6. Proposed control system for quadrotors

In this section, we outline a control strategy for quadrotors and verify that this strategy does not defy the vehicle's limits given by its controllability and underactuation.

#### 6.1. Proposed control strategy

<sup>M</sup>Tð Þþ <sup>t</sup> <sup>M</sup><sup>g</sup> <sup>ϕ</sup>ð Þ<sup>t</sup> ; <sup>θ</sup>ð Þ<sup>t</sup> � � <sup>þ</sup> <sup>M</sup>ð Þ¼ <sup>ω</sup>ð Þ<sup>t</sup> <sup>m</sup>Q<sup>r</sup>

0; 0; Ω ̇ P,i ð Þt h i<sup>T</sup>

<sup>C</sup> <sup>F</sup><sup>g</sup> <sup>ϕ</sup>; <sup>θ</sup> � �, <sup>ϕ</sup>; <sup>θ</sup> � �<sup>∈</sup> � <sup>π</sup>

P<sup>4</sup>

þ I<sup>P</sup> X 4

86 Adaptive Robust Control Systems

respect to A. The terms I<sup>P</sup>

F(�) in the reference frame I.

I A � � �

Proof: It follows from (26) that

for all vA; <sup>ϕ</sup>; <sup>θ</sup>;<sup>ψ</sup> � �<sup>∈</sup> <sup>R</sup><sup>3</sup> � � <sup>π</sup>

(ϕ, θ,ψ) =R<sup>T</sup>

� �

aerodynamic force is given by

<sup>∥</sup>vA<sup>∥</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup> <sup>ϕ</sup>; <sup>θ</sup>;<sup>ψ</sup> � �r\_<sup>A</sup> �

FI

ð Þ¼� vA r\_

tion, R�<sup>1</sup>

velocity v<sup>I</sup>

namic forces acting on a quadrotor. It holds that

FI

FI

�<sup>R</sup> <sup>ϕ</sup>; <sup>θ</sup>;<sup>ψ</sup> � �KFR<sup>T</sup> <sup>ϕ</sup>; <sup>θ</sup>; <sup>ψ</sup> � �r\_

M<sup>g</sup> ϕ; θ � �≜r�

i¼1

�

X 4

½ � <sup>0</sup>; <sup>0</sup>; <sup>Ω</sup>P,ið Þ<sup>t</sup> <sup>T</sup>

, t ≥ t0, and ω�ð Þt I<sup>P</sup>

i¼1

where MT(t)=[u2(t), u3(t), u4(t)]<sup>T</sup> denotes the moment of the forces induced by the propellers,

<sup>2</sup> ; <sup>π</sup> 2

weight with respect to <sup>A</sup>, and <sup>M</sup> : <sup>R</sup>3!R<sup>3</sup> denotes the moment of the aerodynamic force with

known as inertial counter-torque and gyroscopic effect, respectively. In this chapter, we refer to

F vð Þ¼� <sup>A</sup> <sup>∥</sup>vA∥KFvA, vA <sup>∈</sup> <sup>R</sup><sup>3</sup>

<sup>M</sup>ð Þ¼� <sup>ω</sup> <sup>∥</sup>ω∥KMω, <sup>ω</sup> <sup>∈</sup> <sup>R</sup><sup>3</sup>

where KF,KM ∈ R<sup>3</sup> � <sup>3</sup> are diagonal, positive-definite, and unknown; for details, refer to [23]. The aerodynamic force (26) is expressed in the reference frame J. The next result allows expressing

Proposition 5.1 Consider the translational kinematic equation (21) and let (26) capture the aerody-

<sup>A</sup>, <sup>r</sup>\_A; <sup>ϕ</sup>; <sup>θ</sup>;<sup>ψ</sup> � �<sup>∈</sup> <sup>R</sup><sup>3</sup> � �

� � � ½ Þ <sup>0</sup>; <sup>2</sup><sup>π</sup> , and it follows from (21) that

<sup>2</sup> ; <sup>π</sup> 2

� <sup>¼</sup> <sup>∥</sup>r\_A<sup>∥</sup> ([24], p. 132). □

I

<sup>2</sup> ; <sup>π</sup> 2

ð Þ¼� vA <sup>∥</sup>vA∥<sup>R</sup> <sup>ϕ</sup>; <sup>θ</sup>;<sup>ψ</sup> � �KFR�<sup>1</sup> <sup>ϕ</sup>; <sup>θ</sup>;<sup>ψ</sup> � �r\_

