Introduction to Multicopter Design and Controlrfly.buaa.edu.cn/course/en/Lesson_05_Coordinate_System_and_Attit… · oz ee o ee y ox b oz b b ox b b o b y b Subscript e represents

Introduction to Multicopter Design and Control

Quan Quan , Associate [email protected]

BUAA Reliable Flight Control Group, http://rfly.buaa.edu.cn/Beihang University, China

Lesson 05 Coordinate System and Attitude Representation

2016/12/25 2

Whatarethethreeattituderepresentationmethodsandtherelationshipbetweentheirderivatives

andtheaircraftbody’sangularvelocity?

Preface

Outline

1. Coordinate System

2. Attitude Representation– Euler Angles– Rotation matrix– Quaternion

3. Conclusion

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Right-Hand Rule

As shown in the figure above, the thumb of the right hand points to the positivedirection of the ox axis, the first finger points to the positive direction of the oy axisand the middle finger points to the positive direction of the oz axis. Furthermore, asshown in the figure above, in order to determine the positive direction of a rotation,the thumb of the right hand points the positive direction of the rotation axis and thedirection of the bent fingers is the positive direction of rotation.

Fig 5.1 Coordinate axes and the positive direction of a rotation using the right-hand rule

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EFCF and ABCF

eoex

ey

ez

bo

bxby

bz

Fig 5.2 The relationship

between the ABCF and the EFCF

The Earth-Fixed Coordinate Frame (EFCF) is used to studymulticopter’s dynamic state relative to the Earth’s surfaceand to determine its 3D position. The Earth’s curvature isignored. The initial position of the multicopter or the centerof the Earth is often set as the coordinate origin , theaxis points to a certain direction in the horizontal planeand the axis points perpendicularly to the ground. Then,the axis is determined according to the right-hand rule.The Aircraft-Body Coordinate Frame (ABCF) is fixed to themulticopter. The Center of Gravity (CoG) of the multicopteris chosen as the origin . The axis points to the nosedirection in the symmetric plane of the multicopter. Theaxis is in the symmetric plane of the multicopter, pointingdownwards, perpendicular to the axis. The axis isdetermined according to the right-hand rule.

eo

bo

e eo xe eo z

e eo y

b bo x

b bo zb bo yb bo x

Subscript e represents the Earth, subscript b represents the Body.


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EFCF and ABCF

eoex

ey

ez

bo

bxby

bz

1 2 3

1 0 00 , 1 , 00 0 1

e e e

Define the following unit vectors

In the EFCF, the unit vectors along the axis, axis andaxis are expressed as , respectively. In the ABCF, the

unit vectors along the axis, axis and axis satisfy thefollowing relationship (Superscript b represents the expressionin ABCF of a vector)

b b b1 1 2 2 3 3, , b e b e b e

In the EFCF, the unit vectors along the axis, axis andaxis are expressed as , respectively. (Superscript

e represents the expression in EFCF of a vector).

e eo x e eo y

e eo z 1 2 3, ,e e eb bo x b bo zb bo y

b bo x b bo y

b bo z e e e1 2 3, ,b b b

Fig 5.2 The relationship

between the ABCF and the EFCF


o o o

3n

3k

2 2( )k n

1k

1n

2n

2b

3n

1 1( )n b

3b

1e

1k

3 3( )e k

2e

2k

(a) Yaw angle (b) Pitch angle (c) Roll angle

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Fig 5.3 Euler angles and frame transformation

The rotation from the EFCF to the ABCF is composed of threeelemental rotations about axes by separately.3 2 1, ,e k n , ,

2. Attitude Representation

Euler Angles(1) Definition


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Euler Angles

Fig 5.4 Intuitive representation of the Euler angles (x axis is orange , y axis is green, z axis is blue)

(1) Definition

exeo

ey

ez

bo by

bz

bx

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Euler Angles

Fig 5.5 Representation of Euler angles

(1) Definition

The angles between the EFCF and theABCF are attitude angles, namely Eulerangles.Pitch angle ：the angle between the bodyaxis and the horizon plane. The pitch angleis positive when the aircraft nose pitches up.Yaw angle ： the angle between theprojection of the body axis in the horizonplane and the earth’s axis. The yaw angle ispositive when the aircraft body turns to right.Roll angle ： the rotation angle of theaircraft symmetry plane around the bodyaxis. The roll angle is positive when theaircraft body rolls to right.


eoex

ey

ez

bo

bxby

bz

bobzo

bxo

e bx xo

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Fig 5.6 Diagram of the Euler angles

The pitch angle shown in theleft figure is positive;The roll angle shown in the leftfigure is negative;The yaw angle shown in theleft figure is positive.

