2. The First Manned Quadrotor
• Quadrotor is a kind of unmanned aerial vehicle
(UAV)
• 29/9/1907: Louis Bréguet & Jacques Bréguet,
under the guidance of Professor Charles
Richet, demonstrated the first flying quadrotor
named Bréguet-Richet Gyroplane No. 1
3. Advantages of quadrotor
• Quadrotor is a rotary wing UAV
• Its advantages over fixed wing UAVs:
– Vertical Take Off and Landing (VTOL)
– Able to hover
– Able to make slow precise movements.
– Four rotors provide a higher payload capacity
– More flexible in maneuverability through an
environment with many obstacles, or landing in
small areas.
5. Quadrotor structure
• Front motor (Mf) (+x)
• Back motor (Mb) (-x)
• Right motor (Mr) (+y)
• Left motor (Ml) (-y)
• Mfand Mb rotates CW
• Mr and Mlrotates CCW
• This arrangement can
overcome torque effect to
prevent on the spot
spinning of the structure
• Each spinning motor
provides
– thrust force (T) for lifting
– torque () for rotating
7. Reference Frames
• There are a few reference frames to model
the kinematics and dynamics of a quadrotor
– Inertia Frame (Global frame), Fi
– Vehicle Frame, Fv
– Vehicle-1 Frame, Fv1
– Vehicle-2 Frame, Fv2
– Body Frame (Local frame), Fb
8. The inertia frame
• For the context of quadrotor, the Earth is a
flat surface
• The starting position of the quadrotor is the
origin of the global frame or the inertia frame
(Fi)
• Fi: x-y-z axis is right hand system with x
pointing to North, y pointing to East and z
pointing to Down, it is also known as the NED
system
X (North)
Y (East)
Z (Down)
9. The Vehicle Frame
• Fv is the vehicle frame
• It is the inertia frame, Fi, linear shifted to the
centre of gravity (COG) for the quadrotor
• The coordinates of the COG for the quadrotor
wrt Fi is (xc, yc, zc).
Xi (North)
Yi (East)
Zi (Down)
Xv
Yv
Zv
F v
F i
14. , , are known as Euler angles. They are measured from different frames (Roll in
Fv2 frame, Pitch in Fv1 frame, Yaw in Fv frame)
15. Gimbal Lock
• This is a fundamental problem when
using sensors to sense Euler angles
• When pitch angle is 90 degrees, roll and
yaw rotation give the same sensor
readings
• Information for 1 dimension is lost and
the actual configuration of the rigid body
is not correctly sensed
• Solution:
– Avoid 90 degree pitch when using Euler
angle sensor
16. Quadrotor State Variables
• Positions in Fi : pn, pe, h
• Velocities in Fb: u, v, w
• Angular velocities in Fb: p, q, r
• Euler angles:
– Yaw angle in Fv: ψ
– Pitch angle in Fv1: θ
– Roll angle in Fv2: ϕ
17. Quadrotor Kinematics
−
−+
+−
=
=
− w
v
u
CCCSS
CSSSCCCSSSSC
SSCSCSCCSSCC
w
v
u
R
h
p
p
dt
d v
be
n
θφθφθ
ψφψθφψφψθφψθ
ψφψθφψφψθφψθ
( ) ( ) ( ) ( ) ( ) ( )
−=
−
−
=
+
+
=
r
q
p
CCCS
SC
TCTS
CCS
CSC
S
r
q
p
RRRRRR
r
q
p
v
v
v
v
b
v
v
v
b
v
b
v
θφθφ
φφ
θφθφ
θφφ
θφφ
θ
ψ
θ
φ
ψ
θ
φ
ψ
ψθφθθφ
φ
φ
0
0
1
and
0
0
01
0
0
0
0
0
0 12
12
2
122
• Relating position (Fi) and velocities (Fb) in
the same frame (Fi):
• Relating angular velocities (Fb) to Euler
angle rates (Fv, Fv1, Fv2)
18. Equation of Coriolis
pp
dt
d
p
dt
d
b
bi
×+= ω
• Inertia frame, Fi looking at Body frame, Fb
• Vector p is moving in Fb and Fb is rotating and
translating with respect to Fi
• Time derivative of p as seen from Fi is
obtained using equation of Coriolis:
19. Quadrotor Dynamics
• Equation of Coriolis:
• m is the mass
• vector v is the velocities
• vector ωb is the angular
velocities in the body frame
• vector f is the applied forces
• In body coordinates:
+
−
−
−
=
z
y
x
f
f
f
m
pvqu
rupw
qwrv
w
v
u
1
fv
dt
vd
m
dt
vd
m b
bi
=
×+= ω
20. Rotational Motion
• Equation of Coriolis for rotational motion:
• vector h is angular momentum, h = Jωb
• J is symmetric inertia matrix
• vector m is the applied torque
• Substitutes into equation of Coriolis:
• Angular acceleration is hence given by:
mhh
dt
d
h
dt
d
b
bi
=×+= ω
=
=
=
ψ
θ
φ
τ
τ
τ
ω m
r
q
p
I
I
I
J b
z
y
x
;;
00
