Leader follower formation control using on board sensors in noisy environment

List of Figures Figure 1.1 Examples of biological systems exhibiting cooperative Figure 1.2 Applications of formation control 16 Figure 1.3 Block diagram of a mobile robot 17 Figure 1.4

TRAN VIET HONG

Advisor: Professor Lee Suk Gyu

December 2010

Ph.D Thesis

LEADER-FOLLOWER FORMATION CONTROL USING ON-BOARD SENSORS IN NOISY ENVIRONMENT

Advisor: Professor Lee Suk Gyu

Presented in Partial Fulfillment

of the Requirements for the Degree of Doctor of Engineering

December 2010

The Graduate School of Yeungnam University

Department of Electrical Engineering Robotics and Control Major

TRAN VIET HONG

Tran Viet Hong’s Ph.D Thesis is approved by

Committee members

Professor Lee, Ki Dong

Professor Lee, Suk Gyu

Professor Lee, Hai Young

Professor Park, Ju Hyun

Professor Lee, Jeh Won

December 2010

The Graduate School of Yeungnam University

Acknowledgments

To be accepted for a PhD position at Robotics and Control Laboratory, Department of Electrical Engineering, Yeungnam University under the supervision of Professor Lee Suk-Gyu is like a fate Right at the first meeting with him, I felt that this is a good chance for my career After four years, I have had a happy time and learnt a lot I have even more experience of a new academic life and a new culture when following PhD course abroad

Looking back on the time that I spent to study and finish this thesis, I must admit that I enjoyed doing this research very much, even though it was not easy

I am glad to recall many wonderful people who accompanied me in that tough road to assist me in various ways I cannot reach this point by only myself

First and foremost, my very special thank gives to Professor Lee Suk-Gyu for giving me a chance to do research on simultaneous localization and mapping, and multi-robot system His expertise, motivation, enthusiasm, understanding, and patience, taken together, make him a great mentor Thank you for directing

me through my research and for all your help during my stay

Many thanks go to Professors at Yeungnam University, in general, and Department of Electrical Engineering, in particular, who gave me valuable lectures and advices Especially, I would like to express my sincere appreciation to Professor Noh Seok-Kyun, my wife’s supervisor, for his valuable and numerous help to our living in Korea

I would like to thank the members of my thesis committee, Professor Lee Ki-Dong, Professor Lee Hai-Young, Professor Jessie Park Ju-Hyun and Professor Lee Jeh-Won to kindly for their time, interest, and helpful suggestions and comments

I consider myself fortunate to be with all of my past and present colleagues in Robotics and Control Laboratory and other laboratories such as Wee Sung-Gil,

Park Je-Yong, Kim Jong-Uk, Joo Jin-Hwan, Lee Ho-Geun, Dilshat Saitov, Choi Kyung-Sik, Choi Yun-Won, Kim Kyung-Dong, Ryu Hee-Rack, Im Sung-Gyu,

Jo Young-rae, Qu Xiaochuan, Xu Zhiguang, Dai YanYan, Liu Fenggang, Lee Tae-Hee, Kim Hon-Hee, and too many others to put all your names here With great appreciation, I shall acknowledge Hochiminh city University of Technology (Vietnam National University, Hochiminh city), Faculty of Mechanical Engineering for the permission to study abroad

I will never forget four years of living with a solidary and affectionate community of Vietnamese students in Yeungnam University You helped me to overcome the difficulties of living abroad Thanks also go to Vietnamese students in Korea for kind help, encouragement, friendship, and happy times And last but not least, I deeply thank Doctor Park Jung-Tae and Doctor Park Jin-Wook for your patience to take care of the health for my family

Mommy and daddy, please receive my gratitude for sacrificing your lives for us, and providing unconditional love and care Viet Hung, my brother, is a wonderful model of scientific passion that gives me more self-confidence It is lucky to have Nhon, my sister-in-law, here with us Her help is invaluable I am also very grateful to my wife’s family who have dealt with my personal issues

in Vietnam, and encouraged me constantly Finally, my wife is one extraordinary person deserving most of the acknowledgements She is always right beside me with listening ears, loving smiles and gives me the feeling of warmth, hope and peace Especially, she is spending the hard time in Vietnam

to prepare for delivering our first baby

December 2010

Tran Viet Hong

Yeungnam University, South Korea

Contents

CHAPTER 1

1.1 Problem overview 14

1.2 Contributions and outline of the thesis 18

CHAPTER 2 Formation and Formation Control 20 2.1 Introduction to formation 20

2.1.1 Applications of formation 21

2.2 Introduction to formation control 25

2.2.1 Formation control structures 26

2.2.2 Formation control approaches 29

2.2.3 A leader-follower formation example and basic tasks to be controlled 32

2.3 Motivation 34

CHAPTER 3 Stable On-board Sensor Based Formation Control in the Presence of Obstacles 38 3.1 Problem Statement 39