Pitch angle

Roll angle

Yaw angle

(1) Definition

Euler Angles


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(2) Relationship between the attitude rate and the aircraftbody’s angular velocity

The angular velocity of the aircraft body’s rotation is

b b b

Tbx y z ω

Then

？

Euler Angles


b

b

b

x

y

z

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b b b b3 2 1 ω k n b

b

b

b

1 0 sin0 cos cos sin0 sin cos cos

x

y

z

Then

Superscript b represents the expression in ABCF of a vector.

The angular velocity of the aircraft body’s rotation is

b b b

Tbx y z ω


Euler Angles


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Furthermore

1 tan sin tan cos0 cos sin0 sin cos cos cos

WSingularity problem

2

T Θwhere

When , one has, 0


Euler Angles


b Θ W ω

b

b

b

x

y

z

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Rotation Matrix(1) Definition

Define the rotation matrix as

The vectors in rotation matrix satisfy

e e e eb 1 2 3 R b b b

Rotation matrix from the ABCF to the EFCF

Superscript b represents the expression in ABCF of a vector.

Superscript e represents the expression in EFCF of a vector.

e eT eT eb b b b 3 R R R R I

ebdet 1R

Note: det() represents the determinant


e e b e e e b e e e b e1 b 1 b 1 2 b 2 b 2 3 b 3 b 3, , b R b R e b R b R e b R b R e

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The rotation from the EFCF to the ABCF is composed of three elemental steps

where

Rotation Matrix(1) Definition


1 1 1 1 1

2 2 2 2 2

3 3 3 3 3

x

y

z

R

R

R

e k n b ne k n k be k e n b

cos sin 0 cos 0 sin 1 0 0sin cos 0 , 0 1 0 , 0 cos sin .0 0 1 sin 0 cos 0 sin cos

z y x

R R R

o o o

3n

3k

2 2( )k n

1k

1n

2n

2b

3n

1 1( )n b

3b

1e

1k

3 3( )e k

2e

2k

(a) Yaw angle (b) Pitch angle (c) Roll angle

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(1) Definition

Rotation Matrix

1e bb e

1 1 1z

=

cos cos cos sin sin sin cos cos sin cos sin sincos sin sin sin sin cos cos sin sin cos cos sin .

sin sin cos cos cos

y x

z y x

R R

R R R

R R R

1 1 1 1 1

2 2 2 2 2

3 3 3 3 3

x

y

z

R

R

R

e k n b ne k n k be k e n b

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11 12 13eb 21 22 23

31 32 33

r r rr r rr r r

R

21

11

31

32

33

tan

sin

tan

rrr

rr

Calculate Eulerangles from therotation matrix

It is assumed that inthis case.

0

Singularity problem2

21

11

31

32

33

arctan

arcsin

arctan

rr

rrr


(1) Definition

Rotation Matrix

When ,2

e

b

0 sin cos0 cos sin .1 0 0

R

Infinite number of combinations

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The cross product of two vectors and is defined as

where

(2) Relationship between the derivative of the rotation matrix and the aircraft body’s angular velocity


Rotation Matrix

0

00

z y

z x

y x

a aa aa a

a sin a b a b n

a b a b

T

x y za a a a T

x y zb b b b


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If the rigid body’s rotation (without translation) is only considered,then the derivative of a vector satisfies (Similar to the circularmotion)

e 3r e

e eddt

r ω r

where the symbol × represents the vector cross product. One has


Rotation Matrix

e e e1 2 3 e e e e e e

1 2 3

ddt

b b b

ω b ω b ω bFig 5.7 The derivative of a vector presented by the circular motion

eddt

r

er

eω

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According to and the property of the cross product Following property ofthe cross product isused : for a rotationmatrixand any two vectors

, one has

3 3 det 1 R R

3, a b Ra Rb R a b

The use of the rotation matrix can avoid the singularity problem. However, sincehas nine unknown variables, the computational burden of solving equation is heavy.


