DYNAMIC MODELS OFTHE 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

xy

z

CHARACTERISTICS OF THE DRAGANFLYER:SPINNING DIRECTION OF ROTORS

Front

Rear

Given the direction that the motors rotate, gyroscopic effects and aerodynamic torques tend to cancel.

CHARACTERISTICS OF THE DRAGANFLYER:FORCES, TORQUES, & VARIABLES

Front

Rear

M1M2

M3M4

f1f2

f4f3

u

τ1

Q1

τ3

Q3

τ2

Q2

τ4

Q4

mg

d

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 mechanicsq = (x, y, z, , , )

Ttrans mT2

1 JTrotT

2

1

mgzU

UTTqqL rottrans ),(

mgzm TT J2

1

2

1

),,( zyx ),,(

SIMPLIFIED DYNAMIC MODEL

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

),( Fq

L

q

L

dt

d

FRF ˆ

ccsccsssscsc

scccssscsssc

scscc

R

24

1

,,0

0ˆ

iiii

i kffu

u

F

dff

dffi

Mi

)(

)(

13

42

4

1

SIMPLIFIED DYNAMIC MODEL

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

),( Fq

L

q

L

dt

d

mg

Fm 0

0

)(2

1 JJJ T

),(V J

mg

um 0

0

coscos

sincos

sin

SIMPLIFIED DYNAMIC MODEL

Finally, we have

mg

um 0

0

coscos

sincos

sin

),(V J

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 directionA = {Ea

1, Ea2, Ea

3}

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:

Wherev

FRmgevm ˆ3

)( RskR

JJ

aa G

IAR

AF

Iv

Izyx

:

,ˆ,,

),,(

J

DYNAMIC MODEL IN MORE DETAIL

Recall

Additionally

24

1

a3 ,Εˆ

iiii

i kffF

4

1

42

13

)(

)(

iMi

a dff

dff

ccscs

sccssccssscs

sscsccsssccc

R

Collective thrust pointing upwards

ii

raG )e(4

13

J2ii

iiir

bQ

Q

J Mechanics of each of the rotors.

DYNAMIC MODEL IN MORE DETAIL

Finally, we have

v

FRmgevm ˆ3

)(RskR

JJ 2iiir b J

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

THANK YOU

Questions / Comments