3.1.2 Robot model 40

3.1.3 Formation control framework for SLSF scheme 41

3.2 Proposed Control 43

3.2.1 Formation control framework for TLSF scheme 43

3.2.2 Proposed control law 45

3.3 Obstacle Avoidance Algorithm 49

3.3.2 Flowchart 51

3.3.3 Choose new desired position 53

3.3.4 Stability of obstacle avoidance algorithm 54

3.4 Simulations and Analysis 54

3.4.1 First simulation: small-scale robot team, merits of TLSF scheme 55

3.4.2 Second simulation: big-scale robot team, merits of TLSF scheme 61

3.4.3 Third simulation – formation switching 63

3.4.4 Fourth simulation – single obstacle 65

3.4.5 Fifth simulation – multiple obstacles, schemes switching 68

3.5 Summary and Possible Extensions 70

CHAPTER 4 Wavelet-based Methods to Enhance Sonar Measurement 72 4.1 Introduction 73

4.2 Related works 75

4.2.1 Direct Cross Correlation (CC) 75

4.2.2 Generalized Cross Correlation (GCC) 76

4.2.3 Wavelet-based Generalized Cross Correlation 77

4.3 Enhanced Wavelet-based Methods 79

4.3.2 Improved Wavelet Pre-filter GCC (IWP-GCC) 80

4.3.3 Improved Wavelet-domain Inner Product GCC (IWDIP) 82

4.3.4 Computational complexity comparison 85

4.4 Simulation Results 87

4.4.1 Performance analysis 87

4.4.2 Application in formation control 91

4.5 Summary and Possible Extensions 97

CHAPTER 5 Conclusions and Future Research 99 5.1 Summary of contributions 100

5.2 Future research directions 101

Appendix A Calculation of ϕkm 115

Appendix B Proof of Lyapunov stability 117

List of tables

Table 3.1 Parameters of the first simulation 55

Table 3.2 Displacement errors of control law [GH08] 61

Table 3.3 Displacement errors of control law (3.14) 61

Table 3.4 Parameters of the third simulation 63

Table 3.5 Displacement errors of control law [GH08] in third

Table 3.6 Displacement errors of control law (3.14) in third

Table 3.7 Parameters of the fourth simulation for 4-robot team 67

Table 3.8 Parameters of the fifth simulation 68

Table 4.1 Computational time using Matlab 91

Table 4.2 Error range of each method when SNR < –40dB 92

Table 4.3 Displacement errors of control law (3.14) at various error

Table 4.4 Relation between increment of std

0

d d

∆ and increment of e d 96

List of Figures

Figure 1.1 Examples of biological systems exhibiting cooperative

Figure 1.2 Applications of formation control 16

Figure 1.3 Block diagram of a mobile robot 17

Figure 1.4 Outline of the thesis in corresponding with block diagram

Figure 2.1 Formation of UGVs working in a terrain and a field 22

Figure 2.2 Some applications of formation of UAVs and satellites 24

Figure 2.3 Two applications of formation on and under water 25

Figure 2.4 Centralized and decentralized structures 26

Figure 2.5 Motion and formation process of a group of robots with

three basic tasks: forming, maintaining, and obstacle

Figure 3.1 (a) SLSF scheme in diamond formation

(b) SLSF scheme in zigzag formation

(c) TLSF scheme in diamond formation

Figure 3.4 TLSF scheme with detailed information 45

Figure 3.5 An example of TLSF scheme in obstacle avoidance with

Figure 3.6 An example of TLSF scheme in obstacle avoidance with

Figure 3.7 TLSF scheme control without obstacle avoidance 52

Figure 3.8 TLSF scheme control with obstacle avoidance algorithm 52

Figure 3.9 Choose new desired position for follower robot to avoid

Figure 3.10 Performance of (a) control law [GH08] and

Figure 3.11 Trajectory seen from leader robot R0 of (a) R1 and (b) R2 58

Figure 3.12 Relative distance over time between (a) R1 and R0 and

Figure 3.13 Relative bearing angle over time between (a) R1 and R0,

Figure 3.14 Performance of a team of 5 robots using (a) the control

law [GH08] and (b) the control law (3.14) in the TLSF

Figure 3.15 Performance of a team of 3 robots in switching from

a triangular formation to a line formation using

(a) control law [GH08] and (b) control law (3.14) 64

Figure 3.16 Robot team avoids a single obstacle when switching from

Figure 3.17 Robot team avoids a single obstacle in maintaining

Figure 3.18 Robot team avoids obstacles without changing role of

Figure 3.19 Robot team avoids obstacles with changing roles

(formation ΩF before the first obstacle and formation ΩG

Figure 4.1 Direct cross correlator configuration 75

Figure 4.2 A generalized cross correlator configuration [AH84] 76

Figure 4.3 Wavelet Pre-filter GCC configuration 77

Figure 4.4 Wavelet-domain inner product GCC (WDIP) configuration 78

Figure 4.5 Denoise and recognition comparison 80

Figure 4.6 Original WP-GCC process 80

Figure 4.7 Improved WP-GCC process

(a) Delay prediction in the wavelet domain

(b) Calculate the delay by the cross correlation in the

Figure 4.8 Improved WDIP process

(a) Delay time calculation

Figure 4.9 Block diagram of the simulation process 87

