Basic concept of dynamics 1 (Vector, matrix, and...

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Basic concept of dynamics 1(Vector, matrix, and differential calculus)

Vehicle Dynamics (Lecture 3-1)

Vehicle Dynamics Model

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q How to describe vehicle motion? Need Reference frames and Coordinate systems

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How to describe motion of a rigid body?

q Reference frames

• Translational and rotational motion• Position• Velocity• Acceleration

q Coordinate systems

q What is motion of a rigid body?

• Inertial reference frame (global frame)• Body fixed reference frame

• Cartesian coordinate systems• Relative coordinate systems• Orientation parameters

Need vector and matrix representation and differential calculus

Geometric and Algebraic Vector Representation(1)

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q Geometric vectors is represented as shown in the following figure.

• Direction is represented by arrow.• Magnitude is represented by length of vector and detected by or a

rar

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q A geometric vector can be resolved into components and along the x and y axes of a Cartesian reference frame.

ar

xa ya

x ya a i a j= +r rr


q Algebraic vector in Cartesian coordinates

,Tx

x yy

aa a

aé ù

é ù= =ê ú ë ûë û

a

• an algebraic vector is defined as a column matrix.

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q Scalar product in geometric vector representation

cos ( , )a b ab a bq× =r r r r

a b b a× = ×r r r r


, , , , , , , ,TT T T T T

x y x y x y x ya a b b a a b b é ùé ù é ù é ù= = = =ë û ë û ë û ë ûa b d a b

Ta b× « a brr

q Algebraic vector can be extended more than 2 components

q Scalar product in algebraic vector representation

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q Perpendicular vector in geometric vector representation.

y xa a a^= - +i j

r r r

0y x x ya a a a a a^× = - + =

r r


q Perpendicular vector in algebraic vector representation.

0 11 0

y y

x x

a aa a

^ - -é ù é ùé ù= = =ê ú ê úê ú

ë ûë û ë ûa Ra

where is the orthogonal rotational matrix R 1 0,

0 1T é ù

= = = -ê úë û

R R I RR I

Matrix Algebra(1)

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q A matrix is defined as a rectangular array of numbers, taken to be real(here)

[ ]11 12 1 1

21 22 2 21 2

1 2

, , ,

n

nij n

m m mn m

a a aa a a

a

a a a

é ù é ùê ú ê úê ú ê úé ùº = = =ë û ê ú ê úê ú ê úë û ë û

bb

A a a a

b

LL

LM M M M

Lq If matrix has m rows and n columns, the dimension of the matrix is

said to be m x n.q The matrix with only one column is called a column matrix.

q The matrix with only one row is called a row matrix.

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21 2, , ,

j

Tjj j j mj

mj

aa

a a a

a

é ùê úê ú é ù= = ë ûê úê úê úë û

a LM

[ ]1 2, , ,i i i ina a a=b L

Matrix Algebra(2)

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Matrix Algebra(3)

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q A square matrix : number of rows = number of columnsq A diagonal matrix : [ ]11 22, , , nndiag a a aºA L ( 0 )ija for i j= ¹

q The n x n identity matrix or is a diagonal matrix with I nI 1iia =q Matrix addition and subtraction

= +C A B : must have the same dimension and for all and ,A B ij ij ijc a b= +

= -D A B : must have the same dimension and for all and ,A B ij ij ijd a b= -

( ) ( )+ + = + + = + +A B C A B C A B C

i j

i j

Matrix Algebra(4)

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Matrix Algebra(5)

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q Matrix multiplication

[ ]1

21 2,ij ij nm p p n

m

a b´ ´

é ùê úê úé ù é ù= = = =ë û ë ûê úê úë û

dd

A B b b b

d

LM

1 2 1,i i i ip p

d d d´

é ù= ë ûd L1 2 1

T

i i i pi pb b b

´é ù= ë ûb L

=C AB1

p

ij ik kj i jk

c a b=

= =å d b

¹AB BA

( )+ = +A B C AC BC

( ) ( )= =AB C A BC ABC

, ,m p p n´ ´Ì ÌA B R C R

, ,m p p q q n´ ´ ´Ì Ì ÌA R B R C Rifa=C A ij ijc aa= for all and i j

Matrix Algebra(6)

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Matrix Algebra(7)

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q A symmetric matrix : q A skew symmetric matrix :

T=A AT= -A A

( )T T T+ = +A B A B

( )T T T=AB B A ,m p p n´ ´Ì ÌA R B Rif

Matrix Algebra (8)

q The row(column) rank of a matrix is defined as the largest number of linearly independent rows(columns) in the matrix.

q The row and column ranks of any matrix are equal.

q The rank of a matrix is also equal to the dimension of the largest square submatrix with nonzero determinant.

q A square matrix with linearly independent rows(column) is said to have full rank.

q Singular matrix is a square matrix which does not have full rank.

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Matrix Algebra (9)

q Inverse matrix

such that

q Inverse matrix properties

q Orthogonal matrix

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1-A 1 1- -= =AA A A I

( ) ( )( )

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1 1 1

T T --

- - -

=

=

A A

AB B A

1

T T

T-

= =

=

A A AA IA A

Transformation of Coordinates

§ can be represented in coordinates.

§ Also the same vector can be represented

in coordinates denoted as .

§ The relationship between and is as follows,where orientation matrix(coordinate transformation matrix).

, is orthogonal matrix :

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srT

x ys sé ù= ë ûs

x y¢ ¢-

x y-

T

x ys s¢ ¢¢ é ù= ë ûss ¢s

¢=s As

( )cos sinsin cos

f ff

f f-é ù

= º ê úë û

A A A 1 1T- -= Þ =AA I A A

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Transformation of Coordinates

Position P vector using coordinate transformation

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'

p p

p

= +

= +

r r sr As

§ To describe a body position, position vector is used.