Eq. (28) now follows from (30), since R(�, � , �) is an orthogonal matrix and hence, per defini-

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

Eq. (26) captures the aerodynamic drag acting on a quadrotor in absence of wind. If the wind

<sup>W</sup> : ½ Þ! <sup>t</sup>0; <sup>∞</sup> <sup>R</sup><sup>3</sup> is not identically equal to zero, then it follows from (28) that the

ð Þ¼� vA ∥vA∥½ � KFvA

(ϕ, <sup>θ</sup>,ψ), <sup>ϕ</sup>; <sup>θ</sup>; <sup>ψ</sup> � � <sup>∈</sup> � <sup>π</sup>

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

þ ω�ð Þt I<sup>P</sup>

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

̇ Pi ð Þt h i<sup>T</sup>

We model the aerodynamic force and the moment of the aerodynamic force as

<sup>i</sup>¼<sup>1</sup> <sup>0</sup>; <sup>0</sup>; <sup>Ω</sup>

(21)–(23) and (25) as the equations of motion of a quadrotor helicopter.

<sup>C</sup> v\_Að Þþ t ω� ½ ð Þt vAð Þt � þ Iω\_ ð Þþ t ω�ð Þt Iωð Þt

� �, denotes the moment of the quadrotor's

<sup>i</sup>¼<sup>1</sup> <sup>0</sup>; <sup>0</sup>; <sup>Ω</sup>Pi

P<sup>4</sup>

π 2 ; π 2 � �

<sup>I</sup> ¼ �∥vA∥<sup>R</sup> <sup>ϕ</sup>; <sup>θ</sup>; <sup>ψ</sup> � �KFvA (29)

� � � ½ Þ <sup>0</sup>; <sup>2</sup><sup>π</sup> , ([22], Def. A.13) and

I

, <sup>ω</sup>ð Þ¼ <sup>t</sup><sup>0</sup> <sup>ω</sup>0, <sup>t</sup> <sup>≥</sup>t0, (25)

ð Þ<sup>t</sup> � �<sup>T</sup> in (25) are

, (26)

, (27)

� � π 2 ; π 2 � �

<sup>A</sup>: (30)

� ½ Þ 0; 2π :

(28)

The configuration of a quadrotor, whose frame is modeled as a rigid body, is uniquely identified by the position in the inertial space of the reference point A, that is, r<sup>I</sup> <sup>A</sup>ðÞ¼ t ½rXð Þt ;rYð Þt ; rZ(t)]<sup>T</sup> , t ≥ t0, and the Euler angles ϕ(t), θ(t), and ψ(t). Observing the equations of motion of a quadrotor (21)–(23) and (25), one can show that the four control inputs u1(�), …, u4(�) are unable to instantaneously and simultaneously accelerate the six independent generalized coordinates rX(�), rY(�), rZ(�), ϕ(�), θ(�), and ψ(�), and hence quadrotors are underactuated mechanical systems ([25], Def. 2.9). However, it follows from (21)–(23) and (25) that the control inputs u1(�), …, u4(�) are able to instantaneously and simultaneously accelerate the independent generalized coordinates rZ(�), ϕ(�), θ(�), and ψ(�), which uniquely capture the vehicle's altitude and orientation dynamics.

In practical applications, quadrotors are employed to transport detection devices, such as antennas or cameras, that must be taken to some specific location and pointed in some given direction. For this reason, one usually needs to regulate a quadrotor's position r<sup>I</sup> <sup>A</sup>ð Þ� and yaw angle ψ(�). To meet this goal despite quadrotors' underactuation, we apply the following control strategy. Let [rX, ref (t),rY, ref (t),rZ, ref (t)]T∈ R<sup>3</sup> , t ≥ t0, denote the quadrotor's reference trajectory, let ψref (t)∈[0, 2π) denote the quadrotor's reference yaw angle, and assume that rX, ref (�), rY, ref (�), rZ, ref (�), and ψref (�), are continuous with their first two derivatives and bounded with their first derivatives. It follows from Example 1.4 of [22] that (21) and (23) are equivalent to

