MIY STUDIO · COBOT PROJECT 03
Size the motor
for the full cycle.
#03 — Continuous & Peak Torque
and RMS Motor Sizing
Peak torque checks the hardest instant. RMS torque describes the copper-loss burden of repeating the motion. This study connects both checks to motor-side requirements through the reducer.
DESIGN → CALCULATE → BUILD → TEST → REPEAT
WORKED SELECTION EXAMPLE / MOTOR SHAFT
0.403 N·m
RMS torque over a 10-second cycle.
Illustrative sizing values. Actual robot duty cycle and component ratings remain to be established.
01 / CONTINUOUS & PEAK TORQUE
Two ratings. Two different checks.
An actuator has to deliver the required torque at the required speed, then repeat that operation within its thermal limits. A large peak rating alone does not establish a continuous operating capability.
CONTINUOUS TORQUE
The repeated-load check
Torque that the motor can sustain under its specified speed, ambient temperature, mounting and cooling conditions.
TRMS, required ≤ Tcontinuous, available
A first thermal screening check, with the continuous rating evaluated for the actual operating conditions.
PEAK TORQUE
The hardest-instant check
Higher torque available for a limited duration. The motor and drive must both support the required torque, speed and overload time.
|T|max, required ≤ Tpeak, available
Check the full torque–speed trajectory and the manufacturer’s overload limits.
02 / COPPER LOSS AT STANDSTILL
A stationary arm can still heat the motor.
When the motor holds an arm against gravity, it produces torque even though the joint is not moving. Shaft power is zero at zero speed, but winding current can remain nonzero.
Pmech = Tω = 0
PCu = I2R
The hold segment belongs in the load cycle. Its torque must be determined from the posture, load and holding arrangement.
SIMPLE EQUIVALENT-CIRCUIT EXAMPLE
I = 3 A · R = 0.8 Ω
7.2 W
PCu = 32 × 0.8 = 7.2 W of winding heat, while the shaft is stationary.
The equivalent resistance and current must use a consistent definition. For a real three-phase BLDC motor, calculate copper loss from the phase currents and phase resistance.
Zero shaft speed does not imply zero electrical loss.
03 / ROOT MEAN SQUARE TORQUE
Replace a changing load with its thermal equivalent.
RMS torque is the constant torque that produces the same average copper loss as a changing torque profile, assuming a linear torque–current relationship and constant resistance.
TRMS = √[(T12t1 + T22t2 + …) / (t1 + t2 + …)]
Because torque is approximately proportional to current and copper loss scales with I2, higher-torque segments receive a squared weighting. Signed torques are squared before averaging; acceleration and braking do not cancel thermally.
CONCEPT EXAMPLE / MOTOR SHAFT
ARITHMETIC AVERAGE
0.280 N·m
(0.60 × 2 + 0.20 × 8) / 10
Describes mean torque, but underestimates the copper-loss equivalent for this cycle.
RMS / COPPER-LOSS EQUIVALENT
0.322 N·m
√[(0.602 × 2 + 0.202 × 8) / 10]
Use this value for the initial comparison with the available continuous torque.
Follow the RMS calculation
Square each torque and multiply by its duration:
0.602 × 2 + 0.202 × 8 = 1.04 (N·m)2·s
Divide by the total cycle time, then take the square root:
√(1.04 / 10) = 0.32249 N·m
RMS torque is a sizing metric. The real motor still follows the original changing torque profile.
04 / JOINT-SIDE TO MOTOR-SIDE
Compare torques at the same shaft.
The joint requirement is measured after the reducer. A bare motor rating is measured at the motor shaft. For the following illustrative motoring segments, use a 50:1 reduction ratio and constant running efficiency of 80%.
Tmotor = Tjoint / (Nη)
Nη = 50 × 0.80 = 40
The two segments below are assumed to be rotating in motoring operation. This fixed running-efficiency assumption is not a model for stationary holding or backdriving.
| Rotating segment | Joint torque | Duration | Motor torque |
|---|---|---|---|
| Higher load | 30 N·m | 2 s | 0.75 N·m |
| Lower load | 10 N·m | 8 s | 0.25 N·m |
MOTOR PEAK REQUIREMENT
0.75 N·m
The largest absolute motor-shaft torque in this simplified cycle.
MOTOR RMS REQUIREMENT
0.403 N·m
√[(0.752 × 2 + 0.252 × 8) / 10]
= 0.40311 N·m
Ratio also sets motor speed: at 60 rpm joint speed, a 50:1 reducer requires 3,000 rpm motor speed. That speed is a separate illustrative checkpoint, not a defined trajectory for this 10-second cycle.
05 / THE SELECTION DECISION
Candidate B passes both torque checks.
These are fictional candidates for the exercise. Required speed and permitted peak duration are assumed to be satisfied by all three; the table isolates the continuous and peak torque checks.
| Candidate | Continuous rating | Peak rating | Torque check | Decision |
|---|---|---|---|---|
| A | 0.30 N·m | 0.90 N·m | Continuous rating below 0.403 N·m | Reject |
| B | 0.50 N·m | 0.90 N·m | Both requirements met | SELECTED |
| C | 0.50 N·m | 0.65 N·m | Peak rating below 0.75 N·m | Reject |
CONTINUOUS TORQUE HEADROOM
24.0 %
(0.50 / 0.40311 − 1) × 100
PEAK TORQUE HEADROOM
20.0 %
(0.90 / 0.75 − 1) × 100
Select for the repeating cycle as well as the maximum load.
These percentages describe torque-rating headroom over the exercise requirements. They are not a predicted temperature margin or a completed hardware qualification.
06 / TURN THE EXERCISE INTO A ROBOT DESIGN
The real duty cycle is the next input.
The method is established. The actual robot still needs a torque-versus-time profile, operating conditions and verified component limits before a motor can be selected.
- Define motion and hold segments.Include acceleration, travel, deceleration, waiting posture and payload. Use zero hold torque only when the holding arrangement actually unloads the motor.
- Evaluate the full torque–speed trajectory.Check available motor and drive torque at each operating speed and supply voltage. Include motor inertia, friction and coupled-joint effects as the model develops.
- Check thermal behavior beyond the average.RMS is an initial copper-loss screening method. Compare cycle duration with thermal time constants and verify winding temperature, ambient temperature, mounting, iron losses and overload protection.
- Check the whole actuator.Confirm drive continuous and peak current, reducer load limits and applicable holding conditions. Torque constant, phase-current convention and winding resistance must be consistent.
Technical references
- Kollmorgen — How to Calculate RMS Torque
- maxon — On the heating of motors
- Kollmorgen — Application Sizing Guide (PDF)
Calculations shown here are hand-calculated educational examples using the stated assumptions. Manufacturer references support the sizing method; they do not certify the fictional candidates or this robot design.