Dynamic Models of the Draganflyer Chayatat Ratanasawanya May 21, 2009

Overview Characteristics of the quad-rotor Kinematics review Coriolis effect Gyroscopic effect Simplified dynamic model More detailed dynamic model Summary Questions/Comments

Characteristics of the Draganflyer: 6 DOF movement z   y x 

Characteristics of the Draganflyer: Spinning direction of rotors Given the direction that the motors rotate, gyroscopic effects and aerodynamic torques tend to cancel. Front Rear

Characteristics of the Draganflyer: Forces, torques, & variables Front τ1 τ2 M1 M2 f3 d f4 Q3 τ3 Q4 Rear τ4 M4 M3 mg

Kinematics review Inertial frame, I Body fixed frame, A Origin is at the c.m. of the rotorcraft Position vector,  Denotes the position of A relative to I Rotation matrix, R Denotes the orientation of A w.r.t. I Euler angles: , , 

Coriolis effect The effect refers to a situation where an object, which initially sits on a rotating body fixed frame and therefore has a rotational motion, undergoes translational motion in the body fixed frame. It appears that there is a force acting on the object. The Coriolis force (a pseudo force) is proportional to the rotation speed. The Coriolis force acts in a direction perpendicular to the rotation axis and to the velocity of the body in the rotating frame and is proportional to the object's speed in the rotating frame

Gyroscopic effect The effect refers to a situation where a rotating object has a rotational motion in the inertial frame. A torque τ applied perpendicular to the axis of rotation, and so perpendicular to the angular momentum L, results in a rotation about an axis perpendicular to both τ and L.

Simplified dynamic model Using Lagrangian mechanics q = (x, y, z, , , )

Simplified dynamic model The model is obtained from the Euler-Lagrange equations with external generalized force

Simplified dynamic model Since the Lagrangian contains no cross-terms in the kinetic energy combining and ,

Simplified dynamic model Finally, we have

Simplified model: summary Six equations are derived using Lagrangian mechanics; 3 for translational motion, 3 for rotational motion. Coriolis and gyroscopic terms (due to translational & rotational motion of the rotating body) appear in the equations of motion automatically. Coriolis and gyroscopic effects due to translational and rotational motions of the rotors are ignored. The mechanics of each of the rotors is ignored.

Dynamic model in more detail e1, e2, e3 are unit vectors in x, y, z direction A = {Ea1, Ea2, Ea3} Consider the mechanics of each of the rotors Account for gyroscopic effect due to rotational motion of the rotors. 

Dynamic model in more detail Using Newtonian mechanics, we have: Where

Dynamic model in more detail Recall Additionally Collective thrust pointing upwards Mechanics of each of the rotors.

Dynamic model in more detail Finally, we have

More detailed model: summary More than 6 equation are derived using Newtonian mechanics. The mechanics of each of the rotors and the gyroscopic effect due to their rotational motions are accounted for. Gyroscopic term is added to the equations. We assumed that the angular velocity of the rotors, ω, is much higher than the angular velocity of the body fixed frame, Ω. This makes the last equation on the previous slide valid. Coriolis effect due to translational motion of the rotors is still ignored.

Summary Characteristics of the quad-rotor Kinematics review Introduction to Coriolis effect Introduction to gyroscopic effect Dynamic model simplified Dynamic model in more detail

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