Modeling the Quad-Rotor Mini-Rotorcraft

Abstract

This chapter presents the modeling of a quad-rotor UAV. A general overview of the quad-rotor helicopter and its operation principle is given. Next, the quad-rotor modeling is addressed using two different approaches: Euler–Lagrange and Newton–Euler. How to derive Lagrange’s equations from Newton’s equations is also shown. Finally, the author presents also the Newton–Euler modeling for an “X-Flyer” quad-rotor configuration.

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The complete dynamics of an aircraft, taking into account aero-elastic effects, flexibility of the wings, internal dynamics of the engine and the whole set of changing variables are quite complex and somewhat unmanageable for the purposes of control. Therefore, it is interesting to consider a simplified model of an aircraft formed by a minimum number of states and inputs, but retaining the main features that must be considered when designing control laws for a real aircraft.

This chapter deals with the modeling of a quad-rotor rotorcraft, and is organized as follows. Section 2.1 gives a general overview of the quad-rotor aerial vehicle and its operation principle. Next, Sect. 2.2 deals with the quad-rotor modeling, presenting two different approaches: Euler–Lagrange in Sect. 2.2.1 and Newton–Euler in Sect. 2.2.2. Subsequently, it is shown in Sect. 2.2.3 how to derive Lagrange’s equations from Newton’s equations. Section 2.2.4 presents a Newton–Euler modeling for an “X-Flyer” quad-rotor configuration. Finally, some concluding remarks are presented in Sect. 2.3.

2.1 The Quad-Rotor Mini-Rotorcraft

The quad-rotor mini-rotorcraft is controlled by the angular speeds of four electric motors as shown in Fig. 2.1. Each motor produces a thrust and a torque, whose combination generates the main thrust, the yaw torque, the pitch torque, and the roll torque acting on the quad-rotor. Conventional helicopters modify the lift force by varying the collective pitch. Such aerial vehicles use a mechanical device known as swashplate. This system interconnects servomechanisms and blade pitch links in order to change the rotor blades pitch angle in a cyclic manner, so as to obtain the pitch and roll control torques of the vehicle. In contrast, the quad-rotor does not have a swashplate and has constant pitch blades. Therefore, in a quad-rotor we can only vary the angular speed of each one of the four rotors to obtain the pitch and roll control torques.

Fig. 2.1

The quad-rotor control input

From Fig. 2.1 it can be observed that the motor M _i (for i=1,…,4) produces the force f _i , which is proportional to the square of the angular speed, that is, $f_{i} = kw_{i}^{2}$. Given that the quad-rotor’s motors can only turn in a fixed direction, the produced force f _i is always positive. The front (M ₁) and the rear (M ₃) motors rotate counter-clockwise, while the left (M ₂) and right (M ₄) motors rotate clockwise. With this arrangement, gyroscopic effects and aerodynamic torques tend to cancel in trimmed flight. The main thrust u is the sum of individual thrusts of each motor. The pitch torque is a function of the difference f ₁−f ₃, the roll torque is a function of f ₂−f ₄, and the yaw torque is the sum $\tau_{M_{1}} + \tau_{M_{2}} + \tau_{M_{3}} + \tau_{M_{4}}$, where $\tau_{M_{i}}$ is the reaction torque of motor i due to shaft acceleration and blades drag. The motor torque is opposed by an aerodynamic drag τ _drag, such that

(2.1)

where I _rot is the moment of inertia of a rotor around its axis. The aerodynamic drag is defined as

(2.2)

where ρ is the air density, the frontal area of the moving shape is defined by A, and v is its velocity relative to the air. In magnitude, the angular velocity ω is equal to the linear velocity v divided by the radius of rotation r

(2.3)

The aerodynamic drag can be rewritten as

(2.4)

where k _drag>0 is a constant depending on the air density, the radius, the shape of the blade and other factors. For quasi-stationary maneuvers, ω is constant, then

(2.5)

Forward pitch motion is obtained by increasing the speed of the rear motor M ₃ while reducing the speed of the front motor M ₁. Similarly, roll motion is obtained using the left and right motors. Yaw motion is obtained by increasing the torque of the front and rear motors (τ _M1 and τ _M3, respectively) while decreasing the torque of the lateral motors (τ _M2 and τ _M4, respectively). Such motions can be accomplished while maintaining the total thrust constant, see Fig. 2.2.

