Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 1

Chapter 2

Coordinate Frames

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 2

Reference Frames •  In guidance and control of aircraft, reference frames

used a lot •  Describe relative position and orientation of objects

–  Aircraft relative to direction of wind –  Camera relative to aircraft –  Aircraft relative to inertial frame

•  Some things most easily calculated or described in certain reference frames –  Newton’s law –  Aircraft attitude –  Aerodynamic forces/torques –  Accelerometers, rate gyros –  GPS –  Mission requirements

Must know how to transform between different reference frames

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 3

Rotation of Reference Frame

p = p0x

i0 + p0y

j0 + p0z

k0

p = p1x

i1 + p1y

j1 + p1z

k1

p1x

i1 + p1y

j1 + p1z

k1 = p0x

i0 + p0y

j0 + p0z

k0

p1 4=

0

@p1x

p1y

p1z

1

A =

0

@i1 · i0 i1 · j0 i1 · k0

j1 · i0 j1 · j0 j1 · k0

k1 · i0 k1 · j0 k1 · k0

1

A

0

@p0x

p0y

p0z

1

A

p1 = R10p

0 R10

4=

0

@cos ✓ sin ✓ 0

�sin ✓ cos ✓ 0

0 0 1

1

Awhere

(rotation about k axis)

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 4

Rotation of Reference Frame Right-handed rotation about j axis:

R10

4=

0

@cos ✓ 0 �sin ✓0 1 0

sin ✓ 0 cos ✓

1

A

Right-handed rotation about i axis:

R10

4=

0

@1 0 00 cos ✓ sin ✓0 �sin ✓ cos ✓

1

A

Orthonormal matrix properties:

P.1. (Rba)

�1 = (Rba)

> = Rab

P.2. RcbRb

a = Rca

P.3. det�Rb

a

�= 1

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 5

Rotation of a Vector

R10 can be interpreted

as left-handed rotation of

vector by angle ✓.

Let and q4= |q|, then

p =

0

@p cos(✓ + �)p sin(✓ + �)

0

1

A

=

0

@p cos ✓ cos�� p sin ✓ sin�p sin ✓ cos�+ p cos ✓ sin�

0

1

A

=

0

@cos ✓ �sin ✓ 0

sin ✓ cos ✓ 0

0 0 1

1

A

0

@p cos�p sin�

0

1

A

where p4= |p|.

Define

q =

0

@p cos�p sin�

0

1

A

then

p = (R10)

>q =) q = R10p

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 6

Inertial Frame and Vehicle Frame

• Vehicle frame has same ori-

entation as inertial frame

• Vehicle frame is fixed at cm

of aircraft

• Inertial and vehicle frames

are referred to as NED frames

• N �! x, E �! y, D �! z

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 7

Euler Angles

•  Need way to describe attitude of aircraft •  Common approach: Euler angles

•  Pro: Intuitive •  Con: Mathematical singularity

–  Quaternions are alternative for overcoming singularity

: heading (yaw)

✓: elevation (pitch)

�: bank (roll)

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 8

Vehicle-1 Frame

Rv1v ( ) =

0

@cos sin 0

� sin cos 0

0 0 1

1

A

gives

pv1= Rv1

v ( )pv

where is the heading.

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 9

Vehicle-2 Frame

Rv2v1(✓) =

0

@cos ✓ 0 � sin ✓0 1 0

sin ✓ 0 cos ✓

1

A

gives

pv2= Rv2

v1(✓)pv1

where ✓ is the pitch angle.

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 10

Body Frame

Rbv2(�) =

0

@1 0 0

0 cos� sin�0 � sin� cos�

1

A

gives

pb= Rb

v2(�)pv2

where � is the roll (bank) angle.

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 11

Inertial Frame to Body Frame Transformation

Define

Rbv(�, ✓, ) = Rb

v2(�)Rv2v1(✓)Rv1

v ( )

=

0

@1 0 0

0 cos� sin�0 � sin� cos�

1

A

0

@cos ✓ 0 � sin ✓0 1 0

sin ✓ 0 cos ✓

1

A

0

@cos sin 0

� sin cos 0

0 0 1

1

A

=

0

@c✓c c✓s �s✓

s�s✓c � c�s s�s✓s + c�c s�c✓c�s✓c + s�s c�s✓s � s�c c�c✓

1

A

to give

pb= Rb

v(✓)pv

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 12

Stability Frame

Stability frame helps us rigorously define angle of attack and is useful for analyzing stability of aircraft

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 13

Wind Frame

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 14

Wind Frame (continued)

•  Wind frame helps us rigorously define side-slip angle

•  Side-slip angle is nominally zero for tailed aircraft

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 15

ground  track  

Airspeed, Wind Speed, Ground Speed

a/c wrt to inertial frame expressed in body frame

wind wrt to inertial frame expressed in body frame

a/c wrt to surrounding air expressed in body frame

Va = Vg �Vw

Vbg =

0

@uvw

1

A

Vbw =

0

@uw

vwww

1

A = Rbv(�, ✓, )

0

@wn

we

wd

1

A

Vwa =

0

@Va

00

1

A

Vba =

0

@ur

vrwr

1

A =

0

@u� uw

v � vww � ww

1

A

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 16

Airspeed, Angle of Attack, Sideslip Angle

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 17

Flight  path  projected  onto  ground  

horizontal  component  of  groundspeed  vector  

Course and Flight Path Angles

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 18

ground  track  

north  Wind Triangle

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 19

Vg

VwVa

�a

�

↵✓

ib

Wind Triangle

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 20

When wind speed and sideslip are zero…

If both the windspeed and the sideslip angles are zero, i.e.,

Vw = 0

� = 0

then we have the following simplifications

Va = Vg Airspeed equals groundspeed

u = ur Velocity equals velocity relative to the air mass

v = vr

w = wr

= � Heading equals course

� = �a Flight path angle equals air-mass-referenced flight path angle

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 21

Differentiation of a Vector �b/i

p = px

ib + py

jb + pz

kb

d

dti

p = px

ib + py

jb + pz

kb + px

d

dti

ib + py

d

dti

jb + pz

d

dti

kb

d

dtb

p = px

ib + py

jb + pz

kb

ib = !b/i

⇥ ib

jb = !b/i

⇥ jb

kb = !b/i

⇥ kb

px

ib + py

jb + pz

kb = px

(!b/i

⇥ ib) + py

(!b/i

⇥ jb) + pz

(!b/i

⇥ kb)

= !b/i

⇥ p

d

dti

p =d

dtb

p+ !b/i

⇥ p

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 22

2

1

3

4

5

6 7

8

10

9

11

14

13

16

15

12

Project Aircraft

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 23

wing_l

fuse_l3

fuse_l2

fuse_l1

tailwing_l

tail_h

fuse_h

Project Aircraft

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: Slide 24

fuse_l1 fuse_l2

fuse_l3

tailwing_l

wing_l

wing_w

tailwing_w

fuse_w

Project Aircraft