**2. System description**

developing effective control algorithms for UAVs [1–4]. In many cases, the development of new control strategies requires the use of software and platforms which are able to simulate the operation of the UAVs in order to perform experimental tests for evaluating the different designs. The use of this kind of tools increases the productivity and reduces the development time. For this purpose, different laboratory test rigs have been specifically designed for teaching and research in flight dynamics and control. One such platform is the laboratory helicopter used in this research, namely the Twin Rotor MIMO System (TRMS) [5]. The TRMS is a nonlinear, multivariable and underactuated system, characterised by a coupling effect between the dynamics of the propellers and the body structure, which is caused by the actionreaction principle originated in acceleration and deceleration of the motor-propeller groups. All these features make the control of the TRMS to be perceived as a challenging engineering problem (note that the TRMS, and other laboratory platforms with similar dynamics are more difficult to control than a real helicopter platform [6]). The achievement of an accurate system dynamics model is a challenging problem, whilst, at the same time, an important issue is to

The development of the dynamic model for the TRMS has been studied by an important number of researches. Ahmad et al. presented mathematical models for the dynamic characterisation of the TRMS, using a black box system identification technique [7] and radial basis function (RBF) networks [8]. Shaheed modelled the dynamics of the TRMS by means of a nonlinear autoregressive process through external input (NARX) approach with a feedforward neural work and a resilient propagation (RPROP) algorithm [9]. Rahideh and Shaheed have also contributed to the study of the TRMS dynamics by using both Newton- and Lagrange-based methods [10], and two models based on neural networks using Levenberg-Marquardt (LM) and gradient descent (GD) algorithms [11]. Toha and Tokhi presented an adaptive neuro-fuzzy inference system (ANFIS) network design, which was deployed and used for the TRMS modelling [12]. Finally, Tastemirov et al. developed a complete dynamic

On the other hand, the design of the control system for the TRMS has been widely discussed through several investigations. Ahmad et al. developed the dynamic model and implemented a feed-forward/open-loop control [14] and a linear quadratic Gaussian control [15]. López-Martínez et al. studied the design of a longitudinal controller based on Lyapunov functions [16], and the application of a nonlinear *L*2 controller [17]. Rahideh et al. presented an experimental implementation of an adaptive dynamic nonlinear model inversion control law using artificial neural networks [18]. Other interesting works are those of Tao et al. who designed a parallel distributed fuzzy linear quadratic regulator (LQR) controller [19]. Studies of Reynoso-Meza et al. developed a holistic multi-objective optimisation design technique for controller tuning [20], or the use of a particle swarm optimisation (PSO) algorithm for the proportional-

integral-derivative (PID) controller optimisation developed by Coelho et al. [21].

The aim of the present research is to develop a nonlinear cascade-based control algorithm in order to locally guarantee an asymptotically and exponentially stable behaviour of the controlled generalised coordinates of the TRMS. Additionally, the effectiveness of the proposed

develop accurate and efficient control systems.

266 Nonlinear Systems - Design, Analysis, Estimation and Control

TRMS model using the Euler-Lagrange method [13].

The TRMS (see **Figure 1**) is a laboratory helicopter platform manufactured by *Feedback Instruments Ltd©*. The TRMS is composed of two propellers that are perpendicular to each other and placed in the extreme of a beam that can rotate freely in both vertical and horizontal planes. Each propeller is driven by a DC motor, thus forming the main and tail rotor of the platform. A main feature of the TRMS is that its movement, unlike a real helicopter, is not achieved by varying the angle of attack of the blades. In this case, the movement of the platform is gotten by means of the variation in the angular velocity of each propeller, which is caused by the change in the control input voltage of each motor.

**Figure 1.** Twin rotor MIMO system.

This constructive simplification in the TRMS model substantially complicates the dynamics of the system, because a coupling effect between rotors dynamics and the body of the model appears. This effect is caused by the action-reaction principle originated in acceleration and deceleration of the motor-propeller groups.

In addition, the TRMS is an underactuated system. This implies that the number of variables that act as control inputs (voltages applied to the main and the tail rotor; and respectively) is lower than the number of degrees of freedom (DoF) of the system. The DoF are: the pitch () and the yaw () angles, both measured by digital encoders, as well as the angular velocities of the rotors ( for the main rotor and for the tail), both measured by DC tachometers. Finally, we have to remark that the laboratory platform is locked mechanically, so it cannot move more than ±2.82 rad in the horizontal plane from −1.05 to +1.22 rad in the vertical plane [22]. In other words, −2.82 rad ≤ ≤ + 2.82 rad and −1.05 rad ≤ ≤ + 1.22 rad.

## **2.1. Dynamic model of the TRMS**

The development of an efficient control algorithm requires a model that represents the dynamic behaviour of the platform under study as accurately as possible. In the particular case of the Twin Rotor MIMO System, the modelling has been addressed from several approaches [7– 13]. However, not all of them provide a model that represents the entire complex dynamic behaviour of this experimental platform. For instance, models based on identification techniques have difficulties in representing the effects of coupling, which are characteristic in this platform [7], and neuronal networks and learning algorithms allow obtaining accurate models, but limited to a range of input values and frequencies [11]. Based on previous works developed for the dynamic model of this platform [13, 22–24], a detailed dynamic model of the TRMS has been developed by dividing the whole dynamics of the system in their electrical and mechanical parts. This approach allows not only to adequately capture the complex dynamics behaviour of the TRMS but also the development of novel control algorithms based on nested feedback loops that offer a higher performance than classical control schemes. Moreover, the use of the Euler-Lagrange method in the modelling of the mechanical structure of the TRMS allows a higher adjustment with the real control laboratory platform in comparison with other analytical methods based on the Newtonian approach [25]. The dynamic modelling has been developed in two stages and validated by our research group by means of experimental identification trials. It is presented in the following subsections. The first subsection illustrates the dynamic model of the electrical part, and the second depicts the dynamic model of the mechanical part of the system.

