1   Introduction

The ocean is a rich source of energy and plays a key role in forecasting variations in the earth's climate. Hence, numerous studies have been performed to explore natural resources and assist with underwater robotic manipulators and vehicles. Autonomous underwater vehicles (AUVs) are generally employed in underwater interventions owing to their excellent maneuvering capabilities and high accuracy in path-following control in spatial space (Guo et al., 2003). Studies on underwater robotic vehicles have increased over the past two to three decades. Owing to improvements in technology and the capacity to create intelligent, smarter robots, autonomous vehicles are employed for more expeditions. This situation enables exploring higher-risk areas, spending more time underwater, and conducting more productive underwater applications compared with human-piloted/tethered underwater vehicles (Fossen, 1994). These vehicles have been engineered to be strong, adaptable, and agile. Consequently, during the past 25 years, substantial research has been devoted to developing AUVs that can perform a wide range of tasks that are much beyond the scope of divers and small submarines; the tasks include long-term sample collection and 3D mapping of underwater tunnels. These vehicles can dive deeper and stay submerged for longer periods of time without risking people (Lakhekar and Waghmare, 2014).

Controlling underwater vehicles is very challenging due to nonlinear, time-varying, and unexpected external disturbances, including sea current fluctuations, which oppose the motion of AUVs. Because the dynamics of an AUV are inherently complicated, nonlinear, and time-varying, it can neither be neglected nor considerably simplified while designing controls. Additionally, depending on the operating circumstances, its mass and buoyancy changes (Elmokadem et al., 2016). The difficulty in precisely simulating the hydrodynamic effect and Coriolis force is a substantial impediment to AUV autonomy. AUV actuators can generate a limited amount of torque. During AUV operation, torque requirements may surpass the AUV's torque limit. Thus, actuator saturation is inevitable. This outcome demonstrates that actuator saturation is a substantial issue in the motion control of AUVs. Because linear controllers function locally in water, where manually tweaking the control settings is challenging, linear controllers may not satisfy the desired requirements, specifically when operating conditions or the system environment change (Sahu and Subudhi, 2014). Despite remarkable progress in the field of control design strategies that enable the vehicle to operate under parametric uncertainty, varying dynamics, unbounded constraints and failures, still need to be developed (Rout and Subudhi, 2016). Two aspects contribute to the challenges in the field of marine robotics: First, the complex dynamics of marine vehicles cannot be disregarded or dramatically minimized during the design of the control scheme. Second, the designed control scheme is dependent on unmeasured states. AUVs pose a control challenge because a majority of them are underactuated, imposing nonintegrable acceleration constraints (Lakhekar and Roy, 2014) by having fewer actuated inputs than degrees of freedom (DOF). To address this issue, researchers have developed advanced control approaches that include intelligence, adaptive capacity, rapid convergence, and resilience (Lakhekar and Saundarmal, 2013). Nevertheless, throughout the last several decades, much attention has been devoted to developing a variety of acceptable control approaches for underwater vehicle motion control in ocean engineering (Nakamura and Savant, 1992). These approaches include traditional PD/PID control (Jalving, 1994), sliding mode control (SMC; Yoerger and Slotine, 1985), quasi-SMC (Shet et al., 2023), self-adjusting control (Goheen and Jefferys, 1990), suboptimal control (Geranmehr and Nekoo, 2015), LQG control (Naeem et al., 2003), gain scheduling-based control (Silvestre et al., 2002), feedback control (Pisano and Usai, 2004), backstepping method (Rout and Subudhi, 2016), linear matrix inequality (LMI) convex approach (Innocenti and Campa, 1999), H ∞ control (Feng and Allen, 2004), and Riccati equation-based control (Naik and Singh, 2007). Here, most controllers were developed with no or very minimal knowledge of hydrodynamic coefficients and maritime instabilities such as waves and currents (Antonelli et al., 2001). Intelligent methods used for control, such as neural networks (NN) (Yuh, 1990), fuzzy logic control (FLC) (DeBitetto, 1995), genetic algorithms (GA) (Moura et al., 2010), self-organizing control (Lakhekar and Waghmare, 2023), and adaptive neuro-fuzzy control (Lakhekar et al., 2020), offer a more practical solution for route tracking control in such situations.

By contrast, well-documented disadvantages of intelligent control include the following: Conventional FLC requires specialist knowledge about vehicle operating conditions and several trial-and-error cycles. Suppose the controlled plant is unpredictable and nonlinear, FLC with fixed scaling factors and tedious rules may fail (Lakhekar and Roy, 2014). While training time is uncertain in the case of NN-based control, it may not be acceptable for practical application. Furthermore, NN has the drawback of requiring a lengthy training period and a sluggish pace, so many systems are incapable of handling it. In addition, NNs fail to meet essential characteristics such as quick response and minimum computation overload (Yuh, 1990). While GA-based control schemes cannot ensure their stability, they are frequently utilized as compensatory in most previous research works (Moura et al., 2010). Nonlinearities in the system are not considered by linear quadratic regulators and LQG, which may lead to poor performance and even instability in maneuvering operations (Naeem et al., 2003). Adaptive control appears more effective for handling model uncertainty (Antonelli et al., 2001), but it requires a large amount of computer power for multivariable higher-order nonlinear systems. Of these, the SMC robust control scheme provides a decent path-tracking performance with an easy-to-understand layout, but it has undesired high control activity in the steady-state zone. To compensate for the effect of chattering and the unmodeled dynamics of the vehicle, researchers have revised and extended the robust sliding mode technique to adaptive terminal SMC (Lakhekar and Waghmare 2018). The integral SMC technique is also used to reject and overcome the effects of unpredictable disturbance due to ocean waves, tides, and currents to stabilize an AUV in ocean space (Kim et al., 2015). The second-order sliding mode control (SOSMC) technique requires the upper bound of uncertainty, which is challenging to determine accurately in advance for practical applications. The reduction of the chattering effect is remarkably influenced by the most suitable choice of hitting gain, which is facilitated by salp swarm optimization (SSWO) (Cheng and Wang, 2020). With fast vehicle convergence to the predetermined planar path, the super twisting algorithm (STA) proposed in this paper aims to overcome the flaws and improve the SOSMC approach. STA – SMC creates a smooth, continuous control signal, thus resolving the chattering issue and achieving high accuracy under the effect of quick disturbances (Chalanga et al., 2016).

The main contributions of this paper compared with the related investigations are listed as follows:

• An adaptive STA-based second-order SMC scheme is designed for the stabilization and path-following control of the nonlinear AUV model.

• An appropriate adaptive parameter-tuning law is suggested for hitting gain and sliding surface slope through the SSWO algorithm to dominate the perturbations of the system without the information of their upper bounds.

• A path-following control algorithm is proposed to establish a chatter-free robust performance and finite-time convergence for the AUV model.

• The effectiveness of the proposed technique is verified by demonstrative simulation results and experimental assessments.

This paper is organized as follows: In Section 2, the AUV model is expressed in nonlinear form. In Section 3, the proposed overall control method is described, and optimization is explored. In Sections 4 and 5, various simulation studies and experimental hardware-in-the-loop (HIL) validation demonstrate the efficacy of the recommended efficient control technique, respectively. In Section 6, a conclusion is drawn.

2   Mathematical modeling of vehicle

Directing an AUV along a defined path at a constant forward speed is formulated in this section by discussing the kinematic and dynamic models of an AUV in the horizontal plane.

2.1   Controlled path following in a horizontal plane

The path tracking or following problem is described in X-Y plane through inertial reference frame {I}, Serret-Frenet (S-F) frame {F}, and body-fixed frame {B}, as illustrated in Figure 1. The center of mass (CM) of the AUV, is well defined along the vehicle's principal axes (such as x longitudinal axis and y transverse axis in the inertial reference frame or along axes (x_e, y_e) in the S-F frame), serves as the origin of the body-fixed frame. U is the AUV's net velocity, $U=\sqrt{u^2+v^2}$. The line l indicates the distance between the origin of the global coordinate system {I} and the origin of the body coordinate system {B}, k represents the distance between the origin of the global coordinate system {I} and point S as a reference point the on S-F frame.

Figure  1  Serret-frenet frame parameters: Path tracking

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The AUV heading angle is denoted by ψ, the drift angle $\beta=\tan ^{-1}\left(\frac{u}{v}\right)$ is expressed as the angle between the net and sway velocities in the {B} coordinate system; the angle formed by the S-F frame and the inertial frame is denoted as θ_s. Considering a desired path Ω is followed by an AUV with its body frame attached to its center of gravity and c_c is expected curvature at point S with their length of arc s^* in horizontal plane. The undersea vehicle's velocities as indicated in {I} and {B} are u for forward motion, v for transverse motion, and r for heading angular motion. The surge motion equation along the x-direction, the sway motion equation along the y-direction, and the yaw motion equation, which is rotational movement along the z-direction for the route following control in the x-y domain, comprise the vehicle model.

A vehicle model's corresponding kinematic equations must also be considered. The INFANTE AUV model described in (Lapierre and Soetanto, 2007) is the basis for the model presented in this paper, and the reader is referred to it for more information. The kinematic and dynamic equations of AUV can be stated as follows.

2.2   Kinematics of AUV

An underactuated AUV's kinematic equations can be represented by rotating motion along the z-axis as well as linear motion along the x, and y axes.

where (x, y) represents an AUV's linear coordinates in a fixed frame on the ground, ψ is the vehicle's yaw angle, u is the surge, v is the sway and r is the yaw velocity along the x, y, and z axes, respectively.

2.3   Dynamics of AUV

The elements that contribute to motion along the z-axis, i.e., pitch and heave motion equations, are neglected because of the motion of the AUV in the x-y plane. because the AUV in this paper is of the flat-fish variety, the roll equation of motion is ignored. The following motion equations are used as follows.

Surges equation of motion:

Sway equation of motion:

Yaw equation of motion:

where, F_u and F_v represent the control forces of vehicle's surge and sway motions, respectively. Yaw motion equation is unregulated in vehicle dynamics, hence the AUV is reported as a dynamic model that is underactuated. The extra mass terms such as $\left(X_{\dot{u}}, Y_{\dot{v}}\right)$ and total rigid body mass m of the vehicle comprise the parameters m₁₁ = m − $X_{\dot{u}}$ and m₂₂ = m − $Y_{\dot{v}}$, respectively. The collective rigid body mass of m and the additional moment of inertia about the z_b axis, denoted as $N_{\dot{r}}$, are other constants, and their values are expressed as m₂₂ = m − $N_{\dot{r}}$. The coefficients of linear and quadratic drag are X_u, X_u|u|, Y_v, Y_v|v| and N_r. Unknown environmental disturbances affecting the AUV include τ_Ev and τ_Er. To guarantee the trajectory tracking in a horizontal plane x-y plane, a 3-DOF model with input force vector is established as τ = [ F_u, F_v, 0 ]^T, while its velocity and attitude vector remain v = [ u, v, r ]^T and η = [ x, y, ψ ]^T, respectively

Generalized mathematical model of AUV is represented in standard form as follows:

where, M(v) is inertial matrix including the added mass, C (v) is the Coriolis matrix, D (v) is the damping matrix, g(η) is the vector of gravitational/buoyancy forces, and moment τ is the vector of control inputs.

2.4   Serret-frenet frame

In general, the parameters in the Serret-Frenet (S-F) frame are used to investigate the path following control problem of underactuated AUVs. Incorporating the global coordinate transformation and S-F frame, the convergence problem of the vehicle is transformed into the actuation system control problem, simplifying the controller design. The established reference path functions as the foundation for the LOS guidance, which is built in a moving S-F frame along the virtual target on the path with tangential velocity. Independent of movements, the S-F formulas convey the kinematic characteristics of a particle on a differentiable path in three dimensions. The formulas that describe the position and orientation error dynamics (route following error space model) more clearly show the derivatives of the tangent, binormal unit vectors, and normal.

2.5   Path tracking error

Using the coordinate transformation provided, a path following/tracking error dynamic model (Guo et al., 2003) is developed to simplify the construction of the controller. To follow the reference trajectory, a desired heading angle must be determined for the vehicle and is as follows

where, in an earth-fixed frame, x, y, and ψ_d denote the intended positions and orientations for the vehicle. The definition of the orientation error is

Its differentiation is represented as follows:

Orientation error between total velocity U and ṡ^* is expressed below,

When used with the AUV model, the orientation tracking error and positional variables derivatives are shown below,

2.6   Problem formulation

An underatuated AUV, represented by point B travling along the desired path Ω, which is continuously parameterized by the variable S, which in the S-F coordinate system is known as a virtual reference guidance point. The position of the variable S and the elevation angle of the point are denoted by (x_d, y_d, ψ_d). In addition, the derivatives of the reference velocity (u_d, v_d, r_d) with respect to time are constrained.

The AUV center of mass specified by (x, y, ψ) is represented by point B, which is necessary for them to converge and track the predetermined reference path. The positional error between the point S in frame {F} and AUV is denoted by the variables x_e and y_e. The longitudinal force F_u, sway force F_v, and changing rate position are all sought after by the AUV. The tracking errors x_e and y_e become zero as the longitudinal speed u converges to the goal speed u_d.

For position tracking errors (x_e, y_e) to converge by stabilizing u_e and v_e, the desired surge and sway velocity depends on the data of (x_d, y_d) and (x_e, y_e). Also satisfies the following condition proposed in (Rout and Subudhi, 2016)

where the positive constant parameters k₁ and k₂ are present. The desired position tracking errors and planar path determine the required surge velocity and sway velocity. The derivative of (u_d, v_d) is calculated as shown below,

These variables are considered during the design of control for AUV's guidance system for path-planning algorithm to minimize tracking errors and prevent issues in real-world applications.

Remark 1: Because the nonzero reference surge velocity is positive or negative, the condition |u_{d (t)}| ≥ u_dmin, ∀t ≥ 0, incorporates path tracking in forward and backward directions. In fact, the persistently exciting condition on the yaw reference velocity is substantially more restricted than the condition |u_{d (t)} | ≥ u_dmin, ∀t ≥ 0. Because of high resisting force and lack of control input during motion in the yaw axis for AUVs, the condition |v_{d (t)} | < |u_{d (t )} |, ∀t ≥ 0 means the AUV system cannot track a large spiral curvature with twist.

Assumption 1. (ⅰ) The center of buoyancy (CB) and center of mass (CM) are congruent. (ⅱ) The mass distribution of vehicle body is uniform. (ⅲ) Heave, pitch, and roll motions, as well as terms higher than second order hydrodynamic drag, are disregarded.

Assumption 2. The reference signals u_d, v_d, and r_d are bounded. u_dmin is strictly positive constant, such that |u_{d (t)} | ≥ u_dmin, ∀t ≥ 0. The sway velocity satisfies |v_{d (t)} | < |u_{d (t)} |, ∀t ≥ 0.

Assumption 3. The undiscovered environmental disturbances are constrained, namely, | τ_Eu | ≤ τ_Eumax, | τ_Ev | ≤ τ_Evmax, and | τ_Er | ≤ τ_Ermax. where, τ_Eumax, τ_Evmax, and τ_Ermax are constants and |.| indicates a variable's absolute value.

Assumption 4. The AUV system's unmodeled dynamics are constrained. Positive constants ε_u, ε_v, and ε_r are positive constants such that $\overline{\varepsilon_u}$, $\overline{\varepsilon_v}$, and $\overline{\varepsilon_r}$ satisfy the relationships | ε_u | ≤ ε_u, | ε_v | ≤ ε_v and | ε_r | ≤ ε_r.

