1 Introduction

In the last years, a growing number of marine robotic vehicles including unmanned surface vehicles (USVs), autonomous underwater vehicles (AUVs), remotely operated vehicles (ROVs), and underwater gliders (UGs) have been developed for civil, military, and scientific research applications [7, 12, 78, 87, 101, 128, 162, 164, 170, 187, 195]. For instance, these vehicles prove their capabilities in three-dimensional reconstruction of seabed surface [197], automatic underwater sampling [175, 194, 217], detection and monitoring of marine gas seeps [6], subsea pipeline/cable tracking and inspection [3, 166, 171, 184], monitoring and assisting human divers [96], flow field mapping [13], and mine countermeasures [29].

For such vehicles to be capable of undertaking these missions, they require advanced, intelligent, reliable and adaptable modeling, planing, navigation, guidance, and control system [38, 40, 63, 79, 89, 98, 100, 143, 151, 159, 191,192,193, 200, 201, 212, 214]. As a result, various methods such as proportional–integral–derivative (PID) control [11, 41, 83, 188], feedback linearizing control [10], backstepping control [47, 72, 73, 80, 167, 174, 216], \({\mathcal {L}}_1\) [92], sliding mode control [25, 26, 111, 208], neural network control [24, 44, 105, 106, 108, 196, 209, 215], robust control [153], model predictive control [45], and fuzzy logic control [109, 149] are being widely applied to guidance and control of marine robotic vehicles.

Particularly, the fuzzy logic control based on fuzzy set theory by Zadeh exhibits excellent immunity to system nonlinearity and uncertainties [61, 145, 155, 190]. Based on the differences of fuzzy control rules and their generation methods, approaches to fuzzy logic control can be roughly classified into the following categories: (1) conventional fuzzy control (CFLC); (2) adaptive fuzzy control (AFLC); (3) hybrid fuzzy control including fuzzy PID control (FPIDC), fuzzy sliding mode control (FSMC), and neuro-fuzzy control (NFLC).

In this paper, we aim to survey state of the art of various fuzzy controllers for the guidance and control of marine robotic vehicles as well as the design and analysis. Hence, we hope to provide a valuable guide for learning guidance and control of marine robotic vehicles based on fuzzy logic control.

The rest of the paper is organized as follows. The guidance and control concept of marine robotic vehicles is defined in the next section. Various fuzzy logic controllers in guidance and control application are reviewed as well as the design and analysis in Sect. 3. In Sect. 4, the performance comparison between hybrid fuzzy control and other control is shown. Trends of the future of fuzzy-logic-based guidance and control in marine robotic vehicles are summarized in Sect. 5, while conclusions are included in Sect. 6.

2 Guidance and Control

According to [38], the terms guidance and control are defined as: (1) Guidance is the action of determining the course, attitude, and speed of a marine robotic vehicle, relative to some reference frame; (2) Control is the development and application to a marine robotic vehicle of appropriate forces and moments for operating point control, tracking and stabilization, which is related to design the feedforward and feedback control laws. A guidance and control system for automatic weather routing of a ship is shown in Fig. 1.

Fig. 1
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Guidance and control system for automatic weather routing of ships

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The control of marine robotic vehicles can be divided into single degree of freedom (1-DOF) maneuvering, point stabilization, path following, and trajectory tracking [31, 94, 174, 213], as shown in Fig. 2.

Fig. 2
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Basic control tasks for marine robotic vehicles: a maneuvering; b point stabilization; c path following; d trajectory tracking

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1-DOF maneuvering, as shown in Fig. 2a: 1-DOF maneuvering includes zigzag maneuver, heading control, tuning motion, and roll stabilization.

Point stabilization, as shown in Fig. 2b: The point stabilization problem usually means that a vehicle is stabilized in a desired goal posture (position and orientation) from a given initial configuration.

Path following, as shown in Fig. 2c: In the path following task, the assigned Cartesian path is usually given in a parameterized form expressing the desired motion in terms of a path parameter, i.e., the arc length s along the path. A vehicle is required to follow this path with spatial convergence alone, without any temporal specification.