## *2.1.1. Dynamics of the electrical part*

The electrical part of the system is formed by the interface circuit and the DC motors of the main and tail rotors. The interface circuit is the internal electrical circuit that adapts the input control voltages, applied in MATLAB®/Simulink® ( for the main rotor and for the tail rotor), to the actual voltage value of the DC motors ( for the main rotor and for the tail rotor). This interface can be modelled as a linear relationship [13], obtaining the following result:

This constructive simplification in the TRMS model substantially complicates the dynamics of the system, because a coupling effect between rotors dynamics and the body of the model appears. This effect is caused by the action-reaction principle originated in acceleration and

In addition, the TRMS is an underactuated system. This implies that the number of variables that act as control inputs (voltages applied to the main and the tail rotor; and respectively) is lower than the number of degrees of freedom (DoF) of the system. The DoF are: the pitch () and the yaw () angles, both measured by digital encoders, as well as the angular velocities

Finally, we have to remark that the laboratory platform is locked mechanically, so it cannot move more than ±2.82 rad in the horizontal plane from −1.05 to +1.22 rad in the vertical plane

The development of an efficient control algorithm requires a model that represents the dynamic behaviour of the platform under study as accurately as possible. In the particular case of the Twin Rotor MIMO System, the modelling has been addressed from several approaches [7– 13]. However, not all of them provide a model that represents the entire complex dynamic behaviour of this experimental platform. For instance, models based on identification techniques have difficulties in representing the effects of coupling, which are characteristic in this platform [7], and neuronal networks and learning algorithms allow obtaining accurate models, but limited to a range of input values and frequencies [11]. Based on previous works developed for the dynamic model of this platform [13, 22–24], a detailed dynamic model of the TRMS has been developed by dividing the whole dynamics of the system in their electrical and mechanical parts. This approach allows not only to adequately capture the complex dynamics behaviour of the TRMS but also the development of novel control algorithms based on nested feedback loops that offer a higher performance than classical control schemes. Moreover, the use of the Euler-Lagrange method in the modelling of the mechanical structure of the TRMS allows a higher adjustment with the real control laboratory platform in comparison with other analytical methods based on the Newtonian approach [25]. The dynamic modelling has been developed in two stages and validated by our research group by means of experimental identification trials. It is presented in the following subsections. The first subsection illustrates the dynamic model of the electrical part, and the second depicts the dynamic model of the

The electrical part of the system is formed by the interface circuit and the DC motors of the main and tail rotors. The interface circuit is the internal electrical circuit that adapts the input control voltages, applied in MATLAB®/Simulink® ( for the main rotor and for the tail rotor), to the actual voltage value of the DC motors ( for the main rotor and for the tail

[22]. In other words, −2.82 rad ≤ ≤ + 2.82 rad and −1.05 rad ≤ ≤ + 1.22 rad.

for the tail), both measured by DC tachometers.

deceleration of the motor-propeller groups.

268 Nonlinear Systems - Design, Analysis, Estimation and Control

of the rotors ( for the main rotor and

**2.1. Dynamic model of the TRMS**

mechanical part of the system.

*2.1.1. Dynamics of the electrical part*

$$
\omega\_m = k\_{u\_m} u\_m \tag{1}
$$

$$
\sigma\_t = k\_{u\_t} u\_t \tag{2}
$$

where and denote the constant gains for the main and tail rotors, respectively. With regard to the DC motors, there are two identical permanent magnet motors, one in each rotor of the TRMS, with the only difference of the mechanical loads (the propellers). Bearing in mind that the dynamics of the motor´s current can be neglected [13], the DC motor dynamics for the main rotor and the tail rotor are the following ones:

$$
\upsilon\_m = R\_m i\_m + k\_{\upsilon\_m} o \rho\_m \tag{3}
$$

$$
\sigma\_t = R\_t i\_t + k\_{\upsilon\_t} o\_t \tag{4}
$$

where and are the motor currents (the subscripts *<sup>m</sup>* and *<sup>t</sup>* mean "main" and "tail"), and represent the motor resistances, and and denote the electromotive forces of each motor ( and represent the angular velocities of the each motor). On the other hand, the electromechanical balance of the torques acting on each motor is expressed as:

$$I\_{m\_l} \dot{o}\_m = k\_{t\_m} i\_m - f\_{v\_m} o\_m - C\_{\underline{Q}\_m} o\_m \left| o\_m \right| \tag{5}$$

$$I\_{t\_l} \dot{o}\_l = k\_{t\_l} i\_l - f\_{\upsilon\_l} o\_l - C\_{\underline{Q}\_l} o\_l \left| o\_l \right| \tag{6}$$

being 1 and 1 are the moment of the inertia rotors, and denote the electromechanical torques generated by the DC motors, and are the friction torques and and illustrate the aerodynamic torques.

After substituting the expression for the current intensity of the respective motors [obtained from Eqs. (3) and (4)] and the linear relationships for the interface circuit Eqs. (1) and (2), in Eqs. (5) and (6), and after operating and rearranging terms, the following two equations are yielded for the main and tail rotors of the TRMS:

$$I\_{m\_l} \dot{\alpha}\_m = \frac{k\_{l\_m}}{R\_m} k\_{u\_m} u\_m - \left(\frac{k\_{l\_m} k\_{v\_m}}{R\_m} + f\_{v\_m}\right) \phi\_m - C\_{Q\_m} \phi\_m \left| \phi\_m \right| \tag{7}$$

$$I\_{t\_l} \dot{o}\_l = \frac{k\_{t\_l}}{R\_t} k\_{u\_l} u\_t - \left(\frac{k\_{t\_l} k\_{v\_l}}{R\_t} + f\_{v\_l} \right) o\_l - C\_{\underline{Q}\_l} o\_l \left| o\_l \right| \tag{8}$$

The dynamics of the electrical part of the TRMS is now expressed in a matrix form, using the following compact notation:

$$\dot{\boldsymbol{\alpha}}(t) = \mathbf{N}\boldsymbol{\mu}(t) + \boldsymbol{\Gamma}\left(\boldsymbol{\alpha}(t)\right) \tag{9}$$

where *<sup>ω</sup>* <sup>=</sup> , and *<sup>u</sup>* <sup>=</sup> , represent the vector of angular velocities and the input control voltages, respectively, and, *N*<sup>=</sup> (, ) and () <sup>=</sup> , are defined by:

$$\mathbf{N} = \begin{bmatrix} n\_m & \mathbf{0} \\ \mathbf{0} & n\_t \end{bmatrix} = \begin{bmatrix} \frac{k\_{l\_m} k\_{u\_m}}{I\_{m\_l} R\_m} & \mathbf{0} \\\\ \mathbf{0} & \frac{k\_{l\_t} k\_{u\_t}}{I\_{l\_t} R\_t} \end{bmatrix} \tag{10}$$

$$\Gamma\left(o\boldsymbol{o}(t)\right) = \begin{bmatrix} \Gamma\_m\\ \Gamma\_t \end{bmatrix} = \begin{bmatrix} -\left(\frac{k\_{t\_m}k\_{\boldsymbol{\nu}\_m}}{R\_m} + f\_{\boldsymbol{\nu}\_m}\right)\frac{o\boldsymbol{o}\_m}{I\_{m\_l}} - \frac{C\_{\mathcal{Q}\_m}}{I\_{m\_l}}o\boldsymbol{o}\_m|o\boldsymbol{o}\_m|\\ -\left(\frac{k\_{t\_l}k\_{\boldsymbol{\nu}\_l}}{R\_l} + f\_{\boldsymbol{\nu}\_l}\right)\frac{o\boldsymbol{o}\_l}{I\_{l\_l}} - \frac{C\_{\mathcal{Q}\_l}}{I\_{l\_l}}o\boldsymbol{o}\_l|o\boldsymbol{o}\_l| \end{bmatrix} \tag{11}$$

Finally, in order to complete the dynamic model of the electrical part of the TRMS, **Tables 1** and **2** show the parameters used in the model, indicating the description of the parameters, their values and their corresponding units. These values, which are based on the data presented in [13], have been experimentally tuned and validated in the dynamics identification tests that we have performed during our research.


