*3.1. Dynamic Compensators for Robust Control Design*

In order to realize the stable control design, the quadrotor disturbed tracking error dynamics from Equations (4) and (5) are simplified as follows

$$\dot{\varepsilon}\_{\mu} = \upsilon\_{\mu} + \dot{\mathfrak{g}}\_{\mu}(t) \tag{9}$$

Moreover, *ξμ*(*t*) are assumed to be bounded time-varying disturbance signals locally approximated into a self-adaptive small interval of time around a given time instant *t*<sup>0</sup> > 0, say [*t*0, *t*<sup>0</sup> + *ε*], by *r*-th order Taylor polynomials as

$$\xi\_{\mu}^{x}(t) \approx \sum\_{n=0}^{r} \frac{\xi\_{\mu}^{(n)}(t\_0)}{n!} (t - t\_0)^n = \sum\_{n=0}^{r} \sigma\_{n,\mu}(t - t\_0)^n \tag{10}$$

where the superscript (*n*) stands for *n*-th order time derivative. Furthermore, to avoid velocity measurements, from Equation (9) structural estimates—known as integral reconstructors as well [26]—for time derivatives of velocity tracking errors are computed by

$$
\hat{\mathcal{E}}\_{\mu} = \int\_{t\_0}^{t} v\_{\mu} \, dt \tag{11}
$$

Here, initial conditions of the non-linear dynamic system, as well as the polynomial disturbance signal parameters are assumed to be completely unknown. Then, the polynomial relationship between integral reconstructors \$*e*˙*<sup>μ</sup>* and actual velocity tracking error signals *e*˙*μ* is given by

$$\dot{\mathfrak{e}}\_{\mu} = \widehat{\dot{\mathfrak{e}}}\_{\mu} + \sum\_{n=0}^{r+1} a\_{n,\mu} (t - t\_0)^n \tag{12}$$

where parameters *αn*,*<sup>μ</sup>* are assumed to be unknown as well.

In this fashion, the following family of controllers based on dynamic compensators to actively compensate polynomial disturbances can be synthesized as follows

$$
\omega\_{\mu} = -\beta\_{r+3,\mu}\dot{\varepsilon}\_{\mu} - \beta\_{r+2,\mu}\varepsilon\_{\mu} - \delta\_{r+1,\mu} \tag{13}
$$

with

$$\begin{aligned} \dot{\delta}\_0 &= \beta\_{0,\mu} \mathfrak{e}\_{\mu} \\ \dot{\delta}\_1 &= \delta\_{0,\mu} + \beta\_{1,\mu} \mathfrak{e}\_{\mu} \\ &\vdots \\ \delta\_r &= \delta\_{r-1,\mu} + \beta\_{r,\mu} \mathfrak{e}\_{\mu} \\ \delta\_{r+1,\mu} &= \delta\_{r,\mu} + \beta\_{r+1,\mu} \mathfrak{e}\_{\mu} \end{aligned} \tag{14}$$

Substitution of Equation (13) into Equation (9), closed-loop tracking error dynamics is then described by

$$
\mathfrak{e}\_{\mu}^{(r+4)} + \sum\_{n=0}^{r+3} \mathfrak{z}\_{n,\mu} \mathfrak{e}\_{\mu}^{(n)} = 0 \tag{15}
$$

Thus, closed-loop system stability criteria is fulfilled by selecting the control gains *βk*,*<sup>μ</sup>* for *k* = 0, 1, ... ,*r* + 3, such a way the characteristic polynomial of Equation (15) is stable (Hurwitz). By using the family of Hurwitz polynomials

$$P\_{\mathbb{C}L\_{\mu}}(s) = \left(s + \gamma\_{\mu}\right)^{r+4}, \quad \gamma\_{\mu} \gg 0 \tag{16}$$

the control design parameters can be then computed by

$$\beta\_{k,\mu} = \frac{(r+4)!}{k!(r+4-k)!} \gamma\_{\mu}^{r+4-k} \tag{17}$$

In the present study, three layers B-spline artificial neural networks and particle swarm optimization are properly implemented to compute adaptive control gains in order to avoid possible undesirable high-gain control effects. Furthermore, first order Taylor polynomial expansions for approximation of disturbance signals are selected. Nevertheless, higher order polynomial expansions can be also chosen for applications where a much better approximation of disturbances is demanded. In this work, it is shown that first order polynomial disturbance approximations yield an acceptable motion trajectory tracking performance under significantly perturbed operating conditions.

