Flight Formation Control E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 407

Veröffentlichungsjahr: 2012

Beschreibung

In the last decade the development and control of Unmanned Aerial Vehicles (UAVs) has attracted a lot of interest. Both researchers and companies have a growing interest in improving this type of vehicle given their many civilian and military applications. This book presents the state of the art in the area of UAV Flight Formation. The coordination and robust consensus approaches are presented in detail as well as formation flight control strategies which are validated in experimental platforms. It aims at helping students and academics alike to better understand what coordination and flight formation control can make possible. Several novel methods are presented: - controllability and observability of multi-agent systems; - robust consensus; - flight formation control; - stability of formations over noisy networks; which generate solutions of guaranteed performance for UAV Flight Formation. Contents 1. Introduction, J.A. Guerrero. 2. Theoretical Preliminaries, J.A. Guerrero. 3. Multiagent Coordination Strategies, J.A. Guerrero, R. Lozano, M.W. Spong, N. Chopra. 4. Robust Control Design for Multiagent Systems with Parametric Uncertainty, J.A. Guerrero, G. Romero. 5. On Adaptive and Robust Controlled Synchronization of Networked Robotic Systems on Strongly Connected Graphs, Y.-C. Liu, N. Chopra. 6. Modeling and Control of Mini UAV, G. Flores Colunga, J.A. Guerrero, J. Escareño, R. Lozano. 7. Flight Formation Control Strategies for Mini UAVs, J.A. Guerrero. 8. Formation Based on Potential Functions, L. García, A. Dzul. 9. Quadrotor Vision-Based Control, J.E. Gomez-Balderas, J.A. Guerrero, S. SALAZAR, R. Lozano, P. Castillo. 10. Toward Vision-Based Coordination of Quadrotor Platoons, L.R. García Carrillo, J.A. Guerrero, R. Lozano. 11. Optimal Guidance for Rotorcraft Platoon Formation Flying in Wind Fields, J.A. Guerrero, Y. Bestaoui, R. Lozano. 12. Impact of Wireless Medium Access Protocol on the Quadrotor Formation Control, J.A. Guerrero, Y. Challal, P. Castillo. 13. MAC Protocol for Wireless Communications, A. Mendez, M. Panduro, O. Elizarraras, D. Covarrubias. 14. Optimization of a Scannable Pattern for Bidimensional Antenna Arrays to Provide Maximum Performance, A. Reyna, M.A. Panduro, A. Mendez.

Leseprobe

Table of Contents

Chapter 1. Introduction

1.1. Motivation

1.2. Historical background

1.3. Flight control

1.4. Flight formation control

1.5. Outline of the book

1.6. Bibliography

Chapter 2. Theoretical Preliminaries

2.1. Passivity

2.2. Graph theory

2.3. Robustness problems

2.4. Bibliography

Chapter 3. Multiagent Coordination Strategies

3.1. Introduction

3.2. Controllability and observability of interconnections

3.3. Formation leader tracking

3.4. Time-varying trajectory tracking

3.5. Linear high-order multiagent consensus

3.6. Conclusion

3.7. Bibliography

Chapter 4. Robust Control Design of Multiagent Systems with Parametric Uncertainty

4.1. Introduction

4.2. Robust control design

4.3. Robust stability analysis

4.4. Robust stability of time-delay systems

4.5. Application to multiagent systems

4.6. Conclusions

4.7. Bibliography

Chapter 5. On Adaptive and Robust Controlled Synchronization of Networked Robotic Systems on Strongly Connected Graphs

5.1. Summary

5.2. Introduction

5.3. Problem formulation

5.4. Adaptive controlled synchronization on strongly connected graphs

5.5. Robust controlled synchronization on strongly connected graph

5.6. Numerical examples

5.7. Conclusions

5.8. Appendix

5.9. Bibliography

Chapter 6. Modeling and Control of Mini UAV

6.1. Introduction

6.2. General model

6.3. Control of a mini tailsitter

6.4. Quad-tilting rotor convertible MAV

6.5. Concluding remarks

6.6. Bibliography

Chapter 7. Flight Formation Control Strategies for Mini UAVs

7.1. Introduction

7.2. Formation geometry

7.3. Communication network

7.4. Dynamic model

7.5. Formation flying control based on coordination

7.6. Formation flying control based on nested saturations

7.7. Trajectory-tracking control

7.8. Simulation results

7.9. Conclusions

7.10. Bibliography

Chapter 8. Formation Based on Potential Functions

8.1. Introduction

8.2. Dynamical model

8.3. Formation control

8.4. Position control

8.5. Simulation results

8.6. Conclusions

8.7. Bibliography

Chapter 9. Quadrotor Vision-Based Control

9.1. Introduction

9.2. Quadrotor dynamic model and control

9.3. Computer vision preliminaries

9.4. Tracking of a visual target

9.5. Tracking of a visual line

9.6. Embedded architecture

9.7. Experimental results

9.8. Conclusions

9.9. Bibliography

Chapter 10. Toward Vision-Based Coordination of Quadrotor Platoons

10.1. Introduction

10.2. Problem statement

10.3. Dynamic model and control of a quadrotor

10.4. Vision-based position estimation

10.5. Coordination position control of two quadrotors

10.6. Architecture of the experimental platforms

10.7. Experimental results

10.8. Conclusions and future work

10.9. Bibliography

Chapter 11. Optimal Guidance for Rotorcraft Platoon Formation Flying in Wind Fields

