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GearLab gears & robot gearboxes
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This is a static copy of the chapter for search engines. The interactive version has animated figures, check questions, and buttons that load the example into the calculator.

1. Why a robot joint needs a gearbox

An electric motor is a device that is good at spinning fast and bad at pushing hard. A robot joint wants the opposite: it moves slowly and has to hold a payload against gravity all day. The gearbox is what converts one into the other, and in a robot arm it is usually the single most expensive part of the joint.

MOTOR JOINT 3000 rpm 0.5 N·m i = 100 η ≈ 0.8 30 rpm 40 N·m torque ×100 · speed ÷100 · inertia ÷10000
The gearbox is not an accessory. A motor that is efficient at 3000 rpm produces a torque a robot joint cannot use, and the reflected inertia — which is what the controller has to fight — falls with the square of the ratio.

Three things change at once

Put a ratio i between the motor and the joint and, ignoring losses:

Tout = i · η · Tin     ωout = ωin / i     Jreflected = Jload / i²

The first two are the obvious ones. The third is the one that decides whether the joint can be controlled at all: the inertia the motor has to accelerate falls with the square of the ratio. A 0.5 kg·m² arm behind a 1:100 gearbox feels like 50 g·cm² at the motor shaft — which is the same order as the motor's own rotor, and that balance is exactly what a servo loop wants.

Inertia matching. The classic rule is to aim for a reflected load inertia of roughly one to three times the rotor inertia. Below that the motor is dragging its own rotor around for nothing; far above it, the loop gets slow and the joint feels soft.

And three things get worse

  • Efficiency. Every mesh loses something. A spur pair is 97–99 % efficient, a planetary stage similar, a harmonic drive 70–85 % depending on ratio and speed.
  • Backlash. The gap between the teeth becomes dead travel at the joint, and it is multiplied by nothing — but it is seen at the output, where it matters most.
  • Compliance. A gearbox is a spring. Its torsional stiffness sets the resonance that limits how high the position loop can be tuned — the two-mass problem.
Where the numbers come from. This chapter is definitions and first principles. The gear geometry and the flank rating in the chapters that follow are sourced — each formula carries its reference, and chapter 8 lists them all.

Formulas in this chapter

i, η — what a ratio does
T_out = i η T_in, ω_out = ω_in/i, J_ref = J/i² [—] definition
J_ref — reflected inertia
J_reflected = (J_load + J_gear) / i² [kg·m²] definition