Figure 4.10 The transmitted signal 88

Figure 4.11 The received signal at SNR = –10 dB 89

Figure 4.12 The delay error rate versus the SNR 90

Figure 4.13 Trajectories of 5 robots when e d = 42% 92

Figure 4.14 Trajectory of robot R2 when no error, e d = 30%

List of Abbreviations

CC Cross Correlation

DSP Digital Signal Processing

DWT Discrete Wavelet Transform

EERUF Error Eliminating Rapid Ultra-sonic Firing FFT Fast Fourier transform

GCC Generalized Cross Correlation

IWDIP Improved Wavelet-Domain Inner Product GCC IWP-GCC Improved Wavelet Pre-filter GCC

PHAT Phase Transform

SCOT Smoothed Coherent Transform

SLSF Single Leader – Single Follower

SNR Signal-to-Noise Ratio

Sym8 Symplet whose vanishing moment is 8

TLSF Two Leaders – Single Follower

UAV Unmanned Air Vehicle

UGV Unmanned Ground Vehicle

UUV Unmanned Underwater Vehicle

WDIP Wavelet-Domain Inner Product GCC

WP-GCC Wavelet Pre-filter GCC

Abstract

This thesis addresses three problems of leader-following formation control for multiple non-holonomic mobile robot system: stable control using only on-board sensors, obstacles avoidance, and noise’s effect reduction Specifically, via kinematic analysis, we estimate the leaders’ translational and angular accelerations to build a stable controller whose inputs are only distance and angle information acquired from on-board sensors (do not need to measure velocities of leader robot) In addition, the controller is common for both single leader – single follower (SLSF) and two leaders – single follower (TLSF) schemes in order to have an ability of flexible switching between those schemes Taking full advantage of this ability, we also extend the function of the controller by an obstacle avoidance algorithm to help the formation overcome harassment of static obstacles in the environment Moreover, because

of high error from distance measurement by using ultrasonic sensor, the stability property is not enough for disturbance rejection Therefore, we present two enhanced wavelet-based method as a supplement to ability of reducing effect of noise Although, the controller is such multi-functional and effective,

it is still simple for quick processing, so that the time delay is kept small

Theoretical and simulation analysis show that all the functions of the controller work very well and rapidly The controller can work with any scale of the robot team, but it shows an advantage in large scale where TLSF scheme can suppress the oscillation and damping and increase convergence rate of third, fourth, and succeeding follower robots Even in the presence of obstacles, the formation is kept as close as required form and reform when there is no obstacle In noisy environment, although the effect of noise is not able to be fully rejected, even a small measurement error decrement is valuable to improve the performance of formation control

of multi-robot systems In these challenging application domains, multi-robot systems can often deal with tasks that are difficult, if not impossible, to be accomplished by an individual robot A team of robots may provide redundancy and contribute cooperatively to solve the assigned task, or they may perform the assigned task in a more reliable, faster, or cheaper way beyond what is possible with single robots For instance,

• it is usually more cost-effective to manufacture and deploy a number of cheap robots rather than a single expensive one

• higher number yields better potential for a system resilient to individual robot failures

• smaller robots have obviously better mobility in tight and confined spaces, and

• a group can survey a larger area than an individual robot, even if the latter is equipped with better sensors

The field of cooperative autonomous mobile robotics is still new enough that

no topic area within this domain can be considered mature Some areas have been explored more extensively, however, and the community is beginning to understand how to develop and control certain aspects of multi-robot teams [TEL02]

Therefore, nowadays control and coordination of multi-agent systems has emerged as a topic of major interest [LC+08] This is partly due to broad applications of multi-agent systems in cooperative control of unmanned vehicles, formation control of swarms, where collective motions may emerge from groups of simple individuals through limited interactions The world around us is teeming with examples of this emergent behavior, from a flock of birds to a school of fish, a herd of wildebeest to a swarm of locusts In physics,

a flock can be defined as the coherent motion of a group of self-propelled particles emerging from a single set of interactions between the constituents of that group Some examples of biological systems exhibiting cooperative behaviors are shown in Fig 1.1