Thus, from Equation (10), Taylor polynomial expansions for disturbance signals are described in this work as

$$\mathcal{J}\_{\mu}(t) \approx \sigma\_{1,\mu} + \sigma\_{2,\mu}(t - t\_0) \tag{18}$$

where coefficients *σ*1,*<sup>μ</sup>* and *σ*2,*<sup>μ</sup>* are assumed to be uncertain. Moreover, the structural estimated variables and actual velocity tracking error signals are related by

$$
\dot{\varepsilon}\_{\mu} = \widehat{\dot{\varepsilon}}\_{\mu} + \kappa\_{0,\mu}(t - t\_0) + \kappa\_{1,\mu}(t - t\_0)^2 \tag{19}
$$

where parameters *αi*,*<sup>μ</sup>* are unknown.

In this sense, we proposed the following family of auxiliary controllers for robust quadrotor motion control

$$
\omega\_{\mu} = -\beta\_{4,\mu}\widehat{\dot{\epsilon}}\_{\mu} - \beta\_{3,\mu}\varepsilon\_{\mu} - \beta\_{2,\mu}\delta\_{1,\mu} - \beta\_{1,\mu}\delta\_{2,\mu} - \beta\_{0,\mu}\delta\_{3,\mu} \tag{20}
$$

with

$$\begin{aligned} \delta\_{1,\mu} &= \mathfrak{e}\_{\mu} \\ \dot{\delta}\_{2\mu\_{\prime}} &= \delta\_{1,\mu} \\ \dot{\delta}\_{3,\mu} &= \delta\_{2,\mu} \end{aligned} \tag{21}$$

Thence, from Equations (9) and (20) the closed-loop error dynamics is governed by

$$
\varepsilon^{(5)}\_{\mu} + \beta\_{4,\mu} \varepsilon^{(4)}\_{\mu} + \beta\_{3,\mu} \varepsilon^{(3)}\_{\mu} + \beta\_{2,\mu} \ddot{\epsilon} + \beta\_{1,\mu} \dot{\epsilon} + \beta\_{0,\mu} \varepsilon\_{\mu} = 0 \tag{22}
$$

The control gains *βk*,*<sup>μ</sup>* for *k* = 0, 1, ... , 4 should be properly selected in order to the associated characteristic polynomials

$$P\_{\mathbb{C}L\_{\mu}}(\mathbf{s}) = \mathbf{s}^{5} + \beta\_{4,\mu}\mathbf{s}^{4} + \beta\_{3,\mu}\mathbf{s}^{3} + \beta\_{2,\mu}\mathbf{s}^{2} + \beta\_{1,\mu}\mathbf{s} + \beta\_{0,\mu} \tag{23}$$

are Hurwitz polynomials. In this fashion, reference trajectory tracking can be achieved:

$$\lim\_{t \to \infty} e\_{\mu} = 0 \quad \Rightarrow \quad \lim\_{t \to \infty} \mu = \mu^{\star} \tag{24}$$

with *μ* and *μ* standing for the real and planned references for translational and rotational trajectories, respectively.

Notice from (5) that the rotational dynamic model can be also be expressed as follows:

$$\ddot{\boldsymbol{\eta}} = \mathbf{J}^{-1} \left( \boldsymbol{\pi}\_{\boldsymbol{\eta}} - \mathbf{C}(\dot{\boldsymbol{\eta}}, \boldsymbol{\eta}) \dot{\boldsymbol{\eta}} \right) + \mathbf{J}^{-1} \mathbf{\mathcal{J}}\_{\boldsymbol{\eta}} \tag{25}$$

which can be expressed matching the structure in (9). Therefore, from (21) it is observed that the synthetic controllers drive the system closed-loop dynamics. Finally, by analyzing the full non-linear dynamics, the control inputs nature and the robustness of the synthesized robust scheme, a suitable selection of the control inputs is given as follows

$$\begin{array}{rcl} \mu &=& \frac{1}{\cos\phi\cos\theta}(mv\_z + mg) \\ \tau\_\Psi &=& I\_z v\_\Psi \\ \tau\_\theta &=& I\_y v\_\theta \\ \tau\_\phi &=& I\_x v\_\phi \end{array} \tag{26}$$
