Modeling and design of quadcopter under six degrees of freedom

Quadcopters are rotary wing unmanned aerial vehicles (UAVs) who’s entire six axis flight envelope is governed by differentially varying the speeds of four fixed-pitch motors. They are inherently open-loop unstable and require continuous automatic feedback control to hover and maneuver. This paper presents a complete model-based design study of a quadcopter operating under six degrees of freedom (6-DOF): the full Newton-Euler equations of motion, the ZYX Euler rotation matrix, the 4×4 motor mixing matrix, and moments of inertia estimated from measured component weights (m = 0.890 kg, L = 0.300 m, Ixx = Iyy = 0.0317 kg·m², Izz = 0.0431 kg·m²). A PD controller is designed and implemented using the laboratory-measured gain values Kp = 1.5 and Kd = 2.5 for all attitude axes, and Kp_z = 6.0, Kd_z = 1.5 for altitude. Numerical simulation confirms that the attitude loop is over-damped (ζ = 5.74) with settling time Ts ≈ 5 s, while the altitude loop is underdamped (ζ = 0.325) with 34% overshoot and Ts ≈ 0.74 s. Circular trajectory tracking achieves an RMS position error of 0.015 m. A binary (0/1) motor command framework, verified during laboratory bench testing, is documented as the practical commissioning protocol for the mixing matrix.