Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: 1/21

Chapter 2

Coordinate Frames

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

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: 3/21

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: 4/21

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 0

0 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: 5/21

Rotation of a Vector Left-handed rotation of p

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

q =

0

@q cos�q sin�

0

1

A p4= |p| = q

4= |q|

p =

0

@cos ✓ �sin ✓ 0

sin ✓ cos ✓ 0

0 0 1

1

Aq

= (R10)

>q

q = R10p

R10 can be interpreted

as left-handed rotation of

vector by angle ✓

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: 6/21

Inertial Frame and Vehicle Frame

•  Vehicle frame has same orientation as inertial frame

•  Vehicle frame is fixed at cm of aircraft •  Inertial and vehicle frames are referred

to as NED frames •  Nàx, Eày, Dàz

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: 7/21

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: 8/21

Vehicle-1 Frame

: heading

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: 9/21

Vehicle-2 Frame

✓: elevation (pitch)

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: 10/21

Body Frame

�: bank (roll)

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: 11/21

Inertial Frame to Body Frame Transformation

pb = Rbv(✓)p

v

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: 12/21

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: 13/21

Wind Frame

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: 14/21

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: 15/21

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

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: 16/21

Airspeed, Angle of Attack, Sideslip Angle

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: 17/21

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: 18/21

ground  track  

north  Wind Triangle

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: 19/21

Vg

VwVa

�a

�

↵✓

ib

Wind Triangle

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: 20/21

When wind speed is zero…

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

Differentiation of a Vector �b/i

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 2: 22/21

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: 23/21

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: 24/21

fuse_l1 fuse_l2

fuse_l3

tailwing_l

wing_l

wing_w

tailwing_w

fuse_w

Project Aircraft