*5.1.1 Elevation axis model*

The free body diagram of 3DOF helicopter system based on elevation axis is shown in **Figure 5**. The movement of the elevation axis is governed by the following differential equations:

$$J\_c \ddot{\varepsilon} = \left(J\_p - J\_\tau\right) \dot{\varepsilon}^2 \cos\left(\varepsilon\right) \sin\left(\varepsilon\right) + l\_1 F\_m \cos\left(\rho\right) - l\_1 F\_m - M\_G - M\_{\hat{f}, \varepsilon} \tag{17}$$

**Figure 4.** *Prototype model of 3DOF helicopter system.*

*A Hybrid Control Approach Based on the Combination of PID Control with LQR Optimal Control DOI: http://dx.doi.org/10.5772/intechopen.94907*

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 becomes:

$$J\_{\epsilon}\ddot{\epsilon} = l\_1 F\_m - M\_{w,\epsilon} + M\_{f,\epsilon} \tag{18}$$

$$J\_e \ddot{\epsilon} = l\_1 \left( F\_f + F\_b \right) - M\_{w,\epsilon} + M\_{f,\epsilon} \tag{19}$$

$$F\_i = K\_c V\_i \ i = f, b \tag{20}$$

$$J\_c \ddot{\epsilon} = K\_c l\_1 \left( V\_f + V\_b \right) - M\_{w,c} + M\_{\hat{f},c} \tag{21}$$

$$J\_{\epsilon}\ddot{\epsilon} = K\_{\epsilon}l\_{1}V\_{\epsilon} - M\_{w,\epsilon} + M\_{f,\epsilon} \tag{22}$$