Fig. 2.2

Pitch, roll and yaw torques of the quad-rotor

2.2 Quad-Rotor Dynamical Model

The quad-rotor model is obtained by representing the aircraft as a solid body evolving in a three dimensional space and subject to the main thrust and three torques: pitch, roll and yaw.

2.2.1 Euler–Lagrange Approach

Let the generalized coordinates of the rotorcraft be expressed by

(2.6)

where ξ=(x,y,z)∈ℝ³ denotes the position vector of the center of mass of the quad-rotor relative to a fixed inertial frame . The rotorcraft’s Euler angles (the orientation of the rotorcraft) are expressed by η=(ψ,θ,ϕ)∈ℝ³, ψ is the yaw angle around the z-axis, θ is the pitch angle around the y-axis and ϕ is the roll angle around the x-axis (see [33] and [5]). An illustration of the generalized coordinates of the rotorcraft is shown in Fig. 2.3. Define the Lagrangian

(2.7)

where is the translational kinetic energy, is the rotational kinetic energy, U=mgz is the potential energy of the rotorcraft, z is the rotorcraft altitude, m denotes the mass of the quad-rotor, Ω is the vector of the angular velocity, I is the inertia matrix and g is the acceleration due to gravity. The angular velocity vector ω resolved in the body-fixed frame is related to the generalized velocities (in the region where the Euler angles are valid) by means of the standard kinematic relationship [38]

(2.8)

where

(2.9)

then

(2.10)

Define

(2.11)

where

(2.12)

so that

(2.13)

Thus, the matrix acts as the inertia matrix for the full rotational kinetic energy of the quad-rotor, expressed directly in terms of the generalized coordinates η.

Fig. 2.3

The quad-rotor in an inertial frame. f ₁, f ₂, f ₃, f ₄ represent the thrust of each motor, ψ, θ and ϕ represent the Euler angles, and u is the main thrust

The model of the full rotorcraft dynamics is obtained from Euler–Lagrange equations with external generalized forces

(2.14)

where $\mathbf{F}_{\xi} = R\hat{\mathbf{F}} \in \mathbb{R}^{3}$ is the translational force applied to the rotorcraft due to main thrust, τ∈ℝ³ represents the yaw, pitch and roll moments and R denotes the rotational matrix. R(ψ,θ,ϕ)∈SO(3) represents the orientation of the aircraft relative to a fixed inertial frame:

(2.15)

where c _θ stands for cosθ and s _θ for sinθ. From Fig. 2.1, it follows that

(2.16)

where u is the main thrust directed out of the bottom of the aircraft and expressed as

(2.17)

and, for i=1,…,4, f _i is the force produced by motor M _i , as shown in Fig. 2.1. Typically $f_{i} = k \omega_{i}^{2}$, where k _i is a constant and ω _i is the angular speed of the ith motor. The generalized torques are thus

(2.18)

where ℓ is the distance between the motors and the center of gravity, and $\tau_{M_{i}}$ is the moment produced by motor M _i , for i=1,…,4, around the center of gravity of the aircraft.

Since the Lagrangian contains no cross terms in the kinematic energy combining with , the Euler–Lagrange equation can be partitioned into dynamics for ξ coordinates and η coordinates. The Euler–Lagrange equation for the translational motion is

(2.19)

then

(2.20)

As for the η coordinates, it can be written

(2.21)

or

(2.22)

Thus one obtains

(2.23)

Defining the Coriolis-centripetal vector

(2.24)

one writes

(2.25)

but can be expressed as

(2.26)

where is referred to as the Coriolis term and contains the gyroscopic and centrifugal terms associated with the η dependence of $\mathbb{J}$. This yields

(2.27)

(2.28)

To simplify let us take

(2.29)

Finally one obtains

(2.30)

(2.31)

(2.32)

(2.33)

(2.34)

(2.35)

where x and y are coordinates in the horizontal plane, z is the vertical position, and $\tilde{\tau}_{\psi}$, $\tilde{\tau}_{\theta}$ and $\tilde{\tau}_{\phi}$ are the yawing moment, pitching moment and rolling moment, respectively, which are related to the generalized torques τ _ψ , τ _θ , τ _ϕ .

