1. Autonomous Landing of
Unmanned Aerial Vehicles
A Thesis
Submitted For the Degree of
Master of Science
in the Faculty of Engineering
by
Shashiprakash Singh
Department of Aerospace Engineering
Indian Institute of Science
BANGALORE – 560 012
February 2009
3. DECLARATION
I declare that the thesis entitled Autonomous Landing of Unmanned Aerial
Vehicles submitted by me for the M.Sc.[Engg.] degree of the Indian Institute of Science
did not form the subject matter of any other thesis submitted by me for any outside
degree and the original work done by me and incorporated in this thesis is entirely done
at Indian Institute of Science, Bangalore.
Bangalore Shashiprakash Singh
February 2009
4. Dedicated to
“Late Dr. S. Pradeep”
For his guidance and teaching me the basics of research.
5. Acknowledgements
I would like to thank my research advisor Dr. Radhakant Padhi for his constant guidance,
discussions, support and constructive criticism. I am also thankful to Prof. M. S. Bhat and
Prof. D. Ghose for their valuable advice at various instances. I also want to thank all my
course instructors for having enriched my knowledge on various research topics.
My thanks to Mr. V. Surendranath and all the staff at the wind tunnel facility, for carrying
out the wind tunnel test and for letting us use their facility at UAV lab. I also want to thank
Prof. S. P. Govindaraju for answering many of my queries related to aerodynamic data.
I am grateful to all my lab members for creating a constructive research environment in
the lab. It is through the discussions with lab members that I have gained the knowledge on
various problems in the areas of aerospace and control design.
I would like to take this opportunity to thank Prof. B. N. Raghunandan for all the research
facilities at department and his motivating talks. I will not miss to thank the staff at aerospace
main office who have been most humble and have always shown readiness to help.
I am grateful to my family for their patience and support of my decision to pursue higher
studies. They have always understood my position in spite of my less frequent visits to home.
In the last I thank all the trees, plants and flowers of IISc for making the institute a beautiful
and silent place, where the researchers can think freely in the peaceful air.
i
6. Publications based on this thesis
1. Shashiprakash Singh and Radhakant Padhi,
Automatic Landing of Unmanned Aerial Vehicles Using Dynamic Inversion,
Proceedings of International Conference on Aerospace Science and Technology, Banga-
lore, India, 26-28 June 2008.
2. Shashiprakash Singh and Radhakant Padhi,
Automatic Path Planning and Control Design for Autonomous Landing of UAVs
using Dynamic Inversion, Accepted for 2009 American Control Conference, Missouri,
USA, 10-12 June 2009.
ii
7. Abstract
In this thesis the problem of autonomous landing of an unmanned aerial vehicle named
AE-2 is addressed. The guidance and control technique is developed and demonstrated
through numerical simulation results. The complete work includes Mathematical mod-
eling, Control design, Guidance and State estimation for AE-2, which is a fixed wing
vehicle with 2m wing span and 6kg weight.
The aerodynamic data for AE-2 is available from static wind tunnel tests. Functional
fit is done on the wind tunnel data with least squares method to find static aerody-
namic coefficients. The aerodynamic forces and moment coefficients are highly nonlinear
some of them are partitioned in two zones based on the angle of attack. The dynamic
derivatives are found with Athena Vortex Lattice software. For the validation of vortex
lattice method the static derivatives obtained by the wind tunnel tests and vortex lattice
method, are compared before finding dynamic derivatives. The dynamics of the servo
actuators for the aerodynamic control surfaces is incorporated in the simulation.
The nonlinear dynamic inversion technique has been used for the guidance and control
design. The control is structured in two loops, outer and inner loop. The goal of outer
loop is to track the guidance commands of altitude, roll angle and yaw angle by converting
them into body rate commands through dynamic inversion. The inner loop than tracks
these commanded roll rate, pitch rate and yaw rate by finding the required deflection of
control surfaces. The forward velocity of the vehicle is controlled by varying the throttle.
A controller for actuator is also designed to reduce the lag.
iii
8. Abstract iv
The guidance for landing consists of three phases approach, glideslope and flare.
During approach the vehicle is aligned with the runway and guided to a specified height
from where the glideslope can begin. The glideslope is straight line path specified by
a flight path angle which is restricted between 3 to 4 degree. At the end of glideslope
which is marked by flare altitude the flare maneuver begins which is an exponential curve.
The problem of transition between the glideslope and flare has addressed by ensuring
continuity and smoothness at transition. The exponential curve of flare is designed to
end below the ground so that it intersects the ground at a prespecified point. The sink
rate at touchdown is also controlled along with the location of touchdown point.
The state estimation has been done with Extended Kalman Filter in continuous dis-
crete formulation. The external disturbances like wind shear and wind gust are accounted
by appending them in state variables. Further the control design with guidance is tested
from various initial conditions, in presence of wind disturbances. The designed filter has
also been tested for parameter uncertainty.
17. Notation and Abbreviations
English Alphabet
ax − Acceleration in body axis X direction
ay − Acceleration in body axis Y direction
az − Acceleration in body axis Z direction
b − Wing span
c − Mean aerodynamic chord
Cl − Rolling moment coefficient
Cm − Pitching moment coefficient
Cn − Yawing moment coefficient
CX − Axial force coefficient
CY − Side force coefficient
CZ − Normal force coefficient
d − Offset of the thrust line from CG in the Z axis
h − Altitude above ground
Ixx, Iyy, Izz − Moment of inertia around body fixed X, Y and Z axis
Ixz − Product of inertia between body fixed x and z axis
kh − proportional gain for altitude error
kp − proportional gain for rolling moment error
kq − proportional gain for pitching moment error
kr − proportional gain for yawing moment error
ku − proportional gain for forward velocity error
xiii
18. Notation and Abbreviations xiv
kv − proportional gain for sideward velocity error
ky − proportional gain for sideward distance error
kφ − proportional gain for roll angle error
kθ − proportional gain for pitch angle error
kψ − proportional gain for heading angle error
La − Rolling moment due to aerodynamic effects
m − Mass of the aircraft
M − Mach number
Ma − Pitching moment due to aerodynamic effects
Na − Yawing moment due to aerodynamic effects
p − Roll rate
P − Error covariance matrix
q − Pitch rate
Q − Process noise covariance matrix
ˆq − Dynamic pressure = 1
2
ρV 2
t
r − Yaw rate
R − Measurement noise covariance matrix
Re − Reynolds number
S − Surface area of the wing
u − Forward velocity in body axis X direction
U − Control vector.
Ua − Aerodynamic controls (elevator, aileron and rudder).
Uc − Control variables (throttle, elevator, aileron and rudder).
v − Sideward velocity in body axis Y direction
w − Downward velocity in body axis Z direction
x − Forward distance in inertial frame
X − State vector
Xa − Axial force due to aerodynamic effects in body X axis
Xt − Thrust force in body X axis direction
19. Notation and Abbreviations xv
Xd − Wind disturbance state vector
y − Sideward distance in inertial frame
Y − Output vector
Ya − Side force due to aerodynamic effects
Za − Normal force due to aerodynamic effects
Greek Alphabet
α − Angle of attack
β − Sideslip angle
γ − Flight path angle
δa − Aileron deflection
δe − Elevator deflection
δr − Rudder deflection
σt − Throttle setting (0-1)
ρ − Density of air
φ − Roll angle
θ − Pitch angle
ψ − Yaw angle
χ − Combined state vector of system and disturbance
Abbreviations
six-DOF − Six degrees of freedom
AE-2 − All Electric airplane - 2
AVL − Athena Vortex Lattice Software
20. Chapter 1
Introduction
The capabilities of Unmanned Aerial Vehicles(UAVs) as flying machines can be exploited
to carry out surveillance missions and remote operations. The UAVs are becoming more
promising with their applications to the scenarios of disaster management, forest fire
detection, frontier surveillance, power line monitoring and many others. The recovery
of UAVs in landing is one of the key operations in flight which define the overall success
of the mission. The UAVs are recovered through net on naval ships and also on ground,
while some UAVs employ parachutes to decrease its descent rate during landing. Other
methods are hard landing on belly or landing on runway with wheels.
The complete autonomous flight for an UAV without human intervention will include
autonomous take off, waypoint navigation and autonomous landing. The waypoint nav-
igation and take-off are comparatively easier problem than autonomous landing because
they involve lesser constraints and uncertainties. The goals for a successful landing will
require it to touchdown with a bound on sink rate and the pitch angle at touchdown,
also the touchdown location on runway should be within some specified zone. The un-
certainties which make landing a challenging problem the presence of wind disturbances
and ground effects. The ground effects have not been considered in this thesis, because
1
“Flying is the second most adventure known to mankind, first is landing”. Unknown
1
21. Chapter 1. Introduction 2
of the unavailability of reliable models for ground effects for small class of vehicles like
UAVs. However, the uncertainties in forces and moments have been considered.
1.1 Motivation
The dynamics of low speed and light weight vehicles such as UAVs are different compared
to that of conventional aircrafts. The reasons being the coupling between longitudinal
and lateral modes and low Reynolds number effects which is under study [1]. The
wind tunnel tests can give the aerodynamic coefficients for forces and moments. Which
further need to be modeled in global function form [2], [3], [4] in order to reduce the
burden of interpolation from large number of tables. It is found that linearized models
of the aircraft have been used in the literature using separate longitudinal and lateral
dynamics. However, linear system based approaches have a strong limitation that they
work within a small operating range. Gain scheduling can perhaps be used to overcome
this limitation to a limited extent. However gain scheduling is a tedious process and
there is no guarantee that the interpolated gains can assure stability of the closed loop
system [5].
Various control strategies have been adapted to tackle the problem of automatic land-
ing both for manned and unmanned aircrafts. Linear control theory has been heavily
investigated. The concept of Linear Parameter Varying representation with piecewise
affine dependence on the parameters is adopted [6] for modeling and linear matrix in-
equality method for control design . Modern control methods like H2/H∞ have also been
used for landing of UAVs [7], [8]. Dynamic inversion is a popular method of nonlinear
control which relies on the philosophy of feedback linearization [9], [10]. This feedback
control structure cancels the nonlinearities in the plant such that the closed loop plant
behaves like a stable linear system. This method has several advantages, like simplicity in
the control structure, ease of implementation, global exponential stability of the tracking
error etc. A feedback linearization technique has been successfully demonstrated through
22. Chapter 1. Introduction 3
simulation for automatic landing of a high performance aircraft [11]. The dynamic in-
version has also been used for the design of longitudinal landing [12] and for the control
of micro aerial vehicles [13]. The robustness with nonlinear control [14], [15] have also
been studied and reported.
The autonomous landing requires an intelligent path planning and guidance. Landing
trajectories of aerial vehicles typically consists of approach, glideslope and flare [11]. A
successful landing would depend upon the good selection of landing trajectory and closely
following it. The problems associated with following glideslope and flare is the change in
slope at the transition. Which put requirement of different gains for following glideslope
and flare. To overcome this blending function [16] for gains at the transition of glideslope
and flare has been reported in literature. It is found in literature that the desired landing
trajectory can be made function of time [11] or forward distance from runway [6], [17].
