Vision-Based Estimation and Tracking Using Multiple Unmanned Aerial Vehicles. Mingfeng Zhang

Transcription

1 Vision-Based Estimation and Tracking Using Multiple Unmanned Aerial Vehicles by Mingfeng Zhang A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Institute for Aerospace Studies University of Toronto Copyright c 213 by Mingfeng Zhang

2 Abstract Vision-Based Estimation and Tracking Using Multiple Unmanned Aerial Vehicles Mingfeng Zhang Doctor of Philosophy Graduate Department of Institute for Aerospace Studies University of Toronto 213 Small unmanned aerial vehicles (UAVs) have very limited payload capacity and power supply, while they are envisioned to be deployed in complicated environments to perform challenging tasks; therefore, it is imperative to improve their overall autonomy and energy efficiency. This thesis presents a vision-based target surveillance system for small fixedwing UAVs, which provides them with the capability of autonomously localizing and tracking an uncooperative ground moving target (GMT) using passive vision sensors such as monocular cameras. This system could greatly improve the autonomy of UAVs since target estimation and tracking are critical functions involved in many UAV missions. In this thesis, the UAV-GMT tracking problem is studied from various perspectives, and three different tracking algorithms are developed. The first tracking algorithm is based on sliding mode control, and it enables a fixed-wing UAV to maintain a constant distance relative to a GMT with a speed up to the UAV s cruising speed. The second tracking algorithm is designed to deal with an evasive GMT. It is derived in the framework of the pursuit-evasion game theory and guarantees persistent observation of the GMT. The third one, based on visual servoing control, directly uses visual measurements as feedback signals to close the control loop, so it does not require the GMT s states to assist tracking. The second part of this thesis presents a formation flight algorithm which enables multi-uav cooperation in target tracking and target motion analysis. The motion of the formation is solely determined by its virtual leader; therefore, tracking algorithms developed for a single UAV can be directly extended to a multi-uav formation ii

3 through implementations on the virtual leader. The third part of this thesis develops a cooperative nonlinear filter in order to recover the states of a GMT using multiple vision-enabled UAVs. Through fusing visual measurements from multiple sources, the observability of this bearing-only estimation problem is ensured. Special attention is taken in the filter design in order to accommodate the GMT s unknown dynamics. The tracking, estimation, and formation algorithms developed in this thesis are verified in extensive numerical simulations. iii

4 Acknowledgements This thesis would not have been possible without the support of many people. I would like to express my deepest gratitude to my supervisor Prof. Hugh H.T. Liu for his guidance, support, encouragement, and patience throughout my research period at UTIAS. I also want to thank my doctoral examination committee members Prof. Peter R. Grant and Prof. Christopher J. Damaren for valuable feedback and insightful suggestions they provided for my research progress. I am thankful to colleagues and friends in the Flight Systems and Control group, especially Ms. Chen Gao, Mr. Moe Elnabelsy, Mr. Denys Bohdanov, Dr. Sohrab Haghighat, Dr. Keith Leung, and Mr. Yoshitsugu Hitachi, for their generous help. I am grateful to my family and friends, who have always believed in me and helped me reach my goals. I could not have come this far without their continuous support and encouragement over the years. iv

5 Contents 1 Introduction Background and Motivation Literature Review Vision Sensors in UAV Applications Vision-Based Estimation and Tracking Visual Servoing Control Motion Control of Fixed-Wing UAVs Vehicle and Sensor Modeling UAV Model Target Model Vision Sensor Thesis Contributions and Outline Loitering-Based Tracking Introduction Problem Formulation Tracking Law Based on Sliding Mode Control Simulation and Results Summary Persistent Tracking Introduction Problem Formulation Relative Motion Model Definition of Persistent Tracking Viability Set and Tracking Algorithm v

6 3.3.1 Viability Set Persistent Tracking Algorithm Simulation with a Unicycle UAV Implementation with a 6DOF Nonlinear UAV UAV Dynamic Model Flight Controller Design Simulation Results Experiments with Ground Vehicles Experiment Setup Results and Analysis Summary Image-Based Target Tracking Introduction Problem Formulation System Models and Tracking Geometry Relative Motion in Image Space Image-Based Tracking Law Tracking Stationary Targets Tracking a Moving Target Simulation and Results Simulation with Stationary Targets Simulation with a Moving Target Summary Cooperative Tracking Using Multiple UAVs Introduction Problem Formulation Formation Control Law Simulation and Results Multi-UAV Path Following Cooperative Persistent Tracking Cooperative Loitering-Based Tracking Summary vi

7 6 Cooperative Estimation Using Multiple UAVs Introduction Preliminaries Vehicle Model Measurement Model Cooperative Nonlinear Filter Filter Formulation Filter Design Filter Synthesis Simulation and Analysis Simulation Results Performance Comparison Summary Conclusions and Future Work 121 A Projection Operator 124 Bibliography 125 vii

8 List of Tables 2.1 Gains and parameters of the loitering-based tracking algorithm Physical parameters of a flying wing UAV Parameters in the persistent tracking simulation Speed and turning rate of robots used in persistent tracking experiments Initial states of UAVs in path following Gains of the formation control algorithm Initial states of UAVs in cooperative persistent tracking viii

9 List of Figures 1.1 System architecture for vision-based tracking and estimation The FSC flying wing UAV Camera perspective projection model FOV of an onboard camera Tracking a moving target by a UAV The signum and saturation functions GMT speed and heading angle UAV and GMT 2D trajectories in an inertial frame The distance error and the UAV heading angle error UAV heading angle and angle λ UAV turning rate The sliding manifold variable UAV and GMT motion model in different frames Simplified relative motion model in UAV frame Configuration of the semipermeable surface Persistent tracking algorithm Control flow of the persistent tracking algorithm Simulation result with GMT running away from UAV at full speed Simulation result with GMT running away from UAV at half speed Simulation result with constant velocity GMT Simulation result with stationary GMT UAV dimensional turning rate in simulation Speed control loop Inner loop of altitude controller Outer loop of altitude controller ix

10 3.14 Turning rate control loop Augmented UAV in persistent tracking Simulation result with 6DOF UAV and GMT running away from UAV at full speed Simulation result with 6DOF UAV and GMT running away from UAV at half speed Simulation result with 6DOF UAV and constant velocity GMT Simulation result with 6DOF UAV and stationary GMT Ground robots and the Vicon motion capture system Experimental result with GMT running away from UAV at full speed Experimental result with GMT running away from UAV at half speed Experimental result with constant velocity GMT Experimental result with stationary GMT Experimental result with manually-operated GMT UAV 3D trajectory in tracking stationary targets UAV 2D trajectory in tracking stationary targets UAV turning rate in tracking stationary targets GMT speed and heading angle UAV and GMT 3D trajectories in image-based tracking UAV and GMT 2D trajectories in image-based tracking Relative distance between the UAV and the GMT UAV turning rate in tracking a GMT Geometry of virtual structure formation in an inertial frame Trajectories of a three-uav formation in path following Trajectories of a three-uav formation in path following (-1 sec) Speeds of a three-uav formation in path following Heading angles of a three-uav formation in path following Distance error and angle separation of three UAVs in formation A four-uav formation deployed in persistent tracking Target tracking using four UAVs in formation Speeds of four UAVs in formation while tracking a GMT Heading angles of four UAVs in formation while tracking a GMT x

11 5.11 Distance error and angle separation of four UAVs in formation while tracking a GMT Distance between the GMT and the formation center GMT speed and heading angle UAV trajectories in reaching formation (-1 sec) Tracking a GMT by a three-uav formation UAV trajectories in maintaining formation ( sec) UAV speeds and heading angles in reaching formation Formation variables in tracking a GMT UAV turning rates in tracking a GMT Distance between the GMT and UAVs Dependence of λ min (J(p t, p a )) on UAV formation GMT s true acceleration GMT s true and estimated 3D trajectories and estimation errors GMT s true and estimated speeds Parameters in assumptions related to the measurement model Performance comparison of different estimation methods xi

12 Notation x p v a (x, y) v u ψ ω C CI α N f vehicle state vector vehicle position vector vehicle velocity vector vehicle acceleration vector vehicle position on a horizontal plane vehicle linear speed vehicle linear acceleration vehicle heading angle vehicle turning rate rotation matrix from an inertial frame to a camera frame GMT-UAV speed ratio number of UAVs camera focal length Subscripts: a UAV t GMT Superscripts: c camera frame v vehicle-carried frame xii

13 Acronyms 6DOF CA CNF CV EKF FIM FOV GMT GPS IBVS IE IMM LMI NED PBVS PI PID RMS ROS SLAM UAS UAV UP Six degrees of freedom Constant acceleration Cooperative nonlinear filter Constant velocity Extended Kalman filter Fisher information matrix Field of view Ground moving target Global positioning system Image-based visual servoing Input estimation Interacting multiple model Linear matrix inequality North-east-down Position-based visual servoing Proportional-Integral Proportional-Integral-Derivative Root-mean-square Robot Operating System Simultaneously localization and mapping Unmanned aircraft system Unmanned aerial vehicle Usable part xiii

14 Chapter 1 Introduction 1.1 Background and Motivation The research interest in unmanned aircraft systems (UASs) has experienced an explosive growth in the past two decades due to their great potential in both military and civil applications. Compared with manned aircraft, unmanned aerial vehicles (UAVs) are more suited to perform dull, dirty, and dangerous missions in complex and dynamic environments, such as battle fields and devastated areas, without exhausting aircrew or risking the life of human pilots [1, p. 5-6]. Besides, the initial cost of UAVs can be considerably reduced since UAVs are much smaller than manned aircraft used in the same role. The operating cost of UAVs is also much less than that of manned aircraft because maintenance costs, fuel costs, and labor costs are much less [1, p. 7-8]. UAVs have been successfully applied in a wide range of missions, such as surveillance, reconnaissance, battle field assessment, target designation and monitoring, search and rescue, traffic monitoring and pipeline inspection, to name a few [1, 2, 3, 4]. Recently, much effort has been devoted to the research on mini-uavs because of their great versatility and excellent portability. Mini-UAV systems are defined as small UAV systems that have a mass under 2 kg and can be back-packed, assembled and deployed by no more than two operators [1, p. 4]. They can be controlled through a portable computer which runs a ground control station and displays telemetry data and aerial videos. Thanks to recent advances in electronic and electric technologies, compact and energy-efficient sensing, computation, and communication devices are commercially available to accommodate stringent payload and power restrictions imposed by mini- UAVs. For monitoring and surveillance missions, vision sensors, especially off-the-shelf 1

15 Chapter 1. Introduction 2 digital cameras, are generally considered an ideal payload option for mini-uavs. Vision sensors are small, light, and passive, so they suit the payload and power restrictions. More importantly, vision sensors can provide rich information of surrounding environments and targets of interest, which are usually crucial to ensure the safety and success of UAV missions. One central task of the current UAS research focuses on developing new theories and methodologies in control, sensing, and communication to enhance the sensing capability and improve the autonomy of UASs [5]. Current UAVs typically require several operators per aircraft, even when performing simple tasks. In the meantime, the environments where UAVs are deployed in are becoming increasingly complicated, and a priori knowledge about the environment is usually unavailable or limited. To overcome these challenges, it is imperative to provide UAVs with the capabilities of detecting, assessing, and adapting to environmental changes and potential threats. This demands a higher level of autonomy and intelligence for UAVs to improve their adaptability and survivability in realistic scenarios. Researchers from robotics, aerospace, and control communities have devoted a great deal of effort to tackling this challenge from different aspects at different levels. The ultimate goal of the current UAS research is to transform UASs from primarily tele-operated platforms under designated way-points into multi-mission systems with a higher level of autonomy. A critical function commonly required in UAV missions is target surveillance, which involves localizing and tracking moving targets detected by onboard sensors [5]. This task usually presents a great challenge for human operators if manually performed. It is difficult for human operators to keep an uncooperative target within the field of view (FOV) of onboard sensors; it might be impossible for human operators to conduct this task using multiple UAVs as coordinating multiple UAVs and maintaining tracking might be conflicting with each other. The limited maneuverability of fixed-wing UAVs further complicates this problem since they need to maintain persistent forward motion to stay in the air. The goal of this thesis is to develop vision-based target surveillance methodologies for fixed-wing mini-uavs to autonomously track an unknown, uncooperative ground moving target (GMT) and to estimate its states. In this work, we only consider fixed-wing UAVs because of several advantages they can offer, including their adaptability in adverse weather, high fuel efficiency, high durability in harsh environments, and relatively larger payload capacity and higher speed compared to helicopters. This study is motivated by the usage of mini-uavs in aerial surveillance missions such as traffic monitoring, boarder

