nonlinear motion equation

7/31/2019 Nonlinear Motion Equation

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Nonlinear Motion Equation for Tilting Duct Vehicle in Considering Rotation of Ducts

1 Convention (Right-handed)

2 Notations

2.1 Parameters(MKS system of units)

Tilting angle

m Mass

T d PWM duty ratio(DR)

xxf J Moment of inertia of a fan

Atmospheric density

DxC Drag Coefficient about x -direction

G COM of the vehicle

Gd COM of a duct assembly

xS Cross-sectional area of the vehicle about x -axis

K Motor torque to DR(Duty Ratio) coefficient

eK Back-EMF Constant

T C Fan thrust to2 ratio

QC Fan torque to2

f ratio

p Position vector, [ ]T x y z

+

k d k b

i b jb

k b

jb r


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v Velocity vector, [ ]Tu v w r Position vector

/ bb i

Angular velocity of a vehicle w.r.t. the inertia frame, [ ]T p q r

/ bb i

Cross-Product Matrix,

0

0

0

r q

r p

q p

- - -

, / / / / i bb i i b b i b i= C C

Euler-angle vector, [ ]T f

Angular velocity of a fan

C Rotation Matrix

F Force

G Gravity

J Inertia Tensor

M Moment

Q Torque

T Thrust

2.2 Indices

i Inertial reference frame

b Vehicle body frame

d Duct f Fan

act Actuator

ad Aerodynamic drag

/ A Bp Position vector of point A with respect to point B

/ A iv Velocity of point A in frame i

/ b c

A iv Components in coordinate system c of the derivative in frame b

/ b aC Matrix that transforms the components from system a to b .C are orthogonal matrices 1 / / /

T b a b a a b

-= =C C C


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3 Euler Angles, NED Navigation System, and Duct Tilting Angle

3.1 Euler Rotations

/

1 0 0 0 0

0 0 1 0 0

0 0 0 0 1b i

c s c s

c s s c

s c s c

- = = - -

C C C C

(c , s : abbreviations for cos , sin each.)

3.2 NED Navigation System

Orientation follows NED system.

3.3 Duct Tilting Angle

/

00 1 0

0d b

c s

s c

- = =

C C

4 Transport Theorem

/ / b

b i b ii b b

d d d dt dt dt

= + = +

5 Assumption

5.1 The fans are symmetric in form about d z -axis. xxf yyf J J = .

5.2 All the axes about which the ducts rotate lie on COM of each duct fan assembly.

5.3 COM of each fan assembly are coincident with COM of its duct.


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6 Kinematics

6.1 Angular Motion

6.1.1 Angular Momentum h .=h ( ){ } / / , / , / ,b b d d b b i d f k Gd i k d d i k f f i k

k

m m+ + + +J r v J J

Where

( )[ ] [ ]( )

/ , / /

/ , / , / , / , / , /

T T

/ / / / /

T

/

/

0 0 , 0 0

0 0

by symmetry.

b b bGd i k G i b i k

d d d b bd i k d b k b i k d b k d b k b i

b d b bd b d b d b d b d b

d f d f

b

k G ik

= + = + = + = = = = = - =

v v r

C

C

r v 0

( ) ( ){ }( )( ){ }

/ /

/ , / , / , / / ,+

b b bb b i d f k b i k

k

b b d b d k d f d b k d b k b i zzf f d k

k

m m

J

= + +

+ + +

h J r r

C J J C

6.1.2 Effective Moment

/ b i b b b b bG b i= = + M h h h

6.1.3 Euler Kinematic Equation

( ) 1 / 0 0

0 0

0 0

bb i

-

= + + =

C C H

6.2 Translational Motion

6.2.1 Acceleration

/ / / / i b b b b b

G i G i b i G i= + v v v

6.2.2 Effective Force

/ b i b

G im= F v

6.2.3 Position

/ / / i i b

G O i b G i=p C v

6.3 State VectorT T T T

/ / / b b

G O G i b i = x p

v


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7 Dynamics

7.1 Moment

7.1.1 Aerodynamic DragT

ad x y zC p p C q q C r r = - M

7.1.2 Fan Torque and ThrustT2

,0 0d f Q f k C = Q ,

b br T whereT2

,0 0d

T f k C = - T

7.1.3 Total Moment

( ){ } / , / ,b b d d G k b d k k b d k k ad = + + M r C T C Q M

7.2 Force

7.2.1 Aerodynamic Drag

12

Dx x

ad Dy y

Dz z

C u u S

C v v S

C w w S

= -

F

7.2.2 Thrust and GravitybT , [ ]T0 0i g=G

7.2.3 Total Force

/ , / b d i

b d k k b i ad k

m= + + F C T C G F

7.3 DC Motor System

2m m e zzf f f Q f

K K K J u C

R R = - - for low motor inductance ( )1 L .

7.4 Servo System

( ){ } ( ) [ ]( ){ } / / / 0 0 T b b d d f b d d f d b d b zzf f act y yd d

q J dt dt

+ = + + + =

h h C J J C

act t act J K u b = - -

nonlinear motion equation

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