A Quadrotor includes 4 rotors arranged in a cross-like shape as displayed in Figure 1. Rotor blades are equipped with propellers powered by one-by-one DC motors. Rotors 1 and 3 rotate within side the equal route at the same time as rotors 2 and 4 rotate in a contrary route in order to balance the whole machine torque and cancel the gyroscopic and aerodynamics torques.

The motion of quadcopters can be achieved through six distinct maneuvers, which involve a combination of translational and rotational motion as illustrated in Figure 2. There are four inputs on the quadcopter:

Thrust (z): This input generates a vertical force that enables the quadcopter to hover or move up and down through the air. Increased or decreased speed of all four rotors by a similar amount, greater or lesser than gravitational force, creates the force.

First, let us begin by precising the reference coordinate frame and the coordinate frame of the vehicle body, in order to construct the quadrotor’s mathematical model. Let us denote by B = [Bx, By, Bz]^T the body-fixed frame (BFF) and (EFF) E = [E_x, E_y, E_z]^T the ground fixed reference coordinate frame also called inertial frame. The center of the BFF origin is aligned with the center of mass of the quadcopter, and its origin is connected to the body of the vehicle.

In the 3-dimensional Euclidean space, we denote B = (x,y,z)^T. The coordinate vector I B → describes the distance between the Inertia frame and the body frame. The three Euler angles 𝜙, 𝜃 and 𝜓 (so called: roll, pitch, and yaw) describing the quadrotor orientation are represented by the vector Γ = [𝜙,,𝜓]^𝑇.

The vehicle body frame BFF is considered the base for reference frame analysis. To catch the 3D space’s movements and create our mathematical references, Equations 1–5 are considered:

- Body frame rotated in 𝜓⁺:

The movement from one frame to another in this case is defined by Equation 4: (4) R B I ψ θ ϕ = R B I z ψ R B I y θ R B I x ϕ

The rotational matrix is then derived as (5) R B I = cos ψ cos θ sin θ sin ϕ cos ψ − cos θ sin ψ sin θ cos ϕ cos ψ + sin ϕ sin ψ sin ψ cos θ sin θ sin ϕ sin ψ + cos ϕ cos ψ sin θ cos ϕ sin ψ − sin ϕ cos ψ − sin θ sin ϕ cos θ cos ϕ cos θ

The actuator dynamic refers to the correlation between the voltage applied to the actuator, which serves as the real control input, and the speed of the rotor. Designers are particularly interested in the speed of the actuator’s response, as it plays a crucial role in the performance of the system. Based on Kirchhoff laws and the law of rotation, we derive the following simplified actuator dynamic model described by Equation 6: (6) d w i d t = − 1 τ w i + K w τ u i , ∀ i = 1 , .. , 4 where 𝜏 and K w denotes the delay and gain coefficient, respectively (Bouabdallah et al., 2004). Furthermore, d w i d t and u i represent the speed and the input voltage of actuator 𝑖, = 1 , .. , 4 . Actuator delay assumes a critical role, especially in cases where the attitude control loop is functioning at a low frequency.

Let us move on to the kinematics side of the conducted study. Herein, the rigid “+” symmetric quadrotor is shown in Figure 1 with four rotors producing (Th1, Th2, Th3, Th4) forces. Quadrotor structure center of mass and origin of body coordinate system are the same. The center of gravity of the system coincides with the origin of the body-fixed frame (BFF). It is assumed that propellers are rigid and that thrust and drag are proportional to the square of the propeller’s rotational speed. The difference between the body rates p, q, r in BFF and the angle rates expressed in EFF should be calculated following Equations 7, 8. (7) p q r = 1 0 − sin θ 0 cos θ sin ϕ cos θ 0 − sin θ cos ϕ cos θ ϕ ˙ θ ˙ ψ ˙ (8) v = M Γ ˙

Where v and Γ ˙ represent the angular velocity vectors in both frames the B-frame and E-frame, respectively. Furthermore, the inverse of the matrix M is expressed in Equation 9 as (9) M − 1 = 1 tan θ sin ϕ tan θ cos ϕ 0 cos ϕ − sin ϕ 0 sin ϕ cos θ cos ϕ cos θ

During position control, the quadrotor’s body must be compensated for rotation. Compensation is obtained by transposing the rotation matrix.

