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 terms

Inverted 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 instability
TIP	triple inverted pendulum: three bodies; external and both internal joints operate around instability
QIP	quadruple inverted pendulum: four bodies, foot replaces external joint; all three internal joints operate around instability
Eigenfrequency	characteristic frequency that a physical system is oscillating at when externally excited at a limited energy level
DOF	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 DOF
Bi-articular stiffness	coefficient of proportionality between mechanical displacement of a DOF and restoring force/torque component in another DOF
GRF	ground reaction force
HAT	segment including head, arms and trunk
COM	centre of mass
COP	centre of pressure in the plane of the force platform surface

Keywords:

• Biomechanics
• direct dynamics
• posture control

Acknowledgements

During October, November and December 2011, MG was cordially supported by Westfälische-Wilhelms-Universität Münster as a ‘Gastwissenschaftler’. Lying his head in the Wagner-Keller was not exactly the least joy. Owing to Alexandra, this place will always remind of one of the best laughing outbursts ever. Many thanks to Heiko Wagner and Anne Dalhaus in Münster for making this real. MG sends many thanks to you, Syn, for your invaluable backing and faith during the last five years that have been pretty rough!

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

MG was also supported by grants to Syn Schmitt from the DFG Cluster of Excellence in Simulation Technology (Universitat Stuttgart: EXC310/1), the ‘Berufsgenossenschaft Nahrungmittel und Gastgewerbe, Geschaftsbereich Pravention, Mannheim, Germany’ and the DFG [grant number SCHM2392/5-1].