A nonsingular fast terminal sliding mode control for autonomous underwater vehicle

Autonomous underwater vehicles (AUVs) have become vital and cost-effective tools in various marine applications, such as underwater mining, underwater rescue and marine research (Li et al., 2023; Chen et al., 2023; Zhu et al., 2023). Their significant importance has been greatly enhanced by their outstanding capability to navigate perilous oceanic environments and gather dependable, long-term data. To guarantee the successful execution of AUV tasks and missions, it is essential to provide a tracking control system that offers an optimal tracking performance (Karkoub et al., 2017).

To date, various control strategies have been utilized for AUVs to guarantee efficient mission execution despite external disturbances (Yan et al., 2023; Bessa et al., 2010; Guerrero et al., 2023; Wen et al., 2023; Wang et al., 2023a; González-García et al., 2020; Liu et al., 2023). Among these control schemes, siding mode control has been extensively studied for AUV control design because of its robustness, simplicity and insensitivity to parameter variations, as mentioned in references Qin et al. (2023) and Wang et al. (2023b). Linear sliding mode control is commonly used in the development of control schemes for AUV owing to its simplicity. An example of a proposed trajectory tracking control scheme for underwater vehicles based on the linear sliding mode surface, where external disturbances were approximated using an adaptive Takagi–Sugeno–Kang fuzzy model (Li et al., 2018). However, these linear sliding mode control schemes only ensure asymmetrical stability, not finite-time stability. To achieve improved tracking performance for AUV, terminal sliding mode (TSM) control is employed to design the control in this field owing to its ability to accelerate the convergence speed near the equilibrium point. An innovative terminal sliding mode control was proposed to effectively tackle the trajectory tracking challenge of AUV in the horizontal plane (Elmokadem et al., 2017). Yu and colleagues Yu and Zhihong (2002) introduced fast terminal sliding mode (FTSM) control, which demonstrates faster convergence speeds than TSM when the initial position is either distant from or close to the equilibrium point. However, upon analyzing the control design, it was discovered that both TSM and FTSM face the issue of singularity. To solve this challenge, a novel concept called nonsingular terminal sliding mode (NTSM) was implemented to eliminate singularity without requiring any additional procedures (Feng et al., 2002). Furthermore, Elmokadem et al. (2017), Hou et al. (2020) and Yang and Yang (2011) introduced a nonsingular fast terminal sliding mode (NFTSM) control that adapts the terminal sliding mode surface to tackle singularities.

Apart from singularity avoidance, chattering reduction is also a crucial concern in the design of sliding mode control (Li et al., 2024). Several algorithms have been developed to reduce chattering. The boundary layer technique, combined with a saturation function, is commonly employed to suppress the effects of chattering.

In this paper, we propose a novel finite-time NFTSM control, along with its corresponding control law. This control scheme effectively minimizes the impact of chattering and mitigates the influence of uncertain internal parameters and external disturbances on AUV pose convergence. By utilizing the Lyapunov function, we illustrate that the proposed NFTSM guarantees the finite-time convergence of the AUV system while ensuring a nonsingular controller.

The remainder of this paper is organized as follows. In the next section, a brief overview of the dynamic AUV model is provided, followed by a presentation of the problem statement. Section 3 presents a finite-time tracking control law for an AUV, utilizing NFTSM control. The proposed control scheme is validated through the simulation results, presented in Section 4, followed by concluding remarks and potential avenues for future research.

The main features and contributions of this article are summarized as follows:

1. This paper proposes a nonsingular fast terminal sliding mode control with a nonsingular sliding mode surface. The key is to restructure the sliding mode surface function so that its derivatives no longer contain negative fractional powers. This ensures that the control input is smooth and bounded throughout the state space, especially near the equilibrium point.

2. A novel sliding mode control method is proposed for an underwater vehicle system. This method takes into account the characteristics of the underwater vehicle dynamic model and designs a controller that ensures finite-time convergence.

3. An adaptive coefficient is proposed that dynamically adjusts the switching gain in the control input, reducing system jitter and eliminating its interference.

4. The proposed nonsingular fast terminal sliding mode control is applied to a six-degree-of-freedom underwater robot system. Compared with previously proposed control methods, the proposed method achieves better control performance.

This section first explains the AUV model, followed by an explanation of the relevant lemmas, consistencies and definitions.

A general description of the six-DOF nonlinear equation of the AUV motion is described as follows (Geranmehr and Nekoo, 2015):

where η ∈ R^6×1 = $[η1,η2]T$ = [x,y,z,ϕ,θ,ψ]^T represents the position and attitude of AUV in the earth frame; ν ∈ R^6×1 = $[ν1,ν2]T=[u,v,w,p,q,r]T$ represents the velocity and angular velocity in the body-fixed frame; J(η)is the transformation matrix between the inertial and body-fixed frames, and it is assumed that the spatial transformation matrix and its inverse exist, which is reasonable for many AUV applications (Fischer et al., 2014), especially for the fully actuated AUV; $τ∈R6×1=[Fe,Me]T=[X,Y,Z,K,M,N]T$ represents the control forces and moments, which includes surge thrust X, sway thrust Y, heave thrust Z, roll thrust K, pitch thrust M, yaw thrust N.M ∈ R^6×6 is the inertia matrix, C(ν) ∈ R^6×6 represents the centripetal and Coriolis terms for the rigid-body and hydrodynamics, D(ν) is the hydrodynamic damping matrix, G(η) is the gravity and buoyancy forces and moments and d(η, ν, t) represents the external disturbances acting on AUV.

The position and attitude of AUV are usually described in Earth's frame. The transformation matrix between the body-fixed and earth frames is represented by Guerrero et al. (2024).

Where

The two equations in Eq.(1) can be expressed as follows:

where M(η) = MJ₁, C(η) = −MJ₂J⁻¹(η) + C(ν)J⁻¹(η) + D(ν)J⁻¹(η),

To conveniently design a finite-time tracking controller, Eq.(3) can be rewritten in a concise form:

Where, $A=(MJ1)−1(MJ2J−1(η)−C(ν)J−1(η)−D(ν)J−1(η))$⁠, $B=(MJ1)−1,D=−(MJ1)−1d(η,t)$⁠, $f=Aη̇−BG(η)$⁠.

The objective of this study is to propose an NFTSM and provide a control law based on a nonsingular sliding mode surface for an AUV to guarantee that the tracking error converges to a small region near zero in finite time.

The objective of this paper is to propose an NFTSM and provide a control law based on a nonsingular sliding mode surface for an AUV to guarantee that the tracking error converges to a small region near zero in a finite time. This section introduces the symbol representation, related lemmas and definitions that will be used in the subsequent analysis and proof.

Lemma: Shen and Huang (2009) and Plestan et al. (2010) assume that there exists a continuously differentiable and positive definite function V(x), which satisfies:

where c > 0, b > 0, 0 < α < 1. Then V(x) reaches V(x) = 0 in finite time and the stable time T_x(x₀) satisfies

The FTSM is described by the following differential equation (Mishra et al., 2016):

where: $e=η−ηd,ė=η̇−η̇d,ηd$ and $η̇d$ are the desired position vector and desired velocity vector, respectively, m = diag(m₁, …, m₆), n = diag(n₁, …, n₆), m_i, n_i > 0, i = 1, …, 6, p and q are positive odd integers, which satisfy the following condition: p > q.

When the system state variable reaches the sliding mode surface (s = 0), for properly chosen q and p, given an initial state e(0) ≠ 0, dynamics (7) will reach e = 0 in finite time, as shown in Eq.(8). The physical interpretation is that when e is far away from zero, the approximate dynamics become $ė=−me$ whose fast convergence when far away from zero is well understood. When close to e = 0, the approximate dynamics become $ė=−neqp$ which is a terminal attractor.

From any initial state e(0) ≠ 0, the system state converges to the equilibrium point in time (Mishra et al., 2016):

To make dynamical model (4) reach equilibrium quickly along the given surface, the control τ is chosen as:

where γ₁, γ₂, m, n and $qp$ represent the parameter diagonal matrices and all diagonal parameters are positive. When $ė≠0$ and e = 0, $eq−ppė$ becomes extremely large for the control input τ that, leading to a singular problem. In ideal sliding mode control, as defined in Definition 1, as long as the system enters the sliding mode (s = 0) and q < p < 2p, then $eq−ppė=−meqp−ne2q−pp$ and no singular problem occurs.

In the control design of FTSMs, a singularity problem often exists. This is because the designed controllers have nonlinear terms e^α (α < 0), which results in unboundness of the control input when x diminishes to zero (Xu et al., 2015). To overcome this difficulty, the NFTSM is proposed as follows:

Definition NFTSM is described as

where: α = diag(α₁, …, α₆), β = diag(β₁, …, β₆) and α_i, β_i > 0, i = 1, …, 6.0 < q_i < p_i < 2q_i for represent internal parameters, which are odd integers,

Based on the requirement of finite-time stability, the control law is designed as:

where: α, β, p and q are defined in Equation (10). $σ=[σ1,…,σ6]T$ represents internal parameters with σ_i > 0, |D_i| < L_i, i = 1, …, 6, where L_i represents the upper bound of D in the i-th dimension.

This subsection demonstrates the finite-time stability of both the sliding mode surface and tracking error e through Lyapunov stability analysis.

Theorem: For system (4), when the sliding surface is selected as (10) and its control law τ is expressed in (11), the system trajectory rapidly converge to zero within a finite time. In addition, the occurrence of singularity is guaranteed to be avoided throughout the entire process.

Proof: Consider the Lyapunov function candidate as

whose time derivative is

Let $k=pq(1β(ė+αe))p−qq$ then

Because 0 < p < 2q and p is an odd integer, p − q and p + q are even integers,and $(1β(ė+αe))p−qq>0$⁠,which implies k > 0 and $sq+pp$⁠. Therefore, when $V̇≤0$⁠, the condition for Lyapunov stability is satisfied.

Eq.(13) can be rewritten as

The Eq.(14) is further reformulated as

where $α′=2α,k′=2k$ and $α′>0,k′σ>0,q+p2p∈(0,1)$⁠.This inequality has the same form as Lemma 1, and the coefficients satisfy the corresponding coefficient requirements of Lemma 1. This implies that the NFTSM converges in finite time. Once the system enters the sliding mode manifold, we have $e+(1β(ė+αe))pq=0$⁠,and the tracking error of the AUV converges to the equilibrium point in the same time as in Equation (6). This indicates that the tracking error converges in a finite time. Therefore, Theorem 1 is proved.

To analyze the convergence of the tracking error, consider the following Lyapunov function:

from Eq.(8),when the system reaches the equilibrium point, i.e s = 0, it lead to

Eq.(16) is reorganized as follows:

The time derivative of Eq.(15) is

It is evident from the above that V₂ ≥ 0 and $V̇2≤0$⁠, therefore, it is guaranteed that e asymptotically converges to zero, that is, η = η_d. Thus, it can be concluded that the convergence of the AUV’s state to the desired state ensures the asymptotic convergence of the position tracking errors e to zero.

In order to verify the effectiveness of the proposed control scheme, this paper established a kinematic and dynamic model in Matlab, added control input and disturbance and used the Euler method to update the model iteratively. In order to more convincingly illustrate the performances of the proposed finite-time tracking control scheme, two additional finite-time tracking control scheme are employed for comparison: classical NFTSM control (Hou et al., 2020) and nonsingular terminal sliding mode control (NTSMC; Mu and He, 2018). The simulation results, were recorded as [18] and [29].

In these simulations, the initial states of the system are set as $η0=[x0,y0,z0,ϕ0,θ0,ψ0]T$ = [10,5,5,1,1,0]^T, The desired trajectory in the earth-fixed frame is described by

where x_d = 4(1 − cost); y_d = 2sint; z_d = −0.2 t; ϕ_d = 0; θ_d = 0 and ψ_d = 90. The relevant parameters of the control method proposed in this paper are as follows: α = diag(10, 10, 10, 1, 1, 1), β = diag(10, 10, 10, 1, 1, 1) and q = [3,3,3,3,3,3]^T, p = [5,5,5,5,5,5]^T. The relevant parameters of the control method proposed in Hou et al. (2020) are as follows: $α=diag(0.1,0.1,0.1,0.1,0.1,0.1),β=diag(1,1,1,1,1,1),K1=diag(2,2,2,2,2,2)T,K2=diag(1,1,1,1,1,1),γ2=[53,53,53,53,53,53]T,γ1=[2,2,2,2,2,2]T$⁠. The relevant parameters of the control method proposed in [29] are as follows: β = diag(10, 10, 10, 1, 1, 1), q = [3,3,3,3,3,3]^T, p = [5,5,5,5,5,5]^T and D accounts for external disturbances and internal parameter uncertainties.

D = [2sin(200t), 2sin(t), − 0.02 t, 0.2sin(200t), 0.2sin(200t), 0.2sin(200t)].

In terms of parameter selection, the singular fast terminal sliding mode control method, expressed as Equation (7), and the NFTSM control proposed in this paper, expressed as Equation (10), have the same convergence characteristics when the sliding surface s reaches 0, as shown in Equations (17) and (18). To compare the convergence characteristics of the two methods before reaching the sliding surface through numerical simulation, the same parameters were used for the simulations.

In Figure 1, we present the convergence trajectories of the initial state of the AUV model towards the desired state for three distinct sliding mode control approaches.It is clear that NFTSM in this paper and NTSM demonstrate remarkable precision in achieving the desired poses. In contrast, the classical NFTSM showed a slightly lower overall convergence accuracy. When the NFTSM in this study and the NTSM are configured with identical parameters, the sliding mode control method proposed in this study shows faster convergence, with a convergence error of nearly zero.

The proposed sliding mode control method demonstrates clear superiority across all six pose variables, except for the pitch angle. The convergence variances of the pitch angle, calculated using Eq.(18), are 1.8139e−05, 5.9779e−04 and 0.0172, respectively. This analysis confirms the advantages of the approach presented in this study.

This paper quantitatively validates the performance of our method by calculating the root mean square error of approximation (RMSEA) of the pitch tracking error from the experimental results. The proposed method achieves a minimum root mean square error (RMSE) value of 1.8139e−05, which is smaller than the other two methods (5.9779e−04 and 0.0172). The proposed method maintains extremely low tracking error in the presence of model uncertainty and external disturbances, demonstrating its robustness. Figures 1 and 2 show that our method converges faster. The lower RMSE value reflects the system’s ability to reach a steady state more quickly, confirming its rapid convergence.

Figure 2 illustrate the trajectory tracking curves for the three sliding mode control methods, and Figure 3 shows the necessary control input profiles. A comparative analysis between the NFTSM and classical NFTSM reveals that the proposed sliding mode control exhibits not only a faster convergence rate but also a lower control input requirement, thereby making it more suitable for practical use.

In Figure 4, when the initial state is not on the target trajectory, the control method proposed in this paper can converge to the target trajectory in a shorter time, while the other two methods converge more slowly, which has the advantage of convergence speed. In the convergence process, the control method proposed in this study has a higher convergence accuracy. The two control methods, classical NFTSM and NTSM, have obvious tracking errors in the figure, which intuitively shows that the method proposed in this study has the advantage of convergence accuracy.

This paper introduces an innovative approach to NFTSM control, which stands out for its robust performance in dynamic environments. The proposed control scheme is characterized by its nonsingular nature, ensuring that the control inputs remain well-defined even in the presence of uncertainties. A comprehensive proof of the finite-time convergence property was established through the application of a Lyapunov function, demonstrating the effectiveness of this method in achieving stability. One of the significant advantages of this approach is its ability to effectively mitigate the adverse effects of imprecise internal parameters and external disturbances, which often challenge control systems. By addressing these issues, this method enhances the reliability and accuracy of system performance.

The practical implementation of this control strategy is demonstrated through experiments conducted on an underwater robot system, specifically designed to achieve precise pose control. The experimental results revealed that the robot can rapidly converge all six degrees of freedom to nearly zero, indicating high levels of accuracy and responsiveness.

Furthermore, a comparative analysis with other representative NFTSM control methods highlights the superior convergence accuracy achieved by this approach. The findings suggest that this method not only improves performance in terms of speed but also ensures a higher degree of precision in control, making it a significant advancement in the field of control systems for underwater robotics. Due to its inherent robustness, nonsingularity, fast convergence and weak chattering characteristics, it will be applied to actual underwater robot systems in the future.