The other requirements on landing are controlling the sink rate at touchdown and the
location of touchdown.
The landing requires sensors for position location and height with the help of which
it can be achieved successfully. The sensors for height include pressure altitude sensors
or laser altimeter [18]. Vision based algorithms for autonomous landing of UAVs have
been also been designed [19]. The vision sensor is used to extract the relative position of
the UAV from a landing pad than this information is used to direct the UAV to land [20].
The use of vision based precision target detection and recognition along with the GPS for
navigation [21] have also been shown. More advance algorithms have combined the data
from optical flow sensors and pressure based sensor to estimate the HAG [22] for relatively
unknown terrains. The other challenges for landing are uncertainties in measurement
and process noise in form of wind disturbances. The state estimation will be required to
account for non availability for all the sensors. The estimation can be carried out with
Extended Kalman filter [23]. The aircraft landing control and estimation in presence of
wind shear [24], [25] has been reported. The estimation with EKF for landing of UAV
is carried out in presence wind disturbance [26].
23. Chapter 1. Introduction 4
1.2 Contribution of present work
In the present research guidance algorithm has been developed for autonomous landing
along with the development of complete mathematical model for the UAV. The nonlinear
control design and estimation have been carried out and demonstrated with simulations.
The aerodynamic coefficients for force and moments are found from the curve fitting
of the wind tunnel data. The nonlinearities found in the functions obtained from curve
fit are presented. The problem of slope discontinuity during transition from glideslope
to flare has been addressed by careful selection of landing trajectory parameters. The
glideslope and flare path is scheduled as a function of forward distance which makes the
trajectory independent of time. The glideslope and flare path parameters are computed
online (i.e. they are not fixed apriori). Further the trajectory parameters are calculated
such that the sink rate at touchdown remains within specified bounds. The control
is designed using dynamic inversion technique, while continuous-discrete formulation of
extended kalman filter has been used for state estimation. The numerical simulations
are carried out using the data of an UAV named AE-2 (All Electric airplane-2) shown in
fig 1.1. The AE-2 was designed and developed [27] at UAV lab of aerospace engineering
department. It is a small fixed wing aircraft with weight of 6kg and wing span of 2m. It
is designed with a capability of autonomous flying and carrying out surveillance missions
over a range of 10 Km.
Figure 1.1: AE-2 (All Electric airplane-2)
24. Chapter 1. Introduction 5
1.3 Organization of thesis
The remainder of the thesis is organized in five chapters followed by conclusion and a
list of bibliography. The content of each of these chapters is given below
Chapter 2: Mathematical modeling
In this chapter the mathematical model is derived for the UAV from wind tunnel
data and analytical software. The trim condition are calculated and open loop
flight simulation is done to test the model developed.
Chapter 3: Control design
In this chapter control is designed using Dynamic Inversion in outer and outer loop.
Further it is tested by giving various commands.
Chapter 4: Path planning and guidance
In chapter 4 the guidance algorithm for various phases of landing is developed.
The guidance design along with control is validated by numerical simulation.
Chapter 5: State estimation
In this chapter a state estimator with Extended Kalman Filter in continuous-
discrete form is designed. The estimator is tested with wind disturbances and
parameter uncertainties.
25. Chapter 2
Mathematical modeling
The dynamics of an object in motion can be given by Newton’s second law which is
applicable in inertial frame. To model the translational dynamics we need the knowledge
of mass and the external forces acting on it. Similarly for the rotational dynamics we
require moment and product of inertias with the external moments acting on it. The
mass can be easily measured with highly accurate weighing machines available. Where as
the inertia properties can be measured in laboratory with bifilar suspension experiment.
The external forces acting on an aircraft are due to gravity, aerodynamics and thrust.
The aerodynamic forces and moments can be measured in wind tunnel by testing the
scaled model or the full model at the expected flight velocities. The thrust force can also
be measured in wind tunnel. The other methods through which aerodynamics forces
and moments can be estimated are empirical calculations or CFD (Computational Fluid
Dynamics) simulation.
The kinematic motion do not involve any parameters but they are also dependent on
the frame of observation. The vectors can be transformed from one frame of reference to
other through three sequential Euler rotations. The three rotations combined form the
transformation matrix between the frames.
2
“As far as the propositions of mathematics refer to reality they are not certain, and so far as they
are certain, they do not refer to reality.” Albert Einstein(1921)
6
26. Chapter 2. Mathematical modeling 7
The dynamic and kinematic equations for rotational and translational motion of
aircraft is discussed in next section.
2.1 Equations of motion
The reference frame for body axis coordinate system is shown in Fig. 2.1. Where positive
X axis is in forward direction through nose of the aircraft. The positive Y axis protrudes
through the right wing. The positive Z axis is perpendicular to X and Y in downward
direction.
Figure 2.1: Body reference frame
The variables defined with respect to body axis frame are given below.
u, v, w are velocity components along the body axis X, Y , Z direction.
p, q, r are roll, pitch and yaw rates respectively about the body axis X, Y , Z direction.
φ, θ, ψ are roll, pitch and yaw angles around the body axis X, Y , Z direction.
The rotational rates and angles are defined with respect to right hand grip rule. The
position coordinates of the aircraft are defined with respect to earth fixed inertial frame.
x, y, h are defined as position coordinates with respect to inertial frame. Where x is
forward distance, y is sideward distance and h is the height above ground.
27. Chapter 2. Mathematical modeling 8
Six-DOF equations
Under the assumptions of airplane to be a rigid body and earth to be flat, the complete
set of six-DOF equations of motion [28] are given by following differential equations.
Translational dynamic equations
˙u = rv − qw − gsinθ +
Xa + Xt
m
(2.1)
˙v = pw − ru + gsinφcosθ +
Ya
m
(2.2)
˙w = qu − pv + gcosφcosθ +
Za
m
(2.3)
Rotational dynamic equations
˙p = c1rq + c2pq + c3La + c4Na (2.4)
˙q = c5pr + c6(p2
− r2
) + c7(Ma + Mt) (2.5)
˙r = c8pq − c2rq + c4La + c9Na (2.6)
Rotational kinematic equations
˙φ = p + qsinφtanθ + rcosφtanθ (2.7)
˙θ = qcosφ − rsinφ (2.8)
˙ψ = qsinφsecθ + rcosφsecθ (2.9)
Translational kinematic equations
˙x = u cosθcosψ + v(sinφsinθsinψ − cosφsinψ) + w(cosφsinθcosψ + sinφsinψ)(2.10)
˙y = u cosθsinψ + v(sinφsinθsinψ + cosφcosψ) + w(cosφsinθsinψ − sinφcosψ)(2.11)
˙h = u sinθ − vsinφcosθ − wcosφcosθ (2.12)
In above equations Xa, Ya, Za are the aerodynamic forces and La, Ma, Na are the
moments about the body axis. m is the mass of the vehicle. Xt is the thrust force in
body axis X direction and Mt is the moment around the Y axis caused by thrust, as it
does not pass through the center of gravity of the aircraft.
28. Chapter 2. Mathematical modeling 9
The constants c1 − c9 in equations 2.4 - 2.6 are function of inertial properties of aircraft
given as
c1
c2
c3
c4
c8
c9
1
IxxIzz−I2
xz
Izz(Iyy − Izz) − IxzIxz
Ixz(Ixx − Iyy + Izz)
Izz
Ixz
IxzIxz + Ixx(Ixx − Iyy)
Ixx
c5
c6
c7
1
Iyy
(Izz − Ixx)
Ixz
1
The inertia and mass properties with physical data of the UAV AE-2 (Fig. 1.1) is
given in the Table 2.1. Where b is wing span, c is chord length and m is mass of the
vehicle.
Table 2.1: Physical data of AE-2
b c m Ixx Iyy Izz Ixz
2.0 m 0.3 m 6.0 kg 0.5062 kgm2
0.89 kgm2
0.91 kgm2
0.0015 kgm2
2.2 Aerodynamic forces and moments
The aerodynamic forces and moments are given by following equations
[Xa Ya Za] = ˆq S [CX CY CZ] (2.13)
[La Ma Na] = ˆq S [bCl cCm bCn] (2.14)
where ˆq is dynamic pressure given as 1
2
ρV 2
t . ρ is the density of air and Vt is total velocity
of aircraft relative to air. S is wing planform area. CX, CY , CZ are axial force, side
force and normal force coefficients respectively. Whereas Cl, Cm, Cn are rolling moment,
pitching moment and yawing moment coefficients respectively. Aerodynamic coefficients
are obtained from curve fitting on wind tunnel data which is given in the next section.
29. Chapter 2. Mathematical modeling 10
2.2.1 Wind tunnel testing
For the study and evaluation of aerodynamic performance, the full scale model AE-
2 was tested [27] in the open circuit wind tunnel (Fig. 2.2) facility available at the
department. The forces and moments were measured and than normalized to find the
coefficients. Only static tests were conducted in wind tunnel with test variables as angle
of attack(α), sideslip angle(β), elevator deflection(δe), aileron deflection(δa) and rudder
deflection(δr). The force and moment coefficients were evaluated as a function of two
variables. During the test one variable was held constant and other varied for its entire
range. For eg. elevator held at -5o
and angle of attack varied from -6o
to 22o
with
incremental angle of +2o
, in this process all the six coefficients were measured. The tests
conducted with variables and their full range is given in the table below.
Table 2.2: Wind tunnel test variables and their range
test tablesize I min (step) max II min (step) max
1. f(α, β)15×5 α -6o
: +2o
: 22o
β -10o
: +5o
: 10o
2. f(β, α)11×5 β -100
: +2o
: 10o
α -5o
: +5o
: 15o
3. f(α, δe)15×7 α -6o
: +2o
: 22o
δe -25o
: +5o
: 5o
4. f(α, δa)15×7 α -6o
: +2o
: 22o
δa -15o
: +5o
: 15o
5. f(α, δr)15×7 α -6o
: +2o
: 22o
δr -15o
: +5o
: 15o
Figure 2.2: AE-2 in the open circuit wind tunnel
30. Chapter 2. Mathematical modeling 11
From the wind tunnel tests the coefficients are available in 2-dimensional table look
up form. The number of tables is large, which makes the finding of coefficient through
interpolation a time taking process. Also there will be discontinuity in the slope as
the required interpolation point progresses from one region of the table to other. To
overcome these difficulties the curve fitting on aerodynamic coefficients was done to find
a global function [2] which is discussed in the next section. The discussion is carried out
by taking example of normal force coefficient CZ. The same approach has been applied
to other coefficients as well, wherever there is any difference it is discussed. The results
of curve fit with functional form is given in the subsequent sections.
