**4. Validation through Simulation Experiments**

In this section, we investigate the applicability of the adaptive robust scheme for enhancing the tracking performance of a quadrotor non-linear system. Thus, several experiments are performed for an aerial vehicle numerically simulated. It is important to mention that the aim of the experiments is to portray some of the main contributions and advantages of implementing the proposed motion control strategy. Additionally, the experiments will demonstrate if the implementation of the proposal can be successfully extended for motion control of different types of autonomous vehicles. During the experiments, it is considered an aerial vehicle characterized by the set of parameters presented in Table 1.


**Table 1.** Parameters of the 6DOF non-linear quadrotor system.

## *4.1. Polynomial Interpolation for Quadrotor Navigation*

Bézier curves have been used widely and properly for path smoothing in robot navigation [31] and in motion control schemes for electric motors [32] and mechanical systems [33]. In the former, curves are expressed, such as parametric equations, where the time *t* is used to determine the values of coordinate pairs of (*x*, *y*) points graphed on the plane. In this work, a cubic Bézier curve is used and is defined by end points: (*X*1,*Y*1) and (*X*4,*Y*4), and control points: (*X*2,*Y*2) and (*X*3,*Y*3) such illustrated in Figure 4. In the second case, Bézier interpolation polynomials are suitably configured as position or

velocity trajectory reference profiles, in order to soft the transition between two operation points for electromechanical and mechanical systems.

It is worthwhile to note that, due to its structure and after a proper selection of endpoints and control points, Bézier curves can be successfully implemented in a quadrotor to online computing the navigation path in cluttered environments, in order to ensure adequate obstacle avoidance manoeuvres while accomplishing a specific mission. On the other hand, it should be noted that derivatives of the trajectory references are not available in advance, and, in consequence, the proposed approach in this paper can be effectively implemented for this experiment.

**Figure 4.** Cubic Bézier curve defined by a couple of pair of endpoints and control points.

During the first experiment, the quadrotor is tasked to perform the following: soft take-off to a height of 3 m; navigation through specific operation points in the space; and finally, soft landing, all of them by means of Bézier curves. It is worthwhile to note that the use of these curves is a viable strategy for solving properly the navigation and obstacle avoidance problems. Thus, in order to obtain smooth transitions between initial and final vertical operation points, the following motion scheme is adopted for take-off and landing tasks:

$$z^\* = \begin{cases} \begin{array}{c} \Gamma\_0 & 0 \le t < T\_1 \\ \Gamma\_0 + (\Gamma\_m - \Gamma\_0)\mathcal{B}\_z(t, T\_1, T\_2) & T\_1 \le t < T\_2 \\ \Gamma\_m & T\_2 \le t < T\_3 \\ \Gamma\_m + (\Gamma\_0 - \Gamma\_m)\mathcal{B}\_z(t, T\_3, T\_4) & T\_3 \le t < T\_4 \\ \Gamma\_0 & t > T\_4 \end{array} \end{cases} \tag{29}$$

where Γ<sup>0</sup> = 0 and Γ*<sup>f</sup>* = 2, given in meters, stand for the desired initial and maximum vertical positions. The time values given in seconds are as follows: *T*<sup>1</sup> = 1, *T*<sup>2</sup> = 3, *T*<sup>3</sup> = 37 and *T*<sup>4</sup> = 40. In addition, B*<sup>z</sup>* is a Bézier polynomial [32] defined as

$$\mathcal{B}\_z(t, T\_{i\prime}, T\_f) = \sum\_{k=0}^n r\_k \left( \frac{t - T\_i}{T\_f - T\_i} \right)^k \tag{30}$$

with *Ti* and *Tf* as the initial and final transition times. Moreover, *n* = 6, and *r*<sup>1</sup> = 252, *r*<sup>2</sup> = 1050, *r*<sup>3</sup> = 1800, *r*<sup>4</sup> = 1575, *r*<sup>5</sup> = 700, *r*<sup>6</sup> = 126.

Subsequently, after the take-off, the rotorcraft is carry to desired positions in the horizontal plane, where the third order parametric equations used for navigation are defined as follows:

$$\begin{aligned} x^\* &= (1 - \mathcal{T})^3 X\_1 + 3(1 - \mathcal{T})^2 (\mathcal{T} X\_2) + 3(1 - \mathcal{T})(\mathcal{T}^2 X\_3) + \mathcal{T}^3 X\_4 \\ y^\* &= (1 - \mathcal{T})^3 Y\_1 + 3(1 - \mathcal{T})^2 (\mathcal{T} Y\_2) + 3(1 - \mathcal{T})(\mathcal{T}^2 Y\_3) + \mathcal{T}^3 Y\_4 \end{aligned} \tag{31}$$

Here, the values of the endpoints and control points are selected for performing a continuous navigation according to the parameters summarized in Table 2. Observe that four Bézier curves are used to define the whole navigation path and which is segmented for purposes of mathematical description.