3   Controller design

The two sections of the derivation of the suggested control method for an AUV's steering motion are 2-SMC and Salp swarm optimization. The following subsections descript each control part.

3.1   Second-order sliding mode control (SOSMC)

First, sliding manifold σ based on tracking error is defined as follows:

where e_c = η_c − η is the velocity error between the underwater vehicle's actual velocity and the virtual velocity created by the velocity controller, σ is the first order sliding manifold. The order of the uncontrolled system is n, and Λ is a positive constant. Raising the order of the sliding mode manifold, SOSMC achieves zero steady-state error and eliminates chattering by adding the integral term to a first-order sliding manifold. The resulting second order sliding surface is given as follows:

where ζ determines the rate of decay for switching surface. The derivative of the given sliding manifold is taken, and then the scond-order manifold is obtained as follows:

The general form of AUV modelling equation is substituted as follows:

An equivalent control regulation can be derived as follows:

where, $\hat{M}$, $\hat{C}$, $\hat{D}$ and $\hat{g}$ are estimated terms of AUV model. To address the resulting chattering issue, an adaptive term is included in the control rule to replace the switching term,

where $\hat{\tau}_{\text {test }}$ is an adaptive term that estimates the lumped uncertainty vector as an unknown dynamic of vehicle. The estimation of the lumped uncertainty vector is approximated as (Wang et al., 2023a)

The overall control law can be defined along with super twisting algorithm as a disturbance observer as follows:

Control inputs τ_n = [ F_u, F_v ] are then applied to the dynamic model of the vehicle to produce the actual surge and lateral velocity in the body fixed frame in the horizontal plane. To solve the chattering issue in conventional SMC and to reduce the influence of quick acting disturbances, STA technique is utilized to generate a smooth and continuous control signal (Chalanga et al., 2016; Kumar et al., 2022; Sadala and Patre, 2020). The STA technique presented in for steady tracking performance uses the control gains μ₀ and μ₁, which are positive constants. The adaptive term included in control part to reduces the effect of parametric variations in sea environment. Salp Swarm optimization is used to search for best fit for K and Λ for reducing magnitude of chattering and speed of response.

3.2   Salp swarm optimization

Cheng and Wang (2020), Wang et al. (2023a), and Wang et al. (2023b) discussed the SSWO bioinspired optimization technique. Salp belongs to the family of Salpidae, which moves like jellyfish by pumping water to move forward, as shown in Figure 2.

Figure  2  Social structure of swarm of salp (Salp chain)

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The salp population has two mathematical groups, namely, the leader, and followers. The leader or head salp directs the salp chain and the swarm in deep water areas, whereas the rest of the salps are considered as followers. Their main objective is to investigate the food source A in the search area. The location of the leader is altered in relation to the food source, which causes the position of the following salp to shift. The position of the food supply is changed frequently in SSWO. Even if the population declines, the salp chain can pursue the food supply in the best way possible. Another benefit is the presence of only one regulating parameter, and the leading salp position is frequently updated in relation to the food source. As the search space is explored and search space exploitation starts, the number of iterations gradually decreases. This method is preferable to the current optimal algorithm because it avoids local solutions to the problem. The social structure of the salp swarm is shown in Figure 2.

The position of salps is defined in swarm-based algorithms in an m-dimensional search space, where m is the number of variables in a particular problem. As a result, a two-dimensional matrix is used to hold the positions of all salps. Presumably, the swarm uses food source A as its goal in the search space. Using the formula below, the leader's position is updated,

where, x_j¹ represents the position of the salp (leader) in the j^th dimension, A_j represents the location of the food supply in the j^th dimension, u_j, l_j represent the upper, lower bounds of the j₀^th dimension, respectively, c₁, c₂, c₃ represent random numbers.

An objective function is used to determine best fit appropriate values of control parameters, such as hitting gain K and sliding surface slope Λ. The cost function is defined such that the selected parameters minimize the error to reduce chattering simultaneously. The cost function is defined as follows

where, e_k is the tracking error function, and W₁ and W₂ are the weighting fators used in optimization technique. According to the cost function, the leader adjusts its position in relation to the food supply. The most crucial SSWO parameter is the coefficient c₁, which balances the specified exploration and exploitation.

where, L represents maximum number of iterations, l represents current iteration. Random numbers in the range [0, 1] are generated evenly for the parameters c₂ and c₃. They specify the step size and whether the succeeding point in j^th dimension should point in either direction of infinity. Newton's law of motion is applied to the following equation to update the position of the followers.

where, b₀ represents initial speed, t is the time, i ≥ 2, x_jⁱ depicts the location of i^th follower salp in j^th dimension, and a = $\frac{b_{\text {final }}}{b_0}$ and b = $\frac{x-x_0}{t}$. Because the time for optimization is iteration, the discrepancy between iterations equals 1, and considering b₀ = 0, the position of follower can be expressed as follows:

where, i ≥ 2, x_jⁱ provides the position of i^th follower salp in j^th dimension. Thanks to Wang, et al., (2023a); Wang, et al., (2023b), simulating the salp chains is possible. The advantages of the SSWO algorithm come from its ability to improve the initial random solutions efficiently and converge to the best result. The adaptive behavior of this algorithm, which makes it superior to other algorithms, is the major cause of SSWO excellence. SSWO outperforms other swarm-based strategies because it engages in more exploration during optimizion. Moreover, SSWO strongly encourages the use of the c₁ parameter in the latter stages of optimization. As a result, SSWO performs better in terms of outcome accuracy than evolutionary algorithms. The flowchart for SSWO is shown in Figure 3.

Figure  3  Flowchart for salp swarm optimization

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This method not only improves the search intensity of salp swarm algorithm but also increases its diversity. The optimization algorithm ensures that the algorithm can find the optimal value of the algorithm and avoid falling into the local optimum, and the algorithm has better global search capabilities due to its enhanced diversity. The optimization effect of hitting gain and sliding surface slope can be improved.

3.3   Stability analysis

The path tracking control problem of an AUV in Equation (8) with assumptions (1)‒(2) is considered. The control action is seletced as super twisting algorithm based SOSMC along with the adaptive PI law, then the path-tracking error vector asymptotically converges to zero.

Proof: The Lyapunov function candidate is as follows:

where, G ∈ R³ is a diagonal positive definite constant matrix, and $ \tilde{\boldsymbol{L}} $ is the estimation error vector of the lumped perturbation and is defined as follows:

In above Equation (33), L_est and L are the estimated perturbation and the unknown actual perturbation vectors, respectively. The derivative of Lyapunov function is given as follows:

For proposed controller, $\left\|\tilde{\boldsymbol{L}}^{\mathrm{T}}\right\| \approx 0$

K_f can be selected such that the value of $\hat{K}_f$ remains negative (i.e., − $\hat{K}_f$ =− Γ, where Γ > 0). Thus, the reaching condition $\sigma \dot{\sigma} < 0$ is satisfied and σ approaches zero as t → ∞.

Remark 2: Chattering free sliding mode control law is proposed for the path following control of an AUV, where a new adaptive term (Meysar and Notash, 2010; Jin and Yang, 2014; Amer et al., 2011) that eliminates the high frequency control action inherent in a conventional SMC is developed. Adaptive PI term removes the need for a prior knowledge of upper bounds of uncertainties in the dynamic vehicle parameters (Amer et al., 2011).

Remark 3: Super twisting algorithm-based disturbance compensation (Chalanga et al., 2016) and PI uncertainty estimator (Meysar and Notash, 2010) replace the discontinuous term of a conventional SMC with an estimation of the perturbations in an adaptive manner.

Remark 4: The proposed control law incorporates adaptive term, continuously compensating for the effect of unknown system dynamics caused by poorly approximated nonlinear hydrodynamics or sudden environmental loads. In addition, adaptive term does not require the parametrization of a regressor matrix and an unknown parameter vector (Amer et al., 2011).

Remark 5:Super twisting algorithm based SOSMC is designed to achieve finite time estimation of the oceanic disturbances and to generate a smooth and continuous steering command signal.

Remark 6: The proposed STA SOSMC algorithm performs precisely under the influence of fast acting disturbances such as ocean currents and tides and its simple design of path tracking response offers a smooth response in minimum reaching time.

4   Simulation results

To demonstrate the path tracking control performance of the proposed controller in the horizontal plane by regulating heading angle, numerical simulation was conducted through the INFANTE AUV model using the MATLAB/ Simulink tool. A flat-fish type AUV with two back thrusters that are similar and symmetrically positioned with respect to the longitudinal axis was considered. Thus, the vehicle is an under actuated system because it lacks a lateral thruster. The general and differential operating modes of the thrusters provide a force along the vehicles longitudinal axis and a torque about its vertical axis, respectively. In this work, only the back thrusters were used to maneuver in the horizontal plane. For the experimental AUV employed in the simulation, Lapierre and Soetanto (2007) provided the vehicle's physical specifications and hydrodynamic coefficients. In this simulation, which was conducted in the AUV's horizontal plane, typical circular and nonlinear path tracking were examined. The AUV's desired states were assumed to be x_d = 80∙sin (0.01t) and y_d = 50∙sin (0.01t), respectively, and their initial states were specified as zero. The path tracking problem of circular and nonlinear tracks was used to illustrate the robustness of the proposed controller by accounting for ambiguous circumstances and unidentified disruptions. To generate the time-varying disturbances and uncertainty indicated in the works of Wang et al., (2023b), Wang and Er (2016), and Lakhekar et al., (2019) without losing generality, the following parameters are given for evaluating control performance, ∆F_u = 0.1(m₁₁ + ∆m₁₁) + δ_d sin (0.01t), ∆F_v = 0.1(m₂₂ + ∆m₂₂) + δ_d sin (0.01t) and ∆F_w = 0.1(m₃₃ + ∆m₃₃)+ δ_d sin (0.01t) for evaluating control performance. In addition, δ_d = 0.070 7 rand (1, length (t)) and (⋅) is the Gaussian random noise signal.

The simulation results in Figure 4 represent circular and nonlinear path-tracking response in presence of oceanic disturbances along with uncertain operating conditions. In addition, the system parameters simultaneously increased by 20% compared with the actual parameter. In the presence of external perturbation and time-varying uncertain situations, terminal SMC and the proposed SOSMC control methodology outperformed other common methods, such as PD and conventional SMC. In the early phase of the simulation results, PD offered a quick response along with overshoot, whereas SMC required more time to reach the desired path without oscillation. While, terminal SMC and the proposed SOSMC method took a short time to reach the desired circular path in oceanic environment.

Figure  4  Path tracking in presence of adverse oceanic conditions

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Compared with existing control strategies, the suggested planar control performed well in simulation results to reach and maintain the target path. Figure 5 displays the path tracking errors in the x-y plane of the circular and nonlinear path, which was used in the quantitative analysis.

Figure  5  Errors under the effect of uncertainties and disturbances

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The simulation results in Figures 6 and 7 demonstrate response of the forward speed, heading angle, sway, and yaw velocity for circular and nonlinear path scenarios, in presence of oceanic disturbances and noises. PD and conventional SMC exhibited a large settling time and were unable to attenuate the oceanic disturbances. In comparison, the proposed control strategy performed well in numerical simulation to reach and maintain the desired path.

Figure  6  Circular path tracking

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Figure  7  Nonlinear path tracking

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To evaluate the tracking accuracy and efficacy of the proposed controller, anaysis was performed through quantitative methods based on performance measures such as Integral Square Error (ISE), Integral Absolute Error (IAE) and Root Mean Square Error (RMSE), as shown in Table 1. The proposed control scheme obtained the minimum tracking error compared with other controllers. Among all the controllers, the PD controller provided the maximum error in the circular and nonlinear path tracking control in adverse oceanic conditions.

Generating the symmetric bulb turn trajectory requires cosinusoidal reference signals. In the initial phase of simulation, The AUV experienceed an impulsive force due to ocean currents and waves. Thereby, some oscillation appeared in the path tracking response of PD and SMC, as depicted in Figure 8. The tracking performance was also evaluated under the influence of adverse oceanic conditions. However, proposed SOSMC needed minimum reaching time period with smooth response compared with other control techniques. Also able to cope with unexpected disturbances and parametric variations. The control performance was confirmed by quantitative analysis, as presented in Table 2.

Figure  8  (a) Response of symmetric bulb turn path tracking; (b) Heading angle; (c) Yaw error

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5   Experimental validation

Experimental validation (Halil and Sümer, 2014) was conducted to confirm the efficacy of the control scheme further. The results from the experimental setup were obtained using National Instruments (NI) myRIO and Raspberry Pi 4 board setup. The setup verified the AUV performance in the presence of constant disturbances and uncertainties.

Figure 9 (a) shows that HIL simulation was designed to demonstrate real-time performance of the proposed control algorithm, as implemented through numerical simulation. Two hardware boards were used, one for AUV model deployment on Raspberry Pi4 and another for AUV controller on NI myRIO DAQ board setup for HIL validation, as shown in Figure 9 (a). The NI myRIO board has an integrated Xilinx Zynq-7010 system-on-a-chip (SoC) technology, dual-core ARM® Cortex^TM-A9 processor, FPGA with 28 000 programmable logic cells, 10 analog inputs, 6 analog outputs, audio I/O channels, and up to 40 lines of digital input/output (DIO), including onboard Wi-Fi, a three-axis accelerometer. Raspberry Pi4 uses a Quad-core ARM Cortex-A72 at 1.5 GHz, 32 kB of data L1 cache and 48 kB of instruction L1 cache, and 1 MB of shared L2 cache. The real time proposed control algorithm designed through Simulink® models was compiled to C-code and deployed in the NI board. The onboard processor transmitted the command signals to Raspberry Pi 4 module, where an AUV model is deployed for path following applications in HIL testing.

Figure  9  (a) A real time implementation of path following control of AUV model via hardware in loop simulation; (b) Response of path following HIL simulation

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A twisted path scenario was considered in experimental validation in presence of constant disturbances and parametric variations. In this complicated scenario, various waypoints were joined through a continuous desired path with wide turning. The experimental results in Figure 9 (b) indicate desired tracking performance of proposed SOSMC in nonlinear twisted path scenario, with minimum devia tion at each turning point and requiring less settling time.

6   Conclusions

In this paper, a robust optimal super twisting SOSMC with uncertainty estimator based planar motion controller is designed for AUV path tracking in horizontal plane by precisely regulating heading angle. The proposed control technique takes advantage of second order sliding mode, uncertainty estimator, and super twisting method as a disturbance observer with SSW optimization, while the weaknesses attributed to these techniques are overcome by one another. The planar tracking control performance is evaluated under the influence of parametric uncertainties and oceanic disturbances for smooth turning behavior. Starting from initial conditions of the AUV states, the output positional variables asymptotically track desired circular path with a quick and robust response. Based on SSWO approach, hitting gain and sliding surface slope are adjusted through their appropriate optimal values. It is robust against unfavorable oceanic conditions with the help of STA and adaptive control term as uncertainty estimator. Therefore, overall smooth tracking performance, quick hitting time and minimum chattering are achieved. The proposed technique is verified by demonstrative simulation results and experimental validation. The planar tracking control performance is quantified through performance indices and also analyzed through error convergence.