2.2.2 Curve fitting on wind tunnel data
The aerodynamic coefficients [28] vary with respect to Mach number, Reynolds number,
air relative angles, control surface deflections and body rotational rates. Which can be
represented in mathematical form as
CZ = f(M, Re, α, β, δa, δe, δr, p, q, r) (2.15)
The effect of mach number can be neglected for very small mach numbers. Also due to
small variation range in the flight velocity of the vehicle there will be small changes in
Reynolds number, hence its effect can also be dropped. So we have
CZ = f(α, β, δa, δe, δr, p, q, r) (2.16)
With the assumption that effects of individual variables are additive, we can separate
static and dynamic effects
CZ = fstat(α, β, δa, δe, δr) + fdyn(p, q, r) (2.17)
= CZstat + CZdyn
(2.18)
With component build up we can write the static effects,
CZstat = CZ0 +
∂CZ
∂α
α +
∂CZ
∂β
β +
∂CZ
∂δa
δa +
∂CZ
∂δe
δe +
∂CZ
∂δr
δr (2.19)
31. Chapter 2. Mathematical modeling 12
The dynamic effects are discussed in Section 2.2.4. There are five tables available from
the static wind tunnel test for the normal force coefficient which are
Table1 Table2 Table3 Table4 Table5
CZstat = f(α, β) CZstat = f(β, α) CZstat = f(α, δe) CZstat = f(α, δa) CZstat = f(α, δr)
Let us consider the Table1 and Table2 which represent the coefficient as a function of
same variables measured at different values. For these two tables the test was conducted
by setting the value of other variables to zero i.e. δe = 0, δa = 0, δr = 0. If we put these
values in Eq. 2.19 than it will get reduced to give the function which represents Table1
and Table2
CZstat = CZ0 +
∂CZ
∂α
α +
∂CZ
∂β
β (2.20)
Now the goal is to find CZ0 , ∂CZ
∂α
, ∂CZ
∂β
such that Eq. 2.20 represents the Table1 and
Table2 with minimum error. The plots of data Table1 are shown in Fig. 2.3 - Fig. 2.5
−10
−5
0
5
10
−10
0
10
20
30
−0.5
0
0.5
1
1.5
β (deg)α (deg)
C
Zstat
Figure 2.3: Plot of CZstat = f(α, β), data Table1
32. Chapter 2. Mathematical modeling 13
−5 0 5 10 15 20
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
α (deg)
CZstat
β=10
β=5
β=0
β=−5
β=−10
Figure 2.4: CZstat vs α for various β
−10 −5 0 5 10 15
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
β (deg)
C
Zstat
α=22
α=20
α=18
α=16
α=14
α=12
α=10
α=8
α=6
α=4
α=2
α=0
α=−2
α=−4
α=−6
Figure 2.5: CZstat vs β for various α
The value of CZ0 is a constant, which can be found from looking in the Table1 and
Table2 for α = 0, β = 0. The two values found were very close so average was taken.
The nature of ∂CZ
∂α
can be found qualitatively looking at Fig. 2.4. It is observed from
the Fig. 2.4 that CZstat varies linearly (constant slope) up to 10o
of angle of attack and
in the region of higher angles of attack it is nonlinear. The stall angle is near 16o
of angle
of attack after which the value of normal coefficient starts decreasing. This observation
on experimental data matches with the theory of lift generation and its behaviour with
angle of attack.
The behaviour of ∂CZ
∂β
can be seen in Fig. 2.5. It is observed that CZstat varies linearly
with β having a small negative slope.
Now to estimate the values ∂CZ
∂α
, ∂CZ
∂β
such that it predicts the measured value with
minimum error, the least squares method is used. The measured data is divided in to
two regions with respect to angle of attack -6o
to 10o
and 10o
to 22o
. This is done in order
to account for linear and nonlinear behaviour of the CZstat . So two different functions
are fit in these regions while maintaining continuity and slope at the transition of 10o
.
33. Chapter 2. Mathematical modeling 14
Least squares estimation
Let us denote the measured value of normal force coefficient as CZmij
. Which is the
measured value of CZstat at αi and βj. The function which predicts the measured value
is given by Eq. 2.20
CZpij
= CZ0 + CZα αi + CZβ
βj (2.21)
where CZpij
is the predicted value of CZstat at measured values of αi and βj. The constants
CZα and CZβ
represent ∂CZ
∂α
and ∂CZ
∂β
respectively. Here CZα and CZβ
are constants as we
have observed above that the slopes of normal force coefficient vary linearly with α and
β in the range of -6o
to 10o
of angle of attack. Where the slopes vary, the derivatives are
expanded in power series of the depending variables.
The error in measured value and predicted value can be written as
error = C0
Zmij
− C0
Zpij
(2.22)
where C0
Zmij
CZmij
− CZ0 and
C0
Zpij
CZpij
− CZ0 = CZα αi + CZβ
βj (from Eq. 2.21)
The value of CZ0 is subtracted as it is found directly from the table as discussed
above and assumed that is measured to a very good accuracy. The total sum of square
errors will be
J = Σ(C0
Zmij
− C0
Zpij
)2
(2.23)
Taking the derivative of the above cost function with free variables CZα and CZβ
and
equating them to zero for minimum
∂J
∂CZα
= 2Σ(C0
Zmij
− C0
Zpij
)αi = 0 (2.24)
∂J
∂CZβ
= 2Σ(C0
Zmij
− C0
Zpij
)βj = 0 (2.25)
34. Chapter 2. Mathematical modeling 15
Carrying out the necessary algebra we get
Σ C0
Zmij
αi = CZα Σ α2
i + CZβ
Σ αiβj (2.26)
Σ C0
Zmij
βj = CZα Σ αiβj + CZβ
Σ β2
j (2.27)
Which can be written in matrix notation as
Σ C0
Zmij
αi
Σ C0
Zmij
βj
=
Σ α2
i Σ αiβj
Σ αiβj Σ β2
j
CZα
CZβ
(2.28)
Performing the inverse matrix operation we get
CZα
CZβ
=
Σ α2
i Σ αiβj
Σ αiβj Σ β2
j
−1
Σ C0
Zmij
αi
Σ C0
Zmij
βj
(2.29)
Above equation will give the values of derivatives CZα and CZβ
with least square error
from the measured values of coefficients. A similar operation is carried out in the nonlin-
ear zone i.e. 10o
to 22o
of angle of attack for normal coefficient. In the nonlinear region
the higher order functions from power series are taken in the predicting function.
Now let us consider the Table3 which gives the normal force coefficient as a function
of angle of attack and elevator deflection. This test was conducted by setting the value
of other variables to zero i.e. β = 0, δa = 0, δr = 0. If we put these values in Eq. 2.19
than it will get reduced to give the predicting function which represents Table3
CZpij
= CZ0 +
∂CZ
∂α
αi +
∂CZ
∂δe
δej
(2.30)
We have found the value of CZ0 and ∂CZ
∂α
. To find the value of ∂CZ
∂δe
we redefine the
predicting function by transferring the known quantities to right hand side
C0
Zpij
CZpij
− CZ0 − CZα αi = CZδe
δej
(where CZδe
∂CZ
∂δe
)
The measured value of normal force coefficient will be defined as C0
Zmij
given by
C0
Zmij
CZmij
− CZ0 − CZα αi
Now we can use least squares method as above to find out the the value of CZδe
.
35. Chapter 2. Mathematical modeling 16
The plots of data Table4 and Table5 are shown in Fig. 2.6 and Fig. 2.7
−15 −10 −5 0 5 10 15 20
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
δr (deg)
C
Zstat
α=22
α=20
α=18
α=16
α=14
α=12
α=10
α=8
α=6
α=4
α=2
α=0
α=−2
α=−4
α=−6
Figure 2.6: CZstat vs δr for various α
−15 −10 −5 0 5 10 15 20
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
δa (deg)
C
Zstat
α=22
α=20
α=18
α=16
α=14
α=12
α=10
α=8
α=6
α=4
α=2
α=0
α=−2
α=−4
α=−6
Figure 2.7: CZstat vs δa for various α
It is observed from Fig. 2.6 that there is no variation in the value of CZstat with
rudder deflection. In other words the derivative ∂CZ
∂δr
is nearly equal to zero, hence it can
be neglected from the function. From Fig. 2.7 it is seen that there is small variation
in the CZstat with aileron deflection. The observed behaviour with aileron deflection
is not symmetric on either side of zero. Also measurements at -10o
aileron deflection
are contrary to the behaviour, so the effect of aileron is not considered in the normal
coefficient prediction.
Validation of least square
To validate the derivatives found from least squares the measured data from wind tunnel
is divided into sets of 70% and 30% called as training set and testing set respectively. The
training set is used for least squares to estimate the derivatives. The testing set is used
to test the accuracy of the predicting function, if the error in measured value is higher
than 10% in linear range and more than 30% in nonlinear range than the predicting
function is not accepted. The process of least square is again carried with a higher order
predicting function until the error criteria is satisfied.
36. Chapter 2. Mathematical modeling 17
2.2.3 Static stability derivatives
The results of curve fitting on wind tunnel data is presented in this section. First the
results for normal force coefficient is presented followed by axial force, pitching moment,
side force, rolling moment and yawing moment coefficient. Some of the derivatives are
partitioned with respect to angle of attack into linear region and nonlinear region. The
α value for partition was found to be 10 degree. In expanded derivative functions
α10 0 if α <= 10
α − 10 if α > 10
The values of derivatives and their constants found is given in Table 2.4. The result
plots are shown below where circles in plots show the wind tunnel data and solid lines
are the curves obtained by function fit.
CZstat : Normal force coefficient
Fig 2.8 and Fig. 2.9 show the results for curve fit on CZstat with angle of attack.
−5 0 5 10 15 20
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
α (deg)
C
Zstat
β=10
β=5
β=0
β=−5
β=−10
Figure 2.8: Curve fit on CZstat (α, β) vs α
−5 0 5 10 15 20
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
α (deg)
C
Zstat
δ e=5
δ e=0
δ e=−5
δ e=−10
δ e=−15
δ e=−20
δ e=−25
Figure 2.9: Curve fit on CZstat (α, δe) vs α
37. Chapter 2. Mathematical modeling 18
−10 −5 0 5 10 15
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
β (deg)
C
Zstat
α=22
α=20
α=18
α=16
α=14
α=12
α=10
α=8
α=6
α=4
α=2
α=0
α=−2
α=−4
α=−6
Figure 2.10: Curve fit on CZstat (α, β) vs β
−25 −20 −15 −10 −5 0 5 10 15
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
δe (deg)
C
Zstat
α=22
α=20
α=18
α=16
α=14
α=12
α=10
α=8
α=6
α=4
α=2
α=0
α=−2
α=−4
α=−6
Figure 2.11: Curve fit on CZstat (α, δe) vs δe
The Fig. 2.10 shows the fit on variation with respect to sideslip angle. Further Fig.
