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CHAPTER 6

DESIGN OF SIX DEGREES OF FREEDOM

AIRCRAFT MODEL AND LONGITUDINAL AUTOPILOT

FOR AUTONOMOUS LANDING

6.1 INTRODUCTION

This chapter deals with the development of six degrees of freedom

(6-DOF) aircraft model. This 6-DOF model can be used to design the

longitudinal autopilot for autonomous landing. Glide slope and flare

autopilots are designed and implemented using the 6-DOF model. Finally the

results are verified using X-Plane Flight simulator.

6.2 DEVELOPMENT OF 6-DOF AIRCRAFT MODEL

The block implements a nonlinear 6-degree-of-freedom aircraft

dynamic model, using blocks provided in the AeroSim library (Aerosim –

Aeronautical Blockset - User’s Guide). The equations of motion are

implemented in geodetic-frame. The model parameters are read from a user-

configurable mat-file. The 6 DOF aircraft model block is shown in Figure 6.1.

The various parameters given as inputs and the outputs obtained are discussed

in the Section 6.2.1. The complete aircraft model is shown in Figure 6.2.

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Figure 6.1 6-DOF Aircraft Model Block

6.2.1 Block characteristics

The Parameters of 6-DOF Aircraft Model Block are given below:

Aircraft configuration file: The path and name of the aircraft parameter mat-

file, provided as a string. For example, if the mat-file is someairplane.mat,

and it is in the current directory, then we would use ’someairplane.mat’.

Initial position: The 3×1 vector of initial aircraft location [Lat Lon Alt ]T , in

[rad rad m].

Initial velocities: The 3×1 vector of initial aircraft velocity components in

geodetic-frame [VN VE VD ].

Initial attitude: The 4×1 vector of initial aircraft attitude provided as Euler-

Rodrigues quaternion’s [e0 ex ey ez ] .

Initial angular rates: The 3×1 vector of initial aircraft angular rates (in body

axes) [ p q r ] .

Initial fuel mass: The initial mass of the fuel quantity available on-board the

aircraft, in kg.

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Initial engine speed :The initial engine shaft rotation speed, in rad/s.

Ground altitude: The altitude of the terrain relative to mean-sea-level, at

aircraft location, in meters.

WMM coefficient file: The complete path to the magnetic model coefficient

file,

Simulation date: 3×1 vector of the calendar date in the format [Day Mon

Year].

Sample time: The sample time at which the aircraft model will run.

The inputs of 6-DOF Aircraft Model Block are given below:

Controls: The 7×1 vector of aircraft controls [flap elevator aileron rudder

throttle mixture ignition ] in [rad rad rad rad frac ratio bool].

Winds: The 3×1 vector of background wind velocities, in navigation frame

[WN WE WD], in m/s.

RST: The integrator reset flag

The outputs of 6-DOF Aircraft Model Block are given below:

States: The 15×1 vector of aircraft states [VN VE VD p q r e0 ex ey ez Lat Lon

Alt mfuel Weng ] .

Sensors: The 18×1 vector of sensor data [Lat Lon Alt VN VE VD ax ay az p q r

pstat , pdyn OAT Hx Hy Hz ] .

VelW: The 3×1 vector of aircraft velocity in wind axes [Va b a] in [m/s rad

rad].

Mach: The current aircraft Mach number.

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Angular Acc: The 3×1 vector of body angular accelerations [p q r ] .

Euler: The 3×1 vector of the attitude of the aircraft given in Euler angles

[f q y], in radians.

AeroCoeff : The 6×1 vector of aerodynamic coefficients [CD CY CL Cl Cm

Cn ], in rad.

PropCoeff : The 3×1 vector of propeller coefficients [J CT CP ] .

EngCoeff: The 5×1 vector of engine coefficients [MAP m air m fuel BSFC

P]T given in [kPa kg/s kg/s g/(W*hr) W].

