DC Gearmotor Joint Actuator: Back-EMF & Torque-Speed

Drive Mode

Manual Voltage

PWM duty cycle 55%

Position Hold

Target joint angle 45°

Controller gain K_p 1.4

Actuator & Load

Gear ratio N 30:1

Payload mass 1.5 kg

Live Readouts

Armature current I

0.00 A

Back-EMF

0.00 V

Joint torque

0.00 N·m

Joint speed ω

0.00 rad/s

Joint angle θ

45.0°

Applied voltage

13.2 V

How it works

A real robot joint is not a kinematic hinge — it is an electric motor fighting gravity through a gearbox. Each animation frame integrates the coupled electrical and mechanical equations of a permanent-magnet DC gearmotor:

V = I·R + Kb·ωm        (armature loop, ωm = motor speed)
τm = Kt·I               (motor torque from current)
ωm = N·ωj,  τj = N·τm   (gear reduction, N:1)
J·dωj/dt = τj − m·g·L·cosθ − b·ωj

V is the applied armature voltage (duty cycle × 24 V supply), R the winding resistance, and Kb the back-EMF constant — the faster the motor spins, the more voltage it generates against itself, which is why current (and torque) drop as speed rises. Kt is the torque constant (equal to Kb in SI units). The gearbox multiplies motor torque by N but divides motor speed by N, so a high ratio buys holding torque at the cost of top speed. J is the combined inertia (arm + payload + reflected motor inertia N²·Jm) and b is joint friction.

PWM duty cycle — sets the average armature voltage directly (open loop); negative duty reverses the motor.
Position Hold — switches to a proportional controller, duty = clamp(Kp·(θ_target − θ), −1, 1), so the joint drives itself toward the target angle and you can watch current spike while accelerating and settle once gravity torque is balanced.
Gear ratio — higher N raises stall torque (τ_max = N·Kt·V/R) but also raises reflected inertia N²·Jm, slowing the response.
Payload mass — raises the gravity torque the motor must overcome at the horizontal, m·g·L, which is exactly where current draw peaks.

This is the same model — armature resistance, back-EMF, gear reduction, gravity load — used to size and tune the actuator at every joint of a real robot arm.

Arm + payload Current-lit motor housing

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