Rotation Matrix


e e bb ω R ω

ee b e e b e e b ebb b 1 b b 2 b b 3

e b e b e bb 1 b 2 b 3

e b b bb 1 2 3

e bb

ddt

R R ω R e R ω R e R ω R e

R ω e R ω e R ω e

R ω e ω e ω e

R ω

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Quaternion(1) Definition

Fig 5.8 Quaternion plaque on Brougham(Broom) Bridge, Dublin, and the image is fromhttps://en.wikipedia.org/wiki/Quaternion

It reads “Here as he walked by on the 16thof October 1843 Sir William RowanHamilton in a flash of genius discoveredthe fundamental formula for quaternionmultiplication & cutit on a stone of this bridge.”

Quaternions are normally written as

0

v

q

qq

where is the scalar part of andis the vector part.

For a real number , the correspondingquaternion is defined as . For avector , the corresponding quaternion is

.

0 q 4q T 3

v 1 2 3q q q q

s T1 3q 0 s

3vTT0q v

i2 = j2 = k2 = ijk = −1


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(2) Quaternions’ basic operation rules0 0 0 0

v v v v

p q p q

p qp q p q

T0 0 0 0 v v

v v v v 0 v 0 v

p q p qp q

q pp q

p q p q q p

• Addition and subtraction

• Multiplication

Multiplication properties (Note: q, r, m are quaternions, s is a scalar, u, v are column vectors)

q r m q r q m

q r m q r m q r m0

v

sqs s

s

q qq

T0 0

u vu v

q qu v u v

Quaternion


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• Conjugate

0

v

q

qq

0

v

q

qq

q q

p q q p

p q p q

Some properties:

• Norm2

2 T0 v v

2 2 2 20 1 2 3

q

q q q q

q q q q q

q q

p q p q

q q

Some properties:

(2) Quaternions’ basic operation rules Quaternion


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• Inverse1

3 1

1

q q

0

According to the definition of , one has q 1

qqq

• Unit quaternionFor a unit quaternion , it satisfies . Let . Thenq 1q

(2) Quaternions’ basic operation rules

1p q

1

1p q

q q

Quaternion


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(3) Quaternions as rotationsAssume that represents a rotation process and represents a vector. Then under the action of , the vector is turned into . This process is expressed as

q 31 v

q1v

1

1 1

0 0

q qv v

The first rowalways stands

cos2

sin2

qv

A unit quaternion can always be written in the form

Fig 5.9 Physical meaning of unit quaternionsReaders can further refer to:[1] Shoemake K. Quaternions. Department of Computer and Information Science,University of Pennsylvania, USA, 1994

31v

Quaternion


v

x

y

z

1v

1v

o

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T0 1 cos

2

v v

0 1 0 1 0 1

0 10 1 sin sin

2 2

v v v v v vvv v v v 0 1 sin

2

v v v

Define two unit vectors with being the angle between them. Therefore, one has

0 1 1 0, v v v v 2

Define a unit quaternion, one has

T0 1

010 1

cos 002

sin2

v vq

vvv vvFig 5.10 Rotation represented by quaternions

(3) Quaternions as rotations Quaternion


0 1

0 1

v v

vv v

22v

2

1v0v

0 1

0 1

v v

vv v

22v

2

1v0v

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Fig 5.10 Rotation represented by quaternions

1

02

00

q q

vv* *

1

02 1 1

* *

0 0 1 1

* *

0 0 1 1

3 1 3 1

1

00 0 0

0 0 0 0

0 0 0 0

1 1

0

q qvv v v

qv v v v

qv v v v

q0 0

q

v

*

0

0

v

lies in the same plane as and , as shownin the figure below, and also forms an anglewith .

2v 1v2

Why can quaternionsrepresent the rotation?

0v

1v



2016/12/25 28

1

1 1

0 0

q qv v

• Coordinate frame rotation• Vector rotation

Pay attention to the difference !



1

1 1

0 0

q qv v

v

x

y

z

1v

1v

o

1 1( )v v

v

x

y

z

x

yz

o

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It is assumed that the rotation from the EFCF to the ABCF is represented by

the quaternion , one has (Coordinate frame rotation) Tbe 0 1 2 3q q q qq

1e eb be b

1b be eb

0 0

0

q qr r

q qr

0 1 2 3 0 1 2 3

1 0 3 2 1 0 3 2e b

2 3 0 1 2 3 0 1

3 2 1 0 3 2 1 0

0 0q q q q q q q qq q q q q q q qq q q q q q q qq q q q q q q q

r r

2 2 2 20 1 2 3 1 2 0 3 1 3 0 2

b 2 2 2 2e 1 2 0 3 0 1 2 3 2 3 0 1

2 2 2 21 3 0 2 2 3 0 1 0 1 2 3

2( ) 2( )( ) 2( ) 2( )