$$\boldsymbol{\bar{r}}\_{A}^{\mathbb{II}}(t) = \begin{bmatrix} \boldsymbol{u}\_{X}(t) \\ \boldsymbol{u}\_{Y}(t) \\ \boldsymbol{u}\_{Z}(t) \end{bmatrix} + \boldsymbol{m}\_{\mathbb{Q}}^{-1} \boldsymbol{F}^{\mathbb{I}}(\boldsymbol{v}\_{\mathcal{A}}(t), \boldsymbol{\omega}(t)) - \boldsymbol{\bar{r}}\_{\mathbb{C}}^{\mathbb{I}}(t) + \boldsymbol{\mu}^{\mathbb{I}}(t), \qquad \begin{bmatrix} \boldsymbol{r}\_{A}^{\mathbb{I}}(t\_{0}) \\ \boldsymbol{v}\_{A}^{\mathbb{I}}(t\_{0}) \end{bmatrix} = \begin{bmatrix} \boldsymbol{r}\_{A,0}^{\mathbb{I}} \\ \boldsymbol{v}\_{A,0}^{\mathbb{I}} \end{bmatrix}, \qquad t \succeq \boldsymbol{t}\_{0}. \tag{33}$$

where [26]

$$\mu^1(t) \triangleq m\_Q^{-1} \mu\_1(t) \left[ \mathcal{R} \left( \phi(t), \theta(t), \psi(t) \right) - \mathcal{R} \left( \phi\_{\text{ref}}(t), \theta\_{\text{ref}}(t), \psi\_{\text{ref}}(t) \right) \right] \Sigma,\tag{34}$$

$$\phi\_{\rm ref}(t) \triangleq \sin^{-1} \frac{\mu\_X(t) \sin \psi\_{\rm ref}(t) - \mu\_Y(t) \cos \psi\_{\rm ref}(t)}{\sqrt{\mu\_X^2(t) + \mu\_Y^2(t) + \mu\_Z^2(t)}},\tag{35}$$

uXð Þt sinψrefð Þ� t uYð Þt cosψrefð Þt ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Figure 2. Proposed control scheme for a quadrotor.

6.2. Strong accessibility of quadrotors

� for some t

∗ .

time-invariant dynamical systems [30, 34].

time-varying dynamical system

<sup>Y</sup>ð Þþ <sup>t</sup> <sup>u</sup><sup>2</sup>

Position Control

Eq. (32)

Control System

Attitude Control

Eqs. (22), (24)

Eqs. (34), (35)

u*X*, u*<sup>Y</sup>* , u*<sup>Z</sup>*

φref, θref

q ¼

<sup>Z</sup>ð Þt

to the horizontal plane. Specifically, it is well-known that <sup>θ</sup>ð Þ<sup>t</sup> <sup>∈</sup> � <sup>π</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>Y</sup>ð Þt

<sup>Z</sup>ð Þt

<sup>∗</sup> ≥ t0, then the quadrotor may not be controllable, that is, there may

cos ψrefð Þþ t αð Þt � � <sup>≤</sup> <sup>1</sup>,

(20), (21), (22), (24)

Eqs.

Robust Adaptive Output Tracking for Quadrotor Helicopters

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

r*A*

89

φ, θ, ψ

<sup>2</sup> ; <sup>π</sup> 2

� �, t ≥ t0, is a necessary

<sup>Y</sup>ð Þþ <sup>t</sup> <sup>u</sup><sup>2</sup>

(uY(t)/uX(t)). Alternative control strategies, which rely on the assumption

u2 <sup>X</sup>ð Þþ <sup>t</sup> <sup>u</sup><sup>2</sup>

<sup>T</sup>1, T2, T3, T<sup>4</sup> Eq. (31)

u2 <sup>X</sup>ð Þþ <sup>t</sup> <sup>u</sup><sup>2</sup>

that the roll and pitch angles are small, are provided by Islam et al. [27]; Kotarski et al. [28];

