7. Nonlinear robust control of quadrotors

x t \_ðÞ¼ f tð Þþ ; x tð Þ Gxt ð Þ ð Þ u tð Þ, xtð Þ¼ <sup>0</sup> x0, t ≥ t0, (38)

<sup>0</sup> ; t<sup>0</sup> � �<sup>T</sup>

, <sup>f</sup> : ½ Þ� <sup>t</sup>0; <sup>∞</sup> <sup>D</sup> � <sup>R</sup><sup>4</sup> ! <sup>R</sup>4, <sup>D</sup> <sup>¼</sup> ½ Þ� � <sup>0</sup>; <sup>∞</sup> <sup>π</sup>

3 7

�1

, t ≥ 0, (39)

, and recall that the controllability

, <sup>~</sup><sup>x</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup> � ½ Þ <sup>t</sup>0; <sup>∞</sup> , (40)

<sup>5</sup>; (41)

<sup>2</sup> ; <sup>π</sup> 2 � ��

, <sup>t</sup> <sup>≥</sup> <sup>t</sup>0, <sup>u</sup>(t)<sup>∈</sup> <sup>R</sup><sup>m</sup> is continuous, and both <sup>f</sup> : [t0, <sup>∞</sup>) � <sup>R</sup>n! <sup>R</sup><sup>n</sup> and <sup>G</sup> : <sup>R</sup>n!R<sup>n</sup> � <sup>m</sup>

Definition 6.1 ([15]). Consider the nonlinear time-varying dynamical system (38), let M be a real analytic manifold of dimension n, and let y ∈M and t1, t<sup>2</sup> ≥ t0. The reachable set R(y, t1, t2) of (38) from (y, t1) at t<sup>2</sup> is the set of all states that can be reached at time t<sup>2</sup> by following the solution of (38) with initial condition y, initial time t1, and some continuous control input u(�). The nonlinear time-varying dynamical system (38) is strongly accessible at y ∈M at time t<sup>1</sup> if R(y, t1,t2) has a non-empty interior in M for every t<sup>2</sup> > t1. The nonlinear timevarying dynamical system (38) is strongly accessible on M if it is strongly accessible at every

In practice, Definition 6.1 states that if the nonlinear time-varying dynamical system (38) is strongly accessible on M, then for every point in the reachable set of (38), there exists a continuous control input such that the system's trajectory is contained both in the reachable set and the manifold M at all times. The next theorem provides sufficient conditions for the strong accessibility of the nonlinear dynamical system (38). For the statement of

, <sup>G</sup><sup>~</sup> ð Þ <sup>~</sup><sup>x</sup> <sup>≜</sup> <sup>G</sup><sup>T</sup>ð Þ<sup>x</sup> ; <sup>0</sup><sup>n</sup>�<sup>1</sup>

Theorem 6.1 ([15]). Consider the nonlinear dynamical system (38). If rank Cð Þ¼ ~x n for all

It follows from (21)–(23) and (25) that a quadrotor's altitude and orientation are captured by

04�<sup>1</sup> 04�<sup>3</sup> cos ϕ cos θ 01�<sup>3</sup>

the explicit expression for f(�, �) is omitted for brevity. In this case, the controllability matrix Cð Þ�

of the fully actuated, augmented time-invariant dynamical system (39) is such that

<sup>03</sup>�<sup>1</sup> <sup>m</sup>Q<sup>Γ</sup> <sup>ϕ</sup>; <sup>θ</sup> � �<sup>I</sup>

� �<sup>T</sup>

this result, consider the augmented time-invariant dynamical system

Cð Þ ~x ≜ ~g1ð Þ ~x ;…; ~gmð Þ ~x ; ad<sup>~</sup><sup>f</sup> ~g1ð Þ ~x ;…; ad<sup>~</sup><sup>f</sup> ~gmð Þ ~x h i

<sup>T</sup>ð Þ<sup>x</sup> ; <sup>1</sup> � �<sup>T</sup>

x tðÞ¼ <sup>~</sup><sup>f</sup> ð Þþ <sup>~</sup>x tð Þ <sup>G</sup><sup>~</sup> ð Þ <sup>~</sup>x tð Þ u tð Þ, <sup>~</sup>xð Þ¼ <sup>0</sup> <sup>x</sup><sup>T</sup>

matrix of the augmented time-invariant dynamical system (39) is defined as [15]

where x(t) ∈ R<sup>n</sup>

90 Adaptive Robust Control Systems

are continuously differentiable.

y∈M and every t<sup>1</sup> ≥ t0.

where ~x ≜ x<sup>T</sup>; t � �<sup>T</sup>

� π <sup>2</sup> ; <sup>π</sup> 2

~\_

where <sup>G</sup><sup>~</sup> ð Þ¼ <sup>~</sup><sup>x</sup> <sup>~</sup>g1ð Þ <sup>~</sup><sup>x</sup> ;…~gmð Þ <sup>~</sup><sup>x</sup> � �.

� � � ½ Þ� <sup>0</sup>; <sup>2</sup><sup>π</sup> <sup>R</sup> � <sup>R</sup><sup>3</sup>

, <sup>~</sup><sup>f</sup> ð Þ <sup>~</sup><sup>x</sup> <sup>≜</sup> <sup>f</sup>

~x ∈M � ½ Þ t0; ∞ , then (38) is strongly accessible.

(38) with <sup>n</sup> = 8, <sup>m</sup> = 4, <sup>x</sup> <sup>¼</sup> rZ; <sup>ϕ</sup>; <sup>θ</sup>;ψ;r\_Z; <sup>ω</sup><sup>T</sup> � �<sup>T</sup>

, and

G xð Þ¼ <sup>m</sup>�<sup>1</sup> Q

2 6 4 In this section, we apply the results presented in Sections 3–6 to design control laws so that a quadrotor can follow a given trajectory with bounded error. Specifically, we design a control law for u(�) so that a quadrotor can track both the given reference trajectory [rX, ref(t),rY, ref(t), rZ, ref(t)]<sup>T</sup> , t ≥ t0, and the reference yaw angle ψref(t). In practice, we design control laws both for the virtual control input [uX(t), uY(t), uZ(t)]<sup>T</sup> , t ≥ t0, and the moment of the propellers' thrust [u2(t), u3(t), u4(t)]<sup>T</sup> , so that a quadrotor tracks [rX, ref(t),rY, ref(t),rZ, ref(t)]<sup>T</sup> , the reference roll angle (35), the reference pitch angle (36), and the reference yaw angle ψref(t).