16 Chapter 1. Introduction 3 patrolling, and wildlife tracking, in which the capability of autonomously localizing and tracking uncooperative targets is an integral part to the mission s success. In some scenarios, the estimated states of a target could be relayed to the ground, allowing quick response of ground personnel and units to engage it. In a broad sense, this system could play a significant role in improving the overall autonomy level of UASs for its capability of integrating vision sensors with the flight control and navigation systems of UAVs. 1.2 Literature Review The problem of target surveillance using vision-enabled UAVs has several related areas of research. This section will review some representative applications of vision sensors in UAV missions and provide details of some related work, including vision-based estimation and tracking, visual servoing control, and UAV motion control Vision Sensors in UAV Applications Vision sensors mimic the most powerful sensing system of human being by providing an enormous amount of information of the environment. They have become extremely popular in robotic applications thanks to recent advances in sensing technology and sensor development, image processing, and computer vision technologies. Compact and energy-efficient digital vision sensors are becoming widely used as the primary observation sensor for small UAVs because they are well suited to the stringent power and payload restrictions imposed by these small vehicles [6]. In indoor, confined, or other environments where the global positioning system (GPS) is not available, vision sensors are one of a few options that can ensure the safe operation of small UAVs. Examples of applications of vision sensors in UAV missions include vision-aided navigation [7, 8], obstacle avoidance [9], target detection, localization and tracking [1], and aerial surveillance [11]. Vision-based navigation employs vision sensors to determine a vehicle s position with respect to objects in its environment such as landmarks, targets, and obstacles. It was widely used in autonomous navigation for unmanned ground vehicles, and its success can be best exemplified by the Mars Exploration rover missions [12]. The latest Mars rover, Curiosity, was launched on November 26, 211 for Martian climate and geology investigation. It relies on a set of twelve engineering cameras, four for navigation and eight for hazard avoidance, to facilitate operational activities in an unmapped environment,

17 Chapter 1. Introduction 4 including ground navigation, localization, hazard detection, and robotic arm positioning [13]. Applications of vision-based navigation for aircraft include vision-assisted automatic landing and vision-augmented inertial navigation [14]. Manual landing is proven to be accident-prone and costly for small UAVs [15, 16], so inexpensive, passive, and reliable automated landing systems are highly desired to ensure the safety of UAVs during landing, especially in environments where GPS or other external aids are not available. An early example of vision-assisted landing system for fixed-wing aircraft was described in [17]. This system applies the Kalman filter approach to estimate the aircraft s position with respect to a runway during the approach and landing stages. It integrates position and attitude information provided by GPS and inertial sensors with airport images acquired by an on-board camera. A vision-only landing system for a rotor-craft proposed by Proctor et al. relies a set of known features on a runway to perform relative position estimation [18]. Huh and Shim experimentally demonstrated a vision-based landing system for a small fixed-wing UAV [19], which extracts pitch and yaw deviations of the UAV with respect to an airbag on the ground directly from real-time images obtained by an onboard camera and then adjusts the UAV s attitude accordingly until it flies into the airbag or an arresting net in front of it. Wenzel et al. developed an automatic landing system which allows a quad-rotor to automatically take off and land on a ground moving vehicle using an infrared camera [2]. A vision-based landing system which enables an unmanned helicopter to land in an unknown environment where the landing site is not previously determined was described in [21]. This system fuses information obtained by an onboard GPS receiver, inertial sensors, and a stereo camera pair to build a digital elevation map of the surrounding terrain in real-time, based on which a safe landing site is selected and the automatic landing is conducted. In some vision-assisted landing systems, the vision sensor is employed to complement onboard inertial sensors to estimate the motion of UAVs. This vision-inertial fusion technique was also applied in other UAV missions, such as indoor flights and obstacle avoidance. Lind et al. proposed a vision-based Kalman filter that estimates the velocity and angular rate of a micro aerial vehicle based on real-time images of a set of natural feature points in its environment [22]. A vision-aided inertial navigation system described in [7] fuses image data with inertial measurements to estimate a UAV s position relative to a window. Simulations and indoor flight tests demonstrate that the UAV can safely fly through the window using the estimated relative position. Attempts have also been made to develop vision-only navigation systems for UAVs. Such systems use vision

18 Chapter 1. Introduction 5 sensors solely to assist UAV flight, and they are regarded as an attractive solution for a completely isolated back-up system. A vision-only flight system developed by Johnson et al. can determine a glider s states, its flight plan, and control commands using only a single vision sensor and enable it to fly from a starting point to a predefined destination [23, 24]. A truly autonomous robot needs to create a map of its environment while moving and localizing to explore the environment. This is known as the simultaneously localization and mapping (SLAM) problem in the robotics community [25], and it has received extensive research attention in recent years [26, 27, 28]. Following the success of using active sensors such as laser range-finders and sonar in SLAM, vision-based SLAM or visual SLAM, which uses vision sensors as the primary observation sensor, also attracted great research interest. A visual SLAM system using stereo vision was developed by Davison [29, 3], and the first visual SLAM system using pure monocular vision was reported in his later work [31, 32]. The visual SLAM technique was also extended to UAVs. The first experimentally demonstrated airborne SLAM system was presented by Kim and Sukkarieh [33, 34]. This system used monocular vision in combination with accurate inertial data to map ground artificial landmarks while estimating the states of a fixed-wing UAV. Visual information was also successfully used as a source of direct feedback signals for aircraft attitude stabilization and obstacle avoidance [35, 36, 37]. A vision-aided flight system described in [36] calculates a UAV s altitude from real-time images using a horizon detection algorithm and then feeds it back to the flight control loop to stabilize the UAV s altitude. Flight tests showed that such a self-stabilizing flight control system can maintain steady and uninterrupted autonomous flight in certain outdoor environments. An obstacle avoidance system described in [38] employed a strategy called sense-andavoid for a fixed-wing UAV to avoid uncooperative obstacles in a cluttered environment without explicitly estimating the relative position between the UAV and obstacles. Most of the preceding examples involve a static environment, in which the UAV s dynamics is not tightly coupled with its environment or vice versa. Nevertheless, the success of these applications clearly demonstrates the great potential of vision sensors for UAV autonomous navigation and other missions. It motivates the investigation of the possibility of using vision sensors in more challenging UAV tasks such as target surveillance.

19 Chapter 1. Introduction Vision-Based Estimation and Tracking In the literature, the term target tracking could refer to two related but different tasks. It can be narrowly defined as the process of regulating the relative position between an observer and a target; it could also refer to the task of estimating the states of a target, which is commonly termed as target motion analysis. Sometimes, it also broadly refers to the process of performing both tasks concurrently. In this thesis, the narrow definition of target tracking is adopted in the UAV-GMT tracking problem, and target motion analysis will be used when referring to the estimation of target states. Target motion analysis involves estimating both the position and velocity of a target, while a closely related term called target geo-localization refers to the process of estimating a target s position only [39]. The vision-based target estimation problem is a type of bearing-only estimation problem, which originates from the applications of passive sonar in maritime vehicles. It is well known that the relative range between the observer and the target is unobservable in this problem because bearing-only measurements only contain line-of-sight angles [4, 41, 42]. Due to this bearing-only observability issue, vision-based estimation algorithms using a single camera are usually restricted to cases with stationary targets or mobile targets moving on a flat surface. It is fairly easy to overcome this issue in the case of stationary targets because an accurate estimate of their position can be recovered through the triangulation of multiple bearing measurements obtained from different vantage points. When a fixed-wing UAV is used in a target geolocalization task, a circular path is a natural choice for the UAV to keep the target in the camera s FOV and ensure the estimation observability. Pachter, Ceccarelli, and Chandler developed a linear regression-based geolocation algorithm using this multi-view strategy [43], which simultaneously reduces the geolocation uncertainty and estimation errors of the UAV s attitude. Flight data showed that the multi-view algorithm can reduce the geolocation error to around 3 meters, while the geolocation accuracy achieved by the one-shot scheme is around 1 meters. Barber et al. implemented the same strategy in a UAV geolocation system [39], in which a camera-equipped fixed-wing UAV was guided to fly a circular path around a designated target. The optimal altitude and orbit radius of the circular path were found to minimize the estimation error, and real flight tests reported a geolocation error under 3 meters. This bearing-only problem becomes unobservable in the case of moving targets, and early research in this field mainly focused on deriving an analytical observability condition

20 Chapter 1. Introduction 7 for this estimation process. The first observability criterion was proposed by Lindgren and Gong [44] and Nardone and Aidala [45] for the two-dimensional (2D) bearing-only estimation problem involving a constant velocity (CV) observer and a CV target. Hammel [41] and Levine [42] later derived similar observability conditions for the three-dimensional (3D) case. The observability criterion for the bearing-only estimation problem involving a target with Nth-order dynamics was also identified by Fogel and Gavish [46] and Becker [47, 48]. Cadre and Jauffret derived a parallel observability condition in the discrete-time domain by using a simple formalism of linear and multi-linear algebra [49]. The general conclusion regarding the observability condition indicates that the observer must take certain maneuvers and out-maneuver the target in order to ensure observability. Therefore, many researchers have devoted a great amount of effort in designing optimal observer maneuvers and trajectories to achieve observability for bearing-only target estimation problems and to improve the estimation performance as well. The Fisher information matrix (FIM) [5] and the Cramér-Rao lower bound [51] are the two most common tools used in formulating objective functions in these motion or trajectory optimization problems. Passerieux and Cappel applied the optimal control theory to determine the course of a constant speed observer to ensure observability and estimation accuracy by minimizing a criterion deduced from the FIM in the case of a CV target in a 2D space [5]. Ponda et al. studied the 3D trajectory optimization problem for a small fixed-wing UAV to estimate stationary targets, dynamic targets, and multiple targets [52]. Numerical solutions were obtained by minimizing an objective function derived from the FIM, and they were shown to increase the amount information collected by the UAV measurements and improve the overall estimation observability. In addition to the limited FOV of vision sensors, the restricted maneuverability of fixed-wing aircraft makes it quite difficult to apply the aforementioned strategies to them. It is extremely challenging to design an optimal trajectory for a fixed-wing UAV to ensure estimation observability while keeping a target within its sensor s FOV simultaneously, especially in the case of evasive targets. Therefore, researchers explored alternative strategies to ensure observability for UAV platforms in vision-based estimation problems. One of these approaches to attain observability without exerting special maneuvers is to use some additional information of the target being tracked, such as its physical size [7, 53, 54, 55]. A target s size on the image plane and its physical size relate the depth between itself and the observer, thus rendering this bearing-only estimation problem observable. For example, the maximum angle subtended by a target was used as an extra

21 Chapter 1. Introduction 8 measurement to relate the relative range and the target s size; consequently, the relative range becomes observable if the target geometry is given as prior knowledge [53]. Adaptive estimation methods were adopted to estimate the target s size, along with its states, when its geometry is not provided in advance [54]. An underlying condition of this approach is that the target should be close enough to the UAV so that multiple feature points of the target can be detected and tracked by an onboard camera. In target tracking using a fixed-wing UAV, the target s physical appearance is likely negligible in visual measurements because the UAV s altitude is usually larger than the target s size by one or two orders of magnitude. As a result, the target s size cannot be used as an additional measurement in this scenario, and this strategy becomes inapplicable. Obviously, another alternative solution to this problem is to use stereo cameras. However, the accuracy of stereo vision is proportional to the separation distance between the camera pair, which is unfortunately negligible in the case of small UAVs. Another strategy is to use multiple observers. Multiple observers provide multiple bearing measurements from different vantage positions, so the observability property can be ensured through the triangulation of all bearings. Bethke, Valenti, and How developed and experimentally validated an extended Kalman filter (EKF) based distributed cooperative target estimation system for multiple quad-rotors to estimate the position and velocity of moving targets in a 3D space [56]. Through the cooperation of multiple camera-equipped quad-rotors, this system can provide accurate target state estimation without the need for a single vehicle to execute special maneuvers to ensure observability. In addition, by treating the point that minimizes its distance to all line-of-sight bearings as a pseudo-measurement of the target position, this method approximates this cooperative vision-based estimation problem as a linear estimation problem, thus significantly reducing its computational load. Campbell et al. proposed a cooperative estimator for multiple fixed-wing UAVs to estimate moving targets using vision sensors [57, 58]. Based on the square root sigma point information filter, this estimator incorporates a new cooperative data fusion method in order to provide more accurate, discounted information in the case of communication delays and dropouts. Further, this vision-based estimator was experimentally verified in flight tests using two SeaScan UAVs, which lately evolved into the ScanEagle UAV [59]. The cooperative estimation approach using multiple vision-enabled UAVs will be adopted in this thesis [6]. Another challenge involved in the bearing-only estimation problem arises from the nonlinear nature of the measurement model of vision sensors [61], thus dictating the