In this section, we prove that quadrotors are not controllable whenever the z(�) axis is parallel

condition for a 3-2-1 rotation sequence to uniquely identify a quadrotor's orientation in space and guarantee finiteness of the yaw rate for finite angular velocities ([22], p. 19). In the following, we verify for the first time the conditions for quadrotors' strong accessibility [15], which is a weaker form of controllability for nonlinear dynamical systems [30–32], and prove

not exist a continuous control input that is able to regulate the vehicle's altitude and orienta-

To the authors' best knowledge, the controllability of quadrotors' altitude and orientation dynamics has been studied considering simplified models, which assume that the vehicle's pitch and roll angles are small at all times [5, 33]. Moreover, existing results on the controllability of quadrotors neglect the fact that, as discussed in Remark 5.1, these vehicles are time-varying dynamical systems and rely on sufficient conditions for the controllability of

In the following, we recall the notions of reachable set and strong accessibility for the nonlinear

s

u1

Eq. (36)

u2, u3, u<sup>4</sup>

u2 <sup>X</sup>ð Þþ <sup>t</sup> <sup>u</sup><sup>2</sup>

where α(t) ≜tan�<sup>1</sup>

r*X,*ref, r*Y,*ref, r*Z,*ref

ψref

Liu & Hedrick [29].

that if ϕ t

<sup>∗</sup> ð Þ¼ <sup>π</sup> 2 � � �

tion dynamics at t = t

$$\theta\_{\text{ref}}(t) \triangleq \tan^{-1} \frac{\mu\_X(t) \cos \psi\_{\text{ref}}(t) + \mu\_Y(t) \sin \psi\_{\text{ref}}(t)}{\mu\_Z(t)}.\tag{36}$$

Thus, a feedback control law for the virtual control input [uX(�), uY(�), uZ(�)]<sup>T</sup> is designed so that, after a finite-time transient, <sup>r</sup>A(�) tracks [rX, ref(�),rY, ref(�),rZ, ref(�)]<sup>T</sup> with bounded error. Furthermore, a feedback control law for the control input [u2(�), <sup>u</sup>3(�), <sup>u</sup>4(�)]<sup>T</sup> is designed so that, after a finite-time transient, [ϕ(�), <sup>θ</sup>(�),ψ(�)]<sup>T</sup> tracks [ϕref(�), <sup>θ</sup>ref(�),ψref(�)]<sup>T</sup> with bounded error. Since the quadrotor's mass m<sup>Q</sup> is unknown, we compute the component of the quadrotor's thrust along the z(�) axis of the reference frame J as

$$
\mu\_1(t) = \hat{m}\_Q \sqrt{\mu\_X^2(t) + \mu\_Y^2(t) + \mu\_Z^2(t)}, \qquad t \ge t\_0. \tag{37}
$$

#### Figure 2 provides a schematic representation of the proposed control strategy.

Eqs. (35) and (36) constrain the nonlinear dynamical system given by (33), (22), and (25) and enforce its underactuation. Note that (36) is well-defined, since uZ(t) 6¼ 0, t ≥ t0, is a necessary condition for a quadrotor to fly, and (35) is well-defined since u1(t) 6¼ 0, t ≥ t0, is a necessary condition to fly and

Figure 2. Proposed control scheme for a quadrotor.

rX(�), rY(�), rZ(�), ϕ(�), θ(�), and ψ(�), and hence quadrotors are underactuated mechanical systems ([25], Def. 2.9). However, it follows from (21)–(23) and (25) that the control inputs u1(�), …, u4(�) are able to instantaneously and simultaneously accelerate the independent generalized coordinates rZ(�), ϕ(�), θ(�), and ψ(�), which uniquely capture the vehicle's altitude and orienta-

In practical applications, quadrotors are employed to transport detection devices, such as antennas or cameras, that must be taken to some specific location and pointed in some given direction.

meet this goal despite quadrotors' underactuation, we apply the following control strategy. Let

denote the quadrotor's reference yaw angle, and assume that rX, ref (�), rY, ref (�), rZ, ref (�), and ψref (�), are continuous with their first two derivatives and bounded with their first derivatives. It follows