It follows form (33) that if the aerodynamic force is modeled as in (31), then a quadrotor's translational kinematic and dynamic equations are given by

$$
\dot{\mathbf{x}}\_{\mathbf{p},\mathbf{P}}(t) = A\_{\mathbf{p},\mathbf{P}} \mathbf{x}\_{\mathbf{p},\mathbf{P}}(t) + B\_{\mathbf{p},\mathbf{P}} \Lambda\_{\mathbf{P}} \left( \begin{bmatrix} \boldsymbol{\mu}\_{\mathbf{X}}(t) \\\\ \boldsymbol{\mu}\_{\mathbf{Y}}(t) \\\\ \boldsymbol{\mu}\_{\mathbf{Z}}(t) \end{bmatrix} + \Theta\_{\mathbf{P}}^{\mathrm{T}} \Phi(\mathbf{x}\_{\mathbf{p},\mathbf{P}}(t)) \right) + \widehat{\xi}\_{\mathbf{P}}(t),
$$

$$
\mathbf{x}\_{\mathbf{p},\mathbf{P}}(t\_0) = \left[ r\_{\mathbf{A},0}^{\mathrm{T}}, \left( \boldsymbol{\sigma}\_{\mathbf{A},0}^{\mathrm{T}} \right)^{\mathrm{T}} \right]^{\mathrm{T}}, t \ge t\_0. \tag{43}
$$

$$
\dot{y}\_{\text{P}}(t) = \varepsilon \mathbb{C}\_{\text{P},\text{P}} \mathbf{x}\_{\text{P},\text{P}}(t) - \varepsilon y\_{\text{P}}(t), \qquad y\_{\text{P}}(t\_0) = \mathbb{C}\_{\text{P},\text{P}} \mathbf{x}\_{\text{P},\text{P}}(t\_0). \tag{44}
$$

where xp,PðÞ¼ t r<sup>I</sup> <sup>A</sup>ð Þ<sup>t</sup> � �<sup>T</sup> ; r\_ I <sup>A</sup>ð Þ<sup>t</sup> � �<sup>T</sup> h i<sup>T</sup> , <sup>t</sup> <sup>≥</sup> <sup>t</sup>0, <sup>A</sup>p,<sup>P</sup> <sup>¼</sup> <sup>03</sup>�<sup>3</sup> <sup>1</sup><sup>3</sup> 03�<sup>3</sup> 03�<sup>3</sup> � �, <sup>B</sup>p,<sup>P</sup> <sup>¼</sup> <sup>03</sup>�<sup>3</sup> 13 � �, <sup>Λ</sup><sup>P</sup> <sup>¼</sup> <sup>m</sup>�<sup>1</sup> <sup>Q</sup> 13, <sup>C</sup>p, P = [13, 03 � 3], <sup>b</sup>ξPð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>6</sup> , ½ � 13; 03�<sup>3</sup> bξPðÞ¼ t 03�3, and

$$
\begin{split}
\left[\mathbf{0}\_{3\times 3}, \mathbf{1}\_3\right] \widehat{\boldsymbol{\xi}}\_{\mathcal{P}}(t) &= \boldsymbol{m}\_{\mathcal{Q}}^{-1} \left\| \boldsymbol{v}\_{\mathcal{W}}^{\mathbb{I}}(t) - \boldsymbol{\mathring{r}}\_{A}^{\mathbb{I}}(t) \right\| \boldsymbol{B}\_{\mathcal{P},\mathcal{P}} \mathcal{R}\{\phi(t), \boldsymbol{\theta}(t), \boldsymbol{\psi}(t)\} \\\\
&\cdot \boldsymbol{\mathcal{K}}\_{\mathcal{F}} \boldsymbol{\mathcal{R}}^{\mathbb{T}}(\phi(t), \boldsymbol{\theta}(t), \boldsymbol{\psi}(t)) \left[ \boldsymbol{v}\_{\mathcal{W}}^{\mathbb{I}}(t) - \boldsymbol{\mathring{r}}\_{A}^{\mathbb{I}}(t) \right] + \boldsymbol{\mu}^{\mathbb{I}}(t) - \boldsymbol{\mathring{r}}\_{\mathcal{C}}^{\mathbb{I}}(t).
\end{split}
\tag{45}
$$

Although Θ<sup>T</sup> <sup>P</sup><sup>Φ</sup> <sup>x</sup>p,Pð Þ� � � follows neither from (33) nor (28), this nonlinear term has been introduced to account for failures of the control system; in this section, we assume that

$$\Phi(z) = \tanh z, \qquad z \in \mathbb{R}^n,\tag{46}$$

y\_

system and erroneous modeling assumptions. Similarly, the term Θ<sup>T</sup>

<sup>u</sup>4(�)]<sup>T</sup> so that the measured output signal <sup>y</sup>P(�) tracks the reference signal

and the measured output signal yA(�) tracks the reference signal

<sup>p</sup>,P; x<sup>T</sup> p,A h i<sup>T</sup>

cmd,P; y<sup>T</sup>

� �, <sup>Θ</sup> <sup>¼</sup> <sup>Θ</sup><sup>P</sup> <sup>06</sup>�<sup>3</sup>

cmd,A h i<sup>T</sup>

, B<sup>p</sup> <sup>¼</sup> <sup>B</sup>p,<sup>P</sup> <sup>06</sup>�<sup>3</sup>

, <sup>y</sup>cmd <sup>¼</sup> <sup>y</sup><sup>T</sup>

, Bp,<sup>A</sup> ¼

where

ϕ\_ 0; θ\_ <sup>0</sup>;ψ\_ 0 � �<sup>T</sup> <sup>¼</sup> <sup>Γ</sup> <sup>ϕ</sup>0; <sup>θ</sup><sup>0</sup>

Ap,<sup>A</sup> ¼

(5) and (6) with <sup>x</sup><sup>p</sup> <sup>¼</sup> <sup>x</sup><sup>T</sup>

] T

, and

<sup>A</sup><sup>p</sup> <sup>¼</sup> <sup>A</sup>p,<sup>P</sup> <sup>06</sup>�<sup>6</sup> 06�<sup>6</sup> Ap,<sup>A</sup> " #

<sup>Λ</sup> <sup>¼</sup> <sup>Λ</sup><sup>P</sup> <sup>03</sup>�<sup>3</sup> 03�<sup>3</sup> Λ<sup>A</sup>

u = [uX, uY, uZ, v<sup>T</sup>

bξ ¼ ξ bT <sup>P</sup>; ξ bT A h i<sup>T</sup> <sup>A</sup>ðÞ¼ t εCp,Axp,Að Þ� t εyAð Þt , yAð Þ¼ t<sup>0</sup> Cp,Axp,Að Þ t<sup>0</sup> , (51)