**Table 1.** Parameters of the main rotor.

Eqs. (5) and (6), and after operating and rearranging terms, the following two equations are

w

w

*I ku f C R R* (7)

 ww

 ww

*ω Nu ω* & (*tt t* ) = + ( ) **Γ**( ( )) (9)

represent the vector of angular velocities and the

) and () <sup>=</sup> ,

are defined

*I ku f C R R* (8)

*v*

*mm u m m Q mm*

*v*

*t t ut t Qt t*

æ ö = -+- ç ÷ è ø

*t t t*

The dynamics of the electrical part of the TRMS is now expressed in a matrix form, using the

1

ê ú é ù ê ú = = ê ú ê ú ë û ê ú

*m m mm <sup>t</sup> t v <sup>t</sup> <sup>Q</sup>*

*m m*

é ù

ê ú ë û

*m m m m*

w

*t v m Q*

*k k C <sup>f</sup> R II*

*k k C <sup>f</sup> R II*

é ù æ ö ê ú -+ - ç ÷ é ù <sup>G</sup> è ø = = ê ú


Finally, in order to complete the dynamic model of the electrical part of the TRMS, **Tables 1** and **2** show the parameters used in the model, indicating the description of the parameters, their values and their corresponding units. These values, which are based on the data presented in [13], have been experimentally tuned and validated in the dynamics identification tests that

*t t t t*

w

*t tt*

ë û è ø

*t u*

*k k*

0

*n k k*

*t t u*

0

( ) 1 1

ë û G æ ö

*m m m*

*n I R*

0

1

*I R*

*t t*

*N* (10)

*t t*

1 1

**Γ** *ω* (11)

*v t t*

*v m m*

w w

w w

0

æ ö = -+- ç ÷

è ø

*m m m*

*v*

*m m k kk*

*v*

*t t k kk*

& *<sup>t</sup> t t*

and *<sup>u</sup>* <sup>=</sup> ,

input control voltages, respectively, and, *N*<sup>=</sup> (,

( )

*t*

we have performed during our research.

*t t*

& *<sup>m</sup> m m*

*t t*

yielded for the main and tail rotors of the TRMS:

1

following compact notation:

where *<sup>ω</sup>* <sup>=</sup> ,

by:

w

270 Nonlinear Systems - Design, Analysis, Estimation and Control

1

w


**Table 2.** Parameters of the tail rotor.

#### *2.1.2. Dynamics of the mechanical part*

In the development of the dynamic model of the mechanical part, we consider the mechanics of the TRMS as an assembly of the following three components explained next. The first component is formed by the two rotors, their shields and the free-free beam that links together both rotors. The second component consists in the counterbalance and counterweight beam, and finally, the third component is the pivoted beam. **Figure 2** helps to clarify the different components considered in the dynamics of the mechanical part of the system. From the previous division, and bearing in mind the notation used in **Figures 3** and **4**, the development of the dynamic model is achieved by means of the application of the Euler-Lagrange formulation. It can be summarised in the following steps:


**Figure 2.** Twin rotor MIMO system (TRMS) prototype platform.

Nonlinear Cascade-Based Control for a Twin Rotor MIMO System http://dx.doi.org/10.5772/64875 273

**Figure 3.** View of the TRMS on a vertical plane.

and finally, the third component is the pivoted beam. **Figure 2** helps to clarify the different components considered in the dynamics of the mechanical part of the system. From the previous division, and bearing in mind the notation used in **Figures 3** and **4**, the development of the dynamic model is achieved by means of the application of the Euler-Lagrange formu-

lation. It can be summarised in the following steps:

272 Nonlinear Systems - Design, Analysis, Estimation and Control

**Figure 2.** Twin rotor MIMO system (TRMS) prototype platform.

**2.** Evaluation of the kinetic energy.

**3.** Evaluation of the potential energy.

**4.** Obtaining the equations of motion.

**1.** Resolution of the forward kinematics of the three subsystems.

**Figure 4.** View of the TRMS on a horizontal plane.

#### *2.1.2.1. Resolution of the forward kinematics of the system*

The problem of direct kinematics of the TRMS consists in determining the spatial position of the three subsystems considered, according to the reference system located in the upper part of the platform (see **Figures 3** and **4**). Using the Denavit-Hartenberg method, we can express the position of a point on each subsystem (1, 2, 3) parameterised by 1, 2, 3, which represents the distances between the considerate points and the reference system associated to each subsystem. The results of these positions are expressed in the following three equations (where: <sup>≡</sup> sin, <sup>≡</sup> cos, <sup>≡</sup> sin and <sup>≡</sup> cos):

$$\mathbf{P}\_{\mathbf{l}} = \begin{bmatrix} P\_{\mathbf{l}\_x} & P\_{\mathbf{l}\_y} & P\_{\mathbf{l}\_z} \end{bmatrix}^T = \begin{bmatrix} -R\_{\mathbf{l}}S\_{\phi}C\_{\psi} + hC\_{\phi} & R\_{\mathbf{l}}C\_{\phi}C\_{\psi} + hS\_{\phi} & R\_{\mathbf{l}}S\_{\psi} \end{bmatrix}^T \tag{12}$$

$$\mathbf{P}\_{2} = \begin{bmatrix} P\_{2\_x} & P\_{2\_y} & P\_{2\_z} \end{bmatrix}^T = \begin{bmatrix} -R\_2 S\_\phi S\_\psi + hC\_\phi & R\_2 C\_\phi S\_\psi + hS\_\phi & -R\_2 C\_\psi \end{bmatrix}^T \tag{13}$$

$$P\_3 = \begin{bmatrix} P\_{\mathfrak{Z}\_x} & P\_{\mathfrak{Z}\_y} & P\_{\mathfrak{Z}\_z} \end{bmatrix}^T = \begin{bmatrix} R\_3 C\_\phi & R\_3 S\_\phi & 0 \end{bmatrix}^T \tag{14}$$

