Controlling a quadcopter is a challenging and intriguing task due to its six degrees of freedom, consisting of three rotational and three translational movements. What adds to the complexity is that quadcopters are underactuated systems, with only four independent inputs available (rotor speeds). Consequently, achieving all six degrees of freedom involves coupling between rotational and translational motions.The quadrotor dynamics are highly nonlinear, and the complex aerodynamic effects further add to the challenge. Unlike ground vehicles, quadcopters have very little friction, making it difficult to prevent motion, and they require their own damping to remain stable. These factors make quadcopter control an interesting and complex problem. To tackle this problem, a simplified model of quadcopter dynamics is used, and controllers are designed to follow a designated trajectory. [1]

The control system receives inputs in the form of a position vector denoted by (x, y, z) and a yaw angle, and it produces voltage signals for the quadrotor's four DC motors. To enable full control of the quadrotor in all six degrees of freedom, it is necessary to incorporate the roll and pitch angles into the controller. This is achieved by coupling x and y positions to the pitch (θ) and roll (φ) angles, respectively, through a cascade controller structure for longitudinal and lateral dynamics. 

As illustrated in Figure 1, the outer lateral controller takes in the reference lateral position (y_ref), which originates from the user input command, and the measured lateral position (y_measured) from the sensors. It uses these inputs to calculate the error in lateral position (y_e). Following that, a LimPID controller is established for position control, where, given the small angle assumption [2], the resulting output corresponds to the reference roll angle (φ_ref). Another LimPID is used to control roll angle (φ) and finally generates the roll moment change (M). A similar methodology is applied to the longitudinal control channel.

The yaw channel is controlled with a single LimPID controller and generates the yaw moment change (N) command. 

In the case of the throttle command, the additional thrust increment should be added to the base thrust, which represents the minimum required thrust to overcome the weight of the quadrotor.

Finally, in the Mixer, the roll moment change (L), pitch moment change (M), yaw moment change (N) and throttle command will add up to allocate force and moments to each propeller. The outputs of the mixer and subsequently controller are the voltage of the DC motors.

Figure 1: Controller block diagram.