, Cp,<sup>A</sup> ¼

http://dx.doi.org/10.5772/intechopen.70723

<sup>A</sup><sup>Φ</sup> <sup>x</sup>p,Að Þ� � � has been intro-

, t ≥ t0, (53)

, (54)

� � � <sup>R</sup> � ½ Þ� <sup>0</sup>; <sup>2</sup><sup>π</sup> <sup>R</sup>,

,

, n<sup>p</sup> = 12, m= 6,

<sup>2</sup> ; <sup>π</sup> 2

03�<sup>6</sup> Cp,<sup>A</sup> " #

> � � Φ xp,<sup>A</sup> � � " #:

cmd, 2,A h i<sup>T</sup>

, C<sup>p</sup> <sup>¼</sup> <sup>C</sup>p,P <sup>03</sup>�<sup>6</sup>

� � <sup>¼</sup> <sup>Φ</sup> <sup>x</sup>p,<sup>P</sup>

T

, (52)

93

Robust Adaptive Output Tracking for Quadrotor Helicopters

� �ω0, <sup>Λ</sup> <sup>∈</sup> <sup>R</sup><sup>3</sup> � <sup>3</sup> is diagonal positive-definite, and <sup>Φ</sup>(�) given by (46).

Although Λ<sup>A</sup> = 1<sup>3</sup> [35], we assume that Λ<sup>A</sup> is unknown and accounts for failures of the propulsion

duced to capture matched uncertainties. Since any quadrotor's angular velocity, angular acceleration, and propeller's spin rate are bounded, it follows from (48) that also the unmatched

The next theorem provides feedback control laws both for [uX(�), uY(�), uZ(�)]<sup>T</sup> and [u2(�), <sup>u</sup>3(�),

<sup>y</sup>cmd,AðÞ¼ <sup>t</sup> <sup>ϕ</sup>refð Þ<sup>t</sup> ; <sup>θ</sup>refð Þ<sup>t</sup> ; <sup>ψ</sup>refð Þ<sup>t</sup> � �<sup>T</sup>

where ϕref(�) and θref(�) are given by (35) and (36), respectively, with some bounded error despite model uncertainties, external disturbances, and failures of the propulsion system. For the statement of this result, consider both the nonlinear dynamical system given by (43) and (44) and the nonlinear dynamical system given by (50) and (51), and note that these systems are equivalent to

, <sup>D</sup><sup>p</sup> <sup>¼</sup> <sup>R</sup><sup>3</sup> � <sup>R</sup><sup>3</sup> � � <sup>π</sup>

<sup>2</sup> ; <sup>π</sup> 2 � � � <sup>R</sup> � � <sup>π</sup>

cmd, <sup>2</sup>,P; y<sup>T</sup>

, <sup>y</sup>cmd, <sup>2</sup> <sup>¼</sup> <sup>y</sup><sup>T</sup>

06�<sup>3</sup> Bp,<sup>A</sup> " #

� �, <sup>Φ</sup> <sup>x</sup><sup>p</sup>

06�<sup>3</sup> Θ<sup>A</sup>

uncertainty bξAð Þ� is bounded. Eq. (51) captures the plant sensor's dynamics ([20], Ch. 2).

<sup>y</sup>cmd,PðÞ¼ <sup>t</sup> ½ � rX,refð Þ<sup>t</sup> ;rY,refð Þ<sup>t</sup> ;rZ,refð Þ<sup>t</sup> <sup>T</sup>

which is globally Lipschitz continuous. Since any quadrotor's velocity and acceleration are bounded, it follows from (45) that also the unmatched uncertainty bξPð Þ� is bounded. Eq. (44) captures the plant sensor's dynamics ([20], Ch. 2).

It follows from (22) and (25) that a quadrotor's rotational dynamics is captured by

$$
\begin{bmatrix}
\dot{\phi}(t) \\
\dot{\theta}(t) \\
\dot{\psi}(t) \\
\dot{\omega}(t)
\end{bmatrix} = f(\phi(t), \theta(t), \psi(t), \omega(t)) + \begin{bmatrix} 0\_{3 \times 3} \\
\hat{I}^{-1} \end{bmatrix} \begin{bmatrix} \nu\_2(t) \\
\nu\_3(t) \\
\mu\_4(t) \end{bmatrix} + \hat{\xi}\_A(t),
$$

$$
\begin{bmatrix}
\phi(t\_0), \theta(t\_0), \psi(t\_0), \omega^\mathrm{T}(t\_0) \end{bmatrix}^\mathrm{T} = \begin{bmatrix}
\phi\_0, \theta\_0, \psi\_0, \omega\_0^\mathrm{T} \end{bmatrix}^\mathrm{T}, \qquad t \ge t\_0. \tag{47}
$$

$$\begin{split} \text{where} & f(\boldsymbol{\phi}, \boldsymbol{\theta}, \boldsymbol{\psi}, \boldsymbol{\omega}) = \left[ \boldsymbol{\omega}^{\text{T}} \Gamma^{\text{T}} (\boldsymbol{\phi}, \boldsymbol{\theta}), \left( -\widehat{\boldsymbol{I}}^{-1} \boldsymbol{\omega}^{\times} \widehat{\boldsymbol{I}} \boldsymbol{\omega} \right)^{\text{T}} \right]^{\text{T}}, \widehat{\boldsymbol{\xi}}\_{\text{A}}(t) \in \mathbb{R}^{6}, \left[ \mathbf{1}\_{3}, \mathbf{0}\_{3\times 3} \right] \widehat{\boldsymbol{\xi}}\_{\text{A}}(t) = \mathbf{0}\_{3} \\ & \left[ \boldsymbol{\Omega}\_{3\times 3}, \mathbf{1}\_{3} \right] \widehat{\boldsymbol{\xi}}\_{\text{A}}(t) = \boldsymbol{I}^{-1} \boldsymbol{r}\_{\text{C}}^{\text{T}} \left[ F\_{\boldsymbol{\Theta}} \{ \boldsymbol{\phi}(t), \boldsymbol{\theta}(t) \} - \boldsymbol{m}\_{\text{O}} \boldsymbol{\tilde{r}}\_{\text{A}}^{\text{I}}(t) \right] + \boldsymbol{I}^{-1} \boldsymbol{M} (\boldsymbol{\omega}(t)) + \left[ \widehat{\boldsymbol{I}}^{-1} \boldsymbol{\omega}^{\times} (t) \widehat{\boldsymbol{I}} - \boldsymbol{I}^{-1} \boldsymbol{\omega}^{\times}(t) \boldsymbol{I} \right] \boldsymbol{\omega}(t) + \\ & \left( \boldsymbol{I} - \widehat{\boldsymbol{I}} \right) [ \boldsymbol{u}\_{2}(t), \boldsymbol{u}\_{3}(t), \boldsymbol{u}\_{4}(t) ]^{\text{T}} - \boldsymbol{I}^{-1} \boldsymbol{I}\_{\text{P}} \sum\_{i=1}^{4} \quad \left[ \boldsymbol{0} \\ & \boldsymbol{\Omega}\_{\text{P},i}(t) = \boldsymbol{\Omega}\_{$$

