficulties in stabilizing the double pendulum model in the presence of the large feedback delay, we show that robust and flexible stabilization of the upright posture can be established by an intermittent control mechanism that achieves the goal of stabilizing the body posture according to a "divide and conquer strategy", which switches among different controllers in different parts of the state space of the double inverted pendulum. Remarkably, it is shown that a global, robust stability is achieved even if the individual controllers are unstable and the information exploited for switching from one controller to another is severely delayed, as it happens in biological reality. Moreover, the intermittent controller can automatically resolve coordination among multiple active torques associated with the muscle synergy, leading to the emergence of distinct temporally coordinated active torque patterns, referred to as the intermittent ankle, hip, and mixed strategies during quiet standing, depending on the passive elasticity at the hip joint.

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