2.11 shows the effect of elevator variation on CZstat . The function obtained after curve
fitting on wind tunnel data for CZstat is given as
CZstat = CZ0 + CZα (α) α + CZβ
β + CZδe
δe (2.31)
where, CZα (α) α z10α + z11α2
10 + z11α3
10, CZβ
z20, and CZδe
z30. The constants
z10 - z30 are given in Table 2.4
CXstat : Axial force coefficient
From the wind tunnel results for axial force coefficient it is observed that the variables
strongly affecting it are angle of attack and elevator deflection. The effects of other
variable being marginal it is not considered in curve fitting. The function obtained from
curve fit is given as
CXstat = CX0 + CXα (α) α + CXδe
(α) δe (2.32)
38. Chapter 2. Mathematical modeling 19
where, CXα (α) α x10α + x11α2
+ x12α2
10 + x13α3
10 + x14α4
10
CXδe
x20 + x21α + x22α2
10
The coefficient is divided in two zones on basis of angle of attack as the coefficient
is highly nonlinear after 10o
angle of attack, which can also be observed from Fig. 2.12.
The effectiveness of elevator changes in higher angle of attack region which is accounted
in CXδe
by making it a function of angle of attack.
−5 0 5 10 15 20
−0.2
−0.1
0
0.1
α (deg)
C
Xstat
δ e=5
δ e=0
δ e=−5
δ e=−10
δ e=−15
δ e=−20
δ e=−25
Figure 2.12: Curve fit on CXstat (α, δe) vs α
−25 −20 −15 −10 −5 0 5 10
−0.2
−0.1
0
0.1
δe (deg)
C
Xstat
α=22
α=20
α=18
α=16
α=14
α=12
α=10
α=8
α=6
α=4
α=2
α=0
α=−2
α=−4
α=−6
Figure 2.13: Curve fit on CXstat (α, δe) vs δe
Cmstat : Pitching moment coefficient
It is observed from wind tunnel data of pitching moment coefficient that aileron and rud-
der have negligible effect, hence not considered for curve fitting. The function obtained
from curve fit is given as
Cmstat = Cm0 + Cmα (α) α + Cmβ
(α, β) β + Cmδe
(α) δe (2.33)
where, Cmα m10 + m11α + m12α2
+ m13α3
+ m14α4
Cmβ
m20 + m21β + m22αβ
Cmδe
m30 + m31α
39. Chapter 2. Mathematical modeling 20
The pitching moment coefficient varies smoothly with angle of attack (Fig 2.14 and
Fig. 2.15) for its entire range, therefore a single function a has been fit for Cmα . While
the effect of sideslip (Fig. 2.16) is quadratic in nature and the effectiveness of elevator
(Fig. 2.17) changes with angle of attack.
−5 0 5 10 15 20
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
α (deg)
Cm
stat
beta=10
beta=5
alpha=0
beta=−5
beta=−10
Figure 2.14: Curve fit on Cmstat (α, β) vs α
−5 0 5 10 15 20
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
α (deg)
Cm
stat
δe=5
δe=0
δe=−5
δe=−10
δe=−15
δe=−20
δe=−25
Figure 2.15: Curve fit on Cmstat (α, δe) vs α
−10 −5 0 5 10 15
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
β (deg)
Cm
stat
α=22
α=20
α=18
α=16
α=14
α=12
α=10
α=8
α=6
α=4
α=2
α=0
α=−2
α=−4
α=−6
Figure 2.16: Curve fit on Cmstat (α, β) vs β
−25 −20 −15 −10 −5 0 5 10 15
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
δe (deg)
Cm
stat
α=22
α=20
α=18
α=16
α=14
α=12
α=10
α=8
α=6
α=4
α=2
α=0
α=−2
α=−4
α=−6
Figure 2.17: Curve fit on Cmstat (α, δe) vs δe
40. Chapter 2. Mathematical modeling 21
CYstat : Side force coefficient
The side force coefficient obtained from wind tunnel data had non zero values even when
the sideslip angle was zero along with control surfaces. This non zero side force could
result from asymmetric manufacturing of vehicle or non alignment with wind direction
in wind tunnel. Therefor this error was subtracted from the data table to make the
coefficient symmetric with sideslip.
−5 0 5 10 15 20
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
α (deg)
CYstat
β=10
β=5
β=0
β=−5
β=−10
Figure 2.18: Curve fit on CYstat (α, β) vs α
−10 −5 0 5 10 15
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
β (deg)
C
Ystat
α=22
α=20
α=18
α=16
α=14
α=12
α=10
α=8
α=6
α=4
α=2
α=0
α=−2
α=−4
α=−6
Figure 2.19: Curve fit on CYstat (α, β) vs β
−15 −10 −5 0 5 10 15 20
−0.05
−0.04
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
0.05
δr (deg)
CYstat
α=22
α=20
α=18
α=16
α=14
α=12
α=10
α=8
α=6
α=4
α=2
α=0
α=−2
α=−4
α=−6
Figure 2.20: Curve fit on CYstat (α, δr) vs δr
−15 −10 −5 0 5 10 15 20
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
δa (deg)
C
Ystat
α=22
α=20
α=18
α=16
α=14
α=12
α=10
α=8
α=6
α=4
α=2
α=0
α=−2
α=−4
α=−6
Figure 2.21: Curve fit on CYstat (α, δa) vs δa
41. Chapter 2. Mathematical modeling 22
The side force coefficient is also divided in two regions with angle of attack. The
function obtained from curve fit is given as
CYstat = CYβ
(α) β + CYδr
(α) δr + CYδa
(α) δa (2.34)
where, CYβ
y10 + y11α + y12α2
+ y13α2
10 + y14α3
10
CYδr
y20 + y21α + y22α2
+ y23α2
+ y24α2
10 + y25α3
10 + y26α4
10
CYδa
y30 + y31α2
10 + y32α3
10
Clstat : Rolling moment coefficient
The rolling moment coefficient is also made symmetric with sideslip angle by subtracting
the non zero values from the data. The effect of elevator and rudder being very small,
hence not considered in curve fitting. The function obtained from curve fit is given as
Clstat = Clβ
(α)β + Clδa
(α)δa (2.35)
where, Clβ
l10 + l11α + l12α2
+ l13α2
10 + l14α3
10
Clδa
l20 + l21α + l22α2
+ l23α3
+ l24α2
10 + l25α3
10 + l26α4
10
−5 0 5 10 15 20
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
α (deg)
Clstat
β=10
β=5
β=0
β=−5
β=−10
Figure 2.22: Curve fit on Clstat (α, β) vs α
−5 0 5 10 15 20
−0.05
−0.04
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
0.05
α (deg)
Cl
stat
δa=15
δa=10
δa=5
δa=0
δa=−5
δa=−10
δa=−15
Figure 2.23: Curve fit on Clstat (α, δa) vs α
43. Chapter 2. Mathematical modeling 24
−10 −5 0 5 10 15
−0.025
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
0.025
β (deg)
Cnstat
α=22
α=20
α=18
α=16
α=14
α=12
α=10
α=8
α=6
α=4
α=2
α=0
α=−2
α=−4
α=−6
Figure 2.28: Curve fit on Cnstat (α, β) vs β
−15 −10 −5 0 5 10 15 20
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
δr (deg)
Cn
stat
α=22
α=20
α=18
α=16
α=14
α=12
α=10
α=8
α=6
α=4
α=2
α=0
α=−2
α=−4
α=−6
Figure 2.29: Curve fit on Cnstat (α, δr) vs δr
The effects of elevator and aileron being small they are neglected. Yawing moment
coefficient is also divided into linear and nonlinear zone with respect to angle of attack.
This completes the discussion of static stability derivatives obtained from curve fitting
on wind tunnel test data. The dynamics derivatives are discussed in next section.
2.2.4 Dynamic stability derivatives
The dynamic effects on coefficients can be written from Eq. 2.17 as CZdyn
= fdyn(p, q, r),
Which can be expanded considering the individual effects are additive
CZdyn
=
∂CZ
∂¯p
¯p +
∂CZ
∂¯q
¯q +
∂CZ
∂¯r
¯r (2.37)
Where the rotational rates are non dimensionalized with reference length and velocity
which are given as [¯p ¯q ¯r] = 1
2Vt
[bp cq br].
The dynamics test were not conducted in wind tunnel, hence the dynamics derivatives
are found with AVL (Athena Vortex Lattice) software [29]. AVL is a program for the
aerodynamic and flight-dynamic analysis of rigid aircraft of arbitrary configuration.
44. Chapter 2. Mathematical modeling 25
It employs an extended vortex lattice model for the lifting surfaces, together with a
slender-body model for fuselages. AVL requires the mass properties along with geometric
information (Fig. 2.30) of the vehicle. It also requires the 2-dimensional lift and drag
characteristics of the lifting surfaces.
Figure 2.30: AE-2 modeling in AVL
To validate and have confidence in the results given by AVL its static results are
compared with that of wind tunnel, which is shown in Fig. 2.31. It is seen from figure
that AVL gives a good estimate for the static coefficients.
−5 0 5 10 15 20
−0.2
0
0.2
α (deg)
CXstat
AVL
Wind Tunnel
−5 0 5 10 15 20
−2
0
2
α (deg)
CZstat
−5 0 5 10 15 20
−1
0
1
α (deg)
Cmstat
Figure 2.31: AVL and wind tunnel test results comparison
45. Chapter 2. Mathematical modeling 26
The dynamic derivatives are evaluated with respect to angle of attack. While considering
the dynamics effects it is assumed that only pitch rate effects the longitudinal coefficients
of normal force, axial force and pitching moment. Which can be written as
CZdyn
= CZ¯q ¯q CXdyn
= CX¯q ¯q Cmdyn
= Cm¯q ¯q
where as the coefficients effected by roll rate and yaw rate are side force, rolling moment
and yawing moment coefficient. For them the relation can be written as
CYdyn
= CY¯p ¯p + CY¯r ¯r Cldyn
= CY¯p ¯p + Cl¯r ¯r Cndyn
= CY¯p ¯p + Cn¯r ¯r
The following sections give the results obtained for dynamic derivatives from AVL.
