Disturbance observer based fractional-order sliding mode control for underwater vehicle with variable center of mass

Abstract

To address the challenges of complex underwater environments and model parameter uncertainties, this paper proposes a fractional-order sliding mode control (FOSMC) algorithm integrated with a disturbance observer (DO) for an underwater vehicle-manipulator system (UVMS) with a variable center of mass. First, a conventional dynamic model of the underwater vehicle is established. Considering the motion of the underwater manipulator, an extended dynamic model incorporating the variable center of mass is further derived. A disturbance observer is then designed to estimate both external disturbances and parameter uncertainties in the dynamic model, with a theoretical analysis conducted to ensure its stability. Based on this foundation, a fractional-order sliding mode controller is developed, and a rigorous proof of closed-loop system stability is provided. Finally, numerical simulations are carried out to validate the effectiveness of the proposed control scheme. The results demonstrate that, compared to traditional sliding mode control, the proposed approach achieves superior trajectory tracking accuracy and enhanced robustness under complex hydrodynamic disturbances and parameter uncertainties.

Introduction

The integration of smart subsea technologies, particularly self-navigating unmanned vehicles, has fundamentally transformed methodologies for underwater resource exploitation and marine structural maintenance. ¹ As a type of underwater vehicle, UVMS, equipped with an underwater manipulator, provides significant convenience for underwater tasks.^2,3 To achieve routine inspections and emergency repairs in nuclear power pools, this paper proposes an underwater vehicle with an underwater manipulator. Due to the relatively large weight of the manipulator compared to the total mass of the UVMS, its movement causes significant changes in both the center of gravity and the center of buoyancy of the system, which in turn affects the stability of the vehicle during operation.

The internal dynamic coupling between the underwater manipulator and the underwater vehicle, as well as external disturbances, also add difficulty to the dynamic modeling, and motion control of UVMS, but have attracted significant interest from researchers.^4–6 So far, there have been many achievements in the dynamic modeling and control of UVMS. Dannigan and Russell ⁷ proposed a dynamic coupling model that describes the interaction between an underwater vehicle and a manipulator. They investigated the effect of the manipulator’s movement on the vehicle’s position and orientation, concluding that the vehicle’s yaw angle plays a critical role in minimizing variations in the end effector error. Shah et al. ⁸ derived the dynamic model of UVMS for an underwater vehicle equipped with a two-link flexible joint using the quasi-Lagrange formulation. The numerical analysis demonstrated the impact of the coupling between the underwater manipulator and the underwater vehicle, as well as the coupling between the joints, on the positioning of the end effector. Zhang et al. ⁹ proposed methods for generating both the UVMS disturbance map. They employed a disturbance evaluation function to assess the coupling effects between the underwater vehicle and the underwater manipulator. Londhe et al. and Mohan et al.^10–12 addressing the dynamic coupling issue between the underwater vehicle and underwater manipulator, provided a detailed derivation of the UVMS mathematical model. They compared model reference control with traditional PID control and demonstrated that model reference control offers better robustness against disturbances and parameter uncertainties in the system.

Due to the dynamic characteristics of underwater vehicles, many control algorithms have been applied to them, such as PID, fuzzy control, backstepping control, and model predictive control, etc.^13–18 Among the many advanced control methods, sliding mode control has gained widespread research attention due to its strong robustness. Jesus Guerrero et al. ¹⁹ addressing the impact of ocean currents on an underwater vehicle, proposed an adaptive controller based on high-order SMC. A Lyapunov function candidate was constructed to formally guarantee asymptotic stability while accomplishing prescribed trajectory tracking objectives. Yu et al. ²⁰ proposed an adaptive fractional-order fast terminal SMC to address the control issues of a UVMS under dynamic parameter uncertainties. By integrating an adaptive switching gain mechanism with FOSMC, the approach effectively addresses the path tracking performance degradation typically caused by fixed switching gains. For parameter uncertainties and external disturbances in the system, conventional control algorithms can no longer meet the performance requirements of dynamic systems. Therefore, researchers have introduced disturbance observers to address this issue.^21–25 Huang et al. ²⁶ proposed an SMC with a high-order disturbance observer for an underactuated robot, effectively addressing the issue of robot parameter uncertainties. Zhu et al. ²⁷ proposed an adaptive SMC algorithm with a disturbance observer for a UVMS facing parameter uncertainties and external disturbances, to ensure that the UVMS can maintain accurate tracking of the reference trajectory under external disturbances.

Wang et al. ²⁸ proposed an adaptive fuzzy control method with a fuzzy disturbance observer for UVMS in the presence of model uncertainties and external disturbances. This method utilizes fuzzy logic systems with adaptive capabilities to evaluate the unknown components in the parameters of the underwater vehicle. Hu et al. ²⁹ proposed a model predictive control method with a disturbance observer for a four DOF underwater vehicle, taking into account its nonlinear dynamics, parameter uncertainties, and external disturbances. Simulations demonstrated that the proposed method can effectively suppress unpredictable disturbances.

Most existing studies focus either on integer-order sliding mode control or the independent application of DO, without considering the potential synergy between FOSMC and DO. In systems subject to variable center of mass, hydrodynamic parameter uncertainties, and external disturbances, the integration of FOSMC and DO offers a promising approach to enhance robustness and trajectory tracking accuracy.

In engineering practice, the phenomenon of moving mass is a common occurrence.^30–32 Generally, the dynamic model and control algorithms presented above focus solely on the dynamic coupling between the manipulator and the vehicle, without accounting for the changes in the center of mass of the entire UVMS caused by the manipulator’s movement. Since the underwater manipulator in this study accounts for a significant proportion of the weight of the UVMS, it is necessary to establish a variable center of mass dynamic model for the UVMS. The variable center of mass term in the dynamic model is incorporated as a feedforward component in the controller to improve the control accuracy of the UVMS.

The operations of the UVMS mostly require hovering at a specific location in space while performing tasks such as point stabilization and manipulation. In addition, the system’s nonlinear characteristics, changes in the center of mass, parameter uncertainties, and external disturbances pose challenges for controller design. To address the above issue, this paper proposes an FOSMC with a disturbance observer. The simulation results demonstrate that the proposed method exhibits high robustness and effective disturbance rejection capabilities.

The principal theoretical advancements achieved in this study are formally established through the following key aspects:

• (1) A UVMS dynamic model incorporating a variable center of mass is developed, and the variable center of mass term is explicitly introduced as a feedforward component in the control law. This enhances modeling accuracy and improves trajectory tracking performance under time-varying mass distributions.

• (2) A nonlinear disturbance observer is designed to estimate external disturbances and internal parameter uncertainties in real time. Integrated with the controller, it significantly improves the system’s robustness and adaptability in complex underwater environments.

• (3) A fractional-order sliding mode controller is proposed, leveraging the memory effect of fractional calculus to achieve smoother control actions. Compared with conventional integer-order methods, the proposed controller offers superior disturbance rejection and chattering suppression.

The remainder of this paper is organized as follows: Section 2 introduces the UVMS dynamic model with variable center of mass. Section 3 is dedicated to the DO design and stability proof. Section 4 presents the FOSMC design and stability analysis. Section 5 presents simulation results that validate the effectiveness of the proposed FOSMC algorithm with DO. Section 6 analyzes the key advantages of the control strategy, while Section 7 discusses the implications of this approach for future research.