Figure 1.1 Examples of biological systems exhibiting cooperative behaviors

Many swarm systems, such as flying wild geese, fighting soldiers, and robots performing a task, always form and maintain a certain kind of formation according to overlapping information structure constraints [XC08] In practice, forming and maintaining desired formations would have great benefits for the system to perceive unknown or partially known environment, to perform its tasks Some applications of formation control are shown in Fig 1.2

Figure 1.2 Applications of formation control

For its wide range of applicability, the formation control problem has stimulated a great deal of research in recent years By formation control we simply mean the problem of controlling the relative position and orientation of the robots in a group while allowing the group to move as a whole This thesis focuses on developing a formation controller for a team of mobile robots (wheeled mobile robots with non-holonomic constraints) moving in a 2D space

The robot in the team has limited communications and uses only on-board sensor for sensing The block diagram of a robot in this research is shown in Fig 1.3 This means that the focus is on the formation task control only and is neither going into the details of the formation protocols for coordinating and organizing the grouped robots to accomplish the formation task, nor collision avoidance

Obstacle avoiding controller

Obstacle avoiding controller

Formation controller

Formation controller

Motors

Sensing the environment

Sensing the leaders

Control signal

On-board sensors

Controller

Figure 1.3 Block diagram of a mobile robot

However, the formation control also considers avoiding static obstacles in environment In addition, we also propose methods to improve the accuracy and calculation time for ultrasonic sensor measurement to provide the accurate sensing data and give them to the controller on time In summary, the thesis deals with the problems in three blocks: on-board sensors, obstacle avoiding controller, and formation controller as shown in Fig 1.3

1.2 Contributions and outline of the thesis

As aforementioned in Section 1.1, the proposed methods to solve the problems

in each block in Fig 1.3 will be presented in each chapter The correspondence between chapters and blocks are shown in Fig 1.4

Obstacle avoiding

controller

Formation controller

Motors

Sensing the

environment

Sensing the leaders

Control signal

On-board sensors

Controller

Chapter 3 Chapter 4

Figure 1.4 Outline of the thesis in corresponding with block diagram of the

robot

In Chapter 2, an overview about formation and formation control is presented

Because the formation issue has been studied for a long time, there is a huge amount of information about it This chapter tries to summarize it briefly, but still sufficiently, from general information such as the importance and applications, to specific information such as research directions in formation control, and up-to-date achievements

Chapter 3 deals with two scenarios

In the first scenario, the robot team is assumed to keep a formation in an

obstacle-free environment by leader-follower scheme Every robot has limited communication with other robots in team, and uses only on-board sensor to sense the environment Due to those limitations, it is very difficult for the robot

to measure velocity of its leader robot A TLSF scheme observer-based controller is proposed to overcome this difficulty by approximation of both translational and angular accelerations of the leader robot via kinematic analysis which have not used by any researcher in the literature This control law stability is proved by Lyapunov stability theory

In the second scenario, the group of robots is considered to be able to meet static obstacles when moving To take full advantage of TLSF scheme controller and to keep the easiness in application, an obstacle avoiding algorithm is added to the proposed formation control algorithm The obstacle avoiding algorithm is simple, yet can keep the stability of the formation control algorithm and show good performance

In the above scenarios, there is no noise in the environment Because the formation control is based mainly on the measurement from on-board sensors, and noise affects much to the accuracy of sensing data, the measurement must assure to be accurate, or the performance of the whole system will be decreased

In addition, while a fusion of sensors is often used, it is required that data should be processed and sent to the controller as quick as possible This is like

a chicken and egg problem The more accurate and free of noise data have

come, the more time to process is required In Chapter 4, the improvement in

accuracy and processing time of ultrasonic sensors will be considered As an improvement over the existing literature, two wavelet-based methods using prediction technique are proposed to help ultrasonic sensors getting distance information quickly and precisely

In the last chapter, Chapter 5, major contributions of the thesis are summarized,

and some possible directions for research in the future are highlighted

CHAPTER 2

Formation and Formation Control

2.1 Introduction to formation

Collective robotics studies the different ways of using autonomous robot teams

to efficiently fulfill predefined missions In collective robotics, several new problems have been lately introduced [RB08], such as:

• consensus [CMA08, Mor05, OFM07, Tsi84]

• rendezvous [AO+99, LMA07]

• cyclic pursuit [MB08, MBF04, PF07, SG06]

• coverage and deployment [CM+04, HMS02]

• formation control [AY+08, DF08, FM04, OEH02]

• connectivity/visibility maintenance [DK08, ME07, ZP08, SNB09]

Among them, formation has received a lot of attention In formation problems,

a team of mobile robots establish and maintain predetermined geometrical shapes by controlling the location of each robot relative to the group while allowing the group to move as a whole Geometric formation can be

established from some predetermined initial positions or even random positions

It is maintained during the group movement The robot group formation may need to avoid obstacles, and sometimes be required to perform formation switching during complex tasks [Che09] It is inspired by the swarming of insects, flocking of birds, colonies of bacteria, etc., and vice versa, research on formation control also helps people to better understand behavior of social animals

