1National Kaohsiung University of Applied Sciences, Department of Mechanical Engineering

2005/5/24 計算動力學 1

Euler Parameters

Reading material: Chapter 6

2005/5/24 計算動力學 2

What is Euler Parameters?What is Euler Parameters? Euler parameters can be considered as an example of the

quaternions (四元數) Euler parameters are a normalized form of parameters known

as quaternions (p. 153) Euler parameters are widely used in general purpose multibody

computer programs to avoid the singularities associated with Euler angles to reduce memory in representing rotations , to reduce computation time in actually rotating or updating the

equations of motion

=

=

×

ep 0

143

2

1

0

e

eeee

e0 : angle of rotatione : an unit vector along the axis of rotatione0 : angle of rotatione : an unit vector along the axis of rotation

2National Kaohsiung University of Applied Sciences, Department of Mechanical Engineering

2005/5/24 計算動力學 3

(Review)The configuration of the body axes w.r.t the global axes(Review)The configuration of the body axes w.r.t the global axes

i2

k1

j2

x2

z2

y2

i1

k2

j1

x1

z1

y1

s

PP = x1 i1 + y1 j1 +z1 k1, in CS1 Eq.(a)P = x2 i2 + y2 j2 +z2 k2, in CS2 Eq.(b)

i2 = ( i2．i1) i1 + ( i2．j1) j1 + ( i2．k1) k1j2 = ( j2．i1) i1 + ( j2．j1) j1 + ( j2．k1) k1j3 = ( k2．i1) i1 + ( k2．j1) j1 + ( k2．k1) k1

∴ P = [ x2 ( i2．i1 ) + y2 ( j2．i1 )+ z2( k2．i1 ) ] i1+[ x2 ( i2．j1) + y2 ( j2．j1 )+ z2( k2．j1 ) ] j1+[ x2 ( i2．k1) + y2 ( j2．k1)+ z2( k2．k1) ] k1

.Eq.(c)

Comparing Eqs. (c) and (a) leads to x1= ( i2．i1 ) + y2 ( j2．i1 )+ z2( k2．i1 ) y1 = ( i2．j1) + y2 ( j2．j1 )+ z2( k2．j1 ) z1 = ( i2．k1) + y2 ( j2．k1)+ z2( k2．k1)

2005/5/24 計算動力學 4

(Review)The configuration of the body axes w.r.t the global axes (pure rotation)(Review)The configuration of the body axes w.r.t the global axes (pure rotation)

x1= ( i2．i1 ) + y2 ( j2．i1 )+ z2( k2．i1 ) y1 = ( i2．j1) + y2 ( j2．j1 )+ z2( k2．j1 ) z1 = ( i2．k1) + y2 ( j2．k1)+ z2( k2．k1) 'Ass =

s= [x1,y1,z1]T =coordinate vector in terms of world coordinatess = [x2,y2,z2]T = coordinate vector in terms of body coordinatesA: a rotation matrix representing the orientation of the moving body

Each column of A is a unit vector, and the unit vectors are orthogonal to each other.

Similarly, it can be easily derived that s=ATs sAsAs T== −1'

•••••••••

=)()()()()()()()()(

121212

121212

121212

kkkjkijkjjjiikijii

A

Thus, A is a orthonormal matrix, i.e., A-1=AT

3National Kaohsiung University of Applied Sciences, Department of Mechanical Engineering

2005/5/24 計算動力學 5

Practical ConsiderationPractical Consideration

It is seen that the nine direction cosines in matrix A define the orientation of a body relative to the global coordinate system.

A must be a orthogonal matrix, i.e., A-1=AT , or AAT=I

Ikkjkikkjjjijkijiii

kkkjkijkjjjiikijii

AAM =

•••••••••

•••••••••

==)()()()()()()()()(

)()()()()()()()()(

121212

121212

121212

121212

121212

121212T

00

0))(())(())((111

3223

3113

1212121212122112

33

22

11

====

=••+••+••=====

MMMM

jkikjjijjiiiMMMMM

It is impractical to specify the nine direction cosines to satisfy the six constraints

Only three direction cosines are independent

2005/5/24 計算動力學 6

Express Matrix A in terms of Euler ParametersExpress Matrix A in terms of Euler Parameters

=

=

×

ep 0

143

2

1

0

e

eeee

4National Kaohsiung University of Applied Sciences, Department of Mechanical Engineering

2005/5/24 計算動力學 7

Express Matrix A in terms of Euler ParametersExpress Matrix A in terms of Euler Parameters

m

radius m=ssinθ=| u×s |=NP

θ

φφ cos'''cos NP

NPNPmNQ ==

φφ sin'''sin su

susumQP rrrr

rr

×=×

×=

幾何向量

代數矩陣

2005/5/24 計算動力學 8

Transform Euler Parameters to Matrix ATransform Euler Parameters to Matrix A

Thus, matrix A can be easily determined from specified Quaternion/Euler parameters

'Ass =

=

=

=

=

×

2sin

2sin

2sin

2cos

2sin

2cos

0

143

2

1

0

φ

φ

φ

φ

φ

φ

z

y

x

u

u

ue

eeee

uep

5National Kaohsiung University of Applied Sciences, Department of Mechanical Engineering

