Modeling of 2 DOF Helicopter

Introduction

The 2 DOF Helicopter experiment imparts an economical test at a basic level to design modeling and control laws for vehicles with dynamics representative of a related to rigid body helicopter, spacecraft or an underwater vehicle. The figure shows a Quanser 2-DOF Helicopter experiment model with its comprising components.

System Description and Modeling

Figure 1: 2 DOF Helicopter

Figure 1 represents a Quanser 2 DOF Helicopter experiment model comprised of a body and a metal base. The main components of the models are encoders, slip rings and DC motor-driven front and back propellers. The whole Helicopter model is mounted on a fixed base and propellers which are driven by DC motors. DC motor driven propellers are mounted perpendicular to each other and serve the purpose of a practical helicopter system with the main rotor and anti-torque tail rotor. Elevation of helicopter nose about pitch axis and side to side motion about yaw axis is controlled by front and back propeller respectively. Encoders with high-resolution measure both axes (i.e. Pitch and Yaw axes).The slip ring stimulates the body to rotate the body about yaw angle and nullify the need of wires to connect motors and encoders to the base. The inherent torque generates due to the rotation of the front propeller causes the body to rotate which must be compensated by tail rotor, similar to full-sized helicopters which lead us to the modeling and design of the control scheme of the model.

Figure 2: Simple Free-Body diagram of 2-DOF helicopter

Figure 2 represents the free body diagram of the system used in the Lagrangian modeling process. Convention used for modeling are as follows:

•  The helicopter is horizontal when the pitch angle equals$\theta$=0.
• The pitch angle increases positively,$\dot{\theta}(t)=0 $, when the nose is moved upwards and the body rotates in the counter-clockwise (CCW) direction.
• The yaw angle increases positively,$\dot{\psi}(t)=0$ when the body rotates in the clockwise (CW) direction.
• Pitch increases, $\theta>0$, when the pitch thrust force is positive $F_p> 0$.
• Yaw increases, $\psi>0$, when the yaw thrust force is positive, $F_y > 0$.

The figure shows a dynamic model of a helicopter. $F_p$ and $F_y$ are the thrust forces act at across the pitch and yaw respectively. $r_p$ and $r_y$ are the distances from center. The gravitational force $F_g$ acts against lifting force generated by propeller to lift the helicopter nose. The center of a mass of the body is at a distance of $l_{cm}$ towards the front propeller from the intersection of the yaw axis and pitch axis along the body length. The coordinates of the center of mass are obtained by transformation as below:

Potential energy (P)

Kinetic Energy (K)

Where $K_{r,p}$= Kinetic Energy due to a propeller

           $K_{r,y}$= Kinetic Energy due to a propeller

           $K_{t}$= transition kinetic energy.

By using the Euler-Lagrange formula for equation (1),(2) and (3)

From equation (10),(12) and (14)

Total Kinetic energy

Putting value of $\dot{\theta}^2$ , $\dot{\psi}^2$ , $\dot{X}_m$ ,  $\dot{Y}_m $ , $\dot{Z}_m$

Lagrange variable defined as

Euler-Lagrange equation of motion is defined as

From equation (22),(23) & (24)

 The generalized force equations are

where,
             $K_{pp}$ = Thrust force constant of yaw motor/propeller,
             $K_{yy}$ = Thrust force constant of pitch motor/propeller,
             $K_{py}$ = Thrust torque constant acting on pitch axis from yaw motor/propeller and
                   $K_{yp}$ = Thrust torque constant acting on yaw axis from pitch motor/propeller

By comparing equation of  $Q_1$ and $Q_2 $, we get

 Let $\theta$ = $\theta_1$ and $\psi$ = $\psi_1$
 $\dot{\theta}_1 = \theta_2$ and $\dot{\psi}_1 = \psi_2$

Here, Equation (31), (32) Shows modeling of  2DOF Helicopter.