11.1. Introduction

11.2. Preliminaries

11.3. Path planning

11.4. Quadrotor formation control scheme

11.5. Quadrotor trajectory-tracking control

11.6. Simulation results

11.7. Conclusions and future work

11.8. Bibliography

Chapter 12. Impact of Wireless Medium Access Protocol on the Quadrotor Formation Control

12.1. Introduction

12.2. Multiquadrotor consensus

12.3. Multiagent consensus over wireless networks

12.4. Quadrotor consensus over wireless networks

12.5. Simulation results

12.6. Conclusions and future work

12.7. Bibliography

Chapter 13. MAC Protocol for Wireless Communications

13.1. Introduction

13.2. Protocols of medium access control

13.3. Proposed MAC protocol

13.4. Experimental setup and results

13.5. Conclusions

13.6. Acknowledgments

13.7. Bibliography

Chapter 14. Optimization of a Scannable Pattern for Bidimensional Antenna Arrays to Provide Maximum Performance

14.1. Introduction

14.2. Design of planar antenna arrays

14.3. Design of concentric ring arrays

14.4. Discussions and open problems

14.5. Conclusions

14.6. Acknowledgments

14.7. Bibliography

List of Authors

Index

First published 2012 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

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:

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

Library of Congress Cataloging-in-Publication Data

Flight formation control / edited by Jose A. Guerrero, Rogelio Lozano.p. cm.Includes bibliographical references and index.ISBN 978-1-84821-323-41. Airplanes--Control systems. 2. Airplanes--Automatic control. 3. Drone aircraft--Control systems. 4. Drone aircraft--Automatic control. 5. Stability of airplanes. I. Guerrero, Jose A. (Jose Alfredo), 1977- II. Lozano, R. (Rogelio), 1954-TL589.4.F55 2012629.132'6--dc23

2011052449

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN: 978-1-84821-323-4

Chapter 1

Introduction1

1.1. Motivation

Multiple spacecraft/aircraft flight formation and coordination control are topics that have received a lot of attention over the past decades. Also, the new developments powered by technological advances have spurred a broad interest in autonomous vehicles. The explosion in computation and communication capabilities as well as the advent of miniaturization technologies has increased the interest in a wide variety of research communities, including robotics, communications, automatic control, etc. On the one hand, cooperative and coordinated behavior of a group of unmanned aerial vehicles can cover a larger operational area than a single autonomous vehicle. On the other hand, the lifting of heavy and/or large structures, underway replenishment (fuel, munitions, goods, and personal transfer from one ship to another while under way) and aerial refueling are operations in which coordination is highly required. Thus, the main motivation of this work can be found in the wide variety of applications of multiautonomous vehicle systems such as in the following examples:

Formation flying have been used in survey operations, homeland security, etc. During World War II the groups of B-17 bombers used to fly in a close formation and be escorted by P-51 Mustang fighters also flying in formation to gain better protection as a group. Piloting for many hours in a close formation and under the enemy fire has been proven to be tiring and stressful. Current fighters and bombers fly much faster than those during WWII which may increase the stress and induce nerve-racking experiences on pilots. In Figure 1.1 a group of nine aircraft is shown doing a fly pass during the French Bastille Day Military Parade.

Figure 1.1.Aircraft formation flying

Heavy and/or large load transportation vehicles such as the Helistat and the Skyhook projects that combine features of a blimp and a quadrotor helicopter. The Helistat was planned to be capable of carrying big loads for the US Forest Service. It consisted of a blimp and four Sikorsky Helicopters joined by a metallic structure. All the four helicopters were controlled by a human pilot. The Skyhook is planned to carry up to a 40 ton load with an operational range of 320 km without refueling. Figure 1.2 shows a virtual scene of a Skyhook carrying a heavy load in remote zone.

Figure 1.2.SkyHook heavy lifter vehicle. Courtesy of SkyHook Intl. Inc.

Aerial refueling is a task in which an aircraft (tanker) transfers fuel to another aircraft (receiver). This operation is used when an aircraft needs to take off with a greater payload of weapons, cargo, or personnel. It requires a good coordination between the tanker and the receiver. It is a fact of history that a rescue mission helicopter — UH-60L has made more than 20 attempts to make contact with a tanker with no success. This gives an insight into the difficulty and importance of this type of operation. Figure 1.3 shows a USAF KC-135R Stratotanker, two F-15s and two F-16s, on an aerial refueling operation.

Figure 1.3.Aerial refueling of an F-15 Aircraft. Courtesy of the U.S. Air Force

Spacecraft formation flying is an important project of the National Aeronautics and Space Administration (NASA) in its search for Earth-like planets. Figure 1.4 shows a virtual image of a scheme of multiple spacecraft formation. A spacecraft formation requires a tighter level of precision, slower displacements, and automated control rather than human control.