<sup>Q</sup> <sup>u</sup>1ð Þ<sup>t</sup> <sup>R</sup> <sup>ϕ</sup>ð Þ<sup>t</sup> ; <sup>θ</sup>ð Þ<sup>t</sup> ;ψð Þ<sup>t</sup> � � � <sup>R</sup> <sup>ϕ</sup>refð Þ<sup>t</sup> ; <sup>θ</sup>refð Þ<sup>t</sup> ;ψrefð Þ<sup>t</sup>

<sup>ϕ</sup>refð Þ<sup>t</sup> <sup>≜</sup> sin �<sup>1</sup> uXð Þ<sup>t</sup> sin <sup>ψ</sup>refð Þ� <sup>t</sup> uYð Þ<sup>t</sup> cosψrefð Þ<sup>t</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 <sup>X</sup>ð Þþ <sup>t</sup> <sup>u</sup><sup>2</sup>

<sup>θ</sup>refð Þ<sup>t</sup> <sup>≜</sup> tan �<sup>1</sup> uXð Þ<sup>t</sup> cosψrefð Þþ <sup>t</sup> uYð Þ<sup>t</sup> sinψrefð Þ<sup>t</sup>

Thus, a feedback control law for the virtual control input [uX(�), uY(�), uZ(�)]<sup>T</sup> is designed so that, after a finite-time transient, <sup>r</sup>A(�) tracks [rX, ref(�),rY, ref(�),rZ, ref(�)]<sup>T</sup> with bounded error. Furthermore, a feedback control law for the control input [u2(�), <sup>u</sup>3(�), <sup>u</sup>4(�)]<sup>T</sup> is designed so that, after a finite-time transient, [ϕ(�), <sup>θ</sup>(�),ψ(�)]<sup>T</sup> tracks [ϕref(�), <sup>θ</sup>ref(�),ψref(�)]<sup>T</sup> with bounded error. Since the quadrotor's mass m<sup>Q</sup> is unknown, we compute the component of the quadrotor's thrust along

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Eqs. (35) and (36) constrain the nonlinear dynamical system given by (33), (22), and (25) and enforce its underactuation. Note that (36) is well-defined, since uZ(t) 6¼ 0, t ≥ t0, is a necessary condition for a quadrotor to fly, and (35) is well-defined since u1(t) 6¼ 0, t ≥ t0, is a necessary

<sup>Y</sup>ð Þþ <sup>t</sup> <sup>u</sup><sup>2</sup>

<sup>Z</sup>ð Þt

h i � �

€I <sup>C</sup>ð Þþ <sup>t</sup> <sup>μ</sup><sup>I</sup>

, t ≥ t0, denote the quadrotor's reference trajectory, let ψref (t)∈[0, 2π)

<sup>A</sup>ð Þ t<sup>0</sup> vI <sup>A</sup>ð Þ t<sup>0</sup> <sup>¼</sup> <sup>r</sup><sup>I</sup> A, 0 vI A,0

" #

" #

ð Þ<sup>t</sup> , <sup>r</sup><sup>I</sup>

<sup>Y</sup>ð Þþ <sup>t</sup> <sup>u</sup><sup>2</sup>

<sup>Z</sup>ð Þt

<sup>q</sup> , (35)

uZð Þ<sup>t</sup> : (36)

, t ≥ t0: (37)

<sup>A</sup>ð Þ� and yaw angle ψ(�). To

, t ≥ t0, (33)

Z, (34)

For this reason, one usually needs to regulate a quadrotor's position r<sup>I</sup>

ð Þ� vAð Þt ; ωð Þt r

from Example 1.4 of [22] that (21) and (23) are equivalent to

tion dynamics.

88 Adaptive Robust Control Systems

r €I <sup>A</sup>ðÞ¼ t

where [26]

[rX, ref (t),rY, ref (t),rZ, ref (t)]T∈ R<sup>3</sup>

uXð Þt uYð Þt uZð Þt

μI

ð Þ<sup>t</sup> <sup>≜</sup> <sup>m</sup>�<sup>1</sup>

the z(�) axis of the reference frame J as

condition to fly and

<sup>u</sup>1ðÞ¼ <sup>t</sup> <sup>m</sup><sup>b</sup> <sup>Q</sup>

u2 <sup>X</sup>ð Þþ <sup>t</sup> <sup>u</sup><sup>2</sup>

Figure 2 provides a schematic representation of the proposed control strategy.

