5. Quadrotors' equations of motion

In this section, we present the equations of motion of quadrotors. Specifically, a quadrotor's translational kinematic equation is given by ([22], Ex. 1.12)

$$
\dot{r}\_A^\perp(t) = R\{\phi(t), \theta(t), \psi(t)\} \upsilon\_A(t), \qquad r\_A^\parallel(t\_0) = r\_{A,0'}^\parallel \qquad t \ge t\_{0'} \tag{21}
$$

where

the vehicle's mass and g denotes the gravitational acceleration; the X and Y axes are chosen arbitrarily. The axis z(�) points down and the axis x(�) is aligned to one of the quadrotor's arms;

The attitude of the reference frame J with respect to the reference frame I is captured by the roll, pitch, and yaw angles using a 3-2-1 rotation sequence ([22], Ch. 1). In particular, we denote

respectively. The angular velocity of <sup>J</sup> with respect to <sup>I</sup> is denoted by <sup>ω</sup> : [t0, <sup>∞</sup>) !R<sup>3</sup> ([22], Def. 1.9). The position of the point A with respect to the origin O of the inertial reference frame I is denoted by rA : [t0, <sup>∞</sup>)! <sup>R</sup><sup>3</sup> and the velocity of <sup>A</sup> with respect to <sup>I</sup> is denoted by vA : [t0, <sup>∞</sup>)! <sup>R</sup><sup>3</sup>

The position of the quadrotor's center of mass C with respect to the reference point A is

respect to A is denoted by I ∈ R<sup>3</sup> � <sup>3</sup> and the matrix of inertia of each propeller with respect

i = 1, …, 4. In this chapter, we model the quadrotor's frame as a rigid body and propellers as thin disks. Moreover, we assume that the vehicle's inertial properties, such as the mass mQ, the inertia matrix I, and the location of the center of mass rC, are constant, but unknown. The quadrotor's estimated mass is denoted by <sup>m</sup><sup>b</sup> <sup>Q</sup> <sup>&</sup>gt; 0 and the quadrotor's estimated matrix of

Ω<sup>P</sup>*,*<sup>1</sup>

z

X

ψ

inertia with respect to <sup>A</sup> is given by the symmetric, positive-definite matrix <sup>b</sup><sup>I</sup> <sup>∈</sup> <sup>R</sup><sup>3</sup>�<sup>3</sup>

T4

r*A*

T1

<sup>2</sup> ; <sup>π</sup> 2

. The matrix of inertia of the quadrotor, excluding its propellers, with

A

T<sup>3</sup> r*<sup>C</sup>* C

mQgZ

Payload

. The spin rate of the ith propeller is denoted by ΩP, <sup>i</sup> : [t0, ∞) ! R,

� � the roll and pitch angles,

.

.

Ω<sup>P</sup>*,*<sup>2</sup>

T2

θ

y

Ω<sup>P</sup>*,*<sup>3</sup>

by <sup>ψ</sup>: [t0, <sup>∞</sup>) ![0, 2π) the yaw angle and <sup>ϕ</sup>, <sup>θ</sup> : ½ Þ! � <sup>t</sup>0; <sup>∞</sup> <sup>π</sup>

Ω<sup>P</sup>*,*<sup>4</sup>

O

Z

Figure 1. Schematic representation of a quadrotor helicopter.

x

φ

see Figure 1.

84 Adaptive Robust Control Systems

denoted by rC∈ R<sup>3</sup>

Y

to A is denoted by I<sup>P</sup> ∈ R<sup>3</sup> � <sup>3</sup>

$$\mathcal{R}(\phi,\theta,\psi) \triangleq \begin{bmatrix} \cos\psi & -\sin\psi & 0\\ \sin\psi & \cos\psi & 0\\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \cos\theta & 0 & \sin\theta\\ 0 & 1 & 0\\ -\sin\theta & 0 & \cos\theta \end{bmatrix} \begin{bmatrix} 1 & 0 & 0\\ 0 & \cos\phi & -\sin\phi\\ 0 & \sin\phi & \cos\phi \end{bmatrix}.$$