§ To represent the orientation of a body, body fixed reference frame is introduced.

§ The orientation of the frame with respect to plane is represented by angle which is the angle between x-axis and x'-axis.

§ Thus, if we describe then a body in the plane can be fully described.

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T

x yr ré ù= ë ûr

f, ,

T

x yr r fé ùë û

x y¢ ¢- x y-

Position P vector using coordinate transformation

Relative coordinate transformation

§ is the transformation matrix from the to frame.

§ is the transformation matrix from the to frame.

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iA' 'i ix y- x y-

jA' 'j jx y- x y-

' '

' '

' ' '

i i j j

T Ti i i i j j

Ti i j j ij j

= =

=

= º

s A s A s

A A s A A s

s A A s A s

§ is orthogonal matrixijA

( ) ( )TT T Tij ij i j i j

T T Tj i i j j j

=

= = =

A A A A A A

A A A A A A I( ) ( )( ) ( )

( )

cos sincos sinsin cossin cos

cos sin

sin cos

j ji iTij i j

j ji i

j i j i

j i

j i j i

f ff ff ff f

f f f ff f

f f f f

-- é ùé ù= = ê úê ú

ë û ë ûé ù- - -ê ú= = -ê ú- -ë û

A A A

A

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Vector and matrix differentiation(1)

q Why do we need vector and matrix differentiation?Kinematic relationship such as position, velocity and accelerationis related to the time derivative of position vector.

Let in a stationary Cartesian reference frame.

Definition of time derivate of vector

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( ) ( ), ( )T

x yt t té ùº = ë ûa a a a

( ), ( ) ,T

T

x y x yd d da t a t a adt dt dt

é ù é ù= = º ë ûê úë ûa a& & &


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( )

( )

( )

( ) ,

T T T

T

x y

ddtddtddt

d d t a adt dt

a a a

+ = +

= +

= +

æ ö é ù= =ç ÷ ë ûè ø

a b a b

a a a

a b a b a b

a a

&&

& &

&&

&& && &&

Product rule of differentiation

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§ Consider matrix

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( ) ( )ijt b té ù= ë ûB

( )

( )

( )

ijd d bdt dt

ddtddtddt

a a a

é ùº = ê úë û

+ = +

= +

= +

B B

B C B C

BC BC BC

B B B

&

&&

&&

&&


§ Let be a vector of real variable, be a scalardifferentiable function of and be an vectorof differentiable function of .

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[ ]1T

kq q= ×××q k ( )a qq [ ]1( ) ( ) ( ) Tn= F ××× FΦ q q q nq

1 2 1

1 1 11

1 2

2 2 22

1 2

1 2

k k

k

k

n n n n

k n k

a a a aaq q q

q q q

q q q

q q q

´

´

é ù¶ ¶ ¶ ¶º = ×××ê ú¶ ¶ ¶ ¶ë û

¶F ¶F ¶F¶F é ùé ùê úê ú ¶ ¶ ¶¶ ê úê úê ú¶F ¶F ¶Fê ú¶F

¶ ê úê ú ¶ ¶ ¶¶º = = ê úê ú¶ ê úê úê úê ú

¶F ¶F ¶F ¶Fê úê úê úê ú¶ ¶ ¶ ¶ë û ë û

q

q

q

q

Φ qΦq

q

K

L

M M M M M

L

Jacobian matrix


If and are vectorfunction of vector variables,

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[ ][ ]

( ) ( )

1

1

( ) ( ) ( )

( ) ( ) ( )

Tn

Tn

T T T T T T T T

g g

h h

=

=

¶ ¶ ¶= = + = + ¹ +

¶ ¶ ¶ q q q qq

g q q q

h q q qg hg h g h h g h g g h g h g h

q q q

L

L

[ ]1( ) ( ) ( ) Tm= F FΦ g g gL

If is a constant matrix and and are and vectors of variables, respectively, B m n´ qp m n

[ ]1( ) ( ) ( ) Tng g= =g g q q qK

¶ ¶º

¶ ¶qΦ gΦg q

( )

( )

( )

T T T

T T T Tddt

¶=

¶¶

=¶

= +

Bq Bq

p Bq q Bp

p Bq q B p p Bq& &

Velocity and acceleration of P point fixed in a moving frame

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'p p= +r r As

§ Velocity vector of point P'

'

p p

pf

= +

= +

r r Asr Bs

&& &&&

sin coscos sin

cos sin 0 1 sin cossin cos 1 0 cos sin

dd

f ff f f

f ff

f f f ff f f f

- -é ù= = ºê ú-ë û

- - - -é ù é ù é ù= = =ê ú ê ú ê ú-ë û ë û ë û=

A A B

AR B

AR RA

& & & &

' ' '

'

p p p p

p p

f f

f f^ ^

\ = + = + = +

= + = +

r r As r Bs r ARsr As r s

& & && & & && && &

§ Position vector of point P


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cos sinsin cos

dd

f ff f f

f ff-é ù

= = = -ê ú- -ë ûB B A& & &&

§ Acceleration of point P' '

' 2 '

' 2 '

2

p p p

p p

p p

p p

f f

f f

f f

f f^

\ = + +

= + -

= + -

= + -

r r Bs Bsr Bs Asr ARs Asr s s

&& & &&& &&&& &&&&& &&&&& &&&

where

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Basic concept of dynamics 1 (Vector, matrix, and...

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