#### *2.1.2.2. Evaluation of the kinetic energy*

In order to carry out the evaluation of the total kinetic energy of the TRMS, it is necessary to calculate the kinetic energy corresponding to each of the three subsystems previously defined. Starting with the first subsystem, its kinetic energy, *T*1, yields:

$$T\_1 = \frac{1}{2} \int \left| \mathbf{v}\_1 \right|^2 dm \left( R\_1 \right) = \frac{1}{2} J\_1 \left( C\_\psi \,^2 \dot{\phi}^2 + \dot{\psi}^2 \right) + \frac{1}{2} h^2 m\_{T\_1} \dot{\phi}^2 - h S\_\psi J\_{T\_1} m\_{T\_1} \dot{\phi} \dot{\psi} \tag{15}$$

$$\left| \left| \boldsymbol{\nu}\_{1} \right|^{2} = \left( R\_{\mathrm{l}}^{\,^{2}} C\_{\mathrm{yr}}^{\,^{2}} + h^{2} \right) \dot{\phi}^{2} + R\_{\mathrm{l}}^{\,^{2}} \dot{\boldsymbol{\nu}}^{\,^{2}} - 2 \, R\_{\mathrm{l}} h S\_{\mathrm{yr}} \dot{\phi} \dot{\boldsymbol{\nu}} \tag{16}$$

where and represent the yaw and the pitch angle, respectively, and *mT1* , *lT1* , and *J*<sup>1</sup> are obtained from the following expressions:

$$\int dm(R\_{\rm l}) = m\_{\rm m} + m\_{\rm mr} + m\_{\rm ms} + m\_{\rm t} + m\_{\rm tr} + m\_{\rm ts} = m\_{T\_{\rm l}} \tag{17}$$

$$I\_{T\_1} = \frac{\int R\_l dm(R\_1)}{\int dm(R\_1)} = \frac{\left(\frac{m\_{\rm t}}{2} + m\_{\rm tr} + m\_{\rm ts}\right)l\_t - \left(\frac{m\_{\rm m}}{2} + m\_{\rm mr} + m\_{\rm ms}\right)l\_m}{m\_{T\_1}}\tag{18}$$

$$J\_1 = \left(\frac{1}{3}m\_\text{t} + m\_\text{tr} + m\_\text{ts}\right)l\_\text{f}^{\text{-}2} + \left(\frac{1}{3}m\_\text{m} + m\_\text{mr} + m\_\text{ms}\right)l\_\text{m}\text{ }^2 + m\_\text{tr}r\_\text{ts}\text{ }^2 + \frac{1}{2}m\_\text{ms}r\_\text{ms}\text{ }^2\tag{19}$$

On the other hand, the kinetic energy for the second subsystem, *T*2, results in:

$$T\_2 = \frac{1}{2} \int \left| \mathbf{v}\_2 \right|^2 dm \left( R\_2 \right) = \frac{1}{2} J\_2 \left( S\_\psi \,^2 \dot{\phi}^2 + \dot{\psi}^2 \right) + \frac{1}{2} h^2 m\_{T\_2} \dot{\phi}^2 + h C\_\psi J\_{T\_2} m\_{T\_2} \dot{\phi} \dot{\psi} \tag{20}$$

$$\left| \left| \boldsymbol{\nu}\_{2} \right|^{2} = \left( R\_{2} \, ^{2}S\_{\psi} \, ^{2} + h^{2} \right) \dot{\phi}^{2} + R\_{2} \, ^{2} \dot{\boldsymbol{\nu}}^{2} + 2 \, R\_{2} h C\_{\psi} \dot{\phi} \dot{\nu} \tag{21}$$

in which the terms 2 , 2 and *J*2 are the following:

$$\int dm(R\_2) = m\_\text{b} + m\_\text{cb} = m\_{T\_2} \tag{22}$$

$$I\_{T\_2} = \frac{\int R\_2 dm(R\_2)}{\int dm(R\_2)} = \frac{m\_{\rm b}\frac{l\_{\rm b}}{2} + m\_{\rm cb}l\_{mcb}}{m\_{T\_2}}\tag{23}$$

$$J\_2 = \frac{1}{3} m\_b l\_b \,^2 + m\_{cb} l\_{cb} \,^2 \tag{24}$$

On the other hand, the kinetic energy for the third subsystem, *T*3, gives the following result:

$$T\_3 = \frac{1}{2} \int \left| \mathbf{v}\_3 \right|^2 dm \left( R\_3 \right) = \frac{1}{2} J\_3 \dot{\phi}^2 \tag{25}$$

$$\left|\left|\upsilon\_{3}\right|^{2} = R\_{3}{}^{2}\dot{\phi}^{2}\tag{26}$$

being <sup>3</sup> <sup>=</sup> <sup>1</sup> 3ℎ ℎ 2 .

*2.1.2.1. Resolution of the forward kinematics of the system*

274 Nonlinear Systems - Design, Analysis, Estimation and Control

(where: <sup>≡</sup> sin, <sup>≡</sup> cos, <sup>≡</sup> sin and <sup>≡</sup> cos):

Starting with the first subsystem, its kinetic energy, *T*1, yields:

11 1 22 2 = = ++ y

y

1 1 11

obtained from the following expressions:

1

*T*

*l*

1

( )

*2.1.2.2. Evaluation of the kinetic energy*

The problem of direct kinematics of the TRMS consists in determining the spatial position of the three subsystems considered, according to the reference system located in the upper part of the platform (see **Figures 3** and **4**). Using the Denavit-Hartenberg method, we can express the position of a point on each subsystem (1, 2, 3) parameterised by 1, 2, 3, which represents the distances between the considerate points and the reference system associated to each subsystem. The results of these positions are expressed in the following three equations

*T T*

= =- + + é ù é ù ê úë ë û <sup>û</sup> *<sup>P</sup>* (12)

*T T*

*T T*

<sup>=</sup> é ù =- + + - é ù ë û ê úë <sup>û</sup> *<sup>P</sup>* (13)

f

 fy

 fy

 f= = é ù é ù ê úë û ë û *<sup>P</sup>* (14)

 f

 f

> y

f

> y

<sup>1</sup> 1 m mr ms t tr ts ( ) *<sup>T</sup> dm R m m m m m m m* = + + ++ + = ò (17)

*t m*

<sup>ò</sup> (18)

1

2 22 2

æ öæ <sup>ö</sup> = ++ + + + + + ç ÷ç <sup>÷</sup> è øè <sup>ø</sup> (19)

*T*

 fy2 & & *v R C h R R hS* & & (16)

 y

> y

fy

, *lT1*

, and *J*<sup>1</sup> are

 f

 f

In order to carry out the evaluation of the total kinetic energy of the TRMS, it is necessary to calculate the kinetic energy corresponding to each of the three subsystems previously defined.

