Sizing Joint Actuators for a 7-DOF Robot Arm: Worked Example

The mistake that costs the most

A first robot arm build usually starts with one number: the peak torque of the hardest joint. That number goes into a spreadsheet, the team buys seven modules of roughly the same class, and one of two things happens.

Either the shoulder stalls during the first full-reach horizontal move — because nobody computed gravity torque at 0.63 m with payload — or every wrist joint carries half a kilogram of harmonic gearbox it will never use. That second failure is worse than it looks, because the wasted mass sits distal to every joint upstream of it. Mass at the wrist raises the torque the elbow needs. Mass at the elbow raises the torque the shoulder needs. You pay for the mistake three times.

The fix is not sophisticated. It is four steps of arithmetic, run per joint, distal end first. Below is the version we use, applied to a real configuration so you can see where every number lands.

Step 1 — Fix the envelope before you look at hardware

You cannot size anything until five things are frozen, because each one changes the torque answer by more than the difference between two neighbouring joint actuator frames:

• Link lengths and reach — torque scales linearly with reach, and the lever arm is the whole game.

• Payload in kilograms, at what distance — "5 kg" without a distance is meaningless.

• Mass of each link assembly — structure, neighbours, cabling, and the distal joints themselves.

• Target move time — this sets angular acceleration, which sets the inertia term.

• Mounting envelope — the maximum frame diameter each joint can physically accept. This constraint is discovered last and invalidates everything most often.

Step 2 — The gravity term (static torque)

Worst case is always the arm fully extended horizontally: every lever arm is maximal and nothing is partially supported. Sum each distal mass multiplied by its horizontal distance from the joint axis.

Joints whose axis is vertical — a humeral yaw, a wrist roll near the tool — have a gravity term near zero. That does not make them free: they still need to carry structural bending moment, pass cabling, and close the kinematic chain. Size them on stiffness and interface, not on torque.

Step 3 — The inertia term (dynamic torque)

Rotational inertia about each joint axis is the sum of each distal mass times the square of its distance. Note the square — this is why reach hurts twice, once as mass and once as distance.

Realistic angular accelerations by joint role, from constraints we have seen hold up across build reviews:

Your required working torque for that joint is then simply:

Step 4 — Duty class and safety factor

This is where most mis-specification actually happens, and it happens because of one ambiguity: the number printed as "rated torque" on a harmonic module datasheet is usually a thermal steady-state figure, not a mechanical ceiling. It tells you what the unit can sustain indefinitely without cooking the windings. It does not tell you what it survives.

One vocabulary fix first, because it ends most arguments before they start. Duty class is the IEC code (S1 continuous, S2 short-time, S3 intermittent) that a rating was measured under. Duty cycle is the percentage of on-time within your own motion cycle — the number you actually have. Suppliers quote the class; you think in cycle percentage. Convert one to the other explicitly, or you will compare two joint actuators that were never measured the same way.

With that settled, the multiplier you apply to the continuous rating depends on how forgiving the application is:

Two derates people forget. First, ambient temperature: a rating measured at 25 °C is not the rating at 45 °C inside a sealed arm cavity. Second, bus voltage sag: low-voltage modules lose available torque when the supply droops under simultaneous multi-joint moves, which is exactly when you need the torque.

The frame you're choosing from

Before the worked example, the six reference frames. These are not a recommendation — they are the space you are searching.

Representative values for configured builds. Mass changes with brake and torque-sensor options — always request the mass of the exact unit number you are buying, not the frame family. Torque density here is continuous torque divided by reference mass.

Read the last column carefully, because it changes how you think about upsizing. Torque density improves with frame size across this range. Adding six grams of nothing matters at the wrist; moving up one frame at the shoulder buys you roughly 95 N·m for three-quarters of a kilogram. Uniform upsizing is not automatically the safe choice, and uniform downsizing is not automatically the efficient one.

Worked example: a 7-DOF research arm

Now the whole thing end to end. The requirement: 0.63 m reach, 4 kg payload at the tool flange, mounted to a fixed base, target 90° shoulder moves in about 1.2 s.

The frozen envelope

Joint 1 — shoulder pitch (the one that decides everything)

Gravity term, arm horizontal:

Inertia term at α = 4 rad/s²:

Working torque = 42.3 + 9.3 = 51.6 N·m. At a research-grade 2.0× margin, the target continuous rating is about 103 N·m.

R68 gives 33 N·m — out by a factor of three. R83 gives 53 N·m — only 1.03×, no headroom at all. R102 delivers 124 N·m continuous: a 2.4× margin. It is also 0.74 kg heavier than the R83 — which is exactly why Step 1 told you to freeze the mass budget before this moment rather than after.

Joint 2 — shoulder roll

A horizontal axis at the same station, carrying the same distal mass with the same moment arms. Same 51.6 N·m working torque, same answer: R102. Two identical shoulder units also gives you a spare-parts and cable-standardization win, which is worth real money later.

Joint 4 — elbow pitch

Now only what sits distal to the elbow counts: the forearm assembly at 0.12 m past the joint, and the gripper plus payload at 0.33 m.

At 1.5–2.0× you need 32–42 N·m continuous. R68 at 33 N·m gives 1.56× — acceptable for a research arm, and it is one-third the mass of an R83 at the same station.

Joint 6 — wrist pitch

R48's 6.5 N·m continuous sits at 1.05× — numerically enough, with no margin for a revised gripper. R58 at 20 N·m gives 3.2× for 0.28 kg more. That is the cheapest insurance in the whole arm.