00
00
{ }
{ }
{ }
+−
+−
+−
=
=
×
+
zyx
yxz
xzy
IpqII
IprII
IqrII
r
q
p
r
q
p
J
r
q
p
r
q
p
J
ψ
θ
φ
ψ
θ
φ
τ
τ
τ
τ
τ
τ
)(
)(
)(
21. Summary of Equation Set
−
−+
+−
=
− w
v
u
CCCSS
CSSSCCCSSSSC
SSCSCSCCSSCC
h
p
p
e
n
θφθφθ
ψφψθφψφψθφψθ
ψφψθφψφψθφψθ
−=
r
q
p
CCCS
SC
TCTS
θφθφ
φφ
θφθφ
ψ
θ
φ
0
0
1
{ }
{ }
{ }
+−
+−
+−
=
zyx
yxz
xzy
IpqII
IprII
IqrII
r
q
p
ψ
θ
φ
τ
τ
τ
)(
)(
)(
+
−
−
−
=
z
y
x
f
f
f
m
pvqu
rupw
qwrv
w
v
u
1
22. Thrust Force and Gravity Force
• fx , fy , fz are total forces acting on the body frame, Fb
• there are two components:
– quadrotor thrust force (produced by propeller)
– gravity force
• Total thrust in Fb: T = Tf + Tb +Tl + Tr
• Gravity force in Fi : (0,0,mg)
• In Fb:
−
+
−
=
+
−
=
φθ
φθ
θ
CmgC
SmgC
mgS
Tmg
RR
Tf
f
f
v
i
b
v
z
y
x
0
0
0
0
0
0
23. Torque / Moment
• Roll : τϕ = l (Tl - Tr)
• Pitch : τθ = l (Tf- Tb)
• Yaw : τψ = τr+ τl- τf- τb
• The drag of the propellers produces a
yawing torque on the body of the
quadrotor (Newton's 3rd Law)
• The direction of the torque is int he
opposite direction to the motion of the
propeller
• The thrust and torque of each motor is
controlled by its angular speed in rpm:
– Ti = kf ωi
2
– τi = km ωi
2
• i can take the value 1 (front motor), 2 (right
motor), 3 (back motor) and 4 (left motor)
24. Simplified model
• Use vehicle 1 frame
for position estimate
• Small Euler angles
(sin, tan -> 0)
• Ignore Coriolis terms
(qr, pr, pq)
−−
−+
+−
=
w
v
u
CCCSS
CSSSCCCSSSSC
SSCSCSCCSSCC
h
p
p
e
n
θφθφθ
ψφψθφψφψθφψθ
ψφψθφψφψθφψθ
−
+
−
=
=
+
−
−
−
=
φθ
φθ
θ
CC
SC
S
g
T
m
f
f
f
m
f
f
f
m
pvqu
rupw
qwrv
w
v
u
z
y
x
z
y
x
0
0
111
=
−=
r
q
p
r
q
p
CCCS
SC
TCTS
θφθφ
φφ
θφθφ
ψ
θ
φ
0
0
1
=
=
z
y
x
I
I
I
r
q
p
ψ
θ
φ
τ
τ
τ
ψ
θ
φ
−
−=
=
w
v
u
CCCSS
SC
SCSSC
w
v
u
RR
p
p
p
v
b
v
v
z
y
x
θφθφθ
φφ
θφθφθ
021
2
−
+−
−−
+
=
−
−=
θφ
ψφψθφ
ψφψθφ
θφθφθ
φφ
θφθφθ
CC
CSSSC
SSCSC
m
T
gw
v
u
CCCSS
SC
SCSSC
p
p
p
z
y
x
0
0
0
−
+
−
=
+
−
=
φθ
φθ
θ
CmgC
SmgC
mgS
Tmg
RR
Tf
f
f
v
i
b
v
z
y
x
0
0
0
0
0
0
=
=
z
y
x
I
I
I
r
q
p
ψ
θ
φ
τ
τ
τ
ψ
θ
φ
−
+−
−−
+
=
θφ
ψφψθφ
ψφψθφ
CC
CSSSC
SSCSC
m
T
gp
p
p
z
y
x
0
0
25. State Estimates
• States to be measured or estimated:
– p, q, r (from sensors)
– ,θ,ψϕ
– dot p, dot q, dot r
– px, py, pz (from sensors)
– u, v, w
– dot u, dot v, dot w
• From rate gyroscopes, we can get (p,q,r)
• Integrating and differentiating (p,q,r) to get
( ,θ,ψ) and angular accelerationϕ
• From position sensor (usually external
camera), we get (px, py, pz)
• Differentiating position to get (u,v,w)
• From accelerometer, we get T/m, toggether
with Euler angles, we can get position
acceleration
=
=
z
y
x
I
I
I
r
q
p
ψ
θ
φ
τ
τ
τ
ψ
θ
φ
−
+−
−−
+
=
θφ
ψφψθφ
ψφψθφ
CC
CSSSC
SSCSC
m
T
gp
p
p
z
y
x
0
0
=
−=
r
q
p
r
q
p
CCCS
SC
TCTS
θφθφ
φφ
θφθφ
ψ
θ
φ
0
0
1
−
−=
w
v
u
CCCSS
SC
SCSSC
p
p
p
z
y
x
θφθφθ
φφ
θφθφθ
0
=
∫
∫
∫
rdt
qdt
pdt
ψ
θ
φ
26. Case study: From rest to hover in z
• A quadrotor is resting at its vehicle frame
• There is no rotational movement
• It starts to climb to a certain height and hovers
• From rest, the thrust is incrased
• T > mg, a is positive vertically, v increases, h increases
• T < mg, a is negative vertically, v decreases to zero, h increases
• T = mg, a is zero, v is zero, h maintains
• Exercise: From hovering to the ground
27. Case study: From hover to x and hover
• From hovering,
• Negative pitch, ax is positive , vx increases, x
increases
• Positive pitch, ax is negative, vx decreases to
zero, x increases
• No pitch, T = mg, a is zero, vx is zero, x is
maximum
• Exercise: How to maintain h during these
operation?
28. Jerk free planning
• Jerk is the time derivative of acceleration
• Physically, it is sudden start or stop
• Maximum force is upon the quadrotor with jerk
• Exercise: Qualitatively design a jerk profile such that
quadrotor is climbing up to a height to hover with jerk-free
movement.
3
3
2
2
dt
zd
dt
vd
dt
ad
jz
===
