Unmanned Aerial Vehicles E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 370

Veröffentlichungsjahr: 2013

Beschreibung

This book presents the basic tools required to obtain the dynamical models for aerial vehicles (in the Newtonian or Lagrangian approach). Several control laws are presented for mini-helicopters, quadrotors, mini-blimps, flapping-wing aerial vehicles, planes, etc. Finally, this book has two chapters devoted to embedded control systems and Kalman filters applied for aerial vehicles control and navigation. This book presents the state of the art in the area of UAVs. The aerodynamical models of different configurations are presented in detail as well as the control strategies which are validated in experimental platforms.

Leseprobe

Table of Contents

Chapter 1. Aerodynamic Configurations and Dynamic Models

1.1. Aerodynamic configurations

1.2. Dynamic models

1.3. Bibliography

Chapter 2. Nested Saturation Control for Stabilizing the PVTOL Aircraft

2.1. Introduction

2.2. Bibliographical study

2.3. The PVTOL aircraft model

2.4. Control strategy

2.5. Other control strategies for the stabilization of the PVTOL aircraft

2.6. Experimental results

2.7. Conclusions

2.8. Bibliography

Chapter 3. Two-Rotor VTOL Mini UAV: Design, Modeling and Control

3.1. Introduction

3.2. Dynamic model

3.3. Control strategy

3.4. Experimental setup

3.5. Concluding remarks

3.6. Bibliography

Chapter 4. Autonomous Hovering of a Two-Rotor UAV

4.1. Introduction

4.2. Two-rotor UAV

4.3. Control algorithm design

4.4. Experimental platform

4.5. Conclusion

4.6. Bibliography

Chapter 5. Modeling and Control of a Convertible Plane UAV

5.1. Introduction

5.2. Convertible plane UAV

5.3. Mathematical model

5.4. Controller design

5.5. Embedded system

5.6. Conclusions and future works

5.7. Bibliography

Chapter 6. Control of Different UAVs with Tilting Rotors

6.1. Introduction

6.2. Dynamic model of a flying VTOL vehicle

6.3. Attitude control of a flying VTOL vehicle

6.4. Triple tilting rotor rotorcraft: Delta

6.5. Single tilting rotor rotorcraft: T-Plane

6.6. Concluding remarks

6.7. Bibliography

Chapter 7. Improving Attitude Stabilization of a Quad-Rotor Using Motor Current Feedback

7.1. Introduction

7.2. Brushless DC motor and speed controller

7.3. Quad-rotor

7.4. Control strategy

7.5. System configuration

7.6. Experimental results

7.7. Concluding remarks

7.8. Bibliography

Chapter 8. Robust Control Design Techniques Applied to Mini-Rotorcraft UAV: Simulation and Experimental Results

8.1. Introduction

8.2. Dynamic model

8.3. Problem statement

8.4. Robust control design

8.5. Simulation and experimental results

8.6. Conclusions

8.7. Bibliography

Chapter 9. Hover Stabilization of a Quad-Rotor Using a Single Camera

9.1. Introduction

9.2. Visual servoing

9.3. Camera calibration

9.4. Pose estimation

9.5. Dynamic model and control strategy

9.6. Platform architecture

9.7. Experimental results

9.8. Discussion and conclusions

9.9. Bibliography

Chapter 10. Vision-Based Position Control of a Two-Rotor VTOL Mini UAV

10.1. Introduction

10.2. Position and velocity estimation

10.3. Dynamic model

10.4. Control strategy

10.5. Experimental testbed and results

10.6. Concluding remarks

10.7. Bibliography

Chapter 11. Optic Flow-Based Vision System for Autonomous 3D Localization and Control of Small Aerial Vehicles

11.1. Introduction

11.2. Related work and the proposed 3NKF framework

11.3. Prediction-based algorithm with adaptive patch for accurate and efficient optic flow calculation

11.4. Optic flow interpretation for UAV 3D motion estimation and obstacles detection (SFM problem)

11.5. Aerial platform description and real-time implementation

11.6. 3D flight tests and experimental results

11.7. Conclusion and future work

11.8. Bibliography

Chapter 12. Real-Time Stabilization of an Eight-Rotor UAV Using Stereo Vision and Optical Flow

12.1. Stereo vision

12.2. 3D reconstruction

12.3. Keypoints matching algorithm

12.4. Optical flow-based control

12.5. Eight-rotor UAV

12.6. System concept

12.7. Real-time experiments

12.8. Bibliography

Chapter 13. Three-Dimensional Localization

13.1. Kalman filters

13.2. Robot localization

13.3. Simulations

13.4. Bibliography

Chapter 14. Updated Flight Plan for an Autonomous Aircraft in a Windy Environment

14.1. Introduction

14.2. Modeling

14.3. Updated flight planning

14.4. Updates of the reference trajectories: time optimal problem

14.5. Analysis of the first set of solutions S1

14.6. Conclusions

14.7. Bibliography

List of Authors

Index

First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Adapted and updated from Objets volants miniatures : Modélisation et commande embarquée published 2007 in France by Hermes Science/Lavoisier © LAVOISIER 2007

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK

John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA

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© ISTE Ltd 2010

The rights of Rogelio Lozano to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Unmanned aerial vehicles : embedded control / edited by Rogelio Lozano.

p. cm.

“Adapted from Objets volants miniatures : Modélisation et commande embarquée published 2007.”

Includes bibliographical references and index.

ISBN 978-1-84821-127-8

1. Drone aircraft--Automatic control. 2. Embedded computer systems. I. Lozano, R. (Rogelio), 1954-TL589.5.U56 2010

629.132’6--dc22

2010005983

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-84821-127-8

Chapter 1

Aerodynamic Configurations and Dynamic Models1

1.1. Aerodynamic configurations

In this chapter, we present the aerodynamic configurations commonly used for UAV (unmanned aerial vehicles) control design. Our presentation is focused on mini-vehicles like the airplane (fixed wing models), the flapping wing UAV aircrafts, and the rotorcrafts (rotary wing models). The rotorcrafts will also be classified according to the number of rotors they are equipped with: 1, 2, 3 or 4.

A UAV, also called drone, is a self-descriptive term commonly used to describe military and civil applications of the latest generations of pilotless aircraft. UAVs are defined as aircrafts without the onboard presence of human pilots, used to perform intelligence, surveillance, and reconnaissance missions. The technological objective of UAVs is to serve across the full range of missions cited previously. UAVs present several basic advantages compared to manned systems that include better maneuvrability, lower cost, smaller radar signatures, longer endurance, and minor risk to crew.

Usually, people, and also we ourselves, tend to use the terms airplane and aircraft as synonymous. However, dictionary defines an aircraft as any craft that flies through the air, whether it be an airplane, a helicopter, a missile, a glider, a balloon, a blimp, or any other vehicle that uses the air to generate lift for flight. On the other hand, the term airplane is more specific and refers only to a powered vehicle with fixed wings to generate lift.

Each type of mini-aerial vehicle presents advantages and disadvantages but scenarios are used to represent different types of UAV. For instance, fixed-wing UAVs can easily achieve high efficiency and long flight times compared to other UAVs, consequently they are well suited to operating during required extended loitering times. Nevertheless they are usually unable to enter buildings since they cannot hover or make the tight turns required. In opposition to fixed-wing UAVs, rotary-wing UAVs (like vertical take-off and landing aircrafts — VTOL or short take-off and landing aircrafts — STOL) can easily hover and move in any direction during a shorter flight time [HIR 97]. The last flapping-wing configuration offers the best potential in terms of miniaturization and maneuvrability compared to fixed- and rotary-wing UAVs, but are usually very inferior to fixed- and rotary-wing MAVs (micro air vehicles (MAVs)).

Single rotor configuration

This type of aerodynamic configuration is composed of a single rotor and ailerons to compensate the rotor torque (yaw control input). Since the rotor does not hold swashplate, it has extra ailerons to produce pitch and roll torques.

This type of flying machine is particularly difficult to control even for experienced pilots. The single rotor configuration is mechanically simpler than standard helicopters but it does not have as much control authority. In both cases, a significant amount of energy is used in the anti-torque, i.e. to stop the fuselage turning around the vertical axis. However, due to its mechanical simplicity, this configuration seems more suitable for micro-aircraft than the other configurations. In this type of configuration, we firstly find the planes called 3D or STOL, see Figure 1.1.

Figure 1.1.The 3D plane

Twin rotor configuration

In this type of configuration, we can distinguish those that use one or two swashplates (i.e. collective pitch) and those that use fixed pitch. Among the configurations that use cyclic plates, we can quote the following: the classic helicopter, the tandem helicopter and the coaxial helicopter.

The aircraft configuration with two rotors without swashplate: we find in this category the twin rotors aircraft with ailerons, i.e. two rotors placed on different axes (or in coaxial rotor configuration) and the ailerons orientated in direction of the rotor air flow in order to obtain the required torques to control the vehicle in 3D. Let us note that the rotors can turn in opposite direction or in the same direction. In order to better describe this configuration, we can mention for example the T-Wing of the University of Sydney, see Figure 1.2 [STO 01].

It is also possible to get two counter-rotating rotors on the same axis and ailerons in the air flow direction of the rotors. This last configuration is very compact but difficult to control. Finally, we can have two rotors which tilt on two axes (bi-rotor with counter-rotating propellers in tandem). In this configuration, the propellers do not have a swashplate and the rotors can tilt in two different directions to generate the pitch and the roll torque. The roll torque is obtained by the speed difference between the two rotors.

Figure 1.2.The classic helicopter and the T-Wing aircraft

Multi-rotors

In this category, we find the three-rotor rotorcrafts, the four-rotor rotorcrafts and the rotorcrafts with more than four rotors.

The four-rotor rotorcraft or quad-rotor is the most popular multi-rotor rotorcraft (Figure 1.3). With this type of rotorcraft we can attempt to achieve stable hovering and precise flight by balancing the forces produced by the four rotors. One of the advantages of using a multi-rotor helicopter is the increased payload capacity. Therefore, with higher lift heavy weights can be carried.

Quad-rotors are highly maneuverable and enable vertical take-off and landing, as well as flying into tough conditions to reach specified areas. The main disadvantages are the heavy weight of the aircraft and the high consumption of energy due to extra motors. The quadrotor is superior to the others rotor configurations, from the control authority point of view. Controlled hover and low-speed flight has been successfully demonstrated. However, further improvements are required before demonstrating sustained controlled forward-flight. When internal-combustion engines are used, multiple-rotor configurations have disadvantages compared to single-rotor configurations because of the complexity of the transmission gear.

Figure 1.3.Helicopter with four rotors

Airship

An airship or dirigible is a lighter-than-air aircraft that can be steered and propelled through the air using rudders and propellers or other thrust. In opposition to other aerodynamic aircrafts such as fixed-wing aircrafts and helicopters that produce lift by moving a wing or airfoil through the air, the aerostatic aircrafts (airships, hot air balloons, etc.) stay aloft by filling a large cavity like a balloon with a lifting gas. Major types of airship are non-rigid (or blimps), semi-rigid and rigid. Blimps are small airships without internal skeletons but semi-rigid airships are a bit larger and have any forms of internal support such as a fixed keel.

Airplane

An aeroplane, or airplane, is a kind of aircraft which uses wings in order to generate lift. The body of the plane is called the fuselage. It is usually a long tube shape. The wing surfaces are smooth and their shape helps to push the air over the top of the wing more rapidly than the air travel, as it approaches the wing. As the wing moves, the air flowing over the top has further to go and moves faster than the air underneath the wing. So the pressure of the air above the wing is lower creating a depression which produces the upward lift. The design of the wings determines how fast and high the plane can fly. The wings are called airfoils.

Figure 1.4.The LSC’s airship

The hinged control surfaces are used to steer and control the airplane. The flaps and ailerons are connected to the backside of the wings. The flaps may move backward and forward modifying the surface of the wing area, but also may tilt downward increasing the curve of the wing. There are also slats, located at the top of the wing, which move out to create a larger wing space. It helps to increase a lifting force to the wing at slower speeds like take-off and landing. The ailerons are hinged on the wings and move downward to push the air down and make the wing tilt up. This moves the plane to the side and helps it turn during flight. After landing, the spoilers are used like air brakes to reduce any remaining lift and slow down the airplane.

The tail at the rear of the plane provides stability and the fin is the vertical part of the tail. The rudder at the back of the plane moves left and right to control the left or right movement of the plane. The elevators are placed at the rear of the plane and they can be raised or lowered to change the direction of the plane’s nose. The plane will go up or down, depending on the direction toward which the elevators are moved.

Flapping-wing UAV

A new trend in the UAV community is to take inspiration from flying insects or birds to achieve unprecedented flight capabilities. Biological systems are not only interesting for their smart way of relying on unsteady aerodynamics using flapping wings, they are increasingly inspiring engineers for other aspects such as distributed sensing and acting, sensor fusion and information processing. Birds demonstrate that flapping-wing flight (FWF) is a versatile flight mode, compatible with hovering, forward flight and gliding to save energy. However, design is challenging because aerodynamic efficiency is conditioned by complex movements of the wings and because many interactions exist between morphological (wing area, aspect ratio) and kinematic parameters (flapping frequency, stroke amplitude, wing unfolding) [RAK 04].

Figure 1.5.The plane configuration

Figure 1.6.The dragonfly

1.2. Dynamic models

The dynamic representation of a flying object is of course one of the main goals to be solved before the control strategy development. In this chapter, three approaches to modeling a flying object will be presented (Newtonian, Lagrangian and quaternion approaches). The flying object is considered to be a solid object moving in a 3D environment, submitted to forces and torques applied to the body depending on the type of flying object considered [CAS 05, LOZ 00]. The dynamic model is then used to express and represent the behavior of the system over time. At the end of this chapter, we will present the dynamic model of a helicopter with four rotors.

Figure 1.7.Geometrical representation of a rigid body

1.2.1. Newton-Euler approach

A rigid body is a system of particles in which the distances between the particles do not vary. We find in literature [GOL 80] different ways of presenting rigid body dynamics moving in a 3D space. Newton-Euler and Euler-Lagrange approaches are the most prominent. The Newton-Euler approach is used, firstly, to develop the dynamics body and to represent it in the body frame and then in the inertial frame [KOO 98]. After these manipulations, we express these dynamics using the Euler-Lagrange approach.

Then, the dynamic model of a rigid object evolving in SE(3) and using Newton’s classic equations of motion is

(1.1)

(1.2)

(1.3)

(1.4)

where the vector F represents the gravitational force and all other forces applied to the body relative to the frame . For the helicopter, these forces are produced by the rotation of the rotors. The vector represents the speed of the center of mass of the rigid body relative to the frame , m represents the total mass of the body and g is the constant of gravity. describes the angular velocity and matrix represents the inertia body, both relative to the frame . The matrix represents the skew-symmetric matrix of Ω and is given by

An angular velocity in the body fixed frame is related to the generalized velocities (in the region where the Euler angles are valid) via the standard kinematic relationship

Defining

then

To represent the dynamic model of the rigid body C in the inertial frame , it is necessary to specify the F coordinates in . Thus, we use

(1.5)

Define as the body speed relative to frame . Therefore, the complete model dynamics of a rigid body relative to the inertial frame is given by the following equations:

(1.6)

(1.7)

(1.8)

(1.9)

We expressed the rotation dynamic of the model in the frame because the rotation velocity measurements are always obtained in this frame.

1.2.2. Euler-Lagrange approach

Another way to represent a dynamic model is by using Euler-Lagrange equations of motion. Define the generalized coordinates of the helicopter as

(1.10)

where ξ and η represent the position and orientation of the helicopter with respect to the inertial-fixed frame respectively (see Figure 1.7).

The translational and rotational kinetic energy of the helicopter are

(1.11)

(1.12)

where

We consider that the potential energy of the body consists of the gravitational potential energy

(1.13)

and thus the Lagrangian function is defined as

(1.14)

which satisfies the Euler-Lagrange equation

(1.15)

where FL represents the forces and torques applied to the fuselage.

After some algebraic steps, we can obtain the following standards equations:

(1.16)

where is the symmetric positive definite inertia matrix, is the matrix of centrifugal and Coriolis forces. Finally, is the gravity force vector. Moreover, the matrices M and C verify the passivity property necessary if where P denotes an antisymmetric matrix.

1.2.3. Quaternion approach

The quaternions represent another way of describing the dynamics of a mobile vehicle. This type of representation is used as an alternative to model the attitude dynamics in order to avoid the singularities given by a classic 3D representation (Euler angles or Rodrigues parameters) [CHO 92]. Quaternion is based on a four-parameter representation that gives a more global parametrization; however, it fails for a 180º rotation about some axis [FJE 94]. Then, the minimal number of parameters avoiding any singularity are five.

A unit quaternion is composed of four real numbers (q0,q1,q2,q3) giving a rotation representation with the constraint:

(1.17)

where the parameters belong to a complex number:

(1.18)

from which q0 represents the scalar part and

(1.19)

denotes a vector part. In the context of a dynamic representation for an object’s orientation, in a 3D space, all unit quaternion can form a matrix [ISI 03]

(1.20)

that satisfies , as a consequence SO(3), that can be considered as a rotation matrix. Thus, if we have a rotation matrix ∈ SO(3), then a unit quaternion can be created such that

(1.21)

In [CHO 92, ISI 03], we can find the next relations:

(1.22)

where

(1.23)

(1.24)

(1.25)

with 0 ≤ θ < π.

Using simple calculus, it is easy to show that

(1.26)

where

(1.27)

(1.28)

(1.29)

which is well defined for all positive values of “1 + r11 + r22 + r33”. If this value is negative, then we can use some algorithms which solve this problem [PAI 92, SHE 78].

One goal of the matrix rotation using quaternion is linked to the possibility of expressing the solution of the equation

(1.30)

in terms of the solution of an associated differential equation (1.28)–(1.29), which is defined on the set of unit quaternions [ISI 03]. In order to obtain this equation, we start with the quaternion propagation rule

(1.31)

and using quaternion algebra [CHO 92], we obtain

(1.32)

where

(1.33)

(1.34)

The inverse of (1.32) is given by

(1.35)

Thus, the dynamic model evolving in a 3D space, related with an inertial frame, can be represented by

(1.36)

(1.37)

(1.38)

(1.39)

1.2.4. Example: dynamic model of a quad-rotor rotorcraft

In this section we derive a dynamic model of the quad-rotor helicopter. This model is obtained by representing the aircraft as a solid body evolving in a 3D space and subject to the main thrust and three torques (see Figure 1.8).

Figure 1.8.Helicopter with four-rotor scheme

Define the Lagrangian

Define

(1.40)

where

The model of the full rotorcraft dynamics is obtained from Euler-Lagrange’s equations with external generalized forces

(1.41)

(1.42)

where

The generalized torques are thus

(1.43)

where l is the distance between the motors and the center of gravity, and is the moment produced by motor Mi.

Since the Lagrangian contains no cross terms in the kinetic energy combining with , the Euler-Lagrange equation can be partitioned into dynamics for ξ coordinates and the η coordinates.

Figure 1.9.Forces scheme

The Euler-Lagrange equations for the translation motion are

Finally, we obtain

(1.44)

For the η coordinates, we have

or

Thus, we get

Define the Coriolis/centripetal vector as

we may write

(1.45)

but we can rewrite as

(1.46)

where is referred to as the Coriolis term and contains the gyroscopic and centrifugal terms associated with the η dependence on .

From (1.44), (1.45) and (1.46), we can obtain the dynamic model of the helicopter with four rotors

(1.47)

Note that the η-dynamic can be written in the general form as

(1.48)

where

To obtain the Coriolis matrix, we use

then

However,

where

Differentiating we obtain

On the other hand, we have

thus

where

The Euler-Lagrange equation for the torques is

where

From the above and (1.48), M(η) and C(η) are

(1.49)

and

where

1.3. Bibliography

[CAS 05] CASTILLO P., LOZANO R. and DZUL A., Modelling and Control of Mini-Flying Machines, Springer-Verlag, London, 2005.

[CHO 92] CHOU JACK C. K., “Quaternion kinematic and dynamic differential equations”, IEEE Transactions on Robotics and Automation, vol. 8, no. 1, pp. 53–64, 1992.

[FJE 94] FJELLSTAD O.-E., Control of unmanned underwater vehicles in six degrees of freedom: a quaternion feedback approach, PhD thesis, The Norwegian Institute of Technology, University of Trondheim, November 1994.

[GOL 80] GOLDSTEIN H., Classical Mechanics, Addison-Wesley Publishing, Reading, MA, 2nd ed., 1980.

[HIR 97] HIRSCHBERG M. J., “V/STOL: the first half-century”, Vertiflite, vol. 43, no. 2, pp. 34–54, 1997.

[ISI 03] ISIDORI A., MARCONI L. and SERRANI A., Robust Autonomous Guidance: An Internal Model Approach, Springer-Verlag, Berlin, 2003.

[KOO 98] KOOT J. and SASTRY S., “Output tracking control design of a helicopter model based on approximate linearization”, IEEE Conference on Decision and Control CDC’98, Tampa, USA, pp. 3635–3640, December 1998.

[LOZ 00] LOZANO R., BROGLIATO O., EGELAND B. and MASCHKE B., Dissipative Systems Analysis and Control: Theory and Applications, Springer-Verlag, Berlin, 2000.

[MUR 94] MURRAY R. M., LI Z. and SASTRY S., A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL, 1994.

[PAI 92] PAIELLI R. A., “Global transformation of rotation matrices to Euler parameters”, Journal of Guidance, Control, and Dynamics, vol. 15, no. 5, pp. 1309–1311, 1992.

[RAK 04] RAKOTOMAMONJY T., LE MOING T. and OULADSINE M., “Développement d’un modèle de simulation d’un microdrone à ailes vibrantes”, Conférence Internationale Francophone d’Automatique, Douz, Tunisia, November 2004.

[SCI 00] SCIAVICCO L. and SICILIANO B., Modelling and Control of Robot Manipulators, Springer-Verlag, Berlin, 2nd ed., 2000.

[SHE 78] SHEPPERD S. W., “Quaternion from rotation matrix”, Journal of Guidance and Control, vol. 1, no. 3, pp. 223–224, 1978.

[STO 01] STONE H. and CLARKE G., “The T-Wing: a VTOL UAV for defense and civilian applications”, UAV Australia Conference, Melbourne, Australia, February 2001.

1 Chapter written by Pedro CASTILLO and Alejandro DZUL.

Chapter 2

Nested Saturation Control for Stabilizing the PVTOL Aircraft1

2.1. Introduction

Within the research activities on the aerial vehicle control, we have been particularly interested in the PVTOL aircraft model for several reasons. The planar vertical take-off and landing (PVTOL) aircraft system is indeed based on a simplified real aircraft model which possesses a minimal number of states and inputs but which retains the main features that must be considered when designing control laws for a real aircraft. It also serves as a support for researchers interested in helicopter aerodynamics, because it represents a simplified mathematical model of a helicopter. It then represents a good test-bed for researchers, teachers and students working on flying vehicles.

Let us recall that the PVTOL aircraft system is an underactuated system, since it possesses two inputs u1, u2 and three degrees of freedom [x, y, θ] (see equations (2.1)), and it moves on a plane (see Figure 2.1). The PVTOL aircraft is composed of two independent motors which produce a force and a moment on the vehicle. The main thrust is the sum of each motor thrust. The roll moment is obtained by the difference of motor’s angular velocities.

The control of the PVTOL aircraft represents a real challenge because of its simplicity, its nonlinear nature and its particular properties. Indeed, the system has a nonminimum phase as the linearized system possesses an unstable zero dynamic because of the coupling between the roll moment and the lateral acceleration of the aircraft [HAU 92]. The existing control methodologies for stabilization and trajectory following the PVTOL aircraft are numerous. In section 2.2, a description of the main existing works in the literature (but not exhaustive) is given.

We have proposed several control laws to stabilize the PVTOL aircraft, based on embedded saturations and on Lyapunov theory. In this chapter, we present the PVTOL aircraft model in section 2.3 and we develop in detail one of the stabilization algorithms with stability analysis in section 2.4. In the presented method, the altitude of the aircraft is first stabilized and then the roll angle and the horizontal position converge to zero which is more reliable for experiments. The control law is obtained from the simplified model of the aircraft and is tested on experimental platforms described in section 2.6 and specially built for that purpose. The other techniques that we have proposed are also summarized in section 2.5.

2.2. Bibliographical study

The first control ideas for the PVTOL aircraft have been focused on the extension of the input-output (I-O) linearization procedure developed by Hauser et al. [HAU 92] in 1992. For nonminimum phase systems, the usual state feedback control method using the I-O linearization technique was not applicable. Hauser and his co-authors then used an approximate I-O linearization procedure which resulted in bounded tracking and asymptotic stability for the V/STOL (vertical/short take-off and landing) aircraft.

Martin et al. [MAR 96] in 1996 presented an extension of the result proposed by Hauser et al. [HAU 92]. Their idea was to find a flat output for the system and to split the output tracking problem in two steps. Firstly, they designed a state tracker based on exact linearization using the flat output and secondly, they designed a trajectory generator to feed the state tracker. They thus controlled the tracking output through the flat output. In contrast to the approximate-linearization-based control method proposed by Hauser et al., their control scheme provided output tracking of nonminimum phase flat systems. They have also taken into account, in the design, the coupling between the rolling moment and the lateral acceleration of the aircraft (i.e. ε ≠ 0).

In the same year, Teel [TEE 96] illustrated his nonlinear small gain theorem by stabilizing the PVTOL aircraft system with a perturbation on the input. His theorem provided a formalism in the performance analysis of the control systems with saturations. He established a stabilization algorithm for nonlinear systems known as feedforward systems which included the PVTOL aircraft.

The developed technique by Mazenc and Praly [MAZ 96], combining the Lyapunov analysis with an integration addition in the controller, has been used for stabilizing a VTOL (vertical take-off and landing aircraft) using only the position measure.

In 1999, Lin et al. [LIN 99] studied robust hovering control of the PVTOL using nonlinear state feedback based on optimal control.

Fantoni et al. [FAN 02a] proposed a control algorithm for the PVTOL aircraft using a forwarding technique. This approach has enabled the design of a Lyapunov function insuring asymptotic stability. Other techniques based on linearization have also been described in [FAN 01].

A paper on an internal-model-based approach for the autonomous vertical landing on an oscillating platform has been proposed by Marconi et al. [MAR 02]. They presented an error-feedback dynamic regulator that is robust with respect to uncertainties of the model parameters, and they provided global convergence to the zero-error manifold. The platform movement was created as the superimposition of a fixed number of sinusoidal functions of time, frequency, amplitude and unknown phase.

Saeki et al. [SAE 01] offered a new design method which makes use of the center of oscillation and a two-step linearization. In fact, they designed a controller by applying a linear high gain approximation of backstepping to the model and provided experimental results for a helicopter model with twined rotors.

Olfati-Saber [OLF 02] proposed a configuration stabilization for the VTOL aircraft with a strong input coupling using a smooth static state feedback.

In [FAN 02b, FAN 02c, ZAV 02, ZAV 03, LOZ 04], control strategies taking into account (arbitrary) bounded inputs have been developed, using embedded saturation functions. Some of them obtain global asymptotic stability of the origin in closed-loop. A summary of these techniques is given in section 2.5.

Wood et al. [WOO 05] have developed an extension of the approaches of [SEP 97, OLF 02], with an optimal state feedback, for the case where the aerodynamic forces cannot be neglected, i.e. when the velocities are high. They have considered the case where the PVTOL aircraft represents a model of a “Hovering Rocket” with three degrees of freedom.

Recently, Wood and Cazzolato [WOO 07] proposed a nonlinear control scheme using a feedback law that casts the system into a cascade structure and proved its global stability. Global stabilization was also achieved by Ye et al. [YE 07] through a saturated control technique by previously transforming the PVTOL dynamics into a chain of integrators with nonlinear perturbations. In addition, a nonlinear prediction-based control approach [CHE 08] is proposed for the stabilization problem. The control method is based on partial feedback linearization and optimal trajectory generation to enhance the behavior and the stability of the system’s internal dynamics. Robustness towards parameters uncertainties (especially on ε) has been addressed and shown only through simulations. Tracking and path following controllers have also been developed. Indeed, on the one hand, an open-loop exact tracking for the VTOL aircraft with bounded internal dynamics via a Poincaré map approach was presented in [CON 07]. On the other hand, a path following controller was proposed in [NIE 08] that drives the center of mass of the PVTOL aircraft to the unit circle and makes it traverse the circle in a desired direction. Instead of using time parametrization of the path, they use a nested set stabilization approach.

Some authors have also been interested in designing observers when the full state of the PVTOL system is not completely measurable. Indeed, Do et al. [DO 03] proposed an output feedback tracking controller considering no velocity measurements in the system and Sanchez et al. [SÁN 04] presented a nonlinear observer design for the PVTOL aircraft in order to estimate the angular position of the system.

In the following section, we will recall the dynamic equations of the PVTOL aircraft.

2.3. The PVTOL aircraft model

The dynamics of the PVTOL aircraft, depicted in Figure 2.1, is modeled by the following equations [HAU 92]:

(2.1)

Figure 2.1.PVTOL aircraft front view

For the case when the value of ε is accurately known, several authors have shown that by an appropriate coordinate transformation, we can obtain a representation of the system where such a coupling effect (ε ≠ 0) does not explicitly appear [OLF 02, SAE 01, SET 01]. For instance, Olfati-Saber [OLF 02] proposed the following change of coordinates:

(2.2)

Under these new coordinates, the system dynamics become

(2.3)

(2.4)

(2.5)

(2.6)

which means that either ε has been neglected (ε close to zero), or the system (2.4)–(2.6) is the result of a coordinates transformation (2.2), when ε is known.

2.4. Control strategy

The control strategy that will be developed uses saturation functions proposed by Teel [TEE 92] and is based on a Lyapunov analysis [LOZ 04]. The method is simple and the control law has been tested on an experimental platform.

This section is divided in two parts. In the first part, we are interested in the stabilization of the altitude and in the second part, we propose to control the roll angle and the horizontal displacement x by means of u2.

2.4.1. Control of the vertical displacement y

The vertical displacement will be controlled by forcing the altitude to behave as a linear system. This is done by using the following control strategy:

(2.7)

where 0<p< and ση is a saturation function for some η > 0:

(2.8)

and

(2.9)

where yd is the desired altitude. a1 and a2 are positive constants such that the polynomial s2 + a1s + a2 is stable. Let us assume that after a finite time T2, θ(t) belongs to the interval

(2.10)

for some >0 so that cos θ(t)≠ 0. Introducing (2.7) and (2.9) into (2.3), we obtain for t > T2,

(2.11)

From equations (2.11), it follows that → yd and r1 → 0 when t → ∞. This means that the altitude is stabilized at the origin.

2.4.2. Control of the roll angle 9 and the horizontal displacement x

We now propose u2 to control . The control algorithm will be obtained step by step. The final expression for u2 will be given at the end of this section (see (2.53)). Roughly speaking, for 9 close to zero, the ,subsystem is represented by four integrators in cascade.

2.4.2.1. Boundedness of

In order to establish a bound for we define u2 as

(2.12)

where a > 0 is the desired upper bound for |u2| and z1 will be defined later. Let us propose the following positive function:

(2.13)

Differentiating V1 with respect to time, we obtain

(2.14)

Notice that if for some b > 0 and for all δ > 0 arbitrarily small, then . Therefore, after some finite time T1, we will have

(2.15)

and is then bounded, for t ≥ T1.

Let us assume that b verifies

(2.16)

Then, from (2.11) and (2.12) we obtain for t ≥ T1,

(2.17)

2.4.2.2. Boundedness of θ

In order to establish a bound for θ, we define z1 as

(2.18)

with z3 that will be defined later and

(2.19)

Differentiating z2 and using (2.17)−(2.19), we obtain

(2.20)

We propose the following positive function:

(2.21)

By differentiating V2 with respect to time, we have

(2.22)

Notice that if |z2| >c+ δ for some δ arbitrarily small and for some c > 0, then . Therefore, it follows that after some finite time T2≥ T1, we obtain

(2.23)

From (2.19) we obtain that for t ≥ T2,

(2.24)

Therefore, it follows that there exists a finite time T3 such that for t ≥ T3> T2 we have

(2.25)

If

(2.26)

then , see (2.10), fort≥T2.

Assume that b and c also satisfy the following condition:

(2.27)

Then, in view of (2.23), (2.20) reduces to

(2.28)

for t ≥ T3.

Note that the following inequality holds for |θ| < 1:

(2.29)

We will use the above inequality in the following development.

2.4.2.3. Boundedness of

In order to establish a bound for , let us define z3 as

(2.30)

where z4 is defined as

(2.31)

and z5 will be defined later. From (2.11), (2.19) and (2.28) and the above it follows that

(2.32)

Let us propose the following positive function:

(2.33)

Differentiating V3, we obtain

(2.34)

Since r1 tan(θ) → bn 0 (see (2.9) and (2.11)), there exists a finite time T5> T4 large enough that

and

(2.35)

for one δ arbitrarily small and d > 0. This implies that Therefore, after some finite time T6 > T5, we have

(2.36)

Let us assume that d and c verify

(2.37)

Thus, after a finite time T6,(2.32) reduces to

(2.38)

Note that in view of (2.19), (2.31) and (2.36), it follows that x is bounded.

2.4.2.4. Boundedness of

The last step of the control algorithm is to establish a bound for . We then define z5 as

(2.39)

From (2.11), (2.19), (2.31) and (2.38) we obtain

(2.40)

We propose the following positive function:

(2.41)

Differentiating V4 with respect to time, it follows that

(2.42)

Since r1 tan(θ) → 0, it follows that there exists a finite time T7> T6, large enough that if for some δ arbitrarily small and

(2.43)

then . Therefore, after some finite time T8> T7, we have

(2.44)

So, after time T8,(2.40) reduces to

(2.45)

We can notice that from (2.36), (2.39) and (2.44) it follows that is bounded.

Let us rewrite all the constraints on the parameters a, b, c, d and :

(2.46)

From the above equations, we obtain

(2.47)

2.4.2.5. Convergence of to zero

Therefore, the chosen c and 5 should be small enough to satisfy (2.47). The parameters a, b and d can then be calculated as a function of c as above.

From (2.45) it follows that for a large enough time,

(2.48)

for δ arbitrarily small. From (2.38) and (2.48) we have that for a large enough time,

(2.49)

for δ arbitrarily small. From (2.30) and (2.49), we obtain

(2.50)

Similarly, from (2.28),

(2.51)

and finally for a large enough time and an arbitrarily small δ, from (2.24) and (2.51) we get

(2.52)

(2.53)

The amplitudes of the saturation functions should satisfy the constraints in (2.47).

In section 2.6, we will describe experimental platform and the validation of our control algorithm. First, we will summarize other control strategies similarly developed in order to stabilize the PVTOL aircraft system. They all use saturation functions either in an embedded way or with separated functions. They represent interesting alternatives for the PVTOL aircraft stabilization problem.

2.5. Other control strategies for the stabilization of the PVTOL aircraft

The proposed technique in [PAL 05] is one of the first methods developed for which the objective was its applicability on an experimental platform. In this method, a desired dynamics for the altitude variable y has first been designed using the input u1. By applying this input u1, an expression for the horizontal movement x has been deduced. Taking tan 0 as the artificial input of the horizontal movement, we have arranged so that this input converges to the ideal input so that the x dynamics is equal to the desired dynamics. It has then been shown that the system state remains bounded and converges to constant values that correspond to hovering flight. This control strategy followed the methodology used in [FAN 02b].

In the strategy developed in [FAN 03], a global stabilization algorithm for the PVTOL aircraft with bounded inputs has been presented. In a similar way, the altitude of the aircraft has been first stabilized, followed by the stabilization of the horizontal position and the roll angle. The control strategy has been based on the use of nonlinear combinations of linear saturations functions delimiting both the thrust input and the roll moment by arbitrary saturation limits. The global convergence of the state to the origin has been obtained. Recall that this type of control requires that the altitude of the aircraft is first stabilized, which is more reliable for experimental realizations.

In [ZAV 03], the proposed algorithm also considers inputs u1 and u2 arbitrary bounded and takes the positive nature of the thrust u1 into account. The global convergence of the state to the origin has been proven using Lyapunov theory.