UVMS dynamic model with variable center of mass

The UVMS consists of a framed underwater vehicle carrying a three DOF mobile platform (manipulator), and its 3D model is shown in Figure 1.

Figure 1.

UVMS model and its coordinate systems.

To accurately describe the characteristics of the UVMS with the variable center of mass, it is necessary to establish an accurate model that takes into account the movement of the three DOF mobile platform. To facilitate the analysis of the six DOF motion of the UVMS, coordinate systems were established, including the Earth coordinate system, the body coordinate system, and the 3-DOF mobile platform coordinate system, they are established as shown in Figure 1. The Earth coordinate system {E} is fixed on the Earth, the body-fixed coordinate system {B} is established at the center of mass of the vehicle, and the three DOF mobile platform coordinate system {P} is established at the center of mass of the Y-axis guide rail.

UVMS dynamic model

The differential form of the motion equations for an underwater vehicle can be expressed as ³³ :

$$\begin{array}{c}m({\hat{\mathit{\nu }}}_0+\mathit{\omega }\times {\mathit{\nu }}_0+\hat{\mathit{\omega }}\times {\mathit{r}}_G+\mathit{\omega }\times (\mathit{\omega }\times {\mathit{r}}_G))\hfill \\ ={\mathit{f}}_0{\mathit{I}}_0\hat{\mathit{\omega }}+\mathit{\omega }\times \left({\mathit{I}}_0\mathit{\omega }\right)+m{\mathit{r}}_G\times ({\hat{\mathit{\nu }}}_0+\mathit{\omega }\times {\mathit{\nu }}_0)={\mathit{m}}_0\hfill \end{array}$$

(1)

In the equation, $\widehat{·}$ represents the derivative of a vector in the body-fixed coordinate system, ${·}_0$ denotes a vector or matrix in an arbitrary coordinate system (where the coordinate system does not coincide with the center of mass), ${\mathit{r}}_G$ represents the center of mass vector. The variables and their physical significances are listed in Table 1.

Table 1.

The motion variables of the UVMS and their physical significances.

Variables	Physical significance
${\mathit{f}}_0={\mathit{\tau }}_1={[X,Y,Z]}^T$	The forces acting on the UVMS.
${\mathit{m}}_0={\mathit{\tau }}_2={[K,M,N]}^T$	The torques acting on the UVMS.
$\mathit{\tau }={[{\mathit{\tau }}_1^T,{\mathit{\tau }}_2^T]}^T$	The driving force/torque acting on the UVMS.
${\mathit{\nu }}_0={\mathit{\nu }}_1={[u,v,w]}^T$	The linear velocity of the UVMS.
$\mathit{\omega }={\mathit{\nu }}_2={[p,q,r]}^T$	The angular velocity of the UVMS.
$\mathit{\nu }={[{\mathit{\nu }}_1^T,{\mathit{\nu }}_2^T]}^T$	The velocity vector of the UVMS in the body-fixed coordinate system.
${\mathit{r}}_G={[x_G,y_G,z_G]}^T$	The center of mass coordinates of the UVMS.
${\mathit{\eta }}_1={[x,y,z]}^T$	The position coordinate of the UVMS.
${\mathit{\eta }}_2={[\phi ,\theta ,\psi ]}^T$	The orientation coordinate of the UVMS.
$\mathit{\eta }={[{\mathit{\eta }}_1^T,{\mathit{\eta }}_2^T]}^T$	The position/orientation vector of the UVMS in the inertial coordinate system.

According to the equation (1), the vector form of the UVMS six DOF dynamics equation in space can be expressed as:

$$\mathit{M}\stackrel{·}{\mathit{\nu }}+\mathit{C}\left(\mathit{\nu }\right)\mathit{\nu }+\mathit{D}\left(\mathit{\nu }\right)\mathit{\nu }+\mathit{g}\left(\mathit{\eta }\right)=\mathit{\tau }$$

(2)

The symbol $\mathit{S}\left(\right)$ is defined as the skew-symmetric matrix of a vector, as shown in the equation (3).

$$\mathit{S}\left({[x,y,z]}^T\right)=\left[\begin{array}{ccc}0\hfill & -z\hfill & y\hfill \\ \multicolumn{1}{c}{z}& 0\hfill & -x\hfill \\ \multicolumn{1}{c}{-y}& x\hfill & 0\hfill \end{array}\right]$$

(3)

Then, the inertia matrix of the UVMS is given by:

$$\mathit{M}=\left[\begin{array}{cc}m{\mathit{I}}_{3\times 3}\hfill & -m\mathit{S}\left({\mathit{r}}_G\right)\hfill \\ \multicolumn{1}{c}{m\mathit{S}\left({\mathit{r}}_G\right)}& {\mathit{I}}_0\hfill \end{array}\right]$$

(4)

The Coriolis and centripetal force matrix as:

$$\begin{array}{c}\mathit{C}\left(\mathit{\nu }\right)=\hfill \\ \left[\begin{array}{cc}{0}_{3\times 3}\hfill & -m\mathit{S}\left({\mathit{\nu }}_1\right)-m\mathit{S}\left(\mathit{S}\left({\mathit{\nu }}_2\right){\mathit{r}}_G\right)\hfill \\ \multicolumn{1}{c}{-m\mathit{S}\left({\mathit{\nu }}_1\right)-m\mathit{S}\left(\mathit{S}\left({\mathit{\nu }}_2\right){\mathit{r}}_G\right)}& m\mathit{S}\left(\mathit{S}\left({\mathit{\nu }}_1\right){\mathit{r}}_G\right)-\mathit{S}\left({\mathit{I}}_0{\mathit{\nu }}_2\right)\hfill \end{array}\right]\hfill \end{array}$$

(5)

The viscous hydrodynamic coefficient matrix as:

$$\begin{array}{c}\multicolumn{1}{c}{\mathit{D}\left(\mathit{v}\right)={\mathit{D}}_L+{\mathit{D}}_{\mathrm{NL}}\left(\mathit{v}\right)}\\ =\mathrm{diag}{\left[\begin{array}{c}{X}_u+X_{u\left|u\right|}\left|u\right|,Y_v+Y_{v\left|v\right|}\left|v\right|,Z_w+Z_{w\left|w\right|}\left|w\right|,\hfill \\ \multicolumn{1}{c}{K_p+K_{p\left|p\right|}\left|p\right|,M_q+M_{q\left|q\right|}\left|q\right|,N_r+N_{r\left|r\right|}\left|r\right|}\end{array}\right]}^T\hfill \end{array}$$

(6)

The restoring force matrix as:

$$\begin{array}{c}\mathit{g}\left(\mathit{\eta }\right)=\hfill \\ \left[\begin{array}{c}(G-B)\sin \theta \hfill \\ \multicolumn{1}{c}{-(G-B)\cos \theta \sin \varphi }\\ \multicolumn{1}{c}{-(G-B)\cos \theta \cos \varphi }\\ \multicolumn{1}{c}{-(y_{G}G-y_{B}B)\cos \theta \cos \varphi +(z_{G}G-z_{B}B)\cos \theta \sin \varphi }\\ \multicolumn{1}{c}{(x_{G}G-x_{B}B)\cos \theta \cos \varphi +(z_{G}G-z_{B}B)\sin \theta }\\ \multicolumn{1}{c}{-(x_{G}G-x_{B}B)\cos \theta \sin \varphi -(y_{G}G-y_{B}B)\sin \varphi }\end{array}\right]\hfill \end{array}$$

(7)

When the manipulator moves, the parameters in the UVMS dynamic model will change. Below, based on the center of mass dynamics model, the influence of the manipulator’s motion on the UVMS model parameters is fully considered, and the derivation of the UVMS’s model with variable center of mass is presented.