Trajectory tracking, as shown in Fig. 2d: In the trajectory tracking task, the vehicle must track a time-parameterized trajectory, i.e., a geometric path with an associated timing specification t. Hence, it inherently mixes the time and space assignments into one assignment.

Remark, Maneuvering trials are often performed to assess the path keeping and path changing ability of a marine robotic vehicle. Point stabilization task can often be achieved for a fully actuated marine robotic vehicle where the number of control inputs is the same as that of degrees of freedom. For path following, time dependence is not relevant because one is concerned only with the geometric displacement between the vehicle and the path. In this sense, the time evolution of the path parameter is usually free. Yet, the evolution of the trajectory parameter in trajectory tracking is time dependent [21, 30, 31, 39, 58, 66, 122, 154, 174, 205, 207].

3 Fuzzy Logic Application and Analysis

This section reviews the application of three kinds of fuzzy control in guidance and control of marine robotic vehicles.

3.1 Conventional Fuzzy Control (CFLC)

Generally, the CFLC can be divided in two types: Mamdani fuzzy control [95] and Takagi–Sugeno–Kang (TSK) fuzzy control [132], which need to deal with fuzzification, fuzzy inference, and defuzzification operations, as shown in Fig. 3. Here, the inputs of CFLC are usually composed of the error e and the error rate \(\dot{e}\) and the output can be directly given to the onboard actuator.

Fig. 3
figure 3

Basic structure of a CFLC system

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3.1.1 Mamdani Fuzzy Control

In terms of guidance, the fuzzy logic system was used to generate a new safe heading angle, yaw rate, or rudder angle for an AUV in order to avoid obstacle [8, 9, 127]. It was also applied in a submarine to evade the attack from a torpedo [124]. Moreover, an outer-loop Mamdani fuzzy logic controller with triangular membership functions for the inputs and output was designed to generate the desired rudder angle of path following [42]. Besides its application for safe maneuvering and command guidance, the fuzzy comprehensive evaluation method was applied to evaluate the motion performance of an AUV, which can provide a valuable and scientific guide for layout scheme decision-making at the preliminary design stage [85].

Mamdani fuzzy control also plays an important role in motion control of marine robotic vehicles. A Mamdani fuzzy logic controller with triangular membership functions was designed for fixed and varying heading control of a tank [102, 147]. In [69], the Mamdani fuzzy control system was developed for classifying the transverse of terrain and performing pipeline following by a ROV. In addition to the above heading control and path following, a hierarchical closed-loop fuzzy control system was applied for horizontal-plane trajectory tracking of an under-actuated AUV [113]. The single Mamdani fuzzy logic controller was also used in horizontal-plane trajectory tracking of an USV [74] and vertical-plane trajectory tracking of an AUV [90, 117].

3.1.2 TSK Fuzzy Control

The difference is that unlike the Mamdani method, the output membership functions of the TSK method are only constants (singletons/zero-order) or have linear relationship (first-order) with respect to the input, which obviously simplifies the defuzzification process. In this sense, the TSK fuzzy control seems to be more popular in guidance and control fields of marine robotic vehicles.

The zero-order TSK fuzzy controller was designed for dynamic positioning [137], vertical-plane variable depth trajectory tracking [112, 118], and three-dimensional path following control [49, 182]. Actually, the zero-order TSK fuzzy control is a special case of the first-order TSK fuzzy control which is more general. For instance, the first-order TSK fuzzy logic was used to generate the desired heading angle in the underwater docking mission [135, 136]. Besides the guidance application, it was also applied to the heading, pitch and depth control of an AUV [123]. Compared with the above 1-DOF control, the first-order TSK fuzzy design for the coupled motion is relatively complicated, such as point stabilization of an AUV [15], vertical-plane trajectory tracking and path following of an AUV [64, 65]. Different form these two linear output functions, the yaw control of an AUV is based on a TSK fuzzy controller where the output of each rule is a nonlinear Gaussian function with respect to double inputs [130]. In [160], the nonlinear fuzzy output combined a smoothing function with a switch function.