The formation problem is interesting because of its advantages:

• carry out tasks that are too difficult or simply inefficient for a single mobile robot to perform alone

• provide redundancy, reconfiguration ability and structure flexibility for the system [CW05]

• reduce the system cost

• increase the robustness and efficiency of the system [CW05]

• better sensor coverage (each team member concentrates its sensor across a portion of the environment, while its partners cover the rest ) [BA98]

• less power consumption

2.1.1 Applications of formation

Due to aforementioned a plenty of advantages, formation has a wide applications include the coordination of multiple mobile robot/unmanned ground vehicles (UGVs), unmanned air/underwater vehicles (UAVs/UUVs), satellites, aircraft and spacecraft

a Formation on the ground

Figure 2.1 Formation of UGVs working in (a) a terrain, and (b) a field

Formation of UGVs is the major application area of formation which covers many domains such as search, surveillance, reconnaissance and cooperative transport, agricultural coverage tasks, security patrols, etc Two examples are shown in Fig 2.1 Some typical specific applications are and not limited to:

• in the rendezvous application, multiple mobile robots simultaneously arrive at a common a priori unknown location determined through team negotiation

• in the axial alignment application, multiple mobile robots collectively align their final positions along a line

• in military missions, a group of autonomous vehicles are required to keep in a specified formation for area coverage and reconnaissance

• in automated highway system, the throughput of the transportation network can be greatly increased if vehicles can form to platoons at a desired velocity while keeping a specified distance between vehicles

• in agriculture, screening solar energy in a greenhouse [FKF09]

• cooperative robot reconnaissance [BA98]

• the first very large-scale artificial swarm with a swarm size of up to 1,000 heterogeneous micro-robots which all are equipped with limited, pre-rational on-board intelligence Such a robot swarm can then be employed for a variety of applications, including micro assembly, biological, medical or cleaning tasks [SS+05]

b Formation in the air

This is the second important application area of formation Both the Air Force and NASA have identified autonomous formation of spacecraft as key technological milestones for the 21st century [BLH00] Over the past decade, numerous formation flying missions have been conceived These missions were driven by scientific and programmatic objectives ranging from sparse-aperture imaging of extra-solar planets to lunar gravitometry A group of researchers at Stanford University, led by Ilan Kroo, suggested one of the most interesting advantages of airlines flying in formation: cutting jet-fuel use They found that the three aircrafts flying in V formation consumed as much as 15 percent less fuel, with a concomitant reduction in CO2 output, and NO2 emissions fell by around 25 percent Aeronautics expert Peter Lissaman has suggested that a formation of 25 birds might enjoy a range increase of 71 percent [Web09] NASA has proposed many formation flying missions

Some typical applications in formation of UAVs, satellites, aircraft and spacecraft are shown in Fig 2.2 and listed below:

• the TechSat-21 concept [Web1] was a revolutionary space architecture

of collaborating clusters of similar, agile, capable microsatellites that could be adapted on-demand to perform a variety of missions

• CLUSTER [Web2] comprises four identical spacecraft launched into large, highly elliptical polar orbits around the Earth to measure subtle changes in the interaction between the Earth and the Sun

• The GRACE is another mission that implements a formation flying technology – measurement of inter-spacecraft range [Web3] The GRACE features two identical satellites in a leader/follower formation (GRACE A and GRACE B) orbiting the Earth on the same orbital plane to generate high-fidelity modeling of Earth’s gravitational field

• PRISMA4 [Web4] is a Swedish-led satellite project with the objective

to develop and qualify new technology necessary for future formation flying science missions

• cooperative fire monitoring with multiple UAVs [CK+06]

• cooperative surveillance with multiple UAVs [BM+06]

• formation flight control [MH01]

• satellite clustering [McI95]

Figure 2.2 Some applications of formation of UAVs and satellites

c Formation on and under water

There are also applications of formation for vehicles working on and under water but not as many as on the ground and in the air [EB+04, Fos94] The most popular applications are school of ships and maritime navigation Some applications are shown in Fig 2.3

Figure 2.3 Two applications of formation on and under water

2.2 Introduction to formation control

Formation control of multiple autonomous vehicles poses significant theoretical and practical challenges

• First, the research objective is to develop a system of subsystems rather than a single system

• Second, the communication bandwidth and connectivity of the team are often limited, and the information exchange among vehicles may be unreliable It is also difficult to decide what to communicate and when and with whom the communication takes place

• Third, arbitration between team goals and individual goals needs to be negotiated

• Fourth, the computational resources of each individual vehicle will always be limited