q

3 7 <sup>5</sup> <sup>þ</sup> <sup>m</sup>�<sup>1</sup> <sup>Q</sup> F<sup>I</sup>

2 6 4

$$\frac{\mu\_X(t)\sin\psi\_{\text{ref}}(t) - u\_Y(t)\cos\psi\_{\text{ref}}(t)}{\sqrt{\mu\_X^2(t) + \mu\_Y^2(t) + \mu\_Z^2(t)}} = \sqrt{\frac{\mu\_X^2(t) + \mu\_Y^2(t)}{\mu\_X^2(t) + \mu\_Y^2(t) + \mu\_Z^2(t)}}\cos\left(\psi\_{\text{ref}}(t) + a(t)\right) \le 1/\lambda$$

where α(t) ≜tan�<sup>1</sup> (uY(t)/uX(t)). Alternative control strategies, which rely on the assumption that the roll and pitch angles are small, are provided by Islam et al. [27]; Kotarski et al. [28]; Liu & Hedrick [29].

#### 6.2. Strong accessibility of quadrotors

In this section, we prove that quadrotors are not controllable whenever the z(�) axis is parallel to the horizontal plane. Specifically, it is well-known that <sup>θ</sup>ð Þ<sup>t</sup> <sup>∈</sup> � <sup>π</sup> <sup>2</sup> ; <sup>π</sup> 2 � �, t ≥ t0, is a necessary condition for a 3-2-1 rotation sequence to uniquely identify a quadrotor's orientation in space and guarantee finiteness of the yaw rate for finite angular velocities ([22], p. 19). In the following, we verify for the first time the conditions for quadrotors' strong accessibility [15], which is a weaker form of controllability for nonlinear dynamical systems [30–32], and prove that if ϕ t <sup>∗</sup> ð Þ¼ <sup>π</sup> 2 � � � � for some t <sup>∗</sup> ≥ t0, then the quadrotor may not be controllable, that is, there may not exist a continuous control input that is able to regulate the vehicle's altitude and orientation dynamics at t = t ∗ .

To the authors' best knowledge, the controllability of quadrotors' altitude and orientation dynamics has been studied considering simplified models, which assume that the vehicle's pitch and roll angles are small at all times [5, 33]. Moreover, existing results on the controllability of quadrotors neglect the fact that, as discussed in Remark 5.1, these vehicles are time-varying dynamical systems and rely on sufficient conditions for the controllability of time-invariant dynamical systems [30, 34].

In the following, we recall the notions of reachable set and strong accessibility for the nonlinear time-varying dynamical system

$$\dot{\mathbf{x}}(t) = f(t, \mathbf{x}(t)) + G(\mathbf{x}(t))u(t), \qquad \mathbf{x}(t\_0) = \mathbf{x}\_0, \qquad t \ge t\_0. \tag{38}$$

det Cð Þ¼ ~x

<sup>2</sup> ; <sup>π</sup> 2

7. Nonlinear robust control of quadrotors

the virtual control input [uX(t), uY(t), uZ(t)]<sup>T</sup>

xp,Pð Þ¼ t<sup>0</sup> r

y\_

<sup>A</sup>ð Þ<sup>t</sup> � �<sup>T</sup>

and it follows from Theorem 6.1 that <sup>ϕ</sup>ð Þ<sup>t</sup> <sup>∈</sup> � <sup>π</sup>

from (35) that <sup>ϕ</sup>refð Þ<sup>t</sup> <sup>∈</sup> � <sup>π</sup>

rZ, ref(t)]<sup>T</sup>

[u2(t), u3(t), u4(t)]<sup>T</sup>

where xp,PðÞ¼ t r<sup>I</sup>

<sup>C</sup>p, P = [13, 03 � 3], <sup>b</sup>ξPð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>6</sup>

cos ϕ m2 <sup>Q</sup>det<sup>2</sup> I

conditions for strong accessibility of quadrotors' altitude and orientation dynamics.

