Look for sources of how the reflected inertia for two stage planetary is calculated

[Hyh2001]

Thanks for open sourcing this awesome project! I am quite interested in the equation of how inertia for two stage planetary is calculated. May somebody points me to some resources about this since it's a little unintuitie.
def reflected_inertia_from_two_stage_planetary( rotor_inertia: tuple[float, float, float], gear_ratio: tuple[float, float, float], ) -> float: """Compute reflected inertia of a two-stage planetary gearbox.""" assert gear_ratio[0] == 1 r1 = rotor_inertia[0] * (gear_ratio[1] * gear_ratio[2]) ** 2 r2 = rotor_inertia[1] * gear_ratio[2] ** 2 r3 = rotor_inertia[2] return r1 + r2 + r3

[kevinzakka]

The reflected inertia equation for the two-stage planetary gearbox is based on how inertia "reflects" through gear reductions. When a rotating mass is connected through a gear reduction, the inertia seen at the input shaft is: J_reflected = J_actual × (gear_ratio)^2. Or tldr; inertia scales with the square of the gear ratio.

In a two-stage planetary gearbox, you have three rotating components:

1. Motor rotor (rotor_inertia[0]): Sees the full gear reduction through both stages so multiplied by (gear_ratio[1] × gear_ratio[2])^2
2. First stage output / Second stage input (rotor_inertia[1]): Only sees the second stage reduction, so multiplied by gear_ratio[2]^2
3. Final output (rotor_inertia[2]): Already at the output, no reduction so multiplied by 1

The total reflected inertia at the motor shaft is the sum of all three components.

Regarding gear_ratio[0] == 1: that assertion ensures consistent indexing. The motor shaft has no reduction "before" it, so gear_ratio[0] corresponds to rotor_inertia[0] with a 1:1 ratio.

And finally, here's a numerical example:

rotor_inertia = (0.01, 0.02, 0.03)  # kg·m^2
gear_ratio    = (1, 5, 4)           # Motor=1, Stage1=5:1, Stage2=4:1

# Compute each reflected component:
r1 = 0.01 * (5 * 4)^2 = 0.01 * 400 = 4.00   # Motor rotor through both stages
r2 = 0.02 * (4)^2     = 0.02 * 16  = 0.32   # Stage 1 output through stage 2
r3 = 0.03 * 1        = 0.03                 # Final output (no reduction)

# Total reflected inertia at motor shaft:
total = 4.00 + 0.32 + 0.03 = 4.35 kg·m^2

[Hyh2001]

Thank you for explanation, it looks more reasonable now!