**Table 2.** Control and endpoint values for the Bézier curves.

On the other hand, external vibrating disturbance forces have been included after 12 s for robustness assessment purposes of the introduced motion control scheme, and are given by

$$\mathcal{S}\_{\dot{\jmath}} = \mathcal{A}\_{\dot{\jmath}} \sin(\omega\_{\dot{\jmath}} t) \tag{32}$$

with *j* = *x*, *y*, *z*, A*<sup>x</sup>* = A*<sup>y</sup>* = 1 N, A*<sup>z</sup>* = 2 N, and *ω<sup>x</sup>* = *ω<sup>y</sup>* = *ω<sup>z</sup>* = 10 rad/s.

In Figure 5, it is presented the quadrotor flight performance by implementing the proposed controller, where a proper path following is exhibited. Throughout the manuscript, the use of solid and dashed lines for representing real and desired trajectories is adopted, respectively. As observed in Figure 6, the Bézier curves are successfully implemented for navigation between operation positions, and as a consequence of the proposed controller, a proper trajectory tracking of the planned references is achieved. Moreover, according to this figure, angular tracking of the online computed references *φ* and *θ* is achieved in spite of there is not information about the derivatives of these references since a properly integration of integral reconstructors and neural networks within the robust motion control approach is achieved.

Furthermore, it is evident the satisfactory performance of the quadrotor tracking motion control scheme even though the quadrotor is subjected to undesired harmonic forces. Notice that regulation around *ψ*- = 0 rad is performed in this experiment. Additionally, Figure 7 portrays the controlled vertical quadrotor dynamics, the height control, and yaw motion regulation. From this figure, the utility of the Bézier polynomial curve, where a soft take-off and landing are achieved thanks to the mathematical framework introduced by Equations (30) and (52) is appreciated. In the next section, the ground effect is included within the analysis in order to assess the control scheme robustness for controlling the quadrotor vertical motion.

**Figure 5.** Quadrotor navigation on the plane and space.

**Figure 6.** Lateral and longitudinal motion tracking in experiment 1.

**Figure 7.** Vertical motion tracking for experiment 1.

For this experiment the following desired Hurwitz polynomial has been selected,

$$P\_d(\mathbf{s}) = (\mathbf{s} + \gamma^2)^5 \tag{33}$$

where, in order to ensure close-loop stability and the properly tracking of the planned trajectory, the control gains in (23) should match the following

$$\begin{aligned} \beta\_{4\_i} &= 5\gamma\_i\\ \beta\_{3\_i} &= 10\gamma\_i^2\\ \beta\_{2\_i} &= 10\gamma\_i^3\\ \beta\_{1\_i} &= 5\gamma\_i^4\\ \beta\_{0\_i} &= \gamma\_i^5 \end{aligned} \tag{34}$$

where *γi*, for *i* = *x* , *y* , *z* , *φ* , *θ* , *ψ*, is the unique online computed control parameter. To improve and ease the parameter selection process in this experiment, each of these control parameters are suitably derived by the adaptive framework introduced in Figure 2, where the output of each individual neural network is the value for the control parameter *γi*. As it is presented in Figure 8, dynamical updating, as well as a successful parameter computation of the control gains, is achieved by using the adaptive B-spline artificial neural networks.

**Figure 8.** Adaptive *γ<sup>i</sup>* control parameters, for *i* = *x*, *y*, *z*.

In Figure 9, it has been included results considering both perturbed and unperturbed cases in order to contrast the compensation action of the adaptive robust control scheme. It is worthwhile to note, from Figure 9b, that it is possible to track, satisfactorily, the references, as well as being demonstrated in Figure 9a. Nevertheless, the vibrating disturbance compensation is not present in the unperturbed case. By analyzing Figure 9b, it is evident the reachability of the control commands which benefits the non-saturation of the actuators. It is also important to mention that similarly as the oscillations due to the control compensation action, in Figure 6 it is appreciated the compensation of the vibrating disturbance forces affecting translational dynamics since are related with the rotational trajectory tracking trough the under-actuation property.