The dynamic equations for the UVMS considering the variable center of mass

In the UVMS model diagram, the underwater manipulator is connected to the underwater vehicle using a fixed connection, with the Y-axis guide rail of the underwater manipulator fixed to the body of the underwater vehicle. The slider on the Y-axis guide rail drives the underwater manipulator’s X and Z axes to perform linear motion. The slider on the Y-axis also drives the Z-axis to perform linear motion. The center of mass of the Y-axis guide rail is selected as the origin of the manipulator coordinate system. For the convenience of calculation, the following simplifications are made:

1. The three linear guide rails of the manipulator in the Cartesian coordinate system are simplified into three rectangular prisms. Let $[l,w,h]$ be the dimensions of the guide rails along the x, y, and z directions.

2. The range of motion of the Z-axis is very short, so it is assumed that the center of mass of the Z-axis remains unchanged.

3. The initial position of the guide rails is the center position of each guide rail, and the positions at both ends of the guide rails represent the maximum values of the forward and reverse motion travel. The maximum forward and reverse travel of the X-axis and Y-axis guides are both 250 mm.

We define the centroid coordinates of the Y-axis guide rail in the body-fixed coordinate system as ${\mathit{r}}_b^{\mathrm{yc}}$ , with the expression given below:

$${\mathit{r}}_b^{\mathrm{yc}}={[i_b,j_b,k_b]}^T$$

(8)

We define the centroid coordinates of the X-axis and Z-axis guide rail in the manipulator coordinate system as ${\mathit{r}}_p^{\mathrm{xc}}$ and ${\mathit{r}}_p^{\mathrm{zc}}$ , respectively. The expressions are given below:

$$\begin{array}{c}{\mathit{r}}_p^{\mathrm{xc}}={[0,Y_c-\frac{w_y}{2},\frac{h_x+h_y}{2}]}^T\hfill \\ {\mathit{r}}_p^{\mathrm{zc}}={[X_c-\frac{l_x}{2},Y_c-\frac{w_y}{2},\frac{h_y+h_z}{2}]}^T\hfill \end{array}$$

(9)

Where, $X_c$ and $Y_c$ represent the movement commands for the X-axis and Z-axis guide rails, respectively.

Therefore, the centroid coordinates ${\mathit{r}}_b^{*c}$ of the three linear guide rails in the body-fixed coordinate system are shown in the equation (10).

$${\mathit{r}}_b^{*c}={\mathit{J}}_b^p{\mathit{r}}_p^{*c}+{\mathit{r}}_b^{\mathrm{yc}}$$

(10)

Where, ${\mathit{J}}_b^p$ represents the transformation matrix from the manipulator coordinate system to the body-fixed coordinate system, * represents the coordinate axes x, y, z. Therefore, the equivalent inertia ${\mathit{I}}_c$ of the three guide rails around their center can be expressed as:

$${\mathit{I}}_c=\left[\begin{array}{ccc}\frac{m(w^2+h^2)}{12}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \frac{m(l^2+h^2)}{12}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & \frac{m(l^2+w^2)}{12}\hfill \end{array}\right]$$

(11)

According to the parallel axis theorem, ³⁴ the equivalent inertia about any point O can be defined as:

$${\mathit{I}}_o={\mathit{I}}_c-m(\mathit{r}{\mathit{r}}^T-{\mathit{r}}^T\mathit{r}{\mathit{I}}_{3\times 3})$$

(12)

The equivalent inertia of the three guide rails can be expressed as:

$${\mathit{I}}_{*}={\mathit{I}}_{*}^c-m_{*}[{\mathit{r}}_b^{*c}{\left({\mathit{r}}_b^{*c}\right)}^T-{\left({\mathit{r}}_b^{*c}\right)}^T{\mathit{r}}_b^{*c}{\mathit{I}}_{3\times 3}]$$

(13)

Therefore, the equivalent inertia of the UVMS as:

$$\mathit{I}\left(\mathit{\Theta }\right)={\mathit{I}}_{\mathrm{body}}+{\mathit{I}}_x+{\mathit{I}}_y+{\mathit{I}}_z$$

(14)

In the equation (14), the equivalent inertia of the underwater vehicle is represented by ${\mathit{I}}_{\mathrm{body}}$ , therefore, the centroid of the entire UVMS can be expressed as:

$${\mathit{r}}_G\left(\mathit{\Theta }\right)=\frac{\sum ^{m_i}{\mathit{r}}_i}{\sum ^{m_i}}=\frac{m_{\mathrm{ROV}}{\mathit{r}}_b^G+m_x{\mathit{r}}_b^{\mathrm{xc}}+m_y{\mathit{r}}_b^{\mathrm{yc}}+m_z{\mathit{r}}_b^{\mathrm{zc}}}{m_{\mathrm{ROV}}+m_x+m_y+m_z}$$

(15)

Therefore, the equivalent inertia matrix of the UVMS can be obtained as:

$$\mathit{M}\left(\mathit{\Theta }\right)=\left[\begin{array}{cc}m{\mathit{I}}_{3\times 3}\hfill & -m\mathit{S}\left({\mathit{r}}_G\left(\mathit{\Theta }\right)\right)\hfill \\ \multicolumn{1}{c}{m\mathit{S}\left({\mathit{r}}_G\left(\mathit{\Theta }\right)\right)}& \mathit{I}\left(\mathit{\Theta }\right)\hfill \end{array}\right]$$

(16)

The Coriolis force and centripetal force matrix as:

$$\begin{array}{c}\mathit{C}(\mathit{\nu },\mathit{\Theta })=\hfill \\ \left[\begin{array}{cc}{0}_{3\times 3}\hfill & -m\mathit{S}\left({\mathit{\nu }}_1\right)-m\mathit{S}\left(\mathit{S}\left({\mathit{\nu }}_2\right){\mathit{r}}_G\left(\mathit{\Theta }\right)\right)\hfill \\ \multicolumn{1}{c}{-m\mathit{S}\left({\mathit{\nu }}_1\right)-m\mathit{S}\left(\mathit{S}\left({\mathit{\nu }}_2\right){\mathit{r}}_G\left(\mathit{\Theta }\right)\right)}& m\mathit{S}\left(\mathit{S}\left({\mathit{\nu }}_1\right){\mathit{r}}_G\right)-\mathit{S}\left({\mathit{I}}_0{\mathit{\nu }}_2\right)\hfill \end{array}\right]\hfill \end{array}$$

(17)

Therefore, the dynamic equation of the UVMS with variable center of mass can be expressed as:

$$\mathit{M}\left(\mathit{\Theta }\right)\stackrel{·}{\mathit{\nu }}+\mathit{C}(\mathit{\nu },\mathit{\Theta })\mathit{\nu }+\mathit{D}\left(\mathit{\nu }\right)\mathit{\nu }+\mathit{g}(\mathit{\eta },\mathit{\Theta })=\mathit{\tau }$$

(18)

The relationship between the velocity of the UVMS in the body-fixed coordinate system and the inertial reference frame is as follows:

$$\stackrel{·}{\mathit{\eta }}=\mathit{J}\left(\mathit{\eta }\right)\mathit{\nu }$$

(19)

Where $\mathit{J}\left(\mathit{\eta }\right)$ is the UVMS rotational transformation matrix related to the attitude angles. By differentiating both sides of the above equation, we can obtain:

$$\stackrel{··}{\mathit{\eta }}=\mathit{J}\left(\mathit{\eta }\right)\stackrel{·}{\mathit{\nu }}+\stackrel{·}{\mathit{J}}\left(\mathit{\eta }\right)\mathit{\nu }$$

(20)

By combining the above equations (19) and (20), we get:

$$\stackrel{·}{\mathit{\nu }}={\mathit{J}}^{-1}\left(\mathit{\eta }\right)[\stackrel{··}{\mathit{\eta }}-\stackrel{·}{\mathit{J}}\left(\mathit{\eta }\right){\mathit{J}}^{-1}\left(\mathit{\eta }\right)\stackrel{·}{\mathit{\eta }}]$$

(21)

Substituting the above equation into the variable center of mass dynamic model, we can derive the UVMS dynamic model in the inertial reference frame, as shown in the equation (22):

$${\mathit{M}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta })\stackrel{··}{\mathit{\eta }}+{\mathit{C}}_{\mathit{\eta }}(\mathit{\nu },\mathit{\eta },\mathit{\Theta })\stackrel{·}{\mathit{\eta }}+{\mathit{D}}_{\mathit{\eta }}(\mathit{\nu },\mathit{\eta })\stackrel{·}{\mathit{\eta }}+{\mathit{g}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta })=\mathit{\tau }\left(\mathit{\eta }\right)$$

(22)

Where:

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{M}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta })={\mathit{J}}^{-T}\left(\mathit{\eta }\right)\mathit{M}\left(\mathit{\Theta }\right){\mathit{J}}^{-1}\left(\mathit{\eta }\right){\mathit{C}}_{\mathit{\eta }}(\mathit{\nu },\mathit{\eta },\mathit{\Theta })}\\ ={\mathit{J}}^{-T}\left(\mathit{\eta }\right)[\mathit{C}(\mathit{\nu },\mathit{\Theta })-\mathit{M}\left(\mathit{\Theta }\right){\mathit{J}}^{-1}\left(\mathit{\eta }\right)\stackrel{·}{\mathit{J}}\left(\mathit{\eta }\right)]\hfill \\ {\mathit{J}}^{-1}\left(\mathit{\eta }\right){\mathit{D}}_{\mathit{\eta }}(\mathit{\nu },\mathit{\eta })={\mathit{J}}^{-T}\left(\mathit{\eta }\right)\mathit{D}\left(\mathit{\nu }\right){\mathit{J}}^{-1}\left(\mathit{\eta }\right){\mathit{g}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta })\hfill \\ ={\mathit{J}}^{-T}\left(\mathit{\eta }\right)\mathit{g}(\mathit{\eta },\mathit{\Theta }){\mathit{\tau }}_{\mathit{\eta }}\left(\mathit{\eta }\right)={\mathit{J}}^{-T}\left(\mathit{\eta }\right)\mathit{\tau }\hfill \end{array}$$

We assume that the control inputs are 0, meaning no control is applied to the UVMS, and analyze the impact of the manipulator movement on the UVMS position and orientation. The initial positions of the X-axis and Y-axis are at the origin, and the X-axis and Y-axis move at a speed of 10 mm/s in the positive direction. The analysis is conducted considering whether external disturbances exist, with the results shown in Figures 2 to 5.

Figure 2.

The position variation of the UVMS in the absence of disturbances.

Figure 3.

The orientation variation of the UVMS in the absence of disturbances.

Figure 4.

The position variation of the UVMS under disturbances.

Figure 5.

The orientation variation of the UVMS under disturbances.

Through Figures 2 to 5, we can observe that when the controller does not control the UVMS, adjusting the manipulator alone affects the UVMS’s orientation. Due to the UVMS having a certain center of gravity height, it initially self-adjusts to control its attitude. However, as the center of gravity shifts, stability is gradually lost. When external disturbances are present, both the position and orientation of the UVMS will change. This requires the adjustment of the controller to achieve UVMS’s stable orientation dynamic hovering.

Disturbance observer design and stability proof

Disturbance observer is a method used to estimate and compensate for external disturbances and parameter uncertainties in system. Its main function is to estimate the disturbances in the system in real-time and then adjust the controller’s output, thereby enhancing the system’s robustness against disturbances and ensuring its stability and accuracy.

Disturbance observer design

Considering the parameter uncertainties and external disturbances of the UVMS, we rewrite equation (18) as:

$$\begin{array}{c}\mathit{M}\left(\mathit{\Theta }\right)\stackrel{·}{\mathit{\nu }}+\mathit{C}(\mathit{\nu },\mathit{\Theta })\mathit{\nu }+\mathit{D}\left(\mathit{\nu }\right)\mathit{\nu }+\mathit{g}(\mathit{\eta },\mathit{\Theta })=\mathit{\tau }+\mathit{dM}\left(\mathit{\Theta }\right)\hfill \\ =\hat{\mathit{M}}\left(\mathit{\Theta }\right)+\mathrm{\Delta }\mathit{M}\hfill \end{array}$$

(23)

Where, $\hat{\mathit{M}}\left(\mathit{\Theta }\right)$ is the estimated values, $\mathrm{\Delta }\mathit{M}$ is the error between the actual values and the estimated values, and $\mathit{d}$ represents the external disturbance.

In the inertial reference frame, we rewrite the dynamic model as:

$$\begin{array}{c}\multicolumn{1}{c}{{\hat{\mathit{M}}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta })\stackrel{··}{\mathit{\eta }}=\mathit{\tau }\left(\mathit{\eta }\right)-{\mathit{C}}_{\mathit{\eta }}(\mathit{\nu },\mathit{\eta },\mathit{\Theta })\stackrel{·}{\mathit{\eta }}}\\ -{\mathit{D}}_{\mathit{\eta }}(\mathit{\nu },\mathit{\eta })\stackrel{·}{\mathit{\eta }}-{\mathit{g}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta })+{\mathit{f}}_{\mathit{d}}\hfill \end{array}$$

(24)

Where, ${\mathit{f}}_{\mathit{d}}$ represents the observed quantity.

$${\mathit{f}}_{\mathit{d}}=\mathit{d}\left(\mathit{\eta }\right)-\mathrm{\Delta }{\mathit{M}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta })\stackrel{··}{\mathit{\eta }}$$

(25)

