Dynamic Models of Different VTOLs

ByMuhammad Abdullah

Quad RotorAn under actuated system with 6 Degrees of

Freedom.

It is composed of 4 rotors. Each rotor provides necessary thrust to take-off (Four Input Forces).

The Input (u1) is the sum of thrust of four rotors.

Pitch Movement obtained by increasing (reducing) the speed of rear motor and reducing (increasing) the speed of front motor.

The roll movement is obtained by increasing (reducing) speed of right motor while reducing (increasing) speed of left motor.

The yaw movement is obtained by increasing (decreasing) speed of front and rear motors while decreasing (increasing) speed of lateral motors simultaneously. This should be done while keeping total thrust constant.

Therefore there are 4 Input Forces and 6 Output States (x, y, z, θ, φ, ψ).

The dynamic model of the quad rotor can be obtained via a Lagrange approach and a simplified model using force and moment equation is given as:

The Euler angles equations are given as:

Where θ, φ and ψ pitch, roll and yaw angles respectively. Ki’s are the drag coefficients. Ii’s moment of inertia.

Single Tilting Rotor VTOLThe dynamic model of this system consists of

the translational dynamics and rotational dynamics. The rotational dynamics include actuator torques, gyroscopic torques and weight torques.

However incorporating gyroscopic and control torques and other aerodynamic effects in system modeling may lead to a very complex dynamic system.

where

The above system looks very tedious and complex. So we will neglect gyroscopic and weight torques and assume mass and inertial matrix is normalized.

Now the above system is reduced and converted into three set of equations:

Lateral Dynamics (θ=0, ψ=0):

Longitudinal Dynamics (φ=0, ψ=0, γ=0):

Axial Dynamics(φ=0, θ=0, γ=0):

where γ is the rotor tilt angle.

Single Rotor VTOLThe dynamics consists of translational and

rotational components as described in case of tilting rotor case.

Using previous concepts we use gyroscopic torques, drag torques and actuator torques to get the following dynamical system: (letting m=1)

Now neglecting the gyroscopic torques, drag torques and normalizing inertial matrix we get simplified model.

Lateral Dynamics (θ=0, ψ=0):

Longitudinal Dynamics (φ=0, ψ=0):

Axial Dynamics(φ=0, θ=0):

ReferencesAtheer L. Salih, M. Moghavvemi, Haider A. F.

Mohamed and Khalaf Sallom Gaeid, “Flight PID controller design for a UAV quadrotor”.

O. Garcia, A. Sanchez, J. Escareño, R. Lozano, “Tail-sitter UAV having one tilting rotor: Modeling, Control and Real-Time Experiments”.

J. Escareño, S. Salazar and R. Lozano , “Modeling and Control of a Convertible VTOL Aircraft”.

Afshin Manouchehri, Hamid Hajkarami and Mohammad Saleh Ahmadi, “Hovering Control of Ducted Fan VTOL UAV Based on PID Control”.