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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
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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