Chattering suppression and hydrodynamic disturbance estimation of underwater manipulators using adaptive fuzzy sliding mode control

Abstract

Robotic manipulators are nowadays widely used in various underwater scenarios, but their motion control remains a challenging task due to hydrodynamic effects. This article proposes a novel adaptive fuzzy sliding mode control (AFSMC) strategy for precise and robust control of underwater manipulators. To simulate the movement of the robotic manipulators in the underwater environment, the Unified Robot Description Format (URDF) file of a custom-designed electric underwater manipulator is imported into the Simscape Multibody and the hydrodynamic disturbance is modeled according to Morison’s equation. Furthermore, the proposed control strategy takes advantage of the universal approximation capability of fuzzy systems to avoid chattering and observe disturbances by adjusting the control gains of classical sliding mode control (CSMC). And adaptive laws are designed to update the parameters of the fuzzy systems. The strong friction caused by the seal is also compensated by actual test data. In the simulation experiments, a special environment of water flow and variable loads is considered. The results demonstrate that the AFSMC strategy can achieve high precision and strong robustness against disturbances for trajectory tracking. More importantly, the chattering caused by CSMC can be eliminated and the hydrodynamic disturbance can be estimated with high precision through the proposed control strategy.

Introduction

Robotic manipulators have started playing pivotal roles in the underwater world to protect workers from workplace hazards in complex and unpredictable environments. In recent years, many underwater manipulators have been developed for different tasks, including inspection (Wang et al., 2019), hull cleaning (Long et al., 2022), marine products grasping (Cai et al., 2020), sea mining exploration (Di Vito et al., 2021), and cooperative object transportation (Heshmati-Alamdari et al., 2021). Compared with traditional industrial manipulators, there are many technical difficulties in controlling an underwater manipulator because of the hydrodynamic influence, unknown external disturbances, and uncertainty in the dynamical model. For more precise control of underwater manipulators, many studies have been carried out in modeling and controller design.

A crucial component of precise automatic control for underwater manipulators is developing the hydrodynamic model. Morison’s equation was first proposed in 1950 and has since been extensively used to calculate the drag force and virtual mass force caused by the wave (Morison et al., 1950). Recently, a novel dynamic model for multi-link underwater manipulators has been developed using Lagrangian mechanics and Morison’s equation (Yuguang and Fan, 2019). This simplified model can well simulate the disturbance of the robotic manipulators in the underwater environment. In addition, flow visualization, theoretical analysis, experimental measurements, and computational fluid dynamics (CFD) are also gradually being employed to analyze the hydrodynamic forces and coefficients of robotic manipulators (Kolodziejczyk, 2018; McLain and Rock, 1998).

The most widely used control methods for underwater manipulators are proportional–integral–derivative (PID) control, model-based control, sliding mode control (SMC), neural control, and fuzzy control (Sivčev et al., 2018). SMC has been proven to be a powerful nonlinear control method because of its strong robustness and high convergence speed (Utkin, 1977). However, various problems, especially the chattering problem, make the application of SMC challenging in real-world scenarios (Young et al., 1999). Therefore, many methods have been proposed by researchers to investigate the improvement of SMC in the corresponding robotic fields, which include reaching law (Tang et al., 2021), terminal SMC (Hao et al., 2021), fuzzy SMC (Guo and Woo, 2003; Xiuxiang et al., 2018), neural networks (Zhu et al., 2020), reinforcement learning (Xie et al., 2020), and model predictive control (Nicolis et al., 2020). Among them, thanks to the universal approximation capability of fuzzy systems, fuzzy SMC is an effective strategy to avoid chattering caused by classical sliding mode control (CSMC).

For dynamic modeling and hydrodynamic coefficient determination of underwater manipulators, accurate estimation of hydrodynamic disturbance is meaningful. The combination of SMC and disturbance estimation methods is a popular and promising field due to the effective utilization of disturbance information. Disturbance observer is a classical and effective method for estimation (Dou and Yue, 2022; Yin et al., 2022). Time-delay control is also a robust and accurate scheme to estimate the dynamics of robotic manipulators with uncertainties (Baek et al., 2017; Lee et al., 2017). Furthermore, some researchers utilized the properties of fuzzy systems in combination with SMC for disturbance estimation (Ravandi et al., 2018; Van and Ge, 2021).

However, the greater friction of electric underwater manipulators due to joint sealing has not been treated in much detail in previous studies. And existing studies rarely consider the operation of underwater manipulators with water flow. In addition, very little attention has been paid to the role of the adaptive fuzzy sliding mode control (AFSMC) strategy in hydrodynamic disturbance estimation for underwater manipulators. Inspired by the above discussion, this paper is dedicated to developing an AFSMC that enables manipulators to operate efficiently and accurately in underwater moving water environments.

Major contributions can be listed as follows:

1. A hydrodynamic model applicable to underwater manipulators in water tunnels with water flow is developed and verified by simulation experiments.

2. Based on CSMC, fuzzy system, adaptive law, and friction compensation, an AFSMC for electric underwater manipulators is designed to overcome the chattering problem and estimate hydrodynamic disturbance.

3. A visual simulation environment for underwater manipulators is constructed through Simscape Multibody, and the effectiveness of the proposed control strategy is verified by simulation experiments with water flow and variable loads.

The remaining part of the paper proceeds as follows: In section “Modeling,” the underwater manipulator and hydrodynamic disturbance are modeled in MATLAB/Simulink environment. In section “Controller design,” the detailed design process of the proposed AFSMC strategy is provided and the stability of the system is proved mathematically. In section “Experiments and results,” numerical simulations are carried out and the results are presented to prove the superiority of the proposed control strategy. In section “Conclusion,” the overall works and findings of this paper are summarized.

