Rotary vs Linear Humanoid Actuators: Reducers, Roller Screws and How to Calculate Output

The bundle of parts that moves one humanoid joint is called an actuator. There are two broad families: rotary actuators that turn a joint directly, and linear actuators that push and pull a link. Because they are built differently, they differ in how you compute their output, in backdrivability and in cost. This article confirms their structure from public sources, gives the formulas to calculate output yourself, and suggests where each belongs.

How rotary and linear actuators differ

A rotary actuator places a reducer after the motor and turns the joint axis directly. The reducer type determines size, precision and shock tolerance; the three main types are compared in Harmonic, cycloidal and planetary reducers.

A linear actuator converts motor rotation into straight-line motion with a screw mechanism. Like muscles in human limbs, it lengthens and shortens between two links to move a joint. Roller screws are the usual choice when large forces are needed.

Real products mix the two. According to IEEE Spectrum’s coverage of Tesla AI Day 2022, Tesla said Optimus would use six different actuators — three rotary and three linear units. The same report says each hand has six cable-driven actuators moving the fingers and thumb, with 11 degrees of freedom per hand, designed to carry a 9 kg bag.

Roller screws: the heart of a linear actuator

In a roller screw, rollers rather than balls carry the load between screw and nut. The rollers orbit the screw like planet gears, touching both screw and nut; turning the nut moves the screw axially.

According to the same source, roller screws fit more bearing points into the same volume than ball screws, which improves load rating, stiffness, lifetime and acceleration, with efficiency of about 75–90% at low to moderate speeds. The drawback is cost: their complexity can make them up to roughly an order of magnitude more expensive than ball screws. That is why the screw is such a large share of a linear actuator’s cost.

Calculating output yourself

Joint torque from a rotary actuator is motor torque multiplied by the reduction ratio and efficiency:

Thrust from a linear actuator depends on the lead $L$, the distance the screw advances per revolution:

import math

def rotary_output_torque(motor_torque_nm, ratio, efficiency):
    return motor_torque_nm * ratio * efficiency

def linear_output_force(motor_torque_nm, lead_m, efficiency):
    return 2 * math.pi * efficiency * motor_torque_nm / lead_m

print(rotary_output_torque(1.0, 100, 0.7))    # 70.0 N·m
print(linear_output_force(1.0, 0.002, 0.8))   # about 2513 N

The example values are assumptions chosen to show the method; real efficiencies come from the reducer or screw datasheet. To turn linear force into joint torque you must also multiply by the moment arm — the distance from the joint axis to the line of force — which changes with joint angle and has to be checked across the whole range of motion.

Backdrivability and force sensing: the case for low ratios

A high ratio lets a small motor produce large torque, but forces pushed in from the output side barely reach the motor. That property, backdrivability, matters for proprioceptive actuation, where external forces are estimated from motor current alone. The Walker3 humanoid study reports that joint friction in the reducers degrades torque tracking in proprioceptive actuation; by modeling that friction, the robot kept its balance under external impulses and foot disturbances.

Work on low-cost, high-performance actuators points the same way. Katz’s 2018 MIT thesis developed a low-cost modular actuator for dynamic robots, built a 12-degree-of-freedom quadruped with it, and reports a full 360° backflip from standing on flat ground. Pairing high-torque motors with low ratios to survive impacts and sense force is a common starting point for leg joints.

Continuous vs peak torque: heat sets the limit

Actuator specifications usually list continuous and peak torque separately. Motor heating grows with current, so the torque a motor can hold indefinitely (continuous) is lower than what it can deliver for a few seconds (peak). Standing in one pose for a long time or holding an object should be checked against continuous torque; jumping, landing and fast acceleration against peak torque.

• Define the duty cycle first. You cannot estimate heating until you know how long and how often peaks occur.
• Check the heat path. A motor buried inside a joint has little room to shed heat, so its practical continuous torque can fall below the datasheet value.
• Look at the power supply too. If several joints hit peak torque at the same moment, the battery and power electronics may reach their current limit before any single actuator does.

These limits often decide the actuator choice more than the mechanism does. A linear actuator with a high-lead screw may meet the force target on paper but run hot in a knee that carries body weight all day, while a rotary design with better cooling may meet the same continuous requirement with a smaller motor.

The two families side by side

Which family to evaluate first, by joint

1. Joints that need wide rotation (shoulder, waist, wrist rotation): start with rotary.
2. Bending joints that need high force and stiffness (knee, elbow, ankle): compute linear thrust together with the moment arm at each joint angle.
3. Joints that must sense and absorb external forces: evaluate low-ratio rotary designs.

A practical tip for early design reviews: put the torque–speed requirement of each joint on one chart with the candidate actuators’ continuous and peak curves. Gaps show up immediately, and so do joints where a cheaper actuator would do. Once the architecture is set, it is worth checking what shipping robots chose in Humanoid platforms compared.