Adaptive path-following control for high-underactuated underwater glider under hydrodynamic coefficient uncertainties

Abstract

Underwater gliders (UGs) play a prominent role in collecting data within a specific underwater depth range. Due to the unmanned nature of these vehicles, the path-following control significantly influences their other systems and energy consumption. In this study, the performance of the path-following control was improved under hydrodynamic coefficient uncertainties using the adaptive control structure. The main control system, using the adaption law, autonomously updates itself to mitigate the adverse effects of uncertainties. The stability of the control law was proven through the utilization of a Lyapunov-function candidate. In the study, we take into account uncertainties in hydrodynamic coefficients, actuators, and position and orientation estimation. The impact of uncertainties on performance was investigated using the Monte Carlo simulation technique, which stochastically selected parameters. The results ultimately demonstrate that the uncertainties adversely affect the control of states. However, the adaptive structure exhibits robust performance compared to traditional controllers such as PID and LQR controllers with fixed gains. The adaptive path-following control structure reduced actuator usage, leading to decreased power consumption by the controller. Furthermore, the proposed structure was involved in an experimental case study, which was tested in a pool to validate its performance.

Introduction

Autonomous underwater marine vehicles possess attributes such as compact size, low cost, autonomy, and intelligence. These vehicles conduct numerous endurances in deep water without requiring human intervention [1]. Over the last few decades, the development of sensory, control, and computing technology has significantly advanced research in the open sea [2]. Various underwater missions, including environmental sampling for hydrological research and seabed surveys, necessitate the operation of autonomous underwater marine vehicles within a specific range of depth and attitude angle to collect high-quality data [3].

The principle of intelligent marine systems based on control, navigation, and guidance holds significant potential. A comprehensive review article has discussed the methods employed in the domains of control, navigation, and guidance for marine intelligent vehicles [4]. In certain underwater robots, such as UGs, the controller is not able to directly control the forward speed and heading angle [5, 6]. Typically, the number of actuators in UGs is fewer compared to autonomous underwater vehicles (AUVs), posing additional challenges in motion and path-following control. As a result, these robots fall into the category of "high-underactuated".

Studies on UG motion control can be categorized into two main groups: vertical motion control and horizontal motion control. Pitch angle and depth are the primary states in vertical motion control. The vertical motion typically refers to a steady-state gliding motion in the vertical plane, resembling a sawtooth pattern. In general, the main actuators that control the gliding motion state are buoyancy driven and pitch driven [7], executed at the desired minimum and maximum depths (Fig. 1). In horizontal motion control, the main control state is the heading angle of the vehicle. Traditional gliders are steered using a laterally movable mass or rudders for heading control. Optimizing the geometry of the rudders through advanced methods can improve the performance of heading control and, consequently, path-following control. Some studies [8, 9] have used computational fluid dynamics (CFD) techniques to investigate the effect of hydrofoil section profiles on the rudder's performance, aiming to increase efficiency in course changing and maneuverability. Additionally, in UGs equipped with a rudder and control surface, CFD techniques can be employed to optimize the hydrodynamic characteristics.

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Fig. 1

The schematic of UG motions

The heading angle serves as the main control state in intelligent path-following control for underactuated vehicles. In recent years, noteworthy advancements have been achieved within the field of path-following control for underwater robots. Through the use of the canonical perturbation theory, it derived an approximately analytical expression of the steady-state steering motion, from which the rudder angle was determined [10]. Various control approaches have been applied in path-following control, such as classical PID controllers [11], backstepping [12, 13], sliding mode control [14], and perturbation theory [10]. The control gains in traditional controllers are fixed throughout the control process, like the classic PID controller. Traditional controllers with constant gains efficiently have acceptable performance, while system parameters remain fixed. Adjusting the control gains is initially straightforward and can be done empirically. However, traditional controllers cannot guarantee appropriate performance under significant variations in the system parameters. HUAUGs are also susceptible and may experience instability in dynamic changes during endurance. Therefore, it is notable to utilize robust controls to address these challenges.

The challenges of model uncertainty, actuator saturation, and position estimation error are fundamental in the design of path-following control [15]. Model uncertainties result in variations in system characteristics and dynamic behavior. Adaptive controls, initially proposed for flight systems, have been explored to address these challenges [16]. Consequently, efforts have been dedicated to applying adaptive controls to systems dealing with model uncertainty and actuator saturation [17]. Model uncertainty is not limited to underwater robots; it is also considered in the path-following control design for submarines [18]. The tracking control of an AUV was evaluated using Port–Hamiltonian theory, considering external disturbances and unmodeled dynamics [19]. Due to the unpredictable nature of chaotic dynamics, saturations, and uncertainties in real-time applications, developing effective control techniques for underactuated chaotic systems like robot manipulators is challenging. In [20], a model-free digital adaptive control approach has been proposed to address these difficulties. Novel terminal sliding surfaces and several finite-time adaption laws are employed to design finite-time control inputs for an AUV [21]. Another approach, the robust adaptive control (RAC) system, was introduced for trajectory tracking [22]. This method utilized the benefits of both robust sliding mode controller (SMC) and adaption law. The suggestion was made to develop an adaptive asymptotic backstepping-based tracking control method for AUVs operating in the presence of non-vanishing uncertainty and input saturation [23]. Additionally, in another paper, an adaptive backstepping sliding mode control (ABSMC) scheme for AUVs subject to dynamic uncertainty was proposed [24]. Also, three-dimensional (3-D) trajectory tracking using the model predictive control (MPC) was demonstrated [25, 26].

Although some studies have explored control and path following for both traditional AUVs and UGs in the literature review, a notable gap exists as the hydrodynamic coefficient uncertainties have not been investigated in the context of path-following control. The hydrodynamic coefficients are crucial parameters that directly influence the dynamic behavior and control systems. In this study, we address the impact of model uncertainty on the performance of path-following control and propose improvements through an adaptive structure. The adaptive path-following control structure, using an adaption law, mitigates the adverse effect of model uncertainty. The main innovations of this study include:

1. The effect of uncertainties related to the hydrodynamic coefficients and model is specifically examined on the path-following control.