According to the results, the proposed control method is robust and able to efficiently reduce induced oscillations. Additionally, it is demonstrated that Bézier polynomial interpolation can be widely and satisfactorily exploited in quadrotor motion control systems: path and trajectory tracking. The experiment presented in this section illustrates that the complex quadrotor non-linear system is motion controlled in an acceptable way. As no information is required about derivatives of the trajectory references and from the external disturbances the control process is simplified significantly.

**Figure 9.** Computed control inputs in experiment 1. (**a**) Unperturbed. (**b**) Perturbed.

### *4.2. Improved Robust Quadrotor Autonomous Landing*

One of the most essential requirements for a VTOL vehicle is to ensuring a safe landing flight phase. Rotorcraft are subjected to significant variations in motion control during take-off and landing stages due to the increase in lift force when they are close to the ground. Such phenomena are known as the ground effect [34]. The aim of this experiment is to assess the capabilities of the proposed controller for dealing with the ground effect in simulation. Therefore, the Cheeseman and Bennett modified ground effect model, proposed for quadrotors by authors in [35], are used, which state the following:

$$\frac{u}{u\_r} = 1 - \rho \left(\frac{r}{4z\_r}\right)^2\tag{35}$$

where the ratio *<sup>u</sup> ur* is equal to one outside of the ground-effect. In addition, *r* is the propeller radius, *zr* represents the distance from the rotor to the ground, *u* and *ur* is the input thrust commanded and the generated real thrust, respectively. Notice, the third expression of equations set (4) is affected by the introduced model representation of the ground effect phenomenon, where it is evident that

$$
\mu\_r = \mu + \mu\_l \rho \left(\frac{r}{4z\_r}\right)^2 \tag{36}
$$

and referring to the above equation and using the real generated input thrust in the nominal mathematical model it yields the following

$$m\ddot{z} = \mu\_T \cos\theta \cos\phi - mg\tag{37}$$

or

$$m\ddot{z} = \mu \cos\theta \cos\phi + \mu\_r \rho \cos\theta \cos\phi \left(\frac{r}{4z\_r}\right)^2 - mg\tag{38}$$

Thereafter, without loss of generality

$$m\ddot{z} = \mu \cos\theta \cos\phi - mg + \lg\_z \tag{39}$$

with

$$\zeta\_z^\chi = \mu\_r \rho \cos \theta \cos \phi \left(\frac{r}{4z\_r}\right)^2 \tag{40}$$

where *ξ<sup>z</sup>* should be compensated by the adaptive robust motion control approach. In addition, the following data have been used during the simulation: *ρ* = 10, *r* = 0.1 m, and *zr* = 0.1 m.

On the other hand, in Figure 10 the quadrotor landing is illustrated. Here, it is used two different values for the learning rate and for the weighting vector for vertical motion **w***<sup>z</sup>* = [*w*1,*<sup>z</sup>* , *w*2,*<sup>z</sup>* , *w*3,*<sup>z</sup>* , *w*4,*z*], in order to illustrate two cases where the effect of increasing or decreasing the parameter values within the adaptive framework defines the quadrotor operation. Moreover, it is observed that a better tracking performance of the closed-loop system is achieved when a suitably selection of the parameters is done. In Table 3 are showcased the respective values for the aforementioned parameters in each case.

It is relevant to mention that in this experiment it is adopted the same set up outlined by expressions (33) and (34) defined in the previous section. Thus, as corroborated by the dynamic behavior of *γ<sup>z</sup>* in Figure 10, online computation of the control parameters is accomplished dynamically by the adaptive framework. From the same figure, it is also appreciated that the magnitude of the control effort is modified in function of the disturbance force exerted as a consequence of the ground effect. Nevertheless, a significant deviation of the actual motion from the planned reference is observed in the first case. In contrast, in the second case, acceptable attenuation levels of induced oscillations is attained by a proper selection of the parameters presented in Table 3.

**Figure 10.** Quadrotor autonomous landing under the ground effect phenomenon: (**a**) First case. (**b**) Second case.

The key for a successfully performance of the adaptive scheme depends on a properly selection of the adaptive parameters during the design process. Note that the selection of the initial weights within the offline training procedure, different operational conditions can be take into account for improving the initial system response, and, in this way, leading the quadrotor non-linear system to stable scenarios. In the next section, a different setup is introduced for selection of the control parameters: a desired Hurwitz polynomial where three parameters will be computed and a optimized selection by means of particle swarm theory.

**Table 3.** Parameters for the adaptive framework in experiment 2.

