Sliding mode control design for the attitude and altitude of the quadrotor UAV

The quadrotor UAV comprises of four rotors to produce the forces (F₁, F₂, F₃, F₄). The rotors are fixed in a cross structure and symmetric shape, as illustrated in Figure 1.

Figure 1:

The quadrotor motions are controlled by changing the speed of the rotors. The front and the rear of the quadrotor are represented by the rotors 1 and 3, and the left and the right are represented by the rotors 2 and 4, as depicted in Figure 1. By agreement, the rotors 1 and 3 turning in the clockwise direction, whereas rotors (2 and 4) turning in to the counterclockwise direction.

The quadrotor moves in a vertical direction by increasing or decreasing the angular velocities of all rotors with equal speed, which generates a total lift force (thrust) against the gravitational force. Consequently, the quadrotor takes-off or lands, as illustrated in Figure 2a and b, respectively.

Figure 2:

The right direction movement of the quadrotor is achieved by increasing the rotational speed of the rotor (2) and decreasing the rotational speed of the rotor (4); subsequently, the quadrotor moves in the right direction as depicted in Figure 2c. Likewise, the left direction movement is achieved by increasing the rotational speed of the rotor (4) and decreasing the rotational speed of the rotor (2); as a result, the quadrotor moves to the left direction as shown in Figure 2d.

The forward movement of the quadrotor is achieved by increasing the rotational speed of the rotor (3) and decreasing the rotational speed of the rotor (1); subsequently, the quadrotor moves to the forward direction as illustrated in Figure 2e. Similarly, the backward movement of the quadrotor is achieved by increasing the rotational speed of the rotor (1) and decreasing the rotational speed of the rotor (3); as a result, the quadrotor moves to the backward direction as shown in Figure 2f. The anti-clockwise and clockwise movements of the quadrotor are controlled by changing the yaw angle (ψ) as shown in Figure 2g and h, respectively.

The quadrotor kinematics are represented into two frames. The earth fixed, or the reference frame (E-frame), is denoted by E = (x_e, y_e, z_e) and the body-fixed (B-frame) is represented by B = (x_b, y_b, z_b) as shown in Figure 1.

Consider that q = (x, y, z, ϕ, θ, ψ) ∈ R⁶ denotes the generalized coordinates, with (x, y, z) representing the position the quadrotor and ϕ, θ, ψ representing the quadrotor’s orientation. Accordingly, the quadrotor mathematical model can be divided into two subsystems: the position and the attitude subsystems, and the associated coordinates are given as follows: (1) q = [ ξ , η ] T , (1)where: (2) ξ = [ x , y , z ] T , (2)and: (3) η = [ ϕ , θ , ψ ] T .

Therefore, the quadrotor kinematics are obtained as follows: (4) ξ = RV , where V denotes the linear velocity in the B-frame, whereas ξ represents the linear velocity in the E-frame with respect to B-frame, and R is the rotation matrix: (5) R = [ cos ψ cos θ sin θ sin ϕ cos ψ − cos ϕ cos ψ sin θ sin ϕ cos ψ + cos ϕ sin ψ sin ψ c o s 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 quadrotor rotational motions are obtained as the following: (6) η ̇ = T ω , where ω denotes the angular velocity for the B-frame, whereas η ̇ denotes the angular velocity for the E-frame with respect to the B-frame, and T is the transfer matrix (Olfati-Saber, 2001): (7) T = [ 1 tan θ sin ϕ tan θ cos ϕ 0 cos ϕ − sin ϕ 0 sin ϕ cos θ cos ϕ cos θ ] .