<sup>F</sup>g(�, �) is given by (24), <sup>r</sup>A(�) verifies (43), and <sup>M</sup>(�) is given by (27). Let <sup>x</sup>p,<sup>A</sup> <sup>¼</sup> <sup>ϕ</sup>; <sup>ϕ</sup>\_ ; <sup>θ</sup>; <sup>θ</sup>\_ ;ψ;ψ\_ � �<sup>T</sup> , η xp,<sup>A</sup> � � <sup>¼</sup> <sup>ϕ</sup>\_ ; <sup>θ</sup>\_ ;ψ\_ � �<sup>T</sup> , β xp,<sup>A</sup> � � <sup>¼</sup> <sup>L</sup><sup>2</sup> <sup>f</sup> ϕ; L<sup>2</sup> <sup>f</sup> θ; L<sup>2</sup> <sup>f</sup> ψ; h i<sup>T</sup> , and v∈ R<sup>3</sup> ; the explicit expression of β(�) is omitted for brevity. By proceeding as in Example 6.3 of [35], one can prove that the nonlinear dynamical system (47) is feedback linearizable ([31], Ch. 5). Specifically, (47) with

$$\begin{aligned} \left[\boldsymbol{\mu}\_{2}, \boldsymbol{\mu}\_{3}, \boldsymbol{\mu}\_{4}\right]^{\mathrm{T}} &= \widehat{\boldsymbol{I}} \Gamma^{-1} \left(\boldsymbol{\phi}, \boldsymbol{\Theta}\right) \left[\boldsymbol{\eta}\left(\mathbf{x}\_{\mathrm{p}, \mathrm{A}}\right) - \boldsymbol{\beta}\left(\mathbf{x}\_{\mathrm{p}, \mathrm{A}}\right) + \boldsymbol{v}\right], \\\\ \left(\mathbf{x}\_{\mathrm{p}, \mathrm{A}}, \boldsymbol{v}\right) &\in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \times \mathbb{R} \times \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \times \mathbb{R} \times \left[0, 2\pi\right) \times \mathbb{R} \times \mathbb{R}^{3}, \end{aligned} \tag{49}$$

is equivalent to

$$\dot{\mathbf{x}}\_{\mathbf{p},\mathbf{A}}(t) = A\_{\mathbf{p},\mathbf{A}} \mathbf{x}\_{\mathbf{p},\mathbf{A}}(t) + B\_{\mathbf{p},\mathbf{A}} \Lambda\_{\mathbf{A}} \left[ \mathbf{v}(t) + \Theta\_{\mathbf{A}}^{\mathrm{T}} \Phi(\mathbf{x}\_{\mathbf{p},\mathbf{A}}(t)) \right] + \widehat{\xi}\_{\mathbf{A}}(t),$$

$$\mathbf{x}\_{\mathbf{p},\mathbf{A}}(t\_0) = \left[ \phi\_0, \dot{\phi}\_0, \theta\_0, \dot{\theta}\_0, \psi\_0, \dot{\psi}\_0 \right]^{\mathrm{T}}, \qquad t \ge t\_{0\prime} \tag{50}$$

Robust Adaptive Output Tracking for Quadrotor Helicopters http://dx.doi.org/10.5772/intechopen.70723 93

$$
\dot{y}\_A(t) = \varepsilon \mathbb{C}\_{\mathbb{P}, \mathcal{A}} \mathbf{x}\_{\mathbb{P}, \mathcal{A}}(t) - \varepsilon y\_A(t), \qquad y\_A(t\_0) = \mathbb{C}\_{\mathbb{P}, \mathcal{A}} \mathbf{x}\_{\mathbb{P}, \mathcal{A}}(t\_0). \tag{51}
$$

where

Although Θ<sup>T</sup>

92 Adaptive Robust Control Systems

<sup>P</sup><sup>Φ</sup> <sup>x</sup>p,Pð Þ� � � follows neither from (33) nor (28), this nonlinear term has been intro-

03�<sup>3</sup> bI �1

� �<sup>T</sup>

, <sup>b</sup>ξAð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>6</sup>

3 7 <sup>5</sup> � <sup>I</sup> �1

, and v∈ R<sup>3</sup>

� � � <sup>β</sup> <sup>x</sup>p,<sup>A</sup> � � <sup>þ</sup> <sup>v</sup> � �,

Mð Þþ ωð Þt bI

" # u2ð Þt

2 6 4

u3ð Þt u4ð Þt

0

3 7 <sup>5</sup> <sup>þ</sup> <sup>b</sup>ξAð Þ<sup>t</sup> ,

, t ≥ t0, (47)

�1 ω�ð Þt I

> 3 7 <sup>5</sup>; (48)

, (49)

ωð Þþt

,

, ½ � 13; 03�<sup>3</sup> bξAðÞ¼ t 03,

ω�ð Þt bI � I

X 4

i¼1

h i

0 0 ΩP,ið Þt

; the explicit expression of β(�) is

, t ≥ t0, (50)

2 6 4

�1

ω�ð Þt I<sup>P</sup>

� <sup>R</sup> � ½ Þ� <sup>0</sup>; <sup>2</sup><sup>π</sup> <sup>R</sup> � <sup>R</sup><sup>3</sup>

<sup>A</sup><sup>Φ</sup> <sup>x</sup>p,Að Þ<sup>t</sup> � � � � <sup>þ</sup> <sup>b</sup>ξAð Þ<sup>t</sup> ,

<sup>Φ</sup>ð Þ¼ <sup>z</sup> tanh z, z<sup>∈</sup> <sup>R</sup>n, (46)

duced to account for failures of the control system; in this section, we assume that

It follows from (22) and (25) that a quadrotor's rotational dynamics is captured by

� �<sup>T</sup> <sup>¼</sup> <sup>ϕ</sup>0; <sup>θ</sup>0;ψ0; <sup>ω</sup><sup>T</sup>

�1 ω�bIω

�1 IP X 4

<sup>f</sup> ϕ; L<sup>2</sup> <sup>f</sup> θ; L<sup>2</sup> <sup>f</sup> ψ;

h i<sup>T</sup>

dynamical system (47) is feedback linearizable ([31], Ch. 5). Specifically, (47) with

