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When a rigid body is hanging freely from a fixed pivot point and is displaced, it oscillates similar to a simple pendulum and is known as a physical pendulum. The period and angular frequency of a physical pendulum are obtained by using the small-angle approximation and drawing parallels with a spring-mass system. The small-angle approximation (sinθ=θ) is valid up to about 14°.
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Related Experiment Video

Updated: Sep 19, 2025

Experimental Methods to Study Human Postural Control
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Adjoint sensitivity method for parameter estimation: applications to inverted pendulum and human standing balance.

Jingtian Chen1, Shaoyi Lu1, Li Zhang1

  • 1State Key Laboratory of Mechanics and Control for Aerospace Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, People's Republic of China.

Journal of the Royal Society, Interface
|June 18, 2025
PubMed
Summary
This summary is machine-generated.

Human balance control is modeled using a time-delayed proportional-derivative (PD) feedback controller. This study identifies key parameters, control gains, and time delays, revealing an energy-efficient human balancing strategy.

Keywords:
adjoint sensitivity analysishuman balance controlinverted pendulumparameter identificationtime delay system

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Area of Science:

  • Biomechanical Engineering
  • Control Systems Theory
  • Human Physiology

Background:

  • The inverted pendulum is a classic model for stability and control studies.
  • Human standing balance can be effectively modeled as an inverted pendulum with time-delayed proportional-derivative (PD) feedback control.
  • Understanding human balance control mechanisms is crucial for biomechanical research.

Purpose of the Study:

  • To investigate human balance control strategies using an adjoint sensitivity analysis method.
  • To directly determine system parameters, control gains, and time delays in a human balancing model.
  • To validate the accuracy of an optimizer for systems with non-smooth dynamics and time delays.

Main Methods:

  • Implementation of an adjoint sensitivity analysis method and an optimizer.
  • Numerical simulations and experimental verification using a physical inverted pendulum on a cart model.
  • Identification of parameters and control gains from human balance data.

Main Results:

  • The optimizer accurately identifies system parameters, control gains, and time delays.
  • Experimental results confirm the algorithm's performance on systems with non-smooth dynamics and time delays.
  • Human balance data analysis indicates that a time-delayed PD feedback controller effectively represents human balance control.

Conclusions:

  • The time-delayed proportional-derivative (PD) feedback controller is an effective model for human balance control.
  • Human balance control strategy tends towards optimal control, minimizing energy consumption.
  • The identified control gains are located in the lower-left region of the stability diagram, suggesting energy efficiency.