Pitch rate derivatives
The pitch rate derivatives are found at various angle of attack. Then a function is fit
which will be able to predict derivative for entire range of angle of attack. The results
are shown in Fig. 2.32 and the functions obtained from curve fit are given as
CZ¯q z40 + z41α CX¯q x30 + x31α Cm¯q m40 + m41α + m42α2
−5 0 5 10 15 20
4
6
8
α (deg)
C
Zq
AVL
Curve fit
−5 0 5 10 15 20
−5
0
5
α (deg)
C
Xq
−5 0 5 10 15 20
−14
−13.5
−13
α (deg)
C
mq
Figure 2.32: Pitch rate derivatives
46. Chapter 2. Mathematical modeling 27
Roll rate derivatives
The results for roll rate derivatives are shown in Fig. 2.33 and the functions obtained
from curve fit are given as
CY¯p y40 + y41α Cl¯p l30 + l31α + l32α2
Cn¯p n30 + n31α + n32α2
−5 0 5 10 15 20
−0.5
0
0.5
α (deg)
C
Yp
AVL
Curve fit
−5 0 5 10 15 20
−0.6
−0.5
−0.4
α (deg)
C
lp
−5 0 5 10 15 20
−0.5
0
0.5
α (deg)
C
np
Figure 2.33: Roll rate derivatives
Yaw rate derivatives
The results for yaw rate derivatives are shown in Fig. 2.34 and the functions obtained
from curve fit are given as
CY¯r y50 + y51α Cl¯r l40 + l41α + l42α2
Cn¯r n40 + n41α + n42α2
47. Chapter 2. Mathematical modeling 28
−5 0 5 10 15 20
0
0.2
0.4
α (deg)
C
Yr AVL
Curve fit
−5 0 5 10 15 20
0
0.2
0.4
α (deg)
C
lr
−5 0 5 10 15 20
−0.4
−0.2
0
α (deg)
C
nr
Figure 2.34: Yaw rate derivatives
The aerodynamic coefficients with static and dynamic effects can be summarized as
CZ = CZ0 + CZα (α) α + CZβ
β + CZδe
δe + CZ¯q (α) ¯q (2.38)
CX = CX0 + CXα (α) α + CXδe
(α) δe + CX¯q (α) ¯q (2.39)
Cm = Cm0 + Cmα (α) α + Cmβ
(α, β) β + Cmδe
(α) δe + Cm¯q (α) ¯q (2.40)
CY = CYβ
(α) β + CYδa
(α) δa + CYδr
(α) δr + CY¯p (α)¯p + CY¯r (α) ¯r (2.41)
Cl = Clβ
(α) β + Clδa
(α) δa + Cl¯p (α) ¯p + Cl¯r (α) ¯r (2.42)
Cn = Cnβ
(α) β + Cnδr
(α) δr + Cn¯p (α) ¯p + Cn¯r (α) ¯r (2.43)
The Table 2.3 gives the value of average of absolute percentage error in the value obtained
by curve fitting and measured data.
Table 2.3: Average of absolute percentage error in curve fitting
α CZ CX Cm CY Cl Cn
−6o
to 10o
6% 6% 16% 13% 11% 7%
10o
to 22o
23% 26% 27% 30% 25% 25%
48. Chapter 2. Mathematical modeling 29
The values of coefficients of the functions obtained by curve fit is given in Table 2.4.
Table 2.4: Derivatives function coefficient values
CZ0 0.1653 m12 -1.7853e-005 y24 -5.2196e-5 l30 -0.44336
z10 0.087138 m13 -2.1109e-006 y25 8.8682e-6 l31 0.00075577
z11 -0.0091867 m14 1.1346e-007 y26 -3.2717e-7 l32 -0.00013921
z12 0.00024242 m20 0.00024049 y30 -0.0016884 l40 0.076582
z20 -0.0020001 m21 -7.8566e-006 y31 -1.3637e-05 l41 0.010019
z30 0.0039823 m22 1.0663e-006 y32 1.3214e-06 l42 1.1783e-005
z40 6.9303 m23 -7.8866e-007 y40 -0.14504 n10 -0.0015474
z41 -0.047657 m30 -0.0145 y41 0.013516 n11 6.1309e-005
CX0 0.0386 m31 9.2552e-006 y50 0.13784 n12 -1.8989e-006
x10 -0.0040376 m32 9.0437e-006 y51 0.0035514 n13 -5.5706e-006
x11 -0.0010525 m40 -13.954 l10 0.0022856 n20 0.00077238
x12 0.0027887 m41 0.0017379 l11 6.4827e-005 n21 1.1379e-006
x13 0.00010917 m42 0.0016743 l12 -3.0529e-006 n22 -4.1705e-008
x14 -5.3586e-6 y10 0.0099319 l13 -2.7687e-005 n30 -0.015512
x20 -0.00035832 y11 0.00029462 l14 1.7713e-006 n31 -0.011325
x21 -2.2061e-5 y12 1.7831e-005 l20 0.0029091 n32 9.8251e-005
x22 -5.7342e-6 y13 -0.00030969 l21 9.0047e-006 n40 -0.085307
x30 -0.18476 y14 1.6759e-005 l22 -7.4562e-006 n41 0.00080338
x31 -0.10227 y20 0.0022145 l23 3.0423e-007 n42 -0.00026197
Cm0 0.0346 y21 0.00041878 l24 -2.5531e-005
m10 -0.013841 y22 1.3117e-5 l25 4.1263e-006
m11 -0.00026206 y23 -1.1549e-6 l26 -2.0918e-007
2.3 Thrust force and moment
The AE-2 uses electric motor and propeller for thrust generation with lithium polymer
battery as its power source. The thrust force and moment are given as
Xt = (Tmax σt) (2.44)
Mt = −d (Tmax σt) (2.45)
where Tmax is the maximum thrust(15N) which can be produced by the electric motor
and propeller assembly. σt is throttle control which varies from 0 to 1. It is assumed
that thrust produced has linear relation with throttle input. d is offset (0.26m) of the
thrust line from the CG of the vehicle.
49. Chapter 2. Mathematical modeling 30
2.4 Actuator dynamics
The aerodynamic control surfaces are deflected by actuators. AE-2 employs electrome-
chanical servos for the control surface deflection, all the surfaces have the similar servo.
Here the actuator dynamics [30] for elevator servo is given
˙δe = bact δe + aact uδe (2.46)
where bact = −9.5, aact = 6.7 and uδe is the width of pulse width modulated (PWM)
signal. The step response of actuator is shown in the Fig. 2.35. The settling time for
the actuator is 0.54 seconds.
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
X: 0.54
Y: 0.7011
time (sec)
Servodeflection(deg)
Figure 2.35: Step response of actuator
The thrust model can be implemented with first order lag. Where this lag is due
to time delay in increase or decrease of rotational speed of the propeller. This lag is
found from the manual of electric motor and speed controller through which the throttle
command is passed. The equation for thrust generation with first order lag can be
written as
˙T =
1
τ
(T − T∗
) (2.47)
where T is actual thrust, T∗
is the commanded thrust and τ is the time constant.
50. Chapter 2. Mathematical modeling 31
2.5 Trim condition
The trim condition for steady and level flight can be found at a given velocity and
altitude. In order to find trim conditions we equate the dynamic equations to zero
( ˙X = 0) and solve for the unknown variables. Here the given conditions are velocity
and altitude. The steady and level flight condition demand that the body angular rates
and roll angle be equal to zero (p = q = r = φ = 0). The unknown quantities to be
found are angle of attack, sideslip angle, pitch angle, throttle value and deflections of
elevator, aileron and rudder control. The equations used to solve for the unknowns are
[ ˙u ˙v ˙w ˙p ˙q ˙r ˙h] = 0, taken from the six-DOF equation of motion. Since the equations are
nonlinear the fsolve function of Matlab is used for finding solution. The trim conditions
at three different velocities and same altitude are found and given in table below.
Table 2.5: Trim conditions at different velocities
Vt h α β θ σt δe δa δr
m/s m deg deg deg (0-1) deg deg deg
15 100 7.21 0 7.21 0.28 -9.81 0 0
20 100 3.15 0 3.15 0.36 -3.29 0 0
25 100 1.29 0 1.29 0.36 -1.17 0 0
The trim condition found can be verified with numerical simulation, by initializing the
system states with trim condition and integrating it for some time. The states of the
system will remain in the trim condition (Fig. 2.36 - 2.37) if the solution found is correct.
2.6 Simulation results
The open loop simulation and analysis is done in this section. There are twelve states
in the simulation and four controls
X = [u v w p q r φ θ ψ x y h] Uc = [σt δe δa δr]
The states u, v, w are replace by Vt, α, β in simulation results for better insight and
correlation. The results of the simulation have been clubbed in to three groups.
→ Longitudinal states = (Vt α q θ x h) → Lateral states = (β p r φ ψ y)
→ Control variables = (σt δe δa δr)
51. Chapter 2. Mathematical modeling 32
2.6.1 Open loop simulation
The states are initialized with trim condition and simulated for 60 seconds. It is seen
from Fig. 2.36 and Fig. 2.37 that states continue to remain in trim condition.
0 20 40
19
20
21
t (sec)
V
t
(m/s)
0 20 40 60
2
3
4
t (sec)
α(deg)
0 20 40 60
−1
0
1
t (sec)
Q(deg/s)
0 20 40 60
2
3
4
t (sec)
θ(deg)
0 20 40 60
0
500
1000
t (sec)
x(m)
0 20 40 60
99
100
101
t (sec)
h(m)
Figure 2.36: Longitudinal states in trim condition
0 20 40 60
−1
0
1
t (sec)
β(deg)
0 20 40 60
−1
0
1
t (sec)
P(deg/s)
0 20 40 60
−1
0
1
t (sec)
R(deg/s)
0 20 40 60
−1
0
1
t (sec)
φ(deg)
0 20 40 60
−1
0
1
t (sec)
ψ(deg)
0 20 40 60
−1
0
1
t (sec)
y(m)
Figure 2.37: Lateral states in trim condition
52. Chapter 2. Mathematical modeling 33
2.6.2 Perturbation around trim condition
To study the stability of the system various perturbations are given around trim condi-
tion. A stable aircraft should return to the same or a near by equilibrium point after
oscillations. Fig. 2.38 and Fig. 2.39 show the stability of the vehicle with respect to
velocity, angle of attack, pitch angle and pitch rate perturbations.
0 50 100
18
20
22
t (sec)
Vt
(m/s)
0 50 100
2
4
6
t (sec)
α(deg)
Vt
(+2 m/s)
α(+2 deg)
0 50 100
−10
0
10
t (sec)
Q(deg/s)
0 50 100
−20
0
20
t (sec)
θ(deg)
0 50 100
0
2000
4000
t (sec)
x(m)
0 50 100
90
100
110
t (sec)
h(m)
Figure 2.38: Effect of velocity and angle of attack perturbation on longitudinal states
0 50 100
19
20
21
t (sec)
V
t
(m/s)
0 50 100
2.5
3
3.5
t (sec)
α(deg)
θ(+3 deg)
Q(−5 deg/s)
0 50 100
−5
0
5
t (sec)
Q(deg/s)
0 50 100
0
5
10
t (sec)
θ(deg)
0 50 100
0
2000
4000
t (sec)
x(m)
0 50 100
98
100
102
t (sec)
h(m)
Figure 2.39: Effect of pitch angle and pitch rate perturbation on longitudinal states
53. Chapter 2. Mathematical modeling 34
Fig. 2.38 and Fig. 2.39 show the effect of sideslip and roll angle perturbation on the
states. It can be observed from the figures that the perturbation effects of sideslip are
highly damped compared to roll angle and they return to equilibrium very fast.
0 50 100
−5
0
5
t (sec)
β(deg)
0 50 100
−20
0
20
t (sec)
P(deg/s)
β(+5 deg)
φ(+5 deg)
0 50 100
−20
0
20
t (sec)
R(deg/s)
0 50 100
−5
0
5
t (sec)φ(deg)
0 50 100
0
100
200
t (sec)
ψ(deg)
0 50 100
−2000
0
2000
t (sec)
y(m)
Figure 2.40: Effect of sideslip and roll angle perturbation on lateral states
0 50 100
19
20
21
t (sec)
V
t
(m/s)
0 50 100
2
3
4
t (sec)
α(deg)
β(+5 deg)
φ(+5 deg)
0 50 100
−2
0
2
t (sec)
Q(deg/s)
0 50 100
2
3
4
5
t (sec)
θ(deg)
0 50 100
0
1000
2000
t (sec)
x(m)
0 50 100
99
100
101
t (sec)
h(m)
Figure 2.41: Effect of sideslip and roll angle perturbation on longitudinal states
Summary: In this chapter the mathematical model for AE-2 has been developed. The
next chapter discusses the control design for inner and outer loop to track the guidance
commands along with velocity control.
