Robotic Joint PID Controller

PID Gains

Proportional K_p 6.0

Integral K_i 0.5

Derivative K_d 1.5

Target & Disturbance

Target angle 60°

Live Readouts

Joint angle θ

0.0°

Error e = θ_t−θ

60.0°

Overshoot

— %

Settling time

— s

Control torque τ

0.0 N·m

Status

Settling

How it works

This is one revolute joint of a robot arm — a rigid link of moment of inertia I driven by a motor, resisted by viscous friction b and, since the arm is horizontal against gravity, a gravity torque. Its rotational dynamics obey Newton's second law for rotation:

I·θ̈ + b·θ̇ + m·g·L·cos(θ) = τ_control

The motor is driven by a real industrial PID (proportional–integral–derivative) controller, the same control law used in essentially every closed-loop robot joint, from a factory arm to a robot vacuum's wheel motor. It computes the error e(t) = θ_target − θ(t) and outputs a torque:

τ_control = Kp·e(t) + Ki·∫e(t)dt + Kd·(de/dt)

K_p — proportional gain. Bigger K_p reacts harder to a given error and speeds up the response, but too high a value causes overshoot and oscillation.
K_i — integral gain. Accumulates past error to eliminate the small steady-state offset gravity leaves behind; too high causes slow, growing oscillation ("integral windup").
K_d — derivative gain. Reacts to the rate of change of the error, damping oscillation and reducing overshoot, at the cost of amplifying sensor noise in a real system.
The integral term is anti-windup clamped and motor torque is saturated at a realistic limit, exactly as a real servo drive current-limits its motor.

Try it: set K_i = K_d = 0 and raise K_p — the arm oscillates forever around the target instead of settling, because nothing damps it or removes the steady offset. Add K_d back and the ringing dies out quickly; add a little K_i and the final resting angle locks exactly onto the target instead of sagging under gravity.

Robot link Target angle

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θ(t) step responsetarget 60°