2. The uncertainty associated with position measurements is investigated in the context of the path-following control.

3. An adaptive path-following control structure is proposed to improve performance in the presence of uncertainties.

The study is organized as follows: Sect. 2 introduces a non-linear mathematical model for UGs, including hydrodynamic added mass and damping terms. Section 3 addresses the problem statement and introduces uncertainty factors in the context of path following and Monte Carlo simulation as a method for evaluation. Section 4 describes the structure of the adaptive path-following control to enhance performance for UGs. Section 5 represents the results of both numerical simulation and experimental tests. Finally, Sect. 6 provides the conclusions of this study.

Mathematical model for underwater glider (UG)

In the design of the path-following control, a mathematical model serves a dual purpose, not only forming the basis for control algorithm development, but also enabling simulations [27]. In this paper, a six-degree-of-freedom (six-DOF) model is used as the mathematical representation of UG motion. Seif et al., have described the appropriate non-linear dynamic model for underwater vehicles, employing the Newton–Euler motion equation, which is utilized in some studies [28, 29–30]. It decomposes the hydrodynamic damping and added mass acting on the wings and body. Two coordinate systems, as shown in Fig. 2, are defined to describe UG motion.

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Fig. 2

Defined coordinate systems

The desired positions and paths are defined based on the Earth-fixed frame. In contrast, the forces and moments are described based on the body-fixed frame. Vectors (1–3) indicate the vehicle's velocities, Euler angles, and position of the center of gravity and the buoyancy in the body-fixed frame.

1

$$\nu ={[\begin{array}{cccccc}u& v& w& p& q& r\end{array}]}^T,$$

2

$$\eta ={[\begin{array}{ccc}\varphi & \theta & \psi \end{array}]}^T,$$

3

$$r_G={[\begin{array}{ccc}{x}_g& y_g& z_g\end{array}]}^T;r_B={[\begin{array}{ccc}{x}_b& y_b& z_b\end{array}]}^T.$$

Also, Vector (4) indicates the vehicle's velocities in the Earth-fixed frame.

4

$$\lambda ={\left[\dot{X}\dot{Y}\dot{Z}\right]}^T.$$

The coordinate transformation establishes the relationship between translational velocities in the body-fixed and Earth-fixed frames, as indicated in Eq. 5.

5

$$\lambda =J\left(\eta \right){\left[\begin{array}{ccc}u& v& w\end{array}\right]}^T,$$where

6

$$J\left(\eta \right)=\left[\begin{array}{ccc}\begin{array}{c}c\psi c\theta \\ s\psi c\theta \\ -s\theta \end{array}& \begin{array}{c}-s\psi c\varphi +c\psi s\theta s\varphi \\ c\psi c\theta +s\varphi s\theta s\psi \\ c\theta s\varphi \end{array}& \begin{array}{c}s\psi s\varphi +c\psi c\varphi s\theta \\ -c\psi s\varphi +s\theta s\psi c\varphi \\ c\theta c\varphi \end{array}\end{array}\right],$$$$\ast c=cos,\ast s=sin,\ast t=tan.$$

six-DOF equations of motions

The non-linear mathematical model used consists of several parts, including rigid body dynamics, hydrodynamic damping terms, hydrodynamic added mass terms, restoring terms, Coriolis and centripetal terms, and actuators, as shown in Eq. 7. To simulate UG's motions, the non-linear mathematical model is used, with three equations representing translational motion and three equations representing the rotational motion of the vehicle. This approach allows for a comprehensive understanding and analysis of the vehicle's dynamic behavior in both translational and rotational aspects for path-following control.

7

$$M_{\mathit{RB}}\left(r_G\right)\dot{v}+M_A\dot{v}+C_{\mathit{RB}}\left(v\right)v+C_A\left(v\right)v+D\left(v\right)v+G\left(\eta ,r_B,r_G,B\right)={\tau }_{\mathit{xt}},$$where ${\tau }_{\mathit{ext}}\in {\mathbb{R}}^{6\times 1}$, and $M_{\mathit{RB}},M_A,C_{\mathit{RB}},C_A,D\in {\mathbb{R}}^{6\times 6}$. Finally, the equations of motion for UGs are represented in six-DOF by Eqs. 8–13 [28].

8

9

10

11

12

13

The coefficients in Eqs. (8–13) represent hydrodynamic derivatives, e.g., $Z_{\mathit{vp}}={\partial }^{2}Z/\partial v\partial p$ and $N_{\dot{r}}=\partial N/\partial \dot{r}$, as known hydrodynamic coefficients. To determine the hydrodynamic coefficients, the algorithm developed by Seif et al. was utilized based on strip theory [28]. The values of the coefficients are provided in Appendix.

Validation of the mathematical model

The mathematical model used in this study was validated through experimental tests, enhancing the credibility of the numerical simulations. For this, the vertical zigzag maneuver, shown in Fig. 1, was conducted in the laboratory. The Sharif University of Technology glider (SUT glider) was the case study in this test. Hydrodynamic coefficients for the SUT glider were calculated using strip theory, and the values of coefficients and SUT characteristics are attached in Appendix. This validation process ensures that the mathematical model accurately represents the dynamic behavior of the underwater glider in path-following control. The advanced maneuver was carried out in a pool setting to compare the experimental and mathematical results. The experiments were conducted in a pool with a length of 32 m, a width of 16 m, and a maximum depth of 3 m. Efforts were made to conduct the tests far from the pool's sides to minimize the impact of the walls.

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Fig. 3

Comparison of the experimental and simulation results of the depth

A comparison was conducted between the simulation and experimental results to validate the mathematical model. In the simulation, Eqs. (8–13) were used to simulate the vertical zigzag maneuver (Fig. 1). Figures 3 and 4 indicate the depth and pitch angle of the SUT glider in both the simulation and experimental procedures. The results indicate a good match between the two methods. As a result, the mathematical model has the credibility to investigate and simulate the path-following control.

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Fig. 4

The pitch angle of the SUT glider in the experimental and simulation tests

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Fig. 5

Flowchart of the Monte Carlo simulations in this study

Problem statement

Uncertainty parameters

The uncertainty parameters fall into two groups: computable and uncomputable. Unmodeled parameters in the mathematical model contribute to an increase in uncertainty parameters, potentially leading to instability in path-following control. To simplify vehicle modeling, various assumptions are often made, and these assumptions introduce uncertainty parameters. The assumptions and identified uncertainty parameters in path-following control are outlined below:

• Frictional resistance, a significant component of the hydrodynamic damping term, is a function of the Reynolds number and the local flow regime. In the mathematical model, the Reynolds number and drag coefficient value are considered as constant.

