The Force-Driven Harmonic Oscillator as a Model for Human Locomotion Kenneth G. Holt, Joseph Hamill, and Robert O. Andres

Humans as self-optimizing machine Biological systems coordinated “…to produce the ultimate in performance at a minimal energy cost” Examples: wheelchair locomotion, manual tire pumping, and arm ergometry Graph of metabolic cost (O2 consumptions) as a function of frequency

Ambulation Human behaviors as a complex oscillatory processes Cycles Walking is periodic in nature Selection of stride frequencies that result in minimal metabolic energy costs Modeling locomotion as a pendulum Human gait and quadrupedal gait Animation

Harmonic Oscillator A harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x:

Force-Driven Harmonic Oscillator FDHO requires periodic forcing function to maintain its oscillations Takes into considerations gravitational, damping and stiffness forces Resonant frequency is the frequency at which the minimum amount of force is required to maintain its oscillation

Determining driving force for a FDHO

Theory “If humans are sensitive to the minimal force required to drive their limbs, and if those limbs behavior as FDHO, the actual frequency will be equal to the predicted frequency” Frequency : Period: Kugler and Tuvey (1987) completed experiments with quadrupeal gait and suggest multiplying g by 2

Purpose of study Determine if the resonant frequency of the force driven oscillator predicts the freely chosen frequency adopted in walking

Application to antropometric data Thigh-shank-foot can be assumed to be a single rigid body attached at the hip joint by a frictionless pin joint Added mass is assumed to be a point mass located at malleolus (ankle) L (distance from axis of rotation to CM of pendulum system) Draw picture

Method 24 adults Measured extremities Each person completed 5 walks in each of 4 mass conditions (0 kg- 6.82 kg) Counted cycles and timed them to generate period Period: time for a complete cycle

Statistical Procedures Actual period Predicted period 1 Predicted period 2

Results Prediction 2 (with n factor of 2) was closer to the actual period Stride period increased as a function of the added mass

Predictions and actual period

Force (Energy expenditure) with different lengths and masses Stride period increases as a function of the added mass. Essentially making the L longer Similar to arm demonstration in class

Conclusions The resonant frequency of a harmonic oscillator can accurately predict the frequency chosen by subjects when a multiple of 2 is applied to the gravitational constant The multiple can help account for other forces at play Humans engage in self-optimization, specifically in walking gait Motor control parameters emerge from physical attributes of the system Stiffness of muscles Do not want leg to swing without some control Concept of a posterioir rather a priori