The Harmonic Drive

100 : 1 in One Stage

The motor turns at 2,000 rpm; the joint’s output flange turns at 20 rpm, with about 100 times the torque. All of it happens in a gear set about 85 mm across.

For a gear of this size the catalogue gives a positioning accuracy of 1 arc-minute, about 0.017°. At the end of a 1 m arm that is 0.3 mm.

It has next to no backlash: when the input stops, the output stops, without rattling in the gaps between teeth.

200 teeth / 202 teeth → 100 : 1 · accuracy 1 arc-min (catalogue, size 20 and up)

Fast Motors, Slow Joints

An electric motor works best at thousands of rpm, with little torque. A joint is the opposite: ten or twenty rpm, lifting a whole arm.

An ordinary pair of gears reduces speed only a few times per stage; 100 : 1 takes several stages, heavy and long. Each pair of teeth also leaves a little play, and over several stages the output rattles.

A robot joint needs a large reduction in one stage, light, short and without play.

Three Parts

The wave generator: an elliptical cam inside a thin, flexible ball bearing. It sits on the motor shaft and is the input.

The flexspline: a thin-walled alloy-steel cup with 200 teeth round the outside of its open end. Its wall is thin enough to flex. Its base carries the output flange: it is the output.

The circular spline: a rigid steel ring with 202 teeth on the inside, two more than the flexspline. It is fixed to the housing.

Wave generator (input) · flexspline (output) · circular spline (fixed)

Flexed into an Ellipse

The wave generator pushed into the flexspline’s open end bends it into an ellipse. At both ends of the major axis its teeth sit deep in the circular spline’s tooth spaces; at both ends of the minor axis the two rings of teeth are clear of each other.

The bend is small: at the major axis the wall moves out by one module, here about 0.3 mm, one hundredth of the radius. The model bends by the true amount; the fine line drawn on it shows the bend 20 times larger.

As the wave generator turns, the ellipse turns with it, and the mesh runs round the ring. The flexspline itself hardly moves.

Radial bend at the major axis = 1 module ≈ 0.32 mm · line drawn 20× larger

One Turn, Two Teeth Back

The circular spline has two more teeth than the flexspline. In one turn of the ellipse every flexspline tooth meshes twice, and because the ring has two teeth more, the flexspline ends up two teeth behind.

So for each turn of the wave generator the flexspline turns 2/200 of a turn the other way. Ratio = −200 / (202 − 200) = −100; the minus sign means output and input turn in opposite directions.

The red tooth is marked. Watch it against the line on the ring beside it: each turn of the ellipse, it moves back two teeth.

i = −N_flex / (N_circ − N_flex) = −200 / 2 = −100

Many Teeth at Once

An ordinary pair of gears has one or two pairs of teeth in contact at any moment. In a strain-wave gear, many teeth sit in their spaces at once, on an arc either side of the major axis. The maker says up to about 30 % of the teeth mesh at the same time: about 60 of 200.

With the load shared by so many teeth, each carries little, and a gear of a given size can pass a larger torque.

In the mesh the teeth enter their spaces radially, pressed by the ellipse against the ring, leaving no room to rattle. The catalogue lists the backlash as zero, and a hysteresis of about 1 arc-minute that comes from the flexspline’s elasticity.

In mesh at once: up to about 30 % (maker) · hysteresis about 1 arc-min (catalogue)

The Joint Module, Cut Open

A collaborative robot’s joint, from the back: rear cover, output encoder, brake, motor encoder, frameless torque motor, strain-wave gear, cross-roller bearing, output flange.

A hollow tube runs through the middle from end to end; the cables pass through it to the next joint.

The housing is black anodised aluminium. The cut faces are painted red, as on museum cutaways.

Frameless Torque Motor

The motor has no case or bearings of its own, only two pieces: a stator of stacked steel sheets with copper windings in its slots, pressed into the joint’s housing, and a rotor, a ring of magnets on the hollow motor shaft.

So the motor can be short and wide. A larger diameter gives a longer lever arm and more torque from the same current, and leaves room in the middle for the cables.

The front of the motor shaft is the wave generator’s elliptical cam.

Two Encoders and a Brake

The motor encoder reads the rotor’s angle, for switching the current and controlling speed. The output encoder sits on the hollow tube that turns with the output and reads the joint’s angle directly, without the gear’s flex or errors.

With the power off, springs press the brake’s friction disc and hold the motor shaft; with power on, the coil pulls the armature away. If the power fails, the arm does not drop.

Cross-Roller Bearing

The output flange runs on a cross-roller bearing: one ring of short cylindrical rollers, each turned 90° from the next, so one carries load in one direction and the next in the other.

One bearing takes radial and axial loads and tilting moments together. The weight and swing of the arm stay off the thin flexspline, which only carries torque.

Where It Works

In industrial and collaborative robots’ joints. A common six-axis collaborative arm uses a single-stage 100 : 1 strain-wave gear in every joint.

In space. Each of the Apollo Lunar Roving Vehicle’s four wheels was driven by a motor through an 80 : 1 harmonic drive. The Mars rovers Spirit and Opportunity had them in their arms, wheels and steering; JPL says the arms and legs of robots such as Perseverance rely on them.

In telescope mounts and film camera heads: small, a large reduction in one stage, no play, for turning that must be slow, precise and steady.

Lunar rover: an 80 : 1 harmonic drive at each wheel (NASA)

Real Numbers

A size-25 strain-wave component set (ratio 100): about 85 mm across, about 0.42 kg. Rated torque 67–87 N·m, repeated peak 157–204 N·m, momentary peak 284–369 N·m (two series).

By repeated peak torque, the bare gear set gives about 300–380 N·m per kilogram.

As a complete hollow-shaft actuator (motor, encoders, bearing and housing included), sizes 20 and 25 give about 40–50 N·m per kilogram: 107 N·m for 2.6 kg, 204 N·m for 3.95 kg.

A size 25 takes up to 5,600 rpm in (grease) or 7,500 rpm (oil).

Harmonic Drive CSF / CSG catalogue, SHA actuator catalogue

Taken Apart

The joint module’s parts slide apart along its axis. Rest the pointer on one and it is lifted out on its own and named.