Figure 1.4.NASA’s formation flying for which the levels of precision are much tighter. Courtesy of NASA

1.2. Historical background

Man’s dream of flying can be traced back to ancient times and illustrated by Daedalus’ wings made of feathers and wax in Greek mythology. However, the idea of a device capable of horizontal and/or vertical flight was first developed in China. They made the first steps toward flight around 400 BC with the Chinese “tops”, a toy made of feathers at the end of a stick which may be considered as one of the first unmanned aerial vehicles (UAV). A UAV can be defined as an aircraft with no onboard human pilot that can be reused and capable of controlled flight, carrying a payload, etc. The UAV has been a feature of aviation history for many years. The origin of the UAV is closely related to cruise missiles; the main difference is that a UAV has been designed to be used in multiple missions and a cruise missile has been designed to destroy itself along with its target. Therefore, a cruise missile cannot be considered as a UAV, while a UAV can be considered as an evolved form of an almost autonomous aircraft.

1.2.1. Aviation history

Throughout time, man-made flying machines have been evolving in many different ways such as balloons, dirigibles, autogyros, helicopters, airplanes, etc. [NEW 04, AIA 09]. A timeline documenting the evolution of aviation is shown in Table 1.1.

Early in 1754, Mikhail Lomonosov built a mechanical spring-based device, shown in Figure 1.5, capable of vertical takeoff and hover for few moments.

Although, man has been flying for centuries, perhaps the most important advances in aviation started in 1900s when the Wright brothers first successfully flew their glider in 1902 (see Figure 1.6). The Wright brothers’ glider was based on the work of Sir George Cayley and other pioneers of 19th Century aviation. Other pioneers of aviation working in parallel were Gustave Whitehead, Samuel P. Langley, Lyman Gilmore, Richard Pearse, among others. Most of the airplanes developed during the 20th Century were based on the successful glider of the Wright brothers.

Another interesting moment in aviation history is the first flight of a manned helicopter, known to have risen from the ground in France in 1907. The Cornu helicopter, shown in Figure 1.7, was an experimental helicopter developed by Paul Cornu, and it was reported to have made several short hops, rising no more than 2 meters.

However, the first successful rotorcraft was not a true helicopter but an autogyro developed by Juan de la Cierva in 1919. Later, Sikorsky introduced several helicopter configurations from the early 1930s to the present (see Figure 1.8).

Table 1.1.Aircraft evolution

Figure 1.5.Lomonosov’s helicopter. Courtesy of Aviastar [WOR 11]

Figure 1.6.Wright brothers first flight. Courtesy of Wikipedia [WIK 11c]

Figure 1.7.Cornu’s hopper

Figure 1.8.Sikorsky VS-300. Courtesy of Aviastar [WOR 11]

During and after World War I (WWI) there was an explosion in helicopter and airplane development all around the world. A more recent type of aircraft is the tailsitter which is an aircraft capable of vertical takeoff and landing (VTOL) as well as being capable of flying as a classic airplane. After WWII, in 1951, Lockheed and Convair were awarded the contract by the US Army and the US Navy to build the XFV (also referred as “Salmon”) and the XFY (also known as “Pogo”), tailsitters. Figure 1.9 shows a Convair XFY-1 tailsitter. This concept of VTOL was abandoned due to many design and operational problems, e.g. the pilot had to look over his shoulder to properly stabilize the aircraft for landing. Also, it is considered that the XFV and the XFY VTOLs did not contribute to the development of modern VTOLs. Nowadays, there are many efforts to improve actual designs of helicopters and airplanes, to make them more stable, more reliable, more comfortable, etc.

Figure 1.9.Convair XFY “Pogo”. Courtesy of Aviastar [WOR 11]

The advent of new technologies and miniaturization have spurred the design and development of manned and unmanned aerial vehicles. Military and civil aviation stepped up the development and production of aircraft and helicopters. In civil applications, man-piloted aircraft systems have been used to transport people and cargo; unmanned aerial vehicles have been used mainly for surveillance. In military applications, UAVs have been used in a wide variety of missions such as target and decoy, reconnaissance, surveillance, etc.

1.2.2. Evolution of UAVs

The history of unmanned aerial vehicles began around 1849. On August 22, the Austrians attacked the Italian city of Venice using unmanned balloons loaded with explosives. The next important advance in this domain happened during and after WWI. In November 1917, the US Army started the project to build the Kettering Bug that first flew in 1918 (see Figure 1.10). This unmanned aircraft was intended to be used as an aerial torpedo against Zeppelins.

The first French UAV was designed, built and tested in 1923. In the 1930s, the UK and the US developed the Radioplane OQ-2, a small teleoperated airplane. The German army, in 1938, started the development of a radio-controlled antiship flying bomb. The German V1 unmanned airplanes, shown in Figure 1.11, and the V2 missiles were flying bombs rather than UAVs. However, the V1 wing has been a base model for target drones.