2.2.1 Formation control structures

There are two main structures of the formation controller to solve the problem

of motion planning: centralized and decentralized There are many other structures besides those two, such as ring, hierarchical, distributed There are also various definitions and categorizations of these structures, but in our opinion, it should be like in Fig 2.4

hierarchical distributed

Figure 2.4 Centralized and decentralized structures

a Centrialized

Centralized systems have been in use for a long time such as in master and slaves based systems A centralized coordination scheme, for example [LF01, TK05], relies on the assumption that each member of the team has the ability to communicate to a central location or share information via a fully connected network

Advantages

• simplicity

:

• easily managed and have no questions of data consistency or coherence

• The centralized control scheme can provide a complete solution

Disadvantages

• tuning constants in several papers (e.g [RK90, RK92, TLK03]), which are crucial to guarantee that the only desired equilibrium points are asymptotic stable and that the other critical points are unstable, are extremely difficult to obtain for practical implementation

:

• require high computational power and are not robust due to the heavy dependence on a single controller [RK90]

• susceptible to bandwidth limitation as well as external disturbances

• not scalable for a team having large number of mobile agents

• may result in a catastrophic failure of the overall system due to its single point of failure

• real-world communication topologies are usually not fully connected

In addition, wireless communication channels are subject to multipath, fading and drop-out Therefore, cooperative control in the presence of real-world communication constraints becomes a significant challenge

b Decentralized

The concept of decentralized system was developed after centralized systems

In decentralized system, one authority controls others directly below it and becomes controlled by the one directly above it In doing so, the central authority can control the entire system The decentralized scheme, see for instance [SI+04, TK05], have almost the exact opposite characteristics as centralized scheme

Advantages

• the superior robustness of decentralized control laws with respect to plant and controller uncertainties and, in particular, unpredictable structural perturbations whereby the subsystems are disconnected and again connected in various ways during operation [Sil91]

:

• require less computational effort than centralized manner

• relatively more scalable to the team size

Disadvantages

• unable or extremely difficult to predict and control the critical points Basically, the closed loop system under a controller designed by the decentralized approach has multiple equilibrium points It is rather difficult to design a controller such that all the equilibrium points except for the desired equilibrium one are unstable/saddle points for a group of many robots

:

A decentralized system is not always better or worse than a centralized system The choice depends entirely on the needs of the application The simplicity of centralized systems makes them easier to manage and control, while decentralized systems grow better and are more resistant to failures

As for scalability, the story is not clear Centralized systems have limited scale, but that limit is easy to understand In contrast, decentralized systems offer the possibility of massive scalability, but in practice that can be very hard to achieve

2.2.2 Formation control approaches

Because formation control is an interested topic, there are many approaches have been proposed by researchers from over the world It is difficult to analyze all of methods because each one has its own merits and demerits We just review some approaches which attracted the interest of researchers most

a Behavioral approach

In the behavior-based control [FM02, LBY03, MB02, Rey87], several desired behaviors (e.g collision avoidance, formation keeping, target seeking) are prescribed for each robot, and the final control is derived from a weighting of the relative importance of each behavior The formation control is obtained from a weighted summation of each behavioral output

Advantages

• This approach allows a convenient generation of control strategies in the face of multiple competing objectives, and allows for an explicit feedback through communication between neighbors

:

Disadvantages

• For schemes based on local behaviors and agent reactions, the group behavior cannot be explicitly defined, and formal mathematical analysis on group stability is generally difficult

:

• With this method, it might be difficult to describe the dynamics of the group and guarantee the stability of the whole system [TNO04]

b Leader-follower approach

In a leader-follower formation control approach, the leader robot moves along a predefined trajectory while the other robots, the followers, are to maintain a desired distance and orientation to it [DF+02, DOK98, GH08, VSS03, Wan91]

In this case, a follower uses the state of its leader(s) to compute their control signals so that predefined separations and orientations are maintained

Disadvantages

• there is no explicit feedback from the followers to the leader, then if the follower is perturbed, the formation cannot be maintained

:

• the leader is a single point of failure for the formation

• require that the full state of the leader be communicated to each member of the formation

• hard to take into account the functioning capabilities of different robots

• does not tolerate leader faults and exhibits poor disturbance rejection properties

c Virtual-structure approach

By treating the entire formation as a single entity, the virtual structure-based approach is comparatively more amenable to mathematical analysis [BLH01,

EHS01, LF97, LT97] Desired trajectories are not assigned to each single robot

desired trajectories for each vehicle in the group to track Some similar ideas based on the perceptive reference frame, the virtual leader, and the formation reference point are given in [KY02, LF01]

Advantages

stability analysis can be easily performed

:

• fairly easy to prescribe the coordinated behavior for the group, and the formation can be maintained very well during the maneuvers, i.e the virtual structure can evolve as a whole in a given direction with some given orientation and maintain a rigid geometric relationship among multiple vehicles

d Graph-based approach

A formation is defined as a directed graph where the vertices represent the individual robots and the labeled edges represent geometric constraints that are maintained by the robots [BK04, Des02, FD02, Olf06] There can be a number

of formation graphs that result in the same geometric arrangement These graphs all belong to the same equivalence class Compare with leader-follower approach, the unconstrained nodes of a formation graph are known as formation leaders