<sup>2</sup> ; <sup>π</sup> 2

that a quadrotor's altitude rZ(t) and orientation [ϕ(t), θ(t),ψ(t)]<sup>T</sup> can be regulated by some continuous control input <sup>u</sup>(t), while <sup>r</sup>\_Zð Þ<sup>t</sup> ; <sup>ϕ</sup>\_ð Þ<sup>t</sup> ; <sup>θ</sup>\_ð Þ<sup>t</sup> ; <sup>ψ</sup>\_ð Þ<sup>t</sup> � �<sup>T</sup> remain bounded at all times; in practice, to preserve controllability, a conservative control law for quadrotors must prevent rotations of an angle of �π/2 about the x(�) axis of the reference frame J. Note that it follows

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),

It follows form (33) that if the aerodynamic force is modeled as in (31), then a quadrotor's

2 6 4

B@

uXð Þt uYð Þt uZð Þt

, <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>

<sup>W</sup>ð Þ� t r\_ I <sup>A</sup>ð Þt h i <sup>þ</sup> <sup>μ</sup><sup>I</sup>

<sup>W</sup>ð Þ� t r\_ I <sup>A</sup>ð Þt

, ½ � 13; 03�<sup>3</sup> bξPðÞ¼ t 03�3, and

<sup>Q</sup> v<sup>I</sup>

� � �

3 7 <sup>5</sup> <sup>þ</sup> <sup>Θ</sup><sup>T</sup> <sup>P</sup><sup>Φ</sup> <sup>x</sup>p,Pð Þ<sup>t</sup> � � <sup>0</sup>

<sup>P</sup>ðÞ¼ t εCp,Pxp,Pð Þ� t εyPð Þt , yPð Þ¼ t<sup>0</sup> Cp,Pxp,Pð Þ t<sup>0</sup> , (44)

03�<sup>3</sup> 03�<sup>3</sup>

� � �

, so that a quadrotor tracks [rX, ref(t),rY, ref(t),rZ, ref(t)]<sup>T</sup>

(35), the reference pitch angle (36), and the reference yaw angle ψref(t).

translational kinematic and dynamic equations are given by

x\_p,PðÞ¼ t Ap,Pxp,Pð Þþ t Bp,PΛ<sup>P</sup>

T A, <sup>0</sup>; v<sup>I</sup> A,0 � �<sup>T</sup> � �<sup>T</sup>

; r\_ I <sup>A</sup>ð Þ<sup>t</sup> � �<sup>T</sup> h i<sup>T</sup>

½ � 03�<sup>3</sup>; <sup>1</sup><sup>3</sup> <sup>b</sup>ξPðÞ¼ <sup>t</sup> <sup>m</sup>�<sup>1</sup>

�KFR<sup>T</sup> <sup>ϕ</sup>ð Þ<sup>t</sup> ; <sup>θ</sup>ð Þ<sup>t</sup> ;ψð Þ<sup>t</sup> � � <sup>v</sup><sup>I</sup>

, t ≥ t0, and the reference yaw angle ψref(t). In practice, we design control laws both for

, <sup>~</sup><sup>x</sup> <sup>∈</sup> <sup>R</sup><sup>8</sup> � ½ Þ <sup>t</sup>0; <sup>∞</sup> , (42)

Robust Adaptive Output Tracking for Quadrotor Helicopters

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

91

� �, t ≥ t0, is a sufficient condition to guarantee

, t ≥ t0, and the moment of the propellers' thrust

1

, t ≥ t0, (43)

� �, <sup>B</sup>p,<sup>P</sup> <sup>¼</sup> <sup>03</sup>�<sup>3</sup>

<sup>B</sup>p,P<sup>R</sup> <sup>ϕ</sup>ð Þ<sup>t</sup> ; <sup>θ</sup>ð Þ<sup>t</sup> ; <sup>ψ</sup>ð Þ<sup>t</sup> � �

ð Þ� t r €I CA <sup>þ</sup> <sup>b</sup>ξPð Þ<sup>t</sup> ,

13

� �, <sup>Λ</sup><sup>P</sup> <sup>¼</sup> <sup>m</sup>�<sup>1</sup>

<sup>C</sup>ð Þt : (45)

<sup>Q</sup> 13,

, the reference roll angle

� �, t ≥ t0, and hence the reference roll angle verifies sufficient

where x(t) ∈ R<sup>n</sup> , <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> are continuously differentiable.

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 y∈M and every t<sup>1</sup> ≥ t0.