We design a disturbance observer to estimate ${\mathit{f}}_{\mathit{d}}$ , define an expanded state variable ${\hat{\mathit{f}}}_{\mathit{d}}$ , and construct the following observer:

$$\{\begin{array}{c}\stackrel{.}{\widehat{\mathit{\eta }}}=\widehat{\stackrel{.}{\mathit{\eta }}}+{\mathit{L}}_1\left(\mathit{\eta }-\widehat{\mathit{\eta }}\right)\\ \stackrel{.}{\widehat{\stackrel{.}{\mathit{\eta }}}}={\widehat{\mathit{M}}}_{\mathit{\eta }}^{-1}\left(\mathit{\eta },\mathit{\Theta }\right)(\mathit{\tau }\left(\mathit{\eta }\right)-{\mathit{C}}_{\mathit{\eta }}\left(\mathit{\nu },\mathit{\eta },\mathit{\Theta }\right)\stackrel{.}{\mathit{\eta }}-{\mathit{D}}_{\mathit{\eta }}\left(\mathit{\nu },\mathit{\eta }\right)\stackrel{.}{\mathit{\eta }}\\ -{\mathit{g}}_{\mathit{\eta }}\left(\mathit{\eta },\mathit{\Theta }\right)+{\widehat{\mathit{f}}}_{\mathit{d}})+{\mathit{L}}_2\left(\mathit{\eta }-\widehat{\mathit{\eta }}\right)\\ {\stackrel{.}{\widehat{\mathit{f}}}}_{\mathit{d}}=-\gamma {\Vert \mathit{\eta }-\widehat{\mathit{\eta }}\Vert }^a·sa{t}_{\delta }\left(\mathit{\eta }-\widehat{\mathit{\eta }}\right)\end{array}$$

(26)

Where, $\hat{\mathit{\eta }}$ and $\widehat{\stackrel{·}{\mathit{\eta }}}$ represent the estimated values of $\mathit{\eta }$ and $\stackrel{·}{\mathit{\eta }}$ , respectively. ${\stackrel{·}{\hat{\mathit{f}}}}_{\mathit{d}}$ represents the observed value of the observer. ${\mathit{L}}_1$ and ${\mathit{L}}_2$ are the observer gains. $\gamma$ and $0<a<1$ are the observer’s nonlinear gains. $\mathrm{sa}{t}_{\delta }\left(e\right)=\frac{e}{\max (\delta ,\Vert e\Vert )}$ is a saturation function used to avoid high-frequency oscillations.

Stability proof

To prove the stability of the observer, define the observation error as:

$$\{\begin{array}{c}\multicolumn{1}{c}{{\mathit{e}}_1=\mathit{\eta }-\hat{\mathit{\eta }}}\\ \multicolumn{1}{c}{{\mathit{e}}_2=\stackrel{·}{\mathit{\eta }}-\widehat{\stackrel{·}{\mathit{\eta }}}}\\ \multicolumn{1}{c}{{\mathit{e}}_1={\mathit{f}}_{\mathit{d}}-{\hat{\mathit{f}}}_{\mathit{d}}}\end{array}$$

(27)

Based on the above definition, the observer error dynamics can be expressed as:

$$\{\begin{array}{c}{\stackrel{·}{\mathit{e}}}_1={\mathit{e}}_2-{\mathit{L}}_1{\mathit{e}}_1\hfill \\ {\stackrel{·}{\mathit{e}}}_2\approx {\mathit{M}}_{\mathit{\eta }}^{-1}(\mathit{\eta },\mathit{\Theta })(-{\mathit{C}}_{\mathit{\eta }}(\mathit{\nu },\mathit{\eta },\mathit{\Theta }){\mathit{e}}_2-{\mathit{D}}_{\mathit{\eta }}(\mathit{\nu },\mathit{\eta }){\mathit{e}}_2+{\mathit{e}}_3)-{\mathit{L}}_2{\mathit{e}}_1\hfill \\ {\stackrel{·}{\mathit{e}}}_3=\stackrel{·}{\mathit{d}}+\gamma {\Vert {\mathit{e}}_1\Vert }^a·{\mathrm{sat}}_{\delta }\left({\mathit{e}}_1\right)\hfill \end{array}$$

(28)

Design the following Lyapunov function:

$$V=\frac{1}{2}{\mathit{e}}_1^T\mathit{P}{\mathit{e}}_1+\frac{1}{2}{\mathit{e}}_2^T\mathit{Q}{\mathit{e}}_2+\frac{1}{2\gamma }{\mathit{e}}_3^T{\mathit{e}}_3$$

(29)

Where, P and Q are positive definite matrices.

Taking the derivative of V, we get:

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}={\mathit{e}}_1^T\mathit{P}{\stackrel{·}{\mathit{e}}}_1+{\mathit{e}}_2^T\mathit{Q}{\stackrel{·}{\mathit{e}}}_2+\frac{1}{\gamma }{\mathit{e}}_3^T{\stackrel{·}{\mathit{e}}}_3={\mathbf{e}}_1^T\mathit{P}({\mathit{e}}_2-{\mathit{L}}_1{\mathit{e}}_1)+{\mathit{e}}_2^T\mathit{Q}}\\ ({\mathit{M}}_{\mathit{\eta }}^{-1}(\mathit{\eta },\mathit{\Theta })(-{\mathit{C}}_{\mathit{\eta }}(\mathit{\nu },\mathit{\eta },\mathit{\Theta }){\mathit{e}}_2-{\mathit{D}}_{\mathit{\eta }}(\mathit{\nu },\mathit{\eta }){\mathit{e}}_2+{\mathit{e}}_3)-{\mathit{L}}_2{\mathit{e}}_1)\hfill \\ +\frac{1}{\gamma }{\mathit{e}}_3^T(\stackrel{·}{\mathit{d}}+\gamma {\Vert {\mathit{e}}_1\Vert }^a·{\mathrm{sat}}_{\delta }\left({\mathit{e}}_1\right))=-{\mathbf{e}}_1^T\mathit{P}{\mathit{L}}_1{\mathit{e}}_1+{\mathit{e}}_1^T\mathit{P}{\mathit{e}}_2-{\mathit{e}}_2^T\mathit{Q}{\mathit{L}}_2{\mathit{e}}_1\hfill \\ -{\mathit{e}}_2^T{\mathit{QM}}_{\mathit{\eta }}^{-1}(\mathit{\eta },\mathit{\Theta })({\mathit{C}}_{\mathit{\eta }}(\mathit{\nu },\mathit{\eta },\mathit{\Theta })+{\mathit{D}}_{\mathit{\eta }}(\mathit{\nu },\mathit{\eta })){\mathit{e}}_2+{\mathbf{e}}_2^T{\mathit{QM}}_{\mathit{\eta }}^{-1}(\mathit{\eta },\mathit{\Theta }){\mathit{e}}_3\hfill \\ +\frac{1}{\gamma }{\mathit{e}}_3^T\stackrel{·}{\mathit{d}}+{\mathit{e}}_3^T\Vert {\mathit{e}}_1{\Vert }^a·{\mathrm{sat}}_{\delta }\left({\mathit{e}}_1\right)\hfill \end{array}$$

(30)

From the equation (30), it can be seen that the first and fourth terms are negative definite terms. Therefore, by choosing appropriate ${\mathit{L}}_1$ , ${\mathit{L}}_2$ and $\gamma$ , the cross terms can be absorbed.

Under the condition that the disturbance $\stackrel{·}{\mathit{d}}$ is bounded, the derivative of the above Lyapunov function satisfies:

$$\stackrel{·}{V}\le -{\beta }_1\Vert {\mathit{e}}_1{\Vert }^2-{\beta }_2\Vert {\mathit{e}}_2{\Vert }^2+\epsilon$$

(31)

According to LaSalle’s Invariance Principle, ³⁵ the system is asymptotically stable, and when $\epsilon =0$ , the system error converges to 0.

