4. Modeling assumptions on quadrotors' dynamics

Let I ¼ f g O; X;Y;Z denote an orthonormal reference frame fixed with the Earth and centered at some point O, and let J ¼ f g A; x tð Þ; y tð Þ; z tð Þ , t ≥ t0, denote an orthonormal reference frame fixed with the quadrotor and centered at some point A, which is arbitrarily chosen. The axes of the reference frames I and J form two orthonormal bases of R<sup>3</sup> and if a vector a∈ R<sup>3</sup> is expressed in I, then this vector is denoted by a<sup>I</sup> . Alternatively, if a∈ R<sup>3</sup> is expressed in J, then no superscript is used. In this chapter, we consider the reference frame I as an inertial reference frame; quadrotors move at subsonic velocities and are usually operated at altitudes considerably lower than 10 kilometers and hence, the error induced by this modeling assumption is negligible ([21], Ch. 5). The Z axis is chosen so that the quadrotor's weight is given by F<sup>I</sup> <sup>g</sup> ¼ mQgZ, where m<sup>Q</sup> > 0 denotes 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; see Figure 1.

5. Quadrotors' equations of motion

r\_ I

> 2 6 4

2 ; π 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>

2 6 4

2 6 4

<sup>F</sup>Tð Þþ <sup>t</sup> <sup>F</sup><sup>g</sup> <sup>ϕ</sup>ð Þ<sup>t</sup> ; <sup>θ</sup>ð Þ<sup>t</sup> � � <sup>þ</sup> F vð Þ¼ <sup>A</sup>ð Þ<sup>t</sup>

<sup>F</sup><sup>g</sup> <sup>ϕ</sup>; <sup>θ</sup> � � <sup>¼</sup> <sup>m</sup>Q<sup>g</sup> � sin <sup>θ</sup>; cos <sup>θ</sup> sin <sup>ϕ</sup>; cos <sup>θ</sup> cos <sup>ϕ</sup> � �<sup>T</sup>

that allows a quadrotor to hover,

R ϕ; θ;ψ � �≜

Γ ϕ; θ � �≜

equation is given by [4]

<sup>ϕ</sup>; <sup>θ</sup>;<sup>ψ</sup> � �<sup>∈</sup> � <sup>π</sup>

where

where

translational kinematic equation is given by ([22], Ex. 1.12)

cosψ � sin ψ 0 sinψ cosψ 0 00 1

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

3 7

<sup>A</sup>ðÞ¼ <sup>t</sup> <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> � �vAð Þ<sup>t</sup> , r<sup>I</sup>

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

<sup>5</sup> <sup>¼</sup> <sup>Γ</sup> <sup>ϕ</sup>ð Þ<sup>t</sup> ; <sup>θ</sup>ð Þ<sup>t</sup> � �ωð Þ<sup>t</sup> ,

1 sin ϕ tan θ cos ϕ tan θ 0 cos ϕ � sin ϕ 0 sin ϕ sec θ cos ϕ sec θ

3 7 5

2 6 4

In this section, we present the equations of motion of quadrotors. Specifically, a quadrotor's

<sup>A</sup>ð Þ¼ t<sup>0</sup> r I

> ϕð Þ t<sup>0</sup> θð Þ t<sup>0</sup> ψð Þ t<sup>0</sup>

<sup>5</sup>, <sup>ϕ</sup>; <sup>θ</sup> � �<sup>∈</sup> � <sup>π</sup>

<sup>m</sup><sup>Q</sup> <sup>v</sup>\_Að Þþ <sup>t</sup> <sup>ω</sup>�ð Þ<sup>t</sup> vAð Þþ <sup>t</sup> <sup>ω</sup>\_ �ð Þ<sup>t</sup> rCþω�ð Þ<sup>t</sup> <sup>ω</sup>� <sup>½</sup> ð Þ<sup>t</sup> rC�, vAð Þ¼ <sup>t</sup><sup>0</sup> vA, <sup>0</sup>, t <sup>≥</sup> <sup>t</sup>0, (23)

, <sup>ϕ</sup>; <sup>θ</sup> � � <sup>∈</sup> � <sup>π</sup>

2 ; π 2 � � � �

ϕ0 θ0 ψ0

2 ; π 2 � � � �

3 7

2 6 4

2 6 4

3 7

Under the modeling assumptions outlined in Section 4, a quadrotor's translational dynamic

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

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]

3 7 5

2 6 4

10 0

Robust Adaptive Output Tracking for Quadrotor Helicopters

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

0 cos ϕ � sin ϕ 0 sin ϕ cos ϕ

cos θ 0 sin θ 0 10 � sin θ 0 cos θ A, <sup>0</sup>, t ≥ t0, (21)

3 7 5, 85

<sup>5</sup>, (22)

π 2 ; π 2 � �:

> π 2 ; π 2 � �, (24)

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 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> <sup>2</sup> ; <sup>π</sup> 2 � � the roll and pitch angles, 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 denoted by rC∈ R<sup>3</sup> . The matrix of inertia of the quadrotor, excluding its propellers, with respect to A is denoted by I ∈ R<sup>3</sup> � <sup>3</sup> and the matrix of inertia of each propeller with respect to A is denoted by I<sup>P</sup> ∈ R<sup>3</sup> � <sup>3</sup> . The spin rate of the ith propeller is denoted by ΩP, <sup>i</sup> : [t0, ∞) ! R, 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 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> .

Figure 1. Schematic representation of a quadrotor helicopter.