54. Chapter 3
Control design
Various control strategies have been attempted to achieve the flight stabilization and
autonomous control for small unmanned aerial vehicles. Linear control theory has been
heavily investigated. Linear control theory has limitation of operation in linearization
range. To encompass the whole flight regime many such linear models would be required
with gain scheduling. Due to the limitations of linear control theory and the approxima-
tions involved in it, there has been a lot of research interest in nonlinear control design.
A popular method of nonlinear control design for tracking is the technique of dynamic
inversion. Which is based on the philosophy of feedback linearization. In this approach
an appropriate coordinate transformation is carried out followed by the application of
linear control theory. Here the feedback control cancels the nonlinearities in the plant
and closed loop plant behaves as a linear system. The limitation of this method is that
it requires an accurate knowledge of the plant model, in the absence of which the track-
ing will not be perfect. Dynamic inversion concept also involves the presence of hidden
zero dynamics. They must be examined separately to make certain that they are sta-
ble and well-behaved (analysis included at the end of this chapter). The mathematical
background for dynamic inversion [9], [10] is discussed in next Section.
3
“The battlefield is a scene of constant chaos. The winner will be the one who controls that chaos,
both his own and the enemies.” Napoleon Bonaparte(1769-1821)
35
55. Chapter 3. Control design 36
3.1 Dynamic inversion
Let us consider a nonlinear dynamical system which is affine in control and given by
following equations
˙X = f(X) + g(X)U (3.1)
Y = h(X) (3.2)
where X ∈ n
, U ∈ m
, Y ∈ p
are the state, control and output vectors of the nominal
system respectively. We assume the system is pointwise controllable. The objective is
to design a control U so that Y → Y ∗
as t → ∞, where Y ∗
is the commanded signal for
the Y to track. We assume Y ∗
is bounded, smooth and slowly varying.
To achieve the above objective, it is noticed from Eq. 3.2, that using the chain rule
of derivative the expression for ˙Y can be written as
˙Y = fY (X) + gY (X)U (3.3)
where fY [ ∂h
∂X
]f(X) and gY [ ∂h
∂X
]g(X). Next, defining E (Y − Y ∗
) the
controller is synthesized such that the following stable linear error dynamics is satisfied
˙E + KE = 0 (3.4)
The solution of the above differential equation is given as E = E0 exp−Kt
. Where
K is chosen to be positive definite matrix. It can be chosen as diagonal matrix with
positive elements, where elements represent the ‘settling time constant’. Next, using the
definition of E and substituting the expression for ˙Y from 3.3 in 3.4, we can obtain by
carrying out the algebra
gY (X)U = −fY (X) − K(Y − Y ∗
) + ˙Y ∗ (3.5)
56. Chapter 3. Control design 37
If p = m i.e. system has same number of controls as output and gY (X) is nonsingular
than we can obtain the control solution as
U = [gY (X)]−1
[−fY (X) − K(Y − Y ∗
) + ˙Y ∗] (3.6)
The salient features of this method are that it provides a close form solution for the
controller and it can be implemented online without any computational difficulties. The
control solution also ensures that E → 0 as t → ∞, i.e. the asymptotic tracking
is achieved. However, the approach looks simple and it results in powerful nonlinear
controller, there are some important issues with this technique. First, is that it requires
p = m which may not hold good for all the dynamical systems. The second limitation
requires an accurate knowledge of the plant model, in the absence of which the tracking
will not be perfect. This deficiency can however be overcome by using the concepts of
robust control [31] or adaptive control [32].
The implementation of this method will require appearance of control term in dy-
namics after differentiation of output. The number of times output is differentiated to
get control term is known as ‘relative degree’. Sometimes this relative degree can be of
higher order. The other method to achieve this is through forming first order dynamics
in cascaded form using the concept of ‘time scale separation’. Here, the system is divided
into an inner and an outer loop (Fig. 3.1), where the dynamics in the inner loop has to
be faster than the dynamics in the outer loop.
Figure 3.1: Inner and outer loop structure
57. Chapter 3. Control design 38
A cascaded form of inner loop and outer loop also cancels the dynamics of a system,
but the cancellation is not exact. It relies on the approximation that the inner loop is so
fast, that the tracking in inner loop is achieved by the time new commands come from
outer loop. This can achieved by choosing the settling time constants of inner loop lower
than outer loop. The outer and inner loop control for the UAV is discussed in following
sections.
3.2 Outer loop control
The goal of outer loop control (Fig. 3.2) is to track the commands generated by guidance
loop. Where the guidance commands come from path planning which is discussed in next
Chapter. The commands from guidance are desired pitch angle (θ∗
), desired roll angle
(φ∗
) and desired heading angle (ψ∗
). The outer loop achieves the tracking by converting
the guidance commands into desired body rates which are roll rate (p∗
), pitch rate (q∗
)
and yaw rate (r∗
). These desired body rate signals are than fed to inner loop for tracking.
Figure 3.2: Outer loop control
The transformation of guidance commands to body rate commands is achieved by
enforcing first order error dynamics, which is discussed in following sections.
58. Chapter 3. Control design 39
3.2.1 Roll angle control
The roll angle can be controlled through roll rate command. The relative degree between
roll rate and roll angle is one. Hence to generate the roll rate command from error in
roll angle, we enforce the following first order error dynamics for roll angle error
( ˙φ − ˙φ∗) + kφ(φ − φ∗
) = 0 (3.7)
where error = φ − φ∗
and φ∗
is the desired roll angle. kφ is chosen positive, given in
Table 3.1. Rearranging above equation we get
˙φ = ˙φ∗ − kφ(φ − φ∗
) (3.8)
From six-DOF equation of motion we have
˙φ = p + qsinφtanθ + rcosφtanθ (3.9)
Substituting ˙φ equation in Eq. 3.8 and rearranging we get
p∗
= ˙φ∗ − kφ(φ − φ∗
) − (qsinφ + rcosφ)tanθ (3.10)
We have p∗
which is the required roll rate to achieve the desired roll angle.
3.2.2 Pitch angle control
The pitch angle can be controlled through pitch rate command. The relative degree
between pitch rate and pitch angle is one. Hence to generate the pitch rate command
from error in pitch angle, we enforce the following first order error dynamics for pitch
angle error
( ˙θ − ˙θ∗) + kθ(θ − θ∗
) = 0 (3.11)
where error = θ − θ∗
and θ∗
is the desired pitch angle. kθ is chosen positive, given in
Table 3.1.
59. Chapter 3. Control design 40
Rearranging previous equation, we get
˙θ = ˙θ∗ − kθ(θ − θ∗
) (3.12)
From six-DOF equation of motion we have
˙θ = qcosφ − rsinφ (3.13)
Substituting ˙θ equation in Eq. 3.12 and rearranging we get
q∗
= secφ( ˙θ∗ − Kθ(θ − θ∗
) + rsinφ) (3.14)
We have q∗
which is the required pitch rate to achieve the desired pitch angle.
3.2.3 Heading angle control
The heading angle can be controlled through yaw rate command. The relative degree
between yaw rate and heading angle is one. Hence to generate the yaw rate command
from error in heading angle, we enforce the first order error dynamics for heading angle
( ˙ψ − ˙ψ∗) + kψ(ψ − ψ∗
) = 0 (3.15)
where error = ψ − ψ∗
and ψ∗
is the desired heading angle. kψ is chosen positive, given
in Table 3.1. Rearranging above equation we get
˙ψ = ˙ψ∗ − kψ(ψ − ψ∗
) (3.16)
From six-DOF equation of motion we have
˙ψ = qsinφsecθ + rcosφsecθ (3.17)
Substituting ˙ψ equation in Eq. 3.16 and rearranging we get
r∗
= secφ cosθ( ˙ψ∗ − kψ(ψ − ψ∗
)) − qtanφ (3.18)
We have r∗
which is the required yaw rate to achieve the desired heading angle.
60. Chapter 3. Control design 41
3.3 Inner loop control
The goal of inner loop (Fig. 3.3) is to track the body rate commands generated by outer
loop. The inner loop achieves the tracking by transforming the body rate commands
into aerodynamic controls. The body rate commands generated by outer loop control
are desired roll rate (p∗
), desired pitch rate (q∗
), and desired yaw rate (r∗
). Where as
the aerodynamic controls are aileron (δa), elevator (δe) and rudder (δr) deflections.
Figure 3.3: Inner loop control
There is a separate loop for velocity control where velocity is maintained constant
through throttle control. The body rates control and velocity control are discussed in
following sections.
3.3.1 Body rates control
The aerodynamics controls should be calculated such that it tracks the body angular
rates desired by the outer loop. The relative degree between aerodynamic controls and
body rates is one. Hence enforce first order error dynamics for error in body rates
˙p − ˙p∗
˙q − ˙q∗
˙r − ˙r∗
+
kp 0 0
0 kq 0
0 0 kr
p − p∗
q − q∗
r − r∗
= 0 (3.19)
where error = [(p − p∗
) (q − q∗
) (r − r∗
)]T
. kp, kq and kr are chosen positive (Table 3.1).
61. Chapter 3. Control design 42
Rearranging previous equation we get
˙p
˙q
˙r
=
˙p∗ − kp(p − p∗
)
˙q∗ − kq(q − q∗
)
˙r∗ − kr(r − r∗
)
(3.20)
From six-DOF equation of motion we have
˙p = c1rq + c2pq + c3La + c4Na (3.21)
˙q = c5pr + c6(p2
− r2
) + c7(Ma + Mt) (3.22)
˙r = c8pq − c2rq + c4La + c9Na (3.23)
Separating the state and control terms in ˙p, ˙q, ˙r equations and rearranging Eq. 3.20 we
can write
fr + grUa = br (3.24)
where Ua = [δa δe δr]T
and other terms are defined as follows
fr
c1rq + c2pq + c3Lax + c4Nax
c5pr + c6(p2
− r2
) + c7(Max − Mt)
c8pq − c2rq + c4Lax + c9Nax
gr
c3Lau 0 c4Nau
0 c7Mau 0
c4Lau 0 c9Nau
br
˙p∗ − kp(p − p∗
)
˙q∗ − kq(q − q∗
)
˙r∗ − kr(r − r∗
)
Lax ˆqSb[Clβ
(α) β + Clp (α) ¯p + Clr (α) ¯r] Lau ˆqSbClδa
Max ˆqSc[Cm0 + Cmα (α) α + Cmβ
(α, β) β + Cmq (α) ¯q] Mau ˆqScCmδe
Nax ˆqSb[Cnβ
(α) β + Cnp (α) ¯p + Cnr (α) ¯r] Nau ˆqSbCnδr
Carrying out the necessary algebra in Eq. (3.24), the control solution is
Ua = g−1
r (br − fr) (3.25)
The above solution for aerodynamic controls will track the desired body rates generated
by outer loop control.
