The document describes the modeling and control of a quadrotor unmanned aerial vehicle (UAV). It first outlines various modeling assumptions. It then derives the nonlinear model, including kinematic and dynamic equations, by defining reference frames, rotational matrices, forces and moments. Linear models are developed by linearizing the nonlinear model. Finally, it discusses control design, analysis and simulation for the quadrotor.

1. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Modeling and Control of Quadrotor UAV
Aniket Shirsat
April 2, 2015

2. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Table of contents
1 Non-Linear Model
Modeling Assumptions
Reference Frames
Kinematics
Dynamics
2 Linear Models
State Space Model
Nominal Model Parameters
Linear Model Analysis
3 Control
Design
Analysis
Simulation
4 Conclusion

3. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Assumptions
Aircraft is a rigid body.

4. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Assumptions
Aircraft is a rigid body.
Propellers are rigid.

5. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Assumptions
Aircraft is a rigid body.
Propellers are rigid.
Quadrotor frame is symmetrical.

6. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Assumptions
Aircraft is a rigid body.
Propellers are rigid.
Quadrotor frame is symmetrical.
Mass center and geometric center coincide.

7. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Assumptions
Aircraft is a rigid body.
Propellers are rigid.
Quadrotor frame is symmetrical.
Mass center and geometric center coincide.
Motor inertia small and neglected.

8. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Assumptions
Aircraft is a rigid body.
Propellers are rigid.
Quadrotor frame is symmetrical.
Mass center and geometric center coincide.
Motor inertia small and neglected.

9. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Reference Frames
Necessity of different frames
Actuator inputs and forces act on the body. (Body frame)
IMU, accelerometers measures quantities in the body
frame.
GPS measures position in the inertial frame. (Inertial
frame)
Model development is carried out in the inertial frame.

10. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Kinematics
Position of the quadrotor in the inertial frame
η =[XYZ]
Attitude of the quadrotor is represented by
φ : Roll ,θ : Pitch, ψ : Yaw
Angular rates in the body frame are
ν =[pqr]

11. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Rotation Matrix
Rotation matrix sequence to go from earth to body is the
Yaw-Pitch-Roll euler angle sequence
Rotation about z axis by ψ.
Rψ =



cos(ψ) sin(ψ) 0
− sin(ψ) cos(ψ) 0
0 0 1


 (1)

12. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Rotation Matrix
Rotate about the new y-axis positive Pitch(θ).
Rθ =



cos(θ) 0 − sin(θ)
0 1 0
sin(θ) 0 cos(θ)


 (2)

13. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Rotation Matrix
Rotate about the new x axis positive Roll(φ).
Rφ =



1 0 0
0 cos(φ) sin(φ)
0 − sin(φ) cos(φ)


 (3)

14. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Rotation Matrix
The complete rotation matrix is
RBody
Earth = Rotz,ψ ∗ Roty,θ ∗ Rotz,φ (4)

15. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Angular Velocity
The angular velocity matrix is given by
ΩBody
Earth = ˙φ + Rφ
˙θ + RφRθ
˙ψ = Ω ˙η (5)
where
Ω =



1 0 − sin(θ)
0 cos(φ) sin(φ) cos(θ)
0 − sin(φ) cos(φ) cos(θ)


 (6)

16. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Euler rates
The euler rates are given by



˙φ
˙θ
˙ψ


 = Ω−1



p
q
r


 (7)
where
Ω−1
=



1 tan(θ) sin(φ) tan(θ) cos(φ)
0 cos(φ) − sin(φ)
0 sin(φ)
cos(θ)
cos(φ)
cos(θ)


 (8)

17. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Frames representation
Inertial and body reference frame for quadrotor
Figure: Inertial and Body reference frame for a Quadrotor

18. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Forces
Motor force
fi = kω2
i (9)
Thrust
T =
4
i=1
fi = k
4
i=1
ω2
i (10)
Thrust in the body frame
TB =



0
0
T


 (11)

19. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Moments
Roll Moment
Figure: Roll moment about the x axis
Roll torque
τφ = lk(ω2
4 − ω2
2) (12)

20. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Pitch Moment
Pitch moment
Figure: Pitch moment about the y axis
Pitch Torque
τθ = lk(ω2
3 − ω2
1) (13)

21. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Yaw Moment
Yaw Moment
Figure: Yaw movement about the z axis
Yaw Torque
τψ = b(ω2
1 + ω2
3 − ω2
2 − ω2
4) (14)

22. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Moment of Inertia
Schematic
Figure: Schematic for inertia calculation
Since the quadrotor is assumed to be symmetrical Ixx = Iyy
The inertia matrix is
I =