<sup>T</sup> <sup>¼</sup> <sup>b</sup>IΓ�<sup>1</sup> <sup>ϕ</sup>; <sup>θ</sup> � � <sup>η</sup> <sup>x</sup>p,<sup>A</sup>

� R � �

<sup>0</sup>; <sup>θ</sup>0; <sup>θ</sup>\_

� �<sup>T</sup>

<sup>0</sup>; <sup>ψ</sup>0;ψ\_ 0

€I <sup>A</sup>ð Þt

i¼1

þ I �1

0 0 Ω̇ P,i ð Þt

2 6 4

<sup>F</sup>g(�, �) is given by (24), <sup>r</sup>A(�) verifies (43), and <sup>M</sup>(�) is given by (27). Let <sup>x</sup>p,<sup>A</sup> <sup>¼</sup> <sup>ϕ</sup>; <sup>ϕ</sup>\_ ; <sup>θ</sup>; <sup>θ</sup>\_ ;ψ;ψ\_ � �<sup>T</sup>

omitted for brevity. By proceeding as in Example 6.3 of [35], one can prove that the nonlinear

π 2 ; π 2 � �

� �<sup>T</sup> � �<sup>T</sup>

<sup>¼</sup> <sup>f</sup> <sup>ϕ</sup>ð Þ<sup>t</sup> ; <sup>θ</sup>ð Þ<sup>t</sup> ;ψð Þ<sup>t</sup> ; <sup>ω</sup>ð Þ<sup>t</sup> � � <sup>þ</sup>

<sup>ϕ</sup>ð Þ <sup>t</sup><sup>0</sup> ; <sup>θ</sup>ð Þ <sup>t</sup><sup>0</sup> ;ψð Þ <sup>t</sup><sup>0</sup> ; <sup>ω</sup><sup>T</sup>ð Þ <sup>t</sup><sup>0</sup>

<sup>C</sup> <sup>F</sup><sup>g</sup> <sup>ϕ</sup>ð Þ<sup>t</sup> ; <sup>θ</sup>ð Þ<sup>t</sup> � � � <sup>m</sup>Q<sup>r</sup>

½ � <sup>u</sup>2ð Þ<sup>t</sup> ; <sup>u</sup>3ð Þ<sup>t</sup> ; <sup>u</sup>4ð Þ<sup>t</sup> <sup>T</sup> � <sup>I</sup>

, β xp,<sup>A</sup>

� � <sup>¼</sup> <sup>L</sup><sup>2</sup>

½ � u2; u3; u<sup>4</sup>

2 ; π 2 � �

<sup>x</sup>\_p,AðÞ¼ <sup>t</sup> <sup>A</sup>p,Axp,Að Þþ <sup>t</sup> <sup>B</sup>p,AΛ<sup>A</sup> v tðÞþ <sup>Θ</sup><sup>T</sup>

<sup>x</sup>p,Að Þ¼ <sup>t</sup><sup>0</sup> <sup>ϕ</sup>0; <sup>ϕ</sup>\_

<sup>x</sup>p,A; <sup>v</sup> � �<sup>∈</sup> � <sup>π</sup>

h i

captures the plant sensor's dynamics ([20], Ch. 2).

<sup>ϕ</sup>\_ð Þ<sup>t</sup> <sup>θ</sup>\_ð Þ<sup>t</sup> <sup>ψ</sup>\_ð Þ<sup>t</sup> ω\_ ð Þt

where <sup>f</sup> <sup>ϕ</sup>; <sup>θ</sup>;ψ; <sup>ω</sup> � � <sup>¼</sup> <sup>ω</sup><sup>T</sup>Γ<sup>T</sup> <sup>ϕ</sup>; <sup>θ</sup> � �; �b<sup>I</sup>

�1 r �

½ � 03�<sup>3</sup>; 1<sup>3</sup> bξAðÞ¼ t I

η xp,<sup>A</sup>

I �bI � �

� � <sup>¼</sup> <sup>ϕ</sup>\_ ; <sup>θ</sup>\_ ;ψ\_ � �<sup>T</sup>

is equivalent to

which is globally Lipschitz continuous. Since any quadrotor's velocity and acceleration are bounded, it follows from (45) that also the unmatched uncertainty bξPð Þ� is bounded. Eq. (44)

$$A\_{\mathrm{p},\mathrm{A}} = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}, \qquad B\_{\mathrm{p},\mathrm{A}} = \begin{bmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \qquad \mathsf{C}\_{\mathrm{p},\mathrm{A}} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}, \tag{52}$$

ϕ\_ 0; θ\_ <sup>0</sup>;ψ\_ 0 � �<sup>T</sup> <sup>¼</sup> <sup>Γ</sup> <sup>ϕ</sup>0; <sup>θ</sup><sup>0</sup> � �ω0, <sup>Λ</sup> <sup>∈</sup> <sup>R</sup><sup>3</sup> � <sup>3</sup> is diagonal positive-definite, and <sup>Φ</sup>(�) given by (46). Although Λ<sup>A</sup> = 1<sup>3</sup> [35], we assume that Λ<sup>A</sup> is unknown and accounts for failures of the propulsion system and erroneous modeling assumptions. Similarly, the term Θ<sup>T</sup> <sup>A</sup><sup>Φ</sup> <sup>x</sup>p,Að Þ� � � has been introduced to capture matched uncertainties. Since any quadrotor's angular velocity, angular acceleration, and propeller's spin rate are bounded, it follows from (48) that also the unmatched uncertainty bξAð Þ� is bounded. Eq. (51) captures the plant sensor's dynamics ([20], Ch. 2).

The next theorem provides feedback control laws both for [uX(�), uY(�), uZ(�)]<sup>T</sup> and [u2(�), <sup>u</sup>3(�), <sup>u</sup>4(�)]<sup>T</sup> so that the measured output signal <sup>y</sup>P(�) tracks the reference signal

$$\mathcal{Y}\_{\rm cmd,P}(t) = \left[r\_{\rm X,ref}(t), r\_{\rm Y,ref}(t), r\_{\rm Z,ref}(t)\right]^\mathrm{T}, \qquad t \ge t\_0. \tag{53}$$

and the measured output signal yA(�) tracks the reference signal

$$y\_{\rm cmd,A}(t) = \left[\phi\_{\rm ref}(t), \theta\_{\rm ref}(t), \psi\_{\rm ref}(t)\right]^{\rm T},\tag{54}$$

