Decentralized Cooperative Control Methods for Multiple Mobile Robotic Vehicles

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Decentralized Cooperative Control

Methods for Multiple Mobile Robotic

Vehicles

Ravinder Praveen Kumar Jain

Supervisors: Dr. António Pedro Aguiar

Dr. João Borges de Sousa

Department of Electrical and Computer Engineering

Faculty of Engineering, University of Porto

This dissertation is submitted for the degree of

Doctor of Philosophy

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The work presented in this thesis was funded by the European Union’s Horizon 2020 research and innovation programme MarineUAS under the Marie Sklodowska-Curie grant agreement No 642153 and project IMPROVE-POCI-01-0145-FEDER-031823 - funded by FEDER funds through COMPETE2020 - POCI and SYSTEC - Research Center for Systems and Technologies - Research Unit (UID/EEA/00147/2019) funded by FCT/MCTES (PIDDAC). First and foremost, my sincere gratitude to my thesis supervisors Prof. António Pedro Aguiar and Prof. João Borges de Sousa - who, needless to say have played a vital role in shaping not just the work presented in this thesis, but also my career as a researcher. My sincere gratitude to Prof. Tor Arne Johansen, Center for Autonomous Marine Operations and Systems, Norwegian University of Science and Technology (NTNU) for hosting me as a visiting PhD researcher at NTNU and enabling me to work on an interesting problem that forms an important part of the thesis. Special thanks to Andrea Alessandretti, and all the people who influenced my work through collaborations, discussions and help in conducting the experimental results presented in this thesis. I would also like to thank Ana Cristina, Paulo Lopes, Catarina Morais and Sara Costa for all the help, allowing me to focus on my research. Sincere gratitude to the members of Underwater Systems and Technologies Laboratory, Jose Pinto in particular for their assistance in experiments. Finally, my acknowledgments to everyone who directly or indirectly influenced me and my work during the PhD training. Most importantly, thanks to those that stood by me at various stages in my life. I fail by words to express my gratitude.

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Cooperative control of Multi-Robot Systems (MRS) have received immense attention over the past decade, with envisaged applications spanning search and tracking, wildlife monitoring, cooperative estimation, cooperative transport, ocean sampling, automated highway systems, crowd and social movement monitoring. The advent of robotic systems such as fixed-wing Unmanned Aerial Vehicles, Autonomous Surface/Ground/Underwater vehicles, multi-rotors and their increasing use in these applications, demand investigation of advanced cooperative control strategies that goes well beyond static formations, while taking into account the general non-linear dynamics of robotic systems. Consequently, the research focus needs a shift wherein the cooperative control paradigm pushes the envelope and enables cooperative maneuvers or formations of the robotic vehicles that are dynamic in nature. This thesis aims to achieve the aforementioned goal by developing a cooperative control framework that allows execution of multitude of dynamic maneuvers, which was not previously possible with existing methods.

Specifically, a novel decentralized cooperative control method termed the Cooperative Moving Path Following (CMPF) framework is developed that is composed of two intercon-nected subsystems, namely i) Moving Path Following (MPF) system that enables the robotic vehicle to follow a priori specified, geometric path that is moving with respect to a frame of reference, and ii) Dynamic Event-triggered Cooperative Control system that achieves cooperation between the robotic vehicles wherein, a dynamic event-triggered communication scheme is developed that aims to reduce the instances of information exchange between the robotic vehicles. Formal stability and convergence guarantees are provided using the concept of Input-to-State Stability for the overall CMPF system. The advantage of the proposed CMPF framework lies in its flexibility to choose generic paths that lead to various multi-robot formations and maneuvers, as opposed to the existing methods in the literature that is specifically designed for rigid formations or maneuvers with circular patterns. Furthermore, it is shown that the existing circular formation control and rigid formation control objectives can be recast in the CMPF framework, thereby generalizing its utility.

The development of CMPF framework led to the development of different MPF control laws for robotic vehicles using the concepts of Geometric control and Nonlinear Model

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Predictive Control that takes into account the kinematic constraints of the robotic vehicle. The proposed MPF control laws seamlessly stitch into the dynamic event-triggered control and communication approaches at the cooperative control level to form a event-triggered CMPF framework. Through extensive simulation and experimental validations in the marine environment, it is shown that the cooperation between the robotic vehicles can be achieved with significant reduction of communication instances between the robotic vehicles when compared to the traditional periodic transmission methods. The CMPF method assume that the information regarding the motion of the target is known to the robotic vehicles. This assumption is relaxed by developing estimation strategies that enable a group of robotic vehicles to accurately estimate the position of an acoustic source using range based measure-ments. Specifically, a globally stable, eXogenous Kalman Filter (XKF) is developed and it is shown to outperform the Extended Kalman Filter (EKF) based estimation methods through experiments.

Keywords – Moving Path Following, Cooperative Control, Event-triggered Control, Model Predictive Control, Geometric Control, Source Localization, Kalman Filtering, Marine robots.

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O controlo cooperativo de Sistemas Multi-Robóticos tem recebido uma atenção especial na última década, nomeadamente em aplicações que abrangem desde procura e salvamento, passando por monitorização da vida selvagem, estimação cooperativa, transporte coopera-tivo, amostragem dos oceanos, sistemas rodoviários automatizados, até monitoramento de multidões e movimentos sociais. Isto deve-se em particular ao desenvolvimento dos veículos aéreos não tripulados de asa fixa e multi-rotores, veículos de superfície/solo/submarinos autónomos e o crescente uso em aplicações. No entanto, estas aplicações exigem cada vez mais uma investigação de métodos avançados de controlo cooperativo que vão muito além da utilização de robôs em formações estáticas, e que terão que ter em consideração a dinâmica geral não linear dos sistemas robóticos. Para esse efeito, é necessário investigar e desenvolver novos algoritmos de controlo cooperativo de movimento que permitem manobras coopera-tivas ou formações dos veículos robóticos que são dinâmicos por natureza. Esta tese visa atingir precisamente este objetivo, propondo uma arquitetura de controlo cooperativo que per-mita a execução de múltiplas manobras dinâmicas, o que não era possível anteriormente com os métodos existentes. Mais especificamente, a tese propõem um novo método de controlo cooperativo descentralizado denominado CMPF (“Cooperative Moving Path Following”) que é composto por dois subsistemas interconectados: i) “Moving Path Following”(MPF) que permite ao veículo robótico seguir um caminho geométrico desejado e que está a mover-se em relação a um referencial de referência, e ii) Sistema de controlo cooperativo acionado por eventos dinâmicos e que tem por objectivo a sincronização entre os veículos robóticos utilizando uma estratégia dinâmica de comunicação por eventos que visa reduzir as trocas de informação entre os veículos robóticos. Garantias formais de estabilidade e convergência são estabelecidas utilizando o conceito de Estabilidade de Entrada - Estado para o sistema global CMPF. A vantagem da arquitetura de CMPF proposta reside na sua flexibilidade de escolher caminhos genéricos que levam a várias formações e manobras multi-robôs, ao contrário dos métodos existentes na literatura que são especificamente projetados para formações rígidas ou manobras com padrões circulares. Além disso, é demonstrado que os objetivos existentes de controlo de formação circular e controlo de formação rígida podem ser reformulados na estrutura do CMPF, generalizando assim sua utilidade. O desenvolvimento do CMPF

