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
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