$$(\phi, \theta, \psi) \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \times \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \times [0, 2\pi)\omega$$

and the rotational kinematic equation is given by ([22], Th. 1.7)

$$
\begin{bmatrix}
\dot{\phi}(t) \\
\dot{\theta}(t) \\
\dot{\psi}(t)
\end{bmatrix} = \Gamma \{ \phi(t), \theta(t) \} \omega(t), \qquad \begin{bmatrix}
\phi(t\_0) \\
\theta(t\_0) \\
\psi(t\_0)
\end{bmatrix} = \begin{bmatrix}
\phi\_0 \\
\theta\_0 \\
\psi\_0
\end{bmatrix} \tag{22}
$$

where

$$\Gamma(\phi,\theta) \triangleq \begin{bmatrix} 1 & \sin\phi\tan\theta & \cos\phi\tan\theta \\ 0 & \cos\phi & -\sin\phi \\ 0 & \sin\phi\sec\theta & \cos\phi\sec\theta \end{bmatrix}, \qquad (\phi,\theta) \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \times \left( -\frac{\pi}{2}, \frac{\pi}{2} \right).$$

Under the modeling assumptions outlined in Section 4, a quadrotor's translational dynamic equation is given by [4]

$$\begin{split} F\_{\mathsf{T}}(t) + F\_{\mathsf{g}}\left(\phi(t), \theta(t)\right) + F(\mathsf{v}\_{\mathsf{A}}(t)) &= \\ m\_{\mathsf{Q}}[\dot{\boldsymbol{v}}\_{A}(t) + \boldsymbol{\omega}^{\times}(t)\boldsymbol{\uptau}\_{A}(t) + \dot{\boldsymbol{\omega}}^{\times}(t)\boldsymbol{r}\_{\mathsf{C}} + \boldsymbol{\omega}^{\times}(t)\boldsymbol{\upomega}^{\times}(t)\boldsymbol{r}\_{\mathsf{C}}], \quad \boldsymbol{\uptau}\_{A}(t\_{0}) = \boldsymbol{\uptau}\_{A,0}, \quad t \ge t\_{0}, \end{split} \tag{23}$$

where FT(t) = [0, 0, u1(t)]<sup>T</sup> denotes the thrust force, that is, the force produced by the propellers that allows a quadrotor to hover,

$$F\_{\mathbf{g}}\left(\phi,\theta\right) = m\_{\mathbf{Q}}\mathbf{g}\left[-\sin\theta,\cos\theta\sin\phi,\cos\theta\cos\phi\right]^{\mathsf{T}}, \qquad \left(\phi,\theta\right) \in \left(-\frac{\pi}{2},\frac{\pi}{2}\right) \times \left(-\frac{\pi}{2},\frac{\pi}{2}\right), \tag{24}$$

denotes the quadrotor's weight, and <sup>F</sup> : <sup>R</sup>3! <sup>R</sup><sup>3</sup> denotes the aerodynamic force acting on the quadrotor [23]. The rotational dynamic equation of a quadrotor is given by [4]

$$\begin{split} M\_{\Gamma}(t) + M\_{\text{g}}\left(\phi(t), \theta(t)\right) + M(\omega(t)) &= m\_{\text{Q}}r\_{\text{C}}^{\times}\left[\dot{v}\_{A}(t) + \omega^{\times}(t)v\_{A}(t)\right] + I\dot{\omega}(t) + \omega^{\times}(t)I\omega(t) \\ &+ I\_{\text{P}}\sum\_{i=1}^{4}\left[0, 0, \dot{\Omega}\_{\text{P},i}(t)\right]^{\text{T}} + \omega^{\times}(t)I\_{\text{P}}\sum\_{i=1}^{4}\left[0, 0, \Omega\_{\text{P},i}(t)\right]^{\text{T}}, \quad \omega(t\_{0}) = \omega\_{0}, \quad t \ge t\_{0}, \end{split} \tag{25}$$