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 this result, consider the augmented time-invariant dynamical system

$$\dot{\tilde{\mathbf{x}}}(t) = \tilde{f}(\tilde{\mathbf{x}}(t)) + \tilde{G}(\tilde{\mathbf{x}}(t))\boldsymbol{\mu}(t), \qquad \tilde{\mathbf{x}}(0) = \left[\mathbf{x}\_0^T, t\_0\right]^T, \qquad t \ge 0,\tag{39}$$

where ~x ≜ x<sup>T</sup>; t � �<sup>T</sup> , <sup>~</sup><sup>f</sup> ð Þ <sup>~</sup><sup>x</sup> <sup>≜</sup> <sup>f</sup> <sup>T</sup>ð Þ<sup>x</sup> ; <sup>1</sup> � �<sup>T</sup> , <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> � �<sup>T</sup> , and recall that the controllability matrix of the augmented time-invariant dynamical system (39) is defined as [15]

$$\mathcal{L}(\tilde{\mathbf{x}}) \triangleq \left[ \tilde{\mathbf{g}}\_1(\tilde{\mathbf{x}}), \dots, \tilde{\mathbf{g}}\_m(\tilde{\mathbf{x}}), \mathbf{a}d\_f \tilde{\mathbf{g}}\_1(\tilde{\mathbf{x}}), \dots, \mathbf{a}d\_f \tilde{\mathbf{g}}\_m(\tilde{\mathbf{x}}) \right], \qquad \tilde{\mathbf{x}} \in \mathbb{R}^n \times [t\_0, \infty), \tag{40}$$

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> � �.

Theorem 6.1 ([15]). Consider the nonlinear dynamical system (38). If rank Cð Þ¼ ~x n for all ~x ∈M � ½ Þ t0; ∞ , then (38) is strongly accessible.

It follows from (21)–(23) and (25) that a quadrotor's altitude and orientation are captured by (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> , <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> <sup>2</sup> ; <sup>π</sup> 2 � �� � π <sup>2</sup> ; <sup>π</sup> 2 � � � ½ Þ� <sup>0</sup>; <sup>2</sup><sup>π</sup> <sup>R</sup> � <sup>R</sup><sup>3</sup> , and

$$G(\mathbf{x}) = m\_{\mathbf{Q}}^{-1} \begin{bmatrix} 0\_{4 \times 1} & 0\_{4 \times 3} \\ \cos \phi \cos \theta & 0\_{1 \times 3} \\ 0\_{3 \times 1} & m\_{\mathbf{Q}} \Gamma(\phi, \theta) I^{-1} \end{bmatrix};\tag{41}$$

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

$$\det \mathcal{C}(\ddot{\mathbf{x}}) = \frac{\cos \phi}{m\_{\mathbb{Q}}^2 \mathbf{d} \mathbf{e}^2 \, I}, \qquad \ddot{\mathbf{x}} \in \mathbb{R}^8 \times [t\_0, \infty) \tag{42}$$

and it follows from Theorem 6.1 that <sup>ϕ</sup>ð Þ<sup>t</sup> <sup>∈</sup> � <sup>π</sup> <sup>2</sup> ; <sup>π</sup> 2 � �, t ≥ t0, is a sufficient condition to guarantee that a quadrotor's altitude rZ(t) and orientation [ϕ(t), θ(t),ψ(t)]<sup>T</sup> can be regulated by some continuous control input <sup>u</sup>(t), while <sup>r</sup>\_Zð Þ<sup>t</sup> ; <sup>ϕ</sup>\_ð Þ<sup>t</sup> ; <sup>θ</sup>\_ð Þ<sup>t</sup> ; <sup>ψ</sup>\_ð Þ<sup>t</sup> � �<sup>T</sup> remain bounded at all times; in practice, to preserve controllability, a conservative control law for quadrotors must prevent rotations of an angle of �π/2 about the x(�) axis of the reference frame J. Note that it follows from (35) that <sup>ϕ</sup>refð Þ<sup>t</sup> <sup>∈</sup> � <sup>π</sup> <sup>2</sup> ; <sup>π</sup> 2 � �, t ≥ t0, and hence the reference roll angle verifies sufficient conditions for strong accessibility of quadrotors' altitude and orientation dynamics.