Achieving seamless tracking performance utilizing traditional gains is the primary goal of the proposed PD-PID controller, while simultaneously ensuring satisfactory performance in position hold mode through gain adjustment aimed at mitigating the effects of wind disturbances. A schematic representation of the proposed PD-PID technique is depicted in Figure 3. The proportional, integral, and derivative gains are optimized utilizing gain scheduling, which is determined by variations in natural frequency and the Ziegler-Nichols methods.

The primary control loop, as proposed, governs four Degrees of Freedom (DOF) in the quadrotor system, namely the altitude z , roll angle ϕ , pitch angle θ , and yaw angle ψ . Given that the quadrotor dynamic system is under-actuated, an outer loop is employed to derive the desired control inputs for roll and pitch, denoted as ϕ d and θ d , respectively, based on the desired positions in the earth frame x e y e . For the linear acceleration, very small angles ϕ and θ are assumed. Inspired from the work of Kotarski et al. (2016), the following desired angles are derived: (38) ϕ d θ d = 1 g sin ψ − cos ψ cos ψ sin ψ T x T y

A configurable PD controller is also envisaged, following the work of Nguyen and Hong (2018). the controller produces virtual accelerations T x and T y along the X and Y axes, respectively, such as (39) T x = k 1 x k 2 x x d − x − x ˙ (40) T y = k 1 y k 2 y y d − y − y ˙ where the coordinates ( x d , y d ) refer to the desired position in the X Y plane, whereas ( k 1 x , k 2 x ) and ( k 1 y , k 2 y ) represent the controller gains along the X and Y axes, respectively.

To summarize, the block diagram shown in Figure 4 represents the roll controller for the X-axis. It is evident that the structure of the roll controller for the Y direction is highly similar.

Now, let ξ and w n denote and the controller’s damping ratio and the natural frequency, respectively. Accordingly, Equation 41 was derived as a transfer function of second order that leads to the controller gains k 1 x and k 2 x , expressed by Equations 42, 43, respectively: (41) T x p X d p = k 1 x k 2 x p 2 + k 1 x p + k 1 x k 2 x (42) k 1 x = 2 ξ w n (43) k 2 x = w n 2 ξ

To facilitate roll control and achieve motion along the Y-axis, the PID controller is utilized to generate small variations in forces which are then applied to rotors 3 and 4. If we assume that the roll parameters of the PID controller are represented by K P ϕ , K I ϕ , and K D ϕ , and that the actual roll angle is ϕ m , then the roll angle error e ϕ can be expressed as ϕ d − ϕ m . Accordingly, Δ T h ϕ can be represented by Equation 44: (44) Δ T h ϕ = K P ϕ e ϕ + K I ϕ ∫ e ϕ + K D ϕ e ˙ ϕ

In order to generate motion along the X-axis, a PID controller is employed to generate a small variation in forces, denoted as Δ T h θ , which is then applied to rotors 1 and 2. If the quadrotor needs to move backwards, the speed of rotor 2 is decreased while the speed of rotor 1 is increased. This differential change in rotational speeds causes the quadrotor to move in a backward direction along the X-axis. This speed variation is controlled by PID which pitch angle parameters are K P θ , K I θ , and K D θ so that Δ T h ϕ is expressed according to Equation 45: (45) Δ T h θ = K P θ e θ + K I θ ∫ e + K D θ e ˙ θ and e θ = θ d − θ m , such that the pitch angle error e θ is associated with the measured pitch angle θ m .

The quadrotor’s counter clockwise and clockwise movements can be controlled by altering the yaw angle. The yaw angle parameters of the considered PD or PID controller are K P ψ , K I ψ , and K D ψ . Additionally, ψ m represents the sensor measured yaw angle, and e ψ represents the yaw angle error. Thus, as with the roll and pitch control, e ψ equals ψ d − ψ m . Equations 46, 47 represent the yaw control equations for the PID and PD controllers, respectively: (46) Δ T h ψ = K P ψ e ψ + K I ψ ∫ e ψ + K D ψ e ˙ ψ (47) Δ T h ψ = K P ψ e ψ + K D ψ e ˙ ψ

A thrust force is created to compensate for the gravitational force. Altering the angular velocities of all rotors at equal speed is the way to achieve the desired altitude of the quadrotor. Let K P z , K I z , and K D z denote the altitude parameters of the PID controller, z m be the sensor measured altitude, and e z be the altitude error. Obviously, e z equals z d − z m . According to these notations, the thrust forces for the PID and PD controllers, respectively, could be expressed by Equations 48, 49 as (48) Δ T h z = K P z e z + K I z ∫ e z + K D z e ˙ z (49) Δ T h z = K P z e z + K D z e ˙ z

For the simulation study, we have adopted a quadrotor which main parameters are displayed in Table 2.