22 Chapter 1. Introduction 9 usage of nonlinear filters. The EKF and its variants are widely employed to formulate nonlinear estimators due to their simplicity; however, the convergence property of the EKF is only proven for a limited class of systems [62]. It is further complicated by a lack of complete information of the target s dynamics. In the literature related to target motion analysis, the target is generally modeled as a CV or piece-wise CV vehicle with its acceleration as a zero-mean Gaussian white process [63, 64, 65]. This model cannot capture a target s maneuvers, so it does not necessarily represent its complete behavior in realistic scenarios, especially when the target takes aggressive, evasive maneuvers. A very popular approach for maneuvering target estimation problems is the so-called multiplemodel estimation method [66, 67]. It uses multiple models to describe a target s different maneuvers and non-maneuvering motion and runs multiple estimators in parallel. Each estimator is associated with one particular target model, and the final estimation is a weighted average of estimates obtained by all estimators based on their residuals. While this method has become a very successful tool in estimating maneuvering targets, it remains a challenge to design effective model sets as its performance relies largely on the quality of model sets. The so-called input estimation (IE) method is also a widely used approach in estimating moving targets with unknown maneuvers [68, 69]. It incorporates a standard estimator such as a Kalman filter and a supplementary estimation procedure, which can detect a target s maneuvers and then estimates the maneuver magnitude. It can directly estimate the target maneuvers from available measurements, but the detection sensitivity remains a critical issue, which could lead to false detection of target maneuvers. The particle filter, which is capable of handling nonlinear systems and non- Gaussian noises, is also applied in the bearing-only estimation problem, but it usually requires more computation [7]. The estimation approach adopted in this thesis falls into the category of nonlinear observer [71], which structurally resembles linear observers for deterministic systems Visual Servoing Control Visual servoing control refers to the technology of using visual information to control the pose of a robot s end-effector relative to a target, and it has long been used in visionbased control of robot manipulators. It can be classified into two major approaches, namely the image-based visual servoing (IBVS) control and the position-based visual servoing (PBVS) control [72]. The two approaches differ depending on whether the visual

23 Chapter 1. Introduction 1 measurement is directly used as feedback in control loops. The IBVS approach directly feeds back visual measurements to close control loops and therefore formulates controllers in the image space, thus reducing computation delays and eliminating errors caused by camera modeling and calibration uncertainties. In contrast, the PBVS approach first estimates the target pose relative to the camera and then feeds back the estimated information to control loops, so the corresponding controller must be formulated in a 3D space. In this thesis, the IBVS approach is adopted to formulate a tracking algorithm for its advantage in overcoming the inherent observability issue, which refers to the fact that visual measurements obtained by a camera do not contain depth information (i.e., the distance from the camera to a target) Motion Control of Fixed-Wing UAVs In this thesis, fixed-wing UAVs are considered as the tracking platform because of their superior performance over rotary-wing aircraft in several respects; however, controlling the motion of a fixed-wing UAV presents several unique challenges. The UAV motion control problem can be found in target tracking, trajectory tracking, path following, and formation flight tasks. These control algorithms are usually developed by considering a kinematic model of fixed-wing UAVs on a horizontal plane, assuming that a low-level altitude-hold, speed-hold, and heading-hold autopilot is available. The planar kinematics of fixed-wing UAVs augmented by a low-level flight controller is commonly described by the well-known unicycle model, which is subject to the nonholonomic constraint. As a result, instantaneous lateral movements are forbidden, and in some cases a slight deviation from a desired position may result in a disproportional restoration effort. For example, a fixed-wing UAV might need to fly a circle in order to correct a small lateral deviation since any lateral movement must be accompanied with some longitudinal movement. In addition, fixed-wing UAVs need to maintain forward motion in order to generate sufficient lift to stay in the air, which means that stop-and-wait and immediate reverse strategies are not applicable. Because of the restricted maneuverability, it is difficult for a fixed-wing UAV to keep an uncooperative moving target within its FOV while exerting special maneuvers to ensure observability. In previous work, the problem of tracking a GMT using fixed-wing UAVs is usually formulated based on a predefined tracking pattern, such as loitering, convoying, or following [73]. In other words, the desired relative motion between the UAV and the GMT

24 Chapter 1. Introduction 11 is prescribed by a given tracking pattern so that a tracking problem can be converted into a stabilization problem, which can be solved by conventional feedback control methods. In the literature, various tracking laws associated with different prescribed tracking patterns were proposed for UAV-GMT tracking problems. An approach called the Lyapunov vector field method was applied for a UAV to perform target loitering, in which the UAV flies a circular path around a stationary target, and later this method was extended to the case of a moving target [74]. Dobrokhodov, Kaminer, and Jones also proposed a nonlinear control algorithm to perform loitering [1]. In the convoying pattern, the UAV trajectory relative to the GMT is a circle lengthened to include two straight-line segments along the travel direction of the GMT. This pattern is usually employed to provide aerial protection or sensor coverage for ground vehicles. The Lyapunov vector field method was also modified to perform target convoying [75]. In the so-called following pattern, the UAV strives to keep the moving target within a certain distance in front. For example, a sinusoidal trajectory was proposed for a UAV to conduct target following [73]. While all of these approaches are able to maintain a relative displacement between the UAV and the GMT under certain conditions, they all suffer from a critical limitation: each of these approaches is only applicable to tracking tasks with the target speed constrained in an appropriate range. In other words, none of these approaches can maintain tracking if the target is allowed to vary its speed over a large range. For instance, while the circular path resulting from the loitering pattern comprises a series of vantage points for the UAV to observe a target, which is generally desired, it can only be applied to tracking tasks with targets that are much slower than the UAV. This is because loitering algorithms reported previously regulate both the separation distance and the circling angular velocity between the UAV and the GMT [76, 77, 75, 64]. When the target speed is close to the UAV s, the UAV will lose tracking as a result of failing to regulate both variables simultaneously. On the other hand, the following pattern can only guide the UAV to track a target with a speed greater than the UAV s stall speed. Researchers recently started to address this issue by proposing more comprehensive tracking algorithms. Fu, Feng, and Gao approached this problem by proposing a simple and effective algorithm which guides a fixed-wing UAV to track a GMT by a sequence of heuristically constructed way-points [78]. These way-points are carefully calculated and recursively corrected according to the relative position and orientation between the UAV and the GMT in order to ensure visibility and minimize the UAV s turning rate. Oliveira and Encarnação developed a switching tracking algorithm which consists of a following-based tracking law and a

25 Chapter 1. Introduction 12 loitering-based tracking law [79]. This algorithm switches between the two patterns depending on the target speed and the relative distance, and it guarantees to keep the target in the vicinity of the UAV even if the target moves at a speed lower than the UAV s stall speed. Numerical methods, such as model predictive control, were also adopted to solve this problem with some success [8, 81]. These improved tracking algorithms were verified through numerical simulations and seemed to offer promising solutions to this problem, but no rigorous stability proof was provided. Game theory was also applied to study this tracking problem in the framework of the pursuit-evasion game theory since the GMT is independently controlled and its motion is usually hard to predict [82]. This thesis proposes three tracking algorithms by addressing different aspects of this UAV- GMT tracking problem. A common feature shared by the three algorithms is that they only regulate the distance between the UAV and the GMT. The cooperation of multiple UAVs in this thesis arises from the cooperative estimation algorithm designed to achieve observability. From the motion control perspective, the multi-uav cooperation problem is to coordinate the motion of multiple UAVs such that their visibility of a target of interest is maintained. The strategy adopted in this thesis for multi-uav motion coordination is called rigid formation [83], which was well studied in the literature. Current research on formation control of mobile vehicles can be roughly categorized into three main approaches, namely behavior-based [84, 85], leader-follower [86, 87, 88], and virtual structure approaches [89, 9, 91]. These approaches have been successfully applied to different types of vehicles, including ground vehicles [86], aerial vehicles [92, 89, 93], surface and underwater vehicles [94], and spacecraft [91]. In addition to various benefits offered by multi-uav formation flight [92, 89, 93], such as improved fuel efficiency, expanded sensor coverage, enhanced detection capability, and increased system redundancy, its main advantage for the vision-based target surveillance task is ensuring the observability through the multi-bearing triangulation principle. Although formation control problems for UAVs and other type of vehicles share similar challenges in many aspects, such as communication constraints and collision avoidance, formation control of fixed-wing UAVs is relatively more challenging because of the aforementioned special kinematic characteristics of this type of vehicle. Formation control algorithms based on simple models such as single-integrator and double-integrator dynamics cannot be directly applied to fixed-wing UAVs. The feedback linearization [95] and other techniques have been used to convert the unicycle model into models with simple dynamics in order to implement formation control algorithms on unicycle vehicles. However, it is

26 Chapter 1. Introduction 13 still required that these vehicles are capable of moving forward and backward (i.e., the forward speed is allowed to be positive and negative). Therefore, these methods cannot be extended to fixed-wing UAVs due to the forward motion constraint. Only recently a leader-follower formation control algorithm for two nonholonomic mobile robots with bounded control inputs was reported in [88]. Heuristically constructed algorithms based on the Proportional-Integral-Derivative (PID) technique were also applied to fixed-wing UAV formation flight control problems [92, 89], but a rigorous stability analysis of these algorithms was not provided. In this thesis, the virtual structure approach is chosen because in this approach the motion of the entire formation can be represented by a virtual leader. Therefore, the single-uav tracking algorithm can be directly extended to a multi-uav formation by implementing it on the virtual leader. 1.3 Vehicle and Sensor Modeling The architecture of the UAV system for the vision-based tracking and estimation task is illustrated in Fig The block labeled UAV represents the physical unmanned aircraft, including the airframe, actuators (control surfaces and propeller), sensors (inertial measurement unit, GPS receiver, and air pressure sensor), flight control computers, and communication devices. The UAV considered in this thesis is a tailless fixed-wing unmanned aircraft shown in Fig The physical parameters of this UAV are available so that an accurate nonlinear UAV model can be developed for simulation purposes. The block labeled autopilot refers to a low-level flight controller that provides altitude-hold, airspeed-hold, and heading-hold functions [96]. The shaded block represents the main contribution of this thesis: vision-based tracking and estimation algorithms. Visual measurements of a target of interest and the estimated states of the UAV are fused to the tracking and estimation algorithms, which in turn generate guidance commands for the autopilot, including ground speed, altitude, and course commands. The autopilot can obtain the desired airspeed, altitude, and heading angle by subtracting the velocity of environmental wind from the given guidance commands UAV Model The dynamics of fixed-wing UAVs has six degrees of freedom (6DOF), three translational and three rotational, and its full mathematical model consists of a set of 12 nonlinear

27 Chapter 1. Introduction 14 equations. The UAV model augmented by the autopilot and the state estimator becomes a high dimensional, extremely complex, and highly nonlinear system, which is formidable to facilitate the development of high-level tracking algorithms. The well-known unicycle model is widely adopted to describe the resulting kinematics of augmented fixed-wing UAVs. It appears to be a rather simple vehicle model, but it is fully capable of capturing the kinematic characteristics of the closed-loop behavior of fixed-wing UAVs. A UAV s motion in the north-east-down (NED) inertial frame can be expressed in terms of its course angle χ and flight-path angle γ as follows ṗ n = v g cos χ cos γ ṗ e = v g sin χ cos γ ṗ d = v g sin γ where (p n, p e, p d ) and v g denote the UAV s inertial position and ground speed. (1.1) UAV autopilots directly control the airspeed v a and yaw angle ψ instead of the ground speed v g and course angle χ. Therefore, we rewrite the UAV s kinematic model given by Eq. (1.1) in terms of v a and ψ to incorporate the wind effect. ṗ n = v a cos ψ cos θ + w n ṗ e = v a sin ψ cos θ + w e (1.2) ṗ d = v a sin θ + w d where (w n, w e, w d ) denotes the wind velocity in the NED inertial frame, and θ is the pitch angle. Aircraft commonly control their heading through rolling maneuvers. Under ground speed, altitude & course commands servo & thrust commands Vision-based Tracking & Estimation Autopilot UAV State Estimator Wind Camera Figure 1.1: System architecture for vision-based tracking and estimation

