Dynamics of quiet human stance: computer simulations of a triple inverted pendulum model

Abstract

For decades, the biomechanical description of quiet human stance has been dominated by the single inverted pendulum (SIP) paradigm. However, in the past few years, the SIP model family has been falsified as an explanatory approach. Double inverted pendulum models have recently proven to be inappropriate. Human topology with three major leg joints suggests in a natural way to examine triple inverted pendulum (TIP) models as an appropriate approach. In this study, we focused on formulating a TIP model that can synthesise stable balancing attractors based on minimalistic sensor information and actuation complexity. The simulated TIP oscillation amplitudes are realistic in vertical direction. Along with the horizontal ankle, knee and hip positions, though, all simulated joint angle amplitudes still exceed the measured ones about threefold. It is likely that they could be eventually brought down to the physiological range by using more sensor information. The TIP systems’ eigenfrequency spectra come out as another major result. The eigenfrequencies spread across about . Our main result is that joint stiffnesses can be reduced even below statically required values by using an active hip torque balancing strategy. When reducing mono- and bi-articular stiffnesses further down to levels threatening dynamic stability, the spectra indicate a change from torus-like (stable) to strange (chaotic) attractors. Spectra of measured ground reaction forces appear to be strange-attractor-like. We would conclude that TIP models are a suitable starting point to examine more deeply the dynamic character of and the essential structural properties behind quiet human stance.Abbreviations and technical termsInverted pendulum body exposed to gravity and pivoting in a joint around position of unstable equilibrium (operating point)SIP single inverted pendulum: one rigid body pivoting around fixation to the ground (external joint)DIP double inverted pendulum: two bodies; external and internal joint operate around instabilityTIP triple inverted pendulum: three bodies; external and both internal joints operate around instabilityQIP quadruple inverted pendulum: four bodies, foot replaces external joint; all three internal joints operate around instabilityEigenfrequency characteristic frequency that a physical system is oscillating at when externally excited at a limited energy levelDOF degree of freedom; in mechanics: linear displacement or angle or combination thereof Mono-articular stiffness: coefficient of proportionality between mechanical displacement of a DOF and restoring force/torque component in the respective DOFBi-articular stiffness coefficient of proportionality between mechanical displacement of a DOF and restoring force/torque component in another DOFGRF ground reaction forceHAT segment including head, arms and trunkCOM centre of massCOP centre of pressure in the plane of the force platform surface