where ϕref(�) and θref(�) are given by (35) and (36), respectively, with some bounded error despite model uncertainties, external disturbances, and failures of the propulsion system. For the statement of this result, consider both the nonlinear dynamical system given by (43) and (44) and the nonlinear dynamical system given by (50) and (51), and note that these systems are equivalent to (5) and (6) with <sup>x</sup><sup>p</sup> <sup>¼</sup> <sup>x</sup><sup>T</sup> <sup>p</sup>,P; x<sup>T</sup> p,A h i<sup>T</sup> , <sup>D</sup><sup>p</sup> <sup>¼</sup> <sup>R</sup><sup>3</sup> � <sup>R</sup><sup>3</sup> � � <sup>π</sup> <sup>2</sup> ; <sup>π</sup> 2 � � � <sup>R</sup> � � <sup>π</sup> <sup>2</sup> ; <sup>π</sup> 2 � � � <sup>R</sup> � ½ Þ� <sup>0</sup>; <sup>2</sup><sup>π</sup> <sup>R</sup>, u = [uX, uY, uZ, v<sup>T</sup> ] T , <sup>y</sup>cmd <sup>¼</sup> <sup>y</sup><sup>T</sup> cmd,P; y<sup>T</sup> cmd,A h i<sup>T</sup> , <sup>y</sup>cmd, <sup>2</sup> <sup>¼</sup> <sup>y</sup><sup>T</sup> cmd, <sup>2</sup>,P; y<sup>T</sup> cmd, 2,A h i<sup>T</sup> , n<sup>p</sup> = 12, m= 6, bξ ¼ ξ bT <sup>P</sup>; ξ bT A h i<sup>T</sup> , and <sup>A</sup><sup>p</sup> <sup>¼</sup> <sup>A</sup>p,<sup>P</sup> <sup>06</sup>�<sup>6</sup> 06�<sup>6</sup> Ap,<sup>A</sup> " #, B<sup>p</sup> <sup>¼</sup> <sup>B</sup>p,<sup>P</sup> <sup>06</sup>�<sup>3</sup> 06�<sup>3</sup> Bp,<sup>A</sup> " #, C<sup>p</sup> <sup>¼</sup> <sup>C</sup>p,P <sup>03</sup>�<sup>6</sup> 03�<sup>6</sup> Cp,<sup>A</sup> " #, <sup>Λ</sup> <sup>¼</sup> <sup>Λ</sup><sup>P</sup> <sup>03</sup>�<sup>3</sup> 03�<sup>3</sup> Λ<sup>A</sup> � �, <sup>Θ</sup> <sup>¼</sup> <sup>Θ</sup><sup>P</sup> <sup>06</sup>�<sup>3</sup> 06�<sup>3</sup> Θ<sup>A</sup> � �, <sup>Φ</sup> <sup>x</sup><sup>p</sup> � � <sup>¼</sup> <sup>Φ</sup> <sup>x</sup>p,<sup>P</sup> � � Φ xp,<sup>A</sup> � � " #:

Theorem 7.1 Consider the nonlinear dynamical system given by (43) and (44), the nonlinear dynamical system given by (50) and (51), the reference signals (53) and (54), the augmented dynamical system (8), the reference dynamical model (9), the feedback control law γ(�, � , �) given by (10), and the adaptation laws (11)–(13). If there exist Kx∈ R<sup>18</sup> � <sup>6</sup> and Kcmd∈ R<sup>6</sup> � <sup>6</sup> such that (15) and (16) and satisfied, then (8) with u = γ(t, xp, x) is uniformly ultimately bounded. Furthermore, there exist b > 0 and c > 0 independent of t0, and for every a∈(0, c), there exists a finite-time T = T(a, c) ≥ 0, independent of t0, such that if ∥yP(t0) � ycmd, P(t0) ∥ ≤ a and ∥yA(t0) � ycmd, A(t0) ∥ ≤ a, then

$$\|y\_{\mathbf{P}}(t) - y\_{\mathbf{cmd},\mathbf{P}}(t)\| \le b, \qquad t \ge t\_0 + T,\tag{55}$$

$$\|y\_{\mathcal{A}}(t) - y\_{\text{cmd},\mathcal{A}}(t)\| \le b. \tag{56}$$

Consider a quadrotor of mass m<sup>Q</sup> = 1 kg and matrix of inertia I = 1<sup>3</sup> kg m2

excessive for quadrotors equipped with conventional autopilots.

2 6 4

sor's dynamics be characterized by ε = 10. We assume that the vehicle's mass and matrix of inertia are unknown and estimated to be <sup>m</sup><sup>b</sup> <sup>Q</sup> <sup>¼</sup> <sup>1</sup>:25 kg and <sup>b</sup><sup>I</sup> <sup>¼</sup> <sup>0</sup>:<sup>8</sup> � <sup>1</sup><sup>3</sup> kg m2, respectively. Moreover, we assume that the aerodynamic force (31) and the aerodynamic moment (27) are characterized by KF =KM = 0.01 � 13, which we assume unknown, and the wind velocity is given

Figure 3 shows the quadrotor's trajectory obtained applying the control laws (57) and (58) to track a circular path of radius 0.3 m at an altitude of 0.75 m despite the fact that the quadrotor's payload of 0.5 kg is dropped at t ≥ 40 s and one of the motors is turned off at t = 90 s. These results have been obtained by setting σ<sup>1</sup> = σ<sup>2</sup> = σ<sup>3</sup> = 2, Γcmd = 100 � 16, and Γ<sup>x</sup> and ΓΘ as blockdiagonal matrices, whose non-zero blocks are Γx, (1, 1) = 1000 � 19, Γx, (2, 2) = 2000 � 19, ΓΘ, (1, 1) = 200 � 19,

Figure 3. Reference trajectory and trajectory followed by the quadrotor implementing the proposed control algorithm.

The vehicle is disturbed by some wind constantly blowing at 16 m/s.

0:025 0 0 0 0:025 0 000:05

m=s, t ≥ t0; it is worthwhile to note that this wind speed is considered as

3 7

Robust Adaptive Output Tracking for Quadrotor Helicopters

http://dx.doi.org/10.5772/intechopen.70723

characterized by the matrix of inertia I<sup>p</sup> ¼

by v<sup>I</sup>

<sup>W</sup>ðÞ¼ <sup>t</sup> ½ � <sup>16</sup>; <sup>0</sup>; <sup>0</sup> <sup>T</sup>

abd ΓΘ, (2, 2) = 1600 � 19.