where MT(t)=[u2(t), u3(t), u4(t)]<sup>T</sup> denotes the moment of the forces induced by the propellers, M<sup>g</sup> ϕ; θ � �≜r� <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> <sup>2</sup> ; <sup>π</sup> 2 � � � � <sup>π</sup> <sup>2</sup> ; <sup>π</sup> 2 � �, denotes the moment of the quadrotor's 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 respect to A. The terms I<sup>P</sup> P<sup>4</sup> <sup>i</sup>¼<sup>1</sup> <sup>0</sup>; <sup>0</sup>; <sup>Ω</sup> ̇ Pi ð Þt h i<sup>T</sup> , t ≥ t0, and ω�ð Þt I<sup>P</sup> P<sup>4</sup> <sup>i</sup>¼<sup>1</sup> <sup>0</sup>; <sup>0</sup>; <sup>Ω</sup>Pi ð Þ<sup>t</sup> � �<sup>T</sup> in (25) are known as inertial counter-torque and gyroscopic effect, respectively. In this chapter, we refer to (21)–(23) and (25) as the equations of motion of a quadrotor helicopter.

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

$$F(\upsilon\_A) = -\|\upsilon\_A\| K\_F \upsilon\_{A\prime} \qquad \upsilon\_A \in \mathbb{R}^3,\tag{26}$$

FI

MT(�). One can verify that ([1], Ch. 2)

u1ð Þt u2ð Þt u3ð Þt u4ð Þt

arm, and c<sup>T</sup> > 0 denotes each propeller's drag coefficient.

namic force (27).

defined as r<sup>T</sup>

<sup>A</sup>; vT

relate the gyroscopic effect ω�ð Þt IP

6.1. Proposed control strategy

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

<sup>A</sup>; <sup>ϕ</sup>; <sup>θ</sup>; <sup>ψ</sup>; <sup>ω</sup><sup>T</sup> � �<sup>T</sup>

nonlinear time-varying dynamical system.

6. Proposed control system for quadrotors

ð Þ¼� <sup>t</sup>; vA <sup>v</sup><sup>I</sup>

� � �

<sup>w</sup>ð Þ� t r\_ I A

<sup>t</sup>;r\_A; <sup>ϕ</sup>; <sup>θ</sup>;<sup>ψ</sup> � �<sup>∈</sup> ½ Þ� <sup>t</sup>0; <sup>∞</sup> <sup>R</sup><sup>3</sup> � �

A quadrotor's control vector is given by u(t)=[u1(t), u2(t), u3(t), u4(t)]<sup>T</sup>

� � �

R ϕ; θ; ψ � �KFR<sup>T</sup> ϕ; θ;ψ � � v<sup>I</sup>

T1ð Þt T2ð Þt T3ð Þt T4ð Þt

π 2 ; π 2 � � � �

It is reasonable to assume that the wind velocity does not affect the moment of the aerody-

component of the thrust force FT(�) and the moment of the forces induced by the propellers

where Ti : [t0, ∞)! R, i = 1, …, 4, denotes the component of the force produced by the ith propeller along the �z(�) axis of the reference frame J, l > 0 denotes the length of each propeller's

Remark 5.1 The state vector for the equations of motion of a quadrotor (21)–(23) and (25) is

t ≥ t0, can be explicitly related through algebraic expressions to neither the state vector x nor the control input u. Thus, the inertial counter-torque must be considered as a time-varying term in a quadrotor's rotational dynamic equations. Furthermore, it is common practice not to

(32) ([1], Ch. 2). Hence, also the gyroscopic effect must be accounted for as a time-varying term in a quadrotor's equations of motion. For these reasons, (21)–(23) and (25) are a

In this section, we outline a control strategy for quadrotors and verify that this strategy does

The configuration of a quadrotor, whose frame is modeled as a rigid body, is uniquely identi-

, 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

<sup>i</sup>¼<sup>1</sup> <sup>0</sup>; <sup>0</sup>; <sup>Ω</sup>Pi ð Þ<sup>t</sup> � �<sup>T</sup>

P<sup>4</sup>

not defy the vehicle's limits given by its controllability and underactuation.

fied by the position in the inertial space of the reference point A, that is, r<sup>I</sup>

∈ R<sup>12</sup> and the inertial counter-torque IP

1 11 1 0 l 0 �l l 0 �l 0 �c<sup>T</sup> c<sup>T</sup> �c<sup>T</sup> c<sup>T</sup>

<sup>w</sup>ð Þ� t r\_ I A h i,

Robust Adaptive Output Tracking for Quadrotor Helicopters

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

(31)

87

, t ≥ t0, that is, the third

, t ≥ t0, (32)

P<sup>4</sup>

, t ≥ t0, with the control input u through

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

̇ Pi ð Þt h i<sup>T</sup>

<sup>A</sup>ðÞ¼ t ½rXð Þt ;rYð Þt ;

,

π 2 ; π 2 � � � ½ Þ <sup>0</sup>; <sup>2</sup><sup>π</sup> :

$$M(\omega) = -\|\omega\|\mathcal{K}\_M \omega, \qquad \omega \in \mathbb{R}^3,\tag{27}$$

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 F(�) in the reference frame I.

Proposition 5.1 Consider the translational kinematic equation (21) and let (26) capture the aerodynamic forces acting on a quadrotor. It holds that

$$F^{\mathbb{I}}(\upsilon\_{A}) = -\left| \| \dot{r}\_{A}^{\mathbb{I}} \| \mathbb{R} \{ \phi, \theta, \psi \} \text{K}\_{\mathbb{R}} \mathbb{R}^{\mathbb{T}} \{ \phi, \theta, \psi \} \dot{r}\_{A'}^{\mathbb{I}} \left( \dot{r}\_{A}, \phi, \theta, \psi \right) \in \mathbb{R}^{3} \times \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \times \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \times [0, 2\pi) . \tag{28}$$

Proof: It follows from (26) that

$$F^{\mathbb{I}}(\upsilon\_A) = -\|\upsilon\_A\| \left[K\_F \upsilon\_A\right]^{\mathbb{I}} = -\|\upsilon\_A\| \mathcal{R}\left(\phi, \theta, \psi\right) \mathcal{K}\_F \upsilon\_A \tag{29}$$

for all vA; <sup>ϕ</sup>; <sup>θ</sup>;<sup>ψ</sup> � �<sup>∈</sup> <sup>R</sup><sup>3</sup> � � <sup>π</sup> <sup>2</sup> ; <sup>π</sup> 2 � � � � <sup>π</sup> <sup>2</sup> ; <sup>π</sup> 2 � � � ½ Þ <sup>0</sup>; <sup>2</sup><sup>π</sup> , and it follows from (21) that

$$F^{\mathbb{I}}(\upsilon\_A) = -\|\upsilon\_A\|\mathbb{R}\{\boldsymbol{\phi}, \boldsymbol{\theta}, \boldsymbol{\psi}\} \mathcal{K}\_F \mathcal{R}^{-1} \{\boldsymbol{\phi}, \boldsymbol{\theta}, \boldsymbol{\psi}\} \dot{r}\_A^{\mathbb{I}}.\tag{30}$$

Eq. (28) now follows from (30), since R(�, � , �) is an orthogonal matrix and hence, per definition, R�<sup>1</sup> (ϕ, θ,ψ) =R<sup>T</sup> (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> <sup>2</sup> ; <sup>π</sup> 2 � � � ½ Þ <sup>0</sup>; <sup>2</sup><sup>π</sup> , ([22], Def. A.13) and <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> � � � � <sup>¼</sup> <sup>∥</sup>r\_A<sup>∥</sup> ([24], p. 132). □