Modeling

Mechanical model

Nowadays, water tunnels are crucial for human society because of the need to transport water to areas with large populations or agriculture (Alp and Apaydin, 2019). To replace workers of dangerous repair work in water tunnels, a remote electric underwater manipulator is designed. As shown in Figure 1, the manipulator has three joints and the end is equipped with a gripper to pick up the working tools. The angle values of these joints are expressed as $q_1$ , $q_2$ , and $q_3$ respectively. The corresponding joint torque values are expressed as ${\tau }_1$ , ${\tau }_2$ , and ${\tau }_3$ respectively. The gripper position is defined by $x$ , $y$ , and $z$ . Aluminum alloys are used as the construction material for the manipulator. And for waterproofing, the rubber O-ring seal and the spring-energized seal are used to implement static sealing and dynamic sealing, respectively.

Figure 1.

Mechanical model of the custom-designed electric underwater manipulator.

The main physical parameters of the underwater manipulator are given in Table 1. $m_i$ and $l_i$ denote the mass and length of link $i(i=1,2,3)$ , respectively. $d_i$ denotes the distance from the center of mass to the corresponding joint and ${\theta }_i$ denotes the center of mass offset angle relative to the corresponding link. The moment of inertia tensor is given by $I_{\mathrm{ixx}}$ , $I_{\mathrm{ixy}}$ , $I_{\mathrm{ixz}}$ , $I_{\mathrm{iyy}}$ , $I_{\mathrm{iyz}}$ , and $I_{\mathrm{izz}}$ .

Table 1.

Physical parameters of the manipulator.

$i$	$m_i$ (kg)	$l_i$ (m)	$d_i$ (m)	${\theta }_i$ (rad)	$I_{\mathrm{ixx}}$	$I_{\mathrm{ixy}}$	$I_{\mathrm{ixz}}$	$I_{\mathrm{iyy}}$	$I_{\mathrm{iyz}}$	$I_{\mathrm{izz}}$
1	2.10	0.09	0.08	0	0.0033	0	0	0.0027	0.0001	0.0026
2	2.71	0.41	0.27	0.05	0.0030	0.0012	− 0.0004	0.0163	− 0.0002	0.0164
3	3.01	0.37	0.19	0.10	0.0029	0.0002	− 0.0003	0.0031	− 0.0001	0.0029

Dynamic model

According to the Lagrangian mechanics, the dynamic model of the 3-degree-of-freedom (DOF) underwater manipulator is given by

$$\begin{array}{c}\mathit{M}\left(\mathit{q}\right)\stackrel{··}{\mathit{q}}+\mathit{C}(\mathit{q},\stackrel{·}{\mathit{q}})\stackrel{·}{\mathit{q}}+\mathit{G}\left(\mathit{q}\right)=\mathit{\tau }\hfill \end{array}$$

(1)

where $\mathit{q}=[q_1,q_2,q_3{]}^T$ is a $3\times 1$ vector of joint position, $\stackrel{·}{\mathit{q}}=[{\stackrel{·}{q}}_1,{\stackrel{·}{q}}_2,{\stackrel{·}{q}}_3{]}^T$ is a 3×1 vector of joint angular velocity, $\stackrel{··}{\mathit{q}}=[{\stackrel{··}{q}}_1,{\stackrel{··}{q}}_2,{\stackrel{··}{q}}_3{]}^T$ is a 3×1 vector of joint angular acceleration, $\mathit{\tau }=[{\tau }_1,{\tau }_2,{\tau }_3{]}^T$ is a 3×1 vector of control input torque, $\mathit{M}(\mathit{q})$ is a 3×3 inertial matrix, $\mathit{C}(\mathit{q},\stackrel{·}{\mathit{q}})$ is a 3×3 matrix of Coriolis and centrifugal forces, and $\mathit{G}(\mathit{q})$ is a 3×1 gravity vector.

In addition, the dynamic model of the manipulator also has some important properties, which is crucial in controller design. First, the inertial matrix $\mathit{M}(\mathit{q})$ is symmetric and positive definite. There exists $m_a$ and $m_b$ such that $m_a\le ||\mathit{M}\left(\mathit{q}\right)||\le m_b$ . Second, $\stackrel{·}{\mathit{M}}(\mathit{q})-2\mathit{C}(\mathit{q},\stackrel{·}{\mathit{q}})$ is a skew symmetric matrix which means

$$\begin{array}{c}{\mathit{x}}^T[\stackrel{·}{\mathit{M}}\left(\mathit{q}\right)-2\mathit{C}(\mathit{q},\stackrel{·}{\mathit{q}})]\mathit{x}=0\hfill \end{array}$$

(2)

where $\mathit{x}$ is a 3×1 nonzero vector.

Joint friction

Nonlinear friction torque is one of the main causes of vibration in manipulators at low speeds, and this torque is further increased by the sealing (Armstrong-Helouvry, 1993). In the early 20th century, Stribeck (1902) investigated the relationship between friction and velocity, and his model has been widely used since then. The joint friction torque ${\tau }_f$ can be described utilizing the classic friction model as

$$\begin{array}{c}{\tau }_f=(f_C+(f_S-f_C)e^{-{\left|\frac{\stackrel{·}{q}}{{\stackrel{·}{q}}_S}\right|}^{{\delta }_S}})\mathrm{sgn}\left(\stackrel{·}{q}\right)+f_v\stackrel{·}{q}\hfill \end{array}$$

(3)

where $f_C$ is the Coulomb friction, $f_S$ is the static friction, ${\stackrel{·}{q}}_S$ is the Stribeck velocity, ${\delta }_S$ is an empirical parameter ranging from 0.5 to 2 depending on the geometry of the application, and $f_v$ is the viscous friction coefficient (He et al., 2021; Olsson et al., 1998).