• The influence of appendages on the performance of other parts is neglected.

• The strip theory was utilized to determine the hydrodynamic coefficients, which neglect the effects of the vehicle's forward velocity and fluid viscosity.

• The state space models are separated into horizontal and vertical planes to define an ideal model, which may not accurately represent the true behavior.

• Mathematical models lack the capability to estimate external disturbances through comparisons with real tests.

• The added mass coefficients are considered independent of velocities.

• Water properties (density, temperature, salinity) with respect to depth vary in the real environment so the constant density assumption is not totally true.

• The possibility exists that the actuators of the vehicle may not accurately execute control commands in the experimental test.

• Hydrostatic pressure can deform the vehicle's strength hull: that assumption of the rigid body may not hold true.

• Navigation instruments and sensors involve noise and uncertainty measurements in estimating position and orientation.

For this, uncertain parameters in path-following control are outlined in Table 1.

Table 1. Uncertain parameters in the mathematical model

Type of parameter	Parameter	Uncertainty	unit
Hydrodynamic coefficients	Added mass terms (Table 4)	25%	n/a
	Damping terms (Table 5)	50%	n/a
Mass properties	m	5%	kg
	xg	5% L	m
	yg	10% d	m
	zg	10% d	m
Buoyancy properties	W-B	25%	N
	xb	5% L	m
	yb	10% d	m
	zb	10% d	m
Inertia properties	$I_{\mathit{xx}}$	20%	kg.m2
	$I_{\mathit{yy}},I_{\mathit{zz}}$	20%	kg.m2
	$I_{\mathit{xy}},I_{\mathit{xz}}$	20%$I_{\mathit{xx}}$	kg.m2
	$I_{\mathit{yz}}$	20%$I_{\mathit{yy}}$	kg.m2
Initial condition	${\varphi }_0,{\theta }_0,{\psi }_0$	5	deg
	$p_0,q_0,r_0$	3	deg /s
Water properties	$\rho$	5%	kg /m3
	uc	0.15	m /s
Actuators	Pitch mechanism offset	2% L	m
	Roll mechanism offset	2	deg
	Delay	0.5	s
	${\dot{\tau }}_{\theta }$	2%	m /s
	${\dot{\tau }}_{\varphi }$	2%	deg /s
Gyroscope	Noise	0.1	deg /s
	Bias	0.05	deg /s
AHRS	Noise	1.5	deg
	Bias	1	deg
Pressure sensor	Noise	0.02	Bar
	Bias	0.02	Bar
Estimation of X position	Noise	2	m
Estimation of Y position	Noise	2	m

Monte Carlo simulation technique

The Monte Carlo simulation technique can be employed to evaluate the control system's performance under uncertainty parameters that are not analyzable with exact analytical methods. Also, the Monte Carlo simulation technique is used in autonomous systems where non-linear dynamics are present, together with various unknown parameters and diverse input–output combinations [31, 32]. This technique facilitates the examination of adaptive controllers by extracting and stochastically applying the system's uncertain parameters in many simulations [33, 34]. In this study, the parameters outlined in Table 1 are stochastically selected using the Monte Carlo simulation technique, as presented in Fig. 5. This approach allows for a comprehensive exploration of the control system's behavior under varied and uncertain conditions.

Adaptive path-following control structure

In practical maritime path following, the guidance, navigation, and control (GNC) framework is commonly employed. In our approach, we integrate adaptive control into the control sector to enhance the performance of path-following control in the presence of uncertainty parameters.

Defining the desired heading and roll angle

The line-of-sight (LOS) guidance method is commonly used to make the vehicle along a path. It requires input from either an operator or a path planner, which provides a set of waypoints. The waypoint database, therefore, consists of a set of n Cartesian coordinates that define the predefined path for an underwater vehicle.

14

$$wp=\left[\begin{array}{cccc}{p}_1^n& p_2^n& \dots & p_k^n\end{array}\right]$$

Equation (14) is the waypoint list that UG must follow during its endurance. For this, guidance according to states ($X,Y,\psi$) continuously generates the heading angle error, as shown in Eq. 15 and Fig. 6.

15

$${\psi }_e=\text{a}tan((Y_d-Y),(X_d-X))-\psi .$$

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Fig. 6

The principle of LOS in path-following control

Then, the desired roll angle is determined based on the heading error, as shown in Fig. 7. This methodology effectively resolves the wind-up problem associated with actuator saturation.

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Fig. 7

The desired roll angle with respect to the heading error

Design of control law

The L1 adaptive controller is a suitable adaptive control for dealing with uncertainties. It has been widely used in uncertainty parameter systems [35, 36]. Due to the update of controller gains, the adaptive control structure can decrease the adverse effect of uncertainties. This controller is mainly applied to linear models, although in some studies, it has also been tested on non-linear systems [37]. The planet model structure in the study is considered as depicted in Eq. 16.

16

$$\begin{array}{c}{\dot{x}}_s\left(t\right)=A_m{x}_s\left(t\right)+B_m\left(\omega \tau \left(t\right)+\gamma \left(t\right)x_s+\sigma \left(t\right)\right)\\ y_s\left(t\right)=c^T{x}_s\left(t\right),\\ \end{array}$$where $\omega ={\left[\begin{array}{cc}{\omega }_{\varphi }& {\omega }_{\theta }\end{array}\right]}^T$, $\gamma ={\left[\begin{array}{cc}{\gamma }_{\varphi }& {\gamma }_{\theta }\end{array}\right]}^T$, $\sigma ={\left[\begin{array}{cc}{\sigma }_{\varphi }& {\sigma }_{\theta }\end{array}\right]}^T$, $x_s={\left[\begin{array}{cccc}\varphi & p& \theta & q\end{array}\right]}^T$, and $\tau ={\left[\begin{array}{cc}{\tau }_{\varphi }& {\tau }_{\theta }\end{array}\right]}^T$. In Eq. 16, all unknown parameters are regarded as bounded [38]. Equation 17 describes the reference model for the roll and pitch motions of UG. The values of the coefficients are attached in Appendix.

17

$$\left[\begin{array}{c}\dot{\varphi }\\ \dot{p}\\ \dot{\theta }\\ \dot{q}\end{array}\right]=\left[\begin{array}{cccc}0& 1& 0& 0\\ \frac{K_{\varphi }}{I_x-K_p}& \frac{K_p}{I_x-K_p}& 0& 0\\ 0& 0& 0& 1\\ 0& 0& \frac{M_{\theta }}{I_y-M_q}& \frac{M_q}{I_y-M_q}\end{array}\right]\underset{x_s}{\underbrace{\left[\begin{array}{c}\varphi \\ p\\ \theta \\ q\end{array}\right]}}+\left[\begin{array}{cc}0& 0\\ \frac{K_{\tau }}{I_x-K_p}& 0\\ 0& 0\\ 0& \frac{M_{\tau }}{I_y-M_{\dot{q}}}\end{array}\right]\underset{\tau }{\underbrace{\left[\begin{array}{c}{\tau }_{\varphi }\\ \tau \theta \end{array}\right]}}.$$

The reference model (Eq. 17) was considered in this study as the ideal model that can generate continuous output. The L1 adaptive control aims to track the reference model to bring the vehicle's output close to that of the reference model. To enhance the estimation of unmodeled parameters, a state predictor is utilized [38]. In this context, Eq. 18 represents the state predictor utilized to predict the roll and pitch angles of the UG.

18

$$\begin{array}{c}{\widehat{\dot{x}}}_s\left(t\right)=A_m{x}_s\left(t\right)+B_m\left(\widehat{\omega }\tau \left(t\right)+\widehat{\gamma }\left(t\right)x_s+\widehat{\sigma }\left(t\right)\right)\\ {\widehat{\dot{y}}}_s=c^T{\widehat{x}}_s\left(t\right).\\ \end{array}$$

${\widehat{x}}_s\in R^{n\times 1}$ is the estimated states. The state predictor, using the estimated parameter and the control command from the previous step, predicts the vehicle output. The adaptation laws section estimated the unknown parameters based on the prediction error, controller command, and the vehicle's output. The structure of the adaptation laws for estimating unknown parameters is provided in Eq. 19.

19

$$\begin{array}{cc}\dot{\widehat{\omega }}\left(t\right)=\mathrm{\Gamma }proj\left(\widehat{\omega }\left(t\right),\tau \left(t\right){\tilde{x}}_s^T\left(t\right)P{B}_m\right),& \dot{\widehat{\omega }}\left(0\right)={\widehat{\omega }}_0\\ \dot{\widehat{\gamma }}\left(t\right)=\mathrm{\Gamma }proj\left(\widehat{\gamma }\left(t\right),-x_s\left(t\right){\tilde{x}}_s^T\left(t\right)P{B}_m\right),& \widehat{\gamma }\left(0\right)={\widehat{\gamma }}_0\\ \dot{\widehat{\sigma }}\left(t\right)=\mathrm{\Gamma }proj\left(\widehat{\sigma }\left(t\right),{\tilde{x}}_s^T\left(t\right)P{B}_m\right),& \widehat{\sigma }\left(0\right)={\widehat{\sigma }}_0\end{array}$$where${\tilde{x}}_s={\widehat{x}}_s-x_s$, $\mathrm{\Gamma }\in \left(0,\infty \right)$, and $P=P^T>0$ is a positive definite matrix that is extracted from the Lyapunov equation. Finally, the projection operation was defined in Eq. 20 [38], estimating the unknown parameters as bounded [36].

20

$$proj\left(\theta ,y\right)\cong \left\{\begin{array}{cc}y& iff\left(\theta \right)<0\\ y& iff\left(\theta \right)\ge 0\text{and}\mathrm{\nabla }{f}^{T}y\le 0\\ y-\frac{\mathrm{\nabla }f}{∥\mathrm{\nabla }f∥}⟨\frac{\mathrm{\nabla }f}{∥\mathrm{\nabla }f∥},y⟩f\left(\theta \right)& iff\left(\theta \right)\ge 0\text{and}\mathrm{\nabla }{f}^{T}y>0\end{array}\right).$$

The system's stability can be proven using the Lyapunov energy function. The Lyapunov-function candidate is considered as Eq. (21).

21

$$V\left({\tilde{x}}_s\left(t\right),\tilde{\omega }\left(t\right),\tilde{\gamma }\left(t\right),\tilde{\sigma }\left(t\right)\right)={\tilde{x}}_s^T\left(t\right)P{\tilde{x}}_s\left(t\right)+\frac{1}{\mathrm{\Gamma }}\left({\tilde{\omega }}^2\left(t\right)+{\tilde{\gamma }}^T\left(t\right)+{\tilde{\sigma }}^2\left(t\right)\right),$$which leads to

22

$$\dot{V}\left(t\right)\le -{\tilde{x}}_s^T\left(t\right)Q{\tilde{x}}_s-\frac{2}{\mathrm{\Gamma }}\left({\tilde{\gamma }}^T\left(t\right)\tilde{\gamma }\left(t\right)+\tilde{\sigma }\left(t\right)\tilde{\sigma }\left(t\right)\right).$$

By employing Barbalat's lemma [36, 38], it is determined that the error signal attains asymptotic convergence. Finally, the control law to control the pitch and roll angles of UG was derived as shown in Eq. 23.

23

$$\tau \left(t\right)=\frac{1}{\widehat{\omega }\left(t\right)}\left(k_g{r}_d\left(t\right)-\widehat{\gamma }\left(t\right)x_s-\widehat{\sigma }\left(t\right)\right)$$where,$r_d={\left[\begin{array}{cc}{\varphi }_d& {\theta }_d\end{array}\right]}^T$, and k_g is $k_g=-1\left./\phantom{-1\left(c^T{A}_m^{-1}{B}_m\right)}\right)\left(c^T{A}_m^{-1}{B}_m\right)$.

Actuator constraints

The actuators involve constraints on rates and maximum/minimum amplitude, which need to be considered in modeling. Physically, actuators are not capable of executing entire control commands. Equations 24, 25 were used to model the constraints.

24

$${\dot{\tau }}_{min}\le \dot{\tau }\left(t\right)\le {\dot{\tau }}_{max}$$

25

$${\tau }_{min}\le \tau (t)\le {\tau }_{max}$$

Finally, the block diagram of the proposed adaptive path-following control structure is indicated in Fig. 8.

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Fig. 8

The block diagram of the proposed adaptive path-following control

Results and discussion

In this section, we present the results of the adaptive path-following control under two conditions: the absence and presence of uncertainty parameters. In the previous sections, the principle of the proposed adaptive path-following control was explained. Simulations and experimental tests were conducted on the SUT glider as a case study. The first part of this section was dedicated to simulation results. The adaptive path-following control, employing the L1 adaptive control, was simulated and compared with two conventional controllers, LQR and PID, under both no uncertainty and uncertainty parameters. The impact of uncertainty parameters on the pitch motion control has been examined through the Monte Carlo simulation technique. In the second part, the experimental test results of the SUT glider have been presented with the proposed adaptive path-following control structure. This comprehensive analysis combines simulated and real-world evaluations to provide a thorough assessment of the adaptive control system's performance.

Numerical simulation

The adaptive path-following control was simulated using MATLAB Simulink, employing two control structures for performance comparison: adaptive and fixed structures. In the fixed structure, LQR and PID controllers were utilized, while the adaptive structure employed L1 adaptive control. The PID controller gains ($k_p,k_i,k_d$) were set to 0.96, 0.0287, and 5.952 for pitch motion and 1.3453, 0.0505, and 14.369 for roll motion, respectively. These gains were determined based on hydrodynamic coefficients, natural frequency, and bandwidth of actuators [39]. On the other hand, for the LQR controller, the weights of inputs and outputs were set to be equal. In the L1 adaptive control, the initial values of the estimation of unknown parameters (${\widehat{\sigma }}_0,{\widehat{\gamma }}_0,{\widehat{\omega }}_0$) were considered as 0, 0, and 1; also, the adaptation rates were $\mathrm{\Gamma }=0.02$ for both pitch and roll motions. The comparison between adaptive and fixed structures allows for a comprehensive assessment of the adaptive control system's effectiveness in addressing uncertainties and enhancing performance.

First scenario: the absence of uncertainty parameters

In this scenario, the SUT glider, using the proposed adaptive path-following control, aimed to track desired waypoints and paths under the condition of the absence of uncertain parameters. The hydrodynamic coefficients were assumed to be specific and constant values, and the performance of actuators and the estimation of position and orientation were considered error-free. The L1 adaptive control was compared with two conventional controllers, LQR and PID, to evaluate its performance.

The trajectories, heading angle errors, and control commands are depicted in Fig. 9. According to this figure, all three controllers exhibit a proportional response to control the glider's angles in the absence of uncertainty parameters. Figures 10 and 11 depict the proper performance of controllers in controlling the pitch and roll angles that can be inferred from control commands with no irrational behavior (Fig. 9). The L1 adaptive control (adaptive structure), maintains stability performance by estimating the unmodeled parameters (Fig. 12). In conclusion, classical controllers can also exhibit proportional performance in cases where the system dynamics are ideal. The root mean square error (RMSE) for tracking error and actuator usage cost function are provided in Table 2.

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Fig. 9

Comparison of the trajectories, heading angle errors, and control commands (scenario I)

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Fig. 10

Comparison control of the pitch angle in scenario I (a PID, b LQR, and c L1)

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Fig. 11

Comparison control of the roll angle in scenario I (a PID, b LQR, and c L1)

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Fig. 12

The estimated unknown parameters in scenario I by adaption law

Table 2. RMSE of tracking error and actuator usage cost function

Condition	Controller	RMSE ${\psi }_d$ (deg)	RMSE ${\theta }_d$ (deg)	RMSE ${\varphi }_d$(deg)	$\sum \left|\mathrm{\Delta },t,\left({\tau }_{\varphi i},-,{\tau }_{\varphi i-1}\right)\right|$	$\sum \left|\mathrm{\Delta },t,\left({\tau }_{\theta i},-,{\tau }_{\theta i-1}\right)\right|$
Ideal condition	PID	24.98	4.30	1.27	0.746	1.869
	LQR	27.32	4.36	1.01	1.257	1.085
	L1	25.91	4.59	1.05	0.137	0.366
Under uncertainty	PID	27.27	4.76	5.81	9.005	3.835
	LQR	35.35	4.85	9.43	11.729	4.787
	L1	32.23	4.68	3.88	2.839	0.502

Second scenario: the presence of uncertainty parameters

In contrast to the previous scenario, the simulations considered the presence of uncertainty parameters. Hydrodynamic coefficients, rigid body characteristics, sensor noise and bias, actuator errors, and water properties were stochastically selected with normal distribution using the Monte Carlo simulation technique in the schedule at times 0, 850, 1700, 2600, and 3500 s. This approach introduces variability and uncertainty into the system, allowing for a more realistic evaluation of the controllers under non-ideal conditions.

Based on the results shown in Fig. 13, it can be inferred that the performance of the SUT glider under LQR and PID controllers was significantly affected by uncertainty parameters and dynamic change. However, the adaptive path-following control presented less overshoot than the other two controllers near the waypoints. The fixed path-following control indicated significant oscillations and irrational behavior in the vehicle's outputs (Figs. 14 and 15), indicating ineffectiveness in the presence of uncertainty parameters. On the other hand, the adaptive path-following control, which estimates the unknown parameters (Fig. 16), maintains its effectiveness under uncertain conditions. As a result, controllers that cannot self-adjust reduced robustness when facing dynamic changes, leading to increased energy consumption (Table 2). These findings emphasize the adaptability and robustness of the adaptive path-following control in the presence of uncertainty parameters and dynamic variations.

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Fig. 13

Comparison of the trajectories, yaw angle errors, and control commands (scenario II)

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Fig. 14

Comparison control of the pitch angle in scenario II (a PID, b LQR, and c L1)

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Fig. 15

Comparison control of the roll angle in scenario II (a PID, b LQR, and c L1)

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Fig. 16

The estimated unknown parameters in scenario II by adaption law

Third scenario: Monte Carlo simulations for the pitch motion

The Monte Carlo simulation technique employed stochastically extracted parameter values, investigating the impact of uncertainty and unmodeled parameters. The algorithm was run 1,000 times to analyze pitch motion. In each iteration, the Monte Carlo simulation randomly selected parameters from Table 1. Figures 17, 18, 19 depict the results of pitch, pitch velocity, and control commands at 200 iterations. This approach provides insights into the robustness of the adaptive path-following control under different parameter combinations, offering a comprehensive understanding of its performance in the face of uncertainty.

[See PDF for image]

Fig. 17

Monte Carlo simulation of pitch motion with uncertainty parameters

[See PDF for image]

Fig. 18

Monte Carlo simulation pitch velocity with uncertainty parameters

[See PDF for image]

Fig. 19

Control commands in various Monte Carlo simulation runs

The Monte Carlo simulation results for the pitch motion are shown in Fig. 17. The simulations exhibited more overshoot in the initial two set points as controller gains were updated. Generally, the system demonstrated insignificant overshoot, motion, and velocity oscillations, as evident in Figs. 17 and 18. Additionally, the ocean current and actuator offsets directly influenced the vehicle's dynamics, reflected on the control command extension, as indicated in Fig. 19. Overall, the controller performed its task effectively, indicating its adaptability and robustness in addressing uncertainties and variations in the system parameters.

Experimental case study

In this section, the results of experimental tests on the SUT glider (Fig. 20) are presented, employing the same adaptive path-following control structure as indicated in Fig. 8. The controllers were tuned based on the simulation section. The path-following algorithms were implemented on an Arduino Mega 2560 processor using a standard C +  + program with an open-source architecture. The algorithms were run in a closed loop on the processor. The sensors used were the Inertial Measurement Unit (IMU) BNO055 and the pressure sensor M5256 to measure the depth and Euler angles of the SUT glider. The time step for running the adaptive path-following control in the processor was set to 0.1 s. AC motors, stepper motors, and servo motors were employed for buoyancy, pitch, and roll drive, respectively, to execute control commands.

[See PDF for image]

Fig. 20

The SUT glider

Due to the inaccessibility of position measurement equipment on the SUT glider in water, the adaptive path-following control aimed to track the heading angle according to the scheduled time. Desired heading angle set points were defined based on the time using an Arduino board in the guidance. The adaptive control controlled the heading angle, employing a roll-driven system. This experimental setup provides valuable insights into the real-world performance of the adaptive path-following control structure.

A pool environment was used to conduct preliminary tests and evaluate the performance of the proposed control system under conditions without ocean disturbances. For this purpose, experimental tests were carried out in the pool, allowing the efficiency to be evaluated in an ideal environment. Although this scenario is less challenging than real-world conditions, it provides a useful initial assessment. The pool measures 16 m in width, and 32 m in length, and has a maximum depth of 3 m. The tests were conducted such that the glider's movement occurred at a distance of more than 2 m from the pool's sides, minimizing any practical impact from the walls. According to the results shown in Fig. 21, the distance between the glider and the bottom of the pool was 0.5 m, indicating that the pool's bottom had a minor effect on its movement.

[See PDF for image]

Fig. 21

The experimental test results of the heading control on the SUT glider

Figure 21 presents the experimental test results for the adaptive path-following control on the SUT glider. These results illustrate the effective performance of the adaptive structure in controlling the heading angle. The control system experienced minimal oscillations and maintained stability throughout the test. The performance of the controlling heading angle, a crucial state in the path-following, was found to be acceptable. The experimental test results validate the reliability of the proposed strategy for path following.

Conclusion

In this paper, an adaptive path-following control has been proposed based on L1 adaptive control to mitigate the adverse effects of hydrodynamic coefficient uncertainties and uncertainty in position estimation in UGs. The initial step involves describing the non-linear mathematical model for UGs to represent the dynamic behavior of simulations. To consider uncertainty parameters, the Monte Carlo simulation technique is utilized, stochastically selecting parameters. The RMSE is utilized to assess controller efficiency. The results highlight that uncertainty parameters can significantly reduce the effectiveness of traditional controllers, such as PID and LQR. However, when system parameters remained fixed, traditional controllers demonstrated appropriate performance. In contrast, the adaptive path-following control, both in the absence and presence of uncertainty parameters, exhibited proper performance compared to traditional controllers. Furthermore, adaptive path-following control did not experience irrational and unstable dynamic behavior. The hydrodynamic coefficient uncertainties predominantly affected actuator usage, directly impacting power consumption. Consequently, in the presence of uncertainty parameters, the adaptive structure consumes less power than the fixed structure. The experimental tests were conducted in the pool with 3-m depth to indicate the efficiency under ideal conditions. The experimental test results on the SUT glider validated the effectiveness of the proposed structure in controlling the heading. Overall, the proposed adaptive path-following control strategy demonstrates resilience in addressing uncertainties, indicating improved performance and energy efficiency compared to traditional controllers.

Data availability

The datasets used and analyzed during the current study are available from the corresponding author on reasonable request.

List of symbols

B

Buoyancy force

x_b

Longitudinal center of buoyancy

y_b

Transverse center of buoyancy

z_b

Vertical center of buoyancy

I_xy

Mass moment of inertia around the x-axis as the object rotates around the y-axis

I_xz

Mass moment of inertia around the x-axis as the object rotates around the z-axis

I_yz

Mass moment of inertia around the y-axis as the object rotates around the z-axis

W_b

Weight of water into buoyancy tank

M_RB

Rigid body inertia

M_A

Hydrodynamic added mass

C_RB

Rigid-body Coriolis and centripetal forces

C_A

Hydrodynamic Coriolis and centripetal forces

G

Restoring forces and moments

${\tau }_{\mathit{ext}}$

External forces and moment

$\varphi$

Roll angle

$\theta$

Pitch angle

$\psi$

Yaw angle

${\psi }_e$

Heading error

$p_i^n$

Waypoints

${\varphi }_d,{\theta }_d,{\psi }_d$

Desired roll, pitch and yaw angles

${\varphi }_0,{\theta }_0,{\psi }_0$

Initial roll, pitch and yaw angles

${\dot{\tau }}_{\varphi },{\dot{\tau }}_{\theta }$

Controller command rates of roll and pitch

${\dot{\tau }}_{\theta max},{\dot{\tau }}_{\theta min}$

Maximum and minimum pitch mechanism rate

${\tau }_{\theta max},{\tau }_{\theta min}$

Maximum and minimum pitch mechanism

${\omega }_{\theta },{\gamma }_{\theta },{\sigma }_{\theta }$

Unknown parameter, time variant unknown parameters and input disturbance for pitch motions

$B_m$

Actuator matrix

V

Lyapunov-function candidate

$\widehat{\omega },\widehat{\gamma },\widehat{\sigma }$

Estimated parameters of $\omega ,\gamma ,\sigma$

$\widehat{\varphi },\widehat{\theta }$

Prediction roll and pitch angles

$\tilde{\varphi },\tilde{\theta }$

Prediction error of roll and pitch angles

xs

States vector

ys

Output vector

R

Radius of virtual circle to find interceptor

L

Length

d

Diameter

$\rho$

Water density

x_g

Longitudinal center of gravity

y_g

Transverse center of gravity

z_g

Vertical center of gravity

I_xx

Mass moment of inertia around the x-axis

I_yy

Mass moment of inertia around the y-axis

I_zz

Mass moment of inertia around the z-axis

W_p

Pitch mass moveable

m

Vehicle's mass

W

Vehicle's weight

D

Hydrodynamic damping

u_c,v_c

Surge and sway velocities of currents in Earth-fixed frame

u

Surge velocity

v

Sway velocity

w

Heave velocity

p

Roll velocity

v

Sway velocity

r

Yaw rate

k_p,k_i,k_d

Proportional, integral, and derivative gains of PID controller

${\tau }_{\varphi },{\tau }_{\theta }$

Controller commands of roll and pitch

${\dot{\tau }}_{\varphi max},{\dot{\tau }}_{\varphi min}$

Maximum and minimum roll mechanism rate

${\tau }_{\varphi max},{\tau }_{\varphi min}$

Maximum and minimum roll mechanism

${\omega }_{\varphi },{\gamma }_{\varphi },{\sigma }_{\varphi }$

Unknown parameter, time variant unknown parameters and input disturbance for roll motions

$A_m$

States matrix

Q

Positive definite matrix

P

Positive definite matrix

$\mathrm{\Gamma }$

Adaptation rate

c

Output matrix

${\widehat{x}}_s$

Prediction states vector

${\tilde{x}}_s$

Prediction error of states vector

$X_d,Y_d$

Position of virtual waypoint

Publisher's Note

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References

1. Wang, YL; Han, QL; Fei, MR; Peng, C. Network-based T-S fuzzy dynamic positioning controller design for unmanned marine vehicles. IEEE Trans Cybern; 2018; 48, 9 pp. 2750-2763. [DOI: https://dx.doi.org/10.1109/TCYB.2018.2829730]

2. Du, P; Yang, W; Wang, Y; Hu, R; Chen, Y; Huang, SH. A novel adaptive backstepping sliding mode control for a lightweight autonomous underwater vehicle with input saturation. Ocean Eng; 2022; 263, [DOI: https://dx.doi.org/10.1016/j.oceaneng.2022.112362] 112362.

3. Mondal, K; Banerjee, T; Panja, A. Autonomous underwater vehicles: recent developments and future prospects. Int J Res Appl Sci Eng Technol; 2019; 7, 11 pp. 215-222. [DOI: https://dx.doi.org/10.22214/ijraset.2019.11036]

4. Paull, L; Saeedi, S; Seto, M; Li, H. AUV navigation and localization: a review. IEEE J Oceanic Eng; 2013; 39, 1 pp. 131-149. [DOI: https://dx.doi.org/10.1109/JOE.2013.2278891]

5. Yamaguchi S, Naito T, Kugimiya T, Akahoshi K, Fujimoto M (2007) Development of a motion control system for underwater gliding vehicle. In ISOPE International Ocean and Polar Engineering Conference (pp. ISOPE-I). ISOPE

6. Eriksen, CC; Osse, TJ; Light, RD; Wen, T; Lehman, TW; Sabin, PL; Ballard, JW; Chiodi, AM. Seaglider: a long-range autonomous underwater vehicle for oceanographic research. IEEE J Oceanic Eng; 2001; 26, 4 pp. 424-436. [DOI: https://dx.doi.org/10.1109/48.972073]

7. Wang, J; Wu, Z; Dong, H; Tan, M; Yu, J. Development and control of underwater gliding robots: a review. IEEE/CAA J Autom Sin; 2022; 9, 9 pp. 1543-1560. [DOI: https://dx.doi.org/10.1109/JAS.2022.105671]

8. Hasanvand, A; Hajivand, A; Ali, NA. Investigating the effect of rudder profile on 6DOF ship course-changing performance. Appl Ocean Res; 2021; 117, [DOI: https://dx.doi.org/10.1016/j.apor.2021.102944] 102944.

9. Hasanvand, A; Hajivand, A. Investigating the effect of rudder profile on 6DOF ship turning performance. Appl Ocean Res; 2019; 92, [DOI: https://dx.doi.org/10.1016/j.apor.2019.101918] 101918.

10. Mahmoudian, N; Geisbert, J; Woolsey, C. Approximate analytical turning conditions for underwater gliders: implications for motion control and path planning. IEEE J Oceanic Eng; 2010; 35, 1 pp. 131-143. [DOI: https://dx.doi.org/10.1109/JOE.2009.2039655]

11. Mahmoudian N, Woolsey C (2008) Underwater glider motion control. In 2008 47th IEEE Conference on Decision and Control (pp. 552–557). IEEE

12. Yuan, J; Wu, Z; Yu, J; Tan, M. Sliding mode observer-based heading control for a gliding robotic dolphin. IEEE Trans Industr Electron; 2017; 64, 8 pp. 6815-6824. [DOI: https://dx.doi.org/10.1109/TIE.2017.2674606]

13. Caiti, A; Calabro, V; Grammatico, S; Munafò, A; Geluardi, S. Switching control of an underwater glider with independently controllable wings. IFAC Proceed Vol; 2012; 45, 27 pp. 194-199. [DOI: https://dx.doi.org/10.3182/20120919-3-IT-2046.00033]

14. Zhang, F; Tan, X. Gliding robotic fish and its tail-enabled yaw motion stabilization using sliding mode control. Dyn Syst Control Conf; 2013; 56130, V002T32A006.

15. Sarhadi, P; Noei, AR; Khosravi, A. Adaptive integral feedback controller for pitch and yaw channels of an AUV with actuator saturations. ISA Trans; 2016; 65, pp. 284-295. [DOI: https://dx.doi.org/10.1016/j.isatra.2016.08.002]

16. Astrom KJ, Wittenmark B (2013) Adaptive control. courier corporation. Courier Corporation, 3

17. Bruckner, M; Galeani, S; Del Re, L. Robustified anti-windup via switching adaptation. IEEE Trans Autom Control; 2012; 58, 3 pp. 731-737.3029468 [DOI: https://dx.doi.org/10.1109/TAC.2012.2209261]

18. Rober, N; Hammond, M; Cichella, V; Martin, JE; Carrica, P. 3D path following and L1 adaptive control for underwater vehicles. Ocean Eng; 2022; 253, [DOI: https://dx.doi.org/10.1016/j.oceaneng.2022.110971] 110971.

19. Jia, Z; Qiao, L; Zhang, W. Adaptive tracking control of unmanned underwater vehicles with compensation for external perturbations and uncertainties using Port-Hamiltonian theory. Ocean Eng; 2020; 209, [DOI: https://dx.doi.org/10.1016/j.oceaneng.2020.107402] 107402.

20. Tutsoy, O; Barkana, DE. Model free adaptive control of the under-actuated robot manipulator with the chaotic dynamics. ISA Trans; 2021; 118, pp. 106-115. [DOI: https://dx.doi.org/10.1016/j.isatra.2021.02.006]

21. Sedghi, F; Arefi, MM; Abooee, A; Kaynak, O. Adaptive robust finite-time nonlinear control of a typical autonomous underwater vehicle with saturated inputs and uncertainties. IEEE/ASME Trans Mechatron; 2020; 26, 5 pp. 2517-2527. [DOI: https://dx.doi.org/10.1109/TMECH.2020.3041613]

22. Tijjani, AS; Chemori, A; Creuze, V. Robust adaptive tracking control of underwater vehicles: design, stability analysis, and experiments. IEEE/ASME Trans Mechatron; 2020; 26, 2 pp. 897-907. [DOI: https://dx.doi.org/10.1109/TMECH.2020.3012502]

23. Zhang, Y; Liu, J; Yu, J. Adaptive asymptotic tracking control for autonomous underwater vehicles with non-vanishing uncertainties and input saturation. Ocean Eng; 2023; 276, [DOI: https://dx.doi.org/10.1016/j.oceaneng.2023.114280] 114280.

24. An, S; Wang, L; He, Y; Yuan, J. Adaptive backstepping sliding mode tracking control for autonomous underwater vehicles with input quantization. Adv Theory Simul; 2022; 5, 4 2100445. [DOI: https://dx.doi.org/10.1002/adts.202100445]

25. Yan, Z; Gong, P; Zhang, W; Wu, W. Model predictive control of autonomous underwater vehicles for trajectory tracking with external disturbances. Ocean Eng; 2020; 217, [DOI: https://dx.doi.org/10.1016/j.oceaneng.2020.107884] 107884.

26. Wang, X; Yao, X; Zhang, L. Path planning under constraints and path following control of autonomous underwater vehicle with dynamical uncertainties and wave disturbances. J Intell Rob Syst; 2020; 99, 3–4 pp. 891-908. [DOI: https://dx.doi.org/10.1007/s10846-019-01146-3]

27. Ridao, P; Batlle, J; Carreras, M. Dynamics model of an underwater robotic vehicle; 2001; Institute of Informatics and Applications, University of Girona: pp. 3-28.

28. Seif, MS; Hasanvand, A. Investigating the geometry and control surface of AUV robots on hydrodynamics performance. J Mar Eng; 2021; 17, 33 pp. 53-64.

29. Hasanvand, A; Seif, MS. Investigation the effect of length-to-diameter ratio on six-DOF helical and zigzag maneuvers of the SUT glider with internal actuators. Ocean Eng; 2024; 295, [DOI: https://dx.doi.org/10.1016/j.oceaneng.2024.116819] 116819.

30. Hasanvand, A; Seif, MS. Development of state space models for underwater gliders equipped with embedded actuators in vertical and horizontal planes. Ocean Eng; 2024; 309, [DOI: https://dx.doi.org/10.1016/j.oceaneng.2024.118344] 118344.

31. Enayati, J; Sarhadi, P; Rad, MP; Zarini, M. Monte Carlo simulation method for behavior analysis of an autonomous underwater vehicle. Proceed Inst Mech Eng, Part M: J Eng Marit Environ; 2016; 230, 3 pp. 481-490.

32. Baumann E, Bahm C, Strovers B, Beck R, Richard M (2008) The X-43A six degree of freedom Monte Carlo analysis. In: 46th AIAA Aerospace Sciences Meeting and Exhibit (p. 203)

33. Ton CT, McCourt M, Mehta SS (2016) Robust tracking control of a quadrotor with time-varying gain in the presence of uncertainty and disturbances. In: AIAA guidance, navigation, and control conference (p. 1389)

34. Banerjee, S; Wang, Z; Baur, B; Holzapfel, F; Che, J; Cao, C. L 1 adaptive control augmentation for the longitudinal dynamics of a hypersonic glider. J Guid Control Dyn; 2016; 39, 2 pp. 275-291. [DOI: https://dx.doi.org/10.2514/1.G001113]

35. Souanef, T. L1 adaptive path-following of small fixed-wing unmanned aerial vehicles in wind. IEEE Trans Aerosp Electron Syst; 2022; 58, 4 pp. 3708-3716. [DOI: https://dx.doi.org/10.1109/TAES.2022.3153758]

36. Patre, PM; MacKunis, W; Kaiser, K; Dixon, WE. Asymptotic tracking for uncertain dynamic systems via a multilayer neural network feed forward and rise feedback control structure. IEEE Trans Autom Control; 2008; 53, 9 pp. 2180-2185. [DOI: https://dx.doi.org/10.1109/TAC.2008.930200]

37. Salahshoor, K; Khaki-Sedigh, A; Sarhadi, P. An indirect adaptive predictive control for the pitch channel autopilot of a flight system. Aerosp Sci Technol; 2015; 45, pp. 78-87. [DOI: https://dx.doi.org/10.1016/j.ast.2015.04.016]

38. Hovakimyan, N; Cao, C; Kharisov, E; Xargay, E; Gregory, IM. L1 adaptive control for safety-critical systems. IEEE Control Syst Mag; 2011; 31, 5 pp. 54-104.2866936 [DOI: https://dx.doi.org/10.1109/MCS.2011.941961]

39. Fossen TI (2011) Handbook of marine craft hydrodynamics and motion control. Sussex/John Willy & Sons Ltd