## 1. Introduction

An autonomous underwater vehicle (AUV) is employed with the aim to reduce the possibility of human accident in a long‐term underwater mission. One of the important parts to be installed on an AUV is an advance control system. Beside the capability to ensure the robustness and efficiency of an AUV, the selected control system must have the capability to minimize the effect of hazardous underwater environment such as sea current and hydrodynamics forces that potentially increase the energy usage since the position of AUV is moved from the desired trajectory. Various controllers are introduced to be adapted on the underwater vehicle. Linear quadratic regulator (LQR), linear quadratic Gaussian (LQG),

Meanwhile, other researchers employed a sliding mode controller (SMC) which is robust against an inaccurate model and the external disturbances [3]. In this case, Cristi et al. designed the SMC from the Lyapunov candidate then applied it to adjust the AUV's maneuver based on the dynamics system and operating condition [4]. The simulation obtained a small error but it did not include the effect of the disturbances. The integrator SMC was also proposed by Hong et al. for the depth control of torpedo‐shaped AUV in Ref. [5]. The SMC was applied as an inner pitch controller, while the effect of buoyancy on pitch and heave dynamics was considered. It was shown that the steady‐state error existed and bounded within *x*‐axis with a constant speed of

Fuzzy logic control (FLC) is well known as an intelligent and adaptive control method [7]. For some cases, FLC is used to solve the chattering problem, thus, Guo et al. superposed SMC with fuzzy tuning technique [8]. Stability and robustness of the control system were guaranteed by selecting the shrinking and dilating factors of the fuzzy membership functions. Two experiments were conducted to observe the efficiency of the proposed controller for a Hai‐Min underwater vehicle under the influence of ocean current. The results confirmed the effectiveness of the proposed scheme, although a poor transient performance was produced when the system tried to achieve precise tracking. At the transition moment, the state and sliding surface were separated by a significant distance. Lakehekar and Saundarmal developed an adaptive fuzzy sliding mode controller with a boundary layer scheme [9]. In the simulation, the SMC was required to manage the vertical position of the AUV. However, the variables inside SMC changed dramatically. Thus, a boundary layer near the switching line was introduced as a new method. To maintain the states inside the layer, two fuzzy approximators were employed. The first approximator was used to update the slope's value in the sliding surface, while the second approximator was used to shape the error tracking. The result showed that the proposed controller reduced the reaching time of 1–

Neural network (NN) is commonly used either as a control plant model or as a controller [7, 10]. There are two kinds of learning processes in the NN, online learning and offline training, and the success of NN depends on selecting the correct learning process. Some cases reported that different responses could be resulted even after the same controller was applied under the same environment [11]. Ji‐Hong et al. compared the conventional SMC with neural network (NN) SMC [12]. When the SMC is widely used as a robust control, NN is used in conjunction to minimize the nonlinearity of the dynamic's error [13]. The results show that the NN produced small errors and the AUV was able to track the trajectory after many learning and adaptation processes [14]. The effect of the disturbances was also not considered. Meanwhile, Van de ven et al. approximated the damping model of an AUV by using the value of velocity and acceleration under offline training process [10]. Noise was added in the second simulation, hence an online learning was adopted to decrease the state prediction error as well as to minimize the influence of the noise. It was shown that NN was used to improve the performance of poor identification of the AUV model. However, Van de ven et al assumed that other parts of the AUV model to be fully known, while in real case, the other parts such as added mass and also colioris and centripetal model were fully uncertain.

Another robust controller, which has been developed has a high order sliding mode controller (HOSMC), works on higher order derivatives of the sliding variable/system deviation [15–19]. The development of this method aimed to minimize the chattering effect produced by the conventional SMC. The second order is widely implemented because of the low information demand. Robust integral of sign of error (RISE) is included in the type of HOSMC. Fischer et al. used the RISE as a robust control of a six‐DOF AUV [20]. The experimental setup was conducted in a swimming pool and an openwater sea trial with

Besides making a robust controller for the AUV, the other problem, which needs to be solved, was reducing the energy consumption. As stated in Ref. [6], the AUV could be damaged if it spends more energy. Li et al. was successful in introducing an adaptive region‐tracking controller to overcome this weakness [24]. This success was followed by Ismail and Dunnigan [25]. The proposed controller in both the research guaranteed the error convergence of the sliding vector because the desired target was determined as a region instead of a point. The results showed that the thrusters were only activated when the AUV was outside the region. Therefore, the AUV reduced the energy consumption. Li and Cheah also used a similar approach for manipulator robot [26]. Li and Cheah proposed a unified objective bound method to merge the set point of the control, the trajectory tracking, and the performance bound. The desired trajectory reduced the conventional trajectory when the error was small, and it also changed to a dynamic region which could be scaled or rotated. Then, the system guaranteed the transient and the steady‐state response of close loop system, as long as the objective was specified as a performance bound. A simulation was conducted to show the energy‐saving properties of the proposed controller. The energy remained zero when the end effector of the arm robot started and stayed inside the bound. The proposed controller required less energy than the standard controller.

Reviewing from the advantages and the disadvantages of previous work, this chapter proposes a super twisting sliding mode control with region boundary for an AUV's tracking trajectory under the influence of perturbation. The proposed control is expected to obtain accuracy and efficiency, which is tracking precisely on the desired trajectory as well as saving energy consumption. The chapter is organized as follows: Section 2 studies about kinematic and dynamic model of a 6 DOF AUV, Section 3 describes the proposed control and comparison control, Section 4 performs results of simulation and analysis, while conclusion is explained in Section 5.

## 2. Kinematic and dynamic model

This section presents the kinematic and the dynamic model of a six‐DOF AUV. Before discussing about the kinematic and the dynamic model, we introduce two types of geometric transformation in a six‐DOF AUV, namely translation and rotations. The translation is represented by sway, surge, and heave movements, while the rotation is represented by roll, pitch, and yaw movements. An origin C, which is located on the center of the mass, a body‐fixed reference and an earth‐fixed reference, is used to describe the geometric transformation. The illustration can be seen in **Figure 1**.

The kinematic model studies about the relationship between inertial position of an AUV and velocity of an AUV. First, define the vector of position, vector of velocity, and vector of force as shown in Eq. (1) [27]:

where

where

Meanwhile, the acceleration during a motion is studied by the dynamics model. The dynamic model is developed from the Newtonian and Lagrangian principles. This has been deeply discussed in Ref. [27]. The general dynamic model of the 6 DOFs AUV can be seen in Eq. (3):

where

where

Eliminate the external forces by the hydrodynamics effect as shown in Eqs. (5–7):

where

The dynamic model in Eq. (8) maintains Property 1, Property 2, and Property 3 [28].

## 3. Proposed control

This section discusses a proposed control which combines the super twisting sliding mode and region boundary scheme for a six‐DOF AUV. Some equations in region boundary are used in super twisting sliding mode control; hence, this method is explained earlier.

### 3.1. Super twisting sliding mode controller

Super twisting is a part of the high order sliding mode control (HOSMC). The basic idea of HOSMC is removing the chattering effect, increasing the accuracy for tracking trajectory, and at the same time maintaining the advantages of conventional SMC [16]. In the conventional SMC, the control law consists of discontinuous system to ensure a sliding regime and the error convergence in a finite time happens when the system is restricted in the sliding surface. However, the high switching frequency known as chattering effect in the output signal is produced, thus, the stability of the control system is disturbed [29]. Furthermore, the value of sliding surface cannot be zero if the switching error exists. For this reason, super twisting SMC is used to preserve the zero value of sliding surface although in the presence of switching error [30].

There are two components in the super twisting SMC, the derivative of the discontinuous sliding surface and the continuous function of the sliding variable. Formulating the continuous function is useful to handle the chattering effect produced by the discontinuous function. The super twisting SMC is shown in Eq. (9):

where

where

where

**Remark 3.1: **Equation (13) reaches the finite time convergence as long as

### 3.2. Region boundary scheme

The region boundary scheme works by replacing line‐based into region‐based trajectory and different shapes of region can be decided by choosing the appropriate function [24]. The objective region in an inequality functions is given as follows,

where **Figure 2**. The inequality function of **Figure 2** is given as

where

To calculate the energy consumption when the AUV tracks on the region, the inequality function in Eq. (14) should be modified by adding the potential energy, given in Eq. (16):

where

where

where

**Remark 3.2:** The region error will trigger an AUV toward the desired region. Once the AUV is inside the region, the gradient of potential energy,

### 3.3. Super twisting sliding mode controller with region boundary scheme

Equation of super twisting SMC with region boundary scheme is shown in Eq. (19):

where

The following equation is the reference vector according to the region error,

where

Then, multiplying

where

Substitute

**Remark 3.3:** Energy saving potential control drives the sliding surface converging to zero throughout the tracking mission, thus the AUV tracks inside the region.

Finally, super twisting SMC based on region boundary scheme is shown in Eq. (26):

**Theorem:** The control input

**Proof:** Propose a positive definite function of a Lyapunov candidate given in Eq. (27):

Define

Differentiating Eq. (28) with respect to time yields

where

where

(32) |

*V* is bounded since

### 3.4. Related control laws for comparative analysis

Because of the similarity of the proposed controller, some controller such as conventional sliding mode control (SMC), fuzzy SMC, and super twisting SMC are selected for the comparison purpose in the simulation. The equation of each comparison control is discussed as follows.

#### 3.4.1. Sliding mode control (SMC)

The function of SMC is shown in Eq. (34) [5],

The sliding surface

where

#### 3.4.2. Fuzzy SMC

Formula of fuzzy SMC is given in Eq. (38) [31]

where **Table 1**, where *gauss2mf* as the type of membership function with range value from 5 to **Figure 3**.

## 4. Results

The simulation utilizes an omni‐directional intelligent navigator (ODIN) type of an AUV developed by Hawaii University as it is easy and widely used for the simulation purpose, as well as a holonomic model so that the singularity can be avoided and capable to move in six degree of freedoms (DOFs) without any preference of direction [33, 34]. The ODIN has eight thrusters to support its movement and does not require a heading angle to achieve a certain position. The horizontal diameter of an ODIN is 0.63 (m), while the vertical diameter is 0.61 (m). Its dry weight is about 125 (kg). There are two types of trajectories, which will be used in the simulation, a conventional line trajectory and a region trajectory. The conventional trajectory is applied on the conventional sliding mode control (SMC), fuzzy SMC, and super twisting SMC, while the region trajectory determined as a spherical shape is applied on the proposed control. An AUV is placed in the initial position at

where **Table 2** shows the parameter's values for each controller.

The first result is shown about tracking performance to see whether the AUV tracks outside the trajectory. As shown in the result of the conventional SMC in **Figure 4(a)**, the AUV reached the desired position after it moved down and gave oscillations of around **Figure 4(b)**, also moved down before it reached the desired position, although the oscillations did not appear. A better performance was shown under the super twisting SMC which can be seen in **Figure 4(c)**. The AUV moved slightly straight toward the start tracking point from the initial position. The oscillations also did not appear under this controller. However, the proposed controller gave the best movement compared to the others. From **Figure 4(d)**, the AUV moved straight from the initial position to the start tracking point. In the perturbation time, it is shown that the higher the value of the perturbation, the further the AUV moved from its desired position. While **Figure 4(a)**–**(c)** show that the AUV was unable to maintain its position on the tracking line, the opposite result is shown in **Figure 4(d)**. The AUV remains inside the region boundary even though the perturbations are presented.

Next, results are discussed about error convergence and analyzed how long the controller takes the AUV to settle from the perturbation's effect. The time is counted from **Figures 5(a)** and **(b)** show that the *X*‐ and *Y*‐axes of conventional SMC and fuzzy SMC converged to zero faster than the *Z*‐axis in the beginning of the time. The *X*‐ and *Y*‐axes of conventional SMC took *Z*‐axis. Meanwhile, the *X*‐ and *Y*‐axes of fuzzy SMC required *Z*‐axis. Conventional SMC and fuzzy SMC took a longer time to converge to zero due to the increasing value of perturbations. The time required for the error convergence in the *X*‐ and *Y*‐axes was *Z*‐axis. Meanwhile, the super twisting SMC required **Figures 5(a)** and **(b)** show the conventional SMC and fuzzy SMC producing the chattering in the presence of perturbation. A small chatter is also shown by super twisting SMC in small graph of **Figure 5(c)**, while proposed control showed a stable signal as performed in small graph of **Figure 5(d)**.

