Distributed Finite-Time Tracking control of Multi-Robot Formation Systems with Nonholonomic Constraint 

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More practical convergence characteristics: fixed-time stability

In the last subsection, we devote special attention to the problem of improving the convergence speed of formation systems, which is an important performance index of the controller for distributed coordination of multi-robot systems. How-ever, based on finite-time stability theory, the convergence time instant (settling time) strongly depends on the initial positions of all the robots, which is usually unavailable in reality for single robot. Note that this is a type of global infor-mation for individual robot. Assume that initial positions are bounded but very large, then the convergence speed will even be slower than exponential stability during the rise time of control systems. In order to improve this type of problem, another important concept, fixed-time stability, was introduced by Polyakov (2012). This kind of stability theory can off-line specify the upper bound of convergence time in advance, which is regardless of the initial positions. In other words, the settling time function of fixed-time stability always has a fixed upper bound. This property affords the engineers the possibility to realize more accurate control for the settling time of systems. Motivated by these facts, various results based on this new stability concept have appeared recently, mainly focusing on designing distributed controller with guaranteed fixed upper bound of settling time for the consensus problem of multi-agent systems Fu & Wang (2016); Parsegov et al. (2013b); Zuo (2015). We will contribute two chapters to introduce this practical theory into the development of a novel class of formation tracking controllers for multi-robot systems. Due to the nonlinearity of the multi-robot formation systems and controllers, it’s not easy to directly extend the existing controllers for linear multi-agent systems to the nonholonomic multi-robot formation systems.
On the other hand, preview studies in this thesis only investigate the kinemat-ics of robots. However, the dynamics of systems can provide more information for the motion characteristics of robots in complex environment. In the mean-time, the disturbances are inevitable in practice, we must consider its negative effect on multi-robot formation systems and incorporate the disturbance rejection property into the controllers. For this point, Chapter 4 will provide a satisfactory solution.

Event-triggered mechanism beyond time-triggered com-munication and control

In addition to the convergence rate of the controller, communication and control actuation frequency also play key roles in the whole performance of multi-robot systems. In most of the work on formation control, the assumption that the communication between neighboring robots is time-triggered, has been widely accepted. From a practical point of view, time-triggered information exchange mechanism may be somewhat conservative. From a theoretical perspective, con-tinuous wireless communication will occupy a large amount of limited bandwidth and uninterrupted control input update will lead to excess energy consumption and mechanical wear. In view of these practical issues, event based (also called event-triggered) control methods were revisited. It has been shown in many pa-pers Dimarogonas & Johansson (2009); Dimarogonas et al. (2012a); Fan et al. (2013a, 2015); Nowzari & Cortĺęs (2014); Seyboth et al. (2013a); Tabuada (2007) on multi-agent systems that event based method can indeed effectively improve the above issues through intermittent communication and controller updates. However, substantially less work has been devoted to designing the event based strategy for nonholononmic multi-robot systems. For this reason, the author intends to link the theory with practical application, and devise an applicable event based formation control system to truly reduce communication and control update frequency. Furthermore, most current controllers are digital in real world, then sampled-data control method is introduced naturally to address the real engineering re-quirements Meng & Chen (2013); Postoyan et al. (2015). In general, sampled-data method can be divided into more conservative periodic sampling and more real aperiodic sampling. Compared to event based only method, the introduction of sampled-data method will further reduce the communication frequency and con-trol input update. Moreover, the Zeno-Behavior (a phenomenon that an infinite number of events accumulate in a finite time interval) can be excluded in theory, this is also the core request except for stability in the design of event based for-mation controller. Hence, we will contribute two successive Chapters 5 and 6 to address these problems.

Algebraic graph theory

Before formally introducing the core concepts of algebraic graph theory, we will first explain the reasons to employ it. Considering how information is exchanged among multiple robots, the mutual interaction can be divided into several classes. Here, we just list some commonly used sensing methods, see Figure.1.8. In mode (A), robots exchange information such as absolute position, absolute velocity, heading angle and so on, through wireless communication techniques. In mode (B), a camera is used by robot to obtain the relative distance and orientation information with respect to the target robot. In mode (C), the robot applies the 360◦ Lidar to measure the distance and bearing associated with the target robot. Then how to describe these information exchange manner is one crucial problem in the research of multi-robot formation systems. Popularly, algebraic graph theory is naturally adopted in most of the existing work. We can model the different manners for information exchange by using undirected and directed graphs or/and fixed and switching graphs. Specifically, an undirected graphs can be used to describe bidirectional information flow between neighbouring robots, whereas directed graphs only allow robot to send/receive information to/from the neighbouring robots. In other words, the information transmission is unidi-rectional and asymmetric in directed graphs. We also recall that in a so-called fixed graph, the information links are fixed over time, whereas a switching graphs allow for the change of inter-robot links due to limited sensing range, environ-mental disturbances, and so forth. We here provide an example for an undirected and fixed sensing graph. Consider the robot group (D) in the Figure.1.8, where the robots exchange information based on certain sensing methods of (A), (B) or (C), in order to generate desired collective behavior by means of local interaction. If robot i can mutually exchange information with robot j , an undirected dot line connecting two robots will be used to describe this kind of information link.

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Table of contents :

Acknowledgements
English Abstract
Table of Contents
List of Figures
1 Introduction 
1.1 Background and motivations
1.2 Literature overview
1.2.1 Formation control problems
1.2.2 Finite-time stability of multi-robot formation systems
1.2.3 More practical convergence characteristics: fixed-time stability
1.2.4 Event-triggered mechanism beyond time-triggered communication and control
1.3 Mathematical tools
1.3.1 Algebraic graph theory
1.3.2 Methods
1.3.2.1 Nonsmooth analysis
1.3.2.2 Finite-time stability theory
1.3.2.3 Event-triggered mechanism
1.3.3 Models of nonholonomic mobile robot
1.3.3.1 Nonholonomic constraint
1.3.3.2 Kinematics and dynamics
1.4 Synopsis
1.5 Notations
chapter1 1 I Multi-Robot Formation Systems: Finite-Time Stability
2 Distributed Finite-Time Tracking control of Multi-Robot Formation Systems with Nonholonomic Constraint 
2.1 Problem setup
2.2 Literature overview
2.3 Contributions
2.4 Preliminaries
2.4.1 Variables transformation
2.4.2 Assumptions and lemmas
2.5 Main results
2.5.1 Development of distributed finite-time observer
2.5.2 Design of distributed finite-time controller
2.5.3 From theory to practice
2.6 Numerical example
2.7 Conclusions
chapter2  Distributed Fixed-Time Tracking Control of Multi-Robot Formation Systems with Nonholonomic Constraint 
3.1 Problem setup
3.2 Literature overview
3.3 Contributions
3.4 Main results
3.5 Numerical Example
3.6 Conclusions
chapter3  Robust Fixed-Time Consensus Tracking with Application to Tracking Control of Unicycles Formation 
4.1 Problem setup
4.2 Literature overview
4.3 Contributions
4.4 Preliminaries
4.5 Main results
4.6 Applications
4.6.1 Formation tracking of unicycles with dynamics
4.6.2 From theory to practice
4.7 Simulations
4.7.1 Example 1
4.7.2 Example 2
4.8 Conclusions
chapter4 
II Multi-Robot Formation Systems: Event-Triggered Comiii munication and Control 
5 Distributed Event-Triggered Tracking Control of Multi-Robot Formation Systems with Nonholonomic Constraint 
5.1 Problem setup
5.2 Literature overview
5.3 Contributions
5.4 Preliminaries
5.5 Main results
5.5.1 Distributed formation tracking under fixed topology
5.5.2 Distributed formation tracking under switching topologies
5.5.3 From theory to practice
5.6 Numerical examples
5.6.1 Example 1
5.6.2 Example 2
5.7 Conclusions
chapter5 
6 Distributed Tracking Control of Nonholonomic Multi-Robot Formation Systems via Periodically Event-Triggered Method 
6.1 Problem setup
6.1.1 Mathematic modeling of nonholonomic mobile robot
6.1.2 Original control objectives
6.1.3 Variables transformation
6.2 Literature overview
6.3 Contributions
6.4 Preliminaries
6.5 Main results
6.5.1 Periodic information exchange
6.5.2 Aperiodic information exchange
6.5.3 Convergence analysis of the entire multi-robot systems
6.6 Numerical examples
6.6.1 Example 1: periodic information exchange
6.6.2 Example 2: aperiodic information exchange
6.7 Conclusions
chapter6 
7 Conclusions and Future Work 
7.1 Thesis summary and contributions
7.1.1 Multi-robot formation systems: finite-time stability
7.1.2 Multi-robot formation systems: event-triggered communi- cation and control
7.2 Future work
7.2.1 Theoretical extension of present work
7.2.2 Engineering applications of present work
Conclusions and Future Work
References 

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