Figure 1.10.Kettetring Bug. Courtesy of the National Museum of the US Air Force

Figure 1.11.German V1. Courtesy of Wikipedia [WIK 11b]

During the Korean and Vietnam wars, the development of UAVs made important advances. The Ryan Firebee was a well-proven platform for a target drone that led to other missions such as reconnaissance UAV. A modified version of the Ryan Firebee, called the Ryan Model 147 Lightning Bug, was used as a reconnaissance UAV to spy on Vietnam, China, and North Korea in the late 1960s and early 1970s. During the late 1970s and throughout the 1980s, the Israeli Air Force, an aggressive UAV developer, pioneered several important new UAVs that have been integrated into the UAV fleets of many other countries.

In the late 1990s, the American UAV RQ-1A predator, shown in Figure 1.12, offered real-time video imagery without the danger of aircrew losses. The predator RQ-1L was used in the Balkans in 1995, Iraq in 1996, and it proved to be very effective. UAVs have been used especially in risky missions to collect intelligence information. More recently, the trend for battlefield UAVs had been emerging before the war in Afghanistan that began in 2001. An unmanned aerial system roadmap 2005–2030 has been published in [CAM 05].

Figure 1.12.American RQ-1A Predator. Courtesy of Wikipedia [WIK 11a]

1.2.3. UAV classification

UAV classification is usually determined by some criteria or features, e.g. use application, range, altitude, endurance, vehicle type, size, etc. We are interested in classifying UAVs due to their configuration as:

— fixed wing;

— rotary wing;

— free wing;

— tilt wing/rotor;

— tailsitter.

Based on this classification, we note that fixed wing conventional or hovering rotary-wing aircraft systems are the most commonly used vehicles. On the one hand, fixed wing conventional aircrafts have proven reliability, long flight time, and cruise efficiency, but they cannot hover or fly at low speeds. On the other hand, hovering platforms have the operational flexibility of being able to take off vertically, hover and land vertically, but they usually have limitations in forward flight, such as low speed and poor endurance. A relatively unexplored configuration is the tailsitter due to the awkward position of the pilot during takeoff, hover, and landing phases.

1.3. Flight control

Spacecrafts, aircrafts, and UAVs are dynamic systems that can be classified as underactuated mechanical systems. It is known that an underactuated mechanical system has fewer control inputs than degrees of freedom. Thus, the UAV represents an important challenge in automatic control. The UAV flight controller is designed to stabilize the altitude of an aircraft by holding a desired orientation and position. A flight controller also provides the means for an aircraft to navigate by tracking a desired trajectory. Different control techniques have been used to design flight controllers ranging from linear to nonlinear control algorithms.

In [HAU 92], an input—output linearization to stabilize a vertical/short takeoff and landing vehicle has been proposed. An extension and improvement of this work has been made in [MAR 96], in which the main idea was to find a flat output for the system.

In [BEN 96], a comparative analysis between different techniques has been presented. Here, the authors present techniques such as linearization, minimum phase, and sampled methods. Trajectory tracking for a Planar vertical takeoff and landing aircraft (PVTOL) has been also presented.

In [BOU 04], proportional-integral-derivative (PID) and linear-quadratic regulator (LQR) control schemes were used to control a mini rotorcraft with four rotors. A small experimental platform was developed and experimental results are provided. It is noted that the robustness of the control is not guaranteed against uncertainties and/or disturbances. In [BAR 07], a computer-vision-based algorithm is proposed and accomplished using several PID loops for altitude control.

In [MET 02], system identification modeling has been used to develop a parameterized model of a small helicopter. Unmodeled dynamics have been handled using an intuitive approach as in [GAV 01]. Also, robust control techniques have been used to stabilize small helicopters [LAC 03, MAR 02, LIN 99]. In [MET 02], a robust H∞ loop-shaping controller has been developed and validated on an experimental helicopter platform performing a robust hover flight. In [MAR 02], an internal-model-based approach for autonomous landing of a VTOL vehicle on an oscillating landing platform on a ship has been presented. An internal-model-based error-feedback regulator has been developed ensuring the global convergence to the zero error manifold and the robustness against uncertainties affecting the system.

In the last decade, UAV altitude stabilization and autonomous hover using bounded input strategies were developed. Several nonlinear saturated flight controllers have been proposed in [FAN 02], [CAS 05], [LOZ 07], and [LOZ 03]. Nested saturations and saturated state techniques have been successfully implemented on real-time platforms to stabilize the PVTOL aircraft and mini rotorcrafts with four rotors.

Nonlinear methods such as sliding modes and backstepping have been proposed in [MAH 04], [OLF 01], [BOU 05], and [ISI 03]. In [ISI 03], a nonlinear adaptive output regulation and robust stabilization of system in feedforward form has been applied.