Three identical actuators are used as joints of the manipulator, so the friction torque of each joint is the same. To identify the joint friction parameters, experimental tests are conducted on unsealed and sealed joints. Friction torque is estimated from the measured current values. Both unsealed and sealed curves are consistent with the Stribeck model. The identified friction model parameters of the joint are given in Table 2. The friction torque in sealed joints is about five times greater than in unsealed joints, so the compensation of joint friction torque is necessary for precise control of the underwater manipulator.

Table 2.

Identified friction model parameters of the joint.

	$f_C$	$f_S$	${\stackrel{·}{q}}_S$	${\delta }_S$	$f_v$
Unsealed model	0.35	0.52	0.015	1.50	0.65
Sealed model	1.08	2.22	0.042	1.11	1.37

Therefore, the dynamic model of the manipulator is extended to

$$\begin{array}{c}\mathit{M}\left(\mathit{q}\right)\stackrel{··}{\mathit{q}}+\mathit{C}(\mathit{q},\stackrel{·}{\mathit{q}})\stackrel{·}{\mathit{q}}+\mathit{G}\left(\mathit{q}\right)+{\mathit{\tau }}_{\mathit{f}}=\mathit{\tau }\hfill \end{array}$$

(4)

where ${\mathit{\tau }}_{\mathit{f}}$ is a 3×1 vector of joint friction torque.

Hydrodynamic influence

To simplify the model, the links of the underwater manipulator are approximated as cylindrical elements. According to the report by Schjølberg and Fossen, the following hydrodynamical forces are acting on the manipulator (Schjølberg and Fossen, 1994).

The buoyancy force is an upward force, opposite to the direction of gravity, and it acts on the center of mass of the links. The equation of the buoyancy force is given by

$$\begin{array}{c}{F}_{B_i}=\rho V_{i}g\hfill \end{array}$$

(5)

where $V_i$ is the volume of the link $i$ , $\rho$ is the density of the water, and $g$ is the gravitational acceleration.

When the manipulator moves underwater, the viscous property of the fluid causes the drag force and the attached accelerating fluid causes the additional mass force. Morison’s equation is widely used to calculate these two forces. The equations are given as follows

$$F_{D_i}=\frac{1}{2}\rho C_D{D}_i\underset{0}{\overset{l}{\int }}{v}_i\left(x\right)\Vert v_i\left(x\right)\Vert \mathrm{dx}$$

(6)

$$F_{M_i}=\rho C_M{A}_i\underset{0}{\overset{l}{\int }}\frac{d{v}_i\left(x\right)}{\mathrm{dt}}\mathrm{dx}$$

(7)

where $C_D$ is the coefficient of drag, $C_M$ is the coefficient of additional mass, $D_i$ is the cylinder diameter of the link $i$ , $A_i$ is the area of the cross section, and $v_i(x)$ is the velocity of cylindrical surface.

In water conveyance tunnels, the velocity of water flow is generally 0–3 m/s, and its direction is perpendicular to the direction of gravity (Nabipour et al., 2020). So $v_i(x)$ can be expressed as

$$\begin{array}{c}\{\begin{array}{c}{v}_1(x)=v_{l_1}+v_f\sin {\theta }_1\hfill \\ \multicolumn{1}{c}{v_2(x)=v_{l_2}+v_f\cos {\theta }_1\sin {\theta }_2}\\ \multicolumn{1}{c}{v_3(x)=v_{l_3}+v_f\cos {\theta }_1\sin ({\theta }_2+{\theta }_3)}\end{array}\hfill \end{array}$$

(8)

where $v_{l_i}$ is the absolute velocity of the link $i$ and $v_f$ is the velocity of water flow.

The lift force is generated because of the airfoil structure of the object. The underwater manipulator does not have such a structure, so the lift force is not considered. The vortex shedding induces oscillatory forces on the links in both the drag and the lift direction. This force is also not considered because of the little effect on objects with small cross sections.

Therefore, the dynamic model of the manipulator is extended to

$$\begin{array}{c}\mathit{M}\left(\mathit{q}\right)\stackrel{··}{\mathit{q}}+\mathit{C}(\mathit{q},\stackrel{·}{\mathit{q}})\stackrel{·}{\mathit{q}}+\mathit{G}\left(\mathit{q}\right)+{\mathit{\tau }}_{\mathit{f}}+{\mathit{\tau }}_{\mathit{h}}=\mathit{\tau }\hfill \end{array}$$

(9)

where ${\mathit{\tau }}_{\mathit{h}}$ is a 3×1 vector of hydrodynamic disturbance caused by buoyancy force, drag force, and the additional mass force. And the hydrodynamic parameters for simulations are given in Table 3.

Table 3.

Hydrodynamic parameters for simulations.

