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.
Highlights
During human quiet standing, the passive stiffness of the ankle joint, arising from visco-elasticity of the muscle-tendon-ligament system, is lower than the growth-rate of the gravitational toppling torque [1,2], leaving an upright unstable equilibrium of saddle type which is characterized by a topology of a system’s phase space spanned by the position and the velocity providing a convergent motion toward the equilibrium in one direction and a divergent motion away from the equilibrium in a different direction like a mountain pass
The distance between the stable manifold of the saddle dynamics that describes the system’s behavior when PD-control is off and the leading edge of the state segment in the phase plane when the PD control is turned off depends on the values of P and a as we have demonstrated for Model 3
The intermittent control strategy explored in this paper is based on the idea that, in order to control the behavior of a system characterized by a saddle-type unstable equilibrium, it is smart to take advantage of the stable manifold of the saddle and focus the active control intervention on the task of keeping the state of the system as close as possible to such manifold by means of a sequence of small, well timed control signals
Summary
The passive stiffness of the ankle joint, arising from visco-elasticity of the muscle-tendon-ligament system, is lower than the growth-rate of the gravitational toppling torque [1,2], leaving an upright unstable equilibrium of saddle type which is characterized by a topology of a system’s phase space spanned by the position and the velocity providing a convergent motion toward the equilibrium in one direction (a stable manifold) and a divergent motion away from the equilibrium in a different direction like a mountain pass (an unstable manifold). The upright standing posture requires to be stabilized by suitable active control strategies. The main challenge is how to compensate the danger of instability induced by the large neural feedback transmission delay, which is of the order of 200 ms [6]. The standard PD model faces a stringent trade-off that leaves narrow margins for the design of the control parameters: the proportional gain must be large enough for supplementing the insufficient ankle stiffness but not too large for avoiding delay-promoted instability. Damping of sway patterns requires rather large values of the derivative gain but again the feedback delay sets a stringent upper bound on this parameter. As we show in the following, the combination of these stability constraints leaves a very narrow area in the P–D parameter space where the standard controller is able to provide stability of the upright posture
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