1.4. Flight formation control

Cooperative control and multiple spacecraft formation control have been intensively investigated during the past decades. Multiple spacecraft formation flying has been identified by NASA as an enabling technology for 21st Century missions such as terrestrial planet finding and deep space exploration. Multiple aerial, ground, or underwater vehicles working cooperatively or in coordination have important applications. The applications of multiautonomous vehicles is currently progressing in multiple fields, e.g. industrial, military, and in the study of biological systems. Missions for these type of systems include exploration and map building, military operations, traffic control, entertainment, biological systems, transport of heavy or large loads, search and rescue operations, surveillance, and aerospace and ocean exploration. In this section, a discussion of the different approaches that have been proposed in the literature for coordinating multiple robot systems is presented. In Scharf’s survey [SCH 04], five approaches have been identified for spacecraft formation flying: multiple-input and multiple-output (MIMO), leader/follower, virtual structure, cyclic, and behavioral. In following sections, a state-of-the-art on multiple spacecraft formation flying is discussed.

1.4.1. Multiple-input and multiple-output

In the MIMO architecture, the formation problem is treated as a MIMO system where a dynamic model of the formation was used to develop a formation controller. MIMO approaches are described in [LAW 00], [HAD 00], [SMI 02], and [DUN 02]. In [HAD 00], an LQR controller is designed using a minimal state realization of the relative error states. In [DUN 02], a model predictive control was derived to solve the nonlinear and constrained model predictive control (MPC) problem for multiple vehicle formation to a set of equilibria.

1.4.2. Leader/follower

In the leader/follower architecture, one agent is designated as leader, while the others are designated as followers that should track the orientation and position of the leader with some offset. Leader/follower approaches are described in [HAD 98], [DES 98], [CHE 06], and [KRI 06]. In [CHE 06], an input-to-state stability (ISS) concept has been used as a tool to develop a formation control. In this approach, saturated controls enforce ISS of the dynamics, thereby avoiding the problem of dealing with locally asymptotically stable zero dynamics. In [HAD 98], an adaptive control strategy was developed considering the presence of constant, but unknown disturbances.

1.4.3. Virtual structure

The virtual structure approach considers every agent as an element of a larger structure [LEO 01, BEA 99, LAW 99]. Usually, the motion of the virtual structure is done through controlling the individual spacecrafts by tracking their reference trajectories. In [BEA 99], a constellation template was proposed to solve the problem of the coordinated motion of space-based interferometers. A constellation template is a virtual structure that defines the desired position and orientation of each spacecraft within the constellation. In [LAW 99], an adaptive control approach was adopted to design a controller that includes saturation constraints.

1.4.4. Behavior-based control

The behavioral control in [BAL 98] and [ARR 06] is based on the decomposition of the main control goal into tasks or behaviors. This approach also deals with behaviors such as collision avoidance, flock centering, obstacle avoidance, and barycenter. In [BEA 01], [TAN 03a], [TAN 03b], and [OLF 06], the authors have used algebraic graph theory in order to model the information exchange between vehicles. By using this technique, several control strategies have been developed. In [OLF 06], a coordination control composed of a velocity consensus term, a gradient-based term was proposed. The gradient term helps the cohesion of the group, while the velocity consensus term synchronizes the velocities of the agents. An extension of this approach to include navigational feedback has also been presented in [OLF 06]. The navigational term is used to change the orientation of the group or to move the formation to a given reference position. Ren [REN 07a] presents a new strategy for consensus in multiagent systems with a time-varying reference. Several cases are presented, such as: all agents have access to the reference, several agents have access to the reference, etc. The analysis presented assumes that agent dynamics are represented by a first-order integrator. A state of the art in consensus algorithms can be found in [REN 07b].

1.4.5. Passivity-based control

In [LEE 03] and [LEE 06], an analysis of multiple agent coordination using a passivity approach to decompose the system into two passive subsystems is presented. The first subsystem, called “shape”, maintains the formation of the group of agents, while the second subsystem, called “lock”, represents the translational dynamics of the group. In [LEE 06], the convergence of velocity and relative position of the agents via passive decomposition is shown. A bilateral teleoperation approach has been used in [HOK 07] to teleoperate a group of agents. The authors provide results to achieve a bilateral teleoperation one-to-many (i.e. one master and many slaves in a leader/follower architecture). The center of mass is used as a virtual master robot which is used to coordinate the slave robots. Trajectory tracking is also considered using an input to state stability analysis. Consensus algorithms allow the coordination of velocities and/or positions of multiple agents. They have been the object of extensive analysis and development [BEA 01, REN 07b, TAN 03a, TAN 03b]. Trajectory tracking of flocks has recently been studied in [REN 07a] and [HOK 07].

1.5. Outline of the book

Chapter 2: Theoretical Preliminaries

In this chapter, some useful results on passivity, graph theory, and robust control are presented. These results will be used through the first half of the book.

Chapter 3: Multiagent Coordination Strategies

In this chapter, a contribution to controllability and observability of multiagent systems is presented. Several approaches to velocity and position forced consensus are presented. It is shown that formation tracking to a time-varying reference can be achieved by using a feedback control based on the center of mass of the multiagent system.

Chapter 4: Robust Control Design for Multiagent Systems with Parametric Uncertainty

In this chapter, we develop an algorithm for robust control design for dynamical systems assuming parametric uncertainty and control input time delay. A robust absolute stability analysis is presented with application to multiagent systems.

