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# The mechanics of standing balance

## Problem

A simple model of human standing is given by an inverted pendulum, a one-dimensional, classical model for the motion of a single, massive particle (Figure 1). A standing human is approximated as a single mass (located at the center of mass) separated from the ground by a massless rod of fixed length $L$. The feet are treated as attached to the ground by a fulcrum, such that the center of mass can only undergo motion along an arc of radius $L$ around the feet---thus the mechanics of the system are described by the tilt angle $\theta$, representing the angular displacement of the center of mass from directly above the feet, with the rod positioned normal to the ground. The pendulum is perfectly balanced when the mass is positioned directly above the pivot point; however, a very slight displacement of the mass from this position will cause the pendulum to tip over.
Figure 1. The analogy between a standing patient and an inverted pendulum.
Humans can overcome this difficulty and maintain standing balance, at which the system is at equilibrium, by maintaining active control of the position of their center of mass relative to the fulcrum formed by their feet. For small displacements, the ankles work to exert a torque that counteracts gravity and prevents the individual from falling over. The timescale that this restoring force must act to recover equilibrium is proportional to the period of a simple pendulum, demonstrated by Equation 1.
T= 2π$\sqrt{\frac{L}{g}}$
Equation 1.
Precise measurements of active balancing can be made by filming an individual and digitally tracking the location of the center of mass after the individual is tilted forward by a known angular displacement at $t=0$. A sample figure showing the angular response is given in Figure 2. Surprisingly, these measurements show that the seemingly inert process of standing consists of many coordinated ankle motions that stabilize the body after it undergoes slight deflections.
Figure 2. Simulated data showing the time-varying tilt angle of a standing individual who was tilted forward by $0.2$ radians at $t=0$, and who is actively returning to their original vertical standing position.
Concept adapted from: Winter, D. A. (1995). Human balance and posture control during standing and walking. Gait & Posture, 3(4), 193-214.
Which of the following correctly describe the difference between an inverted pendulum (as shown in in Figure 1) and a standard pendulum with identical length and mass?
I. An inverted pendulum has minimal kinetic energy when it reaches its equilibrium point
II. An inverted pendulum requires a non-gravitational restoring force to remain in equilibrium
III. An inverted pendulum reaches a higher maximum gravitational torque