Moment of inertia

Summary¶

Moment of inertia (MoI) is the rotational equivalent of mass — it describes how strongly an object resists changes to its rotation. For a drone frame, MoI determines how fast the aircraft responds to attitude commands: a lower MoI means faster response, a higher MoI means slower and more predictable response. Frame geometry and mass distribution are the primary MoI levers. A compact frame with mass concentrated near the centre has lower MoI and responds faster. A wide frame or one with heavy batteries or payloads far from the centre has higher MoI and responds more slowly. This relationship directly affects PID tuning — particularly the D-term — and must be re-evaluated whenever the build geometry or payload changes.

Concept¶

What moment of inertia is¶

Linear inertia (mass) describes resistance to changes in linear velocity. Push a heavy object and it accelerates slowly. Push a light object and it accelerates quickly. Moment of inertia describes the same resistance for rotation: how strongly a body resists changes in angular velocity when a torque is applied.

The key difference from mass: MoI depends not just on how much mass is present, but on where that mass is relative to the rotation axis. Mass far from the rotation axis contributes more to MoI than the same mass near the axis. Specifically, MoI = Σ(m × r²) — every mass element contributes its mass multiplied by the square of its distance from the axis.

This r² dependence is what makes geometry so important. A drone where the battery is at the centre of the frame has a lower MoI than an identical drone where the battery is mounted on extended rails at the frame perimeter — even if both weigh the same. Moving the battery 20mm outward does not increase MoI by 20mm — it increases it by (20mm)² relative to the original distance.

MoI in practice: response time and agility¶

The flight controller's PID loop applies torque to the drone through differential motor speed. The same PID gains apply the same torque regardless of MoI. With lower MoI, that torque produces faster angular acceleration and therefore a faster attitude response. With higher MoI, the same torque produces slower angular acceleration.

For a performance FPV platform, low MoI is desirable: snappy response to stick inputs feels immediate and predictable. For a survey platform carrying a sensor mast or payload, higher MoI is acceptable and may even be beneficial — the drone resists wind gusts better when its inertia is larger, reducing attitude excursions that degrade sensor data quality.

The PID D-term dependency¶

The PID D-term responds to the rate of change of attitude error — it is a braking term that prevents overshoot. The correct D-term value depends on how fast the drone naturally responds to inputs: a faster (lower MoI) drone requires less D-term to prevent overshoot; a slower (higher MoI) drone requires more D.

This is why D-term must be re-tuned whenever MoI changes significantly: - Adding a sensor payload (increased MoI → increase D) - Mounting battery lower and flatter (decreased MoI → reduce D) - Switching from 4-inch to 6-inch props with heavier motors (increased MoI)

The libdrone V2.4.6 geometry lowered the centre of gravity 8–12mm compared to its predecessor by changing the battery mount from a tall stack to a flat tray. This reduced the pendulum arm length (see → pendulum-stability) and also reduced the vertical MoI — the D-term recommendation at maiden was reduced 10–15% from the previous baseline specifically because of this.

MoI and the pendulum effect¶

The pendulum arm (distance from the centre of mass to the propeller plane) and MoI are related but distinct. The pendulum arm determines the natural oscillation frequency of the frame under gravity. MoI determines how fast the frame can be accelerated rotationally by the motors. A short pendulum arm increases natural frequency; a low MoI increases motor-driven responsiveness. Both change together when the battery is moved lower, which is why both the D-term and the natural frequency shifted together in V2.4.6.

Reference¶

MoI sensitivity by component placement¶

Rule of thumb: a 10% MoI increase → approximately 5–8% D-term increase to maintain the same overshoot behaviour. Always verify with blackbox after significant payload changes.

Procedure¶

Identify MoI change after a build modification¶

1. After any modification that changes mass distribution — new battery, new payload, different motor mount height — expect the D-term to need adjustment.
2. Fly a hover, apply a sharp roll input, and observe the step response in the blackbox trace (filtered gyro vs motor output).
3. If the drone overshoots and rings → D-term too low (MoI increased, braking insufficient).
4. If the drone responds sluggishly and never quite reaches the commanded attitude → D-term too high (MoI decreased, braking excessive).
5. Adjust D-term in Betaflight rate profile in 5% increments until step response shows crisp arrival with minimal overshoot.

See → pid-derivative-term for D-term tuning methodology and → blackbox-analysis for how to read the step response.

Rationale¶

Moment of inertia is treated as an implicit concept in most Betaflight tuning guides — builders learn to adjust the D-term when the drone changes, but not why. Making MoI explicit as an atom closes this gap: a student who understands MoI can predict in advance how a design change will affect handling, rather than discovering it empirically after the first flight. The V2.4.6 CG lowering is the concrete libdrone example that makes the concept tangible.

Connections¶

requires: - six-degrees-of-freedom - angular-momentum-multirotors related: - pendulum-stability - pid-derivative-term - pid-tuning-rate-profile - vibration-isolation-theory - floating-motor-mounts leads_to: - pendulum-stability - pid-derivative-term - blackbox-analysis