Controller design and stability analysis

Due to the nonlinearities, coupling effects, and parameter uncertainties present in the dynamics of underwater vehicles, the FOSMC is designed to enhance the vehicle’s robustness against model uncertainties and external disturbances by incorporating fractional-order derivatives. Therefore, we design an FOSMC to control the UVMS to achieve reference trajectory tracking. The tracking error ${\mathit{e}}_t$ is defined as follows:

$${\mathit{e}}_t=\mathit{\eta }-{\mathit{\eta }}_R$$

(32)

Derivation of the error and substitution into equation (22) yields:

$$\begin{array}{c}{\mathit{M}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta })({\stackrel{··}{\mathit{e}}}_t+{\stackrel{··}{\mathit{\eta }}}_R)+{\mathit{C}}_{\mathit{\eta }}(\mathit{\nu },\mathit{\eta },\mathit{\Theta })\stackrel{·}{\mathit{\eta }}+{\mathit{D}}_{\mathit{\eta }}(\mathit{\nu },\mathit{\eta })\stackrel{·}{\mathit{\eta }}\hfill \\ +{\mathit{g}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta })=\mathit{\tau }\left(\mathit{\eta }\right)+\mathit{d}\left(\mathit{\eta }\right)\hfill \end{array}$$

(33)

To facilitate the design of the controller in this paper, a new variable z is introduced and defined as follows:

$$\mathit{z}={\stackrel{·}{\mathit{e}}}_t+\lambda {\mathit{e}}_t,\lambda \in {\mathbb{R}}_{+}$$

(34)

Substituting equation (34) into equation (33), we obtain:

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{M}}_{\mathit{\eta }}\left(\mathit{\eta }\right)\stackrel{·}{\mathit{z}}=\mathit{\tau }\left(\mathit{\eta }\right)+\mathit{d}\left(\mathit{\eta }\right)+\lambda {\mathit{M}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta }){\stackrel{·}{\mathit{e}}}_t}\\ -{\mathit{M}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta }){\stackrel{··}{\mathit{\eta }}}_R-{\mathbf{C}}_{\mathit{\eta }}(\mathit{\nu },\mathit{\eta },\mathit{\Theta })\stackrel{·}{\mathit{\eta }}\hfill \\ -{\mathit{D}}_{\mathit{\eta }}(\mathit{\nu },\mathit{\eta })\stackrel{·}{\mathit{\eta }}-{\mathit{g}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta })\hfill \end{array}$$

(35)

Subsequently, the fractional-order sliding surface is constructed as follows:

$$\mathit{s}=k_p\mathit{z}+D^{\alpha }\mathit{z},0<\alpha <1$$

(36)

Where $\mathit{s}={[s_1,s_2,s_3,s_4,s_5,s_6]}^T$ , $k_p\in {\mathbb{R}}_{+}$ is the gain constant. The corresponding control input is designed as follows:

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{\tau }}_{\mathit{\eta }\mathrm{FOSMC}}=-\lambda {\hat{\mathit{M}}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta }){\stackrel{·}{\mathit{e}}}_t+{\hat{\mathit{M}}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta }){\stackrel{··}{\mathit{\eta }}}_R}\\ +{\mathit{C}}_{\mathit{\eta }}(\mathit{\nu },\mathit{\eta },\mathit{\Theta })\stackrel{·}{\mathit{\eta }}+{\mathit{D}}_{\mathit{\eta }}(\mathit{\nu },\mathit{\eta })\stackrel{·}{\mathit{\eta }}+{\mathit{g}}_{\mathit{\eta }}(\mathit{\eta })\hfill \\ -k_p^{-1}{\hat{\mathit{M}}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta })D^{\alpha +1}\mathit{z}-k_1\mathit{s}-k_2\mathrm{sgn}\left(\mathit{s}\right)\hfill \end{array}$$

(37)

Where $k_1,k_2\in {\mathbb{R}}_{+}$ ,are parameters selected to satisfy specific stability conditions.

The Lyapunov function is defined as follows:

$$V_1\left(t\right)=\frac{1}{2}{\mathit{s}}^T{\mathit{M}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta })\mathit{s}$$

(38)

Taking the derivative of the Lyapunov function leads to the following expression:

$${\stackrel{·}{V}}_1\left(t\right)={\mathit{s}}^T{\mathit{M}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta })\stackrel{·}{\mathit{s}}+\frac{1}{2}{\mathit{s}}^T{\stackrel{·}{\mathit{M}}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta })\mathit{s}$$

(39)

The matrix $[{\stackrel{·}{\mathit{M}}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta })-2{\mathit{C}}_{\mathit{\eta }}(\mathit{\nu },\mathit{\eta },\mathit{\Theta })]$ is a skew-symmetric matrix, that is,

$${\mathit{s}}^T[{\stackrel{·}{\mathit{M}}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta })-2{\mathit{C}}_{\mathit{\eta }}(\mathit{\nu },\mathit{\eta },\mathit{\Theta })]\mathit{s}=0$$

(40)

The hydrodynamical matrix D is always a positive definite matrix, that is,

$${\mathit{s}}^T{\mathit{D}}_{\mathit{\eta }}(\mathit{\nu },\mathit{\eta })\mathit{s}>0,\forall \mathit{s}\ne 0$$

(41)

By combining the above equations, we can obtain:

$$\begin{array}{c}{\stackrel{.}{V}}_1={\mathit{s}}^T(k_p\left({\mathit{\tau }}_{\mathit{\eta }FOSMC}+d+\lambda {\mathit{M}}_{\mathit{\eta }}\left(\mathit{\eta },\mathit{\Theta }\right){\stackrel{.}{\mathit{e}}}_t-{\mathit{M}}_{\eta }\left(\mathit{\eta },\mathit{\Theta }\right){\stackrel{··}{\mathit{\eta }}}_R-{\mathit{C}}_{\mathit{\eta }}\left(\mathit{\nu },\mathit{\eta },\mathit{\Theta }\right)\stackrel{.}{\mathit{\eta }}-{\mathit{D}}_{\mathit{\eta }}(\nu ,\eta )\stackrel{.}{\mathit{\eta }}-{\mathit{g}}_{\mathit{\eta }}(\eta \right))\hfill \\ +{\mathit{M}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta })D^{\alpha +1}\mathit{z})=-k_p{\mathit{s}}^T(k_1\mathit{s}+k_2\mathrm{sgn}\left(\mathit{s}\right)-\mathit{d}-\mathrm{\Delta }\mathit{d})\hfill \end{array}$$

(42)

Where,

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Delta }\mathit{d}=\mathrm{\Delta }{\mathit{M}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta }){\stackrel{··}{\mathit{\eta }}}_R-\lambda \mathrm{\Delta }{\mathit{M}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta }){\stackrel{·}{\mathit{e}}}_t}\\ +k_p^{-1}{\mathit{M}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta })D^{\alpha +1}\mathit{z}\hfill \end{array}$$

(43)