( ) ( ) <sup>1</sup> 1 1

 y

ò & && & & *T dm R J C T TT <sup>v</sup> h m hS l m* (15)

1 111 1 1 1 *xyz*

2 222 2 2 2 *xyz*

3 333 3 3 0 *xyz*

2 22 2 2 2

( ) <sup>2</sup> 2 2 2 2 22 1 1 1 1 = ++-

where and represent the yaw and the pitch angle, respectively, and *mT1*

f y

f

<sup>t</sup> <sup>m</sup> tr ts mr ms 1 1

æ ö æ ö ++ - + + ç ÷ ç ÷ è ø è ø = = <sup>ò</sup>

1 t tr ts m mr ms ts ts ms ms 11 1 33 2 *J m m m l m m m l mr m r t m*

*<sup>m</sup> <sup>m</sup> R dm R m ml m m l*

( ) 2 2

*dm R m*

*P P P RC RS*

*P P P R S C hC R C C hS R S* fy

*P P P R S S hC R C S hS R C* fy

> Finally, the total kinetic energy of the TRMS, *T*, is obtained as the sum of the kinetic energy of each subsystem (Eqs. (15), (20) and (25)). One obtains the following result:

$$\begin{split} T = T\_1 + T\_2 + T\_3 &= \frac{1}{2} \left( J\_1 C\_{\mathcal{V}} \,^2 + J\_2 S\_{\mathcal{V}} \,^2 + J\_3 + h^2 \left( m\_{T\_1} + m\_{T\_2} \right) \right) \dot{\phi}^2 \\ &+ \frac{1}{2} (J\_1 + J\_2) \dot{\psi} \,^2 + h \left( l\_{T\_2} m\_{T\_2} C\_{\mathcal{V}} - l\_{T\_1} m\_{T\_1} S\_{\mathcal{V}} \right) \dot{\phi} \dot{\nu} \end{split} \tag{27}$$

#### *2.1.2.3. Evaluation of the potential energy*

Following a similar procedure to the one used in the computation of the kinetic energy, the total potential energy of the TRMS, *V*, consists of the sum of the potential energy of each of the three subsystems, the free-free beam (including rotors and shields), the counterbalance beam and the pivoted beam. The following result is obtained:

$$V = V\_1 + V\_2 + V\_3 = \mathbf{g} \left( S\_{\psi} l\_{T\_1} m\_{T\_1} - C\_{\psi} l\_{T\_2} m\_{T\_2} \right) \tag{28}$$

where:

$$V\_1 = \mathbf{g} \int r\_{z\_1}(R\_1) \, dm(R\_1) = \mathbf{g} \int R\_{\mathbf{l}\_z} \, dm(R\_{\mathbf{l}}) = \mathbf{g} S\_{\mathbf{y}} J\_{T\_1} m\_{T\_1} \tag{29}$$

$$V\_2 = \mathbf{g} \left[ r\_{\varepsilon\_2} \left( R\_2 \right) dm \left( R\_2 \right) = \mathbf{g} \left[ P\_{2\_\varepsilon} dm \left( R\_2 \right) = -\mathbf{g} C\_\psi J\_{T\_2} m\_{T\_2} \right. \tag{30}$$

$$V\_3 = \mathbf{g} \int r\_{z\_3} \left( R\_3 \right) dm \left( R\_3 \right) = \mathbf{g} \int P\_{\mathfrak{Z}\_z} dm \left( R\_3 \right) = 0 \tag{31}$$

#### *2.1.2.4. Equations of motion of the TRMS*

The last step in the mechanical dynamic model of the TRMS is obtaining the equations of motion of the system. The first step is the computation of the Lagrangian of the system, defined as the difference between the total kinetic energy, defined in Eq. (27), and the total potential energy, defined in Eq. (28), yielding the following:

$$\begin{split} L = T - V &= \frac{1}{2} \Big( J\_1 C\_{\psi}{}^2 + J\_2 S\_{\psi}{}^2 + J\_3 + h^2 \Big( m\_{T\_1} + m\_{T\_2} \Big) \Big) \dot{\phi}^2 \\ &+ \frac{1}{2} (J\_1 + J\_2) \dot{\psi}^2 + h \Big( l\_{T\_2} m\_{T\_2} C\_{\psi} - l\_{T\_1} m\_{T\_1} S\_{\psi} \Big) \dot{\phi} \dot{\nu} - \operatorname{g} \Big( S\_{\psi} l\_{T\_1} m\_{T\_1} - C\_{\psi} l\_{T\_2} m\_{T\_2} \Big) \end{split} \tag{32}$$

Once the Lagrangian function has been obtained, the equations of motion of the TRMS can be derived using Lagrange's formulation:

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\psi}}\right) - \frac{\partial L}{\partial \varphi} = \sum M\_{\text{fv}}\tag{33}$$

Nonlinear Cascade-Based Control for a Twin Rotor MIMO System http://dx.doi.org/10.5772/64875 277

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\phi}}\right) - \frac{\partial L}{\partial \phi} = \sum M\_{i\hbar} \tag{34}$$

where ∑ and ∑ represent the sum of the torques of the external forces along the vertical and horizontal axes, respectively. The following expressions illustrate several partial results necessary to achieve the equations of motion represented by Eqs. (33) and (34):

$$\frac{\partial L}{\partial \dot{\psi}} = \left(J\_1 + J\_2\right)\dot{\psi} + h\left(l\_{T\_2}m\_{T\_2}C\_{\psi} - l\_{T\_1}m\_{T\_1}S\_{\psi}\right)\dot{\phi}\tag{35}$$

$$\frac{\partial L}{\partial \boldsymbol{\nu}} = \left( (\boldsymbol{J}\_{2} - \boldsymbol{J}\_{1}) \boldsymbol{C}\_{\boldsymbol{\nu}} \boldsymbol{S}\_{\boldsymbol{\nu}} \right) \dot{\boldsymbol{\phi}}^{2} - h \left( \boldsymbol{l}\_{T\_{1}} \boldsymbol{m}\_{T\_{1}} \boldsymbol{C}\_{\boldsymbol{\nu}} + \boldsymbol{l}\_{T\_{2}} \boldsymbol{m}\_{T\_{2}} \boldsymbol{S}\_{\boldsymbol{\nu}} \right) \dot{\boldsymbol{\phi}} \dot{\boldsymbol{\nu}} - \mathbf{g} \left( \boldsymbol{l}\_{T\_{1}} \boldsymbol{m}\_{T\_{1}} \boldsymbol{C}\_{\boldsymbol{\nu}} + \boldsymbol{l}\_{T\_{2}} \boldsymbol{m}\_{T\_{2}} \boldsymbol{S}\_{\boldsymbol{\nu}} \right) \tag{36}$$