62. Chapter 3. Control design 43
3.3.2 Forward velocity control
The forward velocity can be controlled by varying the thrust through throttle control.
The relative degree between throttle control and forward velocity is one. Enforcing the
following first order error dynamics for error in forward velocity
( ˙u − ˙u∗
) + ku(u − u∗
) = 0 (3.26)
where error = (u − u∗
) and u∗
is the velocity which has to be maintained during the
flight. ku is chosen positive, given in Table 3.1. Rearranging above equation we get
˙u = ˙u∗
− ku(u − u∗
) (3.27)
From six-DOF equation of motion we have
˙u = rv − qw − gsinθ +
Xa + Xt
m
(3.28)
Substituting ˙u equation in Eq. (3.27) and rearranging
fu + guσt = bu (3.29)
Where σt is throttle control value and other terms are defined as follows
fu rv − qw − gsinθ +
Xa
m
gu
Tmax
m
bu ˙u∗
− ku(u − u∗
)
Rearranging the Eq. (3.29) we get the control solution as
σt = g−1
u [bu − fu] (3.30)
The above solution for throttle control will track the desired velocity.
63. Chapter 3. Control design 44
3.4 Actuator controller
The control solutions found by inner loop are fed to actuator for deflection of aerodynamic
control surfaces. The actuator dynamics is given in Section 2.4 which is same for all
control surfaces. It is observed from the step response (Fig. 2.35) of actuator that it
has a settling time of 0.54 seconds. Which is higher than the desired settling time for
the inner loop, hence a first order controller is designed for the actuator. Enforcing first
order error dynamics for error in actuator deflection
( ˙δe − ˙δ∗
e ) + kδe (δe − δ∗
e ) = 0 (3.31)
where error = (δe − δ∗
e ) and δ∗
e is the desired deflection for elevator. Substituting the
actuator dynamics given by Eq. 2.46 in above equation and rearranging we get
uδe =
1
6.7
[−9.5 δe + ˙δ∗
e − kδe (δe − δ∗
e )] (3.32)
where uδe is the command to be fed to the actuator to achieve the desired deflection.
The response of actuator to 1 deg deflection command is shown in Fig. 3.4.
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1.2
X: 0.22
Y: 0.9931
time (sec)
Servodeflection(deg)
X: 0.54
Y: 1
without control
with control
Figure 3.4: Actuator response for 1 deg command of deflection
64. Chapter 3. Control design 45
3.5 Control structure
The Fig. 3.5 shows the combined inner and outer loop structure. Where roll angle, pitch
angle and heading angle command come from guidance. The commands from guidance
are converted into desired body rates by the outer loop. Further inner loop transforms
these desired body rates into required aerodynamic control surface deflections.
Figure 3.5: Control structure
The velocity control loop maintains the desired velocity by throttle control, not shown
in above figure. The various gains for control loops chosen after iterative simulation are
given in the table below.
Table 3.1: Gains for various control loops
kφ kθ kψ kp kq kr ku kδe
sec−1
sec−1
sec−1
sec−1
sec−1
sec−1
sec−1
sec−1
2 2 2 5 5 5 1 20
To test the control design done in above sections, simulations are carried out by
giving various commands. The simulations and resulting performance are presented in
next section.
65. Chapter 3. Control design 46
3.6 Simulation results
The simulation is initialized with the trim condition obtained in Section 2.5 as initial
conditions. After 5 seconds of simulation a sequence of 3-2-1-1 command is given where
the width of one unit has been kept as 5 seconds. The initial conditions are given below.
Initial states
X0 = [ Vt0 α0 β0 p0 q0 r0 φ0 θ0 ψ0 x0 y0 h0 ]
= [ 20 3.15o
0 0 0 0 0 3.15o
0 − 1200 0 100 ]
Initial control
Ua0 = [ σt0 δe0 δa0 δr0 ]
= [ 0.36 − 3.29o
0 0 ]
3.6.1 Pitch angle command
A 3-2-1-1 sequence for pitch angle command is given with ±5 deg around trim pitch
angle. The velocity control loop maintains the forward velocity as in trim condition.
The Fig. 3.6 shows the response of pitch angle to the command.
0 10 20 30 40 50
−4
−2
0
2
4
6
8
10
t (sec)
θ(deg)
actual
desired
Figure 3.6: Pitch angle variation for 3-2-1-1 command
66. Chapter 3. Control design 47
It is seen from Fig 3.7 that the velocity is maintained constant and the altitude is
increasing at constant rate due to increase in pitch angle from the trim condition. The
Fig. 3.8 shows the control requirements. There is increased demand in negative elevator
for the positive pitching motion. The increase in elevator also increase drag hence more
of throttle is required to compensate for maintaining constant velocity.
0 10 20 30 40
19
20
21
t (sec)
V
t
(m/s)
0 10 20 30 40
0
5
t (sec)
α(deg)
0 10 20 30 40
−10
0
10
t (sec)
Q(deg/s)
0 10 20 30 40
−2
0
2
4
6
8
t (sec)
θ(deg)
0 10 20 30 40
−1200
−1000
−800
−600
−400
−200
t (sec)
x(m)
0 10 20 30 40
100
110
120
t (sec)
h(m)
Figure 3.7: Longitudinal states for pitch angle command
0 10 20 30 40
0
0.2
0.4
0.6
0.8
1
t (sec)
σt
(0−1)
0 10 20 30 40
−6
−4
−2
0
t (sec)
δe(deg)
0 10 20 30 40
−1
−0.5
0
0.5
1
t (sec)
δa(deg)
0 10 20 30 40
−1
−0.5
0
0.5
1
t (sec)
δr(deg)
Figure 3.8: Control variables for pitch angle command
67. Chapter 3. Control design 48
3.6.2 Heading angle command
A 3-2-1-1 sequence command for heading angle is given after 5 seconds of simulation
while the desire roll angle is zero and trim pitch angle is maintained. The Fig. 3.9 shows
the response of heading angle to ±5 deg command. It is seen from Fig. 3.10 that there
is small decrease in pitch angle this is due to reduction in lift during the turn.
0 10 20 30 40 50
−6
−4
−2
0
2
4
6
t (sec)
ψ(deg)
actual
desired
Figure 3.9: Heading angle variation for 3-2-1-1 command
0 10 20 30 40
19
20
21
t (sec)
V
t
(m/s)
0 10 20 30 40
2
3
4
t (sec)
α(deg)
0 10 20 30 40
−1
0
1
t (sec)
Q(deg/s)
0 10 20 30 40
2
3
4
t (sec)
θ(deg)
0 10 20 30 40
−1200
−1000
−800
−600
−400
−200
t (sec)
x(m)
0 10 20 30 40
99
100
101
t (sec)
h(m)
Figure 3.10: Longitudinal states for heading angle command
68. Chapter 3. Control design 49
Fig. 3.11 shows the yaw rate required for the heading angle command. Also the
sideslip angle is non zero for some time as the coordinated turn constraint has not been
enforced. The control plots in Fig. 3.12 show the requirement in rudder deflection and
also aileron in an effort to maintain zero roll angle.
0 10 20 30 40
−5
0
5
t (sec)
β(deg)
0 10 20 30 40
−2
0
2
t (sec)
P(deg/s)
0 10 20 30 40
−10
0
10
t (sec)
R(deg/s)
0 10 20 30 40
−2
0
2
t (sec)
φ(deg)
0 10 20 30 40
−5
0
5
t (sec)
ψ(deg)
0 10 20 30 40
0
10
20
t (sec)
y(m)
Figure 3.11: Lateral states for heading angle command
0 10 20 30 40
0
0.2
0.4
0.6
0.8
1
t (sec)
σ
t
(0−1)
0 10 20 30 40
−5
−4
−3
−2
−1
t (sec)
δe(deg)
0 10 20 30 40
−5
0
5
t (sec)
δa(deg)
0 10 20 30 40
−10
−5
0
5
10
t (sec)
δr(deg)
Figure 3.12: Control variables for heading angle command
69. Chapter 3. Control design 50
3.6.3 Bank angle command
A 3-2-1-1 sequence command for roll angle is given while the desire heading angle is zero
and trim pitch angle is maintained. The response of roll angle to ±5 deg command is
shown in Fig. 3.13. It is seen from Fig. 3.14 though the pitch angle is constant, there is
constant decrease in altitude. It is due to reduction in lift due to non zero roll angle.
0 10 20 30 40 50
−6
−4
−2
0
2
4
6
t (sec)
φ(deg)
actual
desired
Figure 3.13: Roll angle variation for step 3-2-1-1 command
0 10 20 30 40
19
20
21
t (sec)
V
t
(m/s)
0 10 20 30 40
2
3
4
t (sec)
α(deg)
0 10 20 30 40
−1
0
1
t (sec)
Q(deg/s)
0 10 20 30 40
2
3
4
t (sec)
θ(deg)
0 10 20 30 40
−1200
−1000
−800
−600
−400
−200
t (sec)
x(m)
0 10 20 30 40
96
98
100
t (sec)
h(m)
Figure 3.14: Longitudinal states for roll angle command
70. Chapter 3. Control design 51
Fig. 3.15 shows the roll angle is maintained without changing the heading angle.
However, this comes at a price of constant control effort for aileron and rudder deflections.
0 10 20 30 40
−5
0
5
t (sec)
β(deg)
0 10 20 30 40
−10
0
10
t (sec)
P(deg/s)
0 10 20 30 40
−1
0
1
t (sec)
R(deg/s)
0 10 20 30 40
−5
0
5
t (sec)
φ(deg)
0 10 20 30 40
−1
0
1
t (sec)
ψ(deg)
0 10 20 30 40
0
10
20
30
t (sec)
y(m)
Figure 3.15: Lateral states for roll angle command
0 10 20 30 40
0
0.2
0.4
0.6
0.8
1
t (sec)
σ
t
(0−1)
0 10 20 30 40
−4
−3
−2
−1
t (sec)
δe(deg)
0 10 20 30 40
−5
0
5
t (sec)
δa(deg)
0 10 20 30 40
−15
−10
−5
0
5
10
t (sec)
δr(deg)
Figure 3.16: Control variables for roll angle command
Summary: The control has been designed in this chapter for inner and outer loop to
track the guidance commands along with velocity control. Next chapter discusses the
path planning and guidance command generation for autonomous landing.