To ensure system stability, the following condition must be satisfied:

$$\begin{array}{c}\multicolumn{1}{c}{k_2\ge \Vert \mathit{\Delta }{\mathit{M}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta }){\stackrel{··}{\mathit{\eta }}}_R{\Vert }_{\infty }+\Vert \lambda \mathrm{\Delta }{\mathit{M}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta }){\stackrel{·}{\mathit{e}}}_t{\Vert }_{\infty }}\\ +\Vert k_p^{-1}{\mathit{M}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta })D^{\alpha +1}\mathit{z}{\Vert }_{\infty }+\Vert \mathit{d}{\Vert }_{\infty }+\xi ,\xi \in {\mathbb{R}}_{+}\hfill \end{array}$$

(44)

By leveraging the relationship between matrix norms and the properties of quadratic functions, the following result can be derived:

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1\le -k_1{k}_p{\mathit{s}}^T\mathit{s}-k_p\xi \Vert \mathit{s}{\Vert }_1\le -k_1{k}_p{\mathit{s}}^T\mathit{s}-k_p\xi \Vert \mathit{s}{\Vert }_2}\\ \le -k_1{k}_p{\mathit{s}}^T\mathit{s}-k_p\xi \sqrt[\frac{1}{2}]{\frac{2}{\lambda {\left({\hat{\mathit{M}}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta })\right)}_{\max }}}\hfill \end{array}$$

(45)

We can obtain:

$${\stackrel{·}{V}}_1+k_p\xi \sqrt[\frac{1}{2}]{\frac{2}{\lambda {\left({\hat{\mathit{M}}}_{\mathit{\eta }}(\mathit{\eta },\mathit{\Theta })\right)}_{\max }}}$$

(46)

Set the parameter $k_2$ as:

$$k_2=\mathrm{\Delta }M{{({\Vert {\stackrel{··}{\mathit{\eta }}}_R\Vert }_{\infty }+{\Vert \lambda {\stackrel{·}{\mathit{e}}}_t\Vert }_{\infty }+{\Vert k_p^{-1}{D}^{\alpha +1}\mathit{z}\Vert }_{\infty })}_{\max }}_{\max }$$

(47)

When $\mathit{s}=0$ , the subsystem simplifies to $D^{\alpha }\mathit{z}=-k_p\mathit{z}$ , forming a standard fractional-order system. This leads to the following result:

$$\left|\arg (-k_p)\right|=\pi >\frac{\alpha \pi }{2}$$

(48)

Given that $0<\alpha <1$ and $k_p>0$ , the above equation remains bounded. Therefore, the fractional-order subsystem is asymptotically stable and converges to the equilibrium point at a rate proportional to $t^{-\alpha }$ .

When $\mathit{z}\to 0$ , we can obtain ${\stackrel{·}{\mathit{e}}}_t=-\lambda {\mathit{e}}_t$ .

Next, set the Lyapunov function for trajectory tracking error as:

$$V_2=\frac{1}{2}{{\mathit{e}}_t}^T{\mathit{e}}_t$$

(49)

By taking the derivative of the Lyapunov function $V_2$ , we obtain

$${\stackrel{·}{V}}_2={\mathit{e}}^T\stackrel{·}{\mathit{e}}=-\lambda {\mathit{e}}^T\mathit{e}$$

(50)

Based on the Lyapunov stability theorem, ³⁶ it can be concluded that as time t tends to infinity, the tracking error e will converge to 0.

In the proposed control law, the sign function’s characteristics can induce chattering in the system, potentially causing instability or damaging the actuators. To mitigate this, the sign function $\mathrm{sgn}(·)$ is replaced by a saturation function. The saturation function $\mathrm{sat}(·)$ is defined as follows:

$$\mathrm{sgn}\left(s_i\right)\to \mathrm{sat}\left(s_i\right)\{\begin{array}{c}\rho ,s_i>\rho \hfill \\ \multicolumn{1}{c}{\frac{s_i}{\rho },\left|s_i\right|\le \rho ,\rho \in {\mathbb{R}}_{+}}\\ \multicolumn{1}{c}{\rho ,s_i<-\rho }\end{array}$$

(51)

With the integration of the above disturbance observer, the control input of the UVMS is defined as follows:

$${\mathit{\tau }}_{\mathit{\eta }}={\mathit{\tau }}_{\mathit{\eta }\mathrm{FOSMC}}-{\hat{\mathit{f}}}_d$$

(52)

Simulation and analysis

Fractional order selection and analysis in FOSMC

Before conducting the simulation analysis, it is essential to specify the structural and dynamic characteristics of the UVMS considered in this study. The UVMS model comprises a base underwater vehicle equipped with an underwater manipulator. The dynamic behavior of the system is influenced by various factors, including mass distribution, added mass, hydrodynamic damping, and gravitational and buoyant forces. These parameters are crucial for accurately capturing the system dynamics and evaluating the performance of the proposed control scheme.

Table 2 summarizes the key physical and dynamic parameters of the UVMS used in the simulations, including the mass and inertia of the vehicle body, hydrodynamic coefficients, and manipulator link properties.

Table 2.

Structural and dynamic parameters of the UVMS.

rameter	Value	Parameter	Value
$m$	150.3	$X_{\stackrel{·}{u}}$	−100.2
$m_x$	4.8	$Y_{\stackrel{·}{v}}$	−160.09
$m_y$	4.8	$Z_{\stackrel{·}{w}}$	−154
$m_z$	2.1	$K_{\stackrel{·}{p}}$ .	−3.02
$m_b$	139.4	$M_{\stackrel{·}{q}}$	−9.11
${\mathit{I}}_b$	$\mathrm{diag}\{4.27,10.546,11.590\}$	$N_{\stackrel{·}{r}}$	−9.15
$B_x$	1	$X_u$	−17.2
$B_y$	0.6	$Y_v$	−25.55
$B_z$	0.2	$Z_w$	−28.03
${[l_x,w_x,h_x]}^T$	${[0.734,0.074,0.045]}^T$	$K_p$	−11.11
${[l_y,w_y,h_y]}^T$	${[0.107,0.704,0.045]}^T$	$M_q$	−14.39
${[l_z,w_z,h_z]}^T$	${[0.117,0.038,0.228]}^T$	$N_r$	−5.21
$X_{u\left|u\right|}$	−211.24	$K_{p\left|p\right|}$	−102.37
$Y_{v\left|v\right|}$	−498.54	$M_{q\left|q\right|}$	−131.65
$Z_{w\left|w\right|}$	−400.89	$N_{r\left|r\right|}$	−80.83

The parameters of the fractional-order sliding mode controller and the observer are as follows:

$$\begin{array}{c}\lambda =2k_p=10\alpha =0.3k_1=4\rho =0.6\xi =0.3,\hfill \\ k_2=1,\gamma =30,\delta =0.01,a=0.6\hfill \\ {\mathit{\eta }}_0={[0.1,0.1,0.1,0.1,0.2,0.45]}^T,\hfill \\ {\mathit{\nu }}_0={[0,0,0,0,0,0]}^T{\mathit{L}}_2=\mathrm{diag}\{20,22,18,10,10,10\},\hfill \\ {\mathit{L}}_1=\mathrm{diag}\{10,12,12,5,6,6.5\}\hfill \\ \hat{\mathit{M}}=\mathrm{diag}\{145,145,145,4.7,9.28,9.86\},\hfill \\ {\hat{\mathit{f}}}_{d0}={[2,0.5,1.2,0.5,-0.1,-0.8]}^T\hfill \end{array}$$

The external disturbances applied to the UVMS are modeled as a combination of sinusoidal signals and additive white noise.

In the proposed FOSMC scheme, the choice of the fractional-order significantly affects the system’s dynamic performance, including convergence speed, control smoothness, and disturbance rejection capability. A lower order introduces a memory effect, which helps to smooth the control input and improve robustness, while a higher order accelerates convergence but may result in increased switching activity and chattering.

In this study, the order was set to $\alpha =0.3$ as a trade-off between convergence speed and chattering suppression. As shown in Figure 6, we evaluated the step response along the x-direction under different fractional orders. The results indicate that the order of 0.3 yields the most favorable performance.

Figure 6.

Step responses under different fractional orders.

Control system performance analysis

To evaluate the estimation accuracy of the proposed disturbance observer, the nominal model of the UVMS was set to match the actual system, meaning no parameter uncertainties were present. Only an external sinusoidal disturbance was applied. The simulation results, shown in Figure 7, demonstrate that the designed observer achieves fast convergence and accurately estimates the external disturbance.

Figure 7.

Estimation results of the disturbance observer.

To verify the effectiveness of the proposed DO-based FOSMC, a series of trajectory tracking simulations were conducted on the UVMS. The proposed control strategy was compared against the conventional SMC to highlight the improvements in robustness and tracking performance, especially under parameter uncertainties and external disturbances. Figure 8 illustrates the tracking errors of the UVMS in six DOF.

Figure 8.

Comparison of trajectory tracking errors: DO-FOSMC versus SMC.

From Figure 8, it can be observed that the proposed DO-FOSMC demonstrates superior tracking accuracy compared to the traditional SMC across six DOF. This advantage is particularly evident in the position directions, where DO-FOSMC exhibits faster error convergence and smaller steady-state errors. Due to the presence of unmodeled disturbances and uncertainties in hydrodynamic parameters, the conventional SMC suffers from larger overshoots and slower convergence. In contrast, the DO-FOSMC achieves more stable and accurate performance in these directions.

To verify the effectiveness of the proposed disturbance observer, this study conducts simulations to evaluate its ability to estimate the dynamic effects caused by external disturbances and model uncertainties in the underwater robotic system. Figures 9 and 10 show the estimated disturbance forces across all six degrees of freedom. These results reflect the actual impact of unmodeled dynamics, including hydrodynamic parameter uncertainties and environmental disturbances.

Figure 9.

Estimated disturbance forces in the three translational directions by the observer.

Figure 10.

Estimated disturbance torques in the three rotational directions by the observer.

As illustrated in Figures 9 and 10, the observer is able to rapidly and accurately estimate the unknown disturbances after the initial transient phase. By compensating for these dynamic effects, the controller can be designed with a reduced switching gain, which effectively mitigates the chattering problem typically encountered in FOSMC and enhances the overall robustness of the system. The zoomed-in regions in the figures further demonstrate the high estimation accuracy of the observer during disturbance variations, highlighting its crucial role in improving control performance.

Figure 11 shows the time-domain response of the sliding surface under the FOSMC scheme. In the initial stage (t < 0.2 s), the sliding surface rapidly converges from its initial value to the equilibrium point, demonstrating the fast reach property of the fractional-order sliding surface. In the interval t∈ [0.2, 0.6]  s, the oscillation amplitude of the sliding surface significantly decreases, indicating that the compensation effect of the disturbance observer effectively suppresses external disturbances. After t > 0.6 s, the sliding surface enters a steady state, verifying the strong robustness of the FOSMC and the high-accuracy estimation capability of the DO.

Figure 11.

Sliding mode surface of the FOSMC.

To validate the reliability of the superior performance of the control system, we conducted six simulation analyses and compared the trajectory tracking errors of two control algorithms across the six DOF of the UVMS. Different initial conditions were set for each simulation. The average and standard deviation of the errors from the six simulations were then calculated, with the results shown in Tables 3 and 4.

Table 3.

Comparison of the average trajectory tracking errors between the two control algorithms.

Method	x	y	z	ϕ	θ	ψ
SMC	0.07628	0.08292	0.1011	0.01199	0.04	0.09535
DOFOSMC	0.003847	0.002976	0.003419	0.002683	0.007843	0.01875

Table 4.

Comparison of the standard deviation trajectory tracking errors between the two control algorithms.

Method	x	y	z	ϕ	θ	ψ
SMC	0.01061	0.01303	0.02589	0.0236	0.06033	0.13926
DOFOSMC	0.01053	0.0093	0.013129	0.01289	0.02992	0.06723

A comparison of the data presented in Tables 3 and 4 clearly demonstrates that the FOSMC with DO achieves significantly smaller mean and standard deviation in trajectory tracking errors across the UVMS six DOF compared to conventional sliding mode control. This indicates the superior accuracy of the proposed control strategy.

The combination of FOSMC and the DO thus leads to improved steady-state precision, faster dynamic response, and greater robustness under uncertain and disturbed conditions. The simulation results strongly validate that the proposed method maintains high accuracy and stability even in complex environments, showing great potential for practical engineering applications.

Discussion

This study presents a novel control strategy for UVMS by integrating variable center of mass dynamics modeling, a DO, and FOSMC. Traditional control methods often neglect the influence of center of mass shifts induced by manipulator motions, which may lead to degraded performance or instability. By introducing a variable center of mass dynamic model, this work explicitly incorporates the effect of manipulator movement into the control framework, thereby providing a more accurate representation of the system’s behavior.

To cope with external disturbances and modeling uncertainties commonly encountered in underwater environments, a state-augmented disturbance observer is designed to estimate unmodeled dynamics and disturbances in real time. This observer provides effective compensation information for the controller, enhancing both accuracy and disturbance rejection. Furthermore, the application of FOSMC brings improved robustness and smoother control actions compared to traditional integer-order sliding mode control. The non-integer derivative nature of FOSMC enables enhanced flexibility in tuning the sliding surface dynamics, which helps reduce chattering and improves trajectory tracking.

Extensive simulations confirm that the proposed FOSMC-DO method consistently performs well under various initial conditions. In comparison with conventional sliding mode control, the FOSMC-DO strategy significantly reduces tracking errors across all six degrees of freedom. Both the mean and standard deviation of tracking errors are notably lower, verifying that the proposed method offers not only high precision but also strong robustness against nonlinearities, uncertainties, and external disturbances.

Conclusion

This paper addresses the trajectory tracking control problem of the UVMS under conditions of variable center of mass, external disturbances, and model uncertainties. A control strategy based on a DO and FOSMC is proposed. By constructing a variable center of mass dynamic model, the influence of manipulator motions on the UVMS dynamics is accurately captured. The incorporation of a state-augmented DO enables real-time estimation of external disturbances and uncertain dynamics. Furthermore, the use of FOSMC enhances both control precision and system robustness.

Simulation results demonstrate that the proposed control algorithm significantly improves trajectory tracking performance across multiple DOF, with lower steady-state errors and favorable dynamic responses. The method maintains high stability and consistency even under varying initial conditions and uncertain disturbances. Compared to traditional control strategies, the proposed approach exhibits clear advantages in terms of accuracy, robustness, and disturbance rejection capability.

Future work may focus on implementing the proposed method on an experimental UVMS platform to further validate its feasibility and practical applicability in real underwater environments. Additionally, integrating adaptive mechanisms or data-driven techniques such as deep learning may further enhance the intelligence and adaptability of the control framework.