Mass: The current aircraft mass, in kg.

ECEF: The 3 × 1 vector of aircraft position in the Earth-centered, Earth-fixed

frame [X Y Z ]

MSL: The aircraft altitude above mean-sea-level, in m.

AGL: The aircraft altitude above ground, in m.

REarth: The Earth equivalent radius, at current aircraft location, in m.

6.2.2 Complete Aircraft Model – Sub Block

The complete aircraft sub block consists of the aerodynamics block,

propulsion system, aircraft inertia, the atmosphere model, total acceleration,

the total moment sub blocks, the aircraft equations of motion and the earth

model. The earth model implemented here is the WGS-84 model.

The acceleration, velocity, the rates and the position are calculated

using the Equations of Motion sub block. The initial conditions i.e. the flap,

elevator, rudder, throttle, ignition, mixture are provided to the aerodynamics

and the propulsion block. The initial wind condition is given to the

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atmosphere block. The initial velocities, rates and lat, lon, alt position are

given in the Equation of Motion (EOM) block. Hence the complete aircraft

model block is shown in Figure 6.2.

Figure 6.2 Complete Aircraft Model Sub Block

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6.2.3 Aerodynamics Block

Aerodynamics: Section 6.21 of the aircraft con guration script

speci es the aerodynamic parameters of the aircraft. These are explained in

the following Figures 6.3 – 6.5.

Figure 6.3 Aerodynamics Sub Block

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Aerodynamic parameter bounds the limits that the aircraft model

will impose on the airspeed, sideslip, and angle-of- attack, given as 1x2 vector

of min and max values. The purpose of using these limits is to keep the

outputs of the aerodynamic model within the linear region.

6.2.4 Propulsion Block

Figure 6.4 Propulsion Sub Block

Propeller : The second section of the aircraft con guration script speci es

the geometry and aerodynamic performance of the propeller.

Propeller hub location: The position of the propulsion force and moment

application point, given with respect to the body- frame origin. The location is

speci ed as a 1x3 row vector of x, y, and z coordinates.

Advance ratio: The aerodynamic performance of the propeller should be

given as a look-up table of propeller coef cients (CP and CT) as functions of

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the propeller advance ratio. This variable speci es the advance ratio vector

which corresponds to the look-up table.

Coef cient of thrust: The vector of coef cients of thrust for the advance

ratios given above (the vector should have the same size).

Coef cient of power: The vector of coef cients of power for the advance

ratios given above (the vector should have the same size).

Propeller radius: The radius of the propeller is used by the propulsion model

to compute the force and torque from the normalized coef cients.

Engine : The third section of the aircraft con guration scripts allows the user

to specify the engine characteristics. All engine data is given at sea-level. The

engine model will correct the data for altitude effects. For a normally-

aspirated general aviation piston engine, this includes the following

parameters:

RPM: The vector of engine speeds for which the engine data is given, in

rotations-per-minute. All engine parameters are speci ed as 2-D look-up

tables (functions of engine speed and intake manifold pressure).

Fuel ow: The sea-level fuel ow as a function of RPM and MAP. The

number of rows in the matrix should match the size of the RPM vector, the

number of columns should match the size of the MAP vector.

Power: The engine power at sea-level, as a function of RPM and MAP. The

number of rows in the matrix should match the size of the RPM vector, the

number of columns should match the size of the MAP vector.

Sea-level atmospheric conditions: The sea-level atmospheric conditions,

including pressure in Pascals and temperature in degrees Kelvin, for which

the engine data above is given.

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Engine shaft inertia: The moment of inertia of the rotating parts of the

engine. This is added to the propeller inertia and used in the propulsion

equation of motion to compute the current engine speed. Generally, the

engine shaft inertia is signi cantly lower than that of the propeller, and it can

be neglected without any major effects over the aircraft dynamics.