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levou ao estudo de diferentes leis de controlo de MPF para veículos robóticos utilizando os conceitos de controlo geométrico e controlo preditivo não-linear que tem em consider-ação as restrições cinemáticas dos veículos robóticos. Estas leis de controlo são facilmente integradas em abordagens dinâmicas de controlo e comunicação acionadas por eventos ao nível da coordenação para formar uma estrutura CMPF acionada por eventos. Para mais, foi possível constatar através de extensas simulações por computador e validações experimentais em ambiente marinho, que a cooperação entre os veículos robóticos pode ser alcançada com significativa redução de instâncias de comunicação entre os veículos robóticos quando comparados aos métodos tradicionais de transmissão periódica. O método CMPF assume que as informações sobre o movimento do alvo são conhecidas pelos veículos robóticos. Essa suposição é retirada em certas aplicações com a utilização de estratégias de estimação presentes na tese que permitem que um grupo de veículos submarinos robóticos determinem a posição de uma fonte acústica utilizando medidas de distancia. Especificamente, um filtro globalmente estável do tipo “eXogenous Kalman Filter” (XKF) é desenvolvido, onde se atesta utilizando resultados experimentais que este é muito superior em performance em relação aos métodos de estimativa baseados nos filtro de Kalman estendidos.

Palavras-chave - Seguimento do caminho em movimento, controlo cooperativo, controlo acionado por evento, controlo preditivo, controlo geométrico, localização, filtragem de Kalman, robôs marinhos.

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List of figures

1.1 Autonomous Oceanographic Sampling Network (Source: [1]) . . . 5

1.2 Underwater Acoustic Sensor Network (Source: [2]) . . . 6

1.3 Autonomous Mobile Robotic Systems . . . 7

2.1 Coordinate frames for Moving Path Following . . . 24

2.2 Position of the target maneuvering in 3D and an UAV following a lemniscate path around the target . . . 32

2.3 Plot of (a) Target relative desired path and UAV path, (b) MPF error ˜p, and (c) Attitude error function Ψ( ˜R) . . . 33

3.1 Moving Path Following framework . . . 39

3.2 Position of the robotic vehicle, target and the reference moving path for the two scenarios . . . 49

3.3 Error variable and the computed control inputs for the circumnavigation scenario . . . 50

4.1 Three AUVs used in experiments. . . 58

4.2 Event-based CPF framework . . . 59

4.3 Vehicle positions and path following error for the AUVs in straight line formation . . . 68

4.4 Results of Event-based CPF for three AUVs in straight line formation . . . 69

4.5 Results of Event-based CPF for three AUVs in circular formation. . . 71

5.1 Coordinate frames for MPF . . . 82

5.2 Position of the robotic vehicle using MPF controller . . . 95

5.3 CMPF with dynamic event-triggered control and communication for cooper-ative target tracking scenario . . . 97

5.4 Effect of the MPF error correction signal gi(t) on the performance of MPF and CMPF with DETC . . . 98

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5.5 Rigid formation using CMPF with DETC . . . 98

5.6 Effect of tuning parameters of DETC on the number of events . . . 101

5.7 Effect of βi on performance of CMPF: (a). Mean-square value of sig-nal 1_n∑Ni=1∑j∈Ni( ¯γi− ¯γj) 2_{, (b). Mean-square value of MPF error signal} 1 n∑ N i=1e′iei, where n is number of samples of the signals obtained during simulation. . . 102

5.8 Effect of βi= 0.1 on the signal ∑j∈Ni( ¯γi− ¯γj) . . . 103

6.1 Mobile acquatic telemetry system . . . 108

6.2 Extended Kalman Filter architecture . . . 109

6.3 eXogenous Kalman Filter architecture . . . 110

6.4 Source localization scenario with receivers R1 to R4 . . . 111

6.5 (a) Representative image of the Themla Biotel fish-tag of 18 mm diameter and 104 mm length with a mass of 43 gram. (b) Thelma Biotel TBR-700 acoustic receivers of 75 mm diameter and 230 mm length with a mass of 1140 gram. . . 118

6.6 Results for estimation of the fish-tag using XKF and EKF with m = 3 re-ceivers and depth measurement. . . 120

6.7 Position estimates of the fish-tag for an arbitrary initialization of the estimator states . . . 122

7.1 Performance of the Nonlinear Model Predictive controller for target estima-tion and tracking . . . 133

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Chapter 1 Introduction

MULTI-ROBOTSYSTEMS- A SYSTEM CONSISTING OF MULTIPLE ROBOTS! Tremendous

advances have been made in this direction over the last decades. The research on Multi-Robot Systems (MRS) span from achieving cooperative behaviors on one end of the spectrum to achieving non-cooperative behaviors on the other end. The research problems on MRS has different manifestations depending on the type of the robot, i.e., if the robot is fixed such as an industrial manipulator, mobile such as an Autonomous Car, Unmanned Aerial Vehicle (UAV) or a combination of both (heterogeneous systems). Needless to say that the literature on MRS is vast and it would be a seemingly impossible task to compile multiple facets of MRS in one single chapter or the thesis. Therefore, the focus in this chapter and the thesis is on cooperating, mobile, MRS.

1.1 Terminology used in the Thesis

In the vast body of literature on MRS, one comes across many terms such as cooperative, coordinated, decentralized, distributed, etc in the context of MRS. Therefore, a discussion on the terminology used in this thesis is in order.

• Robots, Robotic Vehicles, Agents – The terms robots, robotic vehicles, agents are used interchangeably depending on the context and refer to the mobile robots such as Un-manned Aerial Vehicles (UAVs), Autonomous Ground Vehicles (AGVs), Autonomous Surface Vehicles/Vessels (ASVs) or Autonomous Underwater Vehicles (AUVs). The term agent is generally used in the distributed control literature wherein a generic sys-tem is usually referred to as an agent. Consequently, the term agents are synonymous with the terms robots, robotic vehicles within the scope of this thesis. It is assumed

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that the robots or agents are equipped with computation, communication, actuation and sensing units.

• Multi-Agent Systems (MAS) and Multi-Robot Systems (MRS) – MAS and MRS refer to a system consisting of multiple robots or agents that are mobile.

• Homogeneous and Heterogeneous Systems – A MRS is said to be homogeneous if all the robots within the system are identical, for example, a group of identical AUVs. Non-identical robots in a MRS constitute a heterogeneous system.

• Distributed and Decentralized – A clear distinction between the definitions of ‘de-centralized’ and ‘distributed’ are not available in the literature related to cooperative MRS. However, an interesting discussion on the definition of the terms ‘distributed’ and ‘decentralized’ in the context of Model Predictive Control (MPC), proposed in [3] is reproduced below.

“Even though there is not a universal agreement on the distinction between “decen-tralized” and “distributed”, the main difference between the two terms depends on the type of information exchange:

– decentralized MPC: Control agents take control decisions independently on each other. Information exchange (such as measurements and previous control decisions) is only allowed before and after the decision making process. There is no negotiation between agents during the decision process. The time needed to decide the control action is not affected by communication issues, such as network delays and loss of packets.

