Angular momentum in multirotors

Summary¶

A spinning propeller has angular momentum. On a multirotor with all motors spinning in the same direction, the net angular momentum would cause the airframe to spin in the opposite direction — making it uncontrollable. Counter- rotating motor pairs cancel the net angular momentum at hover. More importantly, this means yaw control on a quadcopter requires no moving parts: it works entirely by adjusting the speed balance between the two pairs of counter-rotating motors, shifting the net angular momentum and forcing the airframe to react.

Concept¶

Angular momentum and why it matters¶

Angular momentum is the rotational equivalent of linear momentum. A spinning object resists changes to its spin axis in proportion to how massive it is and how fast it spins. The angular momentum vector points along the spin axis (by the right-hand rule: curl the fingers of the right hand in the direction of rotation, the thumb points in the direction of the angular momentum vector).

When a propeller spins clockwise viewed from above, its angular momentum vector points downward. To change this angular momentum — to speed the propeller up, slow it down, or tilt the airframe — requires applying a torque. The reaction to that torque acts on the airframe.

Why counter-rotation is a physical necessity, not a convention¶

Consider a drone with all four motors spinning clockwise viewed from above. The total angular momentum of the propeller system points downward. The airframe, reacting to this angular momentum, tends to spin counterclockwise. No amount of PID tuning can correct this — it is a direct physical consequence of all the angular momentum being aligned in one direction. The drone is uncontrollable in yaw.

Counter-rotating motor pairs cancel this. Two motors spin clockwise (angular momentum pointing down), two spin counterclockwise (angular momentum pointing up). At equal speed, the total angular momentum of the system is zero — the airframe experiences no net rotational reaction and does not yaw.

Yaw control by angular momentum differential¶

To yaw the drone clockwise: increase the speed of the two counterclockwise motors (increasing their upward angular momentum) while decreasing the speed of the two clockwise motors (decreasing their downward angular momentum). The net angular momentum of the propeller system shifts toward pointing upward. The reaction in the airframe is to rotate clockwise.

This is elegant: yaw is controlled entirely by adjusting four numbers (motor speeds), with no mechanical linkages and no moving parts beyond the motors themselves. The entire yaw response comes from physics, not from mechanical complexity.

Motor layout and prop direction assignment¶

In a standard quadcopter (Betaflight "Props In" configuration): - Front-left (FL): counterclockwise - Front-right (FR): clockwise - Rear-left (RL): clockwise - Rear-right (RR): counterclockwise

Diagonal pairs share the same direction. This arrangement ensures that all roll, pitch, and yaw commands produce balanced thrust changes across the frame. Installing a propeller in the wrong rotation direction produces incorrect mixing — the drone will attempt to spin uncontrollably on arm.

Reference¶

Right-hand rule for angular momentum direction¶

Yaw mixing (standard X quad, Props In)¶

Increasing CCW motor speed shifts net angular momentum upward → airframe reacts clockwise (Newton's third law).

Why yaw is the weakest axis¶

Yaw torque is proportional to the angular momentum differential between clockwise and counterclockwise pairs. At any given throttle, the maximum yaw torque available is limited by how much speed differential can be created before one motor pair runs out of headroom. Roll and pitch use thrust differential directly — a much larger force. Yaw uses angular momentum differential — a smaller, indirect effect. Result: yaw has inherently lower maximum rate than roll or pitch on a quadcopter.

Procedure¶

Rationale¶

Why this article exists separately from six-degrees-of-freedom¶

six-degrees-of-freedom explains the geometry of control — which actuators produce which motions. This article explains the underlying physics of the yaw axis specifically — why counter-rotation is necessary, why yaw authority is limited, and why the mixing table has the signs it does. A student reading only the geometry article will know that yaw is produced by motor pairs; a student reading this article will understand why that works and why it is weaker than the other axes.

Connections¶

requires: - six-degrees-of-freedom related: - lift-and-thrust - hover-and-forward-flight - moment-of-inertia leads_to: - hover-and-forward-flight - moment-of-inertia