6.2.5 Atmosphere Block

Figure 6.5 Atmosphere Sub Block

The standard atmosphere block provides the air parameters at the

current altitude. The standard atmosphere block is using interpolation through

look-up tables which provide air data for an altitude range of 0 to 86000

meters.

The background wind block computes the background wind

velocity components in body axes. The block is applying a frame

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transformation from inertial (geographic) to body frame, using the rotation

matrix provided. The numerical time derivative of the resulting velocity

vector is then computed.

The turbulence block provides a von Karman turbulence model.

The block is applying von Karman turbulence shaping filters for longitudinal,

lateral, and vertical components to 3 white-noise sources. The filter

parameters depend on background wind magnitude and current aircraft

altitude.

The wind shear block computes the angular rate effects caused by

the variation in time/space of the background wind and turbulence velocities.

The wind shear effects considered are the angular velocities and accelerations

for pitch and yaw.

6.2.6 Earth Model

The Earth library folder includes blocks that model the Earth’s

shape, gravity, and magnetic eld as shown in Figure 6.6.

WGS-84: The block computes the local Earth radius and gravity at current

aircraft location using the WGS-84 Earth model coefficients.

EGM-96: The block computes the sea-level altitude with respect to theWGS-

84 ellipsoid, using the EGM-96 geoid undulation model. The EGM-96 block

computes the altitude difference between the theoretical ellipsoid shape and

the actual mean sea level (geoid undulation). This is caused by the non-

uniformity of Earth’s gravitational potential. The correction is performed

using a 2-dimensional Latitude-Longitude look-up table with a resolution of 1

degree in both directions. The geoid undulation is then added to a 0.53 m

WGS-84 correction and to the WGS-84 altitude computed by the aircraft

equations of motion, to obtain the altitude of the aircraft above sea-level.

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Figure 6.6 Earth Model Sub Block

Ground Detection: The Ground Detection block computes the aircraft

altitude Above Ground Level and sets a flag if it is zero. The ground altitude

should be supplied by the user as a constant or a look-up table of terrain

elevation data. In both cases it should be measured with respect to the MSL

and the unit of measure must match that of the MSL altitude.

WMM-2000: The WMM-2000 block computes the Earth magnetic eld

components at current location using the Department of Defense World

Magnetic Model 2000.

6.2.7 Body Frame EOM

These equations form the centerpiece of an aircraft dynamic model

and implemented in the MATLAB as shown in Figure (6.7). There are two

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formulations for the equations of motion that are commonly used, and they

are provided in two separate sub-folders within the AeroSim library.

Figure 6.7 Body Frame EOM Block

6.3 6- DOF AIRCRAFT SIMULATION

Using the AeroSim blockset a basic 6-DoF model is constructed as

shown in Figure 6.8. In the first case a simple closed loop configuration is

analysed and the airspeed and pitch angle output are obtained as shown in

Figure 6.9. Using this six DoF model the landing autpilot is being

implemented.

160Figure 6.8 AeroSim 6-DOF Simulink Model without PID

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Figure 6.9 Airspeed and Pitch angle output without PID

In the second case, as shown in Figure 6.10, a PID control is

added to the feedback loop which atabilised the airspeed and the pitch angle

outputs.

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Figure 6.10 AeroSim 6-DOF Simulink Model with PID

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Figure 6.11 Airspeed and Pitch angle output with PID

As seen from the Figure 6.10 and Figure 6.11 the response of the

airspeed and pitch angle has improved with the addition of the PID controller

in terms of settling time , overshoot and rise time.

Using this 6 DoF mathematical model developed using Aerosim

blockset in MATLAB/Simulink, an autonomous landing controller is

designed. The requirements to successfully complete an autonomous landing

are: define the glide path and flare path geometry, design the pitch autopilot

and design controllers for glide path and flare.