– distributed MPC: An exchange of candidate control decisions may also happen during the decision making process, and iterated until an agreement is reached among the different local controllers, in accordance with a given stopping criterion.”

The control methods presented in this thesis does not involve optimization (distributed or decentralized) performed by multiple robots, the above mentioned definitions are adapted for practical purposes in the context of this thesis. The cooperative control methods presented in this thesis, utilize first order consensus laws to arrive at an agreement between certain variables of interest. The information exchange between the robots occurs only after computation of the control inputs (decision making). Therefore, the cooperative control methods presented in this thesis are categorized as decentralized control methods.

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• Cooperation and Coordination – It is often the case that the terms ‘cooperation’ and ‘coordination’ are used interchangeably, with no universally accepted definition. The definition of the word cooperation and coordination according to the Oxford dictionary are as follows:

Cooperation – The action or process of working together to the same end.

Coordination – The organization of the different elements of a complex body or activity so as to enable them to work together effectively. Alternatively, the ability to use different parts of the body together smoothly and efficiently.

One of the early reviews on cooperative mobile robots [4] places the cooperative behavior of multiple robots in the system as a subclass of collective behavior. The collective behavior is defined as any behavior that is exhibited by the group of robots in a system. Further, a definition of the cooperative behavior is provided as

“Given some task specified by a designer, a multiple-robot system displays cooperative behavior if, due to some underlying mechanism (i.e., the “mechanism of cooperation”), there is an increase in the total utility of the system”

Other definitions of cooperation include

“joint collaborative behavior that is directed toward some goal in which there is a common interest or reward [5]”

“a form of interaction, usually based on communication [6]”

“[joining] together for doing something that creates a progressive result such as increasing performance or saving time [7]”

Clearly, an universally accepted definition for cooperation and coordination does not exist. Considering the rapid developments in the area of MRS, and given a working definition, would such a definition hold ground? This is a question that would be answered with time. For practical purposes, in the context of this thesis, the terms cooperation and coordination, are defined as follows:

Cooperation –A behavior exhibited by MRS where robots cooperate with one another to achieve an unique central objective with identical information set1 available to them. The objective then maybe executed in a centralized, distributed or decentralized manner. For example, this thesis presents results that utilize decentralized, first order consensus laws to arrive at an agreement over certain variables of interest. The unique central objective in this particular example is to achieve consensus and hence such

1_{Information set is defined as the information available to the controller on every robot obtained through} sensing or communication

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behaviors are classified as a cooperative behavior. The definition provided herein utilizes concepts from Section 2 of [8] that presents an interesting classification of the optimization problems.

Coordination – A behavior exhibited by MRS where robots coordinate with one another to achieve a goal that consists of multiple (possibly conflicting) objectives. Additionally, every robot could have a different information set at its disposal. The term coordination is used to imply complex behaviors that could be exhibited by a MRS and cooperative behaviors can be viewed as a subset of coordinated behaviors. An example could be coordinated planning of a group of UAVs to provide communication coverage over a specific area of interest in a marine environment, that consists of AUVs tasked to perform operations like adaptive sampling.

• [Source] Localization and Position estimation – The terms localization and position estimation are used in this thesis interchangeably and refer to the process of estimating the position of an entity that could possibly be moving with respect to an inertial frame of reference.

Through the use of above mentioned terminology, the methods or techniques presented in this thesis can be classified as

DECENTRALIZED, COOPERATIVE CONTROL METHODS FOR MULTIPLE(POSSIBLY

HETEROGENEOUS)MOBILE ROBOTIC VEHICLES

and

SOURCE LOCALIZATION USING MULTIPLE MOBILE ROBOTIC VEHICLES

1.2 Motivation

Cooperative control of multi-robot systems have received immense attention over the past decade, with envisaged applications spanning terrestrial, aerial and marine domains. Moti-vation for the work presented in this thesis stems from the application of robotic vehicles in the study of marine environments or more specifically in oceanographic studies. Ocean covers most part of the planets surface and yet most of it has remained unexplored or unseen by human eyes. Study of oceans are of particular interest as they are one of the primary indicators of climate change and it is essential to understand the complex biological and environmental processes that occur in the depths of the oceans. The dynamics of these processes vary spatially and temporally making the study of such complex processes an extremely difficult task. The exploration and exploitation of the marine resources is another

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Fig. 1.1 Autonomous Oceanographic Sampling Network (Source: [1])

motivation for extensive study of our oceans. This led to extensive efforts to sample our oceans that was traditionally performed manually using ships and deployment of sensors over a certain area of interest. Such a method of sampling or monitoring the oceans is expensive and inefficient with respect to the time it takes to plan and execute such an operation. The idea of using multiple, marine robotic systems and sensors that are highly integrated to perform oceanographic studies was envisaged in [1] and termed Autonomous Oceanographic Sampling Networks (AOSN) as shown in Figure 1.1. A similar concept of Underwater Acoustic Sensor Networks was proposed in [2, 9] as shown in Figure 1.2. A key enabling technology in systems such as AOSN is the use of a combination of AUVs, UAVs and ASVs in collaboration with manned aircraft, satellite-based remote sensing, buoys, ships to obtain in-situ observations. The advent of marine robotic technologies as illustrated in Figure 1.3 offer potential advantages such as increased endurance, reduced cost, increased flexibility and availability, rapid deployment, higher accuracy or resolution, and reduced risk for humans and negative impact on the environment. Further application of such a robotic based system include mapping and monitoring of marine resources, environmental parameters related to the oil spills, climate indicators, icebergs and sea ice in arctic regions, in addition to inspection of fisheries, surveillance for security, oceanography, data acquisition from buoys, surface ships and underwater vehicles, assistance in search and rescue missions, and situation awareness in marine operations. Investigation of methods and techniques to achieve such a functionality naturally aligns with a more generic research of Cooperative MRS and also

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Fig. 1.2 Underwater Acoustic Sensor Network (Source: [2])

finds additional applications in the aerial and terrestrial domains such as, search and tracking [10–12], wildlife monitoring [13, 14], cooperative estimation [15–17], cooperative transport [18], ocean sampling [19, 20], automated highway systems [21], crowd and social movement monitoring [22].

The rationale behind research on the cooperative MRS is to achieve performance level in terms of execution time, operational cost, etc. to achieve a desired objective that is not possible using a single robotic vehicle. It is expected that the development of cooperative behaviors could enable the use of a group of relatively simple robots to achieve an inherently complex task, thereby avoiding development of a single complex robotic platform. Most of the applications of the cooperative MRS involve observation of the environment (mapping), biological process (adaptive sampling), or a moving target (wildlife monitoring, surveillance). The use of multiple robots for such applications is a complex research problem and poses several challenges on various fronts. The logical approach to achieve such a functionality is to decompose the larger research problem into smaller, modular sub-problems such that the complexity becomes manageable. The cooperative behavior of MRS can then be viewed as a complex interplay between the sub-problems.