71. Chapter 3. Control design 52
Appendix: Zero dynamics
The zero-dynamics is defined to be the internal dynamics of the system when the system
output is kept at zero by the input. The states of the system taken as output are
u, p, q, r, φ, θ, ψ, x, y, h. The remaining internal states are v and w. There dynamics is
given by following equations
˙v = pw − ru + gsinφcosθ +
Ya
m
(3.33)
˙w = qu − pv + gcosφcosθ +
Za
m
(3.34)
The input which keeps the outputs to zero can be calculated from Eq. 3.25 by putting
br = 0
Ua = g−1
r (−fr) (3.35)
Now replacing the outputs as zero and the input given by above equation, we will get
˙v
˙w
= Avw
v
w
Where Avw is linearized matrix around the initial condition and is given as
Avw =
∂f1
∂v
∂f2
∂w
∂f1
∂v
∂f2
∂w
The eigen values of the Avw matrix are tabulated for various operating points
Vt(m/s) h(m) λ1 λ2
15 100 -0.5397 -0.0111
20 100 -0.3148 -0.0125
25 100 -0.0128 -0.0011
It is observed from the above table that the eigen values of Avw matrix lie in left half
plane for the operating range of the UAV, hence the zero dynamics is locally stable.
72. Chapter 4
Path planning and guidance
The autonomous landing of UAVs require an intelligent path planning and guidance.
Here the landing is divided into three phases which are approach, glideslope and flare
(Fig. 4.1). The approach is initial phase of landing where UAV descents to a height from
which glideslope can begin while aligning itself with runway. The next is glideslope, a
straight line path where it looses most of its height. The last segment of landing is flare
which is an exponential curve at the end of which UAV touches down on ground. The
design and selection of the landing trajectory is discussed in next section.
Figure 4.1: Phases during landing
4
“Our scientific power has outrun our spiritual power. We have guided missiles and misguided men.”
Martin Luther King Jr.(1963)
53
73. Chapter 4. Path planning and guidance 54
4.1 Path planning
To guide the UAV it is required to specify the desired trajectory. The desired trajectory
can be made a function of time or distance in space. The trajectory tracking with respect
to time requires UAV to be in a particular location at particular point of time. There will
be problems in case of tracking a trajectory varying with time, when the UAV is flying
into wind disturbances [33]. To overcome this the effects of wind should be accounted
properly to move the desired trajectory slower or faster. To remove the dependence of
time a path in space can be specified, where UAV will be required to be on the path
rather than at a certain point on a particular time. Here the desired trajectory is made
a function of forward distance which is discussed in next sections.
4.1.1 Approach
During approach the vehicle should come and align with the runway at a specified height
from where the glideslope can be started. Fig. 4.2 shows the geometry for alignment
where (x0, y0) is the initial location of UAV at the beginning of landing. (xg, yg) is point
from where glideslope begins. ψ0 is the initial orientation of the vehicle.
y
runway x
(0,0)
(x0
,y0
)
(xg
, yg
)
ψ0
Figure 4.2: Approach geometry in x-y plane (top view)
74. Chapter 4. Path planning and guidance 55
The initial condition (x0) for landing has bounds from which landing can be started
though there is no such constraint for the initial orientation (ψ0) and sideward distance
(y0). This constraint is due to the fact that UAV will require a minimum distance to take
turn and align before reaching the glideslope. The margin for initial condition has been
kept as (x0 − xg) > 400m, i.e. the alignment should begin 400m before the glideslope
initiation point. The equation for path of approach can be written as a straight line
function of forward distance.
y∗
= y0 +
yg − y0
xg − x0
(x − x0) (4.1)
This is the desired value of sideward distance during approach which UAV has to follow.
The desired value of sideward distance during glideslope and flare is zero (y∗
= 0) i.e.
aligned with the runway centerline. The UAV may also require to loose height during
approach to reach the glideslope height.
4.1.2 Glideslope
The glideslope is a straight line path whose slope is defined by the flight path angle. The
desired height is also scheduled as a function of forward distance.
(0, 0)
h
(x
g
, h
g
)
(xf
, hf
)
(x
∞
,h
c
)
xγ∗
(x
g0
, h
g0
)
(x
td
, h
td
)
Figure 4.3: Glideslope and flare geometry in x-h plane
75. Chapter 4. Path planning and guidance 56
In Fig. 4.3 (xg, hg) is the point where glideslope begins. (xf , hf ) is point where flare
begins. (xtd, htd = 0) is the touchdown point. (0, 0) is the origin of inertial frame at the
beginning of runway. (xg0 , hg0 = 0) is a fictitious point considered on ground from where
glideslope is projected. (x∞, hc) is the final point of flare path, this point is chosen to be
below ground so that exponential flare path intersects the ground at touchdown point.
From the geometry of the Fig. 4.3 the required flight path angle which vehicle has to
follow is calculated as follows
γ∗
= tan−1 hg − hg0
xg − xg0
(4.2)
The desired flight path angle (γ∗
) which UAV has to follow is limited between 3o
to 5o
.
At lower flight path angle the position from which the glideslope has to begin (xg) will
become very large for a given glideslope altitude. At higher values of flight path angle
the vehicle will have limitation as the component of gravity will cause an increase in
velocity during landing. The desired height at every point of glideslope can be written
as a function of forward distance given by following equation
h∗
= hg0 + (x − xg0 ) tanγ∗
(4.3)
The UAV will follow the path specified by above equation while maintaining lateral error
to zero until the flare height is reached. The flare height is calculated online and it is
not fixed apriori which is discussed in next section.
4.1.3 Flare
The flare path is an exponential curve shown in Fig. 4.3. The exponential curve needs to
be chosen to end below the ground so that it touches the ground at a finite location. The
problem of slope discontinuity at the transition from glideslope to flare can be addressed
by appropriately choosing the parameters which specify the flare path. The advantage
of scheduling desired height as a function of forward distance is that we can specify the
location of touchdown. It is also required to control the sink rate at the touchdown in
order to avoid a hard landing.
76. Chapter 4. Path planning and guidance 57
The desired height during exponential flare can be scheduled as a function of forward
distance, given by following equation
h∗
= hc + (hf − hc)e−kx(x−xf )
(4.4)
The unknowns in above equation are flare height hf , distance at which to begin flare xf ,
final height below the ground where flare trajectory should end hc, and constant kx. We
can solve for these four unknowns under the following four constraints,
Initial condition: The point where glideslope ends and flare begins be coincident. Sub-
stituting x = xf in (4.3) and (4.4) than equating
hf = −(xf − xg0 ) tanγ∗
(4.5)
Initial slope: The slope at the beginning of flare and at the end of glideslope be same.
Differentiating (4.3) and (4.4) than equating at x = xf
(hf − hc)kx = tanγ∗
(4.6)
Touchdown condition: The flare trajectory should intersect the ground at touchdown
point. Replacing x = xtd and h∗
= 0 in (4.4) we get
0 = hc + (hf − hc)e−kx(xtd−xf )
(4.7)
Sink rate at touchdown: The descent rate at touchdown should be equal to specified sink
rate. Differentiating (4.4) and evaluating at x = xtd. Putting ˙h∗ = ˙h∗
td, where ˙h∗
td is the
desired sink rate at touchdown. The ground velocity at touchdown ( ˙xtd) is taken as equal
to the air velocity controlled at touchdown.
˙h∗
t = −(hf − hc)kx ˙xtde−kx(xtd−xf )
(4.8)
Now we can solve the Eq. 4.5 - 4.8 for the four unknowns of the flare path. The solution
will ensure a smooth transition from glideslope to flare path. We also have the direct
control over the touchdown point and sink rate at touchdown. Which can be the design
parameters and tuned as per the requirement.
77. Chapter 4. Path planning and guidance 58
To closely follow the path specified by path planning, appropriate guidance commands
need to be generated which is discussed in next section.
4.2 Guidance
The desired values from path planning to be tracked are sideward distance (y∗
= 0) and
altitude (h∗
= 0). The kinematic equations for sideward distance and altitude can be
inverted to find the desired values of heading angle and pitch angle required for tracking.
While taking turn during approach the coordinated turn constraint is imposed from
which the desired roll angle can be found.
4.2.1 Sideward distance control
To track the desired sideward distance (y∗
) we enforce a first order error differential
equation as
( ˙y − ˙y∗) + ky(y − y∗
) = 0 (4.9)
where error = y − y∗
. The gain ky is chosen positive, given in Table 4.1. Rearranging
above equation we get
˙y = ˙y∗ − ky(y − y∗
) (4.10)
From six-DOF equations we can write
˙y = ay sinψ + by cosψ (4.11)
where ay u cosθ + v sinφsinθ + w cosφsinθ
by v cosφ − w sinφ
Substituting Eq. (4.11) in Eq. (4.10) than solving for ψ
ψ∗
= sin−1
˙y∗ − ky(y − y∗
)
a2
y + b2
y
− tan−1 by
ay
(4.12)
We have ψ∗
which is the required heading angle to track the path for sideward distance.
78. Chapter 4. Path planning and guidance 59
4.2.2 Altitude control
The altitude can be controlled by commanding pitch angle. The relative degree between
pitch angle and altitude is one. To generate the pitch angle command from error in
altitude, we enforce the first order error dynamics for altitude error
(˙h − ˙h∗) + kh(h − h∗
) = 0 (4.13)
where error = h−h∗
and h∗
is the desired altitude. The gain kh is chosen positive, given
in Table 4.1. Rearranging above equation we get
˙h = ˙h∗ − kh(h − h∗
) (4.14)
From six-DOF equations we can write
˙h = ahsinθ − bhcosθ (4.15)
where ah = u and bh = v sinφ + w cosφ
Substituting Eq. (4.15) in Eq. (4.14) than solving for θ
θ∗
= sin−1
˙h∗ − kh(h − h∗
)
a2
h + b2
h
+ tan−1 bh
ah
(4.16)
Now θ∗
is the desired pitch angle required to track the desired altitude. The UAV should
touchdown on the ground at the end of flare with bounds on pitch angle. This is ensured
by passing the desired pitch angle through a limiter (20
θ∗
80
) when near to ground.
The other approach taken is to find the trim condition for a given pitch angle at the
touchdown. This trim condition will provide us with a desired velocity at the end of flare
which can be fed to velocity control loop for tracking.
It is seen from Eq. 4.12 and Eq. 4.16 that they involve inverse sine functions. The value
of quantities inside the inverse function need to be ensured less than equal to 1, else the
solutions will be imaginary. This can be achieved by careful selection of ky and kh.
79. Chapter 4. Path planning and guidance 60
4.2.3 Coordinated turn constraint
The coordinated turn requires the sideslip angle be equal to zero, equivalently it can
enforced by maintaining side velocity in body frame to be zero. This can be achieved
by finding an equivalent roll angle from the enforced first order error dynamics for side
velocity.
(˙v − ˙v∗) + kv(v − v∗
) = 0 (4.17)
where error = v − v∗
and v∗
= 0 is the desired side velocity. From six-DOF equations
of motion we have
˙v = pw − ru + gsinφcosθ +
Ya
m
(4.18)
Substituting Eq. 4.18 in Eq. 4.17 and than solving for φ, we will get
φ∗
= sin−1
˙v∗ − kv(v − v∗
) + ru − pw − Ya
m
gcosθ
(4.19)
This desired roll angle will ensure the coordinated turn and sideslip angle to be zero.
Schematic of path planning and guidance
Fig. 4.4 shows the complete structure of path planning and guidance.
Figure 4.4: Path planning and guidance schematic