Joints 3, 5, 7 — the low-gravity rotations

J3 (upper arm yaw) spins about its own axis; gravity contributes almost nothing and inertia is small. It does, however, structurally carry everything downstream and route the cable bundle — so it is sized on interface and stiffness. Practically that means sharing a frame with J4, giving you one spare part and one mounting pattern across the upper arm: R68.

J5 (wrist roll) and J7 (wrist yaw) carry the payload about the load axis: a fraction of a newton-metre of inertia torque. R48 is sized correctly here, and importantly, anything larger would be carried inboard by every proximal joint for no benefit.

The resulting build

Seven modules totalling about 6.75 kg. Note that three of the seven joints were not chosen on torque at all — they were chosen on structure, sparing, and the mass they impose on everything upstream.

The recursion nobody budgets for

Seven modules at about 6.75 kg against a 4 kg payload. That ratio — roughly 0.6 payload-to-arm-mass — is normal for a first build, and it is why the calculation has to be iterated rather than done once.

Work the feedback loop concretely. Suppose you had played it safe and specified R102 at the elbow instead of R68. That is +1.17 kg sitting 0.30 m from the shoulder:

A 6% increase at the shoulder, bought by one upstream decision. Go the other way and put three R58s where three R48s would serve, and you have added 0.84 kg at roughly 0.5 m — another 4 N·m at the shoulder before you have moved a single payload.

Every gram you add distal to a joint is paid for again at every joint proximal to it. Size distal joints as small as their own requirements honestly allow, and spend the mass budget at the base, where it costs you nothing but money.

The same recursion is why humanoid leg joints are a sizing problem of their own rather than a bigger arm. A hip–knee–ankle chain is three long levers in series carrying full body mass, so every gram at the ankle is paid for three times over before it reaches the hip. The frames that cover those joints are the same ones listed above; the walk-through is different, and we treat it separately.

Then check speed — and check it here, not earlier

Speed is not a Step 1 input. It is a filter you apply to the shortlist that came out of torque, because ratio trades speed against torque and no amount of cleverness escapes that trade.

Convert everything to degrees per second before comparing anything:

Now audit the build above. The shoulder at 12 rpm gives about 72 °/s — a 90° move in roughly 1.25 s, which lands on the original 1.2 s target. Everything distal is a 101:1 unit at 30 rpm, so 180 °/s at the elbow and wrist, which is more than most manipulation tasks use.

What the datasheet will not tell you

Specifications are usually correct and rarely sufficient. These are the nine questions that change part selection most often, and none of them appear on a one-page datasheet:

Three practical notes that save rebuilds

Brakes are holding brakes. On compact harmonic modules the brake almost always sits on the motor side. It holds position with power removed; it is not a working brake and it is not a safety brake. If a joint collapses under gravity when unpowered, it needs one — and if anything operates near people, add an external safety measure rather than trusting a holding brake with a 160:1 gearbox behind it.

Torque sensing is frame-dependent. It is commonly available on mid frames and commonly absent on both the smallest and the largest units. If force control matters to your control stack, verify availability on the specific frame size before you design the joint around it — retrofitting a strain-gauge path later means a new mechanical design, not a firmware update.

Daisy-chain the bus if you can. Seven home-run cables through four rotating joints is a cable-management problem and a failure-mode problem. A single EtherCAT chain with two power rails through hollow shafts removes both, and it is nearly impossible to retrofit.

Selection checklist

1. Reach, payload mass, and payload distance are written down and agreed.

2. Every link assembly has a mass and a centre-of-mass location — including the joints mounted distally.

3. Gravity torque computed per joint at full horizontal extension.

4. Inertia and angular acceleration computed per joint; α chosen per joint role, not globally.

5. Safety factor applied to the continuous rating, with the duty class stated.

6. Ambient temperature and bus-voltage sag accounted for.

7. Degrees per second checked for each joint after the torque shortlist exists.

8. Mounting envelope (max frame Ø and stack length) confirmed against the physical design.

9. Mass loop iterated once: total module mass fed back into the gravity term.

10. Every datasheet question above answered in writing, for the exact unit number being quoted.

11. Brake requirement decided per joint, with the holding-versus-working distinction understood.

12. STEP model in hand before the mounting plates go to the machine shop.

FAQ

How much torque does a robot arm joint need?

It depends on the joint's position in the chain, not on the arm as a whole. Compute gravity torque first (mass × 9.81 × horizontal distance from the joint axis), add the inertia term from your target move time, then multiply by 1.5–2.0× for a research arm or 2.0–2.5× for a production machine. In the 7-DOF example above, shoulder pitch lands at 51.6 N·m of working torque and wrist pitch at 6.2 N·m — an eight-to-one spread across the same arm.

What safety factor should I use on joint torque?

Size the continuous rating at 1.5–2.0× the computed working torque for research and prototype arms, 2.0–2.5× for production machines. Never treat datasheet peak torque as a working number — it is a short-duration mechanical limit, not a continuous capability.

Should every joint on a robot arm use the same joint actuator?

Usually not. In the example above the shoulder carries over eight times the working torque of the wrist. Reusing a joint actuator frame size is justified when it buys sparing, cable standardization, or structural continuity — not as a default.

Do I need a brake on every joint?

Only at joints that would move under gravity when unpowered, and only as a holding brake. Most compact modules use a motor-side holding brake; if the application needs safety-rated stopping or repeated dynamic braking, that is a different component and should be specified separately.

How do I compare torque density claims between suppliers?

Continuous torque divided by the mass of the configured unit you would actually buy — including brake, connector, and any torque sensor — at the same ambient and duty class. Peak-based density figures can differ from continuous ones by two to three times.

When should I check output speed?

After the torque shortlist. Ratio determines both, and converting to degrees per second early prevents the most expensive class of surprise: a joint that meets every torque requirement and cannot complete the move in the cycle time.