Six degrees of freedom

Summary¶

Any rigid body moving in three-dimensional space has exactly six degrees of freedom: three translational (forward/back, left/right, up/down) and three rotational (roll, pitch, yaw). A quadcopter has four independent actuators — four motor speeds — which means it cannot independently control all six simultaneously. Understanding which freedoms are coupled, and how the flight controller manages that coupling, explains why drones behave the way they do.

Concept¶

Six degrees, four actuators¶

Three translational and three rotational degrees of freedom give six dimensions to control. A quadcopter has exactly four independent inputs: motor 1 speed, motor 2 speed, motor 3 speed, motor 4 speed. With four numbers controlling six dimensions, the system is underdetermined — it cannot achieve arbitrary combinations of position and attitude simultaneously.

In practice, the four actuators control: total thrust (vertical translation), roll (differential thrust left vs right), pitch (differential thrust front vs rear), and yaw (differential angular momentum). That is four controlled outputs from four inputs — exactly determined, no redundancy.

What a quadcopter cannot do: move horizontally while keeping its heading and altitude fully independent. To move left it must roll left — tilting the thrust vector sideways. The reduced vertical component then requires increased throttle to maintain altitude. Every degree of pitch or roll requires simultaneous altitude compensation. Everything couples to everything else.

Why coupling matters for the flight controller¶

Because roll affects altitude, pitch affects altitude, and yaw affects neither translation directly but changes the drag asymmetry, the flight controller runs separate PID loops for roll, pitch, yaw, altitude, and position simultaneously — each feeding corrections into the others. The coupling is the reason the flight controller needs to run at 8,000 loops per second: any slower and the corrections from each loop arrive too late to prevent the cross-axis effects from accumulating into visible oscillation.

Motor redundancy and failure modes¶

A standard quadcopter losing one motor drops to three actuators for six degrees of freedom — catastrophically underdetermined, uncontrollable. This is not a design flaw; it is the explicit cost of using the minimum number of motors. The tradeoff is direct: fewer motors means less weight, complexity, and cost, but also zero redundancy.

A hexacopter has six actuators for six degrees — exactly determined. Lose one motor and it drops to five actuators — still underdetermined, but with enough redundancy to maintain controlled flight with reduced authority. This is why professional survey drones for safety-critical applications use six or eight motors despite the mass penalty: each additional motor buys one level of redundancy.

Yaw authority¶

Yaw is the weakest axis on a quadcopter. Roll and pitch are produced by differential thrust — the primary force the motors generate. Yaw is produced by differential angular momentum between clockwise and counterclockwise motor pairs, which is a much smaller effect. At high throttle, the yaw authority decreases relative to the available roll and pitch authority. This is why aggressive yaw manoeuvres are the hardest to tune in Betaflight and why yaw spins in freestyle flying require much more throttle than roll or pitch rolls of equal angular velocity.

Reference¶

Degrees of freedom summary¶

Motor redundancy by configuration¶

Procedure¶

Rationale¶

Why libdrone uses a quad configuration¶

Four motors is the minimum for controlled flight in three dimensions. Each additional motor adds weight, cost, and complexity. For a research payload platform that flies in controlled conditions with a competent operator, the no-redundancy tradeoff is acceptable. The weight saving of a quad versus a hex directly translates to payload mass budget or flight time — both of which are mission-critical parameters. A hex or octo would extend the redundancy but consume the payload weight budget.

Connections¶

requires: - lift-and-thrust related: - angular-momentum-multirotors - hover-and-forward-flight leads_to: - angular-momentum-multirotors - hover-and-forward-flight