2( ) 2( )

q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q

C q

(4) Quaternions and rotation matrix Quaternion


e b be( )r C q re bb e( )R C q

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be z y x q q q q

T

T

T

cos sin 0 02 2

cos 0 sin 02 2

cos 0 0 sin2 2

x

y

z

q

q

q

According to the rotation order of Euler angles, one has

be

cos cos cos sin sin sin2 2 2 2 2 2

sin cos cos cos sin sin2 2 2 2 2 2

cos sin cos sin cos sin2 2 2 2 2 2

cos cos sin sin sin cos2 2 2 2 2 2

q

(5) Quaternions and Euler angles Quaternion


1b be eb e

1

e

-1 -1-1z e

0 0

0

0=

z y x z y x

x y z y x

q qr r

q q q q q qr

q q q q q qr

o o o

3n

3k

2 2( )k n

1k

1n

2n

2b

3n

1 1( )n b

3b

1e

1k

3 3( )e k

2e

2k

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0 1 2 32 21 2

0 2 1 3

0 3 1 22 22 3

2( )tan1 2

sin 2

2tan

1 2

q q q qq q

q q q q

q q q qq q

2

be

cos2 2

sin2 2 2

2cos

2 2

sin2 2

q

When , the singularity problem occurs.

0 2 1 3 0 2 1 32 1|| 2 1q q q q q q q q

Infinite number of combinations

It is assumed that . 0

0 1 2 32 21 2

0 2 1 3

0 3 1 22 22 3

2( )arctan1 2

arcsin 2

2arctan

1 2

q q q qq q

q q q q

q q q qq q

(5) Quaternions and Euler angles Quaternion


be

cos cos cos sin sin sin2 2 2 2 2 2

sin cos cos cos sin sin2 2 2 2 2 2

cos sin cos sin cos sin2 2 2 2 2 2

cos cos sin sin sin cos2 2 2 2 2 2

q

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According to the composite rotationquaternion in the case of rotatingcoordinate frames, one has

where b b

e et t t q q qAircraft body’s angular velocity

The derivative of is obtained as be tq

Perturbation

(6) Relationship between the derivative of the quaternions and the aircraft body’s angular velocity

Quaternion


Tb T11

2t

q ω

b be eb

e 0

b be e

0

Tb b T be e

0

b Tb b

3 e eb b

0

b Tbeb b

lim

lim

112lim

012

lim

012

t

t

t

t

t t tt

tt t

t

t t t

tt

t tt t

t

t

q qq

q q q

q ω q

ωI q q

ω ω

ωq

ω ω

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b T

b be eb b

012

t t

ωq q

ω ω

T b0 v

bv 0 3 v

12

12

q

q

q ω

q I q ω

In practice, can be measuredapproximately by a three-axisgyroscope, then the equationabove is linear!

bω

Tb Te 0 vq q q

Transformation

e bb e( )R C q

Quaternion


(6) Relationship between the derivative of the quaternions and the aircraft body’s angular velocity

3. Conclusion

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ee bbb

ddt

R R ω

• Relationship between the Euler angles andthe aircraft body’s angular velocity

• Relationship between the rotation matrixand the aircraft body’s angular velocity

• Relationship between the quaternions andthe aircraft body’s angular velocity

Fig 5.11 Mutual transformations among three rotation expressions

Singularity, Nonlinear

Non-singularity, High dimension

Non-singularity, Lower dimension

Most autopilots of multicopters use this representation. Un

ique

Sing

ular

ity

Non-uniqueUnique

b Θ W ω

b T

b be eb b

012

t t

ωq q

ω ω

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Yangguang Cai

Acknowledgement

Deep thanks go to

for material preparation.Jinrui Ren Xunhua Dai

2016/12/25 36

Thank you!All course PPTs and resources can be downloaded at

http://rfly.buaa.edu.cn/course

For more detailed content, please refer to the textbook: Quan, Quan. Introduction to Multicopter Design and Control. Springer,

2017. ISBN: 978-981-10-3382-7.It is available now, please visit http://

www.springer.com/us/book/9789811033810

Documents

Introduction to Multicopter Design and Controlrfly.buaa.edu.cn/course/en/Lesson_05_Coordinate_System_and_Attit… · oz ee o ee y ox b oz b b ox b b o b y b Subscript e represents