ω

mg

θ

The Stable Inverted Pendulum

Deriving the EOMSetup from FBD:

Defining generalized moment of inertia for rod:

Initial Conditions:

Solve for the EOM:

ω

mg

l

θ

Deriving the Mathieu Equation

Inverted Pendulum EOM:

Dimensionless driving frequency,

time, amplitude:

Mathieu Equation:

Hanging

Inverted

-ω +A

Stability

Region

Constant ωC

on

stan

t A

Deriving the Effective PotentialThe EOM is split into rapid oscillation (ε) and slow oscillation (θ0) components:

Solving for the slow component EOM:

Defining the Effective Potential:

Hanging

WellInverted

Well

~70° ~30° ~5°

Viewing Potential Well Strength