6.4 DESIGN OF PITCH AUTO PILOT

To begin the design of the pitch angle autopilot, a transfer function

representative of the UAV in landing conditions is required. The aircraft

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transfer function for pitch angle for the experimental UAV has been found

to be:

3 2

1728 46.81

330 116.9 546.4e

s

s s s

Figure 6.12 Pitch autopilot without PID Controller

Figure 6.13 Pitch autopilot and its response

The response, for a unit step input, shows that the oscillations are

more and is not quite good for landing, since the settling time and rise time

are more. The actual settling time of 17 seconds is much too slow for an

autopilot to control an aircraft on landing. In order to decrease the rise time, a

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proportional controller is needed. In order to reduce the settling time, the

derivative controller is needed. In order to reduce oscillations, the integrator is

needed. So, overall, there is a need for PID controllers as shown in Figure

6.14. The tuning of Kp, Kd, Ki are done by Hessian Modified tuning to get

the optimum response.

Figure 6.14 Pitch autopilot with PID Controller

Figure 6.15 Pitch angle autopilot with PID controllers

From the Figure 6.15, it is concluded that the proportional, integral

and derivative controller gains are tuned to get the optimum response and the

values are found to be Kp= 2.5, Ki= 5, Kd= -1.5. That is, the system will not

undergo unstable region with change in gain. The phase margin is found to be

33.8 degree and a crossover frequency of 1.57 rad/s with a settling time

around 4 seconds.

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6.5 AUTOMATIC GLIDE SLOPE CONTROLLER

6.5.1 Basic Longitudinal Autopilot

The Basic longitudinal Autopilot (John Blakelock 1991) ensures

that the pilot pitch angle command is achieved quickly with fewer

oscillations. It is a modified simple displacement autopilot and shown in

Figure 6.16.

Figure 6.16 Basic Longitudinal Autopilot

It was modified using an inner loop with pitch rate feedback to

improve the damping of the short period oscillations and also to achieve

higher damping in outer loop.

6.5.2 Glide Slope Controller

The Automatic glide slope controller guides the UAV down a

predetermined glide slope of 5.5 . At a pre-selected altitude reduces the rate

of descent and cause the UAV to flare out and touch down with an acceptably

low rate of descent (Kim and Golnaraghi 2004).

The glide path is defined as a line from some starting point to the

end of the runway. For this project, a glide path angle of -5.5o

was used, so the

starting point was defined by the LLA position of the end of the runway and

the desired final approach distance. To simulate this dependence on the range

from the aircraft to the runway, the glide path command signal was defined to

earef

e e

S(amp)A/ CElevator

Servo

Rate

Gyro

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include the range. Figure 6.17 shows the glide path geometry where the

commanded height above ground is a function of the range.

Figure 6.17 Geometry of the glide slope

During the flare maneuver, pilots transition from flying a straight

line to an exponential path to slow the descent rate of the airplane. This can be

simulated by defining an exponentially decaying flight path and using altitude

above ground to generate the error signal to the controller. Figure 6.18 shows

the flare path geometry intended touchdown zone approximately 500 ft. from

the runway threshold.

Figure 6.18 Geometry of the flare path

If the UAV is below the center line of the glide slope, then d is

considered negative, as is, , when the velocity vector is below the horizon,

that is, the UAV descending.

x

H

Runway

H

R

x

Runway

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The component of forward velocity U perpendicular to the glide

slope center line is d and this d for small glide slope angle is given by

o oUd Usin( 5.5) ( 5.5)

57.3(6.7)

If <5.5o, then from Figure 6.17, ( +5.5)

o is positive; therefore d is

positive, and as d initially was negative, the UAV is approaching the glide

path from below.

oUd ( 5.5)

57.3s (6.8)

The glide slope receiver does not measure the perpendicular

distance to the glide slope centerline but the angular error resulting there

from. Thus for a given value of d the angular error increases as the UAV

nears the runway, which has the effect of increasing the system gain as the

range to the runway decreases.