28 Chapter 1. Introduction 15 Figure 1.2: The FSC flying wing UAV the coordinated-turn condition, the heading rate and the roll angle satisfy the following relationship ψ = g v a tan φ where φ stands for the roll angle, and g is the gravitational constant. In the UAV literature, the roll angle is commonly used as the control input for heading control. Here we simply take the heading rate as the control input since the roll angle can be easily inferred from the heading rate. The UAV s airspeed is often used as the other control input, especially when a constant-speed flight is required. In some circumstances, its linear acceleration is used as the control input instead of the airspeed. A UAV s airspeed v a and yaw angle ψ are different from its ground speed v g and course angle χ in the presence of environmental wind, the speed of which could reach as high as 5% of the airspeed of small UAVs. Therefore, it is necessary to take the wind effect into consideration in the development of low-level control and high-level guidance algorithms. In this thesis, we assume that the UAV is equipped with dedicated sensors and software to estimate the wind velocity and the low-level autopilot is capable of compensating the wind effect. Thus, it is reasonable for the high-level tracking algorithms to pass ground speed and course angle commands to the low-level autopilot. Further, it is generally assumed the UAV maintains a fixed altitude in the target tracking mission, so the third entry in Eq. (1.2) regarding the altitude is commonly omitted in the UAV literature. Under these assumptions, the UAV model can be further simplified as Eq. (1.3) using a

29 Chapter 1. Introduction 16 set of new state variables with a subscript a denoting aircraft. ẋ a = v a cos ψ a ẏ a = v a sin ψ a v a = u a ψ a = ω a (1.3) where (x a, y a ), v a, and ψ a denote the UAV s inertial position, ground speed, and course angle, respectively; u a and ω a, denoting the UAV s acceleration and turning rate, are the control inputs. It is generally desired in practice for fixed-wing planes to fly at their cruise speeds for the sake of fuel efficiency. With a constant speed, the UAV model (1.3) is further reduced to the well-known Dubins car model [97, 98], which only uses its turning rate ω a as the control input Target Model Targets considered in this work include stationary objects, animals, humans, and mobile vehicles, and they are assumed to remain on the ground and must be estimated in the 3D world. They could have various motion capabilities and usually exhibit different motion behavior as well. Since it is difficult in practice to determine the motion pattern of an unknown target, stochastic models are widely adopted to describe a target s motion in the literature. The feasibility and performance of a UAV tracking a moving target are significantly affected by the target s maneuverability, which can be quantified by its turning rate, speed, and acceleration limits. Instead of using stochastic models, which are more suitable to describe CV targets, this thesis adopts deterministic models with maneuverability constraints to describe a target s motion. For example, the moving target is modeled as a unicycle vehicle as follows in the tracking problem. ẋ t = v t cos ψ t ẏ t = v t sin ψ t (1.4) ψ t = ω t where (x t, y t ) and ψ t denote the target s inertial position and heading angle, and v t and ω t are its linear speed and turning rate. The speed is constrained as v t < v a to ensure the tracking is feasible. There are constraints imposed on ω t as well, which will

30 Chapter 1. Introduction 17 be discussed in the following chapters. The motion of a GMT is fully quantified by its linear speed and heading rate using this unicycle model. By varying these two variables, this model is capable of capturing the behavior of different types of GMTs, including animals, humans, and vehicles; therefore, we do not distinguish the characteristics of these different objects in the tracking problem studied in this thesis. In the target motion analysis problem, the target is modeled by a second-order system as follows { ṗt = v t v t = a t (1.5) where p t, v t, a t R 3 denote the target s position, velocity, and acceleration in the NED inertial frame, respectively. In this new representation, the maneuverability constraint is imposed on the acceleration a t, which is assumed to be an unknown, time-varying but bounded variable. Since this model eliminates unnecessary assumptions regarding a target s dynamics except its acceleration, it can describe the most general behavior of a moving target. A great challenge in the UAV-GMT tracking problem stems from the maneuverability disparity between these two different types of vehicles. Contrary to the great maneuverability that GMTs usually exhibit, fixed-wing UAVs are subject to various kinematic constraints. GMTs can be very agile and vary their speeds quickly over a large range, while fixed-wing UAVs cannot hover and must maintain a forward motion to stay in the air. The speed of fixed-wing UAVs is constrained between two positive numbers. The lower bound is roughly imposed by the stall condition, and the upper bound is mainly determined by the thrust limit. The speed of GMTs can vary from zero to their maximum speeds. On the other hand, the turning rate of GMTs is usually much higher than fixed-wing UAVs. Because of the unmatched motion capability, a fixed-wing UAV may need to fly at a speed significantly different from the GMT s speed in order to maintain tracking. For a given UAV, the GMT s speed might be only allowed to vary over a very small range so that the tracking can be maintained. This challenge is taken into consideration in the development of different tracking algorithms in this thesis Vision Sensor The vision sensors considered in this work are digital cameras that are commercially available off-the-shelf, and each UAV is equipped with only one monocular camera. As

31 Chapter 1. Introduction 18 X c X b ū v x c y c z c O ψ c θ c Z c Image Plane Y b Z b Y c Focal Length f Figure 1.3: Camera perspective projection model shown in Fig. 1.3, a monocular camera projects an arbitrary point in the 3D world into a pixel point on the 2D camera image plane. The perspective projection model can be expressed as [ ] [ ] ū = f x c (1.6) v z c y c where p c [x c y c z c ] T denotes the target position expressed in the camera frame, (ū, v) is the pixel coordinates of the target centroid on the image plane, and f is the focal length of the camera. Note that z c > because the Z axis of the camera frame is parallel to the optical axis and the target must be in front of the camera. The onboard camera could be either directly installed on the UAV s airframe or mounted on a pan-tilt gimbal that provides two degrees of freedom. A gimbal-pointing control algorithm is assumed to be available and provides very fast dynamics for orientation control. The camera FOV is modeled as a disk on the ground, which is shown in Fig The origins of the camera and the gimbal are assumed to be located at the center of mass of the UAV, so the transformation between the inertial frame and the camera frame only involves rotation, which is determined by the Euler angles, the camera orientation, and the gimbal orientation. Let C CI SO(3) denote the rotation matrix from the inertial frame to the camera frame, the target position expressed in the camera frame is given by p c = C CI (p t p a ) (1.7) where p a R 3 denotes the UAV s position in the NED inertial frame.

32 Chapter 1. Introduction 19 UAV FOV h r r Figure 1.4: FOV of an onboard camera. r denotes the UAV s detection radius, and h is the altitude of the UAV relative to the ground 1.4 Thesis Contributions and Outline This thesis presents several novel contributions in the area of target tracking and localization using vision-enabled fixed-wing UAVs, and they fall into the following three categories. Three tracking algorithms greatly simplify tracking strategies and significantly relax the limit on the GMT s speed [81, 82, 99, 1]; a formation flight algorithm enables multi-uav cooperation in target tracking and target motion analysis [83]; and a cooperative estimation scheme employs multiple vision-enabled UAVs to recover the states of a maneuvering GMT with unknown dynamics [6, 11]. The chapters of this thesis are organized as follows. Chapter 2 presents a loitering-based tracking algorithm which enables a fixed-wing UAV to circle around a GMT, assuming its states are known to the UAV. The proposed algorithm significantly simplifies the tracking strategy by only regulating the separation distance between the UAV and the GMT. The circling angular velocity between the two vehicles is left as a free parameter; as a result, the loitering-based tracking strategy enables a constant speed UAV to maintain a circular motion with respect to the GMT, and the GMT is allowed to be almost as fast as the UAV. The loitering-based tracking algorithm is based on the sliding mode control method, which ensures asymptotic stability for the tracking system.

33 Chapter 1. Introduction 2 Chapter 3 revisits the tracking problem by explicitly considering an evasive GMT which is assumed to be aware of the UAV and strives to escape. This problem is termed as persistent tracking, which requires the UAV to keep the evasive GMT within its detection zone for infinite time, regardless of the GMT s motion. In this chapter, the persistent tracking problem is formulated in the framework of the pursuit-evasion game theory. This study first identifies a bounded and closed region around the UAV in which persistent tracking is feasible, and it subsequently constructs a tracking algorithm for the UAV to achieve persistent tracking. This algorithm is implemented with a unicycle UAV model and a 6DOF nonlinear UAV model in numerical simulations. An experimental investigation using ground robots is also presented to demonstrate the performance of the proposed tracking algorithm. Chapter 4 develops an image-based tracking algorithm, which approaches the UAV- GMT tracking problem from a different perspective using visual servoing control. The proposed tracking algorithm is inspired by the so-called IBVS control technique, which directly uses visual measurements to close control loops. As a result, it eliminates the need of estimating a target s states to facilitate tracking. This algorithm is first developed for a stationary target and then extended to a moving target. Chapter 5 presents a formation flight control algorithm that enables cooperative tracking and estimation using multiple UAVs. The proposed algorithm is based on the virtual structure approach, and it allows multiple fixed-wing UAVs to fly in a circular formation while keeping the whole formation team flying as a rigid entity. The motion of the formation s geometry center can be controlled independently as a virtual UAV; therefore, existing control algorithms involving a single UAV can be directly applied to a UAV team through implementations on the virtual UAV. Moreover, the motion of UAVs in the formation tends to be parallel and uniform once the desired formation is achieved, implying that the formation task does not cause extra requirements on the motion capability of individual UAVs. In this chapter, the proposed formation control algorithm is successfully applied in path following and target tracking tasks. Chapter 6 studies the cooperative vision-based estimation problem, in which multiple vision-enabled fixed-wing UAVs are employed to perform target motion analysis. An observer-based nonlinear filter is proposed in this chapter. It recovers the position and velocity of a GMT by using multiple camera-equipped UAVs, and it overcomes the inherent bearing-only observability issue through fusing visual measurements from multiple sources. The dynamics of a maneuvering target could significantly affect the estimation

34 Chapter 1. Introduction 21 performance, but it is generally unknown. In order to capture the general characteristics of a maneuvering target, this study models the target s acceleration as an unknown, timevarying but bounded variable and employs a complementary estimator to recover this variable. As a result, the proposed filter is capable of handling targets with arbitrary but bounded maneuvers. The filter design task is cast into a linear matrix inequality (LMI) optimization problem, and the solution is shown to ensure estimation stability and to provide some tolerance to nonlinear uncertainties and external disturbances. Chapter 7 provides some concluding remarks and presents some discussions about possible directions for future work.

35 Chapter 2 Loitering-Based Tracking This chapter presents a loitering-based target tracking algorithm which enables a fixedwing UAV to loiter around a GMT, provided that the GMT s states are known to the UAV. In order to simplify the control strategy, the UAV is assumed to fly at its constant cruise speed and only uses its turning rate as the control input. The speed of the GMT is allowed to vary from zero to the UAV s cruise speed as the loitering strategy only regulates the separation distance between them. The tracking algorithm is based on sliding mode control, which was successfully applied in trajectory tracking and motion control problems involving ground robots. It is proven to ensure asymptotic stability for the tracking system. In order to avoid the inherent chattering phenomenon of sliding mode control, the switching component in the original tracking algorithm is replaced by a high-slope saturation function, which is designed to retain asymptotic stability for the system. A simulation investigation is presented to demonstrate the effectiveness and performance of the tracking algorithm. 2.1 Introduction Contrary to the great maneuverability that a GMT usually exhibits, the motion of fixedwing UAVs is limited by various constraints. The planar kinematics of fixed-wing UAVs, augmented by a low-level altitude-hold, speed-hold, and heading-hold autopilot, can be described by the well-known unicycle model, which is subject to the nonholonomic constraint. As a result, instantaneous lateral movements are forbidden for fixed-wing UAVs. In addition, a fixed-wing UAV cannot perform stop-and-wait and immediate reverse maneuvers because it has to maintain forward motion in order to generate sufficient lift to 22

36 Chapter 2. Loitering-Based Tracking 23 stay in the air. It is even more restrictive for fixed-wing UAVs to stay at a constant altitude. By contrast, a GMT may be very agile and can vary its speed from zero to its maximum speed. For stationary or slow-moving targets, the UAV can fly along a circular trajectory around the target to ensure persistent visibility; for fast-moving targets, the UAV needs to take proper maneuvers so that the tracking can be maintained. Many tracking algorithms have been proposed for this UAV-GMT tracking problem, but none of them can maintain tracking if the GMT is allowed to vary its speed over a large range. This chapter proposes a loitering algorithm based on sliding mode control to maintain tracking even if the GMT varies its speed over a large range. Different from existing tracking algorithms which generally regulate both the separation distance and the circling angular velocity between the UAV and the GMT [76, 77, 75, 64], the new loitering algorithm only regulates the separation distance and leaves the circling angular velocity as a free parameter. Due to this significant relaxation, the UAV can maintain a circular motion with respect to the GMT with a speed up to its cruise speed. The UAV circles around the GMT when the GMT is slow, and the circling angular velocity decreases as the GMT s speed increases. When the GMT s speed gets close to the UAV s, the UAV would move almost in parallel to the GMT in order to maintain the desired separation distance. The UAV s motion resulting from the proposed tracking algorithm based on sliding mode control comprises a reaching phrase, during which the UAV adjusts its heading angle to a proper value, and a sliding phrase, during which the UAV maintains the desired heading angle until a desired separation distance is achieved. This tracking algorithm guarantees asymptotic stability for the tracking error, provided the GMT states are known to the UAV. Further, in order to avoid the inherent chattering phenomenon of sliding mode control, the switching component is replaced by a high-slope saturation function, which is designed to retain asymptotic stability for the system. This chapter is organized as follows. In Section 2.2, the kinematic model describing the relative motion between the UAV and the GMT is derived. The tracking algorithm and its stability proof are presented in Section 2.3, followed by numerical simulations in Section Problem Formulation The tracking problem is considered in a planar environment since only the relative motion on the horizontal plane between the UAV and the GMT is of interest. The planar

37 Chapter 2. Loitering-Based Tracking 24 geometry between the UAV and the GMT in the NED inertial frame is depicted in Fig x GMT R λ ψ t θ v t y l ψ a UAV v a Figure 2.1: Tracking a moving target by a UAV The planar kinematics of the UAV and the GMT can be described by the unicycle models given by Eqs. (1.3) and (1.4). The relative motion between them can be described by the following equations. l = v a sin(ψ a λ) v t sin(ψ t λ) (2.1) λ = v a cos(ψ a λ) + v t cos(ψ t λ) (2.2) l + R ψ a = ω a (2.3) where v t and ψ t denote the GMT s speed and heading angle, respectively; v a, ψ a, and ω a denote the UAV s speed, heading angle, and turning rate, respectively; l represents the distance error; R is a positive constant denoting the desired separation distance; λ = θ π/2 with θ [, 2π), and it is wrapped into the range [, 2π). This relative motion model permits a formulation to directly control the distance between the UAV and the GMT. The angle λ defines the orientation of the line connecting the two vehicles. It is not directly controlled because the loitering-based tracking only concerns about the separation distance. In following chapters, the relative motion model will be expressed in the Cartesian space when the relative orientation becomes relevant. The control goal of this tracking problem is to regulate the distance error l to zero. In this work, we assume that the UAV flies at a constant speed and only uses its turning rate ω a as the control input. The GMT speed satisfies the constraint v t < v a so that

38 Chapter 2. Loitering-Based Tracking 25 the tracking problem is feasible. Further, it is assumed that the UAV can sense its own states and knows the GMT s states. 2.3 Tracking Law Based on Sliding Mode Control The dynamics of the distance error l described by Eq. (2.1) indicates that l cannot be directly controlled by the UAV s turning rate ω a. Instead, the UAV s heading angle ψ a must be controlled to track a proper value that can gradually drive the distance error l to zero. This motives the usage of sliding mode control, in which a sliding manifold is defined to generate a proper trajectory for the heading angle to achieve the control goal. Once the heading angle is controlled to stay on the manifold using the UAV s turning rate ω a, the distance error l will converge to zero. Define ψ a = ψ a λ and ψ t = ψ t λ, then the steady value of ψ a, denoted by ψ d a, can be computed by setting l = in Eq. (2.1). The steady state ψ d a is given by ( ψ a d = sin 1 vt sin ψ ) t v a The condition v t < v a ensures that the steady state ψ d a in the foregoing equation is well defined with ψ d a ( π/2, π/2). Define ψ a = ψ a ψ d a and wrap it into the range [ π, π], then the dynamics of the distance error can be expressed as l = v a sin( ψ d a + ψ a ) v t sin ψ t = v a ( sin ψ d a(cos ψ a 1) + cos ψ d a sin ψ a ) (2.4) Substituting the trigonometric identities cos ψ a = 1 2 sin 2 ( ψ a /2) and sin ψ a = 2 sin( ψ a /2) cos( ψ a /2) into Eq. (2.4) yields l = 2v a sin ψ a 2 cos( ψ d a + ψ a 2 ) = 2v a sin ψ a 2 cos ψ a d + ψ a 2 (2.5) The straight line connecting the GMT and the UAV divides the horizontal plane into two regions. We assume the UAV keeps its heading angle towards one region such that ψ a remains in the range [ π/2, π/2], as shown in Fig As a result, the inequality cos( ψ d a/2 + ψ a /2) > is valid since ψ d a ( π/2, π/2) by definition. Subsequently, the tracking error l will converge to zero asymptotically if we take ψ a = φ(l), in which φ(l)

39 Chapter 2. Loitering-Based Tracking 26 is a continuous function such that l φ(l) > for all l and φ(l) = if and only if l =. In the following, we will apply sliding mode control to design a control law to achieve this goal. The sliding manifold is designed as where k 1 and k 2 are positive gains to be designed. s = ψ a + k 1 tan 1 (k 2 l) (2.6) The goal then becomes designing a control law to constrain the motion of the original system (2.1) and (2.2) on the manifold s =, which guarantees that l tends to zero as time tends to infinity. To that end, the following control law is proposed. ω a = λ + ψd a k 1k 2 l 1 + k 2 2l 2 k 3sgn(s) (2.7) with u t = v t and ψ d a = u t sin(ψ t λ) + v t (ω t λ) cos(ψ t λ) v 2 a v 2 t sin 2 (ψ t λ) where k 3 is a positive gain, and sgn(s) is the signum function given by 1, s > sgn(s) =, s = 1, s < The main conclusion can be summarized in the following theorem. Theorem 1. Suppose that there exists a positive number η [, 1) such that v t ηv a, and the positive gain k 1 (, 1] satisfies sin 1 η (1 k 1 )π/2. Then, the state l of the system (2.1) and (2.2) converges to zero under the control law given in Eq. (2.7). Moreover, l = is globally asymptotically stable. Proof. The derivative of the sliding variable is ṡ = ω a λ ψd a + k 1k 2 l 1 + k 2 2l 2 which becomes the following differential equation under the control input given by Eq. (2.7) ṡ = k 3 sgn(s) Consider a Lyapunov function candidate V = s 2 /2, we have V = k 3 sgn(s)s = k 3 s (2.8)

40 Chapter 2. Loitering-Based Tracking 27 Thus, W = 2V = s satisfies the following differential equality D + W = k 3 where D + W stands for the right-hand derivative [12, p ] of the non-differentiable variable W. The solution is given by W (s(t)) = W (s()) k 3 t Therefore, the trajectory will reach the manifold s = in finite time, and moreover, it will remain on the manifold thereafter as shown by V = k 3 s. On the manifold s =, the following equality can be derived from Eq. (2.6). ψ a = ψ d a k 1 tan 1 (k 2 l) Thus the inequality sin 1 η (1 k 1 )π/2 ensures that ψa ψd a + k1 tan 1 (k 2 l) sin 1 η + k 1 π 2 π 2 Note that the reaching phase, during which the sliding variable s approaches zero, is independent of ψ a, so s = can always be reached in finite time regardless of the value of ψ a. On the manifold s =, the distance error s dynamics given by Eq. (2.5) is reduced to l = 2v a cos ψ d a + ψ a 2 sin k 1 tan 1 (k 2 l) 2 where cos( ψ d a/2 + ψ a /2) > because ψ a [ π/2, π/2] on the manifold and ψ d a ( π/2, π/2) by definition. Consequently, the forgoing equation ensures that the tracking error l converges to zero asymptotically, which can be proven by using the Lyapunov function candidate U = l 2 /2. Remark 1. Note that the speed ratio η approaches one as k 1 approaches zero, which means that the UAV can maintain tracking of a target with a speed up to its cruise speed by decreasing k 1. Moreover, k 1 must satisfy k 1 1 as η cannot be negative. It is well known that the sliding mode control method suffers from the chattering phenomenon, which is mainly caused by imperfect switching devices and unmodeled delays. In order to eliminate chattering, the signum function can be replaced by a highslope saturation function, which is defined as { y, y 1 sat(y) = sgn(y), y > 1

41 Chapter 2. Loitering-Based Tracking 28 sgn(y) sat(y/ε) 1 1 y ε y -1-1 Figure 2.2: The signum and saturation functions Therefore, the modified control law is given by ω a = λ + ψd a where ɛ is a positive constant to be designed. k 1k 2 l 1 + k 2 2l 2 k 3sat( s ɛ ) (2.9) The signum and the saturation functions are illustrated in Fig For better accuracy, small ɛ is desired, but a too small value of ɛ will induce chattering in the presence of delays or unmodeled dynamics. A modified sliding mode control law with a saturation component can generally achieve ultimate boundedness with a bound that can be reduced by decreasing ɛ; that is, the modified control law will stabilize the system at a new equilibrium point, resulting in a steady-state error. However, the modified control law given by Eq. (2.9) can retain asymptotic stability for the original system if its parameters are chosen properly. Theorem 2. Suppose that there exists a positive number η [, 1) such that v t ηv a, the positive gain k 1 (, 1] satisfies sin 1 η (1 k 1 )π/2, and the positive constant ɛ satisfies ɛ/k 1 π/2. Then, the state l of the system (2.1) and (2.2) converges to zero under the control law given in Eq. (2.9). Moreover, l = is globally asymptotically stable. Proof. When s ɛ, the control law (2.9) is the same as the original sliding mode control law (2.7), and hence the equality given in Eq. (2.8) remains valid. Therefore, whenever s() > ɛ, s(t) will be strictly decreasing until it reaches the set { s ɛ} in finite time and remains inside thereafter. Because of ɛ/k 1 π/2, we can derive from Eq. (2.6) that ψ a = s k 1 tan 1 (k 2 l) and ψ a ɛ + k1 π/2 π. Then, inside { s ɛ} we have l = 2v a cos ψ d a + ψ a 2 sin s k 1 tan 1 (k 2 l) 2

42 Chapter 2. Loitering-Based Tracking 29 The derivative of U = l 2 /2 satisfies U = 2v a cos ψ d a + ψ a 2 sin k 1 tan 1 (k 2 l) s l <, l > 1 tan ɛ 2 k 2 k 1 Thus, the modified control law (2.9) drives the original system to the set Ω ɛ = { l 1 k 2 tan ɛ k 1, { s ɛ}} in finite time and confines it inside thereafter. Inside Ω ɛ, the saturation component in Eq. (2.9) is reduced to k 3 s/ɛ, and then the closed-loop system can be expressed as where a is defined as ψ a = a sin ψ a 2 k 3 ɛ s ṡ = k 3 ɛ s a = 2k 1k 2 v a 1 + k2l cos ψ a d + ψ a Note that a is a time-varying but positive variable. Inside Ω ɛ, the minimum value of a is found to be a min = 2k 1k 2 v a 1 + tan 2 ɛ cos sin 1 η + π/2 k 1 2 Considering a Lyapunov function candidate V 1 = 4 sin 2 ( ψ a /4) + βs 2 /2 for the closedloop system within Ω ɛ, it can be found that V 1 = β k ( 3 s + 1 ɛ 2β sin ψ ) 2 ( a a k ) 3 sin 2 2 4βɛ which is negative definite if β satisfies β > k 3 /(4ɛa min ). Therefore, within Ω ɛ, both ψ a and s tend to zero as time tends to infinity under the control law (2.9). From Eq. (2.6), it can be concluded that l also converges to zero. Remark 2. Under the control laws given by Eqs. (2.7) and (2.9), a UAV needs to know the position, velocity (i.e., speed and heading angle), and acceleration (i.e., linear acceleration and turning rate) of a GMT in order to maintain loitering. ψ a Simulation and Results In this section, numerical simulations are presented to illustrate the performance of the proposed tracking algorithm given by Eq. (2.9). The gains and parameters used in the tracking algorithm are given in Table 2.1. The gain k 1 =.2 leads to η.95, which

43 Chapter 2. Loitering-Based Tracking 3 Table 2.1: Gains and parameters of the loitering-based tracking algorithm k 1 k 2 k 3 ɛ means the UAV is able to track a target with its speed up to 95% of the UAV s cruise speed. The motion profile of the GMT is depicted in Fig. 2.3, which shows that its speed varies over a rather large range. The GMT starts at (, ) m with a heading angle of 45 degrees, and it makes a sharp left turn at 25 seconds. Its initial speed is 2 m/s, and it accelerates to 1 m/s at 35 seconds and to 12 m/s at 45 seconds. The UAV s cruise speed is fixed at 13 m/s, and its initial position and heading angle are ( 1, 1) m and 51.4 degrees. The desired distance between the GMT and the UAV is set as 15 m. The 2D trajectories of the UAV and the GMT in the tracking mission are sketched in Fig As we can see, the UAV is able to loiter around the GMT at the desired separation distance, even through the highest speed ratio reaches 92.3%. This can also be observed from the distance error and heading angle error history illustrated in Fig The UAV s heading angle and the angle λ are shown in Fig The UAV s turning rate required to maintain loitering is shown in Fig It contains three abrupt changes, which are caused by the speed and heading changes of the GMT. The maximum turning rate in the simulation reaches 18.7 deg/s, and the UAV needs to maintain a bank angle around 23 degrees in order to achieve this turning rate under the coordinated-turn assumption. Such a deep bank angle may cause excessive loading on UAV structures. This issue can be alleviated by increasing the separation distance. The sliding manifold variable, shown in Fig. 2.8, demonstrates that the modified tracking law is effective in eliminating the chattering phenomenon. 2.5 Summary This chapter presented a new target tracking algorithm based on the sliding mode control approach for a fixed-wing UAV to loiter around a GMT. The proposed tracking algorithm significantly simplifies the control strategy by only regulating the distance between the UAV and the GMT and leaving the circling angular velocity as a free parameter. The UAV can fly at a constant speed to maintain loitering by only manipulating its turning

44 Chapter 2. Loitering-Based Tracking Speed v t (m/s) Heading ψ t (deg) Time (sec) Figure 2.3: GMT speed and heading angle 16 GMT Path UAV Path X (m) Y (m) Figure 2.4: UAV and GMT 2D trajectories in an inertial frame. Diamond markers are UAV and GMT starting positions

45 Chapter 2. Loitering-Based Tracking 32 2 Distance error l (m) Time (sec) (a) Distance error 7 6 UAV heading error ψa (deg) Time (sec) (b) UAV heading error Figure 2.5: The distance error and the UAV heading angle error

46 Chapter 2. Loitering-Based Tracking 33 5 UAV heading ψ a (deg) Time (sec) (a) UAV heading angle (unwrapped) 5 5 λ (deg) Time (sec) (b) Angle λ (unwrapped) Figure 2.6: UAV heading angle and angle λ. Both are unwrapped to produce continuous plots

47 Chapter 2. Loitering-Based Tracking 34 5 UAV turning rate ω a (deg/s) Time (sec) Figure 2.7: UAV turning rate Sliding manifold s Time (sec) Figure 2.8: The sliding manifold variable

48 Chapter 2. Loitering-Based Tracking 35 rate, while the GMT is allowed to vary its speed over a wide range and can be almost as fast as the UAV. The inherent chattering phenomenon associated with sliding mode control was eliminated by replacing the switching mechanism by a high-slope saturation component. Both algorithms were shown to ensure asymptotic stability. The performance of the proposed tracking algorithms was validated through numerical simulations.

49 Chapter 3 Persistent Tracking This chapter considers the UAV-GMT tracking problem with an evasive target. In this problem, the GMT is assumed to be aware of the UAV and strives to escape. To ensure persistent tracking such that the UAV is able to confine the GMT within its detection zone for infinite time, regardless of the target motion, this chapter formulates this tracking problem in the framework of the pursuit-evasion game theory. The feasibility condition of persistent tracking is identified, and a persistent tracking algorithm is developed subsequently. Both simulations and experiments are presented to demonstrate the performance of this persistent tracking algorithm. 3.1 Introduction Persistent tracking is of vital importance to UAV missions that require uninterrupted observation of targets of interest. A fixed-wing UAV is much less maneuverable than a GMT because of constraints on its speed and turning rate; therefore, the possibility that an evasive GMT may escape from the UAV s detection zone must be considered. To be more specific, the UAV needs to confine the moving target in its detection zone, in spite of any unpredictable motion of the target, to ensure persistent observation. Inspired by the classical Homicidal Chauffeur game which was originally introduced by R. Isaacs [13, p ], this problem is studied in the framework of the pursuitevasion game theory, in which the UAV strives to confine the GMT within its detection zone, while the GMT endeavors to escape. In this thesis, the UAV s detection zone on the ground is the area covered by an onboard monocular camera, and it is defined as a circular disk right below the UAV s center of mass. The UAV s speed and turning rate 36

50 Chapter 3. Persistent Tracking 37 are constrained, while the GMT is assumed to be slower than the UAV s maximum speed but otherwise is allowed to take arbitrary maneuvers. The resulting game consists of a faster but less maneuverable pursuer and a slower but more maneuverable evader. The solution of such a pursuit-evasion game provides optimal strategies for both the UAV and the GMT. The optimal evading strategy for the GMT poses the worst situation for the UAV. In other words, the optimal pursuing strategy for the UAV is robust with respect to the GMT s maneuvers. Based on the pursuit-evasion game theory, a bounded and closed region encircling the UAV, in which persistent tracking is feasible, is first identified. This region is called the viability set, and the condition that guarantees the existence of a closed viability set is then found in terms of the GMT-UAV speed ratio and the UAV s detection range. In addition, a tracking algorithm is constructed for the UAV to achieve persistent tracking in the viability set. This algorithm is robust with respect to the GMT s maneuvers because it is derived by considering the worst evading strategy that the GMT can take. Furthermore, this algorithm only needs the information of the relative position between the UAV and the GMT; as a result, the burden to estimate the GMT velocity to facilitate tracking is essentially eliminated. In this chapter, the persistent tracking algorithm is derived by only considering the UAV s kinematics, which is described by the unicycle model. In addition to numerical simulations considering a unicycle UAV model, the proposed persistent tracking algorithm is also implemented with a 6DOF nonlinear UAV model in simulations in order to study the impact of UAV nonlinear dynamics on persistent tracking. Furthermore, the achievement of persistent tracking is demonstrated by an experimental investigation using two ground vehicles, one serving as the UAV and the other as the GMT. This chapter is organized as follows. In Section 3.2, a simplified mathematical model describing the relative motion between the UAV and the GMT is presented, and the definition of persistent tracking is also introduced. In Section 3.3, the viability set is determined in the framework of the pursuit-evasion game theory, and a tracking algorithm for the UAV to achieve persistent tracking is formulated. Numerical simulations are presented in Section 3.4 with a unicycle UAV model. In Section 3.5, a 6DOF nonlinear model of a fixed-wing UAV is developed, and the persistent tracking algorithm is implemented with this nonlinear UAV model, which is augmented by a flight controller. The performance of the tracking algorithm is further verified experimentally by using ground vehicles in Section 3.6.

51 Chapter 3. Persistent Tracking Problem Formulation In this section, a simplified model describing the relative motion between the UAV and the GMT is presented. The definition of persistent tracking is also given Relative Motion Model Both of the UAV s and the GMT s planar kinematics can be described by the unicycle models given by Eqs. (1.3) and (1.4). The motion of the UAV and the GMT in an inertial frame is illustrated in Fig. 3.1(a). x ψ t v t x 1 ψ v t GMT GMT ψ a v a v a UAV r y UAV r x 2 (a) UAV and GMT in inertial frame (b) UAV and GMT in UAV frame Figure 3.1: UAV and GMT motion model in different frames. r represents the UAV s detection radius By converting the relative states in the UAV frame as follows x 1 cos ψ a sin ψ a x t x a x 2 = sin ψ a cos ψ a y t y a ψ 1 ψ t ψ a the relative motion between the UAV and the GMT can be expressed as ẋ 1 = v a + ω a x 2 + v t cos ψ ẋ 2 = ω a x 1 + v t sin ψ (3.1) ψ = ω t ω a Consider a scenario in which the UAV and the GMT both move with constant speeds and they only use the turning rates as their control inputs, and normalize the control

52 Chapter 3. Persistent Tracking 39 inputs by letting ω a = u 1 ω a,max with u 1 [ 1, 1] and ω t = w 1 ω t,max with w 1 [ 1, 1]. Define R a = v a /ω a,max, R t = v t /ω t,max, α = v t /v a, and β = ω t,max /ω a,max. Let τ denote the time in Eq. (3.1). By taking coordinate transformation as x 1 /R a x 1, x 2 /R a x 2, and τω a,max t, the relative motion model (3.1) is simplified as ẋ 1 = 1 + u 1 x 2 + α cos ψ ẋ 2 = u 1 x 1 + α sin ψ (3.2) ψ = u 1 + βw 1 where ( x 1, x 2, ψ) = (dx 1 /dt, dx 2 /dt, dψ/dt). Note that t denotes the normalized time, which is dimensionless. The relative motion model expressed in the UAV s frame is shown in Fig. 3.1(b). Assuming the GMT is able to change its heading angle instantaneously (i.e., w 1 (, + )), we can describe the relative motion by a normalized 2D model as follows { ẋ1 = 1 + u 1 x 2 + α cos u 2 ẋ 2 = u 1 x 1 + α sin u 2 (3.3) where u 1 U 1 [ 1, 1], denoting the UAV s turning rate, and u 2 U 2 ( π, π], denoting the GMT s heading angle, are the UAV s and the GMT s control variables, respectively; α is the GMT/UAV speed ratio, and it is constrained as α < 1 to ensure the tracking problem is feasible. x 1 u 2 GMT u 1 UAV x 2 r Figure 3.2: Simplified relative motion model in UAV frame. detection radius r represents the UAV s Remark 3. The assumption of the GMT s heading rate greatly simplifies the formulation of the persistent tracking problem. The direct benefit is that an analytical solution of this

53 Chapter 3. Persistent Tracking 4 tracking problem is tractable. If an upper bound is imposed on the GMT s turning rate, the GMT s initial heading angle will also affect the feasibility of persistent tracking. As a result, this problem can only be solved numerically Definition of Persistent Tracking The simplified relative motion model is graphically illustrated in Fig The area enclosed by the dashed circle with radius r denotes the detection zone. It is referred to as the target set Λ in the following analysis, and it can be defined as Λ = {(x 1, x 2 ) R 2 : x x 2 2 r 2 } Let ẋ = f(x, u 1, u 2 ) denote the relative motion model (3.3) with x (x 1, x 2 ). Given a time horizon T > and a set K Λ, the concepts of tracking and persistent tracking are then defined as follows Definition 1 (Tracking Feasibility). The tracking is feasible in K during time period [, T ] if the following condition is satisfied: x() K, u 2 ( ) U 2,[,T ], u 1 ( ) U 1,[,T ] t [, T ], x(t) K (3.4) Definition 2 (Persistent Tracking Feasibility). Persistent tracking is feasible in K if the tracking defined in Definition 1 is feasible when T =. Set K is then called the viability set. In the single-uav-single-gmt persistent tracking problem, we seek to determine the viability set K Λ and to design a tracking strategy for the UAV such that the GMT can be confined in the viability set K. To be specific, problems of interest that will be studied in this chapter include Feasibility: Identify the condition in terms of motion parameters, such as the GMT- UAV speed ratio and the UAV s detection range, under which persistent tracking can be achieved. Optimization: Determine the minimum detection radius that ensures the feasibility of persistent tracking. Control: Design a tracking strategy for the UAV to achieve persistent tracking.

54 Chapter 3. Persistent Tracking Viability Set and Tracking Algorithm Viability Set In the Homicidal Chauffeur game, the objective of the pursuer is to bring an evader into the target set in minimum time, whereas the evader strives to prevent this from happening. The resulting solution provides the pursuer the best strategy to capture the evader in minimum time. Inspired by this game, we treat the persistent tracking problem as a pursuit-evasion game. In this game, the GMT, acting as an evader, strives to escape from the target set Λ as soon as possible, while the UAV, acting as a pursuer, endeavors to keep the GMT within the target set Λ for as long as possible or forever if possible. Following the procedure of solving the Homicidal Chauffeur game, we first determine a subset of Λ from which the GMT can escape in finite time, and then we will show that the complementary subset is the viability set, in which persistent tracking is feasible. The following derivation procedure is very similar to the pursuit-evasion game solution presented in [14, p ]. Assume the initial state of Eq. (3.3) stays within the set Λ, then the game terminates when the state reaches the boundary of Λ, denoted by Λ. Note that only those points x of Λ that satisfy the following condition are candidates for a terminal position of the game max min ν T f(x, u 1, u 2 ) = min max ν T f(x, u 1, u 2 ) < (3.5) u 1 u 2 u 2 u 1 where ν R 2 is the inward normal vector of Λ at the point x. This condition implies that the velocity vector f(x, u 1, u 2 ) must point outward when it reaches Λ at the point x so that the GMT can penetrate Λ and then escapes from the set Λ. derivation of this condition can be found in [14, p ]. A detailed The set of all points x of Λ that satisfy Eq. (3.5) is called the usable part (UP) of Λ. Substituting Eq. (3.3) into Eq. (3.5) yields max min{ x 1 (u 1 x 2 + α cos u 2 1) x 2 (α sin u 2 u 1 x 1 )} = x 1 αr < (3.6) u 1 u 2 Therefore only those x 1 for which x 1 < αr on the circle x x 2 2 = r 2 constitute the UP, which is the arc ÂCB shown in Fig This implies that it is impossible for the GMT to escape from the set Λ through the arc GMT is in front of the UAV, it cannot escape immediately. ÂB. This is easy to understand: when the The next step is to determine a semipermeable surface S within Λ, which is also referred to as barrier because it divides the set Λ into two subsets with opposite properties.

55 Chapter 3. Persistent Tracking A αr B 3 p(t) 2 Λ 22 1 Λ 21 Λ 23 x 1 1 O N p(t 2 ) p(t) 2 D 3 Λ C x 2 Figure 3.3: Configuration of the semipermeable surface The first subset, which is enclosed by the surface S and the UP, defines a region in which a termination of the game is possible. Starting from this subset, the GMT can always find a way to escape from the set Λ in finite time. In the complementary subset, the escape time of GMT is infinite (i.e., the GMT will not be able to escape if the UAV plays optimally); therefore, this complementary subset is the viability set we strive to find. Consider a point D on S, at which a tangent line of S exists. The inward normal vector at D, denoted by p R 2, is unique, apart from its magnitude. The surface S is then determined by max min p T f(x, u 1, u 2 ) = (3.7) u 1 u 2 which is a special case of the Isaacs equation [14, p ]. On the surface S, the equality p T f(x, u 1, u 2 ) = holds. Differentiating this equality with respect to time yields = ṗ T f(x, u 1, u 2 ) + p T ḟ(x, u 1, u 2 ) = ṗ T f(x, u 1, u 2 ) + p T f(x, u 1, u 2 ) ẋ x ( = ṗ T + p T f(x, u 1, u 2 ) ) f(x, u 1, u 2 ) x

56 Chapter 3. Persistent Tracking 43 which implies that ṗ T + p T f(x,u 1,u 2 ) x condition along the surface S dp dt = ; that is, the vector p satisfies the following = ( f x )T p (3.8) with the boundary condition given by p(t ) = ν. ν is the inward normal vector of Λ at S, and T is a positive number denoting the terminal time. The surface S apparently consists of two branches, which are symmetric with respect to x 1 -axis. The first branch can be constructed by solving Eq. (3.8) in retrogressive time from the end point B(x 1 = rα, x 2 = r 1 α 2 ). The other branch of S can be obtained by solving the same equation starting from the other end point of the UP, which is point A(x 1 = rα, x 2 = r 1 α 2 ). Substituting Eq. (3.3) into Eq. (3.7) yields the following equation max min{p 1 ( 1 + u 1 x 2 + α cos u 2 ) + p 2 ( u 1 x 1 + α sin u 2 )} (3.9) u 1 u 2 which leads to the following optimal solution u 1 = sgn(s) sin u 2 = where s p 1 x 2 p 2 x 1. cos u 2 = p 2 p 2 1 +p2 2 p 1 p 2 1 +p2 2 The differential equations derived from Eq. (3.8) are { ṗ1 = u 1 p 2 ṗ 2 = u 1 p 1 (3.1) The boundary condition of p can be normalized to unit magnitude since only the direction of the vector p matters. Thus we have { p1 (T ) = α sin(η) p 2 (T ) = 1 α 2 cos(η) In order to solve for p(t), the optimal control input u 1 must be determined in advance. We have s = when t = T, so u 1 (T ) cannot be determined directly. For an indirect derivation, we examine the derivative of s with respect to time, which is found to be ds dt = p 2 (3.11)

57 Chapter 3. Persistent Tracking 44 Since p 2 (T ) = cos(η) <, we have ds/dt < at t = T, which leads to s(t) > for t slightly smaller than T, and therefore u 1 = +1 just before the termination. Substituting u 1 = +1 into Eq. (3.1) leads to the following solution { p1 (t) = sin(t T + η) p 2 (t) = cos(t T + η) and the equations determining the branch of S ending at point B become { ẋ1 = 1 + x 2 + α sin(t T + η), x 1 (T ) = rα ẋ 2 = x 1 + α cos(t T + η), x 2 (T ) = r 1 α 2 (3.12) which are valid as long as s(t) >. The solution is found to be { x1 = (r + α(t T )) sin(t T + η) sin(t T ) x 2 = (r + α(t T )) cos(t T + η) + 1 cos(t T ) (3.13) which describes the right barrier BN shown in Fig Now we need to determine the starting point of this barrier. One candidate for the starting point is the one corresponding to the first retrogressive time instant when the sign of u 1 changes from positive to negative. Therefore, s(t) = at this point. The following equation can be derived from Eqs. (3.11) and (3.12) s(t) = sin(t T + η) + sin η and the largest solution t [, T ) of equation s(t) = is t 1 = T π 2η. Therefore the starting point can be calculated by substituting t 1 in to Eq. (3.13). Another candidate for the starting point is the intersection of these two branches. The two branches of barrier will intersect on x 1 -axis, thus forming a closed region within Λ, if the target set radius r satisfies the following condition r r c = α 2 + α(π + sin 1 α) (3.14) which is very similar to Eq. (8.3) in [14, p. 439]. This is the condition that guarantees the existence of a closed viability set in Λ. Here we choose the minimum radius r = r c and the resulting barriers are shown in Fig The two branches intersect at the point N on negative x 1 -axis. It is straightforward to verify that the coordinate of the point N is ( α, ) and the retrogressive time to the point N is t 2 = T π η. The inequality t 2 > t 1 implies that the two branches intersect at N

58 Chapter 3. Persistent Tracking 45 before u 1 reverses its sign in retrogressive time; therefore, we choose N as the starting position, and hence t 2 is the starting time. The upper closed region within Λ enclosed by barriers ÂN and BN and the arc ÂB is the viability set, which will be denoted by Λ 2 in the following analysis. The GMT cannot escape from Λ 2 by any means, provided that the UAV plays optimally. In other words, the UAV can always find a strategy to confine the GMT in Λ 2 forever if the GMT is originally in Λ 2. Based on the preceding analysis, the main conclusion of persistent tracking can be summarized by the following theorem. Theorem 3 (Persistent Tracking). Persistent tracking is feasible in the set Λ 2, i.e., x(t ) Λ 2, u 2 ( ) ( π, π], u 1 ( ) [ 1, 1] t t, x(t) Λ Persistent Tracking Algorithm The proof of Theorem 3 is constructively given by introducing an optimal tracking strategy for the UAV to achieve persistent tracking in Λ 2. We divide the set Λ 2 into three subsets by line OA, OB, and ON and label them as Λ 21, Λ 22, and Λ 23 as shown in Fig The optimal tracking strategy, described by Algorithm 1, is graphically illustrated in Fig Algorithm 1 (UAV Persistent Tracking Algorithm). If x 1 > u 1(t) = 1 if x 2 /x 1 > 1/k u 1(t) = sin ( π x 2 /x 1 δ 2 1/k δ if + δ < x 2/x 1 1/k u 1(t) = if δ < x 2 /x 1 δ u 1(t) = sin ( π x 2 /x 1 +δ 2 1/k+δ if 1/k < x 2/x 1 δ u 1(t) = 1 if x 2 /x 1 1/k (3.15) If x 1 { u 1 (t) = 1 if x 2 u 1(t) = 1 if x 2 < (3.16) where k = α/ 1 α 2, and δ is a small positive number introduced to ensure continuity of this algorithm. Before proving Theorem 3, we present the following lemma without proof as it readily follows the procedure of deriving the barrier equations.

59 Chapter 3. Persistent Tracking A F u 1 = sin( ) u 1 = sin( ) B u 1 = 3 2 x u 1 = 1 u 1 = 1 O N x 2 Figure 3.4: Persistent tracking algorithm for a UAV to confine a GMT within the viability set. The coordinate of point F is (r/ δ 2 + 1, rδ/ δ 2 + 1) Lemma 1 (Tracking Strategy on Barrier). On the left barrier ÂN, the tracking strategy of UAV is u 1 = 1; on the right barrier BN, the tracking strategy of UAV is u 1 = 1. Theorem 3 can then be proven by showing that the algorithm given by Eqs. (3.15) and (3.16) is a candidate of tracking strategies for the UAV to achieve persistent tracking in Λ 2. Proof. The UAV s action is determined by the region it is located in. If x(t) Λ 22, we have x 1 αρ, so we can derive ρ ρ = α(x 1 cos u 2 + x 2 sin u 2 ) x 1 αρ x 1 (3.17) where ρ = x x 2 2 r. Therefore, x(t) will never leave Λ 22 through the arc ÂB. In other words, x(t) will either remain in Λ 22 or enter Λ 21 or Λ 23, regardless of the UAV s control input u 1. If x(t) Λ 21, we have u 1 = 1. x(t) will either enter Λ 22 directly or reach the barrier ÂN. According to Lemma 1, x(t) starting from the barrier will eventually enter Λ 22.

60 Chapter 3. Persistent Tracking 47 Λ 22 u 1 [ 1, 1] u 1 [ 1, 1] u 1 = 1 u 1 = 1 u 1 = 1 u 1 = 1 Λ 21 Λ 23 u 1 = 1 ÂN BN u 1 = 1 Figure 3.5: Control flow of the persistent tracking algorithm If x(t) Λ 23, we have u 1 = 1. x(t) will either enter Λ 22 directly or reach the barrier BN. According to Lemma 1, x(t) starting from the barrier will eventually enter Λ 22. The tracking algorithm given in Eqs. (3.15) and (3.16) allows u 1 to retain its value at a previous instant of time when x(t) Λ 21 Λ 23 enters Λ 22 through the line OA or OB, so an instant change of the UAV control input is avoided. Remark 4. The tracking algorithm given by Eqs. (3.15) and (3.16) only requires the information of the relative position between the UAV and the GMT. This feature might be highly valuable when it is difficult to estimate the GMT s velocity with sufficient accuracy. Remark 5. The conclusions presented above are obtained by assuming that the GMT is able to change its heading angle instantaneously. This assumption greatly exaggerates the motion capability of the GMT under the given speed limit αv a. From the UAV s perspective, this treatment provides the persistent tracking algorithm with some robustness against the GMT s motion since a physical GMT is definitely less maneuverable than the simplified model. As a result, these conclusions derived upon this assumption remain valid in real implementations. This claim will be illustrated by simulations and experiments shortly.

61 Chapter 3. Persistent Tracking Simulation with a Unicycle UAV As shown in preceding sections, the GMT can not escape from the viability set Λ 2 by any means if it is originally within Λ 2. In other words, the GMT does not have an optimal strategy in Λ 2 in the sense of escaping time. In order to fully demonstrate the performance of the proposed tracking algorithm, four autonomous escaping strategies are applied to the GMT in simulations. The strategies are described as follows. Radially running away from the UAV at full speed (u 2 = atan2(x 1, x 2 ), v t = αv a ) Radially running away from the UAV at half speed (u 2 = atan2(x 1, x 2 ), v t = αv a /2) Constant velocity (u 2 = constant, v t = αv a ) Remaining stationary (v t = ) The GMT-UAV speed ratio α is set as.8, and the resulting minimum radius (normalized) of the detection zone is r c = as per Eq. (3.14). The proposed persistent tracking algorithm can ensure tracking in infinite time. In order to demonstrate this feature in simulations in finite time, the simulation time is adjusted in each case until the same tracking pattern repeats a few times. The simulation results associated with full speed and half speed strategies are given in Fig. 3.6 and 3.7, respectively. The four plots in each figure show the normalized relative trajectory in the UAV frame, the UAV and GMT trajectories in the inertial frame, the relative distance, and the UAV control input, respectively. As shown in these figures, the GMT is confined in the viability set in both cases through the simulation. The GMT s third strategy is to move with constant velocity, which means the GMT is moving in a straight line with a constant speed. Indicated by Fig. 3.8, the UAV is able to keep the GMT within the viability set through the simulation. The last autonomous escaping strategy the GMT takes is remaining stationary, which is definitely the worst strategy that the GMT should follow to evade the UAV. The resulting relative trajectory is a circle as shown in Fig. 3.9, and the GMT is confined in the viability set during the simulation. Note that all variables in Fig are dimensionless due to the normalization adopted to derive the simplified relative motion model in Eq. (3.3). Depending on the maximum turning rate of the UAV, one unit of the dimensionless time in these figures may represent several seconds in the physical time. Therefore, the change rates of the control inputs shown in these figure are not as high as they appear to be. Besides, they

62 Chapter 3. Persistent Tracking GMT path UAV path x 1 x x y (a) Trajectory in UAV frame (b) Trajectory in inertial frame Relative Distance Time UAV Turning Time (c) Relative distance (d) UAV control Figure 3.6: Simulation result with GMT running away from UAV at full speed. Diamond markers are the UAV and GMT starting positions are all continuous since the persistent tracking algorithm is designed to be continuous within the viability set. Assuming the maximum turning rate of the UAV is ω a,max =.2 rad/s, one unit of the dimensionless time represents 5 seconds in the physical time, and the dimensional control inputs in the first 7 seconds are displayed in Fig As we can see, the dimensional control inputs are continuous, and their change rates are moderate. Therefore, it is feasible to implement the proposed persistent tracking algorithm on real UAVs.

63 Chapter 3. Persistent Tracking GMT path UAV path x x x y (a) Trajectory in UAV frame (b) Trajectory in inertial frame Relative Distance UAV Turning Time Time (c) Relative distance (d) UAV control Figure 3.7: Simulation result with GMT running away from UAV at half speed. Diamond markers are the UAV and GMT starting positions

64 Chapter 3. Persistent Tracking 51 5 x x 2 x GMT path UAV path y (a) Trajectory in UAV frame (b) Trajectory in inertial frame Relative Distance UAV Turning Time Time (c) Relative distance (d) UAV control Figure 3.8: Simulation result with CV GMT. Diamond markers are the UAV and GMT starting positions

65 Chapter 3. Persistent Tracking GMT path UAV path x 1 x x y (a) Trajectory in UAV frame (b) Trajectory in inertial frame Relative Distance Time UAV Turning Time (c) Relative distance (d) UAV control Figure 3.9: Simulation result with stationary GMT. Diamond markers are the UAV and GMT starting positions

66 Chapter 3. Persistent Tracking 53 ω a (rad/s) ω a (rad/s) ω a (rad/s) ω a (rad/s).2 Full speed Half speed Constant speed Stationary Time (sec) Figure 3.1: UAV dimensional turning rate in simulation 3.5 Implementation with a 6DOF Nonlinear UAV In this chapter, the concept of persistent tracking and the corresponding persistent tracking algorithm are introduced by modeling the UAV as a unicycle vehicle. They might become inapplicable when a 6DOF nonlinear UAV model is considered. In this section, we show that the persistent tracking algorithm remains valid when the UAV is augmented by a properly designed flight controller. Throughout this section, the following notations will be used: (u, v, w) denotes the UAV velocity expressed in the body frame; (p, q, r) represents the angular rate expressed in the body frame; (φ, θ, ψ) are the Euler angles; V denotes the UAV airspeed; H denotes the UAV altitude; δ th denotes the throttle adjustment in percentage of the throttle setting at a trimmed condition; (δ e, δ a ) denotes the deflection of the elevator and aileron. A subscript c denotes the desired value of a variable. A prefix denotes a small perturbation of a variable from its reference value at a trimmed condition. When the reference value is zero, the prefix is omitted UAV Dynamic Model The tracking vehicle considered in the persistent tracking task is a flying wing UAV [15]. This UAV is equipped with a rear-installed brushless electric motor with a 9 6 inch propeller, powered by two lithium-polymer batteries. The control surface comprises two

67 Chapter 3. Persistent Tracking 54 elevons actuated by two standard hobby servos. The characteristic parameters of the UAV are given in Table 3.1. Table 3.1: Physical parameters of a flying wing UAV Span b (m) MAC a c (m) Mass m (kg) Wing Sweep Λ (deg) a MAC stands for mean aerodynamic chord. In order to establish the UAV s mathematical model, predictive methods presented by Jan Roskam in his textbook [16] were used to calculate the aerodynamic derivatives of the UAV. The methods presented by Roskam draw heavily upon the correlated wind-tunnel data of the USAF DATCOM [17]. The data provides various graphical and analytical techniques to predict aerodynamic derivatives based on the geometric parameters of planes, such as aspect ratio, wing sweep, static margin, and tail volume ratio. The literature of similar work shows that such approximation methods can be used to obtain sufficiently accurate estimation of these aerodynamics derivatives for flight control system design in initial stages [18]. The 6DOF equations of motion describing a UAV s nonlinear dynamics can be found in flight dynamics textbooks, such as [19]. For flight controller design, the nonlinear model is linearized through the small disturbance theory at a reference flight condition and decoupled into longitudinal and lateral linear models [19]. In this work, the reference flight condition is steady wings-level cruise with the airspeed V = 13 m/s, the altitude h = 1 meters, and the trimmed angle of attack α = 3.36 degrees. The longitudinal linear model can be described by the following equation in the state space format ẋ long = A long x long + B long u long (3.18) where x long = [ u w p θ] T is the longitudinal state vector, and u long = [ δ e δ th ] T is the longitudinal control input. A long and B long are system matrices of the longitudinal

68 Chapter 3. Persistent Tracking 55 model. Under the given flight condition, they are found to be A long = B long = , The lateral linear model can be similarly described by the following equation in the state space format ẋ lat = A lat x lat + B lat u lat (3.19) where x lat = [v p r φ] T is the lateral state vector, and u lat = δ a is the lateral control input. A lat and B lat are system matrices of the lateral model. Under the given flight condition, they are found to be A lat =, B lat = The control commands δ e and δ a are mixed by a mixer circuit before being executed by the two elevons. The deflection of the right and left elevons are calculated by the following mixing algorithm [ δr ] [ 1 1 ] [ δe ] δ L = δ a Flight Controller Design In order to achieve unicycle-like dynamics for the fixed-wing UAV so that the persistent tracking algorithm can be implemented, the flight controller must provide speed, altitude, and turn rate holding capabilities. To that end, a flight controller constructed of successive PID and Proportional-Integral (PI) control loops is designed. A PID controller is

69 Chapter 3. Persistent Tracking 56 defined by three parameters: proportional gain k P, integral gain k I, and derivative gain k D, and a PI controller is defined by only the first two gains [11]. The flight controller is separated into longitudinal and lateral controllers. The longitudinal controller provides speed and altitude holding functions. The purpose of the speed controller is to maintain the UAV airspeed V at a constant value by adjusting the throttle δ th. As illustrated in Fig. 3.11, the speed controller contains one PI controller to generate the throttle adjustment commands. V c + - k pv +k iv /s δ th Aircraft V Figure 3.11: Speed control loop. V c denotes the commanded speed The altitude-hold controller comprises an inner loop and an outer loop, and their block diagrams are shown in Fig and The inner-loop contains a PI controller and a damping feedback element, and it is responsible for holding the pitch angle θ by generating an elevator deflection command δ e. The outer loop, depicted in Fig. 3.13, employs one PI controller to generate the pitch angle command based on the altitude error. θ c + - k pθ +k iθ /s u 1 + -u - Actuator k dθ δ e Aircraft q θ Figure 3.12: Inner loop of altitude controller. θ c denotes the commanded pitch angle h c + - k ph +k ih /s θ c h/θ c h Figure 3.13: Outer loop of altitude controller. h c denotes the commanded altitude The lateral controller is responsible for tracking the turning rate command ψ generated by the persistent tracking algorithm. Under the coordinated-turn assumption, the

70 Chapter 3. Persistent Tracking 57 relationship between turning rate and roll angle is approximated as ψ = g V φ (3.2) where g is the gravitational constant. Therefore, this loop is essentially responsible for holding a suitable roll angle to achieve the desired turning rate. The structure of the lateral controller is depicted in Fig As can be seen, the lateral controller constitutes a PI controller and a damping feedback element. ϕ c + - k pϕ +k iϕ /s u 1 + -u - Actuator k dϕ δ a Aircraft p ϕ Figure 3.14: Turning rate control loop. φ c denotes the commanded roll angle The longitudinal and lateral flight controllers are designed separately initially and verified with the UAVs linear models in numerical simulations. When the overall performance is satisfactory in linear simulations, the two controllers are incorporated together with the 6DOF nonlinear UAV model, and the PID and damping gains are manually tuned through numerical simulations until the performance of the autopilot is satisfactory Simulation Results Figure 3.15 illustrates the configuration of the persistent tracking simulation using a 6DOF nonlinear UAV model. The control inputs (i.e., the UAV s turning rate) generated by the persistent tracking algorithm are passed as guidance commands to the flight controller described in the preceding subsection. The flight controller then generates actuator commands for the UAV to track these commands. The simulation parameters are given in Table 3.2, which indicates the minimum turning radius of the UAV is m. The GMT-UAV speed ratio α is also set as.8. At this speed ratio, the normalized minimum radius of the detection zone is r c = , and the dimensional minimum radius is m. The four escaping strategies described in the forgoing section are also used here, and the corresponding results are presented in Fig Note that all variables in these

71 Chapter 3. Persistent Tracking 58 GMT Persistent Tracking ψ & c = ω Algorithm UAV states Unicycle Kinematics h c =constant Autopilot V c =constant δ e, a, th UAV Figure 3.15: Augmented UAV in persistent tracking Table 3.2: Parameters in the persistent tracking simulation v a (m/s) ω a,max (rad/s) H (m) α figures are dimensional, except the GMT s trajectories expressed in the UAV frame. As shown in these figures, the UAV is able to confine the GMT within the viability set in all four cases. The simulation results demonstrate that the persistent tracking algorithm can be successfully implemented with the nonlinear UAV model when some augmentation mechanisms are provided. Besides, we can observe very similar behavior of the UAV in each scenario from simulation results with a unicycle UAV model presented in the preceding section. The similarities lie in the similar distance range achieved in each scenario. This implies that the dynamics of the augmented UAV resemble the unicycle model very closely. In addition, the resulting UAV control inputs are all in reasonable ranges. Hence, the proposed tracking algorithm and flight controller are applicable to a real UAV with actuator constraints. 3.6 Experiments with Ground Vehicles Due to the nature of the game theory, the exact behavior of a UAV and a GMT in a persistent tracking task are neither predictable nor repeatable. As a result, the numerical simulations might not be sufficient to demonstrate the performance of the proposed persistent tracking algorithm. Laboratory experiments using two ground robots were conducted in order to further validate this algorithm in a more realistic setting. This section describes the setup of the experiments and presents the experimental results.

72 Chapter 3. Persistent Tracking x x 2 x (m) UAV Path GMT Path y (m) (a) Normalized trajectory in UAV frame (b) Trajectory in inertial frame H (m) Elevator ( ) γ ( ) Aileron ( ) ψ ( ) Thrust Time (sec) Time (sec) (c) UAV states (d) UAV control Figure 3.16: Simulation result with 6DOF UAV and GMT running away from UAV at full speed. Diamond markers are the UAV and GMT starting positions

73 Chapter 3. Persistent Tracking UAV Path GMT Path x 1 x (m) x y (m) (a) Normalized trajectory in UAV frame (b) Trajectory in inertial frame H (m) Elevator ( ) γ ( ) Aileron ( ) ψ ( ) 2 Thrust Time (sec) Time (sec) (c) UAV states (d) UAV control Figure 3.17: Simulation result with 6DOF UAV and GMT running away from UAV at half speed. Diamond markers are the UAV and GMT starting positions

74 Chapter 3. Persistent Tracking x 1 x (m) x 2 2 UAV Path GMT Path y (m) (a) Normalized trajectory in UAV frame (b) Trajectory in inertial frame H (m) 1.2 γ ( ) Elevator ( ) Aileron ( ) ψ ( ) 1 Thrust Time (sec) Time (sec) (c) UAV states (d) UAV control Figure 3.18: Simulation result with 6DOF UAV and CV GMT. Diamond markers are the UAV and GMT starting positions

75 Chapter 3. Persistent Tracking UAV Path GMT Path x 1 x (m) x y (m) (a) Normalized trajectory in UAV frame (b) Trajectory in inertial frame H (m) Elevator ( ) γ ( ) Aileron ( ) ψ ( ) Thrust Time (sec) Time (sec) (c) UAV states (d) UAV control Figure 3.19: Simulation result with 6DOF UAV and stationary GMT. Diamond markers are the UAV and GMT starting positions

76 63 Chapter 3. Persistent Tracking Experiment Setup The two robots used in the experiment are identical in hardware configuration and are built from the irobot Create (two-wheel differential drive) platform. Each robot is equipped with a notebook computer, running the ROS (Robot Operating System) software [111] to perform motion control and data logging. A 1-camera Vicon motion capture system is used to provide ground truth data of robot positions at 2Hz with millimeter accuracy [112]. The ground robots and a Vicon camera are shown in Fig The experiments were carried out in an area of approximately 15 m 8 m. During the experiments, the robots recorded their position and heading angle measurements, as well as control inputs with timestamps. (a) irobot Create (b) Vicon camera Figure 3.2: Ground robots and the Vicon motion capture system The experiment parameters are given in Table 3.3, which indicates the minimum turning radius of the UAV robot is 1 m. The speed ratio α is also set as.8. At this speed ratio, the normalized minimum radius of the detection zone is rc = , and the dimensional minimum radius is m. Note that the GMT robot s maximum turning rate is significantly larger than that of the UAV robot, so it is reasonable to assume that the GMT can change its heading instantly. Table 3.3: Speed and turning rate of robots used in persistent tracking experiments va (m/s) ωa,max (rad/s) α ωt,max (rad/s) In order to avoid possible collisions between the two robots, the persistent tracking