2.2.2 Newton–Euler Approach

The general motion of a rigid body in space is a combination of translational and rotational motions. Consider a rigid body moving in inertial space, undergoing both rotations and translations. Let us define now an earth fixed frame and a body-fixed frame , as seen in Fig. 2.4. The center of mass and the body-fixed frame are assumed to coincide. Using Euler angles parametrization, the airframe orientation in space is given by a rotation R from to , where R∈SO(3) is the rotation matrix. Using the Newton–Euler formalism, the dynamics of a rigid body under external forces applied to the center of mass and expressed on earth fixed frame is

(2.36)

where ξ=(x,y,z)^T denotes the position of the center of mass of the airframe with respect to the frame relative to a fixed origin, denotes the linear velocity expressed in the inertial frame, and denotes the angular velocity of the airframe expressed in the body-fixed frame. The mass of the rigid body is denoted by m, and I∈ℝ^3×3 denotes the constant inertia matrix around the center of mass (expressed in the body-fixed frame ). denotes the skew-symmetric matrix of the vector ω. represents the vector of the principal non-conservative forces applied to the object; including thrusts T _f and drag terms associated with the rotors. is derived from differential thrust associated with pairs of rotors along with aerodynamics effects and gyroscopic effects.

Fig. 2.4

The quad-rotor in an inertial frame. f _i represent the thrust of motor i and T _f is the main thrust

Translational Force and Gravitational Force

The only forces acting on the body are given by the translational force T _f and the gravitational force g. From Fig. 2.4, the thrust applied to the vehicle is

(2.37)

where the lift f _i generated by a rotor in free air can be modeled as $f_{i} k \omega_{i}^{2}$ in the z-direction, where k>0 is a constant and ω _i is the angular speed of the ith motor. Equation (2.37) can be rewritten as

(2.38)

Then

(2.39)

The gravitational force applied to the vehicle is

(2.40)

This yields

(2.41)

Torques

Due to the rigid rotor constraint, the dynamics of each rotor disc around its axis of rotation can be treated as a decoupled system in the generalized variable ω _i , denoting angular velocity or a rotor around its axis. The torque exerted by each electrical motor is denoted by $\tau_{M_{i}}$. The motor’s torque is opposed by an aerodynamic drag $\tau_{\mathrm{drag}} = k_{\tau} \omega_{i}^{2}$. Using Newton’s second law one has

(2.42)

where I _M is the angular moment of the ith motor and k _τ >0 is a constant for quasi-stationary maneuvers in free flight. In steady state, i.e., when $\dot{\omega}_{i} = 0$, the yaw torque is

(2.43)

The generalized torques are thus

(2.44)

where ℓ represents the distance between the motors and the center of gravity. Rewriting (2.44) one has

(2.45)

(2.46)

(2.47)

where τ _ψ , τ _θ and τ _ϕ are the generalized torques (yawing moment, pitching moment and rolling moment, respectively). Each rotor may be thought of as a rigid disc rotating around the axis E _z in the body-fixed frame, with angular velocity ω _i . The rotor’s axis of rotation is itself moving with the angular velocity of the frame. This leads to the following gyroscopic torques applied to the airframe:

(2.48)

This yields

(2.49)

Rewriting (2.36), one has

(2.50)

2.2.3 Newton’s Equations to Lagrange’s Equations

Using the classical yaw, pitch and roll Euler angles (ψ,θ,ϕ) applied in aeronautical applications [5, 33], the rotation matrix can be expressed as

(2.51)

The equations in (2.50) can be separated into the ξ coordinates dynamics and the η dynamics. Rewriting the ξ dynamics one has

(2.52)

where

From Figs. 2.3 and 2.4 one has u=T _f , this yields

(2.53)

(2.54)

(2.55)

From Newton–Euler formalism, one obtains in (2.53)–(2.55) the same equations as obtained in (2.30)–(2.32).