Note that the above TSK fuzzy controller has two inputs. It implies that if the number of membership functions for each input is n, the number of fuzzy rules will be \(n^2\). Based on the following signed distance method

$$\begin{aligned} \begin{aligned} d=\frac{\dot{e}+\lambda e}{\sqrt{1+\lambda ^2}} \end{aligned} \end{aligned}$$
(1)

with \(\lambda \) being a constant slope and d being the signed distance, the single input fuzzy controller using d as its input was proposed in [22] and applied to the control of marine robotic vehicles, such as the heading control of a chemical ship tanker [60], vertical-plane trajectory tracking of an AUV [2, 56, 57], and three-dimensional (3D) path following [183]. As shown in [56], the number of rules of the single input fuzzy controller can be reduced from \(n^2\) to n and the computational time was reduced form 1500 to 10 \(\upmu {\hbox {s}}\).

In summary, the CFLC-based documents in guidance and control of marine robotic vehicles are listed in Table 1.

Table 1 Application classification for conventional fuzzy control
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3.1.3 Stability Analysis

The stability of the Mamdani fuzzy system was rarely reported in the listed documents, while the stability of the TSK fuzzy system can be based on common quadratic Lyapunov functions, piecewise quadratic Lyapunov functions, or fuzzy Lyapunov functions. The authors can refer to [37] for the details and in this paper it is omitted.

3.2 Adaptive Fuzzy Control (AFLC)

Fuzzy control design is composed of three important stages, namely fuzzy rules, scaling factors, and membership functions. Generally, the CFLC has a fixed set of IF-THEN rules, usually derived from experts’ knowledge. The membership functions of the associated input and output linguistic variables are generally predefined on a common universe of discourse. For the successful design, the proper selection of input and output scaling factors is also important.

As we all known, the marine robotic vehicle is a nonlinear second-order system and often has system uncertainties including hydrodynamic modeling inaccuracy and unknown environmental disturbances. Hence, the CFLC with a fixed number of IF-THEN rules, fixed valued scaling factors and predefined membership functions may be not enough sufficient to get a satisfactory control performance. In this subsection, we list three kinds of AFLC: self-tuning fuzzy control, direct adaptive fuzzy control, and indirect adaptive fuzzy control.

3.2.1 Self-tuning Fuzzy Control (SFLC)

If a fuzzy logic controller has self-tuning membership functions, scaling factors, or fuzzy rules, it can be called SFLC. An adaptive fuzzy controller with self-tuning membership functions was proposed for the obstacle avoidance of an AUV [35]. To achieve better track-keeping performance in the presence of external disturbances, the scaling factors of a fuzzy controller for the ship autopilot were changed by an adjustable mechanism with the object distance and the heading angle as its inputs [139]. In [75], a learning control algorithm automatically generated the fuzzy controller’s knowledge base online as new information on how to control the ship, as shown in Fig. 4. However, the stability of the above SFLC was not analyzed.

Fig. 4
figure 4

Basic structure of self-tuning fuzzy control

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3.2.2 Direct Adaptive Fuzzy Control (DAFLC)

Before introducing DAFLC, we should first review the fuzzy approximation theorem, which is the basis of DAFLC and the subsequent indirect adaptive fuzzy control (IAFLC).

For a real continuous function f(x) whose analytic expression is unknown, given a sufficiently large number N of fuzzy rules and any small tolerance \(\bar{\varepsilon }\), there exists an optimal output weight matrix \({\varvec{\varpi }}^*\) such that

$$\begin{aligned} {\varvec{\varpi }}^*=\text {arg}~\min \limits _{{\varvec{\varpi }\in R^n}}\left\{ \sup \limits _{x\in {\varOmega }}\left[{\varvec{\varpi }}^{\top }{\varvec{\xi }}(x)-f(x)\right] \right\} \end{aligned}$$
(2)

and

$$\begin{aligned} f(x)={\varvec{\varpi }}^{*\top }{\varvec{\xi }}(x)+\varepsilon \end{aligned}$$
(3)

with the minimal functional approximation error \(\varepsilon \) satisfying \(|\varepsilon |\le \bar{\varepsilon }\) [62, 146, 157].

