A standing body behaves like an inverted pendulum balanced over the feet: gravity constantly tries to topple it, and the nervous system counteracts that with corrective torque generated at the ankles, hips and inner-ear (vestibular) reflexes. Balance training strengthens exactly this feedback loop.
I·θ̈ = m·g·h·sin(θ) − c·θ̇ − Kp·θ − Kd·θ̇ + τ(t)
θ = body tilt from vertical (both axes)
m·g·h = destabilizing gravity torque (h = height of center of mass)
c·θ̇ = passive damping from muscles/tissue
Kp,Kd = reflex "gains" — proportional + derivative correction
τ(t) = random perturbation (a push, a gust, an uneven step)
- Push strength — how large and frequent the random destabilizing impulses τ(t) are; simulates external pushes or gusts.
- Reflex speed — scales Kp and Kd, i.e. how quickly and firmly the "athlete" corrects tilt; this is precisely what improves with practice.
- Surface — firm ground keeps the effective pivot height low and damping high; a wobble board or BOSU ball raises the effective pivot and lowers damping, so the same push produces far more sway.
- The fading trail under the figure is the center-of-pressure sway path — the same metric physiotherapists record on a force plate (posturography) to score balance.
Falling below the reflex/surface combination needed to counteract θ pushes the system past the base of support — the simulated "fall" — resetting the balance timer, just like stepping off a real wobble board.