2.2.4 Newton–Euler Approach for an X-type Quad-Rotor

The quad-rotor model presented in Sects. 2.2.1 and 2.2.2 considers front and rear motors aligned with the longitudinal axis, and left and right motors aligned with the lateral axis. This section introduces an “X-type” quad-rotor flying configuration, considering two frontal motors and two rear motors. The quad-rotor dynamical model equations are based on Newton–Euler formalism, where the nonlinear dynamics is obtained in North-East-Down (NED) inertial and body-fixed coordinates, see Fig. 2.5. Let {N,E,D} represent the inertial reference frame and {X,Y,Z} represent the body-fixed frame. The position vector of the center of mass of the rotorcraft is denoted by ξ=(x,y,z)^T, representing the position coordinates of the vehicle relative to the NED inertial frame. The orientation vector of the aircraft with respect to the inertial frame is expressed by η=(ψ,θ,ϕ)^T, where ψ, θ and ϕ are the yaw, pitch and roll Euler angles, respectively. The full nonlinear dynamics of the quad-rotor can be expressed as

(2.56)

(2.57)

where R∈SO(3) is a rotation matrix that associates the inertial frame with the body-fixed frame, F denotes the total force applied to the vehicle, m is the total mass, g denotes the gravitational constant, Ω represents the angular velocity of the vehicle expressed in the body-fixed frame, I describes the inertia matrix, and τ is the total torque.

Fig. 2.5

NED diagram of the quad-rotor dynamical model

Let $u=\sum_{i=1}^{4}T_{i}$ be the force applied to the vehicle, which is generated by the four rotors. Assuming that this force has only one component in the Z direction, the total force can be written as F=(0,0,−u)^T. The rotation matrix R is defined as

$$ R=\left [ \begin{array}{c@{\quad}c@{\quad}c} c_{\theta }c_{\psi } & s_{\phi }s_{\theta }c_{\psi } - c_{\phi }s_{\psi } & c_{\phi }s_{\theta }c_{\psi } + s_{\phi }s_{\psi } \\ c_{\theta }s_{\psi } & s_{\phi }s_{\theta }s_{\psi } + c_{\phi }c_{\psi } & c_{\phi }s_{\theta }s_{\psi } - s_{\phi }c_{\psi } \\ -s_{\theta } & s_{\phi }c_{\theta } & c_{\phi }c_{\theta } \end{array} \right ] $$

(2.58)

where c _.=cos(.) and s _.=sin(.). Let us define now an auxiliary vector $\tilde{\tau}$ related to the generalized torque τ and based on (2.57):

(2.59)

where and W is [38]:

(2.60)

Using (2.56)–(2.59), the quad-rotor dynamical model can be represented by

(2.61)

(2.62)

(2.63)

(2.64)

(2.65)

(2.66)

In the “X-type” quad-rotor model, the motors M ₁ and M ₃ rotate clockwise, while motors M ₂ and M ₄ rotate counter-clockwise. Assuming that total thrust approximately counteracts gravity, i.e., the quad-rotor is in hover or near-hover flight conditions, we can consider that each thrust can be modeled as $\tau_{i}=Cw_{i}^{2}$, where C is a constant value depending on the rotor characteristics and w _i denotes the speed of the rotor i [6]. For simplicity, it is also assumed that the torque τ _i generated by each rotor is proportional to its lift force, then τ _i =C _M T _i . Taking into account the previous assumptions, we can obtain the generalized torques as

$$ \left [ \begin{array}{c} \tau_{\psi } \\ \tau_{\theta } \\ \tau_{\phi } \end{array} \right ] =\left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} -C_{M} & C_{M} & -C_{M} & C_{M}\\ -l & -l & l & l \\ -l & l & l & -l \end{array} \right ] \left [ \begin{array}{c} T_{1} \\ T_{2} \\ T_{3} \\ T_{4}\end{array} \right ] $$

(2.67)

where l represents the distance between the center of mass and the center of the rotor.

2.3 Concluding Remarks

In this chapter Euler–Lagrange and Newton–Euler approaches have been applied for obtaining a simplified model of a quad-rotor rotorcraft. The model is formed by a minimum number of states and inputs, but retains the main features that must be considered when designing control laws. Two quad-rotor configurations were analyzed. The first configuration addressed a classical motor arrangement having one pair of motors aligned with the longitudinal axis while the other pair is aligned with the translational axis. The second configuration addressed an “X-Flyer” motor arrangement, having two frontal motors and two rear motors.

The models obtained here will be used in later sections for designing control laws devoted to attitude stabilization and autonomous positioning.