The quadrotor dynamics equations in six degrees of freedom (6 DOFs) are given as follows: (8) x ̈ = ( sin θ cos ϕ cos ψ + sin ϕ sin ψ ) u 4 m y ̈ = ( sin θ cos ϕ cos ψ − sin ϕ sin ψ ) u 4 m z ̈ = − g + cos ϕ cos θ m u 4 ϕ ̈ = θ ̇ ψ ̇ I y − I z I x + θ ̇ Ω d J r I x + 1 I x u 1 θ ̈ = ϕ ̇ ψ ̇ I z − I x I y + ϕ ̇ Ω d J r I y + 1 I y u 2 ψ ̈ = θ ̇ ψ ̇ I x − I y I z + 1 I z u 3 , (8) where u₁, u₂, u₃, u₄ represent the control inputs, which calculated as follows: (9) u 1 = b ( Ω 4 2 − Ω 2 2 ) u 2 = b ( Ω 3 2 − Ω 1 2 ) u 3 = d ( Ω 4 2 + Ω 2 2 − Ω 3 2 − Ω 1 2 ) u 4 = b ( Ω 1 2 + Ω 2 2 + Ω 3 2 + Ω 4 2 ) , (9) while Ω_d denotes the disturbance, and mathematically expressed as follows: (10) Ω d = − Ω 1 + Ω 2 − Ω 3 + Ω 4 .

The control inputs in (9) can be re-written in the matrix as: (11) [ u 1 u 2 u 3 u 4 ] = [ b b b b 0 − b 0 b − b 0 b 0 − d d − d d ] [ Ω 1 2 Ω 2 2 Ω 3 2 Ω 4 2 ] .

The feedback linearization technique is used to transfer the nonlinear systems to the equivalent linear systems, as illustrated in Figure 3 (Eltayeb et al., 2019, 2020).

Figure 3:

Now, consider the general nonlinear system expressed as: (12) x ̇ = f ( x ) + Gu , (12)where f(x) is the nonlinear function; x is the system’s state vector; and u is the control input.

The control input u in (12) can be chosen as: (13) u = G − 1 ( − f ( x ) + v ) .

By substituting (13) into (12) yields to the following linear system: (14) x ̇ = v .

Now, by following the same prior steps from (12) to (14), the control inputs u₁, u₂, u₃ and u₄ in (8) of the quadrotor attitude and altitude can be chosen as in (15): (15) u 1 = 1 b 1 ( − a 1 θ ̇ ψ ̇ − a 2 θ ̇ Ω d + v 1 ) u 2 = 1 b 2 ( − a 3 ϕ ̇ ψ ̇ − a 4 ϕ ̇ Ω d + v 2 ) u 3 = 1 b 3 ( − a 5 θ ̇ ϕ ̇ + v 3 ) u 4 = m cos ϕ cos θ ( g + v 4 ) , (15) where: a 1 = I y − I z I x , a 2 = J r I x , a 3 = I z − I x I y , a 4 = J r I y , a 5 = I x − I y I z , b 1 = l I x , b 2 = l I y , b 3 = 1 I z .

Thus, from (8) and (15), the quadrotor’s attitude and altitude systems are obtained in the linearized version as follows: (16) ϕ ̈ = v 1 θ ̈ = v 2 ψ ̈ = v 3 z ̈ = v 4 . (16)

The obtained linear system in (16) can be represented in the state space as follows: (17) x ̇ = Ax + Bu , (17)where the output of the linear system is: (18) y = Cx , (18)and: A = [ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ] , B = [ 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 ] , and C = [ 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 ] , and the system’s states are: x = [ φ φ ̇ θ θ ̇ ψ ψ ̇ z z ̇ ] .

Now, the objective is to design the PID control strategy to stabilize and control the quadrotor linear system (17). The errors dynamics are defined as follows: (19) e ϕ = ϕ d − ϕ e θ = θ d − θ e ψ = ψ d − ψ e z = z d − z , (19) where e_ϕ, e_θ, e_ψ, and e_z are the errors signals for roll, pitch, yaw, and the altitude, respectively. ϕ_d, θ_d, ψ_d, and Z_d are the desired signals for roll, pitch, yaw, and the altitude, respectively.

Thus, the PID control is designed and implemented for each as follows: (20) v i = K p i e i + K I i ∫ 0 t e i d τ + K D i d d t e i , where i = 1, 2, 3, 4, and K_p⩾0, K_I⩾0, and K_D⩾0 represent the PID controller parameters; v_i is the PID generated control signal.

