Reflected inertia
How it works
1) A gearbox of ratio N couples the speeds: ω_motor = N·ω_output, and the torques: τ_output = N·τ_motor·η (η — efficiency). 2) From conservation of rotational kinetic energy ½·J·ω², the equivalent inertia referred across the gearbox changes with the square of the speed ratio, i.e. with N². 3) Load inertia reflected to the motor shaft: J_reflected = J_load / N² (large N → the motor “feels” a light load). 4) Rotor inertia reflected to the output/joint: J_reflected = J_motor · N² (large N → the joint “feels” a heavy, hard-to-backdrive drive). 5) Total joint output inertia ≈ J_motor·N² + J_link; at high N the first term dominates, which ruins backdrivability and increases impact forces. 6) Quasi-direct-drive (QDD) actuators deliberately use low N (on the order of 6:1–10:1) to keep reflected inertia small, achieving backdrivability, proprioceptive torque control and impact resilience.
Problem solved
It lets engineers compute and design the real inertia “felt” on each side of a gearbox instead of assuming the motor drives the load directly. Without this quantity one cannot correctly size the motor and gear ratio, predict torque-control bandwidth, backdrivability, or the peak forces that arise when a robot joint takes an impact.
Implementation
Assuming the motor drives the link directly and omitting the J_motor·N² term leads to poor motor sizing, unexpectedly high output inertia and incorrect dynamics models.
At high ratios the reflected rotor inertia grows quadratically and starts to dominate the link inertia, drastically increasing peak impact forces and degrading backdrivability.
Evolution
Sangbae Kim’s group formalized design principles for low-ratio quasi-direct-drive actuators, showing that minimizing reflected inertia (J_motor·N²) is key to impact mitigation and high-bandwidth physical interaction in dynamic legged robots.
Hyperparameters (configurable axes)
The gearbox reduction ratio. Rotor inertia reflected to the output grows with N², so this is the single most important parameter shaping joint output inertia, backdrivability and impact forces.
The moment of inertia of the motor rotor. Multiplied by N² it gives the reflected-inertia term at the output; low-rotor-inertia motors reduce reflected inertia.