At the level of an individual robot, the challenges include (but not limited to) motion planning, guidance or control law design in the presence of external disturbances and un-certainties, navigation or estimation of certain variables of interest that could be necessary for control or planning. At the cooperation level, one is confronted with the issues related to decentralized cooperative control, asynchronous local communication, limited

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communi-(a) Light AUV (Source: LSTS, University of Porto) (b) Maribot Ducking ASV (Source: KTH)

(c) X8 Fixed wing UAV (d) Bebop Quadcopter (Source: parrot.com)

Fig. 1.3 Autonomous Mobile Robotic Systems

cation bandwidth, cooperative estimation based on local measurements. From the systems perspective, it is necessary to consider performance and stability guarantees of the complex system. Additionally, real-time implementation of such systems warrant investigation into the fault tolerant capability of the system and efficient use of the on-board limited computation facilities. A methodology that solves all these problems in an unified manner is not yet available and we seek to find answers to the subset of these problems in this thesis.

In a generic theme of cooperative control of MRS, formation control has sparked particular interest within the research community and forms the starting point for motivating the research conducted in this thesis. Early work on formation control generally focused on stabilizing the multi-robot system to a desired rigid, static, geometric formation [23, 24]. Reference [25] provides a comprehensive survey of methods to achieve rigid formations of multi-robot systems. The advent of robotic systems such as fixed-wing Unmanned Aerial Vehicles (UAVs), Autonomous Surface/Ground/Underwater vehicles (ASVs, AGVs, AUVs), multi-rotors and their increasing use in the previously mentioned applications, demand investigation of advanced cooperative control strategies that goes well beyond static formations, while

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taking into account the general non-linear dynamics of robotic systems. Consequently, the research focus needs a shift wherein the cooperative control paradigm pushes the envelope and enables cooperative maneuvers or formations of the robotic vehicles that are dynamic in nature. This thesis aims to achieve the aforementioned goal by developing a cooperative control framework that allows to execute multitude of dynamic maneuvers, which was not previously possible with existing methods. Consequently, three research areas crucial to the goal were identified that are further investigated in the three parts that constitutes this thesis, namely,

• Motion control of single robotic vehicle;

• Cooperative motion control of multiple robotic vehicles;

• Localization and tracking of moving targets.

1.2.1 Motion control of single robotic vehicle

Motion control of robotic vehicles is one of the essential control problems that needs to be addressed in order to execute any given task. The problem is further complicated due to the nonlinear dynamics that govern the motion of the robotic vehicles. Generally, the robot dynamics are always nonlinear and in particular for fixed wing UAVs, most AUVs, ASVs, and AGVs, are also under-actuated. The control design is therefore challenging, further complicated by the unavailability of an accurate model that is essential for the model-based control design approaches. A common approach to solve the motion control problem of a robotic system is to consider an abstract model for the robots such as a rigid body kinematic model. The control objective is then to use the kinematic model to design the linear and angular velocity references such that desired controller performance specifications are achieved. In the case of robotic vehicles mentioned above, it is reasonable to expect that there exists an auto-pilot inner-loop controller, that is responsible to handle the actuator dynamics of the robotic system and tracks the velocity references generated by the motion controller based on an abstract kinematic model. This thesis employs the same strategy and investigates design of motion controllers that are based on this abstract kinematic model2. Specifically, under-actuated robotic vehicles such as fixed-wing UAVs, AUVs and ASVs are considered.

A typical motion control objective is to make the robot track a desired trajectory or follow a given geometric path. Consequently, the motion control problem of the robotic vehicles has been extensively investigated, where the proposed control approaches can be broadly

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classified into Trajectory Tracking and Path Following schemes. Trajectory tracking schemes require the robotic vehicle to converge to a reference trajectory with temporal constraints. On the contrary, the Path Following scheme requires the robotic vehicle to follow a given geometric path at a desired speed without temporal specifications. It has been shown in the literature that the path following schemes have inherent advantages over the trajectory tracking schemes [26]. Therefore, path following schemes and its variants are investigated in further detail. For robots such as fixed-wing UAVs, there is an additional constraint that the robot requires a minimum positive forward speed for its operation. Such a restriction is not required for robots such as an ASV or an AGV. This leads to two different types of path following schemes, where i) the longitudinal speed and angular velocities form the control inputs for the robots with no restrictions on the speed, and ii) angular velocities are the control inputs while the longitudinal speed is held constant to a minimum positive value, for robots with speed restrictions.

The control theoretic tools used to solve the path following control schemes with no restriction on the longitudinal speed include backstepping control [27], robust control [28], nonlinear model predictive control [29], optimization based methods [30], vector field approach [31], to list a few. A detailed review of the state-of-the-art is delegated to the relevant chapters of this thesis. This thesis investigates a generalized path following scheme termed the Moving Path Following (MPF) motion control problem introduced in [32]. The MPF problem differs from the classical path following schemes wherein the desired path is defined with respect to a coordinate frame that can move. Such a problem finds potential applications in target tracking, convoy protection, etc. To this end, the following research objectives are considered in this thesis.

Research Objective 1. Design of a MPF control strategy for robotic vehicles with minimum positive longitudinal speed restriction that extends the state-of-the-art by

1. Providing a MPF formulation in 3D that excludes the stringent restriction on the initial condition of the robotic vehicle;

2. Address the problem of geometric singularities that arise when using local attitude representations such as Euler angles or ambiguities when using quaternions.

The solution to the research problem 1 is presented in Chapter 2. For robotic vehicles (such as ASVs, AGVs) without any restrictions on the longitudinal speed for its operation or equivalently, allows negative or zero speed, the MPF motion control problem in 3D remains an unsolved problem. The problem is further complicated when the state and input (actuator) constraints of the system are to be considered explicitly. Model Predictive Control (MPC),

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owing to its ability to explicitly handle system constraints, has emerged as an attractive alternative for control design of constrained systems. The seminal work in the design of provably stable MPC for nonlinear systems, termed Nonlinear Model Predictive Control (NMPC), presented in [33, 34] has accelerated its application to constrained nonlinear systems. This thesis addresses the MPF motion control problem for robotic vehicles without speed restrictions for both the input-constrained and unconstrained case leading the following research objective.

Research Objective 2. Design of a MPF control strategy for robotic vehicles without speed restrictions to obtain

1. An unconstrained MPF control law in 3D with stability guarantees;

2. Input-constrained MPF control law using Nonlinear Model Predictive Control with stability guarantees.

The research problem 2 is addressed in part in Chapter 3 and Chapter 5. The input-constrained MPF is considered in Chapter 3. The unconstrained 3D MPF problem is considered in Section 5.3 of Chapter 5.

