**5.1 Helicopter structure and modeling**

The conceptual platform of 3DOF helicopter scheme is presented in **Figure 4**. It consists of an arm mounted on a base. The main body of the helicopter constructed of propellers driven by two motors mounted are the either ends of a short balance bar. The whole helicopter body is fixed on one end of the arm and a balance block installed at the other end.

The balance arm can rotate about the travel axis as well as slope on an elevation axis. The body of the helicopter is free to roll about the pitch axis. The system is provided by encoders mounted on these axes used to measure the travel motion of the arm and its elevation and pitch angle. The propellers with motors can generate an elevation mechanical force proportional to the voltage power supplied to the motors. This force can cause the helicopter body to lift off the ground. It is worth considering that the purpose of using a balance block is to reduce the voltage power supplied to the propellers motors. In this study, the nonlinear dynamics of 3DOF helicopter system is modelled mathematically based on developing the model of the system behavior for each of the axes.

Where *Fm* is the thrust force of propeller motor and *Mf*,*<sup>ϵ</sup>* represents the torque component generated from combining the joint friction and air resistance. But the rotation angle of the pitch axis *ρ* ¼ 0, if the elevation angle *ϵ* ¼ 0, then the torque exerted on the elevation axis will be zero. Eq. (17) based on Euler's second law

*A Hybrid Control Approach Based on the Combination of PID Control with LQR Optimal Control*

Consider the pitch schematic diagram of the system in **Figure 6**. It can be seen from the figure that the main torque acting on the system pitch axis is produced from the thrust force generated by the propeller motors. When *ρ* 6¼ 0, the gravitational force will also generate a torque *Mw*,*<sup>ρ</sup>* acts on the helicopter pitch axis. The

Based on the assumption that the pitch angle *ρ* ¼ 0*, Mw*,*<sup>ρ</sup>* ¼ 0, then Eq. (23)

*Jρ*€*ρ* ¼ *l<sup>ρ</sup> F <sup>f</sup>* � *Fb*

*Jϵ*€*ϵ* ¼ *l*<sup>1</sup> *F <sup>f</sup>* þ *Fb*

*Schematic diagram of elevation axis model for 3DOF helicopter system.*

*DOI: http://dx.doi.org/10.5772/intechopen.94907*

*Jϵ*€*ϵ* ¼ *Kcl*<sup>1</sup> *V <sup>f</sup>* þ *Vb*

dynamics of the pitch axis can be modeled mathematically as follows:

Where *Mf*,*<sup>ρ</sup>* is the friction moment exerted on the pitch axis.

*Jϵ*€*ϵ* ¼ *l*1*Fm* � *Mw*,*<sup>ϵ</sup>* þ *Mf*,*<sup>ϵ</sup>* (18)

 � *Mw*,*<sup>ϵ</sup>* <sup>þ</sup> *Mf*,*<sup>ϵ</sup>* (19) *Fi* ¼ *KcVi i* ¼ *f*, *b* (20)

� *Mw*,*<sup>ϵ</sup>* <sup>þ</sup> *Mf*,*<sup>ϵ</sup>* (21)

*Jϵ*€*ϵ* ¼ *Kcl*1*Vs* � *Mw*,*<sup>ϵ</sup>* þ *Mf*,*<sup>ϵ</sup>* (22)

*Jρ*€*ρ* ¼ *F <sup>f</sup> l<sup>ρ</sup>* � *Fbl<sup>ρ</sup>* � *Mw*,*<sup>ρ</sup>* � *Mf*,*<sup>ρ</sup>* (23)

*Mw*,*<sup>ρ</sup>* ¼ *mhglh sin* ð Þ*ρ cos*ð Þ*ϵ* (24)

� *<sup>M</sup> <sup>f</sup>*,*<sup>ρ</sup>* (25)

becomes:

**Figure 5.**

*5.1.2 Pitch axis model*

becomes as follows:

**25**