Eq. (26) captures the aerodynamic drag acting on a quadrotor in absence of wind. If the wind velocity v<sup>I</sup> <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 aerodynamic force is given by

$$\begin{split} F^{\mathbb{I}}(t, \boldsymbol{v}\_{A}) &= -\left\| \boldsymbol{v}\_{\mathrm{w}}^{\mathbb{I}}(t) - \boldsymbol{\dot{r}}\_{A}^{\mathbb{I}} \right\| \mathbb{R} \big( \boldsymbol{\phi}, \boldsymbol{\theta}, \boldsymbol{\psi} \big) \mathbb{K}\_{\mathbb{R}} \mathrm{R}^{\mathbb{T}} \big( \boldsymbol{\phi}, \boldsymbol{\theta}, \boldsymbol{\psi} \big) \Big[ \boldsymbol{v}\_{\mathrm{w}}^{\mathbb{I}}(t) - \boldsymbol{\dot{r}}\_{A}^{\mathbb{I}} \big], \\ & \left( t, \boldsymbol{\dot{r}}\_{A}, \boldsymbol{\phi}, \boldsymbol{\theta}, \boldsymbol{\psi} \big) \in \left[ \boldsymbol{t}\_{0}, \boldsymbol{\omega} \right) \times \mathbb{R}^{3} \times \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \times \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \times \left[ 0, 2\pi \right). \end{split} \tag{31}$$

It is reasonable to assume that the wind velocity does not affect the moment of the aerodynamic force (27).

A quadrotor's control vector is given by u(t)=[u1(t), u2(t), u3(t), u4(t)]<sup>T</sup> , t ≥ t0, that is, the third component of the thrust force FT(�) and the moment of the forces induced by the propellers MT(�). One can verify that ([1], Ch. 2)

$$
\begin{bmatrix} u\_1(t) \\ u\_2(t) \\ u\_3(t) \\ u\_4(t) \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & l & 0 & -l \\ l & 0 & -l & 0 \\ -c\_\Gamma & c\_\Gamma & -c\_\Gamma & c\_\Gamma \end{bmatrix} \begin{bmatrix} T\_1(t) \\ T\_2(t) \\ T\_3(t) \\ T\_4(t) \end{bmatrix}, \qquad t \ge t\_{0,} \tag{32}
$$

where Ti : [t0, ∞)! R, i = 1, …, 4, denotes the component of the force produced by the ith propeller along the �z(�) axis of the reference frame J, l > 0 denotes the length of each propeller's arm, and c<sup>T</sup> > 0 denotes each propeller's drag coefficient.

Remark 5.1 The state vector for the equations of motion of a quadrotor (21)–(23) and (25) is defined as r<sup>T</sup> <sup>A</sup>; vT <sup>A</sup>; <sup>ϕ</sup>; <sup>θ</sup>; <sup>ψ</sup>; <sup>ω</sup><sup>T</sup> � �<sup>T</sup> ∈ R<sup>12</sup> and the inertial counter-torque IP P<sup>4</sup> <sup>i</sup>¼<sup>1</sup> <sup>0</sup>; <sup>0</sup>; <sup>Ω</sup> ̇ Pi ð Þt h i<sup>T</sup> , t ≥ t0, can be explicitly related through algebraic expressions to neither the state vector x nor the control input u. Thus, the inertial counter-torque must be considered as a time-varying term in a quadrotor's rotational dynamic equations. Furthermore, it is common practice not to relate the gyroscopic effect ω�ð Þt IP P<sup>4</sup> <sup>i</sup>¼<sup>1</sup> <sup>0</sup>; <sup>0</sup>; <sup>Ω</sup>Pi ð Þ<sup>t</sup> � �<sup>T</sup> , t ≥ t0, with the control input u through (32) ([1], Ch. 2). Hence, also the gyroscopic effect must be accounted for as a time-varying term in a quadrotor's equations of motion. For these reasons, (21)–(23) and (25) are a nonlinear time-varying dynamical system.