• applicable for only point-mass model of robot

better to use with centralized structure

e Swarming approach

A number of researchers [MRY07, SX+05] have proposed simple heuristic control laws for arranging arbitrarily large numbers of vehicles into regular arrangements based on local information These swarming methods have the advantage that they easily scale to large numbers of vehicles without incurring large communication or computation burdens However, they are typically not fuel-optimal and rarely include provisions guaranteeing collision avoidance

Among those approaches, due to its wide domain of application and easiness to understand and implement, the leader-follower formation control problem has received special attention and has stimulated a great deal of research

2.2.3 A leader-follower formation example and basic tasks to be controlled

We consider a group of non-holonomic mobile robots move along a desired trajectory while maintaining a desired formation In any case and at any time, the group of robots must do three basic tasks: forming, maintaining, and obstacle avoiding Fig 2.5 illustrates an example where a robot team moves

along a road with a requirement to maintain a pyramid formation when the road

is wide enough and a sequential formation when the road is narrow The formation is, therefore, required to switch back and forth between the two configurations In addition, when robots in team meet obstacles, they must avoid and then quickly come back to the form

forming

maintaining

obstacle avoidance

Figure 2.5 Motion and formation process of a group of robots with three

basic tasks: forming, maintaining, and obstacle avoiding

With the leader-follower formation strategy, there is defined a group leader R0

which leads the group bulk motion, and the other robots, labeled as R i (i = 1,

2, n) are the followers that maintain the respective relationships with the group leader R0, in general However, when the number of robots in the group

is large, the relationships of some followers with R0 are hard to define due to the limitation of sensors’ working range Therefore, the definition of whole group relationships is a combination of unit relationships Each unit contains one follower and one leader (SLSF) or two leaders (TLSF) The leaders here are local leaders, which are the robots physically close to the follower for easy sensorial connection Hence, all robots in the group are linked, either directly

or indirectly

Remark 2.1 Ω: R k→ R i = (d ki , φk0 ) is notated for a leader-follower formation

in SLSF scheme where:

R i and R k are leader robot and follower robot, respectively

d ki is the required distance displacement between R i and R k

φk0 is the required bearing angle from the orientation of R k to the d ki -axis

Remark 2.2 Ω: R k→ (R i , R j ) = (d k0 , l k0 , φk0 ) is notated for a leader-follower formation in TLSF scheme where:

R i and R j are major leader robot and minor leader robot, respectively

R k is follower robot

d k0 is the required distance displacements between R i and R k

l k0 is the required distance displacements between R j and R k

φk0 is the required bearing angle from the orientation of R k to the d ki -axis

2.3 Motivation

From the leader-following formation control strategy based on a unicycle model discussed in [DF+02], many other papers, for instance [KX+04, TPK04, VSS03], have also treated formation control of multiple mobile robots with unicycle dynamics A different approach, relying on the neighborhood-based control algorithm, has been used for the formation control law in [OM02, JLM03], and many papers that have followed them; however, this control scheme applies to linear systems only Other researchers have proposed the use

of a second-order model of the robot for SLSF scheme and used feedback, robust and adaptive control methods [LTL07] which analyze the acceleration of the robot in detail, even if the leader has complex trajectories (straight paths, curved paths, circular paths) but the relative orientation between the follower robot and its leader robot cannot be converged to zero

One of the latest research is artificial force based approach [SP10] has many potential real-world applications, but the assumption that agents/members have identical physical properties limits the application of this method

Other recent research, such as [SW+09, WS08], transfers the formation problem to a synchronization control problem, and a synchronous controller is developed to converge both the position and synchronization (formation) errors toward zero in formation switching tasks, but they used a centralized cooperative control scheme which is susceptible to bandwidth limitation as well as external disturbances and hence is not scalable for a team with a large

number of mobile agents The drawback of complexity and resource assumption in aforementioned research is also the disadvantage of using neural

networks as in [CL08]

Another problem in multi-agent networks is that the robots can sense their immediate environment and communicate with their closest neighbors Under

those conditions, problems with time-delay can occur as it takes a while for

information to propagate through the network and reach all agents The delay is

a serious problem in complex system

To reduce the complexity, i.e reject the effect of time-delay, it should reduce the number of measurement data which inputs to the controller, and limit the communications between robots The controller also has to be able to process quickly However, note that there always requires the measurement of the leader’s speed, the reduction as mentioned above will make it difficult to get

this information Because the absolute velocity of the leader is an indirect information which cannot be measured directly by local sensors carried by the

follower robot and it must be estimated by positioning measurements, which

tend to enhance measurement noise dramatically; therefore, the estimation of

absolute speed is difficult to obtain because it is required simultaneously in all the robot’s own speed controllers In addition, it will increase the time-delay

In addition, the measurement of any sensor can causes error due to the unpredicted noise in the environment There are not many research consider effect of noise to formation control performance

The above problems of complexity, limited communications, limited sensing, indirect measurement, short time-delay, and noise reduction motivate the contribution of this research

A novel approach to this problem has recently been developed by [GH08], which presents dynamic feedback controllers that do not require direct measurement of the leader’s speed, but instead a method to predict that speed However, their scheme of SLSF, which theoretically does not depend on the number of robots, is still not scalable for a big group of robots due to the accumulated errors and resulting oscillations In this thesis, a stable leader-following formation control for multiple non-holonomic mobile robot systems working in both SLSF and TLSF schemes using only limited on-board sensor information is proposed

• To reduce the complexity: limit communication and use on-board

sensors

• To overcome the indirect measurement: the controller does not predict

or estimate the exact values of those velocities, but still deals with both translational and angular velocities of the leader We use kinematic analysis to estimate the translational and angular accelerations in order

to compensate the change of those velocities at each time step

• To reduce the time-delay: besides limiting communication, the

calculation of translational acceleration, angular acceleration, and control algorithm is built with simple equations which require only three distance and angular data acquired from on-board sensors In addition, the controller is common for both SLSF and TLSF schemes,

so the unit groups can switch between the two schemes easily without

increasing the complexity of the system

• To reduce the effect of noise: The main controller is simple, yet can

suppress the oscillation and damping in formation of large robot teams

by using TLSF scheme Moreover, two enhanced wavelet-based methods are presented to increase the accuracy of sonar measurement

in high noisy environment while keeping small calculation time

• To avoid obstacles: the controller also include a simple obstacle

avoidance algorithm which fully takes advantage of the ability to flexibly switch between SLSF and TLSF schemes of the formation controller and does not take much time to process

In addition, the control law can be quickly calculated with some basic operations and uses only some information such as distances and angles, which are easily acquired by on-board sensors A novel TLSF scheme is also proposed

to take advantage of the conventional SLSF scheme in order to deal with the unwanted oscillations and the convergence rate of all followers except the first one The algorithm is common to both SLSF and TLSF schemes so that global formation of the local control laws can be formed flexibly and stably This property, in addition, is also useful in obstacle avoidance A simple yet effective algorithm based on the flexible switching back and forth between SLSF scheme and TLSF scheme is added to the formation controller to give it an

ability to avoid obstacles The simulation results prove that the formation is kept as close as required form when a robot avoids obstacles and quickly reform when there is no more obstacles The stability of the controller is maintained

This chapter is organized as follows Section 3.1 gives the mathematical background of the problems studied and Section 3.2 presents the new proposed method along with an examination of its stability and parameter tuning methodology Section 3.3 describes the additional obstacle avoidance algorithm

In Section 3.4, some simulation results are given to show the merits of the proposed control law, with and without obstacles, and this is followed by a summary and conclusions which are provided in Section 3.5

3.1 Problem Statement

This chapter focuses on the formation task control only and is neither going into the details of the formation protocols for coordinating and organizing the grouped robots to accomplish the formation task, nor collision avoidance The environment is also assumed to be obstacle free The problem to be

investigated is formulated as follows: a group of n non-holonomic mobile robots are controlled to follow a group leader R0, which moves along a desired trajectory, and to maintain a desired form implicitly defined by the relative distance and angle between each follower and its leader (in the SLSF scheme)

or the relative distances between a follower and its two leaders, as well as the relative angle with one of those two leaders (in the TLSF scheme) The SLSF and TLSF schemes are shown in Fig 3.1

Figure 3.1 (a) SLSF scheme in diamond formation

(b) SLSF scheme in zigzag formation

(c) TLSF scheme in diamond formation of a four-robot team

As shown in Fig 3.1, in SLSF scheme, the robot R i (i = 1, 2, 3) must keep a relative distance d i and a relative bearing angle ϕi with its leader Robot R2 and

R3 can have many choices of its leader, e.g the leader of R2 is R0 in Fig 3.1(a)

while its leader in Fig 3.1(b) is R1 In TLSF scheme, the follower R2 is required

to follow two leaders at a distance of d20 and d21, respectively, as well as a bearing angle ϕ2 with one of them By changing required relative distances and

a relative bearing angles, it is easily to switch from diamond form to zigzag form and vice versa

3.1.2 Robot model

This section considers a system of n non-holonomic mobile robots labelled as

R0, , R n-1 , where R0 is the global leader The problem of this study deals with wheeled mobile robots with two degrees of freedom and the dynamics of the

ith robot are described by the unicycle model, as follows: