Gaze-Assisted Prescribed Performance Controller for AUV Trajectory Tracking in Time-Varying Currents

Abstract

Trajectory tracking for underactuated autonomous underwater vehicles (AUVs) is challenging due to coupling dynamics, modeling inaccuracies, and unknown disturbances. To tackle this, we propose a decoupling gaze-assisted prescribed performance controller (GAPPC). We first use an error transformation approach to achieve the prescribed performance, incorporating the line-of-sight (LOS) algorithm and an event-triggering mechanism to handle the kinematic characteristics of underactuated AUVs. Next, we develop a control strategy for the transformed error that does not require knowledge of the model parameters, including fast dynamic compensation to reduce steady-state errors. Finally, we analyze the controller’s stability and present simulation results. Simulations, which account for modeling inaccuracies and unknown ocean currents, show that the GAPPC improves stability errors by 67.3% compared to the adaptive robust controller.

1. Introduction

Autonomous underwater vehicles (AUVs) are increasingly used to achieve various mission objectives, requiring precise positioning and control [1,2]. This paper focuses on precise trajectory tracking [3,4], which involves designing control laws to guide the vehicle in following and maintaining parameterized trajectories with specified temporal characteristics [5]. In practical applications, AUVs are often configured in an underactuated mode, similar to torpedoes, due to considerations of cost, weight, and efficiency [6,7].

Designing a precise trajectory tracking law for an underactuated AUV is challenging due to strong coupling, uncertainty, and highly nonlinear dynamics [8]. An AUV is typically equipped with a propeller and four rudder plates arranged 90 degrees apart at the rear. These actuators do not directly control the three-dimensional position, leading to fewer actuators than degrees of freedom [9]. Addressing these complexities requires a highly nonlinear solution, incorporating a model compensation term in the controller that relies on accurate parameter estimation. Additionally, external disturbances in the marine environment are difficult to measure or estimate in real-time, further complicating trajectory tracking [10]. Therefore, control methods for underactuated AUVs must manage coupled dynamics, modeling inaccuracies, and time-varying unknown disturbances [11].

The coupling effects resulting from the underactuated characteristics of AUVs are commonly addressed using several methods, including the line-of-sight (LOS) guidance algorithm [12,13,14], the virtual vehicle approach [3,15], the backstepping method [16,17], and the Nussbaum function method [18]. The LOS guidance law effectively manages the underactuated nature of AUVs by converting three-dimensional tracking into controllable degrees of freedom, which simplifies understanding. However, when ocean currents affect AUV motion, calculating the sideslip angle to counteract drift becomes crucial [19,20]. The virtual vehicle method involves creating a virtual AUV as a tracking target, mirroring the same position and trajectory, and computing its angle and velocity based on dynamics. This method often overlooks the influence of ocean currents, presenting an idealized scenario [3]. In contrast to these methods, the backstepping and Nussbaum function approaches separate the AUV’s kinematic and dynamic differential equations, allowing for more organized control variable calculations according to the actual motion state, even in the presence of ocean currents [17,18]. However, these methods may lack intuitive adjustability and can introduce oscillations that are challenging to explain or correct, especially in stable environments. To combine the strengths of kinematic-based methods with those based on differential equations while addressing their limitations, we have developed a double-loop solution framework that integrates motion characteristic constraints from the LOS method with backstepping principles.

Various control strategies have been explored for AUV applications, ranging from traditional proportional–integral–derivative (PID) control [21,22] to more complex methods such as adaptive control (AC) [23,24], model predictive control [25], reinforcement learning (RL) control [26,27], and sliding mode control (SMC) [3,11,28,29]. PID control is popular in AUV systems due to its simplicity in design and implementation [11]. However, PID control struggles with adaptability to time-varying disturbances [30]. Adaptive control provides asymptotic tracking for large-scale nonlinear systems and handles parameter uncertainties [31], but it can become unstable in the presence of non-modeling disturbances [32]. Model predictive control’s effectiveness depends on accurate model parameter identification. Reinforcement learning control, driven by data, either requires an accurate model parameter simulator for training or extensive real operational data, which can be difficult to obtain. Among these methods, SMC is notable for its robustness against parameter variations, model uncertainties, and disturbances [33]. However, a significant limitation of standard SMC is the high-frequency control actions (chattering) caused by its discontinuous switching term [11]. Therefore, it is crucial to investigate controllers that do not depend on prior model knowledge and can reduce steady-state errors in practical applications.

In this paper, we propose a gaze-assisted prescribed performance controller (GAPPC) for AUV trajectory tracking in time-varying currents without relying on model parameters. We compare the GAPPC to the prescribed performance controller [34] and the adaptive robust controller [17], and validate its performance through numerical simulations.

The main contributions of this paper are outlined as follows:

• We introduce a gaze-assisted prescribed performance error transformation. Unlike the method in [34], our approach uses the LOS technique tailored to AUV motion characteristics to improve convergence behavior. This reduces the maximum and average tracking errors by over 27.2%.
• The proposed controller does not require prior model parameters. It reduces steady-state errors using dynamic compensation terms. Unlike adaptive robust control [17], our method does not rely on model parameter identification, making it more practical for experiments.
• Simulations with unknown time-varying ocean currents show our control strategy improves tracking accuracy by 67.3% compared to adaptive robust control [17]. It ensures both transient and steady-state performance without requiring model parameters.

The remainder of this paper is organized as follows: in Section 2, we introduce the materials and methods. Section 3 discusses the results. Section 4 presents the discussion.

2. Materials and Methods

2.1. Control Framework Design

2.1.1. Kinematics and Dynamics of Underactuated AUVs

Two frames are utilized to model the kinematics of the AUV, as illustrated in Figure 1. The north–east–down (NED) coordinate system is adopted as the frame {W}. The front–right–down frame is adopted as the body-fixed reference frame {A}.

The kinematic characteristics of the AUV in 6 degrees of freedom are described in detail in [8], leading to the derivation of the corresponding kinematic equations as depicted in (1) and (2). Given that the roll motion of the AUV is inherently self-stabilizing, it is disregarded. ${\mathit{R}}_{\mathit{A}}^{\mathit{W}}\left({\mathit{\eta }}_2\right)$ is the transformation matrix from the body-fixed reference frame {A} to the world reference frame {W}, and $\mathit{v}={\left[\begin{array}{c}\begin{array}{ccc}u& \upsilon & w\end{array}\end{array}\right]}^T$ is the body-fixed linear velocity. The body-fixed angular velocity vector $\mathit{\omega }={\left[\begin{array}{cc}q& r\end{array}\right]}^T$ and the Euler rate vector $\dot{{\mathit{\eta }}_2}$ are related through ${\mathit{C}}_{\mathit{A}}^{\mathit{W}}\left({\mathit{\eta }}_2\right)$.

$$\dot{{\mathit{\eta }}_1}=\left[\begin{array}{cc}cos\psi cos\theta & \begin{array}{cc}-sin\psi & cos\psi sin\theta \end{array}\\ \begin{array}{c}sin\psi cos\theta \\ -sin\theta \end{array}& \begin{array}{cc}cos\psi & sin\psi sin\theta \\ 0& cos\theta \end{array}\end{array}\right]\left[\begin{array}{c}u\\ \upsilon \\ w\end{array}\right]={\mathit{R}}_{\mathit{A}}^{\mathit{W}}\left(\left[\begin{array}{c}\theta \\ \psi \end{array}\right]\right)\mathit{v}$$

(1)

$$\dot{{\mathit{\eta }}_2}=\left[\begin{array}{cc}1& 0\\ 0& 1/cos\theta \end{array}\right]\left[\begin{array}{c}q\\ r\end{array}\right]={\mathit{C}}_{\mathit{A}}^{\mathit{W}}\left({\mathit{\eta }}_2\right)\left[\begin{array}{c}q\\ r\end{array}\right]$$

(2)

where $\mathit{v}$ represents the velocity of the AUV in {A} and ${\mathit{\eta }}_1$ represents the position of the AUV in {W}.

Assumption 1.

(1) The vehicle’s shape structure exhibits symmetry across three principal planes. (2) The hydrodynamic drag terms in the dynamic equations are of an order lower than two. (3) The roll motion of the vehicle is disregarded. (4) The vehicle is a neutrally buoyant rigid body with a homogeneous mass distribution.

As per Assumption 1, the dynamic system of the AUV is shown as outlined in [8].

$$M\left[\begin{array}{c}\dot{u}\\ \dot{\upsilon }\\ \begin{array}{c}\dot{w}\\ \dot{q}\\ \dot{r}\end{array}\end{array}\right]=\left[\begin{array}{c}{\mathit{\phi }}_{Au}^T{\mathit{\theta }}_{Au}\\ {\mathit{\phi }}_{Av}^T{\mathit{\theta }}_{Av}\\ \begin{array}{c}{\mathit{\phi }}_{Aw}^T{\mathit{\theta }}_{Aw}\\ {\mathit{\phi }}_{Aq}^T{\mathit{\theta }}_{Aq}\\ {\mathit{\phi }}_{Ar}^T{\mathit{\theta }}_{Ar}\end{array}\end{array}\right]+\left[\begin{array}{c}{\tau }_1\\ 0\\ \begin{array}{c}0\\ -(z_{G}W-z_{B}B)\mathrm{s}\mathrm{i}\mathrm{n}\left(\theta \right)+{\tau }_2\\ {\tau }_3\end{array}\end{array}\right]$$

(3)

The definition of AUV dynamics model parameters can be found in the Appendix A.

Assumption 2.

Let ${\mathit{\eta }}_{1d}$ represent the desired position trajectory, characterized by continuously differentiable functions with a bounded and continuously differentiable derivative denoted by ${\dot{\mathit{\eta }}}_{1d}$.

Assumption 2 facilitates differentiating the tracking trajectory to study the essence of tracking control. In practical applications, the proposed controller may experience oscillations with discontinuous trajectory segments. Setting an upper limit on the trajectory’s differential values can help reduce such oscillations in practice. However, continuous trajectories are assumed in the theoretical derivation for the ease of analysis.

Remark 1.

Due to the size of the mechanical mechanism, the force and torque output by the actuator are bounded. Due to the size of the input, the speed of the vehicle is bounded.

Remark 2.

For narrow and elongated AUVs, the signs of $sign(m_{ii}-m_{11})$ for $i\in 1, 2, 3$ are consistently positive based on the hydrodynamic analysis.

2.1.2. Error Systems

This section adopts the double-loop framework previously examined in our study as the control framework for underactuated AUVs [17].

The desired position trajectory, denoted as ${\mathit{\eta }}_{1d}$, serves as the reference for position tracking. The position tracking error is defined as ${\mathit{\eta }}_{1e}={\mathit{\eta }}_1-{\mathit{\eta }}_{1d}$. In order to decouple the dynamic equations of the AUV, we introduce ${\mathit{y}}_1$ as a substitute variable for the tracking error ${\mathit{\eta }}_{1e}$. The definition of ${\mathit{y}}_1$ with a positive constant ${\mathit{k}}_1={[k_{11},k_{12},k_{13}]}^T$ is as follows:

$${\mathit{y}}_1={{\mathit{R}}_{\mathit{A}}^{\mathit{W}}\left({\mathit{\eta }}_2\right)}^{\mathit{T}}({\dot{\mathit{\eta }}}_1-{\dot{\mathit{\eta }}}_{1d}+diag\left({\mathit{k}}_1\right){\mathit{\eta }}_{1e})$$

(4)

When ${\mathit{y}}_1=0$, we observe that ${\dot{\mathit{\eta }}}_{1e}=-diag\left({\mathit{k}}_1\right){\mathit{\eta }}_{1e}$, leading to the exponentially converging position tracking error ${\mathit{\eta }}_{1e}$.

Referring to Equation (1),

$${\mathit{P}}_1{\dot{\mathit{y}}}_1=\left[\begin{array}{c}{\mathit{\phi }}_{11}^T{\mathit{\theta }}_{11}\\ {\mathit{\phi }}_{12}^T{\mathit{\theta }}_{12}\\ {\mathit{\phi }}_{13}^T{\mathit{\theta }}_{13}\end{array}\right]+diag\left({\mathit{\phi }}_u\right){\mathit{y}}_{2d}-{\mathit{P}}_1\mathit{\sigma }$$

(5)

where ${\mathit{y}}_{2d}$ is the control input of the layer controller. Specifically, it corresponds to the forward force ${\tau }_1$, the expected heading angular velocity $r_d$, and the negative value of the expected pitch angular velocity (${-q}_d$). The last two are the control objectives of the next level controller. ${\mathit{P}}_1$ is the diagonal matrix related to $m_{ii}$ for $i\in 1, 2, 3$. Please refer to the Appendix A for specific values.

Referring to Equation (1),

$${\mathit{P}}_2{\dot{\mathit{y}}}_2=\left[\begin{array}{c}{\mathit{\phi }}_{21}^T{\mathit{\theta }}_{21}\\ {\mathit{\phi }}_{22}^T{\mathit{\theta }}_{22}\end{array}\right]+{\mathit{y}}_{3d}-{\mathit{P}}_2\left[\begin{array}{c}{\dot{q}}_d\\ {\dot{r}}_d\end{array}\right]$$

(6)

$${\mathit{y}}_2={\left[q-\left({-\mathit{y}}_{2d}\left(3\right)\right),r-{\mathit{y}}_{2d}\left(2\right)\right]}^T={[q-q_d,r-r_d]}^T$$

(7)

where ${\mathit{y}}_{3d}$ is the control input of the layer controller. Specifically, it corresponds to the torque in the pitch direction ${\tau }_2$ and the torque in the heading direction ${\tau }_3$. ${\mathit{P}}_2$ is the diagonal matrix related to $m_{ii}$ for $i\in 5, 6$. Please refer to the Appendix A for specific values.

2.1.3. Control Objective

In order to achieve trajectory tracking of the AUV, it is necessary to develop a controller within this decoupling framework. The primary goal is to ensure effective handling of errors ${\mathit{y}}_1$ and ${\mathit{y}}_2$ in the presence of unknown external disturbances.

2.2. Control Design

2.2.1. Gaze-Assisted Error Transformation

The gaze-assisted error transformation for the AUV is derived. Specifically, it provides a quantitative characterization of the tracking performance as follows:

$$-{\mathit{\rho }}_i\left(t\right)<{\mathit{y}}_i\left(t\right)<{\mathit{\rho }}_i\left(t\right),\forall t>0$$

(8)

where ${\mathit{\rho }}_i(i=\mathrm{1,2})$ is a bounded and strictly positive function, referred to as the performance function. Because the AUV is underactuated and continuously approaching the heading angle, it is not feasible to independently minimize the three positions simultaneously. Consequently, the three scalars in ${\mathit{y}}_1$ cannot all converge simultaneously to ${\mathit{\rho }}_1={[{\rho }_{11},{\rho }_{12},{\rho }_{13}]}^T$ that converges over time, particularly when the heading angle significantly deviates from the expected trajectory’s heading angle. Therefore, an event-triggering mechanism is proposed to update ${\mathit{\rho }}_1$ under underactuated influence. ${\mathit{\rho }}_1$ can be defined as

$${\rho }_{1i}\left(t\right)=\left({\rho }_{1i}\left(0\right)-{\rho }_{1i}^{\infty }\right)e^{{-l}_{1i}(t-t_0)}+{\rho }_{1i}^{\infty }$$

(9)

$$t_0=\left\{\begin{array}{cc}\begin{array}{c}t\\ t_0\end{array}& \begin{array}{c}\exists i, y_{1i}>{\rho }_{1i} or y_{1i}<{-\rho }_{1i}\\ \forall i,-{\rho }_{1i}<y_{1i}<{\rho }_{1i}\end{array}\end{array}\right.$$

(10)

where ${\rho }_{1i}>0$ for $i\in 1, 2, 3$ represents the maximum allowable error size for real-time updates. ${\mathit{\rho }}_1\left(0\right)$ is selected as ${\mathit{\rho }}_1\left(0\right)=k_{\rho 1}{\mathit{y}}_1\left(0\right)$, $k_{\rho 1}>1$. The initial value of $t_0$ is $0$. When any value of ${\mathit{y}}_1$ exceeds the specified performance range due to underactuated characteristics, $t_0$ is assigned as the current time $t$, i.e., $t_0=t$, and ${\mathit{\rho }}_1$ is reinitialized to converge from ${\mathit{\rho }}_1\left(0\right)$ to ${\mathit{\rho }}_1^{\mathit{\infty }}={[{\rho }_{11}^{\mathit{\infty }},{\rho }_{12}^{\mathit{\infty }},{\rho }_{13}^{\mathit{\infty }}]}^T$. The constant $l_{1i}>0$ determines the convergence rate of ${\rho }_{1i}\left(t\right)$. ${\rho }_{1i}^{\infty }>0$ is a small enough steady-state performance requirement. The symbol “∃” represents “existential symbol” in mathematical notation.

The inspiration for ${\mathit{\rho }}_1^{\infty }$ comes from the commonly used method of LOS in underactuated AUV trajectory tracking. The LOS method fully considers the underactuated nature and utilizes tracking of the orientation angle to achieve trajectory tracking. In order to dynamically adjust the required lower limit, we use a virtual vehicle following the LOS method to track the trajectory as the value principle of ${\mathit{\rho }}_1^{\infty }$. The initial position ${\mathit{\eta }}_1={[{\eta }_{11},{\eta }_{12},{\eta }_{13}]}^T$ and posture ${\mathit{\eta }}_2={[\theta ,\psi ]}^T$ of the virtual vehicle are the same as the actual one. We set the speed and angular velocity of the virtual vehicle to ${\mathit{v}}_{vir}={\left[u_v,0,0\right]}^T$ and ${\mathit{\omega }}_{vir}={\left[q_v,r_v\right]}^T$, respectively.

$${\psi }_e=\left\{\begin{array}{cc}{\psi }_d-\psi & \left|{\psi }_d-\psi \right|<\pi \\ {\psi }_d-\psi -sign({\psi }_d-\psi )2\pi & \left|{\psi }_d-\psi \right|\ge \pi \end{array}\right.$$

(11)

$${\psi }_v=\left\{\begin{array}{cc}{\psi }_d& \left|{\psi }_e\right|<r_v{t}_1\\ \psi +sign\left({\psi }_e\right)r_v{t}_1& \left|{\psi }_e\right|\ge r_v{t}_1\end{array}\right.$$

(12)

$${\theta }_v=\left\{\begin{array}{cc}{\theta }_d& \left|{\theta }_d-\theta \right|<q_v{t}_1\\ \theta +sign\left({\theta }_d\right)q_v{t}_1& \left|{\theta }_d-\theta \right|\ge q_v{t}_1\end{array}\right.$$

(13)

where in the expected heading angle of a virtual vehicle is ${\psi }_d=atan2({\eta }_{12}-{\eta }_{12d},{\eta }_{11}-{\eta }_{11d})$, ${\psi }_e$ is the error between the desired heading angle and the actual one. The desired pitch angle of virtual vehicle is ${\theta }_d=atan2({\eta }_{13}-{\eta }_{13d},\sqrt{{\left({\eta }_{11}-{\eta }_{11d}\right)}^2+{\left({\eta }_{12}-{\eta }_{12d}\right)}^2})$. $t_1$ is the time interval for issuing control instructions, determined by the control frequency. ${\psi }_v\left(t\right)$ and ${\theta }_v\left(t\right)$ are the heading and pitch angles that can be achieved under LOS law.

By utilizing the virtual vehicle speed ${\mathit{v}}_v$, the virtual vehicle position ${\mathit{\eta }}_{1v}$ at that angle ${\mathit{\eta }}_{2v}={[{\theta }_v,{\psi }_v]}^T$ can be obtained. The error between this position ${\mathit{\eta }}_{1v}$ and the target ${\mathit{\eta }}_d$ is considered the maximum allowable error after convergence ${\mathit{\rho }}_1^{\infty }$, which can be defined as

$${\rho }_{1i}^{\infty }=\left\{\begin{array}{cc}\begin{array}{c}{k}_{\rho i}\left|y_{vi}\right|\\ {\rho }_d^{\infty }\end{array}& \begin{array}{c} k_{\rho i}\left|y_{vi}\right|>{\rho }_d^{\infty }\\ k_{\rho i}\left|y_{vi}\right|\le {\rho }_d^{\infty }\end{array}\end{array}\right. i=1, 2, 3$$

(14)

$${\mathit{y}}_v={{\mathit{R}}_{\mathit{A}}^{\mathit{W}}\left({\mathit{\eta }}_2\right)}^{\mathit{T}}\{{\mathit{R}}_{\mathit{A}}^{\mathit{W}}\left({\mathit{\eta }}_{2vir}\right){\mathit{v}}_v-{\dot{\mathit{\eta }}}_{1d}+diag\left({\mathit{k}}_1\right)\left({\mathit{\eta }}_{1v}-{\mathit{\eta }}_{1d}\right)\}$$

(15)

where the constant $k_{\rho i}>0$.${\rho }_d^{\infty }$ is a constant, representing the minimum convergence value that can be selected. ${\mathit{y}}_v={\left[y_{v1},y_{v2},y_{v3}\right]}^T$ is the error term converted according to Equation (4).

${\mathit{y}}_2$ is controlled by two independent rudder plates, so the two scalars do not affect each other and ${\mathit{\rho }}_2{=[{\rho }_{21},{\rho }_{22}]}^T$ can be set to converge over time. ${\mathit{\rho }}_2$ can be defined as

$${\rho }_{2i}\left(t\right)=\left({\rho }_{2i}\left(0\right)-{\rho }_{2i}^{\infty }\right)e^{{-l}_{2i}t}+{\rho }_{2i}^{\infty }$$

(16)

where ${\rho }_{2i}>0$ for $i\in 1, 2$ represents the maximum allowable error size for real-time updates.${\mathit{\rho }}_2\left(0\right)$ is selected as ${\mathit{\rho }}_2\left(0\right)=k_{\rho 2}{\mathit{y}}_2\left(0\right)$, $k_{\rho 2}>1$. ${\mathit{\rho }}_2$ is initialized to converge from ${\mathit{\rho }}_2\left(0\right)$ to ${\mathit{\rho }}_2^{\mathit{\infty }}={\left[{\rho }_{21}^{\infty },{\rho }_{22}^{\infty }\right]}^T$. The constant $l_{2i}>0$ determines the convergence rate of ${\rho }_{2i}\left(t\right)$. ${\rho }_{2i}^{\infty }>0$ is a small enough steady-state performance requirement.

In order to make the error converge with the specified performance, the following error transformation is devised:

$${\mathit{y}}_i\left(t\right)=W\left({\mathit{\kappa }}_i\right){\mathit{\rho }}_i\left(t\right)$$

(17)

$T\left({\mathit{\kappa }}_i\right)$ needs to meet the following conditions:

(1)

$-1<W\left({\mathit{\kappa }}_i\right)<1$, $\forall {\mathit{\kappa }}_i\in \mathbb{R}$;

(2)

$\underset{{\mathit{\kappa }}_i\to +\mathbf{\infty }}{\lim }W\left({\mathit{\kappa }}_i\right)=1$;

(3)

$\underset{{\mathit{\kappa }}_i\to -\mathbf{\infty }}{\lim }W\left({\mathit{\kappa }}_i\right)=-1$.

This study chooses the hyperbolic tangent-type function as $T\left({\mathit{\kappa }}_i\right)$.

$$W\left({\mathit{\kappa }}_i\right)={\displaystyle \frac{e^{{\mathit{\kappa }}_i}-e^{{-\mathit{\kappa }}_i}}{e^{{\mathit{\kappa }}_i}+e^{{-\mathit{\kappa }}_i}}}$$

(18)

Then, ${\mathit{\kappa }}_i$ can be derived as

$${\mathit{\kappa }}_i={\displaystyle \frac{1}{2}}\ln \left({\displaystyle \frac{1+{\mathit{\xi }}_i}{1-{\mathit{\xi }}_i}}\right)$$

(19)

where ${\mathit{\xi }}_i={\mathit{y}}_i$/${\mathit{\rho }}_i$.

It is evident that when ${\mathit{y}}_i$ tends towards 0, ${\mathit{\kappa }}_i$ also approaches 0.

2.2.2. Control for AUV Trajectory Tracking

In order to control the system represented by Equations (5) and (6) and ensure accurate tracking with specified performance, the control strategy is introduced, leveraging the error transformation discussed earlier. By applying Equation (19), the error ${\mathit{y}}_1$ can be converted into ${\mathit{\kappa }}_1={[{\kappa }_{11},{\kappa }_{12},{\kappa }_{13}]}^T$, with its derivative expressed as

$${\dot{\mathit{\kappa }}}_1={\displaystyle \frac{{\mathit{Q}}_1}{diag\left({\mathit{\rho }}_1\right)}}\{{\mathit{P}}_1^{-1}\left[\begin{array}{c}{\mathit{\phi }}_1^T{\mathit{\theta }}_1\\ {\mathit{\phi }}_2^T{\mathit{\theta }}_2\\ {\mathit{\phi }}_3^T{\mathit{\theta }}_3\end{array}\right]+{\mathit{P}}_1^{-1}diag\left({\mathit{\phi }}_u\right){\mathit{y}}_{2d}-\mathit{\sigma }-diag({\mathit{\kappa }}_1){\dot{\mathit{\rho }}}_1\}$$

(20)

where ${\mathit{Q}}_1=diag(q_{11}, q_{12}, q_{13})$; $q_{1i}$ is defined as

$$q_{1i}={\displaystyle \frac{1}{\left(1-{\kappa }_{1i}\right)\left(1+{\kappa }_{1i}\right)}} for i\in 1, 2, 3$$

(21)

Introducing the variable ${\mathit{y}}_{2d}$ ensures the stability of ${\mathit{\kappa }}_1$. The control strategy comprises a stabilizing feedback control component with a positive constant ${\mathit{k}}_2={[k_{21},k_{22},k_{23}]}^T$ and an additional fast dynamics compensation term denoted as ${\widehat{\mathit{u}}}_1$. The design of ${\mathit{y}}_{2d}$ is formulated as

$$diag\left({\mathit{\phi }}_u\right){\mathit{y}}_{2d}=-{\mathit{k}}_2{\mathit{Q}}_1^T{\mathit{\kappa }}_1-{\widehat{\mathit{u}}}_1$$

(22)

$${\dot{\widehat{\mathit{u}}}}_1\left(i\right)=\left\{\begin{array}{c}\begin{array}{cc}0& if \left|{\widehat{\mathit{u}}}_1\left(i\right)\right|={\mathit{u}}_{U1}\left(i\right) and {\widehat{\mathit{u}}}_1\left(i\right){\mathit{\kappa }}_1\left(i\right)>0\end{array}\\ \begin{array}{cc}{\mathit{\Gamma }}_1\left(i\right){\mathit{Q}}_1\left(i\right){\mathit{\kappa }}_1\left(i\right) & otherwise \end{array}\end{array}\right.for i=1, 2, 3$$

(23)

where ${\mathit{\Gamma }}_1={[{\Gamma }_{21},{\Gamma }_{22},{\Gamma }_{23}]}^T$. This selection rule can ensure that $\left|{\widehat{\mathit{u}}}_1\left(i\right)\right|\le {\mathit{u}}_{U1}\left(i\right)$. ${\mathit{u}}_{U1}$ is the pre-set constant.

The next controller aims to track $r_d$ and $q_d$. Similarly, the error ${\mathit{y}}_2$ can be converted into ${\mathit{\kappa }}_2={[{\kappa }_{21},{\kappa }_{22}]}^T$, with its derivative expressed as

$${\dot{\mathit{\kappa }}}_2={\displaystyle \frac{{\mathit{Q}}_2}{diag\left({\mathit{\rho }}_2\right)}}\{{\mathit{P}}_2^{-1}\left[\begin{array}{c}{\mathit{\phi }}_4^T{\mathit{\theta }}_4\\ {\mathit{\phi }}_5^T{\mathit{\theta }}_5\end{array}\right]+{\mathit{P}}_2^{-1}{\mathit{y}}_{3d}-\left[\begin{array}{c}{\dot{q}}_d\\ {\dot{r}}_d\end{array}\right]-diag({\mathit{\kappa }}_2){\dot{\mathit{\rho }}}_1\}$$

(24)

where ${\mathit{Q}}_2=diag(q_{21}, q_{22})$; $q_{2i}$ for $i\in 1, 2$ is defined as

$$q_{2i}={\displaystyle \frac{1}{\left(1-{\kappa }_{2i}\right)\left(1+{\kappa }_{2i}\right)}}$$

(25)

${\mathit{y}}_{3d}$ includes a stabilizing feedback control action with a positive constant ${\mathit{k}}_3={[k_{31},k_{32}]}^T$ and additional fast dynamics compensation as follows:

$${\mathit{y}}_{3d}=-{\mathit{k}}_3{\mathit{Q}}_2^T{\mathit{\kappa }}_2-{\widehat{\mathit{u}}}_2$$

(26)

$${\dot{\widehat{\mathit{u}}}}_2\left(i\right)=\left\{\begin{array}{c}\begin{array}{c}0 if \left|{\widehat{\mathit{u}}}_2\left(i\right)\right|={\mathit{u}}_{U2}\left(i\right) and {\widehat{\mathit{u}}}_2\left(i\right){\mathit{\kappa }}_2\left(i+1\right)>0\end{array}\\ \begin{array}{cc}{\Gamma }_2{\mathit{Q}}_2\left(i\right){\mathit{\kappa }}_2\left(i\right) & otherwise \end{array}\end{array}\right.for i=1, 2$$

(27)

where ${\Gamma }_2>0$. This selection rule can ensure that $\left|{\widehat{\mathit{u}}}_2\left(i\right)\right|\le {\mathit{u}}_{U2}\left(i\right)$. ${\mathit{u}}_{U2}$ is the pre-set constant.

The stability proof of the control strategy can be found in the Appendix A. Based on the stability proof provided, the proposed control strategy is stable. This control strategy uses a dual-loop framework to decouple the 3D trajectory tracking control of the underactuated AUV. Each loop is designed with a prescribed performance controller. By controlling the intermediate transformed quantities, the actual 3D errors are reduced. The next section presents comparative experiments between the proposed algorithm and existing algorithms.

3. Results

The robustness of the control strategy was confirmed through simulation in an unknown ocean current environment. The AUV parameters utilized can be referenced in [35].

The desired compensation adaptive robust controller (DCARC) [17], designed to address model parameter inaccuracies and time-varying bounded ocean current conditions, is contrasted with the algorithm introduced in this study. The DCARC requires knowledge of the model parameters and their uncertainty range for the AUV. For this comparison, we assume a 20% uncertainty in the model parameters, indicating that the nominal dynamics parameters are estimated to be 20% higher than their true values. In contrast, the control algorithm proposed in this paper operates without demanding model parameters. The parameter configurations are detailed as follows:

${\mathit{k}}_1={[1,1,1]}^T$, $u_v=1 \mathrm{m}/\mathrm{s}$, $q_v=r_v=0.17 \mathrm{rad}/\mathrm{s}$, $k_{\rho 1}=0.4, k_{\rho 2}=0.1, k_{\rho 3}=0.2$, $k_{\rho 1}=k_{\rho 2}=3$, ${\rho }_d^{\infty }=0.001$, ${\mathit{l}}_1={[0.1, 0.2, 0.05]}^T$, ${\mathit{l}}_2={[0.25, 0.5, 0.5]}^T$, ${\mathit{\rho }}_2^{\mathit{\infty }}={\left[0.52,0.52\right]}^T$, ${\mathit{k}}_2={[900,0.45,0.45]}^T$, ${\Gamma }_1={[10,0.03,0.03]}^T$, ${\mathit{d}}_{M1}={[10,0.01,0.01]}^T$, ${\mathit{k}}_3={[1200,2400]}^T$, ${\Gamma }_2=1$, ${\mathit{d}}_{M2}={[5,5]}^T$.

To validate the effectiveness of the gaze-assisted error transformation proposed for addressing underdrive, this paper also selected the error transformation presented in [34] as the predetermined performance controller (PPC) for comparison. The only difference between this controller and the gaze-assisted predetermined performance controller (GAPPC) lies in their respective error transformations, with no other distinctions.

To test the tracking performance of the GAPPC, the desired trajectory selected includes various trajectories such as straight lines and spirals.

To facilitate a quantitative comparison of the tracking outcomes, we will evaluate the performance indices outlined in [36].

(1)

$e_l=MAX\left(\sqrt{e_{Nj}^2+e_{Ej}^2+e_{Dj}^2}\right)$, represents the maximum value of the error after the AUV tracks a straight line and stabilizes, that is, the error range after convergence is within this range. Specifically, this metric identifies the largest error within the time interval [1900 s, 2200 s] along this trajectory segment.

(2)

$e_b=MAX\left(\sqrt{e_{Ni}^2+e_{Ei}^2+e_{Di}^2}\right)$, signifies the maximum value of the error after the AUV tracks a spiral and stabilizes, that is, the error range after convergence is within this range. Specifically, this parameter captures the peak error within the time span [450 s, 650 s] for this trajectory segment.

(3)

$E_2=\sqrt{{\displaystyle \frac{{\sum }_{k=1}^N(e_{Ne}^2+e_{Ee}^2+e_{De}^2)}{N}}}$, represents the average tracking error of the AUV throughout the entire driving process.

(4)

$L_2\left({\tau }_1\right)=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_{k=1}^N\left({\tau }_1^2\right)}$, the average control input of ${\tau }_1$, is used to evaluate the amount of control effort.

(5)

$L_2\left({\tau }_{23}\right)=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_{k=1}^N({\tau }_2^2+{\tau }_3^2)}$, the average control input of ${\tau }_2$ and ${\tau }_3$, is used to evaluate the amount of control effort.

3.1. Simulations with Modeling Inaccuracies

Figure 2 depicts the tracking outcomes for all three control strategies, while Figure 3 displays the position tracking errors associated with each controller.

Upon reviewing

Table 1. Performance indices for DCARC with modeling inaccuracies, PPC, and GAPPC.

Performance Indices	${\mathit{e}}_{\mathit{l}}$ [m]	${\mathit{e}}_{\mathit{b}}$ [m]	${\mathit{E}}_2$ [m]	${\mathit{L}}_2\left({\mathit{\tau }}_1\right)$ [N]	${\mathit{L}}_2\left({\mathit{\tau }}_{23}\right)$ [N/m]
DCARC	0.0757	0.1122	2.2603	354.2057	35.4269
PPC	2.4531	2.3074	3.3937	361.9542	26.6106
GAPPC	0.0443	0.0131	2.4698	353.7242	59.0387

and Figure 3, it becomes apparent that the tracking error of the PPC, which does not consider underactuated characteristics, is notably larger than that of both the GAPPC and the DCARC. During the spiral descent, the steady-state errors are less than 0.1122 m, 2.3074 m, and 0.0131 m. The stability error of the GAPPC has seen an improvement of over 88.3%. The subgraph in Figure 3, covering the time intervals [450 s, 650 s], illustrates the error of the three control strategies during the AUV spiral phase. Despite the control errors in the three directions of the GAPPC and the DCARC constantly fluctuating, they remain within 0.1 m, with the GAPPC exhibiting smaller errors. Meanwhile, during the AUV’s straight phase with time intervals of [1900 s, 2000 s], the steady-state errors of the three control strategies are less than 0.0757 m, 2.4531 m, and 0.0443 m. The proposed control strategy has seen a stability error improvement of over 41.5%. The subgraph in Figure 3, spanning the time intervals [1900 s, 2000 s], displays the error of the three control strategies during the AUV’s straight phase. The control errors in the three directions of the GAPPC and the DCARC demonstrate relative stability, with the GAPPC exhibiting smaller tracking errors in the north direction. Observing Figure 2 and Figure 3 during the initial stage of trajectory tracking, it becomes evident that the GAPPC requires a longer convergence time compared to the DCARC, resulting in larger controller errors in the global error calculation indicator $L_2\left(e_e\right)$. Additionally, Figure 4 shows the driver output, speed, and angle information of the respective control strategy. The GAPPC demonstrates more frequent oscillations in the input ${\tau }_i(i=\mathrm{1,2},3)$. This behavior is attributed to the controller’s lack of model parameters and model identification, making it susceptible to oscillations with negative feedback. Our comparison reveals that the proposed method results in the lowest average force $L_2\left({\tau }_1\right)$ on the propeller, indicating a slight energy saving in propeller speed. However, the average torque $L_2\left({\tau }_{23}\right)$ on the rudder is higher due to some oscillation in the rudder torque, which causes more frequent adjustments in angular velocity. This results in slightly higher energy consumption compared to the other controllers.

Subsequent simulations involve further testing of the algorithm’s robustness by introducing ocean current disturbances.

3.2. Simulations with Modeling Inaccuracies and Unknown Ocean Currents

We address the treatment of unknown ocean currents as bounded external disturbances with a fixed direction angle and time-varying magnitude. The specific disturbances selected for consideration are listed below.

$$\begin{gathered} V_C=V_{max}\cos \left({\omega }_{c}t\right) \\ {u}_C=V_C\left(t\right)\cos \left({\beta }_c-\psi \right) \\ {\upsilon }_C=V_C\left(t\right)\text{sin}\left({\beta }_c-\psi \right) \end{gathered}$$

$V_C$ represents the ocean currents, which is constrained by the maximum value $V_{max}$. ${\beta }_c$ represents the direction of ocean currents. ${\omega }_c$ represents the speed of change in the size of ocean currents. In this study,$V_{max}=0.5 \mathrm{m}/\mathrm{s}$, ${\beta }_c=60°$, and ${\omega }_c=0.01$. $u_C$ is the velocity component of ocean currents in the forward direction of the AUV.${\upsilon }_C$ is the velocity component of ocean currents in the lateral direction of the AUV.

Figure 5 illustrates the tracking outcomes for these control strategies, while Figure 6 displays their position tracking errors. As seen in

Table 2. Performance indices for DCARC, PPC, and GAPPC with unknown ocean currents.

Performance Indices	${\mathit{e}}_{\mathit{l}}$ [m]	${\mathit{e}}_{\mathit{b}}$ [m]	${\mathit{E}}_2$ [m]	${\mathit{L}}_2\left({\mathit{\tau }}_1\right)$ [N]	${\mathit{L}}_2\left({\mathit{\tau }}_{23}\right)$ [N/m]
DCARC	0.2526	0.2451	2.0944	390.1219	34.5640
PPC	2.7805	2.8115	3.3939	397.7357	27.2189
GAPPC	0.0540	0.0802	2.3508	387.9854	44.8443

and Figure 6, the PPC exhibits larger tracking errors compared to the other two controllers. During the AUV’s spiral descent with time intervals of [450 s, 650 s], the tracking errors remain below 0.2451 m, 2.8115 m, and 0.0802 m. The proposed control strategy demonstrates an improvement in stability error of over 67.3%. During the AUV’s straight phase with time intervals of [1900 s, 2000 s], the tracking errors for these control strategies are less than 0.2526 m, 2.7805 m, and 0.0540 m, with the proposed control strategy showing an improved stability error of over 78.6%. Similar to the previous section, the GAPPC exhibits slower convergence compared to the DCARC. Figure 7 shows the driver output, speed, and angle information of the respective control strategy. The sway $\upsilon$ of the AUV fluctuates within the range of [−0.7, 1.3] m/s due to unknown currents. Similar to the previous simulation results, the comparison shows that our proposed method achieves the lowest average force $L_2\left({\tau }_1\right)$ on the propeller compared to the other two controllers. However, due to some oscillation in the rudder torque ${\tau }_2$ and ${\tau }_3$, the average torque $L_2\left({\tau }_{23}\right)$ on the rudder is the highest.

Comparing these two simulation results reveals that our proposed controller achieves a significantly lower steady-state error than the PPC and the DCARC without relying on specific model parameters. Additionally, it results in the smallest average force on the propeller. However, there are areas to address in future work. Our method converges more slowly compared to the DCARC and exhibits greater oscillation in ${\tau }_2$ and ${\tau }_3$, leading to higher average torque compared to the other two controllers. Future efforts will focus on maintaining the current advantages of low steady-state errors and propeller energy consumption while reducing torque oscillations.

4. Discussion

A gaze-assisted prescribed performance controller is designed for 3D trajectory tracking of underactuated AUVs. This controller uses a double-loop control framework with an error transformation prescribed performance. Due to the underactuated nature of the AUV’s north–east–down position, which cannot be minimized simultaneously, the convergence value is dynamically determined using a kinematic-based LOS algorithm. An event-triggering mechanism ensures effective error transformation. We propose a control method that does not require model parameters and includes fast dynamic compensation to reduce steady-state errors. The stability analysis confirms the controller’s performance in both transient and steady-state conditions. In simulations, this control strategy outperforms the double-loop adaptive robust control strategy, which requires model parameters. With parameter uncertainty, the GAPPC improves steady-state errors by over 41.5% compared to adaptive robust control. Under unknown time-varying ocean currents, the GAPPC reduces steady-state errors by over 67.3%. This method guarantees effective transient and stable tracking performance. Although the GAPPC achieves significantly lower steady-state errors and less propeller force than the PPC and the DCARC, it converges more slowly and shows higher oscillations in ${\tau }_2$ and ${\tau }_3$, resulting in higher average torque.

In our future research, we will investigate how to achieve faster convergence under specified performance constraints, explore methods to reduce actuator oscillations, and conduct real-world tests in actual water environments.