DAFLC uses the fuzzy approximation theorem to estimate the optimal control laws dependent on accurate dynamics model. For instance, an optimal control law was first developed based on full-state feedback control. Then, the fuzzy approximation was used to estimate the unknown terms in the optimal control law for a fully actuated USV [134, 148]. The general framework is given as follows:

The kinematic and dynamic models of a marine robotic vehicle can be described in the following vectorial strict-feedback form

$$\begin{aligned} \left\{ \begin{aligned} {\varvec{\eta }}&={\varvec{R}}({\varvec{{\varOmega }}}){\varvec{\upsilon }}\\ \dot{\varvec{\upsilon }}&=-{\varvec{M}}^{-1}\left( {\varvec{C}}({\varvec{\upsilon }}){\varvec{\upsilon }}+{\varvec{D}}({\varvec{\upsilon }}){\varvec{\upsilon }}+{\varvec{g}}({\varvec{\eta }})+{\varvec{d}}+{\varvec{\tau }}\right) \\ \end{aligned}\right. \end{aligned}$$
(4)

where \({\varvec{\eta }}\) denotes the position and orientation vector with coordinates in the earth-fixed inertial frame; \({\varvec{R}}\) is a transformation matrix which is related through the functions of the Euler angles \({\varvec{{\varOmega }}}\); \({\varvec{\upsilon }}\) denotes the linear and angular velocity vector with coordinates in the body-fixed frame; \({\varvec{M}}\) is the inertia matrix; \({\varvec{C}}\) is the matrix of Coriolis and centripetal terms; \({\varvec{D}}\) is the damping matrix; \({\varvec{g}}\) is the vector of gravitational forces and moments; \({\varvec{\tau }}\) is used to describe the forces and moments acting on the vehicle in the body-fixed frame; and \({\varvec{d}}\) denotes the environmental disturbances in the body-fixed frame.

Assume that the desired position and orientation vector is \({\varvec{\eta }}_d\). Define error variables \({\varvec{z}}_1={\varvec{\eta }}-{\varvec{\eta }}_d\) and \({\varvec{z}}_2={\varvec{\upsilon }}-{\varvec{\alpha }}_1\) where \({\varvec{\alpha }}_1\) is a virtual velocity vector. The first step is to consider the Lyapunov function candidate \(V_1=0.5{\varvec{z}}_1^T{\varvec{z}}_1\) and then differentiating \(V_1\) yields the virtual control law \({\varvec{\alpha }}_1={\varvec{R}}({\varvec{{\varOmega }}})^T(\dot{{\varvec{\eta }}}_d-{\varvec{K}}_1{\varvec{z}}_1)\) where \({\varvec{K}}_1\) is a diagonal matrix. The second step is to choose the Lyapunov function candidate \(V_2=V_1\,+\,0.5{\varvec{z}}_2^T{\varvec{M}}{\varvec{z}}_2\) to obtain the optimal control law

$$\begin{aligned} {\varvec{\tau }}^*=-{\varvec{R}}({\varvec{{\varOmega }}})^T{\varvec{z}}_1-{\varvec{K}}_2{\varvec{z}}_2+{\varvec{C}}({\varvec{\upsilon }}){\varvec{\upsilon }}+{\varvec{D}}({\varvec{\upsilon }}){\varvec{\upsilon }}+{\varvec{g}}({\varvec{\eta }})-{\varvec{d}}+{\varvec{M}}{\dot{\varvec{\alpha }}}_1 \end{aligned}$$
(5)

where \({\varvec{K}}_2\) is also a diagonal matrix.