Chapter 5: On Adaptive and Robust Controlled Synchronization of Networked Robotic Systems on Strongly Connected Graphs

In this chapter, a controlled synchronization of networked robotic systems communicating on strongly connected graphs is presented. Adaptive and robust tracking control algorithms are utilized to synchronize heterogeneous robotic systems (with dynamic uncertainty) while following a desired trajectory. The robustness of the control algorithms to constant delays in communication is also demonstrated.

Chapter 6: Modeling and Control of Mini UAV

In this chapter, we present the general dynamic model for mini UAVs considering the aerodynamic moments and forces. The dynamic model of two prototypes are developed, a bi-rotor tailsitter and a convertible quadrotor UAV are studied. The main contribution of this chapter is the modeling of two new designs of mini UAV, a tailsitter using variable pitch propellers and a convertible quadrotor using tilting rotors. The stabilization on vertical mode using linear and nonlinear control laws for stabilizing the attitude and position.

Chapter 7: Flight Formation Control Strategies for Mini UAVs

In this chapter, we introduce two approaches to flight formation control such as nested saturation based nonlinear control and high-order consensus nonlinear control.

Chapter 8: Formation Based on Potential Functions

In this chapter, we address a 2D formation control, using simple potential functions that generate the desired forces and a nested saturation controller to move the vehicles to their goal positions.

Chapter 9: Quadrotor Vision-Based Control

In this chapter, a vision-based control scheme for autonomous hovering and trajectory tracking of a miniature quadrotor is presented. Vanishing points techniques are used to estimate the rotation matrix and translation vector of the camera mounted on the quadrotor. These methods have been tested using real images. The analytic results are supported by experimental tests.

Chapter 10: Toward Vision-Based Coordination of Quadrotor Platoons

This chapter presents a vision-based scheme for position coordination of two camera-equipped quadrotors in hover flight. Applying a homography estimation technique, the aircrafts are capable of estimating their relative position with respect to their corresponding target. Simulations and real-time experiments illustrate the performance of this method.

Chapter 11: Optimal Guidance for Rotorcraft Platoon Formation Flying in Wind Fields

In this chapter, a time-optimal guidance for a platoon of rotorcraft flying in formation through a region of strong winds fields is presented. The main goal is to program the heading for the virtual center of mass in such way as to minimize the flight time between two-way points. The heading program is obtained by using a Zermelo navigation approach.

Chapter 12: Impact of Wireless Medium Access Protocol on the Quadrotor Formation Control

This chapter presents an overview of the medium access protocols’ impact on the average consensus problem over wireless networks for a group of quadrotors. The analysis considers groups of quadrotors communicating over a wireless network considering both directed and undirected graphs of information flow.

Chapter 13: MAC Protocol for Wireless Communications

This chapter deals with the design of a wireless MAC protocol for UAV communication applications. A discussion on the protocols that define and control access to the wireless channel is provided. A new protocol based on carrier sense multiple access—code division multiple access (CSMA—CDMA) is presented.

Chapter 14: Optimization of a Scannable Pattern for Bidimensional Antenna Arrays to Provide Maximum Performance

This chapter presents an antenna array design for multirobot systems. The main objective of this chapter is to show the behavior of radiation for the design of antenna arrays in a uniform rectangular and concentric ring geometry, considering the optimization of a scannable pattern in a wide scanning range.

1.6. Bibliography

[AIA 09] AIAA, Aerospace America, Out of the Past, AIAA, 2009.

[ARR 06] ARRICHIELLO F., CHIAVERINI S., FOSSEN T., “Formation Control of Marine Vessels using the Null-Space-Based Behavioral Control”, LNCIS 336 Group Coordination and Cooperative Control, Springer-Verlag, 2006.

[BAL 98] BALCH T., ARKIN A., “Behavior-based formation control for multirobot teams”, IEEE Transactions on Robotics and Automation, vol. 14, no. 6, pp. 926–939, 1998.

[BAR 07] BARBER D., GRIFFITHS S., MCLAIN T., BEARD R., “Autonomous landing of miniature aerial vehicles”, AIAA Journal of Aerospace computing Information and Communication, vol. 4, no. 5, pp. 770–784, 2007.

[BEA 99] BEARD R., HADAEGH F., “Finite thrust control for satellite formation flying with state constraints”, Proceedings of the IEEE American Control Conference, San Diego, USA, pp. 4383–4387, 1999.

[BEA 01] BEARD R., LAWTON J., HADAEGH F., “A coordination architecture for spacecraft formation control”, IEEE Transactions on Control Systems Technology, vol. 9, no. 6, pp. 777–790, 2001.

[BEN 96] BENVENUTI L., DI GIAMBERARDINO P., FARINA L., “Trajectory tracking for PVTOL aircraft: a comparative analysis”, Proceedings of the 35th CDC, Kobe, Japan, 1996.

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[WIK 11b] WIKIPEDIA, V-1 flying bomb, http://en.wikipedia.org/wiki/V-1_flying_bomb, September 2011.

