Mathematical Modelling and Simulation of a Quadcopter

Abstract

This report develops a complete mathematical and simulation framework for a quadrotor unmanned aerial vehicle with closed-loop attitude and altitude control. The vehicle is represented as a six-degree-of-freedom rigid body with four independently driven rotors. Translational motion is expressed in an inertial North-East-Down frame, while rotational motion is described in the body frame. A unit quaternion is used for attitude propagation and feedback because it avoids the singularity associated with Euler-angle kinematics and supports compact computation of the shortest rotational error. The nonlinear model combines gravitational force, thrust-vector transformation, aerodynamic drag, rigid-body gyroscopic coupling, motor thrust and reaction torque, actuator allocation, sensor perturbations, and ground-contact logic. The control system adopts a cascaded structure. The outer altitude or position loop produces collective thrust and a desired thrust direction. The desired orientation is converted to a quaternion, compared with the measured quaternion, and regulated by an inner attitude loop that generates roll, pitch, and yaw torque commands. A mixer maps collective thrust and body torques into four motor commands, after which saturation, slew-rate limitation, and pulse-width-modulation logic represent practical actuator constraints. The supplied MATLAB/Simulink model is analysed subsystem by subsystem, including Euler-to-quaternion conversion, quaternion-error calculation, nonlinear plant equations, sensor modelling, motor mixing, remote-command processing, and three-dimensional visualization.
Time-domain scope outputs are evaluated for sequential roll, pitch, and yaw commands and for threeaxis position response. The roll and pitch channels track 20-degree pulse commands with smooth, strongly damped responses and negligible visible overshoot. The yaw channel tracks a 45-degree command but exhibits a small peak near 47-48 degrees and a modest negative undershoot after command removal, indicating a more lightly damped yaw loop. The NED vertical coordinate settles near z = -4 in the displayed units and remains bounded during the attitude manoeuvres, while the horizontal coordinates evolve continuously with the commanded tilt. These results support nominal closed-loop stability and coordinated attitude-altitude control, while also identifying yaw damping, cross-axis coupling, exact unit documentation, and numerical performance extraction as the principal areas for further refinement. Beyond subsystem description, the thesis integrates coordinate-frame verification, complete six-degree-of-freedom derivation, hover linearisation, near-hover state-space analysis, altitude and attitude gainselection procedures, actuator-feasibility assessment, quantitative performance metrics, robustness testing, verification and validation traceability, and hardware-in-the-loop deployment considerations within the main chapters rather than isolating them in a separate technical appendix.