, let the propellers be

95

5kg m2, and let that the sen-

Lastly, the thrust force generated by the quadrotor's propellers is such that

$$\mu\_1(t) = \hat{m}\_{\mathcal{Q}} \left\| \left[ \gamma\_{\mathcal{P}}(t, \mathbf{x}\_{\mathcal{P}}(t), \mathbf{x}(t)) \right] \right\|, \qquad t \ge t\_{0\prime} \tag{57}$$

and the moment of the thrust force generated by the quadrotor's propellers is given by

$$\begin{bmatrix} u\_2(t) \\ u\_3(t) \\ u\_4(t) \end{bmatrix} = \hat{\boldsymbol{I}} \Gamma^{-1} \{ \phi(t), \theta(t) \} \left[ \eta \left( \mathbf{x}\_{\mathbb{P}, \mathcal{A}}(t) \right) - \beta \left( \mathbf{x}\_{\mathbb{P}, \mathcal{A}}(t) \right) + \gamma\_{\mathcal{A}} \left( t, \mathbf{x}\_{\mathbb{P}}(t), \mathbf{x}(t) \right) \right],\tag{58}$$

where <sup>γ</sup> <sup>t</sup>; <sup>x</sup>p; <sup>x</sup> � � <sup>¼</sup> <sup>γ</sup><sup>T</sup> <sup>P</sup> <sup>t</sup>; <sup>x</sup>p; <sup>x</sup> � �; <sup>γ</sup><sup>T</sup> <sup>A</sup> <sup>t</sup>; <sup>x</sup>p; <sup>x</sup> � � � � <sup>T</sup> , (t, <sup>x</sup>p, <sup>x</sup>) <sup>∈</sup>[t0, <sup>∞</sup>) � <sup>R</sup><sup>12</sup> � <sup>R</sup>18, <sup>γ</sup>P(t, <sup>x</sup>p, <sup>x</sup>)<sup>∈</sup> <sup>R</sup><sup>3</sup> , and γA(t, xp, x)∈ R<sup>3</sup> .

Proof: Uniform ultimate boundedness of (8) with u = γ(t, xp, x) is a direct consequence of Theorem 3.1. Thus, both the nonlinear dynamical system given by (43) and (44) with [uX, uY, uZ] <sup>T</sup> <sup>=</sup> <sup>γ</sup>P(t, <sup>x</sup>p, <sup>x</sup>), <sup>t</sup>; <sup>x</sup>p; <sup>x</sup> � �∈½ Þ� <sup>t</sup>0; <sup>∞</sup> <sup>D</sup><sup>p</sup> � <sup>D</sup>, and the nonlinear dynamical system given by (50) and (51) with [u2, u3, u4] <sup>T</sup> = γA(t, xp, x) are uniformly ultimately bounded. Consequently, it follows from Definition 2.2 that there exist b > 0 and c > 0 independent of t0, and for every a ∈ (0, c), there exists a finite-time T = T(a, c) ≥ 0, independent of t0, such that if ∥yP(t0) � ycmd, P(t0) ∥ ≤ a and ∥yA(t0) � ycmd, A(t0) ∥ ≤ a, then (55) and (56) are satisfied. Lastly, (57) directly follows from (37), and (58) directly follows from (49).

#### 8. Illustrative numerical example

In this section, we provide a numerical example to illustrate both the applicability and the advantages of the theoretical results presented in this chapter. Specifically, we design a nonlinear robust control algorithm that allows a quadrotor helicopter to follow a circular trajectory, although the vehicle's inertial properties are unknown, one of the motors is suddenly turned off, the payload is dropped over the course of the mission, and the wind blows at strong velocity.

Consider a quadrotor of mass m<sup>Q</sup> = 1 kg and matrix of inertia I = 1<sup>3</sup> kg m2 , let the propellers be characterized by the matrix of inertia I<sup>p</sup> ¼ 0:025 0 0 0 0:025 0 000:05 2 6 4 3 7 5kg m2, and let that the sen-

Theorem 7.1 Consider the nonlinear dynamical system given by (43) and (44), the nonlinear dynamical system given by (50) and (51), the reference signals (53) and (54), the augmented dynamical system (8), the reference dynamical model (9), the feedback control law γ(�, � , �) given by (10), and the adaptation laws (11)–(13). If there exist Kx∈ R<sup>18</sup> � <sup>6</sup> and Kcmd∈ R<sup>6</sup> � <sup>6</sup> such that (15) and (16) and satisfied, then (8) with u = γ(t, xp, x) is uniformly ultimately bounded. Furthermore, there exist b > 0 and c > 0 independent of t0, and for every a∈(0, c), there exists a finite-time T = T(a, c) ≥ 0, independent of t0, such that if ∥yP(t0) �

∥yPð Þ� t ycmd,Pð Þt ∥ ≤ b, t ≥ t<sup>0</sup> þ T, (55)

<sup>5</sup> <sup>¼</sup> <sup>b</sup>IΓ�<sup>1</sup> <sup>ϕ</sup>ð Þ<sup>t</sup> ; <sup>θ</sup>ð Þ<sup>t</sup> � � <sup>η</sup> <sup>x</sup>p,Að Þ<sup>t</sup> � � � <sup>β</sup> <sup>x</sup>p,Að Þ<sup>t</sup> � � <sup>þ</sup> <sup>γ</sup><sup>A</sup> <sup>t</sup>; <sup>x</sup>pð Þ<sup>t</sup> ; x tð Þ � � � � , (58)

∥yAð Þ� t ycmd,Að Þt ∥ ≤ b: (56)

, (t, <sup>x</sup>p, <sup>x</sup>) <sup>∈</sup>[t0, <sup>∞</sup>) � <sup>R</sup><sup>12</sup> � <sup>R</sup>18, <sup>γ</sup>P(t, <sup>x</sup>p, <sup>x</sup>)<sup>∈</sup> <sup>R</sup><sup>3</sup>

<sup>T</sup> = γA(t, xp, x) are uniformly ultimately bounded. Consequently, it

�, t ≥ t0, (57)

, and

ycmd, P(t0) ∥ ≤ a and ∥yA(t0) � ycmd, A(t0) ∥ ≤ a, then

u2ð Þt u3ð Þt u4ð Þt

.

(50) and (51) with [u2, u3, u4]

3 7

> <sup>P</sup> <sup>t</sup>; <sup>x</sup>p; <sup>x</sup> � �; <sup>γ</sup><sup>T</sup> <sup>A</sup> <sup>t</sup>; <sup>x</sup>p; <sup>x</sup> � � � � <sup>T</sup>

follows from (37), and (58) directly follows from (49).