$$\frac{\partial L}{\partial \dot{\phi}} = \left(J\_1 C\_{\psi}{}^2 + J\_2 S\_{\psi}{}^2 + J\_3 + h^2 \left(m\_{T\_1} + m\_{T\_2}\right)\right) \dot{\phi} + h \left(l\_{T\_2} m\_{T\_2} C\_{\psi} - l\_{T\_1} m\_{T\_1} S\_{\psi}\right) \dot{\nu} \tag{37}$$

$$\frac{\partial L}{\partial \phi} = 0\tag{38}$$

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\boldsymbol{\nu}}}\right) = h\left(l\_{T\_2}m\_{T\_2}\mathbf{C}\_{\boldsymbol{\Psi}} - l\_{T\_1}m\_{T\_1}\mathbf{S}\_{\boldsymbol{\Psi}}\right)\ddot{\boldsymbol{\phi}} + \left(J\_1 + J\_2\right)\ddot{\boldsymbol{\nu}} - h\left(l\_{T\_1}m\_{T\_1}\mathbf{C}\_{\boldsymbol{\Psi}} + l\_{T\_2}m\_{T\_2}\mathbf{S}\_{\boldsymbol{\Psi}}\right)\dot{\boldsymbol{\phi}}\dot{\boldsymbol{\nu}}\tag{39}$$

$$\begin{split} \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\phi}} \right) &= \left( J\_1 C\_{\psi}{}^2 + J\_2 S\_{\psi}{}^2 + J\_3 + h^2 \left( m\_{T\_1} + m\_{T\_2} \right) \right) \ddot{\phi} - h \left( I\_{T\_1} m\_{T\_1} S\_{\psi}{} - I\_{T\_2} m\_{T\_2} C\_{\psi}{} \right) \ddot{\psi} \\ &- h \left( I\_{T\_1} m\_{T\_1} C\_{\psi}{} + I\_{T\_2} m\_{T\_2} S\_{\psi}{} \right) \dot{\psi}^2 + 2 \left( \left( J\_2 - J\_1 \right) C\_{\psi} S\_{\psi}{} \right) \dot{\phi} \dot{\psi} \end{split} \tag{40}$$

The sum of the external torques in the vertical axis is shown next:

*2.1.2.3. Evaluation of the potential energy*

276 Nonlinear Systems - Design, Analysis, Estimation and Control

*2.1.2.4. Equations of motion of the TRMS*

energy, defined in Eq. (28), yielding the following:

1 2

y

derived using Lagrange's formulation:

123

y

*L T V JC J S J h m m*

=-= + + + +

2

*dt*

y

æ ö ¶ ¶ ç ÷ - =

 y

( ( ))

22 2 2

where:

and the pivoted beam. The following result is obtained:

Following a similar procedure to the one used in the computation of the kinetic energy, the total potential energy of the TRMS, *V*, consists of the sum of the potential energy of each of the three subsystems, the free-free beam (including rotors and shields), the counterbalance beam

> y

y*T T* ò ò (29)

y*T T* ò ò (30)

( ) ( ) ( ) <sup>3</sup> 3 3 3 33 <sup>0</sup> *<sup>z</sup> V g r R dm R g P dm R* = == *<sup>z</sup>* ò ò (31)

*TT T T* (28)

( ) 11 2 2 *V V V V gSlm Cl m* =+ + = - <sup>123</sup> y

( ) ( ) ( ) <sup>1</sup> 1 1 1 1 1 11 *<sup>z</sup> V g r R dm R g P dm R gS l m* = == *<sup>z</sup>*

( ) ( ) ( ) <sup>2</sup> 2 2 2 2 2 22 *<sup>z</sup> V g r R dm R g P dm R gC l m* = *<sup>z</sup>* = = -

The last step in the mechanical dynamic model of the TRMS is obtaining the equations of motion of the system. The first step is the computation of the Lagrangian of the system, defined as the difference between the total kinetic energy, defined in Eq. (27), and the total potential

( ) ( ) ( )

f

*iv*

&

*J J hl m C l m S g Sl m Cl m*

2 2 11 11 2 2

è ø ¶ ¶ & å (33)

yy

(32)

*T T TT TT T T*

fy

1 2

*T T*

yy

Once the Lagrangian function has been obtained, the equations of motion of the TRMS can be

++ + - - -

& & &

*dL L <sup>M</sup>*

 y

$$\begin{aligned} \sum M\_{\,\,lv} &= M\_{T\_m} - M\_{R\_l} - M\_{F\_\psi} + M\_{I\_l} \\ \sum M\_{\,\,lv} &= C\_{T\_m} \alpha\_m \left| \alpha\_m \right| l\_m - C\_{R\_l} \alpha\_l \left| \alpha\_l \right| - \left( f\_{v\_\psi} \dot{\nu} + f\_{c\_\psi} \operatorname{sgn}(\dot{\nu}) \right) + k\_t \dot{\alpha}\_t \end{aligned} \tag{41}$$

where <sup>=</sup> expresses the aerodynamic thrust torque caused by the rotation of the main propeller, <sup>=</sup> denotes the load torque created by air resistance in the tail rotor, <sup>=</sup> ˙ <sup>+</sup> ˙ represents the load torque as a result of the friction (including the viscous effects and the Coulomb friction), and <sup>=</sup> ˙ represents the inertial counter torque that is caused by the reaction produced by a change in the rotational speed of the tail rotor.

On the other hand, the sum of the external torques in the horizontal axis is as follows:

$$\begin{split} \sum M\_{\,lh} &= M\_{T\_l} - M\_{R\_m} - M\_{F\_\phi} - M\_c + M\_{I\_m} \\ \sum M\_{\,lh} &= C\_{T\_l} \alpha\_l \left| \alpha\_l \right| l\_l C\_\psi - C\_{R\_m} \alpha\_m \left| \alpha\_m \right| C\_\psi \\ &- \left( f\_{v\_\phi} \dot{\phi} + f\_{c\_\phi} \text{sgn}(\dot{\phi}) \right) - C\_c \left( \phi - \phi\_0 \right) + k\_m \dot{\alpha}\_m C\_\psi \end{split} \tag{42}$$

where <sup>=</sup> expresses the aerodynamic thrust torque of the tail propeller, <sup>=</sup> represents the load torque created by air resistance in the main rotor, = ( ˙ <sup>+</sup> (˙)) denotes the load torque as a result of the friction (including the viscous effects and the Coulomb friction), <sup>=</sup> ( 0) is the magnitude of torque exerted by the cable (it has a certain stiffness that allows to model it as a spring)), and finally <sup>=</sup> ˙ represents the inertial counter torque that is caused by the reaction produced by a change in the rotational speed of the main rotor.

Upon merging Eq. (33) to Eq. (42), and after performing some rearrangements, one obtains the following result for the equations of motion:

$$\begin{split} \left(J\_1 + J\_2\right)\ddot{\boldsymbol{\nu}} &+ h \Big(l\_{T\_2} m\_{T\_2} \mathbf{C}\_{\boldsymbol{\nu}} - l\_{T\_1} m\_{T\_1} \mathbf{S}\_{\boldsymbol{\nu}}\Big) \ddot{\boldsymbol{\phi}} + \left(\frac{\left(J\_1 - J\_2\right)}{2} \mathbf{S}\_{2\boldsymbol{\nu}}\right) \dot{\boldsymbol{\phi}}^2 + \mathbf{g} \Big(l\_{T\_1} m\_{T\_1} \mathbf{C}\_{\boldsymbol{\nu}} + l\_{T\_2} m\_{T\_2} \mathbf{S}\_{\boldsymbol{\nu}}\Big) \\ &= C\_{T\_m} \alpha\_m \left|\alpha\_m \left|l\_m - C\_{R\_l} \alpha\_l\right| \alpha\_l\right| - \left(f\_{v\_\varphi} \dot{\nu} + f\_{c\_\varphi} \text{sgn}\left(\dot{\nu}\right)\right) + k\_l \dot{\alpha}\_l \end{split} \tag{43}$$

$$\begin{split} \left( h \left( l\_{T\_1} m\_{T\_2} C\_{\psi} - l\_{T\_1} m\_{T\_1} S\_{\psi} \right) \ddot{\psi} + \left( J\_1 C\_{\psi} \right. \right. \\ \left. \left. \left( J\_1 m\_{T\_1} C\_{\psi} + l\_{T\_2} m\_{T\_2} S\_{\psi} \right) \dot{\psi}^2 + \left( \left( J\_2 - J\_1 \right) S\_{2\psi} \right) \dot{\phi} \dot{\psi} \right. \right. \\ \left. \left. \left( - \left( J\_1 m\_{T\_1} C\_{\psi} + l\_{T\_2} m\_{T\_2} S\_{\psi} \right) \dot{\phi} \dot{\psi} \right. \right. \right. \right. \end{split} \tag{44}$$
 
$$\begin{split} \left. \left( - \left( J\_1 m\_{T\_1} C\_{\psi} + l\_{T\_2} m\_{T\_2} S\_{\psi} \right) \dot{\phi} \dot{\nu} \right. \right. \end{split} \tag{45}$$

If we use matrix notation, the dynamic model of the mechanical part of the TRMS can be expressed in a compact form:

$$\mathcal{M}\{\mathbf{q}(t)\}\ddot{\mathbf{q}}(t) + \mathcal{E}\{\mathbf{q}(t), \dot{\mathbf{q}}(t)\}\dot{\mathbf{q}}(t) + \eta(\mathbf{q}(t), \dot{\mathbf{q}}(t), \dot{\mathbf{w}}(t)) = \mathcal{E}\{\mathbf{q}(t)\}\Omega(t) \tag{45}$$

in which () = (), () is the vector of generalised coordinates of the TRMS, () = (), () is the angular velocity vector, and the matrices (()), ((), ˙()), (()), and the vectors () and ((), ˙(), ˙ ()) are given by:

#### Nonlinear Cascade-Based Control for a Twin Rotor MIMO System http://dx.doi.org/10.5772/64875 279

$$\mathbf{M}\left(\mathbf{q}\left(t\right)\right) = \begin{bmatrix} J\_1 + J\_2 & h\left(l\_{T\_2}m\_{T\_2}C\_{\psi} - l\_{T\_1}m\_{T\_1}S\_{\psi}\right) \\ h\left(l\_{T\_2}m\_{T\_2}C\_{\psi} - l\_{T\_1}m\_{T\_1}S\_{\psi}\right) & J\_1C\_{\psi}\right\\ \end{bmatrix} \tag{46}$$

( *<sup>v</sup>* ( )) ( <sup>0</sup> )


 f  y

 ff

represents the load torque created by air resistance in the main rotor,

represents the inertial counter torque that is caused by the reaction produced

(˙)) denotes the load torque as a result of the friction (including the

*c c m m*

y

( 0) is the magnitude of torque exerted

 y

0

 f f

& & &

 f  y

y

 w (43)

(44)

(45)

(42)

 w

expresses the aerodynamic thrust torque of the tail propeller,

 f

*T m*

*ih T t t t R m m*

*M C lC C C*

y

= - - -+

*M M M M MM*

*ih T R F c I*

f

f

ww

å

278 Nonlinear Systems - Design, Analysis, Estimation and Control

å

 

viscous effects and the Coulomb friction), <sup>=</sup>

by a change in the rotational speed of the main rotor.

1 2 <sup>2</sup> 2

= - -+ +

*C l C f f sgn k*

 ww

( ) (( ) )

+ +-

 y

*hl mC l m S J J S*

2

y

y

> w w

(()), and the vectors () and ((), ˙(), ˙ ()) are given by:

&& &&

yy

following result for the equations of motion:

<sup>=</sup>

˙ <sup>+</sup> 

y

w w

*m t*

11 2 2

*TT T T*

ww

expressed in a compact form:

() = (),

yy

*T m*

y

y

where

<sup>=</sup>

= (

<sup>=</sup> ˙

= -

f

*t m m*

 w w

by the cable (it has a certain stiffness that allows to model it as a spring)), and finally

Upon merging Eq. (33) to Eq. (42), and after performing some rearrangements, one obtains the

2 2 11 11 2 2 1 2 2

è ø

*T T TT TT T T*

 y

> f

& &&

22 2 123

fy

&& &&

f

*T t tt R m m v c c m m*

*C l C C C f f sgn C k C*

 f

If we use matrix notation, the dynamic model of the mechanical part of the TRMS can be

in which () = (), () is the vector of generalised coordinates of the TRMS,

 y


&

 w

( ( )) ( )

() is the angular velocity vector, and the matrices (()), ((), ˙()),

 f

 y

ff

*J J J J hl m C l m S S glmC l m S*

( ) ( ) ( ) ( )

( ( ))

 y

*v*y

*T m mm R t t c t t*

 y

( ) ( ( ))

*T T TT T T*

2 2 11 1 2

 y

 yy

*h l m C l m S JC J S J h m m*

2 12

& & &

= - - + - -+

æ ö - ++ - + ç ÷ + +

& & &

*f f sgn C k C*

$$\mathcal{C}\left(q(t),\dot{q}(t)\right) = \begin{bmatrix} 0 & \frac{1}{2}(J\_1 - J\_2)S\_{2\nu}\dot{\phi} \\\\ -h\left(l\_{T\_1}m\_{T\_1}C\_{\nu\nu} + l\_{T\_2}m\_{T\_2}S\_{\nu\nu}\right)\dot{\psi} & \left(J\_2 - J\_1\right)S\_{2\nu}\dot{\psi} \end{bmatrix} \tag{47}$$

$$E\left(\boldsymbol{q}\left(t\right)\right) = \begin{bmatrix} C\_{T\_m} I\_m & -C\_{R\_l} \\ -C\_{R\_m} C\_{\mathcal{V}} & C\_{T\_l} I\_l C\_{\mathcal{V}} \end{bmatrix} \tag{48}$$

$$\mathbf{\dot{\Omega}}(t) = \begin{bmatrix} \alpha o\_m \left| \alpha o\_m \right| \\ \alpha o\_l \left| \alpha o\_l \right| \end{bmatrix} \tag{49}$$

$$\eta\left(q\left(t\right),\dot{q}\left(t\right),\dot{o}\left(t\right)\right) = G\left(q\left(t\right)\right) + F\left(\dot{q}\left(t\right)\right) + T\left(q\left(t\right),\dot{o}\left(t\right)\right) \tag{50}$$

$$\mathbf{G}\left(\mathbf{g}\left(t\right)\right) = \begin{bmatrix} \mathbf{g}\left(l\_{T\_1}m\_{T\_1}C\_{\mathcal{W}} + l\_{T\_2}m\_{T\_2}S\_{\mathcal{W}}\right) \\ 0 \end{bmatrix} \tag{51}$$

$$F\left(\dot{\boldsymbol{q}}\left(t\right)\right) = F\_{\upsilon}\dot{\boldsymbol{q}}\left(t\right) + F\_{\mathbf{c}}\left(\dot{\boldsymbol{q}}\left(t\right)\right) = \begin{bmatrix} f\_{\upsilon\_{\boldsymbol{\nu}}} & 0\\ 0 & f\_{\upsilon\_{\boldsymbol{\phi}}} \end{bmatrix} \dot{\boldsymbol{q}}\left(t\right) + \begin{bmatrix} f\_{c\_{\boldsymbol{\psi}}}\operatorname{sgn}\left(\dot{\boldsymbol{\nu}}\right)\\ f\_{c\_{\boldsymbol{\phi}}}\operatorname{sgn}\left(\dot{\boldsymbol{\phi}}\right) \end{bmatrix} = \begin{bmatrix} f\_{\upsilon\_{\boldsymbol{\nu}}}\dot{\boldsymbol{\nu}} + f\_{c\_{\boldsymbol{\psi}}}\operatorname{sgn}\left(\dot{\boldsymbol{\nu}}\right)\\ f\_{\upsilon\_{\boldsymbol{\phi}}}\dot{\boldsymbol{\phi}} + f\_{c\_{\boldsymbol{\phi}}}\operatorname{sgn}\left(\dot{\boldsymbol{\phi}}\right) \end{bmatrix} \tag{52}$$

$$\begin{aligned} T\{q(t), \dot{\omega}(t)\} &= \mathcal{M}\_c(q(t)) - \mathcal{M}\_g(q(t))\dot{\omega}(t) = \begin{bmatrix} 0 \\ \mathcal{L}\_c(\phi - \phi\_0) \end{bmatrix} - \begin{bmatrix} 0 & k\_t \\ k\_m \mathcal{C}\_\psi & 0 \end{bmatrix} \dot{\phi}(t) \\ &= \begin{bmatrix} -k\_t \dot{\omega}\_t \\ \mathcal{C}\_c(\phi - \phi\_0) - k\_m \dot{\omega}\_m \mathcal{C}\_\psi \end{bmatrix} \end{aligned} \tag{53}$$

Finally, after substituting Eqs. (51)–(53) into Eq. (50), the following yields:

$$\eta \left( \mathbf{q} \left( t \right), \dot{\mathbf{q}} \left( t \right), \dot{\mathbf{o}} \left( t \right) \right) = \begin{bmatrix} \mathbf{g} \left( l\_{T\_1} m\_{T\_1} C\_{\psi} + l\_{T\_2} m\_{T\_2} S\_{\psi} \right) + f\_{\upsilon\_{\psi}} \dot{\nu} + f\_{c\_{\psi}} \text{sgn} \left( \dot{\nu} \right) - k\_l \dot{o}\_l \\\ f\_{\upsilon\_{\phi}} \dot{\phi} + f\_{c\_{\phi}} \text{sgn} \left( \dot{\phi} \right) + C\_c \left( \phi - \phi\_0 \right) - k\_m \dot{o}\_m C\_{\psi} \end{bmatrix} \tag{54}$$


**Table 3.** Mechanical parameters.


**Table 4.** Parameters of the pitch movement.


**Table 5.** Parameters of the yaw movement.

**Symbol Parameter Value Units**

<sup>t</sup> Length of the tail part of the free-free beam 0.282 <sup>m</sup>

<sup>m</sup> Length of the main part of the free-free beam 0.246 <sup>m</sup>

<sup>b</sup> Length of the counterbalance beam 0.290 <sup>m</sup>

cb Distance between the counterweight and the join 0.276 <sup>m</sup>

ms Radius of the main shield 0.155 <sup>m</sup>

ℎ Length of the pivoted beam 0.06 <sup>m</sup>

mr Mass of the main DC motor and main rotor 0.236 kg

cb Mass of the counterweight 0.068 kg

t Mass of the tail part of the free-free beam 0.015 kg

m Mass of the main part of the free-free beam 0.014 kg

b Mass of the counterbalance beam 0.022 kg

ms Mass of the main shield 0.219 kg

h Mass of pivoted beam 0.01 kg

**Symbol Parameter Value Units**

<sup>+</sup> Thrust torque coefficient of the main rotor ( <sup>≥</sup> <sup>0</sup>) 1.53 × 10−5 N s2 rad−2

<sup>−</sup> Thrust torque coefficient of the main rotor ( < 0) 8.8 × 10−6 N s2 rad−2

Coulomb friction coefficient 5.69 × 10−4 N m

t Coefficient of the inertial counter torque created by the change in 2.6 × 10−5 Nms2 rad−1

Load torque coefficient of the tail rotor 9.7 × 10−8 Nms2 rad−2

Viscous friction coefficient 0.0024 N m s rad−1

Radius of the tail shield 0.1 <sup>m</sup>

Mass of the tail DC motor and tail rotor 0.221 kg

Mass of the tail shield 0.119 kg

280 Nonlinear Systems - Design, Analysis, Estimation and Control

ts

tr

ts

 

**Table 3.** Mechanical parameters.

**Table 4.** Parameters of the pitch movement.

Finally, in order to complete the dynamic modelling for the mechanical part of the TRMS, **Tables 3**–**5** show in detail the parameters used in the model. For each parameter, its description, its value and the corresponding units is included. The initial approximation of these values was based in the developments described in [13]. Additionally, some values of the parameters have been tuned by carrying out several identification trials.