The SMC control technique uses to attract the state variables of the system towards the equilibrium sliding surface and stay on it. The sliding surface can be designed as follows (Vaidyanathan and Lien, 2017): (21) s ( t ) = ( d d t + k ) n − 1 e ( t ) , where s(t) ∈ Rⁿ is the equilibrium surface; e(t) is the error between the desired and actual position or orientation; and k is a positive constant; and n ∈ N is the system’s order. SMC control can be designed as follows.

First, select a Lyapunov function (candidate function) which maps the system’s state variables as follows: (22) V ( s ) = 1 2 s 2 > 0 , ∀ x V ( 0 ) = 0 , x = 0. (22)

Second, the derivative of the selected function (22) must be negative: (23) V ̇ ( s ) < 0 , (23)yields to: (24) V ̇ = s s ̇ < 0.

The sliding mode control law consists of two terms: continuous part and the discontinuous part, which is given as follows (Herrera et al., 2015): (25) u = u e q + u c .

From (24), sliding mode condition given as follows: (26) s ̇ = − k 1 sgn ( s ) − k 2 s , (26)where k₁>0 and k₂>0 are the SMC design parameters.

Chattering is an unwanted phenomenon with a finite-frequency, and finite-amplitude oscillations happened near to the sliding surface (Al-Younes et al., 2010). The chattering is caused by the switching function; therefore, in this work, the switching function (sing(s)) in the SMC control laws has been replaced by an approximated error function, as shown in Figure 4. The error function is obtained by integrating the normalized Gaussian distribution as follows (Eltayeb et al., 2020): (27) e r f ( x ) = x π ∫ 0 x e − t 2 d t , (27)and it has the following properties: e r f ( ∞ ) = + 1 , e r f ( − ∞ ) = − 1.

Figure 4:

The SMC controller is designed to stabilize the quadrotor’s attitude and altitude.

The first step is to design the error dynamics as in (19).

The second step is to choose the sliding surfaces as the following: (28) s ϕ = e ̇ ϕ + k ϕ e ϕ s θ = e ̇ θ + k θ e θ s ψ = e ̇ ψ + k ψ e ψ s z = e ̇ z + k z e z , (28) (28)where s ϕ , s θ , s ψ sϕ, sθ, sψ, and s z sz represent the surfaces of the roll, pitch yaw, and altitude dynamics, respectively. k ϕ , k θ , k ψ kϕ, kθ, kψ, and k z kz are the SMC controller parameters.

The third step recalls the sliding mode condition as in (26) and applies it to (28) to compute the SMC control laws.

The steps, as mentioned above, will be implemented to calculate the SMC control law (u₁) for the roll ( ϕ ) (ϕ) angle as follows: (29) e ̈ ϕ + k ϕ e ̇ ϕ = − k 1 ϕ sgn ( s ϕ ) − k 2 ϕ s ϕ , (29)where k 1 ϕ > 0 k1ϕ > 0 and k 2 ϕ > 0 k2ϕ > 0 are the SMC control parameters of the controlled variable ( ϕ ) (ϕ).

By substituting (8) into (29), the SMC control law for the roll angle ( u 1 u1) is driven as in (30): (30) u 1 = 1 b 1 ( ϕ ̈ d − a 1 θ ̇ ψ ̇ − a 2 θ ̇ Ω d − k ϕ e ̇ ϕ − k 1 ϕ sgn ( s ϕ ) − k 2 ϕ s ϕ ) .