1.2.2 Cooperative Motion Control of Multiple Robotic Vehicles

A two pronged approach is used to solve the cooperative control problem, wherein two different issues with regards to decentralized, cooperative control is addressed. Firstly, the problem of developing a cooperative control framework that enables robotic vehicles to execute multitude of dynamic maneuvers is considered. Secondly, the stringent assump-tion of constant, uninterrupted communicaassump-tion between the robots that is prevalent in the cooperative control literature is addressed. To this end, methods that reduce the instances of communication between the robots without compromising the stability of cooperative control system is investigated through the use of event-triggered control methods. Consequently, we bifurcate the brief review of the state-of-the-art into two parts:

Cooperative Motion Control

As mentioned previously, formation control of robotic vehicles is one of the most investigated research area and most of the work available in the literature focus extensively on achieving rigid formations. The control methodologies range from the use of model predictive control [35–38], Lyapunov based methods [39, 40], geometric control methods [41], passivity based methods [38, 42–44], etc. An overview of recent results on formation control can be found

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in [45, 25]. Applications such as cooperative target tracking, optimal sensor placement for source localization [15], or other similar scenarios require the robotic vehicles to cooperate with one another in order to encircle the target in the case of target tracking or uniformly distribute around the source for accurate estimation and tracking of a moving source. The problem of tracking or observing a moving target is widely investigated in literature. In addition, enclosing a stationary or moving target using multiple robots have also been addressed. Depending on the kinematic or motion constraints of the robotic vehicle such as a minimum positive forward speed restriction in case of a fixed-wing UAVs, different solutions to the target tracking problem have been proposed. A comprehensive review of methods and motivation for cooperative observation of moving targets can be found in [46]. An extensive review is presented in Chapter 5. The notable features of the existing approaches to cooperative target tracking are that; i). they are specifically designed for circular motions, applicable to unicycle robots maneuvering in 2D with a restriction of a minimum positive forward speed in most cases, and ii). they employ a first order consensus law to estimate target velocities [47, 48], exchange distance or bearing measurements from the robotic vehicle to the target [49], or relative heading of the robotic vehicles [50] in order to cooperate with one another.

Among many possible alternatives to solve the cooperative control problem described above, we focus particularly on a method termed Cooperative Path Following (CPF) problem [51]. CPF decomposes the problem into a two layered control structure, where one layer is responsible for the motion control of the individual robotic vehicle, called the Path Following (PF) controller (See [27] and references therein). The other layer, which consists of a consensus law, termed the Cooperative Controller, is responsible for achieving cooperation between the robots. A framework, akin to the CPF, that exploits synergies between the MPF control and cooperative control does not exist in the literature. Consequently, the research objective with regards to the cooperative control framework can be stated as

Research Objective 3. Development of a novel framework termed Cooperative Moving Path Following (CMPF) with an aim to develop a methodology that allows a group of robotic vehicles to achieve multitude of cooperative dynamic maneuvers. The objective includes design of a MPF controller in 3D thereby connecting the cooperative control objective with that of the MPF research objective 2.

The CMPF framework, akin to the CPF, enables multiple robotic vehicles to follow a priori specified reference geometric path that is moving in 3D with respect to a coordinate frame while achieving a desired cooperation objective. Through the CMPF framework, we demonstrate that the cooperative maneuvers that can be executed by the robotic vehicles are not necessarily circular and includes various dynamic maneuvers through the use of

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regular, parameterizable paths. Further, it is shown that the same framework can be used to generate rigid formation control behaviors. The main advantage of CMPF is its ability to chose generic paths thereby forming a generic tool to achieve varied dynamic cooperative behaviors with robotic vehicles. The research problem 3 is addressed in the Chapter 5 of this thesis.

Event-triggered Control and Communication

One of the major critical points concerning the cooperative control of autonomous robotic vehicles is the communication system and its influence on overall performance of the cooperative MRS. Many of the algorithms proposed in the literature do not address this issue explicitly and assume an uninterrupted, usually high-bandwidth communication between the robots to achieve cooperation. Such a requirement is in general not amenable in practice due to limited bandwidth of the underlying communication medium which usually is a shared resource. Further, this fact can lead to degradation of the Quality-of-Service of the communication medium leading to unwanted effects such as data loss and delay. Moreover, such effects could potentially compromise with the stability of the overall system. It is therefore, necessary to develop techniques that judiciously utilize the shared resources such as communication channels in a networked MRS. Furthermore, such methods gain prominence in marine applications where the underwater communication is achieved through acoustic medium, which is notorious for low bandwidth and communication delays. Therefore, it is imperative to develop cooperative control approaches for MRS with special emphasis on techniques that reduce the frequency of transmission between the robots over the network. Majority of cooperative control approaches employ consensus laws [52] to arrive at an agreement between a certain variable of interest which requires constant, uninterrupted communication thus leading to inefficient utilization of communication resources. Such a requirement is impractical and perhaps, even unnecessary. In this thesis, a possible solution to avoid constant communication between the robotic vehicles is investigated wherein the use of event-based sampling techniques namely, the event-triggered control approaches [53] are explored. The event-triggered control methods can further be classified as static or dynamic event-triggered control approaches depending on the use an internal dynamic variable in the design of an event-triggering condition that determines the instance when an event is generated. The use of event-triggered control and communication approaches applied to the CPF and CMPF, remain an unexplored research area. This motivates our research in the second part of the thesis, with regard to event-triggered communication and investigates the following research problem,

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Research Objective 4. Design of a decentralized, static and dynamic event-triggered con-trol and communication methods applied to the CPF and CMPF framework. Further in combination with the objectives of research problem 3, the properties of CMPF framework with event-triggered communication to achieve various cooperative dynamic maneuvers is investigated. This includes the guarantees of stability and convergence of the overall system.

Chapter 4 demonstrates development of static event-triggered control methods applied to the CPF problem. The proposed methods are further validated through experiments. The research problem 3 and 4 are investigated in Chapter 5. It is shown that the CMPF with dynamic event-triggered communication is a generalization of results obtained in Chapter 4.

1.2.3 Localization and Tracking of Moving Targets

As mentioned previously, AUVs and ASVs are being used extensively for marine applications such as ocean sampling [1, 54], seabed mapping [55], ecological studies [56] to name a few. In such applications, the AUVs with various sensors are deployed for large scale data collection [57] and are often faced with tasks which require tracking of a moving target. One possible application is the case where the AUV needs to return to a possibly mobile base station for recharging, maintenance and so forth, usually performed with manual assistance. In order to perform such a task the AUV needs to continuously estimate and track the moving target so that it can approach to its vicinity after which a docking process could be initiated. The third and the final part of the thesis concerns with localizing or estimating the position of the source or target. The source localization algorithms can be used to provide position and velocity estimates of the source or the target, which in turn can be used in conjunction with CMPF, MPF control techniques described previously. The methods to solve the source localization problem in the literature can be roughly classified into approaches that aim at design of optimal action strategies to achieve improved localization [58, 59] and approaches that improve localization accuracy through the design of efficient estimation strategies [60, 12, 61, 62]. A representative result of both these methods are provided in the last part of this thesis. The following research objective is addressed in Chapter 6, that aims to develop a globally stable estimation strategy to localize a possibly moving target (acoustic fish-tag) using Time-of-Arrival (ToA) measurement of the acoustic signal emitted by the tags using multiple ASVs.

Research Objective 5. Design of estimation method with stability and convergence guaran-tees in order to localize an acoustic fish-tag with ToA measurements using an array of mobile acoustic receivers mounted on ASVs.

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Although the above mentioned application is directed towards localization of an acoustic source, it is worthwhile to note that the research problem resonates with any application that requires estimation of the source using range-based measurements. As explained later in the thesis, the research problem 5 can be solved provided there are at least three ASVs and are positioned (for example, uniformly distributed around the source) such that the source is observable. This begs the question, whether it would be possible to localize the source using a single robotic vehicle and hence a single range measurement? The task such as localization of a possibly moving source using a single robotic vehicle bears strong resemblance to the single beacon navigation problem [63–65], except that the objective is to localize the target, assuming that the position of the robot is known. To this end, a robot is often equipped with a ranging sensor that measures the geometric range to the target obtained through time-of-flight measurement of an acoustic signal emitted by the robot or the target, depending on the underlying hardware. The position estimation of the source using single range-only measurements is a highly nonlinear estimation problem. Additionally, it is possible to show that, there exists undesirable control and state trajectories which result in an unobservable system. Therefore, it is desirable to perform some ‘observability based maneuvers’, which steers the robotic system through highly observable trajectories in order to guarantee a good estimate of the position of the target. The following research objective is investigated in Chapter 7 of this thesis, that exploits the advantage of Nonlinear MPC with economic optimization to generate robot trajectories such that the target is observable

Research Objective 6. Design of Nonlinear MPC control strategy to track and estimate the position of the moving source. To this end, explore the use of economic cost within the Nonlinear MPC formulation that is a function of measure of observability of the single, range only localization problem.

1.3 Thesis Organization and Contributions

This thesis is organized into three main parts, each addressing a particular research problem as elaborated in the previous section. The first part of the thesis, discusses results obtained for motion control of single robotic vehicle and consists of the following chapters:

• Chapter 2 presents a 3D MPF control approach for robotic vehicles that require a minimum positive speed for its operation using the concepts of geometric control. The contributions include Theorem 1 that proposes a Geometric MPF control law with stability guarantees. The results exclude stringent restrictions on the initial conditions of the robotic vehicle thereby extending the state-of-the-art. The attitude control

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problem is formulated in Special Orthogonal Group SO(3) that overcomes limitations of local representations of the attitude such as Euler angles. The contents of this chapter is based on the following publication:

– R. Praveen Jain, A. Pedro Aguiar and João Borges de Sousa, “Geometric Moving Path Following Control for Robotic Vehicles”, under preparation.

– R. Praveen Jain, A. Pedro Aguiar, and João Borges de Sousa, “Target Tracking using an Autonomous Underwater Vehicle: A Moving Path Following Approach”, 2018 IEEE OES Autonomous Underwater Vehicle Symposium, Porto, November 2018.

• Chapter 3 presents a Nonlinear MPC formulation of the MPF control problem in 2D for robotic vehicles with no speed restrictions. The contributions include Proposition 2 that presents design of terminal cost and constraint set that is crucial for a stabilizing NMPC design. The result presented in this chapter led to the following publication: – R. Praveen Jain, Andrea Alessandretti, A. Pedro Aguiar and João Borges de Sousa, “Moving Path Following of Constrained Underactuated Systems: A Nonlinear Model Predictive Control Approach”, 2018 AIAA Information Systems, AIAA SciTech Forum, Florida, January 2018.

The second part of the thesis addresses the cooperative motion control problem of multiple robotic vehicles with event-triggered communication schemes and consist of the following chapters:

• Chapter 4 presents CPF control of robotic vehicles with a novel design of the static event-triggered control and communication scheme. The contributions include The-orem 3 that presents the stability of the event-triggered control and communication scheme, and Theorem 4 that provides the result of CPF with static event-triggered communication. The results are experimentally validated using three AUVs executing CPF with circular and linear paths. The results presented in the chapter was published in

– R. Praveen Jain, A. Pedro Aguiar and João Borges de Sousa, “Cooperative Path Following of Robotic Vehicles using an Event based Control and Communication Strategy”, IEEE Robotics and Automation Letters, July 2018.

• Chapter 5 presents a novel cooperative control methodology termed the CMPF. The CMPF method is augmented with dynamic event-triggered control and communication scheme. The contributions include Theorem 5 that presents a MPF controller in 3D for robotic vehicle without restrictions on the longitudinal speed. Theorem 6 presents

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the dynamic event-triggered control and communication scheme extending the results of Theorem 3. Further, CMPF with dynamic event-triggered communication result is presented in Theorem 7. It is shown that the results of Chapter 4 is special case of the proposed dynamic event-triggered CMPF scheme. The results presented in this chapter led to following publications:

– R. Praveen Jain, A. Pedro Aguiar and João Borges de Sousa, “Cooperative Moving Path Following using Dynamic Event-triggered Communication”, Submitted to the IEEE Transactions on Control Systems Technology, 2019.

– R. Praveen Jain, Andrea Alessandretti, A. Pedro Aguiar and João Borges de Sousa, “Cooperative Moving Path Following using Event based Control and Communication”, 13th APCA International Conference on Automatic Control and Soft Computing, Azores, Portugal, June 2018.

The third part of the thesis discusses source localization problem and consists of the following chapters:

• Chapter 6 addresses the problem of localizing an acoustic source using multiple ASVs. The contributions include design of an XKF based estimation method with stability guarantees. The experiment results are provided to validate the proposed approach. The results presented in this chapter was published in:

– R. Praveen Jain, A. Zolich, E. Erstorp, Tor Arne Johansen, Jo Arve Alfredsen, A. Pedro Aguiar, Jakob Kuttenkeuler and João Borges de Sousa, “Localization of an Acoustic Fish-Tag using the Time-of-Arrival Measurements: Preliminary results using eXogenous Kalman Filter”, IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Madrid, October 2018.

• Chapter 7 addresses the problem of localizing a target using single robotic vehicle that is expected to execute maneuvers that makes the source position observable. The contributions of this chapter are use of Nonlinear MPC based approach to generate observable trajectories of the robotic vehicle. The contents of this chapter led to the following publication:

– R. Praveen Jain, Andrea Alessandretti, A. Pedro Aguiar and João Borges de Sousa, “A Nonlinear Model Predictive Control for an AUV to Track and Estimate a Moving Target using Range Measurements”, ROBOT 2017 - Third Iberian Robotics conference, Seville, Spain, November 2017.

The last part consist of Chapter 8 that concludes the thesis by discussing the results and providing further research directions.

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1.4 List of Publications

The results presented in this thesis led to the following publications in reverse chronological order:

Journal Publications:

1. R. Praveen Jain, A. Pedro Aguiar and João Borges de Sousa, “Cooperative Moving Path Following using Dynamic Event-triggered Communication”, Submitted to the IEEE Transactions on Control Systems Technology, 2019.

2. R. Praveen Jain, A. Pedro Aguiar and João Borges de Sousa, “Geometric Moving Path Following Control for Robotic Vehicles”, under preparation, 2019.

3. R. Praveen Jain, A. Pedro Aguiar and João Borges de Sousa, “Cooperative Path Following of Robotic Vehicles using an Event based Control and Communication Strategy”, IEEE Robotics and Automation Letters, July 2018, IEEE International Conference on Robotics and Automation (ICRA), Brisbane, May 2018.

Conference Publications:

1. R. Praveen Jain, A. Pedro Aguiar, and João Borges de Sousa, “Target Tracking using an Autonomous Underwater Vehicle: A Moving Path Following Approach”, 2018 IEEE OES Autonomous Underwater Vehicle Symposium, Porto, November 2018.

2. R. Praveen Jain, A. Zolich, E. Erstorp, Tor Arne Johansen, Jo Arve Alfredsen, A. Pedro Aguiar, Jakob Kuttenkeuler and João Borges de Sousa, “Localization of an Acoustic Fish-Tag using the Time-of-Arrival Measurements: Preliminary results using eXogenous Kalman Filter”, IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Madrid, October 2018.

3. R. Praveen Jain, Andrea Alessandretti, A. Pedro Aguiar and João Borges de Sousa, “Cooperative Moving Path Following using Event based Control and Communication”, 13th APCA International Conference on Automatic Control and Soft Computing, Azores, Portugal, June 2018.

4. R. Praveen Jain, Andrea Alessandretti, A. Pedro Aguiar and João Borges de Sousa, “Moving Path Following of Constrained Underactuated Systems: A Nonlinear Model Predictive Control Approach”, 2018 AIAA Information Systems, AIAA SciTech Forum, Florida, January 2018.

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5. R. Praveen Jain, Andrea Alessandretti, A. Pedro Aguiar and João Borges de Sousa, “A Nonlinear Model Predictive Control for an AUV to Track and Estimate a Moving Target using Range Measurements”, ROBOT 2017 - Third Iberian Robotics conference, Seville, Spain, November 2017.

Conference Publications not discussed in the thesis:

1. Juan Braga, R. Praveen Jain, A. Pedro Aguiar and João Borges de Sousa, “Self-triggered Time Coordinated Deployment Strategy for Multiple Relay UAVs to Work as a Point-To-Point Communication Bridge”, 2017 Workshop on Research, Education and Development of Unmanned Aerial Systems (RED-UAS), Linkoping, 2017.

2. R. Praveen Jain, A. Pedro Aguiar and João Borges de Sousa, “Self-triggered Co-operative Path Following Control of Fixed Wing Unmanned Aerial Vehicles”, 2017 International Conference on Unmanned Aircraft Systems (ICUAS), Miami, USA, 2017.

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Part I

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Chapter 2 Moving Path Following using Geometric

Control

Abstract

This chapter addresses the problem of steering a robotic vehicle along a geometric path specified with respect to a reference frame moving in three dimensions, termed the Moving Path Following (MPF) motion control problem. The MPF motion control problem is solved for a large class of robotic vehicles that require a minimum positive forward speed to operate, which poses additional constraints, and is developed using geometric concepts, wherein the attitude control problem is formulated on Special Orthogonal group SO(3). Furthermore, the proposed control law is derived from a novel MPF error model formulation that allows to exclude the conservative constraints on the initial position of the vehicle with respect to the reference path by enabling the explicit control of the progression of a virtual point moving along the reference path. The task of the MPF control law is then to steer the vehicle towards the moving path and converge to the virtual point. Formal stability and convergence guarantees are provided using the Input-to-State Stability concept. Simulation results are presented to illustrate the efficacy of the proposed MPF control law.

2.1 Introduction

Motion control of robotic vehicle such as Unmanned Aerial Vehicle (UAV), Autonomous Ground/Surface/Underwater Vehicle (AGV, ASV, AUV) is a fairly mature research area, with path following schemes receiving significant attention. The path following schemes require the vehicles to follow a known geometric path at a desired nominal speed without any

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temporal constraints. Consequently, a series of results, solely addressing the path following motion control problem were published starting with the pioneering work in [66, 67, 27] for the case of wheeled mobile robots, [68, 69] and references therein for the marine vehicles, and [39, 70, 71] for the case of UAVs. Early literature on the path following control utilizes a conveniently defined frame such as a Serret-Frenet frame placed at a point (referred to as ‘virtual point’) on the given path, that is closest to the robotic vehicle. The control strategy is then to steer the robotic vehicle towards the virtual point by controlling the attitude while the vehicle’s forward speed tracks a desired speed profile. Such a strategy, places stringent constraints on the initial position of the vehicle with respect to the path. This issue was alleviated in the works of [27, 72, 73], by explicitly controlling the progression of the virtual point along the path. Numerous other results on path following can be found in a recent survey [74]. The path following methods assume that the specified geometric path is stationary. However, in applications such a source seeking, convoy protection, target tracking, it may be useful to consider a path specified with respect to a moving frame, resulting in a Moving Path Following (MPF) motion control problem introduced in [32] for an UAV tracking a ground target. Reference [32] employs a Serret-Frenet frame approach to the 2D MPF problem, akin to the path following method of [66, 67]. The results were extended to the 3D case in [75] where quaternions were used for the attitude representation. These approaches however assume that the virtual point to be followed is located at a point on the path that is closest to the robotic vehicle, i.e., projection of the vehicle position on the path. Consequently, the proposed control law places constraints on the initial position of the vehicle with respect to the moving path. These constraints were overcome in [76] by employing the idea presented in [72, 30] that explicitly control the progression of the virtual target along the path. The results however, are valid in two dimensions with the attitude of the vehicle parameterized by the heading angle. Other control methods such as vector field method [77], nonlinear model predictive control [78] have been proposed to solve the MPF problem for unicycle type robots.

The main contribution of the chapter is the proposed solution to the 3D MPF motion control problem for robotic vehicles such as a fixed-wing UAV or some classes of AUVs wherein the vehicles are constrained to follow a strictly positive speed profile. The constraint on the initial condition of the position of the vehicle is removed by explicitly controlling the progression of the virtual target along the moving path. Further, the concepts of geometric control theory are used to define and analyze the MPF system in order to exclude geometric singularities that arise when using local representations of attitude such as Euler angles or ambiguities when using quaternions. Specifically, assuming that the robotic vehicle tracks a desired speed profile, the error kinematics of the MPF system is obtained and the attitude

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control problem is formulated on Special Orthogonal group SO(3). Formal stability and convergence guarantees are provided using the Input-to-State Stability (ISS) concept along with the corresponding estimate of a region of attraction. In particular, we show that the MPF controller is ISS with respect to imperfect tracking autopilot errors. Simulation results are presented to illustrate the efficacy of the proposed MPF controller design.

The remainder of the chapter is organized as follows. Section 2.2 derives the MPF position and attitude error kinematics and formulates the MPF problem addressed in this chapter. The main result is presented in Section 2.3 that includes Lyapunov based analysis for stability of the proposed control strategy. The simulation results are illustrated in Section 2.4 followed by conclusion in Section 2.5.

Definitions and Notations –Given a matrix A ∈ R3×3, A′ denotes the transpose operation, while λmin(A) and λmax(A) denotes the minimum and maximum eigenvalue of A, respectively.

The configuration manifold over which the attitude of a rigid body evolves is Special Orthogonal Group SO(3) defined as SO(3) = {R ∈ R3×3|R′R= RR′= I, detR = I}, where

I= I3×3 is the identity matrix unless specified otherwise. The corresponding Lie algebra

so(3) is the set of all 3 × 3 skew-symmetric matrices denoted by the hat map c_{(.) : R}3→ so(3) defined as b ω =    0 −ω3 ω2 ω3 0 −ω1 −ω2 ω1 0   

where ω = [ω1 ω2 ω3]′∈ R3. The inverse of the hat map is the vee map defined as (.)∨:

so(3) → R3. The vectors are denoted by boldface letters and matrices are denoted by uppercase letters. The set of strictly positive real numbers are denoted by R>0. Given the

notation ωA_AB, the subscript denotes angular velocity ω of the coordinate frame {A} with respect to frame {B}, expressed in coordinate frame {A} as denoted by superscript. Similarly, xAdenotes the vector x expressed in coordinate frame {A} and RB_Adenotes the rotation matrix from frame {A} to frame {B}.

2.2 MPF Problem Formulation

Consider an inertial frame of reference {I} = {e1, e2, e3} and a Wind frame {W } =

{w1, w2, w3} with its origin attached to the center of mass of the robotic vehicle as

il-lustrated in Figure 2.1. The wind frame, usually defined in the context of aircrafts and UAVs, is defined such that w1 is aligned along the velocity vector with respect to the fluid of the

robotic vehicle. For UAVs, see e.g. [79], the w1axis points along the airspeed vector, which

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Fig. 2.1 Coordinate frames for Moving Path Following

denoted by p(t) ∈ R3and its attitude denoted by the rotation matrix R_WI ∈ SO(3). Assuming that the position and the attitude of the vehicle is known or measured, the kinematic model satisfies,

˙p = vww1 (2.1)

˙

R_WI = R_WI ω_bW_{W I} (2.2)

where vw∈ R>0 is the speed of the vehicle acting along the longitudinal direction w1that

can be considered constant without loss of generality and constrained as

0 < vw,min≤ vw≤ vw,max (2.3)

Such a restriction arises for fixed-wing UAVs and certain classes of AUVs that cannot have a zero speed with respect to the surrounding fluid during their operation. Further, define a Path Transport frame or a target frame {T } = {t1, t2, t3}, fixed to a known moving target with

position pt∈ R3, linear velocity ˙pt∈ R3, attitude RIT ∈ SO(3) and angular velocity ωT I∈ R3

that are assumed to be known a priori. Let pd : R → R3 denote the reference geometric

path, parameterized by s ∈ R, that needs to be followed by the vehicle and is stationary with respect to the Path Transport frame {T }. The parameter s is the arc length along the reference path and for a given s, pd(s) denotes a virtual point on the reference path. Define a Parallel

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Transport Frame (also referred to as Bishop’s frame [80]) {F} = {f1, f2, f3} attached to this

point. The rotation matrix from frame {F} to the frame {T } is denoted by RT_F = [fT₁ fT₂ fT₃] that can be computed from the given path as explained in [81].

Remark1. The advantage of using the Parallel Transport Frame is that it is well defined when the path has a vanishing second derivative. This is in contrast with the Serret-Frenet frame that is undefined when the path curvature vanishes for example in straight line segments.

2.2.1 MPF Position Error Kinematics

Let ˜p denote the MPF position error between the position of the robotic vehicle and the origin of the parallel transport frame {F}. Then, ˜p = p − p_f, where p_f = pt+ pd(s) is the desired

position of the vehicle. The time derivative of the desired position vector p_f is given as,

˙pf = vtt1+ ωT I× pd+ ˙sf1 (2.4)

where vt = ∥ ˙pt∥, t1= _{∥ ˙p}˙p_tt_∥ and vtt1+ ωT I× pd represents the velocity of the desired position

along the path due to the motion of the target frame {T }. The MPF position error kinematics is given by

˙˜p = vww1− ˙sf1− vtt1− ωT I× pd− ωFT× ˜p (2.5)

where ωFT× ˜p represents the contribution of the rotational motion of the parallel transport

frame {F}.

Remark2. The existing literature on MPF [32, 75] requires computation of the virtual point on the path that is closest to the vehicle, which is equivalent to the projection of the robot position onto the moving path. This places stringent constraints on the initial position of the vehicle. In this chapter, these constraints are eliminated by explicitly controlling the progression of the virtual point along the path, thereby treating ˙sas a virtual control input to the system.

2.2.2 MPF Attitude Error Kinematics

Given that the robotic vehicle is assumed to have a constant speed vw, the MPF motion

control problem is solved by shaping its attitude towards the moving path, aided by the the virtual control input ˙sthat enables faster convergence. To this end, two additional coordinate frames are introduced, namely {Wd} = {wd1, wd2, wd3} and {D} = {d1, d2, d3} with rotation

matrices RF_W

d and R

Wd

D respectively. The basis vector wd1 defines the steady state desired

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basis vector d1 defines the desired direction of the velocity vector of the vehicle during

the transient phase. Therefore the basis vector d1 must be defined such that it smoothly

converges to the steady state vector w_d1, thereby shaping the approach attitude of the vehicle to the moving path. Consequently, the objective of the attitude controller is to ensure that the direction of the velocity vector of the vehicle denoted by w1aligns with the desired direction

d1, i.e., w1· d1= 1, by controlling the angular velocities ωW_{W I} of the vehicle.

Therefore, define a real-valued error function Ψ : SO(3) → R [39] as,

Ψ( ˜R) = 1 2tr(I3− Π ′ RΠR)(I3− ˜R) = 1 2 1 − ˜R11 (2.6)

where ˜R= RD_W is the rotation matrix that denotes the attitude error from {W } frame to the {D} frame and ΠR=

"

0 1 0 0 0 1

#

. The matrix ΠR is the selector matrix, that selects

the (1, 1) entry of the rotation matrix ˜R denoted as ˜R11. Notice that ˜R11 is equivalent to

w1· d1 and hence, the attitude error function is positive definite about ˜R11 = w1· d1= 1.

Following similar steps described in [39], the MPF attitude error kinematics is obtained by differentiating the attitude error function (2.6) with respect to time and satisfies

˙ Ψ( ˜R) = e_R˜· ΠRωWW D (2.7) where e_R˜ = 1 2ΠR (I3− Π ′ RΠR) ˜R− ˜R′(I3− Π′RΠR) ∨ (2.8) =1 2 _˜ R13 − ˜R12 ′ (2.9)

defines the attitude error vector. Note that ∥e_R˜∥ → 0 implies, ˜R11 → 1 and consequently

Ψ( ˜R) → 0. The term ΠRωWW Dsatisfies

ΠRωWW D= ΠRωWW I− ΠR ωWT I+ ωWFT+ ωWWdF+ ω

W DWd

(2.10)

that can be written in terms of known quantities as

ΠRωWW D= ΠRωWW I− ΠRR˜′ RD_TωT_{T I}+ RD_FωF_FT+ R_WD_dωW_Wd dD+ ω D DWd (2.11)

Equations (2.5) and (2.7) represent the MPF position error and attitude error kinematics respectively. The term ΠRωWW I implies that the direction of the velocity vector of the

Decentralized Cooperative Control Methods for Multiple Mobile Robotic Vehicles