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1 Chapter written by J.A. GUERRERO.

Chapter 2

Theoretical Preliminaries1

Most of the papers in the literature dealing with multiple agent coordination consider fully actuated agents capable of movement in all directions.

In the following sections, we provide a background on passivity-based control, graph theory, and parametric uncertainty that we will use throughout this book.

2.1. Passivity

Consider a dynamical system represented by the state space model:

[2.1]

[2.2]

where ƒ is locally Lipschitz, h is continuous, and the system has the same number of inputs and outputs.

In this thesis, we limit ourselves to control affine passive systems. To set the background for what follows, consider the following affine nonlinear system:

[2.3]

LEMMA 2.1.– Consider the nonlinear system ∑. The following statements are equivalent.

[2.4]

[2.5]

[2.6]

2.2. Graph theory

A natural way to analyze the relationship and communication between agents is using either directed or undirected graphs. A multiagent dynamic system can be modeled as a group of dynamical systems which has an information exchange topology represented by information graphs. Then, we introduce some basic definitions and results from graph theory [GOD 07].

DEFINITION 2.1.– A graphis a pair consisting of a set of nodes together with their interconnections on . Each pair (n1,n2) is called an edge

If a pair (n1,n2) is an edge then it is said that n2 is a neighbor of n1. We use directed graphs or digraphs in order to model asymmetric relations between agents or nodes. A digraph consists of a node set and an edge set where every directed edge is an ordered pair of distinct nodes, i.e. the ith node can get information from the jth node but not necessarily vice versa. Directed edges are represented by arrows indicating the direction of the edge. An undirected graph is a graph where the ith and the jth nodes can get information from each other. In this case, a mechanical approach can be adopted since every node (agent) in the graph can be considered as a mass M and the edges can be considered as springs and dampers when states and its first derivative are shared with their neighbors. A more complicated problem is to consider unidirectional communication between agents. We then introduce some important definitions on graph theory.

DEFINITION 2.2.– A subgraphof a graphis such thatand

When it is said that is a spanning graph of .

The valency of a node ni is the number of neighbors of ni. When working with directed graphs, the in-valency of ni is the number of edges ending on ni and the out-valency is defined analogously.

DEFINITION 2.3.– Letbe a directed graph, it is said to be balanced, if its in-valency (number of communication links arriving at the node) is equal to its out-valency (number of communication links leaving the node) for all .

A graph is connected if for every pair (n1, n2) of distinct vertices there is a path from n1 to n2. A connected graph allows the communication between all agents through the network. A digraph is said to be strongly connected if any two nodes can be joined by a path and is weakly connected if any two nodes can be joined by a weak path.

A cycle is a graph where every node has exactly two neighbors. Then, a cycle in a graph is a subgraph that is a cycle. Similarly, an acyclic graph is a graph with no cycles. A connected acyclic graph is called a tree and a spanning subgraph with no cycles is called a spanning tree.

2.3. Robustness problems

Many modern control problems involve uncertainty which refers to the difference between a real system and the model that describes its behavior. The robust control problem appeared with the objective of reducing the difference between the dynamic behavior of real systems and their mathematical models. Robust control considers diverse approaches; however, for linear time-invariant systems, there are two fundamental approaches: the robust control considering dynamic uncertainty and the robust control considering parametric uncertainty. Looking back at the literature, control of dynamic systems involving dynamic or parametric uncertainty has been studied in recent decades. Much of the work dealing with dynamic uncertainty can be found in [HEL 98], [ZHO 96], and [GRE 95]. Also, much of the work on control of dynamic systems considering parametric uncertainty can be found in [BAR 94] and [ACK 93]. Although we are interested in dynamic systems considering parametric uncertainty, we first give a brief introduction to dynamic systems involving dynamic uncertainty.

– Dynamic uncertainty: For dynamical systems involving dynamic uncertainty represented in input—output form, the analysis is done in the frequency domain.There are two important approaches for the analysis of this type of system: the analysis of dynamical systems considering additive uncertainty and the analysis of dynamical system involving multiplicative uncertainty. These approaches are shown in Figures 2.1 and 2.2, respectively,

Figure 2.1.Dynamical system with additive uncertainty

Figure 2.2.Dynamical system with multiplicative uncertainty

where G represents the mathematical model of the physical process, K is the controller acting on the process, Δ represents the dynamic uncertainty, and U(s) and Y (s) represent the system’s input and output, respectively.

Generally, when dynamic uncertainty is considered, the problem statement is as follows: to find a controller K that satisfies certain performance specifications for the family of plants obtained from considering the uncertainty Δ in the system as well as to optimize in the sense that it guarantees the desired performance for the biggest possible uncertainty, measured in function of its infinity norm. This problem is known as H∞ control problem (see [ROM 97] and [GRE 95]).

– Parametric uncertainty: The uncertainty in a dynamical system can also be expressed as uncertainty in the system parameters that define its structure as follows:

[2.7]

where the uncertainty is considered in the matrices A(q), b(q), and c(q).

