PID — derivative term

Summary¶

The derivative term (D) acts on the rate of change of error — it applies braking when the drone is approaching the setpoint rapidly, preventing overshoot. D is the most noise-sensitive PID term: because differentiation amplifies high-frequency content, any motor vibration reaching the gyroscope is amplified by D and converted into motor commands. This is why vibration isolation and RPM filtering must be in place before D is tuned — without them, D amplifies noise rather than damping the response. D also damps prop wash oscillation, the characteristic bouncing that occurs when a drone flies through disturbed air from its own previous path.

Concept¶

Predictive braking¶

The derivative term calculates:

D_output = K_D × d(error)/dt

The rate of change of error: how fast the error is decreasing (or increasing). If the error is decreasing rapidly — the drone is snapping back toward the setpoint quickly — D applies a large braking force proportionally. If error is changing slowly, D is small.

The effect: the drone decelerates before it reaches the setpoint, arriving smoothly rather than overshooting. D makes corrections that P would otherwise overshoot. Without D, a well-tuned P still oscillates; with D, the P correction is damped at the right moment.

Physical analogy: car suspension dampers. The spring (P) pushes the wheel back up after a bump. The damper (D) slows the spring's return, preventing the car from bouncing. Without the damper, even a well-sprung car would oscillate after every bump.

Why D amplifies noise¶

Differentiation is the mathematical operation that amplifies high-frequency signal content. A slowly varying signal (attitude changing at 10 Hz) has a large derivative term only when it changes quickly. A high-frequency noise signal (motor vibration at 500 Hz) appears to be changing rapidly at all times — its derivative is always large.

When motor vibration reaches the gyroscope, the gyro faithfully reports it. The D term sees a rapidly changing error signal and generates large, rapid corrective motor commands. These commands create additional motor current changes, which generate additional vibration. The loop closes in a way that amplifies the noise rather than the correction.

This is the fundamental reason why mechanical vibration isolation and the RPM filter must precede D tuning. They remove the high-frequency noise before D sees it, allowing D to act only on genuine attitude dynamics. → See floating-motor-mounts, rpm-filter.

Prop wash¶

Prop wash is the disturbed air that forms behind a drone after a fast forward pass or after a quick deceleration. When the drone's propellers fly through this disturbed air (on the return path, or if the drone decelerates into its own wake), the airflow into each propeller disc is asymmetric and turbulent. The motors experience rapidly changing load, causing rapid attitude disturbances that appear as oscillation.

D is the primary tuning term for prop wash handling. Higher D provides more damping of the rapid oscillations. However, the same higher D amplifies motor noise — the tuning is a direct tradeoff between noise sensitivity and prop wash handling.

The RPM filter resolves this tradeoff by removing motor noise before D acts on it, allowing D to be tuned higher for better prop wash without the noise amplification penalty.

D and the V2.4.6 lower CG¶

The lower CG in V2.4.6 raises the pendulum natural frequency — the drone oscillates more quickly after a disturbance. This means the rate of change of error is higher for the same displacement, making D more sensitive. The documented recommendation to reduce D by 10–15% from V2.14 baseline is a direct consequence: the same physical D gain produces more aggressive braking on the higher-frequency dynamics.

Reference¶

D effects on flight behaviour¶

D tuning with libdrone V2.4.6¶

Starting point: 10–15% below the V2.14 baseline value (as documented in the Betaflight configuration). Adjust from Blackbox evidence: - If gyro trace shows ringing after sharp inputs: raise D slightly - If motor temperature is elevated at rest after hover: lower D slightly - If prop wash is visible (Blackbox: gyro oscillation on roll/pitch axis during descents): raise D slightly

Procedure¶

D tuning sequence (Betaflight)¶

1. Ensure RPM filter is active and functional (BiDi DShot confirmed working — all four motors report eRPM in Betaflight motors tab).
2. Tune P first, then add D.
3. Set D to half the expected value. Fly and observe prop wash: make a fast forward pass, then a quick stop. Oscillation in the hover after stopping indicates insufficient D.
4. Raise D in 5–10% steps. After each increase, check motor temperature with a hand (warm is acceptable, hot is not) and review Blackbox noise floor.
5. Stop raising D when motor temperature increases noticeably or the Blackbox noise floor rises significantly.
6. Target: clean gyro trace with no sustained oscillation, motors warm but not hot after 3–5 minutes hover.

Rationale¶

Why D is tuned after vibration isolation is confirmed¶

D's entire effect depends on the quality of the gyro signal it receives. If significant motor vibration noise reaches the gyro, any D value will partially amplify noise rather than damp genuine dynamics. The result is a D value that appears to work at low settings but generates motor heat and noise at higher settings. Confirming that mechanical isolation and RPM filtering are working before tuning D ensures D responds to attitude dynamics, not noise.

Connections¶

requires: - closed-loop-control - pid-proportional-term related: - pid-integral-term - rpm-filter - floating-motor-mounts - vibration-isolation-theory - pendulum-stability leads_to: - feed-forward-control - rpm-filter