$D_1$ ( ${\mathrm{m}}^2$ )	$D_2$ ( ${\mathrm{m}}^2$ )	$C_D$	$C_M$	$\rho$ ( $\mathrm{kg}/{\mathrm{m}}^3$ )	$g$ ( $\mathrm{m}/{\mathrm{s}}^2$ )
0.06	0.08	1	2	1000	9.81

Controller design

CSMC

In this paper, the controllers are designed to track the desired trajectory. The desired joint position is defined as ${\mathit{q}}_{\mathit{d}}=[{q_d}_1,{q_d}_2,{q_d}_3{]}^T$ , and then ${\stackrel{·}{\mathit{q}}}_{\mathit{d}}$ and ${\stackrel{··}{\mathit{q}}}_{\mathit{d}}$ denote the desired joint angular velocity and the desired angular joint angular acceleration, respectively. The tracking error is defined as

$$\begin{array}{c}e=q-{\mathit{q}}_{\mathit{d}}\hfill \end{array}$$

(10)

The sliding surface is designed as

$$\begin{array}{c}s=\stackrel{·}{\mathit{e}}+\mathrm{\Lambda }e\hfill \end{array}$$

(11)

where $\mathit{\Lambda }=\mathrm{diag}[{\lambda }_1,{\lambda }_2,{\lambda }_3]$ in which ${\lambda }_i>0$ .

Equation (11) ensures

$$\begin{array}{c}\{\begin{array}{c}\underset{t\to \infty }{lim}\mathit{e}=0\hfill \\ \multicolumn{1}{c}{\underset{t\to \infty }{lim}\stackrel{·}{\mathit{e}}=0}\end{array}\hfill \end{array}$$

(12)

when $\mathit{s}=0$ . Therefore, the controller should be designed to make

$$\begin{array}{c}\underset{t\to \infty }{lim}\mathit{s}=0\hfill \end{array}$$

(13)

In CSMC, the control input $\mathit{\tau }$ is designed as

$$\begin{array}{c}\mathit{\tau }=\hat{\mathit{M}}({\stackrel{··}{\mathit{q}}}_{\mathit{d}}-\mathit{\Lambda }\stackrel{·}{\mathit{e}})+\hat{\mathit{C}}({\stackrel{·}{\mathit{q}}}_{\mathit{d}}-\mathit{\Lambda }\mathbf{e})+\hat{\mathit{G}}-\mathit{As}-\mathit{K}\mathrm{sgn}\left(\mathit{s}\right)\hfill \end{array}$$

(14)

where $\hat{\mathit{M}}$ , $\hat{\mathit{C}}$ , and $\hat{\mathit{G}}$ are the estimations of $\mathit{M}$ , $\mathit{C}$ , and $\mathit{G}$ , respectively, $\mathit{A}=\mathrm{diag}$ [ $a_1,a_2,a_3$ ] in which $a_i>0$ , and $\mathit{K}=\mathrm{diag}[K_1,K_2,K_3]$ in which $K_i>0$ .

Substituting equation (14) into equation (9) obtains

$$\mathit{M}\stackrel{.}{\mathit{s}}+\left(\mathit{C}+\mathit{A}\right)\mathit{s}=\mathit{D}-\mathit{K}sgn\left(\mathit{s}\right)$$

(15)

where $\mathit{D}=\mathrm{\Delta }\mathit{M}({\stackrel{··}{\mathit{q}}}_{\mathit{d}}-\mathit{\Lambda }\stackrel{·}{\mathit{e}})+\mathrm{\Delta }\mathit{C}({\stackrel{·}{\mathit{q}}}_{\mathit{d}}-\mathit{\Lambda }\mathit{e})+\mathrm{\Delta }\mathit{G}-{\mathit{\tau }}_{\mathit{f}}-{\mathit{\tau }}_{\mathit{h}}$ , $\mathrm{\Delta }\mathit{M}=\hat{\mathit{M}}-\mathit{M}$ , $\mathrm{\Delta }\mathit{C}=\hat{\mathit{C}}-\mathit{C}$ , and $\mathrm{\Delta }\mathit{G}=\hat{\mathit{G}}-\mathit{G}$ . $\mathit{D}$ is a bounded disturbance and its boundary is assumed as $D_m$ which means $|D_i|\le D_m.$ Then $\mathit{K}$ is chosen to satisfy

$$\begin{array}{c}{K}_i\ge D_m\hfill \end{array}$$

(16)

To prove the stability of the manipulator system, the Lyapunov function candidate is defined as

$$\begin{array}{c}V=\frac{1}{2}{\mathit{s}}^T\mathit{Ms}\hfill \end{array}$$

(17)

Since $\mathit{M}$ is positive definite, $V=0$ if and only if $\mathit{s}=0$ and $V>0$ if and only if $\mathit{s}\ne 0$ .

Using equations (3), (9), and (14), it can be derived that

$$\begin{array}{l}\stackrel{.}{V}={\mathit{s}}^T\left[-\mathit{A}\mathit{s}+\mathit{D}-\mathit{K}\mathrm{sgn}\left(\mathit{s}\right)\right]\\ =-{\mathit{s}}^T\mathit{A}\mathit{s}+\sum _{i=1}^3\left(s_i\left[D_i-K_i\mathrm{sgn}\left(s_i\right)\right]\right)\end{array}$$

(18)

It can be derived that $D_i-K_i\mathrm{sgn}(s_i)\le 0$ for $s_i\ge 0$ and $D_i-K_i\mathrm{sgn}(s_i)\ge 0$ for $s_i<0$ from equation (16). Therefore, $s_i[D_i-K_i\mathrm{sgn}(s_i)]\le 0$ . Since $\mathit{A}$ is positive definite, $-{\mathit{s}}^T\mathit{As}=0$ if and only if $\mathit{s}=0$ and $-{\mathit{s}}^T\mathit{As}<0$ if and only if $\mathit{s}\ne 0$ . Combining the above inferences, it can be proved that $\stackrel{·}{V}<0$ for all values of $\mathit{s}\ne 0$ .

