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This work presents a novel method for 3-D path-following and path-tracking of Autonomous Underwater Vehicles (AUVs) using the concept of a Virtual Reference Point (VRP) and a kinematic guidance principle. The origins of the along-, cross- and vertical-track errors are proven globally exponentially stable (GES) using Lyapunov stability analysis. The kinematic guidance law exploits the design flexibility of a user-defined VRP in conjunction with a feedback linearizing controller. In addition, a novel concept called the Handy Matrix is introduced and applied to shape the kinematic equations such that the AUV’s non-actuated degrees of freedom (DOFs) can be controlled in a 3-D path-following scenario. The case study considers the Remus 100, a torpedo-shaped underactuated AUV, performing a 3-D path-following maneuver. The computer simulations show that the kinematic guidance law shows excellent tracking performance and stability even in the presence of ocean currents and white measurement noise.

The overall goal of the paper is to derive a kinematic model for vehicle path-following systems, where the pitch and yaw angles $(\theta, \psi)$ together with the surge velocity $u$ can be treated as control inputs. Then the outer control loop can be designed as a nonlinear path-following guidance law. At the same time, the inner control loops can be stabilized using commercial autopilot systems to control the pitch and yaw angles. The main result of the paper is a novel kinematic amplitude-phase representation of the North-East-Down (NED) positional rates, which can replace the classical Euler angle rotation matrix representation. The proposed kinematic model gives equivalent NED positional rates as the rotation matrix representation for all combinations of the Euler angles and the linear velocities. The main advantage of the kinematic amplitude-phase representation is the design of vehicle guidance laws using nonlinear control theory and Lyapunov stability analysis. Furthermore, it is shown that nonlinear guidance laws can be designed for path following and that the proposed methods guarantee that the origins are uniformly semiglobally exponentially stable (USGES). A case study of an unmanned surface vehicle (USV) demonstrates how a path-following control system can be designed using the amplitude-phase representation.

Docking is a safety-critical operation for autonomous surface vehicles and requires highly accurate navigation signals. Since Global Navigation Satellite Systems (GNSS) can be unreliable and inaccurate in urban environments, other sensors should be considered for increased redundancy and reliability. To this end, we present a low-cost visual-inertial navigation system that we can use for automatic docking of small vehicles. The proposed system produces state estimates of the vehicle, including position, velocity, and attitude, based on raw image and inertial data. To simplify the navigation task, we use easily identifiable tags as a reference on the dockside. When the vehicle approaches the dock, a visual fiducial system recognizes the tags and estimates the relative pose between the camera onboard the vehicle and the tags at the dockside. The camera-tag pose and inertial data are then fused using an error-state Kalman filter for robust state estimation of the vehicle. For benchmarking, we use an unmanned surface vehicle equipped with a dual-antenna real-time kinematic GNSS receiver for accurate positioning and heading. We show that the proposed method performs well on regular and adverse weather data. Finally, we demonstrate that the proposed method performs well in feedback control through field experiments and can supplement traditional navigation systems for docking operations.

Dynamic models of ships have been widely used for model-based control and short-term prediction in the past. Identifying the parameters of such models has mainly been done through scaled model tests, full scale tests or computational fluid dynamics software. This is a challenging task due to the many aspects that influence the ship dynamic behaviour and thus one would expect a certain degree of mismatch between the actual motion of the ship and the modelled behaviour. The mismatch in the dynamic model may be due to unmodelled effects, but also the lack of measurements of waves and ocean current. To make up for the discrepancies the authors propose to create a co-operative hybrid model consisting of the dynamic model and a neural network, where the neural network predicts the acceleration error of the dynamic model. The approach is tested on real data originating from the Research Vessel (RV) Gunnerus performing a shutdown of thrusters during stationkeeping. The subsequent task is to predict the propagation of position and heading while drifting due to wind, wave and current forces. Comparing the motion of the real ship and the modelled ship, shows the improved prediction accuracy of the hybrid model.

This paper addresses the problem of ship motion estimation using live data from Automatic Identification Systems (AIS). A globally exponentially stable observer for visualization and motion prediction of ships has been designed. Instead of using the extended Kalman filter (EKF) to deal with the kinematic nonlinearities the eXogenous Kalman Filter (XKF) is applied and by this global stability properties are proven. The proposed observer was validated using live AIS data from the Trondheim harbor in Norway and it was demonstrated that the observer tracks ships in real time. It was also demonstrated that the observer can predict the future motion of ships. The motion prediction capabilities are very useful for decision-support systems since this can be used to improve situational awareness e.g. for manned and unmanned autonomous ships that operate in common waters.

Small USVs are usually equipped with a low-cost navigation sensor suite consisting of a global navigation satellite system (GNSS) receiver and a magnetic compass. Unfortunately, the magnetic compass is highly susceptible to electromagnetic disturbances. Hence, it should not be used in safety-critical autopilot systems. A gyrocompass, however, is highly reliable, but it is too expensive for most USV systems. It is tempting to compute the heading angle by using two GNSS antennas on the same receiver. Unfortunately, for small USV systems, the distance between the antennas is very small, requiring that an RTK GNSS receiver is used. The drawback of the RTK solution is that it suffers from dropouts due to ionospheric disturbances, multipath, interference, etc. For safety-critical applications, a more robust approach is to estimate the course angle to avoid using the heading angle during path following. The main result of this article is a five-state extended Kalman filter (EKF) aided by GNSS latitude-longitude measurements for estimation of the course over ground (COG), speed over ground (SOG), and course rate. These are the primary signals needed to implement a course autopilot system onboard a USV. The proposed algorithm is computationally efficient and easy to implement since only four EKF covariance parameters must be specified. The parameters need to be calibrated for different GNSS receivers and vehicle types, but they are not sensitive to the working conditions. Another advantage of the EKF is that the autopilot does not need to use the COG and SOG measurements from the GNSS receiver, which have varying quality and reliability. It is also straightforward to add complementary sensors such as a Doppler Velocity Log (DVL) to the EKF to improve the performance further. Finally, the performance of the five-state EKF is demonstrated by experimental testing of two commercial USV systems.

This paper presents nonlinear, singularity-free autopilot designs for multivariable reduced-attitude control of fixed-wing aircraft. To control roll and pitch angles, we employ vector coordinates constrained to the unit two-sphere and that are independent of the yaw/heading angle. The angular velocity projected onto this vector is enforced to satisfy the coordinated-turn equation. We exploit model structure in the design and prove almost global asymptotic stability using Lyapunov-based tools. Slowly-varying aerodynamic disturbances are compensated for using adaptive backstepping. To emphasize the practical application of our result, we also establish the ultimate boundedness of the solutions under a simplified controller that only depends on rough estimates of the control-effectiveness matrix. The controller design can be used with state-of-the-art guidance systems for fixed-wing unmanned aerial vehicles (UAVs) and is implemented in the open-source autopilot ArduPilot for validation through realistic software-in-the-loop (SITL) simulations.

This paper proposes a novel methodology, based on Fermat's spiral (FS), for constructing curvature-continuous parametric paths in a plane. FS has a zero curvature at its origin, a property that allows it to be connected with a straight line smoothly, that is, without the curvature discontinuity which occurs at the transition point between a line and a circular arc when constructing Dubins paths. Furthermore, contrary to the computationally expensive clothoids, FS is described by very simple parametric equations that are trivial to compute. On the downside, computing the length of an FS arc involves a Gaussian hypergeometric function. However, this function is absolutely convergent and it is also shown that it poses no restrictions to the domain within which the length can be calculated. In addition, we present an alternative parametrization of FS which eliminates the parametric speed singularity at the origin, hence making the spiral suitable for path-tracking applications. A detailed description of how to construct curvature-continuous paths with FS is given.

This paper considers the subject of straight-line target tracking for unmanned surface vehicles (USVs). Target-tracking represents motion control scenarios where no information about the target behavior is known in advance, i.e., the path that the target traverses is not defined apriori. Specifically, this work presents the design of a motion control system which enables an underactuated USV to track a target that moves in a straight line at high speed. The motion control system employs a guidance principle originally developed for interceptor missiles, as well as a novel velocity controller inspired by maneuverability and agility concepts found in fighter aircraft literature. The performance of the suggested design is illustrated through full-scale USV experiments in the Trondheimsfjord.

This paper addresses the problem of joint identification of infinite-frequency added mass and fluid memory models of marine structures from finite frequency data. This problem is relevant for cases where the code used to compute the hydrodynamic coefficients of the marine structure does not give the infinite-frequency added mass. This case is typical of codes based on 2D-potential theory since most 3D-potential-theory codes solve the boundary value associated with the infinite frequency. The method proposed in this paper presents a simpler alternative approach to other methods previously presented in the literature. The advantage of the proposed method is that the same identification procedure can be used to identify the fluid-memory models with or without having access to the infinite-frequency added mass coefficient. Therefore, it provides an extension that puts the two identification problems into the same framework. The method also exploits the constraints related to relative degree and low-frequency asymptotic values of the hydrodynamic coefficients derived from the physics of the problem, which are used as prior information to refine the obtained models.

The dynamics describing the motion response of a marine structure in waves can be represented within a linear framework by the Cummins Equation. This equation contains a convolution term that represents the component of the radiation forces associated with fluid memory effects. Several methods have been proposed in the literature for the identification of parametric models to approximate and replace this convolution term. This replacement can facilitate the model implementation in simulators and the analysis of motion control designs. Some of the reported identification methods consider the problem in the time domain while other methods consider the problem in the frequency domain. This paper compares the application of these identification methods. The comparison is based not only on the quality of the estimated models, but also on the ease of implementation, ease of use, and the flexibility of the identification method to incorporate prior information related to the model being identified. To illustrate the main points arising from the comparison, a particular example based on the coupled vertical motion of a modern containership vessel is presented.

The Marine Systems Simulator (MSS) is an environment which provides the necessary resources for rapid implementation of mathematical models of marine systems with focus on control system design. The simulator targets models¡Xand provides examples ready to simulate¡Xof different floating structures and its systems performing various operations. The platform adopted for the development of MSS is Matlab/Simulink. This allows a modular simulator structure, and the possibility of distributed development. Openness and modularity of software components have been the prioritized design principles, which enables a systematic reuse of knowledge and results in efficient tools for research and education. This paper provides an overview of the structure of the MSS, its features, current accessability, and plans for future development.

This paper presents a computer effective nonlinear time-domain strip theory formulation for dynamic positioning (DP) and low-speed manoeuvring. Strip theory or 2D potential theory, where the ship is divided in 20 to 30 cross sections, can be used to compute the potential coefficients (added mass and potential damping) and the exciting wave loads (Froude-Krylov and diffraction forces). Commercially available programs are ShipX (VERES) by Marintek (Fathi, 2004) and SEAWAY by Amarcon (Journée & Adegeest, 2003), for instance. The proposed method can easily be extended to utilize other strip theory formulations or 3-D potential programs like WAMIT (2004). The frequency dependent potential damping, which in classic theory results in a convolution integral not suited for real-time simulation, is compactly represented by using the state-space formulation of Kristiansen & Egeland (2003). The separation of the vessel model into a low-frequency model (represented by zerofrequency added mass and damping) and a wave-frequency model (represented by motion transfer functions or RAOs), which is commonly used for simulation, is hence made superfluous. Transformations of motions and coefficients between different coordinate systems and origins, i.e. data frame, hydrodynamic frame, body frame, inertial frame etc., are put into the rigid framework of Fossen (1994, 2002). The kinematic equations of motion are formulated in a compact nonlinear vector representation and the classical kinematic assumption that the Euler angles are small is removed. This is important for computation of accurate control forces at higher roll and pitch angles. The hydrodynamic forces in the steadily translating hydrodynamic reference frame (equilibrium axes) are, however, assumed tobe linear. Recipes for computation of retardation functions are presented and frequency dependent viscous damping is included. Emphasis is placed on numerical computations and representation of the data from VERES and SEAWAY in Matlab/Simulink. For this purpose a Simulink add-in to the Marine Systems Simulator (MSS) at the Norwegian University of Science and Technology has been developed (Fossen et al., 2004).

A new strategy for active control in heavy-lift offshore crane operations is suggested, by introducing a new concept referred to as wave synchronization. Wave synchronization reduces the hydrodynamic forces by minimization of variations in the relative vertical velocity between payload and water using a wave amplitude measurement. Wave synchronization is combined with conventional active heave compensation to obtain accurate control. Experimental results using a scale model of a semi-submerged vessel with a moonpool shows that wave synchronization leads to significant improvements in performance. Depending on the sea state and payload, the results indicate that the reduction in the standard deviation of the wire tension may be up to 50

Complete nonlinear dynamic manoeuvering models of ships, with numerical values, are hard to find in the literature. This paper presents a modeling, identification, and control design where the objective is to manoeuver a ship along desired paths at different velocities. Material from a variety of references have been used to describe the ship model, its difficulties, limitations, and possible simplifications for the purpose of automatic control design. The numerical values of the parameters in the model is identified in towing tests and adaptive manoeuvering experiments for a small ship in a marine control laboratory.

Control of fluid flows span a wide variety of specialities. In Part II of this tutorial, we focus on diagnostics of mixing and the problem of enhancing mixing by boundary feedback control. Diagnostic tools from dynamical systems theory are presented that enable detection and quantification of chaotic transport in periodically perturbed systems. However, real systems are generally not periodic, and available measurements or simulations are finite in time. A method for quantifying mixing in finite-time velocity fields is discussed, and applied to data obtained from simulations of the 2D controlled channel flow. Mixing has traditionally been brought on by open-loop control strategies, such as stirring, jet injection or mixing valves. Applications of active feedback to mixing problems are scarce in the literature, but the idea is currently drawing attention from various research groups. Feedback laws for the purpose of mixing enhancement in 2D and 3D pipe flow are presented, and simulations show that they induce strong mixing.

The field of flow control has picked up pace over the past decade or so, on the promise of real-time distributed control on turbulent scales being realizable in the near future. This promise is due to the micromachining technology that emerged in the 1980s and developed at an amazing speed through the 1990s. In lab experiments, so called micro-electro-mechanical systems (MEMS) that incorporate the entire detection-decision-actuation process on a single chip, have been batch processed in large numbers and assembled into flexible skins for gluing onto body-fluid interfaces for drag reduction purposes. Control of fluid flows span a wide variety of specialities. In Part I of this tutorial, we focus on the problem of reducing drag in channel and pipe flows by stabilizing the parabolic equilibrium profile using boundary feedback control. The control strategics used for this problem include classical control, based on the Nyquist criteria, and various optimal control techniques (H2, H-Infinity), as well as applications of Lyapunov stability theory.

In this paper, a nonlinear tracking controller with integral action for ships is presented. The controller is based on state feedback linearization. Exponential convergence of the vessel-fixed position and velocity errors are proven by using Lyapunov stability theory. Since we only have two control devices, a rudder and a propeller, we choose to control the longship and the sideship position errors to zero while the heading is stabilized indirectly. A Virtual Reference Point (VRP) is defined at the bow or ahead of the ship. The VRP is used for tracking control. It is shown that the distance from the center of rotation to the VRP will influence on the stability of the zero dynamics. By selecting the VRP at the bow or even ahead of the bow, the damping in yaw can be increased and the zero dynamics is stabilized. Hence, the heading angle will be less sensitive to wind, currents and waves. The control law is simulated by using a nonlinear model of the Japanese training ship Shiojimaru with excellent results. Wind forces are added to demonstrate the robustness and performance of the integral controller.

The theoretical foundation of nonlinear backstepping designs is presented in a tutorial setting. This includes a brief review of integral backstepping, extensions to SISO and MIMO systems in strict feedback form and physical motivated case studies. Parallels and differences to feedback linearization where it is shown how so-called "good nonlincarities" can be exploited in the design are also made.

This paper addresses the mathematical modelling and controller design of an automatic thruster assisted position mooring system. Such control systems are applied to anchored floating production offloading and storage vessels and semi-subs. The controller is designed using model based control with a LQG feedback controller in conjunction with a Kalman filter. The controller design is in addition to the environmental loads accounting for the mooring forces acting on the vessel. This is reflected in the model structure and in the inclusion of new functionality.

A dynamic positioning (DP) system includes different control functions for automatic positioning and guidance of marine vessels by means of thruster and propeller actions. This paper describes the control functions which provide station-keeping and tracking. The DP controller is designed using model-based control, where a new modified LQG feedback controller and a model reference feedforward controller are applied. A reference model calculates appropriate reference trajectories. Since it is not desirable nor even possible to counteract the wave-frequency movement caused by first-order wave loads, the control action of the propulsion system should be produced by the low frequency part of the vessel movement caused by current, wind and second-order mean and slowly varying wave loads. A Kalman filter based state estimator and a Luenberger observer are used to compute the low-frequency feedback and feedforward control signals. Full-scale experiments with a supply vessel demonstrate the performance of the proposed controller.

A stable minimum phase transfer function from rudder angle to yaw angle is used to design a globally stable adaptive ship autopilot. First-order wave disturbances in yaw are filtered by applying a notch filter. Integral action is introduced by using a reference model technique. Global stability is proven for the total system which include the yaw rate observer, the parameter update law, the feedback controller, the notch filter and the integral part of the controller. The simulation results showed that the performance is excellent, even with no a priori knowledge of the ship parameters.

Todays model-based dynamic positioning (DP) systems require that the ship and thruster dynamics are known with some accuracy in order to use linear quadratic optical control theory. However, it is difficult to identify the mathematical model of a dynamically posititmed (DP) ship since the ship is not persistently excited under DP. In addition the ship parameter estimation problem is nonlinear and multivariable with only position and thruster state measurements available for parameter estimation. The process and measurement noise must also be modeled in order to avoid parameter drift due to environmental disturbances and sensor failure. This article discusses an off-line parallel extended Kalman filter (EKF) algorithm utilizing two measurement series in parallel to estimate the parameters in the DP ship model. Full-scale experiments with a supply vessel are used to demonstrate the convergence and robustness of the proposed parameter estimator.

Robust adaptive control of underwater vehicles in 6 DOF is analysed in the context of measurement noise. The performance of the adaptive control laws of Sadegh and Harowitz (1990) and Slotine and Benedetto (1990) are compared. Both these schemes require that all states are measured, that is the velocities and positions in surge, sway, heave, roll, pitch and yaw. However, for underwater vehicles it is difficult to measure the linear velocities whereas angular velocity measurements can be obtained by using a 3 axes angular rate sensor. This problem is addressed by designing a nonlinear observer for linear velocity state estimation. The proposed observer requires that the position and the attitude are measured, e.g. by using a hydroacoustic positioning system for linear positions, two gyros for roll and pitch and a compass for yaw. In addition angular rate measurements will be assumed available from a 3-axes rate sensor or a state estimator. It is also assumed that the measurement rate is limited to 2 Hz for all the sensors. Simulation studies with a 3 DOF AUV model are used to demonstrate the convergence and robustness of the adaptive control laws and the velocity state observer.

This paper presents a cascade adaptive control scheme for marine vehicles where the non-linear equations of motion include a model of the actuator dynamics. The adaptive controller does not require the parameters of the vehicle dynamics and the actuator time constants to be known a priori. Both the velocity and position tracking errors are shown to converge to zero by applying Barbalat's lemma. Global asymptotic stability is proven for the velocity scheme while the position/attitude controller is only proven to be convergent. Furthermore, all parameter estimates are shown to be bounded. Computer simulations of an ROV speed control system and an autopilot for automatic ship steering are used to illustrate the design methodology.

This paper describes the application of feedback linearization to automatic steering of ships. The flexibility of the design procedure allows the autopilot to be optimized for both course-keeping and course-changing manoeuvres. Direct adaptive versions of both the course-keeping and turning controller are derived. The advantages of the adaptive controllers are improved performance and reduced fuel consumption. The application of nonlinear control theory also allows the designer in a systematic manner to compensate for nonlinearities in the control design.

A direct model reference adaptive controller (MRAC) is derived for an underwater vehicle with significant thruster dynamics and limited thruster power. The reference model decomposition (RMD) technique is used to compensate for the thruster dynamics. A reference model adjustment (RMA) technique modifying the reference model acceleration is used to avoid thruster saturation. The design methods are simulated for the yawing motion of an underwater vehicle.

Sliding control of MIMO (multivariable input multivariable output) nonlinear minimum phase systems is discussed. Stability conditions related to model errors are emphasized. Global asymptotic stability is guaranteed by applying Barbalat's Lyapunov-like lemma. The control law is applied to a simulator of a polymerization reactor.

The problem of controlling underwater mobile robots in 6 degrees of freedom (DOF) is addressed. Uncertainties in the input matrix due to partly known nonlinear thruster characteristics are modeled as multiplicative input uncertainty. This paper proposes two methods to compensate for the model uncertainties: (1) an adaptive passivity-based control scheme and (2) deriving a hybrid (adaptive and sliding) controller. The hybrid controller consists of a switching term which compensates for uncertainties in the input matrix and an on-line parameter estimation algorithm. Global stability is ensured by applying Barbalat's Lyapunovlike lemma. The hybrid controller is simulated for the horizontal motion of the Norwegian Experimental Remotely Operated Vehicle (NEROV).

When applying feedback linearization techniques actuator dynamics is usually neglected. This is crucial for actuators with large time constants. A large class of systems, e.g. chemical processes, belong to this category. In such cases conventional feedback linearization techniques based on adaptive control or sliding control perform poorly. This paper discusses an approach which allows the inclusion of the actuator dynamics in the controller design. The design methodology has validation both for adaptive feedback linearization as well as sliding control. An augmented Lyapunov-like function candidate including the actuator states was used to prove global stability. This is done by applying Barbalat's Lyapunov-like lemma for non-autonomous systems. The control law is simulated for both the nominal and adaptive cases. In the adaptive case both the plant parameters and the actuator time constant are assumed to be unknown.