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

2. 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

3. 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

4. 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:

5. 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

6. Determining driving force for a FDHO

7. 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

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

9. 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

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

11. Statistical Procedures
Actual period
Predicted period 1
Predicted period 2

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

13. Predictions and actual period

14. 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

15. 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