2005/5/24 計算動力學 9

Some Properties about Euler ParametersSome Properties about Euler Parameters

The four Euler parameters are not independent, because

1)2

(sin)2

(cos 2220 =+=+

φφeeTe

Any vector lying along the rotation axis must have the same components in both initial and final coordinate systems (e lies along the rotation axis)

=

=

×

ep 0

143

2

1

0

e

eeee

201 eT −=ee

2005/5/24 計算動力學 10

Determine Euler Parameters from Rotation Matrix ADetermine Euler Parameters from Rotation Matrix A

Assume that the nine direction cosines of a transformation matrix are given as

Define:

∴

6National Kaohsiung University of Applied Sciences, Department of Mechanical Engineering

2005/5/24 計算動力學 11

Determine Euler Parameters from Rotation Matrix A (e0 ≠0)Determine Euler Parameters from Rotation Matrix A (e0 ≠0)

If e0 ≠0, then

The sign of e0 may be selected as positive or negative Changing the sign of e0 does not influence the rotation matrix

2005/5/24 計算動力學 12

Example: Determine Euler Parameters from Rotation Matrix A (e0 ≠0)Example: Determine Euler Parameters from Rotation Matrix A (e0 ≠0)

7National Kaohsiung University of Applied Sciences, Department of Mechanical Engineering

2005/5/24 計算動力學 13

Determine Euler Parameters from Rotation Matrix A (e0 =0)Determine Euler Parameters from Rotation Matrix A (e0 =0)

similarly,

Since e0 =cos(φ/2)=0 ! φ=±π, ±3π Suppose e1, e2, or e3 is nonzero (determined from one of the

above equations), its sign may be selected as positive or negative

Then, the other two parameters can be determined by the following equations:

2005/5/24 計算動力學 14

Example: Determine Euler Parameters from Rotation Matrix A (e0 =0)Example: Determine Euler Parameters from Rotation Matrix A (e0 =0)

8National Kaohsiung University of Applied Sciences, Department of Mechanical Engineering

2005/5/24 計算動力學 15

Identities with Euler ParametersIdentities with Euler Parameters

Define

Each row of G and L is orthogonal to p=[e0,e]T; i.e.,

GTG=LTL=-ppT+I* (6.46)A=GLT (6.49)dA/dt=2(dG /dt)LT (6.56)

TTGp0Gp ==

2005/5/24 計算動力學 16

Identities with Arbitrary Vector s Identities with Arbitrary Vector s

9National Kaohsiung University of Applied Sciences, Department of Mechanical Engineering

2005/5/24 計算動力學 17

Time Derivative of a Vector Attached to a Moving Body (P.172)Time Derivative of a Vector Attached to a Moving Body (P.172)

The global location of a point P that is fixed in the local coordinate system is given by

0

?=A&

'Ass =

2005/5/24 計算動力學 18

Time Derivative of Rotation Matrix A (pp.173-174)Time Derivative of Rotation Matrix A (pp.173-174)

Let

∴Ω is a skew-symmetric matrix, and we can assume that

IAA =TQ

0AAAAAA =Ω+Ω TTTT )(

0=Ω+ΩT

Ω−=ΩT

ωΩ ~

00

0

12

13

23

=

−−

−=

ωωωωωω

AωA ~=∴ &

What is the physical meaning of ω ?

Ω Is an unknown matrix

10National Kaohsiung University of Applied Sciences, Department of Mechanical Engineering

2005/5/24 計算動力學 19

Time Derivative of Any Vector s Attached to a Moving Body (pp.173-174)

Time Derivative of Any Vector s Attached to a Moving Body (pp.173-174)

PP 'sAs && =

∴Any vector s attached to the local coordinate system can be written as

AωA ~=&

PP 'Ass =

sωssωs r&r& ×=⇔= ~ Thus, ω is the angular velocity of the moving body

2005/5/24 計算動力學 20

Velocity of a Point on Moving BodyVelocity of a Point on Moving Body

PP sωs ~=&

11National Kaohsiung University of Applied Sciences, Department of Mechanical Engineering

2005/5/24 計算動力學 21

Express Angular Velocity in terms of Quaternion and Its Time Derivative (P.174)Express Angular Velocity in terms of Quaternion and Its Time Derivative (P.174)

AωA ~=&

ωAAωAA ~~ == TT&

ωGIppG ~)(2 =+− TT&TTGp0Gp ==

2005/5/24 計算動力學 22

Express Angular Velocity in terms of Quaternion and Its Time Derivative (P.175)

Express Angular Velocity in terms of Quaternion and Its Time Derivative (P.175)

global coordinates local coordinates

12National Kaohsiung University of Applied Sciences, Department of Mechanical Engineering

2005/5/24 計算動力學 23

Express Time Derivative of Quaternion in terms Angular Velocity (P.175)

Express Time Derivative of Quaternion in terms Angular Velocity (P.175)

1334012

103

230

321

21

21

××

−−

−−−−

=

=

z

y

x

T

eeeeeeeeeeee

ωωω

ωGp&

1334012

103

230

321

'

21

21

××

−−

−−−−

=

=

ζ

η

ξ

ωωω

eeeeee

eeeeee

TωLp&

global coordinates local coordinates