Results of force and moment are accumulated to obtain the energy spent by the AUV when the mission is executed. Force is an energy to make an object, in this case an AUV, move linearly from its initial position. On the other hand, moment is a force to make an AUV rotate from its center. The value of the force and moment were collected within the perturbed time or when the perturbations started to disturb the AUV. The reason for selecting this specific time was to see the differences in the way each controller handled the same perturbation. High amounts of force and moment indicate the inefficiency of the controller. The total amount was calculated by using the two‐norm function or Euclidean distance and was considered as the energy spent by the AUV to accomplish the desired mission. As shown in **Table 3**, the proposed control spent the least energy at 356.72 N for force and at 65.02

Control | SMC | Fuzzy SMC | Super twisting SMC | Proposed control |
---|---|---|---|---|

Force | 442.45 | 440.13 | 463.45 | 356.72 |

Moments | 449.58 | 463.97 | 172.27 | 65.02 |

The last results discuss about the thrusters’ distribution. These data are collected to analyze the effort of the propeller to maintain the AUV at its desired position. The more effort spent by the propeller, the more power will be consumed. The ODIN had eight thrusters to move the AUV from one place to another. The eight thrusters were divided into two functions. Thrusters 1–4 were used to move the AUV horizontally, while thrusters 5–8 were used to move it vertically. **Figure 6** shows that the oscillation values of all the thrusters for conventional SMC and fuzzy SMC were similar. For super twisting SMC, more effort was required by thrusters 1, 3, 7, and 8 in the presence of disturbances, while the other thrusters showed constant oscillation. In the case of a proposed controller, all thrusters required more effort when the AUV was disturbed by perturbation.

It can be seen from the results that the proposed controller gave the best performance in terms of robustness and energy consumption. The first was referred from the value of the gain selection. The proposed controller needed smaller parameters value,

In terms of the time required for the error convergence, the proposed controller took the shortest time to converge to zero, followed by the super twisting SMC. The position of the AUV at each axis under the conventional SMC and fuzzy SMC could not converge simultaneously. In this case, the time requirement stops had to be calculated when the position of the AUV in all the axes converged to zero. Thus, the conventional SMC took the longest time to converge to zero, followed by the fuzzy SMC. It could be seen from the results of the tracking performance that the AUV managed to move within the region under the proposed controller in the presence of perturbation. On the other hand, the other controllers failed to maintain the position of the AUV on the desired tracking line.

Based on the energy consumption, the super twisting SMC spent the highest force, both under constant and sine wave Gaussian white noise perturbations. The conventional SMC and fuzzy SMC were in the middle position, while the proposed controller spent the least force. The proposed control saved up to 30.2% of linear force consumption. For the moment under the constant perturbations, the highest value was spent by the conventional SMC, followed by the fuzzy SMC, the super twisting SMC, and the proposed controller. Meanwhile, the highest moment value under the sine wave Gaussian white noise perturbation was spent by the fuzzy SMC, followed by the conventional SMC, the super twisting SMC, and the proposed controller. In this case, the proposed controller saved more than 50% of the moment consumption.

More force was spent by the super twisting SMC and the proposed controller during the transition from the initial point. Therefore, the AUV was enabled to move directly to the start tracking point. These controllers also reacted to the presence of perturbations. The higher the value of the perturbation that disturbed the AUV, the greater the force that would be spent. In contrast, the conventional SMC and the fuzzy SMC showed a similar pattern of force and moment in all the times and in all conditions. This meant that these two controllers were not able to adapt well in handling changes in the situation.

The last discussion is about the propulsion or distribution of the thrusters. The proposed controller produced the least effort during the mission, followed by the super twisting SMC. Meanwhile, the conventional SMC and the fuzzy SMC showed the most active propulsion in all conditions. This situation was not good due to the battery consumption of the AUV. The more active thrusters indicated that the AUV would lose battery power easily. The high value of propulsion of the thrusters was also not good for the electrical devices inside the AUV.

## 5. Conclusion

A new robust‐region‐based controller is introduced from a survey of existing robust controls and saving energy approach for an autonomous underwater vehicle (AUV). Lyapunov candidate was used to prove a global asymptotical stability, while some simulations involving conventional sliding mode control (SMC), fuzzy SMC, and the only use of super twisting SMC were conducted under two kinds of perturbations to observe the effectiveness of the proposed controller. It is shown that the use of proposed controller was able to keep the AUV within the desired region under certain value of constant perturbations as well as a sinusoidal perturbation with a Gaussian white noise. From the results, it can be concluded that the proposed controller was able to minimize the chattering effect, provide a good response when overcoming the disturbances, provide a short computational time of error convergence, and save the amount of force and moment.