8. Illustrative numerical example

2 6 4

94 Adaptive Robust Control Systems

where <sup>γ</sup> <sup>t</sup>; <sup>x</sup>p; <sup>x</sup> � � <sup>¼</sup> <sup>γ</sup><sup>T</sup>

γA(t, xp, x)∈ R<sup>3</sup>

strong velocity.

uZ]

Lastly, the thrust force generated by the quadrotor's propellers is such that

<sup>u</sup>1ðÞ¼ <sup>t</sup> <sup>m</sup><sup>b</sup> <sup>Q</sup> <sup>γ</sup><sup>P</sup> <sup>t</sup>; <sup>x</sup>pð Þ<sup>t</sup> ; x tð Þ � � � � �

Proof: Uniform ultimate boundedness of (8) with u = γ(t, xp, x) is a direct consequence of Theorem 3.1. Thus, both the nonlinear dynamical system given by (43) and (44) with [uX, uY,

follows from Definition 2.2 that there exist b > 0 and c > 0 independent of t0, and for every a ∈ (0, c), there exists a finite-time T = T(a, c) ≥ 0, independent of t0, such that if ∥yP(t0) � ycmd, P(t0) ∥ ≤ a and ∥yA(t0) � ycmd, A(t0) ∥ ≤ a, then (55) and (56) are satisfied. Lastly, (57) directly

In this section, we provide a numerical example to illustrate both the applicability and the advantages of the theoretical results presented in this chapter. Specifically, we design a nonlinear robust control algorithm that allows a quadrotor helicopter to follow a circular trajectory, although the vehicle's inertial properties are unknown, one of the motors is suddenly turned off, the payload is dropped over the course of the mission, and the wind blows at

<sup>T</sup> <sup>=</sup> <sup>γ</sup>P(t, <sup>x</sup>p, <sup>x</sup>), <sup>t</sup>; <sup>x</sup>p; <sup>x</sup> � �∈½ Þ� <sup>t</sup>0; <sup>∞</sup> <sup>D</sup><sup>p</sup> � <sup>D</sup>, and the nonlinear dynamical system given by

and the moment of the thrust force generated by the quadrotor's propellers is given by

sor's dynamics be characterized by ε = 10. We assume that the vehicle's mass and matrix of inertia are unknown and estimated to be <sup>m</sup><sup>b</sup> <sup>Q</sup> <sup>¼</sup> <sup>1</sup>:25 kg and <sup>b</sup><sup>I</sup> <sup>¼</sup> <sup>0</sup>:<sup>8</sup> � <sup>1</sup><sup>3</sup> kg m2, respectively. Moreover, we assume that the aerodynamic force (31) and the aerodynamic moment (27) are characterized by KF =KM = 0.01 � 13, which we assume unknown, and the wind velocity is given by v<sup>I</sup> <sup>W</sup>ðÞ¼ <sup>t</sup> ½ � <sup>16</sup>; <sup>0</sup>; <sup>0</sup> <sup>T</sup> m=s, t ≥ t0; it is worthwhile to note that this wind speed is considered as excessive for quadrotors equipped with conventional autopilots.

Figure 3 shows the quadrotor's trajectory obtained applying the control laws (57) and (58) to track a circular path of radius 0.3 m at an altitude of 0.75 m despite the fact that the quadrotor's payload of 0.5 kg is dropped at t ≥ 40 s and one of the motors is turned off at t = 90 s. These results have been obtained by setting σ<sup>1</sup> = σ<sup>2</sup> = σ<sup>3</sup> = 2, Γcmd = 100 � 16, and Γ<sup>x</sup> and ΓΘ as blockdiagonal matrices, whose non-zero blocks are Γx, (1, 1) = 1000 � 19, Γx, (2, 2) = 2000 � 19, ΓΘ, (1, 1) = 200 � 19, abd ΓΘ, (2, 2) = 1600 � 19.

Figure 3. Reference trajectory and trajectory followed by the quadrotor implementing the proposed control algorithm. The vehicle is disturbed by some wind constantly blowing at 16 m/s.

Figure 4. Altitude of a quadrotor implementing the proposed control algorithm and altitude of an identical quadrotor implementing an autopilot based on the classical PD control.

9. Conclusion

Acknowledgements

In this chapter, we presented a robust MRAC architecture, which we employed to design autopilots for quadrotor helicopters. The proposed autopilot is the first to account for the fact that quadrotors are nonlinear time-varying dynamical systems, the exact location of the vehicle's center of mass is usually unknown, and the aircraft reference frame is centered at some point that does not necessarily coincide with the vehicle's barycenter. Moreover, our autopilot does not rely on the assumption that the Euler angles are small at all times and accounts both

Figure 5. Control input for a quadrotor implementing the proposed control algorithm and control input for an identical

Robust Adaptive Output Tracking for Quadrotor Helicopters

http://dx.doi.org/10.5772/intechopen.70723

97

The applicability of our theoretical results has been illustrated by a numerical example and it is clearly shown how the proposed autopilot is able to track a given reference trajectory despite the fact that the payload is dropped during the mission, one of the motors is turned off, and the wind blows at the prohibitive velocity of 16 m/s. It is also shown that quadrotors implementing autopilots based on the classical PD framework crash if one of the propellers stops functioning. Lastly, it is shown that our autopilot requires a control effort that is smaller than the effort

This work was supported in part by the NOAA/Office of Oceanic and Atmospheric Research under NOAA-University of Oklahoma Cooperative Agreement #NA16OAR4320115, U.S. Department

required by conventional MRAC-based autopilots to fly in less strong wind.

for the inertial counter-torque and the gyroscopic effect.

quadrotor implementing a conventional MRAC-based autopilot.

Figure 4 shows both the quadrotor's altitude as function of time and the altitude of an identical quadrotor implementing an autopilot based on the classical PD framework [36] and flying in absence of wind. It is clear how the quadrotor implementing our control algorithm is able to fly at the desired altitude despite the fact that the payload is dropped at t = 40 s and a motor is turned off at t = 90 s. The quadrotor implementing the PD algorithm is unable to reach the desired altitude because of the large error in the vehicle's mass' estimate. Moreover, this quadrotor reaches a considerably higher altitude after the payload is dropped and crashes after one of the propellers is turned off.

The first plot in Figure 5 shows the control inputs (57) and (58). The second plot in Figure 5 shows the control inputs computed using a conventional MRAC framework [6] for a quadrotor tracking the same circular path despite a wind blowing at 6 m/s; numerical simulations show that quadrotors implementing the conventional MRAC framework are unable to fly in the presence of wind gusts faster than 6 m/s. It is clear that our autopilot requires a control effort that is smaller than the effort required by a conventional MRAC-based autopilot to fly in weaker wind.

Robust Adaptive Output Tracking for Quadrotor Helicopters http://dx.doi.org/10.5772/intechopen.70723 97

Figure 5. Control input for a quadrotor implementing the proposed control algorithm and control input for an identical quadrotor implementing a conventional MRAC-based autopilot.