For dynamic systems involving parametric uncertainty, the analysis can also be done in the frequency domain. In this case, the analysis consists of determining polynomial function properties which can be obtained from the characteristic equation of the dynamical system represented in input—output form. The robust stability property is determined in the function of the family of polynomials expressed in the following form:

[2.8]

where is the vector of uncertainty parameters in the dynamical system, which is reflected in the coefficients of the polynomial through ai(q). The parametric uncertainty is classified in diverse forms. In the following section, we will describe the different classes of parametric uncertainty.

2.3.1. Representation of the parametric uncertainty

Parametric uncertainty on a dynamical system is represented as a vector, q, in which every element represents a parameter whose nominal value is considered uncertain. The vector of uncertainty is delimited by the uncertainty conditions such that it forms a region in a vectorial space. There are three types of parametric uncertainty: box, sphere, and diamond [ROM 97].

DEFINITION 2.4.– The uncertainty type “box” is defined as:

[2.9]

DEFINITION 2.5.– Uncertainty type “sphere” is defined as:

[2.10]

DEFINITION 2.6.– Uncertainty type “diamond” is defined as:

[2.11]

We are interested in the analysis of dynamical systems involving box-type parametric uncertainty. Depending on the quantity of uncertain parameters, we call uncertainty box a region formed by two uncertain parameters, we call uncertainty cube a region formed by three uncertain parameters, and finally, we call uncertainty hypercube a region formed by more than three uncertain parameters.

Generally, uncertainty is defined as a set and is important for this set to be a connected set such that we are able to apply the robust stability results that will be presented later in this work. The definition of a connected set is as follows.

DEFINITION 2.7.– A setis said to be connected, if the following condition is satisfied: given any two pointsthere exists a continuous function such that

This definition can be graphically represented by Figure 2.3.

Figure 2.3.Connected set in

2.3.2. Families of polynomials

The union of an uncertain polynomial and its set of uncertainty is called family of polynomials, which is denoted by:

where p(s, q) is the uncertain polynomial and is the set of all possible values that the uncertainty can take, usually called uncertainty box.

Basically, a family of polynomials is the union of an uncertain polynomial and its uncertainty box that results in a set of polynomials with the same structure called family of polynomials. This family of polynomials can be associated with the stability property as follows [BAR 94].

DEFINITION 2.8.– A family of polynomialsis said to be robustly stable if, for allis stable, i.e. for allall the roots of p(s, q) are placed on left half plane of the complex plane.

The previous definition implies that, if we have a family of polynomials, we have to make all the possible combinations of the uncertain polynomial with the uncertainty to show that every element of the family of polynomials has its roots in the left half plane of the complex plane. This means that we have to evaluate an infinite number of polynomials to show that the family of polynomials is robustly stable. It is impossible to evaluate an infinite number of polynomials; hence, Kharitonov’s result, see [KHA 79], allows us to evaluate the robust stability property of a family of polynomials by evaluating four uncertain polynomials that are created based on the Kharitonov’s theorem. In [KHA 79], it is possible to find the demonstration of the theorem as well as the procedure to evaluate families of polynomials; however, Kharitonov’s theorem can be used only on families of interval polynomials which are known to have independent uncertainty structure. However, the uncertainty structure in typical applications is more complicated than the independent uncertainty structure. Then, the hierarchy of the structure of uncertainty is as follows:

where independent uncertainty structure, affine uncertainty structure, multilineal uncertainty structure, and polynomic uncertainty structure.

In [BAR 94], it is possible to find the definitions of the different uncertainty structures. In this section, we include the definition of the polynomic uncertainty structure because this type of uncertainty is considered in our problem, and it is defined as follows.

DEFINITION 2.9.– It is said thathas polynomic uncertainty structure if each coefficient function ai(q) is a multivariable polynomial in the components ofq.

2.4. Bibliography

[ACK 93] ACKERMANN J., Robust Control, Springer-Verlag, 1993.

[BAR 94] BARMISH B., New Tools for Robustness of Linear Systems, Macmillan, 1994.

[BRO 00] BROGLIATO B., LOZANO R., MASCHKE B., EGELAND O., Dissipative Systems Analsis and Control, Springer, 2000.

[GOD 07] GODSIL C., ROYLE G., Algebraic Graph Theory, Springer, 2007.

[GRE 95] GREEN M., LINEBEER D., Linear Robust Control, Prentice Hall, 1995.

[HEL 98] HELTON J. W., MERINO O., Classical Control Using H∞Methods, SIAM, 1998.

[KHA 79] KHARITONOV V., “Asymptotic stability of an equilibrium point position of a family of systems of linear differential equations”, Differential Equations, vol. 14, pp. 1483–1485, 1979.

[ROM 97] ROMERO G., Análisis de Estabilidad Robusta para Sistemas Dinámicos con Retardo, PhD Thesis, FIME-UANL, June 1997.

[ZHO 96] ZHOU K., DOYLE J., GLOVER K., Robust and Optimal Control, Prentice Hall, 1996.

1 Chapter written by J.A. GUERRERO.