For small angles, o(57.3d / R) (6.9)

Through the use of Equations (6.8) and (6.9), the flight path angle

can be related to the angular error of the UAV from the centerline of the glide

slope. The block diagram of the glide slope control system, including the

geometry is shown below in Figure 6.19.

Figure 6.19 Block diagram of glide slope control system

3o

ref = 0comm

o dCoupler UAV &

Autopilot

d 57.3

R

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6.6 DESIGN OF GLIDE PATH CONTROLLER

The glide slope command is designed in such a way that the

controller will generate the error signal by comparing the instantaneous height

of the UAV, obtained from the image, and the required height at that instant.

So the error signal drives the autopilot, thus making UAV to align with the

glide path. Figure 6.20 shows the glide path controller.

Figure 6.20 Glide Path controller

The Range and the instantaneous height are extracted from the

runway image taken at real time. Since the sine of small value is

approximately a small value, so sin(Theta) = Theta. This glide path command

generates the necessary error signal if the UAV misses the actual glide path.

6.7 DESIGN OF FLARE PATH CONTROLLER

As seen earlier, the flare path geometry is an exponential one. After

certain distances, its need for switch over from glide path to flare. The flare

path command is proportional to the difference between horizontal distance at

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the instant of start of flare phase and instantaneous horizontal distance from

runway threshold. Let it be ‘x’. From Equation (6.10),

x

0H H e (6.10)

The desired value of ‘Tau’ can be obtained by specifying the

distance to the touchdown point from the glide slope transmitter. If this

distance is to be 10 ft and if it is also assumed that the aircraft touches down

in single time constant, then the ground distance traveled during the flare will

be greater than 10 ft.

Let the starting flare path geometry range would be 20 feet. Hence

the Tau value been estimated as 1 and the implementation is shown in

Figure 6.21.

Figure 6.21 Flare path controller

Using MATLAB/Simulink basic longitudinal autopilot shown in

Figure 6.22 and a Glide Slope Control system shown in Figure 6.23 are

designed to guide the UAV along a predefined path having a slope of 5.5o and

maintaining the longitudinal attitude.

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theta_in

thetadot_in

theta

phi

psi

thetadot

vision data calc

-10

s+12

elevator Servo

thetafinal

To Workspace5

-1.39s-.4253

s +.805s+1.3252

Thetadot / delta eStep

Scope5 Scope4

Scope3

ScopeProductPID

PID Controller

1

s

Integrator

180/pi

Constant1

Figure 6.22 Basic Longitudinal Autopilot

Figure 6.23 Glide Slope Control System

The vision data calculation module calculates the pitch angle

components and it uses the current pitch angle given by the (s)/ e(s) transfer

function to set the current orientation of UAV as shown in Figure 6.23. Using

this orientation and current position, images are generated with a separation of

0.5 sec (Rives and Azinheira 2002). The final pitch angle estimated is fedback

for the closed loop. The basic longitudinal autopilot is used in the Glide Slope

control system, which guides the UAV along a predefined glide path.

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Figure 6.24 Vision Data Calculation Module

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Figure 6.25 Pitch angle

It is concluded from Figure 6.25 that the UAV is aligned to the

runway and follows the predefined glide angle (5.5o) to safely land the UAV.

6.8 BLENDING FUNCTION AND ITS IMPLEMENTATION

An additional problem in the flare controller design was the method

of switching from glide slope to flare command signals. With the switch at a

range, the aircraft could not be made stable as having a switch in the

simulation caused adverse affects on the input signals to the switch. It was

determined by experiment that constants in Simulink create problems in

solving the algebraic loop and cause the simulation to produce erroneous

results. To compensate, step blocks were used with the step value equal to

that of the desired constant and the step time equal to the first time step of

simulation. Even with this correction, stability problems were still evident at

the switch of control commands.