According to Lyapunov’s second method, the asymptotic stability of the underwater manipulator system can be proved.

AFSMC

To suppress the chatter problem caused by CSMC, the chattering factor $\mathit{K}\mathrm{sgn}(\mathit{s})$ is replaced by a vector $\mathit{k}$ in our strategy. Combined with fuzzy systems, adaptive laws, and friction compensation, the designed AFSMC can be expressed as

$$\begin{array}{c}\mathit{\tau }=\hat{\mathit{M}}({\stackrel{··}{\mathit{q}}}_{\mathit{d}}-\mathit{\Lambda }\stackrel{·}{\mathit{e}})+\hat{\mathit{C}}({\stackrel{·}{\mathit{q}}}_{\mathit{d}}-\mathit{\Lambda }\mathbf{e})+\hat{\mathit{G}}+{\tilde{\mathit{\tau }}}_{\mathit{f}}-\mathit{As}-\mathit{k}\hfill \end{array}$$

(19)

where the element $k_i$ of $\mathit{k}$ is a dynamic value adjusted by adaptive fuzzy systems and ${\tilde{\mathit{\tau }}}_{\mathit{f}}$ is the estimation of ${\mathit{\tau }}_{\mathit{f}}$ for compensation.

So equation (18) is rewritten as

$$\stackrel{·}{V}=-{\mathit{s}}^T\mathit{As}+\sum _{i=1}^3\left(s_i[d_i-k_i]\right)$$

(20)

To ensure $\stackrel{·}{V}<0$ for all values of $\mathit{s}\ne 0$ , the sign of $k_i$ has to be the same as that for $s_i$ . And when $s_i$ is much larger (smaller) than zero, $k_i$ is expected to be much larger (smaller) than zero so that $s_i$ can rapidly arrive at zero. In this section, a fuzzy system is introduced to adjust $k_i$ continuously to satisfy the above requirements, and each $k_i$ has an independent fuzzy system. For each fuzzy system, the input is $s_i$ and the output is $k_i$ . Positive Big (PB), Positive Middle (PM), Positive Small (PS), Zero (ZO), Negative Small (NS), Negative Middle (NM), and Negative Big (NB) are used to describe the magnitude of the values. And the input and output of the fuzzy system are set to have the same kind of membership functions. The set of membership functions for both $s_i$ and $k_i$ are illustrated in Figure 2.

Figure 2.

Membership functions of the fuzzy system.

The fuzzy rule base is designed as follows:

• R1: IF $s_i$ is PB THEN $k_i$ is PB

• R2: IF $s_i$ is PM THEN $k_i$ is PM

• R3: IF $s_i$ is PS THEN $k_i$ is PS

• R4: IF $s_i$ is ZO THEN $k_i$ is ZO

• R5: IF $s_i$ is NS THEN $k_i$ is NS

• R6: IF $s_i$ is NM THEN $k_i$ is NM

• R7: IF $s_i$ is NB THEN $k_i$ is NB.

A fuzzy system is generally composed of fuzzification, fuzzy rule base, fuzzy inference engine, and defuzzification. And the four units can be expressed as a mathematical formula. Using singleton fuzzification, center average defuzzification, and Mamdani implication, the output $k_i$ of the fuzzy system can be expressed as

$$k_i=\frac{{\sum }_{m=1}^3{\theta }_{k_i}^m{\mu }_{A^m}\left(s_i\right)}{{\sum }_{m=1}^3{\mu }_{A^m}\left(s_i\right)}={\mathit{\theta }}_{k_i}^T{\mathbf{\xi }}_{k_i}\left(s_i\right)$$

(21)

where ${\mathit{\theta }}_{k_i}=[{\theta }_{k_i}^1,{\theta }_{k_i}^2,{\theta }_{k_i}^3{]}^T$ is a vector of parameters which have to be updated through an adaptive law, ${\mu }_{A^m}(s_i)$ is the membership function of $s_i$ , and ${\mathbf{\xi }}_{k_i}(s_i)$ is called fuzzy basic vector.

Substituting equation (19) into equation (9), we get

$$\mathit{M}\stackrel{.}{\mathit{s}}+\left(\mathit{C}+\mathit{A}\right)\mathit{s}=\mathit{d}-\mathit{k}$$

(22)

where $\mathit{d}=\mathrm{\Delta }\mathit{M}({\stackrel{··}{\mathit{q}}}_{\mathit{d}}-\mathit{\Lambda }\stackrel{·}{\mathit{e}})+\mathrm{\Delta }\mathit{C}({\stackrel{·}{\mathit{q}}}_{\mathit{d}}-\mathit{\Lambda }\mathbf{e})+\mathrm{\Delta }\mathit{G}+\mathrm{\Delta }{\mathit{\tau }}_{\mathit{f}}-{\mathit{\tau }}_{\mathit{h}}$ , $\mathrm{\Delta }{\mathit{\tau }}_{\mathit{f}}={\tilde{\mathit{\tau }}}_{\mathit{f}}-{\mathit{\tau }}_{\mathit{f}}$ , and the element $k_i$ of $\mathit{k}$ is designed as equation (21).

Define the optimal vector of parameters as

$$\begin{array}{c}{\mathit{\theta }}_{k_i}^{*}=\arg \underset{k_i\in {\mathrm{\Omega }}_i}{min}\left[\sup |d_i-k_i|\right]\hfill \end{array}$$

(23)

where ${\mathrm{\Omega }}_i$ is the set of $k_i$ .

Many different kinds of fuzzy systems have been proven to have the universal approximation property (Zeng and Singh, 1995). And the property is expected to estimate the disturbance $\mathit{d}$ . According to the universal approximation theory, there exists a minimum error ${\omega }_i$ such that

$$\begin{array}{c}[d_i-{{\mathit{\theta }}_{k_i}^{*}}^T{\mathbf{\xi }}_{k_i}\left(s_i\right)]\le {\omega }_i\hfill \end{array}$$

(24)

Thus, equation (21) can be rewritten as

$$\begin{array}{c}{k}_i={\tilde{\mathit{\theta }}}_{k_i}^T{\mathbf{\xi }}_{k_i}\left(s_i\right)+{{\mathit{\theta }}_{k_i}^{*}}^T{\mathbf{\xi }}_{k_i}\left(s_i\right)\hfill \end{array}$$

(25)

where ${\tilde{\mathit{\theta }}}_{k_i}={\mathit{\theta }}_{k_i}-{\mathit{\theta }}_{k_i}^{*}$ .

The adaptive law is designed as

$$\begin{array}{c}{\stackrel{·}{\stackrel{~}{\mathit{\theta }}}}_{k_i}={\stackrel{·}{\mathit{\theta }}}_{k_i}={\gamma }_i{s}_i{\mathbf{\xi }}_{k_i}\left(s_i\right)\hfill \end{array}$$

(26)

To prove the stability of the manipulator system, the Lyapunov function candidate is defined as

$$\begin{array}{c}V=\frac{1}{2}({\mathit{s}}^T\mathit{Ms}+\sum _{i=1}^3\frac{1}{{\gamma }_i}\left({\tilde{\mathit{\theta }}}_{k_i}^T{\tilde{\mathit{\theta }}}_{k_i}\right))\hfill \end{array}$$

(27)

Since $\mathit{M}$ is positive definite and ${\tilde{\mathit{\theta }}}_{k_i}^T{\tilde{\mathit{\theta }}}_{k_i}>0$ , $V>0$ when $\mathit{s}\ne 0$ .

Using equations (20), (25), and (26), it can be derived that

$$\begin{array}{l}\stackrel{.}{V}=-{\mathit{s}}^T\mathit{A}\mathit{s}+\sum _{i=1}^3\left(s_i\left[d_i-k_i\right]\right)+\sum _{i=1}^3\frac{1}{{\gamma }_i}\left({\tilde{\mathit{\theta }}}_{k_i}^T{\stackrel{.}{\stackrel{~}{\mathit{\theta }}}}_{k_i}\right)\\ =-{\mathit{s}}^T\mathit{A}\mathit{s}+\sum _{i=1}^3(s_i[d_i-({\tilde{\mathit{\theta }}}_{k_i}^T{\mathit{\xi }}_{k_i}\left(s_i\right)+{\mathit{\theta }}_{k_i}^{\ast }{\mathit{\xi }}_{k_i}\left(s_i\right))])\\ +\sum _{i=1}^3\frac{1}{{\gamma }_i}\left({\tilde{\mathit{\theta }}}_{k_i}^T{\stackrel{.}{\stackrel{~}{\mathit{\theta }}}}_{k_i}\right)\\ =-{\mathit{s}}^T\mathit{A}\mathit{s}+\sum _{i=1}^3\left(s_i\left[d_i-{\mathit{\theta }}_{k_i}^{\ast }{\mathit{\xi }}_{k_i}\left(s_i\right)\right]\right)\\ +\sum _{i=1}^3\left(s_i\left[-{\tilde{\mathit{\theta }}}_{k_i}^T{\mathit{\xi }}_{k_i}\left(s_i\right)\right]\right)+\sum _{i=1}^3\frac{1}{{\gamma }_i}\left({\tilde{\mathit{\theta }}}_{k_i}^T{\stackrel{.}{\stackrel{~}{\mathit{\theta }}}}_{k_i}\right)\\ =-{\mathit{s}}^T\mathit{A}\mathit{s}+\sum _{i=1}^3\left(s_i\left[d_i-{\mathit{\theta }}_{k_i}^{\ast T}{\mathit{\xi }}_{k_i}\left(s_i\right)\right]\right)\\ +\sum _{i=1}^3\left(\frac{1}{{\gamma }_i}{\tilde{\mathit{\theta }}}_{k_i}^T{\stackrel{.}{\stackrel{~}{\mathit{\theta }}}}_{k_i}-s_i{\tilde{\mathit{\theta }}}_{k_i}^T{\mathit{\xi }}_{k_i}\left(s_i\right)\right)\\ =-{\mathit{s}}^T\mathit{A}\mathit{s}+\sum _{i=1}^3\left(s_i\left[d_i-{\mathit{\theta }}_{k_i}^{\ast }{\mathit{\xi }}_{k_i}\left(s_i\right)\right]\right)\end{array}$$

(28)

From equation (24), the error ${\omega }_i$ can be as small as possible, so there exists $0<{\beta }_i<1$ , which makes

$$[d_i-{{\mathit{\theta }}_{k_i}^{\ast }}^T{\mathbf{\xi }}_{k_i}\left(s_i\right)]\le {\omega }_i\le {\beta }_i\left|s_i\right|$$

(29)

Then

$$\begin{array}{c}\stackrel{·}{V}\le -{\mathit{s}}^T\mathit{As}+\sum _{i=1}^3\left({\beta }_i{s_i}^2\right)=-{\mathit{s}}^T(\mathit{A}-\mathit{\beta })\mathit{s}\hfill \end{array}$$

(30)

where $\mathit{\beta }=\mathrm{diag}[{\beta }_1,{\beta }_2,{\beta }_3]$ , and the element $a_i$ of $\mathit{A}$ is set to satisfy $a_i>{\beta }_i$ .

Therefore, $\stackrel{·}{V}\le 0$ and $\mathit{s}\equiv 0$ when $\stackrel{·}{V}\equiv 0$ . According to LaSalle’s invariance principle, the asymptotic stability of the underwater manipulator system can be proved, which means

$$\begin{array}{c}\{\begin{array}{c}\underset{t\to \infty }{lim}\mathit{q}={\mathit{q}}_{\mathit{d}}\hfill \\ \multicolumn{1}{c}{\underset{t\to \infty }{lim}\stackrel{·}{\mathit{q}}={\stackrel{·}{\mathit{q}}}_{\mathit{d}}}\end{array}\hfill \end{array}$$

(31)

The architecture of the control system is demonstrated in Figure 3. The blue boxes represent the proposed control strategy, consisting of CSMC, fuzzy systems, adaptive laws, and friction compensation. The green box represents the simulation environment. The model of the physical system is established through the Simscape Multibody toolbox of MATLAB/Simulink. The Unified Robot Description Format (URDF) file of the custom-designed underwater manipulator is imported into Simscape Multibody. The friction torque and the hydrodynamic disturbance are both added to the Simscape environment through the S-function.

Figure 3.

Block diagram of the underwater manipulator control system.

Experiments and results

Simulation of hydrodynamic influence

In this part, numerical experiments are conducted to study the hydrodynamic influence and validate the reasonableness of the model. All the simulations in this paper are conducted in the MATLAB/Simulink environment and the sampling time is chosen as 0.001 s. The simulation is performed in three cases, which are model on land, model underwater without the flow, and model underwater with the flow. In the environment of water flow, $v_f$ is set to 2 m/s.

In experiment 1, three models move in the same angular trajectory ( $q_i=\sin (t)$ ) to compare the control torques of the manipulator in different environments. The comparison of joint torques is shown in Figure 4. For joint 1 without the influence of gravity and buoyancy, the underwater movement requires greater torque due to the extra drag force and additional mass force, which are mainly influenced by ${\stackrel{·}{q}}_1$ . For joints 2 and 3, which are mainly affected by gravity and buoyancy, the underwater movement requires smaller torques because the buoyancy counteracts part of the gravity. It can be seen from 0–1 s that the joints require larger torques during the upstream motion of the manipulator compared to the static water case. On the contrary, the joints require smaller torques during the downstream motion compared to the static water case in 3–4 s.

Figure 4.

Experiment 1: joint torques for the same trajectory in three different environments.

In experiment 2, the manipulator moves without torque input in all cases. The three kinds of movement processes in the Simscape environment are shown in Figure 5. The manipulator on land in Figure 5(a) swings back and forth under the effect of gravity until it stops in about 3.72 s due to friction, while the underwater movement in Figure 5(b) and (c) stops earlier because of the extra hydrodynamic influence. Due to the buoyancy, joints 2 and 3 of the manipulator have different convergence angles underwater compared to on land. It can also be seen in Figure 5(c) that joint 2 stops at a smaller angle than the static water case because of the influence of water flow.

Figure 5.

Experiment 2: manipulator movements without control torque: (a) on land, (b) underwater without flow, and (c) underwater with flow.

Based on the above experiments and results, it can be concluded that the proposed hydrodynamic model can simulate the underwater environment effectively.

Simulation of the proposed AFSMC strategy

To demonstrate the effectiveness of the proposed control strategy, comparative experiments in a special environment are conducted on AFSMC and CSMC. The manipulator is required to perform the task of movement with a variable load under the influence of water flow. The whole simulation environment is shown in Figure 6. As is shown in Figure 7, the task consists of two processes: reaching the position of a 1-kg object (the yellow part) within 0–5 s and grasping the object to move to the target position within 5–10 s.

Figure 6.

Overall simulation environment in MATLAB/Simulink.

Figure 7.

Illustration of the task: (a) process 1 without gripping any object and (b) process 2 with gripping a 1-kg object.

To more realistic simulation of the actual manipulator, quintic polynomials are used for trajectory planning to ensure the same initial angle as well as zero velocity and acceleration at the initial and end positions, which can avoid generating large torques. The desired trajectory of the gripper position is expressed as

$$q_1=\{\begin{array}{cc}30{\left(\frac{t}{5}\right)}^3-45{\left(\frac{t}{5}\right)}^4+18{\left(\frac{t}{5}\right)}^5,& 0\le t<5\\ 3-30{\left(\frac{t-5}{5}\right)}^3+45{\left(\frac{t-5}{5}\right)}^4-18{\left(\frac{t-5}{5}\right)}^5,& 5\le t\le 10\end{array}$$

(32)

$$q_2=\{\begin{array}{c}\multicolumn{1}{c}{-5.5{\left(\frac{t}{5}\right)}^3+8.25{\left(\frac{t}{5}\right)}^4-3.3{\left(\frac{t}{5}\right)}^5,0\le t<5}\\ \multicolumn{1}{c}{-0.55-4.5{\left(\frac{t-5}{5}\right)}^3+6.75{\left(\frac{t-5}{5}\right)}^4-2.7{\left(\frac{t-5}{5}\right)}^5,5\le t\le 10}\end{array}$$

(33)

$$q_3=\{\begin{array}{cc}17.1{\left(\frac{t}{5}\right)}^3-25.65{\left(\frac{t}{5}\right)}^4+10.26{\left(\frac{t}{5}\right)}^5,& 0\le t<5\\ -27.1{\left(\frac{t-5}{5}\right)}^3-40.65{\left(\frac{t-5}{5}\right)}^4-16.26{\left(\frac{t-5}{5}\right)}^5,& 5\le t\le 10\end{array}$$

(34)

In water conveyance tunnels, the maximum velocity of water flow varies usually between 2 and 3 m/s (Nabipour et al., 2020). So $v_f$ is given by the Gaussian function

$$\begin{array}{c}{v}_f=\{\begin{array}{c}2+e^{-\frac{{(x-2.5)}^2}{2\times {0.7}^2}},0\le t<5\hfill \\ \multicolumn{1}{c}{2+e^{-\frac{{(x-7.5)}^2}{2\times {0.7}^2}},5\le t\le 10}\end{array}\hfill \end{array}$$

(35)

The parameters selected for controllers are shown in Table 4, and the estimation ${\tilde{\mathit{\tau }}}_{\mathit{f}}$ is chosen to be $0.95{\mathit{\tau }}_{\mathit{f}}$ .

Table 4.

Parameters of CSMC and AFSMC for simulations.

${\lambda }_1$	${\lambda }_2$	${\lambda }_3$	$a_1$	$a_2$	$a_3$	$K_1$	$K_2$	$K_3$	${\gamma }_1$	${\gamma }_2$	${\gamma }_3$
5	5	5	5	5	5	50	20	10	5000	20000	2000

CSMC: control gains of classical sliding mode control; AFSMC: adaptive fuzzy sliding mode control.

Simulation results are given as follows. Figure 8 shows the three-dimensional (3D) trajectory tracking of CSMC and AFSMC in Cartesian space. It can be seen from the detailed diagram that AFSMC has better tracking performance than CSMC. Figure 9(a) and (c) shows the trajectory tracking performances of the joint angle and the gripper position. Both algorithms allow the trajectory tracking to converge, but the accuracy is different. The maximum angular error of CSMC and AFSMC is about 0.02 and 0.002, respectively. And the maximum gripper position error of CSMC and AFSMC is about 0.01 and 0.001, respectively. For the tracking of joint angular velocity in Figure 9(b), there is a long time and wide range of chatter in the curve of CSMC. In contrast, the curve of AFSMC is much smoother, with joint 2 producing only a small range of chatter during load changes. Based on the above, the proposed AFSMC strategy has higher steady-state accuracy and enables the manipulator to better track the trajectory in an environment with variable load and water flow.

Figure 8.

Three-dimensional trajectory tracking of CSMC and AFSMC in Cartesian space.

Figure 9.

Comparison of tracking performances: (a) joint angle and its error, (b) joint angular velocity and its error, and (c) gripper position and its error.

It is worth mentioning the smooth trajectory of CSMC in Figure 8, despite the large chatter in its angular velocity in Figure 9(b). The reason for this is that the angle is the integral of the angular velocity, and the integrator acts as the equivalent of a low-pass filter. This means that the high-frequency part, namely the chattering part, will be filtered out during the process of integration (Tavares, 1966). So after filtering, the angle curve of CSMC is smooth, and the curve obtained in Cartesian space after the angle is calculated by forward kinematics will also be smooth.

From Figure 10, it can be obviously seen that the control torque of CSMC generates a very high-frequency chatter, where the vibration amplitude of joint 1 reaches about 100 N m. In engineering applications, chattering problems will seriously affect the stability and reliability of the manipulator. AFSMC can effectively solve the problem, which can be seen to have a smooth curve, with joints 2 and 3 producing only a small range of chatter during load changes. The chattering suppression is achieved by adaptive fuzzy systems that continuously adjust $k_i$ to approximate the disturbance boundary. The parameter ${\gamma }_1$ determines the rate of adjustment. Therefore, $k_i$ is the estimation of the disturbance, which means AFSMC has the ability to estimate the hydrodynamic disturbance when the errors $\mathrm{\Delta }\mathit{M}$ , $\mathrm{\Delta }\mathit{C}$ , $\mathrm{\Delta }\mathit{G}$ , and $\mathrm{\Delta }{\mathit{\tau }}_{\mathit{f}}$ of manipulator dynamic model are small. As is shown in Figure 11, the curve of estimation tracking for hydrodynamic disturbance can converge within about 0.1 s. And the detailed diagrams show that the estimated value is within about 0.1 N m error compared to the actual value. Based on the above, the proposed AFSMC strategy can effectively perform chattering suppression and hydrodynamic disturbance estimation.

Figure 10.

Joint torques of CSMC and AFSMC.

Figure 11.

Estimation of hydrodynamic disturbance.

Conclusion

In this paper, a hydrodynamic model is established using the Simscape Multibody toolbox to simulate the movement of underwater manipulators, and a new adaptive fuzzy sliding mode controller is designed for precise and robust control. The joint friction torque is modeled by actual test data, and the hydrodynamic disturbance is modeled according to Morison’s equation to make the dynamic model more accurate. Numerical simulations are conducted in a special environment of water flow and variable load to verify the effectiveness of the proposed control strategy. The results demonstrate that the AFSMC strategy has better performance than CSMC for trajectory tracking. Furthermore, the chattering problem of joint angular velocity and control torque caused by CSMC is overcome by the proposed adaptive fuzzy system. And the AFSMC strategy has the ability to estimate the hydrodynamic disturbance according to the universal approximation theory which can be used as a reference for subsequent hydrodynamic analysis and measurements.

In future work, the tracked vehicle will be installed as the base of the underwater manipulator for the convenience of mobile repair work. Then the new dynamic model will be studied. And the experiments will be conducted on the real system to verify the reliability of the proposed control strategy.