Gearing and transmissions

Electric motors are happiest spinning fast with modest torque, while robot joints and wheels need high torque at modest speed. A transmission bridges that gap. Choosing it is one of the most consequential decisions in robot design, because the gear ratio and gearbox type shape the robot's strength, speed, precision, efficiency, backdrivability and how it behaves when it hits something. This lesson develops the basic relationships, introduces reflected inertia and the optimal gear ratio, and compares the gearbox types found in robots: spur and planetary gears, harmonic drives, cycloidal drives, belts, ball screws and capstan drives.

The ideal gear ratio

A gear train with ratio N reduces speed by N and, ideally, multiplies torque by N. Real gearboxes lose some power to friction, described by an efficiency \( \eta \) between 0 and 1.

Gear reduction does three useful things at once. It lets a small, light motor produce a large joint torque. It keeps the motor in its efficient, cool, high-speed region, because the motor current needed for a given joint torque falls by N. And it multiplies the resolution of a motor-side encoder by N (lesson 2). The costs are friction, backlash, added mass and, as we shall see, a large increase in the inertia the joint presents to the outside world.

Reflected inertia

A gearbox changes not only torque and speed but also how heavy the motor's rotor feels from the output side. The motor's rotor inertia \( J_m \), seen from the joint, is multiplied by the square of the ratio. Equivalently, the load inertia \( J_L \) seen from the motor is divided by \( N^2 \).

This squared factor has big consequences. With a 100:1 gearbox, a tiny rotor of \( 5 \times 10^{-5} \) kg·m² appears at the joint as 0.5 kg·m², often larger than the inertia of the arm link itself. A highly geared joint therefore feels heavy when pushed and delivers a hard impact in a collision, since the spinning rotor's kinetic energy must be absorbed. It is also hard to backdrive, that is, to move from the output side, because the person pushing must accelerate the reflected rotor inertia and overcome the gearbox friction multiplied by the ratio. These are exactly the properties that matter when robots work near people, and why collaborative and legged robots use low ratios, joint torque sensing or elastic elements (lesson 10).

The optimal ratio for acceleration

If the goal is to accelerate a pure inertial load as quickly as possible with a given motor torque, there is a best ratio. The load acceleration is \( \alpha_L = N\tau_m/(J_L + N^2 J_m) \). Differentiating with respect to N and setting the result to zero gives the inertia-matching ratio, at which the reflected rotor inertia equals the load inertia.

Gearbox types

Spur and helical gear trains are simple and efficient (about 95 to 98 % per stage) but need several stages for large ratios, and each stage adds backlash, the small free play between meshing teeth. Planetary gearboxes arrange a sun gear, several planet gears and an outer ring gear coaxially; they are compact, efficient (about 90 to 97 % per stage), carry high torque and offer 3:1 to 10:1 per stage. Quality planetary gearboxes have backlash of a few arcminutes, and they are the standard choice for mobile robot wheels, low-ratio legged robot joints and many industrial axes.

Harmonic drives (strain-wave gears) use an elliptical wave generator to flex a thin toothed cup, the flexspline, inside a rigid circular spline with two more teeth. Each revolution of the wave generator moves the flexspline by two teeth, giving ratios of 30:1 to 160:1 in a single compact, light stage with essentially zero backlash. They are the classic choice for robot arm joints, collaborative robots and space manipulators. Their disadvantages are torsional compliance (the flexspline twists under load), modest efficiency that falls at low temperatures and light loads, a periodic error at twice the input frequency, and poor backdrivability.

Cycloidal drives, and the related RV reducers used in large industrial robots, roll a lobed disc around a ring of pins. They give high ratios, very high stiffness and excellent shock-load capacity, at the cost of size and mass, which makes them the usual choice for the base and shoulder joints of heavy industrial arms. Other transmissions complete the toolkit: timing belts move the motor away from the joint and reduce inertia at the distal end; ball screws convert rotation to precise linear motion with high efficiency; lead screws are cheaper and can be self-locking; worm gears give high single-stage ratios and are self-locking but inefficient; and capstan or cable drives, as in some research and haptic arms, give zero backlash and excellent backdrivability.

Backlash, compliance and their effects

Backlash causes a dead band when motion reverses: the motor turns but the output does not move until the teeth re-engage. In a position loop closed on a motor-side encoder, backlash produces positioning error; closed on a load-side encoder, it can produce limit-cycle oscillation as the controller hunts across the gap. Compliance, the elastic twist of gears, shafts and links under load, lowers the natural frequency of the joint and limits how aggressive the control loops can be. Stiffness \( k \) and the load inertia set the natural frequency \( \omega_n = \sqrt{k/J} \), and controllers generally need to keep their bandwidth well below it.

Efficiency also has a direction. Some transmissions, such as most worm gears and fine-pitch lead screws, are self-locking: the load cannot drive the motor backwards. This is useful for holding a load with the power off, but it means the joint cannot be pushed by a person and absorbs collisions badly. Robots that need a holding function without self-locking use spring-applied brakes that engage when power is removed, a feature found in almost every industrial arm joint.

import numpy as np
Jm, JL, tau_m = 4.0e-5, 0.20, 1.2        # rotor and load inertia (kg*m^2), peak motor torque (N*m)
for N in (10, 30, 50, 70.7, 100, 160):
    alpha = N * tau_m / (JL + N**2 * Jm)
    share = N**2 * Jm / (JL + N**2 * Jm)
    print(f"N = {N:6.1f}   load accel = {alpha:6.1f} rad/s^2   rotor share of inertia = {100*share:4.1f} %")
print(f"inertia-matching ratio N* = {np.sqrt(JL/Jm):.1f}")

N =   10.0   load accel =   58.8 rad/s^2   rotor share of inertia =  2.0 %
N =   30.0   load accel =  152.5 rad/s^2   rotor share of inertia = 15.3 %
N =   50.0   load accel =  200.0 rad/s^2   rotor share of inertia = 33.3 %
N =   70.7   load accel =  212.1 rad/s^2   rotor share of inertia = 50.0 %
N =  100.0   load accel =  200.0 rad/s^2   rotor share of inertia = 66.7 %
N =  160.0   load accel =  156.9 rad/s^2   rotor share of inertia = 83.7 %
inertia-matching ratio N* = 70.7