To compensate for the sudden switch in commands, a blending

function was developed to soften the effect of the switch. This function blends

the signals over range values of 50 m to 20 m. These values were selected to

ensure that the aircraft would be established under the flare controller before

reaching the desired switch range of 50 m.

Figure 6.26 shows the Simulink of the braking function. The

saturation blocks normalize the signal multipliers to a value between zero and

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one. 20 m is subtracted from the range and multiplied by a gain to generate a

multiplier for the glide path command. The gain was selected such that at 50

m range, the glide multiplier is one and at 20 m range, the glide multiplier is

zero. The flare multiplier is much the same, but in the reverse direction. These

multipliers directly multiply the glide and flare signals such that when the

range is between 50 m and 20 m, both signals are active. Above 50 m only the

glide path signal is active and below 20 m only the flare signal is active.

Figure 6.26 Glide/Flare Blending Function

The blending of signals at the switch of glide slope and flare

control signals solved the problem of extreme oscillation and instability

during the switch. Because the flare path command is exponential, the aircraft

tends to bounce if the elevator remains under control of the flare controller

after main gear touchdown. To ensure the aircraft remains on the ground, the

elevator is neutralized with a relay at an altitude above ground level equal to

the height of the main gear.

6.9 INTEGRATION OF GLIDE/FLARE CONTROLLERS WITH

PITCH AUTOPILOT

Figure 6.27 shows the complete autonomous landing autopilot.

Here, the autopilot engages with glide controller till the altitude of 50 m.

Between the altitude ranges of 50 m and 20 m, both glide and flare controllers

are effective. Below the altitude range of 20 m, the autopilot engages with

flare controller. The blending function is to smooth out the switching between

glide and flare phase change.

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Figure 6.27 Glide/Flare autopilot

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6.10 BRAKING FUNCTION

Upon main gear touchdown, the elevator must allow the nose of the

airplane to rotate downward to contact the nose gear with the ground and the

brakes must be applied. Because exponentially decaying functions never

actually reach zero, the elevator command must be switched from the flare

path to neutral upon main gear touchdown.The brakes must be applied

smoothly after touchdown or the gear will fail. A rate limiter after a switch

can be used to accomplish this.

Because the flare path command is exponential, the aircraft tends to

bounce if the elevator remains under control of the flare controller after main

gear touchdown. To ensure the aircraft remains on the ground, the elevator is

neutralized with a relay at an altitude above ground level equal to the height

of the main gear. At the same altitude, the throttle command is also

neutralized and the brake command is changed from one to zero (zero to full

braking) with a rate limiter to limit the brake application time to two seconds.

The rate limiter on braking prevents gear failure due to over-braking upon

touchdown.

Figure 6.28 Braking Command

6.11 SIMULATION RESULTS

To check the performance of the INS/Vision algorithm it was

interfaced with the flight simulator ‘X-Plane’ and the snapshots at different

stages of the flight are shown in Figures 6.29 to 6.34.

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Figure 6.29 X-plane output - Airplane to align with the runway

Figure 6.30 X-plane output - Airplane at the starting of the glide slope

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Figure 6.31 Airplane after glide slope and flare – X plane output - Front

view

Figure 6.32 Airplane landed properly after glide slope and flare –

X plane output - Front view

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Figure 6.33 X-plane output showing the actual and the estimated path of

the aircraft

Figure 6.34 X-plane output showing the path of the aircraft

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Figures 6.29 to 6.34 clearly demonstrates that the UAV is

successfully and safely landed with the integration algorithm output.

6.12 CONCLUSION

In this chapter, a non-linear 6-DOF model is developed using the

Aerosim blockset in MATLAB software and the longitudinal autopilot with

glide and flare controllers are designed to control the landing of the UAV.

The autonomous landing is demonstrated with the integrated vision and

SDINS data and validated by interfacing the autopilot output with the X-plane

simulation software. The generated control signals are transferred to X-Plane

through UDP from the Simulink and the landing performance was verified

and found to be acceptable.