Similarly, the SMC control laws for pitch, yaw, and altitude ( u 2 u2), ( u 3 u3), and ( u 4 u4), respectively, are obtained as follows: (31) u 2 | = 1 b 2 ( θ ̈ d − a 3 ϕ ̇ ψ ̇ − a 4 ϕ ̇ Ω d − k θ e ̇ θ − k 1 θ sgn ( s θ ) − k 2 θ s θ ) , (31) (32) u 3 | = 1 b 3 ( ψ ̈ d − a 5 ϕ ̇ θ ̇ − k ψ e ̇ ψ − k 1 ψ sgn ( s ψ ) − k 2 ψ s ψ ) , (32) (33) u 4 | = m cos ϕ cos θ ( z ̈ d + g − k z e ̇ z − k 1 z sgn ( s z ) − k 2 z s z ) .

Therefore, to reduce the chattering effects, which leads to critical problems such as vibration in the mechanical parts of the quadrotor and heat in the onboard electronics kits (Li et al., 2014). The switching function (sign(s)) has been replaced by the error function erf(s) in the proposed SMC control laws as follows: (34) u 1 = 1 b 1 ( ϕ ̈ d − a 1 θ ̇ ψ ̇ − a 2 θ ̇ Ω d − k ϕ e ̇ ϕ − k 1 ϕ e r f ( s ϕ ) − k 2 ϕ s ϕ ) , (34) (35) u 2 = 1 b 2 ( θ ̈ d − a 3 ϕ ̇ ψ ̇ − a 4 ϕ ̇ Ω d − k θ e ̇ θ − k 1 θ e r f ( s θ ) − k 2 θ s θ ) , (35) (36) u 3 = 1 b 3 ( ψ ̈ d − a 5 ϕ ̇ θ ̇ − k ψ e ̇ ψ − k 1 ψ e r f ( s ψ ) − k 2 ψ s ψ ) , (36) (37) u 4 = m cos ϕ cos θ ( z ̈ d + g − k z e ̇ z − k 1 z e r f ( s z ) − k 2 z s z ) .

In this scenario, the performance of the proposed controller is simulated based on the nominal parameter values of the quadrotor model, on the contrary, neither considering any model parameter uncertainties nor external disturbances.

The commends are given to stabilize the attitude of the quadrotor system from the initial values ϕ = 1 ϕ = 1 θ = 1 θ = 1 and ψ = 1 ψ = 1 to the desired values ϕ = 0 ϕ = 0, θ = 0 θ = 0 and ψ = 0 ψ = 0 as depicted in Figure 5. The altitude of quadrotor tracks the desired take-off and landing trajectory, as shown in Figure 6, and the tracking error for the proposed controller compare to the traditional SMC and FBL are illustrated in Figure 7.

Figure 5:

Figure 6:

Figure 7:

In terms of the attitude tracking error, as shown in Figure 7, the proposed controller outperforms the conventional SMC and FBL linearization. The control inputs of the FBL for the attitude and altitude of the quadrotor system are depicted as in Figure 8.

Figure 8:

Furthermore, the proposed controller significantly reduces the unwanted chattering compared to the conventional SMC, as clearly illustrated in Figures 9 and 10.

Figure 9:

Figure 10:

In this scenario, the controller system is operated and evaluated under the uncertainty in the quadrotor’s mass, which has been increased by %100 from 0.65 kg to 1.3 kg. Furthermore, the external pulse disturbances have been considered as well, for both the attitude and altitude, as shown in Figures 11 and 12, respectively.

Figure 11:

Figure 12:

Even though, in the presence of the parameter uncertainty and external disturbance, the proposed controller has still outperformed the traditional SMC and FBL controllers as it can be clearly seen in the trajectory tracking of the attitude as presented in Figure 13 and its tracking errors as shown in Figure 14.

Figure 13:

Figure 14:

The altitude of quadrotor follows the desired take-off and landing trajectory as presented in Figure 15, and the altitude tracking errors for the proposed controller compared to the traditional SMC and FBL are illustrated in Figure 16. The control inputs of the FBL for the attitude and altitude of the quadrotor system are presented as in Figure 17.

Figure 15:

Figure 16:

Figure 17:

Moreover, it can be observed that the proposed SMC controller provided significant performance in terms of the chattering attenuation compared to the conventional SMC, as shown in Figures 18 and 19, respectively.

Figure 18:

Figure 19: