Robust and Optimal Control Designed for Autonomous Surface Vessel Prototypes

It is well known that activities in running water or wind and waves expose the Autonomous Surface Vessels (ASVs) to considerable challenges. Under these conditions, it is essential to develop a robust control system that can meet the requirements and ensure the safe and accurate execution of missions. In this context, this paper presents a new topology for controller design based on a combination of the Successive Loop Closure (SLC) method and optimal control. This topology enables the design of robust autopilots based on the Proportional-Integral-Derivative (PID) controller. The controllers are tuned from the solution of the optimal control problem, which aims to minimize the effects of model uncertainties. To verify the effectiveness of the proposed controller, a numerical case study of a natural ASV with 3 Degree of Freedom (DoF) is investigated. The results show that the methodology enabled the tuning of a PID controller capable of dealing with different parametric uncertainties, demonstrating robustness and applicability for different prototype scenarios.


I. INTRODUCTION
Among autonomous vehicles, Autonomous Surface Vessels (ASVs) are the ones that increasingly attract researchers around the world. Characterized by low cost, high mobility, and a high degree of autonomy, these vehicles offer applications in a wide variety of marine environments [1], [2], [3]. It is possible to find applications in lakes, rivers and open seas. Activities can range from water and harbor monitoring [4], shallow water hydrological survey [5], maritime search and rescue [6], bathymetric mapping [7], and many other applications [8].
Given the great diversity of environments and activities, ASVs are subject to scenarios from calm waters to envi-The associate editor coordinating the review of this manuscript and approving it for publication was Ton Duc Do . ronments with high exposure to disturbances (for example, currents, winds, and waves) [9], [10]. Furthermore, vessels can also be exposed to challenging applications in shallow waters such as swamps, estuaries, mangroves, lagoons, and coral reefs [11], [12]. In these conditions, conventional vessels cannot operate easily due to limitations imposed on maneuverability and care taken to ensure that the propellers are not damaged. These aspects lead to the need of developing ASVs with a greater maneuverability from new propulsion system topologies and control adjustment techniques [13].
Besides, under difficult operating conditions, the unmodeled dynamics can not be covered by the control system, which can arise and compromise the safety and reliability of the operations. Therefore, it also promotes the need for robust control systems in challenging and uncertain scenarios.
However, designing robust control systems is not a trivial task. A control system is considered robust only when it can guarantee robust stability and/or performance in the presence of assumed uncertainties (e.g., parametric variations, neglected nonlinearities, and unmodeled dynamics) [14], [15]. To reach these criteria, it is necessary to deal with complex solution spaces, convergence problems for specific uncertain sets, and difficulties in obtaining sufficiently robust and simple models to develop control techniques [16]. Moreover, there is also a consensus on the need to seek simplicity in design and practical implementation.

A. STATE OF THE ART AND RELATED WORKS
A quick search on the internet shows many and different ASV topologies for different applications.
In spite of these innovative ASVs, there are still situations that those approaches could need improvements or a better performance. At this moment, let's consider the contributions [13] and [17], where an over-actuated catamaran ASV was designed for shallow waters: it is still necessary to implement well-tuned controllers to perform missions as they are requested.
Taking the methodology for controller tuning, many control approaches have been found in the literature to deal with the mentioned problem. Conventional PID controller, High-Gain State Observer (HGSO), adaptive control, Sliding-Mode Control (SMC), intelligent control, and H-infinity technique are some approaches [18].
Within the scope of PID controllers, the works [19], [20], [21] are examples of successful applications in ASVs. Although these approaches have their limitations when dealing with highly coupled nonlinearities and time-varying dynamics, they are often preferred due to their synthesis and simplification [22]. To reduce these initial limitations, several studies have been conducted to overcome them by combining other techniques, such as in paper [23].
Papers [24], [25], [26] present applications of HGSO in marine vehicle control. It is possible to understand from these works the benefits of applying a well-designed HGSO that makes the control system robust to system uncertainties and with high disturbance rejection. Furthermore, the works also demonstrate the success in obtaining the stability proof of the closed-loop system. It can be an obstacle for systems with complex models and make the applied methodology hard in some cases.
Classical adaptive control approaches are found in the contributions [27], [28], [29]. Specifically, work [27] depicts a comparative study between the techniques of gain scheduling, model reference adaptive control, and L 1 adaptive control. Both methods are investigated in the speed control of the SeaFox vessel, seeking to minimize the effects of the propulsion system uncertainties that occur between low and high-speed operating modes. Good results are observed even in transition and high-speed modes. It is also clear that such techniques must be applied carefully, especially in situations of great uncertainty. Under these conditions, in particular, the convergence problems and parametric oscillations can run, impairing the desired stability and robust performance.
Related to methodologies based on SMC, works [30], [31], [32] are interesting examples of application in ASVs. Paper [31] presents a classic SMC application, while [30] and [32] show changes to increase the closed-loop system robustness and reduce control system chattering. Regardless of the approach adopted, all the contributions were devoted to the initial objectives and are satisfactory in the presence of disturbances and parametric uncertainties. However, it is also possible to understand that the SMC must be applied more carefully, as the controlling action of the rigid slip mode can lead to chattering, energy loss, plant damage, and unmodeled dynamic excitation.
In artificial intelligence approaches, it is possible to find applications of artificial neural networks, Bayesian probability, fuzzy logic, machine learning, evolutionary computation, genetic algorithms, or a combination of these methods. Works [18], [33], [34] present some of these applications. Although intelligent controllers have proven to be good control options, typically, this approach requires a long parameter-setting process. They are often used in experimental vehicles, although industrial vehicles are still an opportunity for these control techniques [35].
Finally, regarding robust control techniques, the papers [36], [37] are interesting examples. Both articles propose controllers based on the infinite control theory that could deal with the uncertainties involved. However, it is noteworthy that the approaches provide strong robustness, also requiring complex solution processes, which deserve attention in establishing the problem [38].
Through the mentioned work, it is possible to understand the difficulties in designing a robust controller. A key element in this process is the simplicity of the control topology. Simpler control laws make testing and implementation steps easier. Therefore, avoiding control law synthesis requires sophisticated numerical solution methods and time-consuming hardware implementation [39]. In this sense, the development of robust controllers based on the traditional PID is a promising path, as long as it is combined with other techniques that overcome the deficiencies of the PID controller [23].
In this direction, the combination of the SLC method with optimal control is quite innovative. To the authors' knowledge, there are no robust PID controller tuning algorithms with these characteristics applied to ASVs. A fact that can be explored and later applied to vehicle driving in general, such as the control algorithms developed in [40] and [41].

B. CONTRIBUTIONS
This work is a part of a project where some important publications were previously done, which are: • Citation number [13] The first one is focused on the ASV design, looking at its construction to minimize the effects and interference of its displacements in the water on the Acoustic Doppler Current Profiler (ADCP) sensor, embedded in the ASV. The second one takes the ASV design and control allocation into account, using also the traditional laws for its modeling. The third one looks at the ASV parameters identification, where a new methodology for optimal parameter estimation was applied and proved. Both of these 3 works were experimentally tested.
Under this background, this paper presents a new control design topology from the combination of the Successive Loop Closure (SLC) method and optimal control. This topology allows the designing of robust autopilots based on PID controllers. The controllers are adjusted from the solution of the optimal control problem using the Interior-Point Algorithm (IPA) to minimize the effects of model uncertainties. A numerical case study of a real ASV with 3 DoFs investigates the proposed controller effectiveness, exposing it to uncertainty scenarios based on the vessel's operational movements.
After this summary, the main contributions of this paper can be written below, and thus: • Depicts the modeling and identification of a catamaran type ASVs with aerial propulsion, considering the coupled hydrodynamics parameters, which are not well explored in the literature; • Presents a robust controller tuning methodology based on SLC and optimal control to design PID autopilots in cascade control applied to an ASV; • Makes in-depth a case study about controller tuning of real ASV in solution space R 12 and robust stability analysis of closed-loop control with uncertainty analysis in R 27 .

C. PAPER ORGANIZATION
This paper is organized as follows: Section II presents the robust control problem, showing its characteristics and complexities; Section III deals with the vessel used for the study and development of the robust control technique; Section IV describes the new methodology of robust control; Section V presents the numerical results of the application of the method in different scenarios; and Section VI presents some concluding remarks.

II. PROBLEM STATEMENT
Assuming that the Nonlinear Dynamic Systems (NDS) is satisfactorily approximated by a nonlinear parametric model M( ), Equation 1 is presented: where f and h are the nonlinear functions of the system, x ∈ R n is the state vector, u ∈ R p is the action vector of the PID controller, y ∈ R m is the output vector, ∈ R r is set of uncertain parameters about NDS and ∈ R q represents the PID controller parameters.
Suppose that is unknown, time-variant and contains an uncertainty region ∈ R r . Also, is a region around the priori estimateˆ with radius δ . Therefore, is defined by: In this context, the design problem of a robust PID controller can be defined as: ''find the control law parameters u(t, ) such as the closed-loop system is globally asymptotically stable and has robust performance for all the parametric uncertainty region ''.
The optimal control can also fulfill the controller robustness, establishing an optimization problem whose minimization assures the robustness criteria. In this way, the controller design can be expressed by the following optimization problem: where − is the vector of the PID controller initial parameters, X is the established reference vector to the controller loops, and S is an arbitrary nonlinear optimization algorithm that searches for the best ⊕ ∈ R q . Finally, the posterior parameter set obtained by S is defined by + .

III. THE AUTONOMOUS SURFACE VESSEL PROTOTYPE
The ASV is a vessel developed to autonomously collect hydrological measurements in environments with underwater obstacles, and shallow and fast water flows.
Based on a catamaran-type vessel, this ASV has an innovative air propulsion system with azimuth control. This setting generates an ASV with 3 over-actuated DoFs, highly maneuverable and capable of operating in the situations mentioned above. Fig. 1 shows the vessel developed. More information about it is found in works [13], [17].
For the best presentation of the ASV coordinate frames, Fig. 2 is shown: where roll, pitch and yaw angles (φ, θ, ψ) are measured in the Inertial Frame F I (î I ,ĵ I ,k I ) which considers the Vehicle Frame F υ (translation transformation from F I ) and the Body-

A. MODELING AND IDENTIFICATION
Traditional surface vessel modeling requires its development from the description of 3 DoFs, i.e., x and y axis translation movements (Surge and Sway) and rotation around the z-axis VOLUME 11, 2023 (Yaw). The others DoFs are disregarded due to their small influence over the vessel's dynamics [43].
According to the marine vehicle nomenclature, the state vectors are expressed as follows: η = [x, y, ψ] T representing inertial (x, y) and angular (ψ) positions in the vehicle inertial frame F I ; ν = [u, υ, r] T representing the linear (u, v) and angular (r) velocities in the Rigid-Body frame F b [43].
The general ASV dynamics and kinematics modeling (without disturbance) is represented as follows [44]: where M ∈ R 3×3 is the Inertia Matrix; C(ν) ∈ R 3×3 is the Coriolis and centripetal Matrix; J(ψ) is the Jacobian matrix that relates the velocities in the Rigid-Body and the inertial frame; D(ν) ∈ R 3×3 is the hydrodynamic damping; τ ∈ R 3×1 is the vector of generalized forces and moments applied by the propellant system. Considering all the couplings between 3 DoFs, the matrices M, C(ν) and J(ψ) are presented as [44]: where m is the vehicle total mass, I z is the z-axis moment of inertia, x g is the gravity center displacements on x-axis. Additionally, the elements Xu, Xv, Xṙ , Yu, Yv, Yṙ , Nu, Nv, Nṙ represent the hydrodynamic derivatives of the added mass phenomenon. Moreover, β 1 = mu + [Xuu + Xvv + Xṙ r] and Finally, the hydrodynamic damping D(ν) is expressed by the superposition of linear and nonlinear damping, and this is associated with quadratic damping. Thus, D(ν) is expressed by [44]: N u , N r represent the linear damping, while all the others represent the nonlinear damping of a quadratic order.
The vessel parameters were identified in two stages. First, the inertial characteristics were obtained by direct measurements and Computer-Aided Design (CAD) tools. Later, the parameters of the M, C(ν) and D(ν) were estimated by the rSOESGOPE method, proposed by [42]. This methodology favors robust parametric estimation, which is fundamental to the problem addressed in this work. The results are shown in Tables 1 and 2. It is important to highlight that the parameters from the table above were validated with open-field experiments in the work [42], where this ASV was exhaustively tested in a real and non-controlled environment.

B. STRUCTURE AND CONTROL LEVELS
This section will describe the structure control developed in this work and embedded in the over-actuated ASV.
This vehicle performs maneuvers using differential torque on the motors or tilting their 4 servomotors independently. This type of possibility leads this vehicle to be faster, more versatile, and have a great torque application, among other benefits. In addition, this vehicle performs yaw maneuvers just by adjusting the 4 servomotors tilting angles.
However, these abilities lead to implementing a different control allocation technique for the vehicle's actuators. This is due to the existence of 3 Virtual Control Actions (VCAs) and 8 Real Control Actions (RCAs) (4 propulsion motors and 4 servomotors), which in fact has a non-unique solution. The control allocation technique considered is named Fast Control Allocation (FCA), which was expanded from the technique presented in [45] and [46].
To overall illustrate the control loops in a simplified procedure, Fig. 3 shows the ASV overall control structure, also with the control allocation task: Considering this figure, it is possible to see that the input parameter to set the path is only the sequence of WayPoint  (WP) (W p (x I , y I )). After that, the 3 desired setpoints are obtained inside each loop.
Then, the control loops were implemented considering two parallel levels: High-Level Loops (circle number 1) and Low-Level Loops (circle number 2). The high-level one runs at 80Hz and is responsible for controlling the inertial position loops (x I and y I ), which deliver X b p and Y b P virtual control actions. The low-level loop runs at 400Hz and deals with the angular position loop (ψ), presenting N b p virtual control action.
The circle number 3 represents the control allocation technique described in Subsection III-B1. Furthermore, every control loop level has 2 feedback cascade structures: external and internal. The outer one is a position control loop (with a Proportional (P) controller), and the inner one is a velocity control loop (with a PID controller).
The external controller is in charge of the position stability of the controlled variable through the controller P. Thus, the output control action P will be the internal control loop input PID in cascade [47]. It is essential to highlight that the output from the external loop is in the inertial referential frame, which is translated to the body-fixed frame before being used in the internal one.
In the integral loops, a saturating block was inserted, functioning as an anti-windup action, preventing the final control action saturation obtained only from the integral action.
To finish the control aspect information from Fig. 3, Low-Pass Filters (LPFs) were implemented to mitigate the abrupt VOLUME 11, 2023  derivative control actions, created to pass signals with a frequency lower than the cutoff frequency and inhibit the passage of signals with frequencies higher than this frequency (5 times slower than the cutoff frequency of the respective control loops).
To illustrate it, a block diagram of the external and internal control loops is shown in Fig. 11.
Also for this figure and the ASV's design characteristics, Table 3 presents the minimum and maximum limits used in saturation blocks:

1) FAST CONTROL ALLOCATION
The control allocation task transforms τ VCAs (from controllers) into RCAs u driven to the system actuators. Fur-thermore, it is known that it may occur impossible to solve it only by matrix manipulation, instead more complex and time-consuming algorithms must be used (i. e., such as the primal-dual optimization algorithm [48]). These approaches may make the processing costs prohibitive for some onboard hardware.
Taking these remarks into account, the FCA technique proposed in [46] aims to turn the nonlinear control approach into a faster linear version by breaking the problem into two interconnected subsets of VCAs and RCAs.
The reasons for choosing these rules are: • The nonlinearities were broken into independent problems and solved recursively and iteratively; • The solution speed and solvability were enhanced with a subsystem coupling all the others together; • Although this approach is not as robust as Interior-Point algorithms, it is fast and well established in the literature [49]. For the projected ASV, the VCAs are: where N b p is the yawing torque, X b p and Y b p are the resultant force components on the X b and Y b ASV body-fixed frame, respectively.
As mentioned previously, the solution of the control allocation procedure has 8 RCAs (outputs), as presented in equations 11, 12 and 13.
where f i and γ i are RCAs for each propulsion motor and servomotor, respectively, with u 1 and u 2 representing the individual propulsion signals and angles for each actuator set. Taking these considerations into account, equation 14 is presented:τ where where M † 1 (u 2 ) is the Moore-Penrose inverse matrix of M 1 (u 2 ). where k 1 = −l y cγ 1 + l x sγ 1 , k 2 = l y cγ 2 − l x sγ 2 , k 3 = l y cγ 3 + l x sγ 3 and k 4 = −l y cγ 4 − l x sγ 4 . The servomotor tilting angles are obtained after breaking the nonlinearities, overlapping the previous solution space of τ :τ To get the final servomotors tilting angles: It is important to highlight that lines 4 to 7 from Equation 19 are mathematical constraints to enforce the property sin 2 (γ i )+cos 2 (γ i ) = 1, ensuring the system does not find any sin or cos out of the trigonometric unit circle.
Then, the matrices from Equations 15 to 20 are solved interactively until the convergence criteria are reached.

C. PARAMETRIC UNCERTAINTY
Before presenting the methodology, it is worth expanding the problem related to parametric uncertainties of ASVs and the PID controller design.
Among the difficulties, the first one faced is the difficulty of modeling hydrodynamic phenomena. Since these are VOLUME 11, 2023 highly non-linear characteristics, mathematical representation is not a trivial task [50]. Typical ASV models do not consider couplings and asymmetries, which possibly create uncertainty about the model.
Allied with the complexity of hydrodynamic phenomena, the difficulty of identifying parametric models is also a reality [1]. Usually involving complex solution spaces, the problem requires competent methodologies for excitation signal design and parameter estimation to reduce the sources of uncertainty, and coarse dynamic errors [42].
Disturbances are another undisputed sources of uncertainty, which are hardly measured and require advanced approaches for estimation [51]. It is still valid to mention sources of uncertainty such as dynamic variations in operation, mechanical changes in projects, or even inconsistencies between planned and executed projects.
Regardless of the type or source of uncertainty, it is agreed that they affect the vessel's control performance. Traditionally based on parametric models, PID controllers are very affected since the design is made from a priori parameterization subject to uncertainty. Therefore, ignoring the uncertainties in the PID control design can lead to considerable performance losses [23].
Specifically, in the problem formulated by the ASV modeling, the uncertainties are treated as structural uncertainties or resulting from parametric variations. Considering that the inertial parameters of the vessel are known variables, then the problem lies in the uncertainties about the hydrodynamic derivatives of M, C(ν) and D(ν). Mathematically, it implies that the Robust PID control design consists of determining the controller parameters in the solution space in R 12 , considering parametric uncertainties around M(ˆ − ) in dimensional space R 27 . This constitutes a challenging problem in view of the dimensions and complex phenomena involved.

IV. METHODOLOGY
Seeking a solution to the problem proposed in Section II, this work proposes the synthesis of PID controllers from the combination of the SLC method and optimal control. The PID tuning is performed by exploring an uncertainty region around the system initial estimateˆ − , using well-spaced Subsequently, a single tuning is obtained by minimizing the effects of the uncertainties studied in all the steps defined by the SLC technique. Then, it provides robust PID controllers capable of controlling M( ) for all ∈ . It is emphasized that the definition of is a fundamental step for the method since it establishes the region whose controller should minimize the effects of the uncertainties proposed by the designer.
Mathematically and it is possible to design a PID controller u(t, C ) parameterized by + , able to control M(P ⊕ ). Then, the same u(t, + ) will also be able to control M( ), ∀ ∈ . Therefore, the designed PID controller parameterized by + is considered stable and has robust performance for all ∈ , as long as it minimizes: (i) control efforts, (ii) tracking deviations, and (iii) overshoot situations along the chosen tuning scenarios''. Minimizing these aspects ensures robust stability, as the optimization process naturally penalizes unstable closed-loop behavior and favors stable tuning, it consequently also tends to the stability of the study region.
Based on this hypothesis and translating the robust control to an optimal one, the search for tuning the PID controller can be represented by: where S(·) is the nonlinear optimization algorithm, − is the a priori PID parameters, X are the set points defined by SLC technique and M(P ⊕ ) are the uncertainties needed to find and define u(t, + ).

A. OPTIMIZATION BY INTERIOR-POINTS
The optimization strategy S(·) was developed based on the Interior Points Algorithm described in [52]. The PID parameters + are searched by the controller performance optimization for all uncertainty scenarios proposed by M(P ⊕ ) using the SLC approach. As previously mentioned, the internal loops are 5 times faster than the external ones. Then, by the SLC definition, the outer loops consider the inner ones as unit gains [53].
To evaluate the performance of the designed control law u(t, + ), an objective function was defined from the following metrics: 1) Minimum control effort (J u ): measure the control effort during the experiment described as: where τ ∈ [τ min , τ max ] are the signals produced by the actuators, τ min and τ max are the minimum and maximum physical actuation system limits, respectively. Furthermore, R is a real symmetric positive definite weighting matrix with constant elements during all the interval simulation time [t o , t f ]. 2) Tracking deviations (J e ): compute the system state deviation u(t) from the desired state r(t), defined as: (23) where r(t) is the desired state and Q is a real symmetric positive definite weighting matrix with constant elements for the controlled state x(t).

3) Overshoot ( ): penalize situations where the state x(t)
exceed the desired state r(t). The function was defined as: 9604 VOLUME 11, 2023 where α(t) is defined as: Using the metrics described above, the following objective function is proposed to be minimized: where min and max are the minimum and maximum limits of and k u , k e and k ∈ R ≥0 are constant weightings related to metrics and priorities.

B. METHODOLOGY STEP-BY-STEP
After presenting the details of the optimization technique, it is essential to demonstrate the process, step by step, where every tuned loop was considered independent of each other. Therefore, using the SLC approach within the primary optimization process. First, every DoF was also manipulated separately. It means that the fast loop (the velocity loop) was tuned before the slow loop (position loop). Then, using this velocity controller, the outer loop is tuned.
Continuing, the first DoF tuned was yawing dynamics. The second one tuned was Surge and the last one was Sway.
In sequence, all the steps of the proposed tuning method are exposed in the flowchart of Fig. 5: As previously mentioned, every loop was tuned independently of the other using the Interior Points Algorithm. Yaw dynamics was chosen to be the first for the best ASV nose pointing and energy efficiency (less drag when moving strictly forward). After tuning yawing velocity and consequently its position loop, surge dynamics was settled up for its velocity and position. In the end, sway dynamics was tuned. For this experiment, it was desired that sway displacements should be as low as possible, also for saving energy.
It is important to highlight that after each tuning step, the gains are used in the next one.

V. SIMULATION RESULTS
The simulation results will be presented considering 50 distinct models, generated by normal distribution taking only the non-zero parameters with a maximum of 15% variation. As for the simulation, the 4 th order Runge-Kutta numerical method and a sampling interval T of 10ms were used. Table 4 reflects the mean (µ) and standard deviation (σ ) of the parameters with their a priori values M( O − ): The weights used in the optimization algorithm were: k e = 1.0, k = 10 3 and k u described in Table 5. It is noteworthy that these were defined experimentally to ensure PID tuning that provides stability and robustness to the system.   Table 2.
It is important to highlight that all the 6 controller gains are shown in Table 6 and the evolution of the tuning process (optimization) is displayed in Figs. 6 and 7.      Table 6. Specifically, in Fig. 6, the evolution of the objective function and also the metric referring to J u are observed. In this scenario, the behavior of the minimization of f ( ) is clearly perceived, while the controller's effort in providing the necessary actions is also considered. Figure 7 shows the evolution of each tuning stage that provides feasible values, according to the expected and presented data in Table 6.
To test the proposed controller tuning robustness, 3 different scenarios will be performed, which are: • Scenario 1 (Subsection V-A): The starting point inside the desired path with ASV's parameters within the %15 variation; • Scenario 2 (Subsection V-B): The starting point outside the desired path with ASV's parameters within the %15 variation; • Scenario 3 (Subsection V-C): The starting point inside the desired path with ASV's parameters beyond the %15 variation (critical variation): | ± 20%|, | ± 30%| and | ± 40%|. The chosen path has a square format, considering that the ASV needs to go through it without breaking when the turn is reached. The reason for this choice is to test the controller trackability when the setpoint is unpredictably changed.
The Integral of the Squared Error (ISE) was considered for the scenario quantitative analysis, in which the definition is the error between the desired and the ASV state variables. It is defined by: where N is the total number of iterations, x d is the desired setpoint vector for the 6 controlled variables, represented by Finally, x represents the data vector from the simulation results [ x, y, ψ, u, v, r]. It is also important to mention that all simulation tests used an Intel(R) Core (TM) i7-2600 device with 3.4GHz 8GB of RAM and Windows operational system 7 with 64 Bits. It is important to highlight that the computational capacity of the entire embedded control board was emulated on this i7-2600 computer, to perform the ASV for future experimental tests better.
A. SCENARIO 1 This scenario is created to illustrate a path where the ASV starts inside the desired path also with its modeled parameters within the 15% considered variation.  Taking the a priori model into account, i.e., the model with the pure and estimated values is shown in work [42].
The setpoints x d and y d are always looking to the next WP. The ψ d setpoint is obtained through the angle between x d and y d . Then, the ASV will not follow the dashed blue line, i.e., it will always go straight to the next WP. It is possible to highlight that a 1 meter WP radius was settled, which means when the ASV is inside of a 1 meter radius location to the desired WP, the WP number is changed for the next one and the ASV will start tracking the next location. Figure 8 presents the controlled variables x, y, ψ, u, v and r for the desired path presented in Fig. 9 for Table 7.
The 6 controlled variables (red lines) showed that the WP corners were quickly reached, which means that the proposed P-PID controlled the ASV satisfactorily. This is also clear by analyzing the 2 controlled variables: u, and ψ, for each WP in Table 8. Note that control loops provided stabilization time minor than T u s ≦ 1.75s, T ψ s ≦ 3.42s, and overshoots (%) that are UP u = 0% and do not exceed UP ψ ≦ 22.5%.
Furthermore, it is necessary to highlight the system saturation in the r controlled variable (yawing dynamics), explained due to the ASV's length and z inertial moment. Regarding sway dynamics, it is possible to see that its desired velocity was always 0m/s, i.e., the ASV is not desired to perform sideslip. Therefore, these indices presented reasonable results for the controller design requirements. Now, taking the simulation of the 50 models with the 6 tuned controllers, Table 9 depicts the mean (µ) and standard  Fig. 9, Scenario 1.

deviation (σ ) values of the ISE for the variables x, y,ψ,u, v and r:
Analyzing the data from the table above, it is possible to see higher numbers for yaw position and velocity (ψ and r, respectively), which is explained by the difficulty of sway maneuverability. It is known from Table 1 that mass and z inertial moments are quite expressive for ASVs with the embedded actuator's requirements. As consequence, yaw angular velocity is naturally saturated.
To finish the presentation of the simulation results, Fig. 10 illustrates the information about the following dataset characteristics: location, dispersion, asymmetry, tail length, and outliers. This graph was obtained considering the Root-Mean Square Error (RMSE) with tuned velocity controllers for Scenario 1 (Fig. 8), taking the 50 distinct models into account (Table 4). Table 4 shows that the linear velocities obtained quite reasonable mean values and RMSE distribution. The compact distribution demonstrates the controller's robustness, capable of dealing with the proposed uncertainties for the 50 models in the range | ± 15%|. In addition, average values around 0.44 m/s for Surge and 0.11 m/s for Sway also demonstrate robust results, knowing that the proposed scenario provided conditions of considerable parameter uncertainties. Under these conditions, the errors are within an acceptable range according to the work [42]. As for the angular velocity (yaw speed), an RMSE around 15.5 deg/s was obtained, also acceptable, given the ASV inertial characteristics and the large steering conditions in Scenario 1.

B. SCENARIO 2
The second scenario illustrates a situation where the ASV starts far from the desired path with its modeled parameters within the %15 considered variation. VOLUME 11, 2023 Figure 11 presents the controlled variables for the path presented in Fig. 12 for Table 10.
At the beginning of the simulation it is possible to check the ASV performing movements to go directly to the first WP, which is reached in around 16s. Furthermore, observing the control loop behaviors through Fig. 12, it is noted that the controller provided satisfactory trackability, even in the difficult condition of abrupt changes provided by the corners. According to the results, the ASV obtained dynamics with T u s ≦ 2.14s, T ψ s ≦ 3.30s, and overshoots (%) that are UP u = 0% and do not exceed UP ψ ≦ 25.2%.

C. SCENARIO 3
Here the path of Scenario 1 with 3 variations of parameters out of the permitted range: 20, 30, and 40% beyond the maximum range permitted in the tuning step is considered. The main idea for this choice is to prove the proposed methodology robustness.
The parameters and their values are shown in Table 11. They were chosen due to the difficulty to get them from the real prototype measurements [13]. Also, they were compared with their mean values, from Table 4: Figure 13 shows the paths performed for the 4 different models with the same P-PID controller previously tuned: The above figure brings out the possibility to analyze the proposed methodology for tuning the ASV controllers (gains shown in Table 6). The red line represents the ASV in simulations with their parameters within the considered range in the tuning controller stage. The black, green and cyan lines represent the ASV with parameter out of the permitted range, | ± 20%|, | ± 30%| and | ± 40%|, respectively. Then, the controller's capability to deal with large variations of the 3 most sensible parameters is visible.
Some metrics are shown in Table 12 for quantitative analysis of their controlled responses: The above table shows that the control requirements of the models extrapolating the parameter range were considerably depreciated compared to the model with the permitted parameter ranges in the tuning procedure. These indices were highlighted in red. Even though, the path was performed with no reached boundary limits, demonstrating robustness and reliability.
Still, in Table 12, it is concluded that the methodology is capable of minimizing the effects of parametric uncertainties, even in adverse situations such as the scenarios of | ± 30%| and | ± 40%| variations. The transient indices results also demonstrate that the designed controller provided stability in all uncertainty scenarios. Therefore, it is valid to state that the hypothesis raised in Section IV provides control laws with stability and robust performance to small regions of uncertainty, defined around a priori estimations of the desired system.

VI. CONCLUSION
This work has proposed a new methodology of controller tuning for an ASV prototype considering uncertainties in the 50 distinct models.
The new methodology for controller tuning was formulated as a sequence of multiple optimization procedures, always considering 50 different models with at most 15% of parameter variations. Every loop was tuned independently of the other using the Interior Points Algorithm. The first tuned controller was yaw dynamics, which directly deals with the ASV steering. After tuning yawing velocity and consequently its position loop, surge dynamics was settled up for its velocity and position. In the end, sway dynamics was tuned. Furthermore, after each tuning stage, the gains were used in the next loop, consecutively, until getting all 6 controller gains, as summarized in the flowchart of Fig. 5.
Concerning the control loop assessment, it is possible to remark the presence of overshoots less than 22.5% for yaw position dynamics and settling times less than 3.42 and 1.75s for surge velocity and yaw position dynamics, respectively. These values were within the control requirements in the tuning stage.
Even though the used a priori parameter model was from the work [42], and validated with open-field experiments, the proposition of this work shows the importance of the existence of robust controllers, not based only on one model, but tuned considering uncertainties in the model's parameters.
The results have demonstrated that the proposed methodology has robustness, applicability, and reliability, where the 6 P-PID controllers performed the setpoints for many distinct models with uncertainty parameters, even when considerable variations were tested out of the permitted range in the controller tuning step.
Finally, as future work, the proposed controller tuning will be performed in open-field tests on the developed prototype.