However, since \({\varvec{M}}, {\varvec{C}}({\varvec{\upsilon }}), {\varvec{D}}({\varvec{\upsilon }}), {\varvec{g}}({\varvec{\eta }}), {\varvec{d}}\) are difficult to measure precisely and usually all unknown or part known, the model-based optimal control law is not feasible [150, 152]. To overcome this difficulty, the fuzzy approximation can be used to approximate the unknown terms:

$$\begin{aligned} {\varvec{\varpi }}^{*\top }{\varvec{\xi }}+\varepsilon ={\varvec{C}}({\varvec{\upsilon }}){\varvec{\upsilon }}+{\varvec{D}}({\varvec{\upsilon }}){\varvec{\upsilon }}+{\varvec{g}}({\varvec{\eta }})-{\varvec{d}}+{\varvec{M}}{\dot{\varvec{\alpha }}}_1 \end{aligned}$$
(6)

Combining the control framework in (5) with the fuzzy approximation of unknown dynamics, the final direct adaptive fuzzy controller can be designed as

$$\begin{aligned} {\varvec{\tau }}=-{\varvec{R}}({\varvec{{\varOmega }}})^T{\varvec{z}}_1-{\varvec{K}}_2{\varvec{z}}_2+{\varvec{\widehat{\varpi }}}^{\top }{\varvec{\xi }} \end{aligned}$$
(7)

where the updated law of \(\widehat{\varvec{\varpi }}\) can be resorted to [134, 148]. The same is that tracking errors are proven to be uniformly ultimately bounded due to the existence of the approximation error \(\varepsilon \).

3.2.3 Indirect Adaptive Fuzzy Control (IAFLC)

Different from the DAFLC, the IAFLC uses the fuzzy approximation to estimate the unknown dynamics model. In [4, 179], adaptive fuzzy control was used to approximate the nonlinear unknown terms in order to achieve the Lyapunov stability of the ship roll stabilization system. It was also used to approximate unknown function in the ship steering systems [114, 180]. In addition, the IAFLC identified the unknown nonlinear parts of the vertical-plane dynamic model of a submarine [115].

In summary, the AFLC used in guidance and control of marine robotic vehicles is listed in Table 2.

Table 2 Application classification for adaptive fuzzy control
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3.3 Hybrid Control Combining Fuzzy Control with Other Algorithms

In this subsection, we will present several kinds of hybrid control combining fuzzy control with other algorithms, which can generate a better behavior than either of them.

3.3.1 Fuzzy PID Control (FPIDC)

It is well known that the conventional PID controller is the most widely adopted controller in industry, due to its simple structure, ease of design, and low cost in implementation [37]. Yet, the conventional PID controller might not perform satisfactorily if the system to be controlled is highly nonlinear, coupled, or uncertain. On the other hand, fuzzy control has been well known for its ability to reject nonlinearities and uncertainties by the use of fuzzy set theory. Hence, it can be believed that by integrating these two methods, a better control system called FPIDC can be designed.

Usually, an adaptive PID controller with self-tuning parameters adjusted by fuzzy control is designed to offer the robustness with respect to the system uncertainties, including inaccurate hydrodynamic parameters and time-varying environmental disturbances, such as longitudinal control [133], heading control [76, 93], variable depth tracking control [88], three-dimensional trajectory tracking [68], and three-dimensional path following [173]. Here, the most commonly used control law \(\tau _i\) in certain degree of freedom is

$$\begin{aligned} \tau _i(t)=k_p(t)e(t)+k_i(t)\int _0^te(\tau ) {\hbox {d}}\tau +k_d(t)\dot{e}(t) \end{aligned}$$
(8)

where the self-tuning parameters are updated by

$$\begin{aligned} \left\{ \begin{aligned} k_{p}(t)&=k_{p0}+\Delta k_{p}(e(t),\dot{e}(t))\\ k_{i}(t)&=k_{i0}+\Delta k_{i}(e(t),\dot{e}(t))\\ k_{d}(t)&=k_{d0}+\Delta k_{d}(e(t),\dot{e}(t))\\ \end{aligned}\right. \end{aligned}$$
(9)

with the initial control gains \(k_{p0},k_{i0},k_{d0}\) and the time-varying incremental gains \(\Delta k_{p},\Delta k_{i},\Delta k_{d}\).

In order to get the gains \(\Delta k_{p},\Delta k_{i},\Delta k_{d}\), a fuzzy logic controller is usually adopted. If the fuzzy linguistic variables are defined as NB, NM, NS, ZE, PS, PM, PB, the fuzzy rules for \(\Delta k_{p},\Delta k_{i},\Delta k_{d}\) can be listed in Tables 3, 4 and 5. For each table, the first column represents the different fuzzy subsets of the tracking error e, the first row represents the different fuzzy subsets of the tracking error ratio \(\dot{e}\), and the other cells in the table are outputs of IF-THEN rules in different cases. The subsequent defuzzification step can be achieved by the use of the center of area method.

Table 3 Fuzzy control rules for \(\Delta k_p\)
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Table 4 Fuzzy control rules for \(\Delta k_i\)
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Table 5 Fuzzy control rules for \(\Delta k_d\)
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3.3.2 Fuzzy Sliding Mode Control (FSMC)

As we all know, sliding mode control (SMC) is a robust approach to control a nonlinear system with internal and external uncertainties [28, 43, 138, 169]. Usually, a sliding mode surface is first defined, i.e.,

$$\begin{aligned} s(t)=\dot{e}(t)+k_1e(t) \end{aligned}$$
(10)

or

$$\begin{aligned} s(t)=\dot{e}(t)+k_2e(t)+k_3\int _0^t e(\tau ){\hbox {d}}\tau \end{aligned}$$
(11)

where \(k_1, k_2, k_3\) are control gains. Then, a switching function \(\sigma \text {sgn}(s)\) with a proper gain \(\sigma \) is necessary in the control law design. Yet, it often results in chattering phenomena due to its discontinuous switching function. Actually, the combination of fuzzy control and sliding mode control can weaken the chattering phenomena and realize the advantages of both techniques. For instance, the fuzzy logic is used to adjust the gain (i.e., \(\sigma \)) of the sliding mode switching part in order to suppress chattering. This kind of applications can be seen in roll stabilization [17], heading control [34], and depth tracking control [110].

On the one hand, fuzzy logic control is also used to approximate the system modeling based on the fuzzy approximation and then sliding mode control rejects the rest uncertainty, such as heading control [119, 189], depth control [186], trajectory tracking [5, 71, 149], and path following [48, 67, 81, 82, 158].

3.3.3 Neuro-Fuzzy Control (NFLC)

Since neural network control has strong learning capabilities and high computation efficiency in parallel implementation [27, 46, 50, 52, 99, 104, 120, 142, 156, 172, 177, 178, 203] and fuzzy control has a powerful framework for expert knowledge representation, the combination of these two methods has attracted lots of attention from control community. A typical combination is the so-called NFLC, which is basically a fuzzy control augmented by neural networks to enhance its characteristics like flexibility, data processing capability, and adaptability [37, 51, 51, 55].

In general, the NFLC has three kinds of control frameworks, as shown in Fig. 5. The first one is the linear superposition of them in Fig. 5a. For instance, a fuzzy PD control plus neural network control was designed for an underwater vehicle-manipulator system [176]. The second one is fuzzy control adjusted by neural network control shown in Fig. 5b. For instance, artificial neural network was used to tune the consequent portion of the fuzzy conditional statements [131]. A neuro-fuzzy system was used to reform the membership functions and rules of a collision avoidance system [1]. The weight of NFLC in path following task of the ODIN AUV was adjusted online to minimize the error function by using a simplified learning and retrieving procedure [70]. The third one is the complete integration of them shown in Fig. 5c. A typical example is that the NFLC was used to model the inverse dynamics of the ODIN AUV and the feedback-error-learning method or other method online tuned the parameters of the recurrent neuro-fuzzy controller [77, 144]. Similarly, the unknown dynamic function in trajectory tracking, heading control, and dynamic positioning was identified by the NFLC [23, 84, 97, 165]. In addition, the NFLC was introduced to approximate a backstepping control law [86].

Fig. 5
figure 5

Basic structure of NFLC

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3.3.4 Other Hybrid Fuzzy Control

In addition to the aforementioned FPIDC, FSMC, and NFLC, there are other algorithms combined with fuzzy control, such as parallel distributed compensation [14, 16, 53], genetic algorithm [36, 54, 59], \(H_\infty \) [116], and particle swarm optimization [198]. In summary, the hybrid fuzzy control used in guidance and control of marine robotic vehicles is listed in Table 6.