Ixx 0 0
0 Iyy 0
0 0 Izz


 (15)

23. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Moment of Inertia
Moment of inertia about the x- axis is
Ixx =
2MR2
5
+ 2ml2
(16)
Moment of inertia about y- axis is
Iyy =
2MR2
5
+ 2ml2
(17)
Moment of inertia about z- axis is
Izz =
2MR2
5
+ 4ml2
(18)

24. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Euler- LaGrange Forulation
Lagrangian L is
L(q, ˙q) = Etrans + Erot − Epot (19)
Translational Acceleration
f = RTB = m¨ξ + mg



0
0
1


 (20)

25. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Equation of Motion
Angular Acceleration
τ = τB = J¨η+
d
dt
(J) ˙η−
1
2
∂
∂η
( ˙ηT
J ˙η) = J¨η+C(η, ˙η) ˙η (21)
which can be rearranged to give
¨η = J−1
(τB − C(η, ˙η) ˙η) (22)

26. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Longitudinal state space model
State Space equation
˙XLong = ALong XLong + BLong ULong (23)
State Transition Matrix
ALong =





Z˙z 0 Zθ 0
0 X˙x Xθ 0
0 0 0 1
0 0 Θθ Θ˙θ





(24)

27. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Longitudinal state space model
Input matrix
BLong =





ZT 0
XT 0
0 0
0 Θτθ





(25)
States
Xlong = ˙z ˙xθ ˙θ
T
(26)
Inputs
ULong = Tτθ
T
(27)

28. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Nominal parameters
Parameter Value Unit
m 0.468 kg
Ixx 4.856e-3 kg.m2
Iyy 4.856e-3 kg.m2
Izz 8.801e-3 kg.m2
Ax 0.25
Ay 0.25
Az 0.25
g 9.81 m/sec2
l 0.225 m

29. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Linear Model Analysis : Forward flight
Longitudinal Dynamics : Nominal Plant
P(s) =
g11
s+a
g12
s2(s+a)
g21
s+a
g22
s2(s+a)
(28)

30. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Linear Model Analysis : Forward and Vertical flight
Equilibrium Thrust Variation
Thrust varies significantly with vertical velocity.

31. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Linear Model Analysis
Equilibrium τθ variation
τθ is always zero.

32. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Linear Model Analysis
Expression for θeq
θeq = arctan(
Ax ˙xeq
mg + Az ˙zeq
) (29)
θeq variation
Varies significantly with forward velocity as compared to
vertical velocity.

33. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Linear Model Analysis
Gain 11 : Thrust to ˙Z
Gradual decease with increasing forward velocity,
Little impact of vertical velocity.

34. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Linear Model Analysis
Gain 12 : Thrust to ˙X
At Hover it is zero, validating the Nominal plant is
decoupled.

35. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Linear Model Analysis
Gain 21 : τθ to ˙Z
At hover, it is zero thus ensuring that the model is
decoupled.

36. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Linear Model Analysis
Gain 22
Significantly affected by the vertical climb velocity.
Little impact of forward velocity.

37. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Linear Model Analysis : Impact of mass
Mass = 1.5 kg

38. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Linear Model Analysis : Impact of mass
Mass = 3 kg
The effect of mass on Thrust is more significant at low
forward flight speeds.

39. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Linear Model Analysis : Impact of mass
Gain: Thrust to ˙Z
Mass increase causes the gain to decrease significantly and
is more significant at low forward flight speeds.

40. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Linear Model Analysis : Impact of mass
Gain: Thrust to ˙X
Mass increase causes the gain to decrease and its effect is
more significant with high flight velocities.

41. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Linear Model Analysis : Impact of mass
Gain : τtheta to ˙Z
Mass increase causes the gain to increase very gradually
but its effect becomes more significant at high forward
velocities.

42. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Linear Model Analysis
Gain 22 : τtheta to ˙X
Mass increase causes the gain to increase significantly but
it remains unchanged with forward velocities.

43. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Linear Model Analysis : Impact of mass
Pole Zero map with change in mass
As mass increases the poles move towards the origin
thereby decreasing the stability of the system.

44. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Linear Model Analysis : Impact of Length
Gain: Thrust to ˙Z
No impact of the gain.

45. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Linear Model Analysis
Gain: Thrust to ˙X
No impact on gain.

46. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Linear Model Analysis : Impact of Length
Gain: τtheta to ˙Z
Gain remains unaffected.

47. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Linear Model Analysis : Impact of Length
Gain: τtheta to ˙X
Length variation does not affect the system properties in
forward flight.

48. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Linear Model Analysis : Impact of Length
Pole Zero map: Impact of length

49. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Control design : Controller for vertical velocity
Transfer function : Thrust to ˙Z
P11 =
g11
(s + a)
(30)
Transfer function: τθ to ˙X
P22 =
g22
s2(s + a)
(31)

50. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Control design
Controller : Thrust to ˙Z
Use a PID structure
K11 =
g(s + z)
s
(32)

51. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Control design
Time Domain Specifications: Ts ≤ 5 sec and % Mp ≤
10%
CL Poles : s=-1± j1

52. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Control design : Controller for forward velocity
Controller: τθ to ˙X
Use a Lead lag structure
K22 =
s2+2ζωz s+ω2
z
s2+2ζωps+ω2
p
p
z
s+z
s+p (33)

53. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Control design
where
ωz = ωm(ζ tan(φm) + ζ2 tan(φm)2 + 1) (34)
ωp = ωm(−ζ tan(φm) + ζ2 tan(φm)2 + 1) (35)
p
z
=
1 + sin(φm)
1 − sin(φm)
(36)

54. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Control design
φm : desired phase lead / Phase Margin at the desired
unity gain frequency.
ωm: desired unity gain frequency.
z : Zero of the single lead- lag compensator.
p : Pole of the single lead- lag compensator.

55. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Controller Analysis
Sensitivity
S = [I + L]−1
(37)
Sensitivity : PM 60 deg
As ζ increases |So|peak decreases .
As ω increases |So|peak decreases significantly.

56. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Controller Analysis
Complimentary Sensitivity
T = L[I + L]−1
(38)
Complimentary Sensitivity : PM 60 deg
As ζ increases |To|peak decreases.
As ωg increases |To|peak decreases initially and again
increases with ω .

57. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Controller Analysis
Reference to Control Action
Tru = KS = K[I + PK]−1
(39)
Reference to control action : PM 60 deg
As ζ increases |KS|peak remains unaffected.
As ωg increases |KS|peak increases significantly.

58. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Controller Analysis
Disturbance to output
Tdoy = PS = P[I + PK]−1
(40)
Disturbance to output : PM 60 deg
As ζ increases |PS|peak remains relatively unaffected.
As ωg increases |PS|peak decreases significantly.

59. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Controller Analysis
Overshoot : PM 60 deg
As ζ increases % Mp decreases gradually.
As ωg increases % Mp decreases significantly.

60. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Controller Analysis
Settling Time: PM 60 deg

61. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Controller Analysis
% Mp ≤ 20 % & Ts ≤ 6 sec =⇒ =⇒ ωg ≥ 3 rad/sec
and ζ ≥ 0.8

62. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Controller Simulation
Plant : τθ to ˙X

63. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Controller Simulation
Controller : τθ to ˙X
As ωg ↑ |K| ↑.

64. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Controller Simulation
Open Loop (L) : τθ to ˙X

65. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Controller Simulation
Sensitivity : τθ to ˙X
As ωg ↑ , |So| ↓ at low frequencies.

66. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Controller Simulation
Complimentary Sensitivity : τθ to ˙X
As ωg ↑ , |To| rolls off at higher ωg .

67. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Controller Simulation
Reference to control action : τθ to ˙X
As ωg ↑ , |KS|peak ↑ .

68. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Controller Simulation
Disturbance to output : τθ to ˙X
As ωg ↑ , |PS|peak ↓ .

69. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Linear Simulation
For ˙Z = 1m/s

70. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Linear Simulation
For ˙X = 1m/s

71. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Conclusion
A modeling procedure for the longitudinal dynamics of the
quadrotor.
A control methodology for designing controller using
classical control techniques.
Future work:
Include a procedure for accounting aerodynamic model.
Optimize the controller for large forward velocities
( ˙X = 10 ∼ 20m/s).
Design a Multi-variable controller for aggressive flight
maneuvers.

72. Modeling and
Control of
Quadrotor
UAV
Aniket Shirsat
Non-Linear
Model
Modeling
Assumptions
Reference Frames
Kinematics
Dynamics
Linear Models
State Space Model
Nominal Model
Parameters
Linear Model
Analysis
Control
Design
Analysis
Simulation
Conclusion
Further Reading
Randal W Beard.
Quadrotor dynamics and control.
Brigham Young University, 2008.
Teppo Luukkonen.
Modelling and control of quadcopter.
Independent research project in applied mathematics, Espoo,
2011.
Armando Rodriguez.
Analysis and Design of Multivariable Feedback Control
Systems.
CONTROL3D,L.L.C., Tempe, AZ, 2002.
Brian L Stevens and Frank L Lewis.
Aircraft control and simulation.