For the design of the quadrotor autopilot, a PD/PID controller is implemented as presented in Section 3. As a result of Ziegler-Nichols PID parameters setting rules, the proportion and derivative coefficients of the controller are adjusted. According to this parameter-setting method, parameters are set according to stability analysis. Therefore, K p the proportion coefficient is set such that K D and K I are fixed to 0. Then, an iterative increase in K p is made until the closed-loop system begins to shock, i.e., as soon as the poles of the closed-loop system cross the 𝑗𝑤 axis. Equations 50–52 express the final K p , K D , and K I values as (50) K p = 0.6 K m (51) K D = K P π 4 w m (52) K I = K P w m π where the value of K p at which the system experiences shock is represented by K m , while w m denotes the frequency of oscillation. The root locus method is applied to determine K m and w m . The root locus of the controlled object’s transfer function can be obtained for a given system. K m corresponds to the gain traversing the 𝑗𝑤 axis, with the oscillation frequency of the point corresponding to w m .

In order to assess and validate the PD and PID controller simulator, several numerical experiments were conducted.

The time evolution of the X, Y, Z, and Yaw angle positions of the tuned PD/PID controllers are reported in Figures 5–8. The PD and PID controllers have similar performance in terms of X and Y-axis motion, as shown in Figures 5, 6. It should also be noted that the Z position controller does not affect movement along the X and Y axes. The PID controller proves to be more efficient in reducing altitude position errors (Figure 7), whereas the PD controller only manages to keep a minor, permanent error. Additionally, the PD controller exhibits a marginally quicker rise time compared to its counterpart, the PID controller. The dynamics of the response of both PD/PID yaw controllers to the input reference show comparative behavior, as displayed in Figure 8.

Based on the information presented in Figures 9, 10, we can infer that the PID controlled inner loop has a quick response time as well as high degree of accuracy. The overshoot is well-damped, and the system stabilizes the roll and pitch input angles of the quadrotor. It succeeds to maintain these angles as minimal as possible, assuming negligible values for the ϕ and θ angles. In general, higher roll and pitch angles correlate with greater speeds along the X and Y directions.

A wide range of signal shapes, such as square, circular, and sinusoidal trajectories, have been tested for trajectory trucking. In order to evaluate the performance of the proposed control architecture, a number of time-varying input references were examined and implemented.

Figure 11 illustrates the time history of helical paths. When the quadrotor is being controlled by a PD command, the PD controller produces a small permanent position error. However, the PID controller operates dynamically to ensure precise control. Besides, Figure 12 highlights that the pitch and roll controllers ensure that the quadrotor remains very stable. It can be observed that the trajectory tracking performance is satisfactory with both PD and PIC controllers, particularly when considering the application of the designed autopilot.

As an input signal to the quadrotor model, a wind force model was added to test whether the control response of the quadrotor was affected by wind disturbances. We have implement the wind force inspired from the work of Alexis et al. (2012). We begin by assessing the impact of disturbances along X and Y axes. The corresponding simulation results are reported in Figure 13.

It is worth mentioning that the quadrotor is capable of following any rectangular trajectory accurately in the absence of disturbances. Now, when adding the wind gusts scenario, the controller shows very good robustness. Indeed, by properly compensating for wind gusts, the controller corrects any deviation caused by the wind as displayed by Figure 14.

Furthermore, controlling a circular trajectory yields the same promising results. As can be clearly observed in Figure 15, the controller succeeds to keep the quadrotor’s trajectory around the input reference in a highly accurate and dynamic manner. For both circular and rectangular trajectories, the simulation results confirm that the quadrotor, utilizing the designed autopilot, can deal with disturbances equivalent to more than 30% of its weight without experiencing any significant deviation from its reference trajectory.

At this stage, it is noteworthy that as the parameters of the controllers are adjustable, their determination could be calculated in several ways and using multiple methods. The mathematical approach used in this paper consists of calculating the parameters according to a second order model that corresponds to the system response. In future work, another approach to be considered involves implementing a more intelligent strategy. This entails using machine learning algorithms trained with historical data. In fact, additional IoT sensors could be deployed for example to continuously update the parameters according to weight variation, flight behavior, environment/landscape nature, and weather information.