Table 6 Application classification for hybrid fuzzy control
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4 Comparative Results

In this section, we will give two simulation comparisons between the fuzzy-logic-based hybrid controller and the corresponding single controller. In the authors’ opinion, the comparison of CFLC/AFLC and the other control (i.e., backstepping, neural network) seems not to be very equitable because the performance also depends on the design of control systems and the choice of control gains. From the simulated results, it can be concluded that the fuzzy-logic-based hybrid controller can perform better than the corresponding single controller, which is due to making full use of advantages of fuzzy logic in the hybrid control.

4.1 FPIDC Versus PID

In the first simulation, the algorithms of FPIDC and PID are taken from [173]. All the parameters of the whole simulated system are the same as it. Here, we give the following paths and errors under two different controllers, as shown in Figs. 6 and 7, respectively. Note that the external environmental disturbances and model uncertainties simultaneously act on the AUV from 160 to 350s. Obviously, the added fuzzy logic control makes the following process smoother and the robustness against disturbances stronger.

Fig. 6
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3D Tanh path following under FPIDC and PID

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Fig. 7
figure 7

Path following errors under FPIDC and PID

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4.2 FSMC Versus SMC

In the second simulation, the algorithms of FSMC and SMC are taken from [181]. All the parameters of the whole simulated system are also the same as it. Here, we give the following paths and control inputs under two different controllers, as shown in Figs. 8 and 9, respectively. Although there seems to be no difference between the two actual following paths, the control inputs under FSMC are smoother than those of SMC, which illustrates that the added fuzzy logic control suppresses the inherent chattering of the conventional SMC.

Fig. 8
figure 8

2D Path following under FSMC and SMC

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Fig. 9
figure 9

Control inputs under FSMC and SMC

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5 Trends for the Future

Based on state of the art of fuzzy logic application in marine robotic vehicles, it can be concluded that more attention should be paid to the following aspects:

The first one is the fuzzy application for new control scenarios, such as motion control with thruster fault [129, 161, 163, 210], motion control with input saturation [121, 126, 141, 185, 206, 209, 211], and formation control of multiple vehicles [18,19,20, 32, 33, 91, 103, 107, 125, 140, 168, 202]. The above topics are hotspots of present control fields in marine robotic vehicles, and it is expected that the fuzzy logic theory will play a greater role due to its simple control structure, easy and cost-effective design.

The second one is the fuzzy generalization in a guidance layer. Fuzzy logic control has shown the outperformance in various control scenarios. Yet, the guidance behavior is usually the basis of motion control, especially for an under-actuated marine robotic vehicle. Hence, we believe the fuzzy logic method will be also more applied to the guidance loop of marine robotic vehicles because most of existing marine robotic vehicles are under-actuated.

The third one is the design and analysis of fuzzy logic system itself. From Sect. 3, the design of most fuzzy logic controllers is coupled with other algorithms, i.e., PID. Yet, how to analyze the stability of FPIDC seems a difficult problem. Maybe the switching theory should be introduced.

The fourth one is the implementation of fuzzy-logic-based guidance and control in actual onboard system of marine robotic vehicles. It can be found that most of published papers presented simulation results except for [8, 48, 85, 119, 136, 199, 204]. Hence, the future work should focus on the application of various advanced adaptive fuzzy controllers in field tests.

6 Conclusion

In this paper, we review three major classes of fuzzy logic control, including CFLC, AFLC, and HFLC used in the marine robotic field. Due to its simple control structure, easy and cost-effective design, it can be seen that they are widely used in guidance and control of marine robotic fields, especially in control field. Subsequently, two comparative results between fuzzy-logic-based hybrid control and the corresponding single control are given to illustrate the superiority of fuzzy-logic-based hybrid control. Besides the review, trends of the fuzzy future in marine robotic vehicles are summarized, which can provide some potential research topics for the readers.