# Adaptive Tracking Control of an Autonomous Underwater Vehicle

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## Abstract

This paper presents the trajectory tracking control of an autonomous underwater vehicle (AUV). To cope with parametric uncertainties owing to the hydrodynamic effect, an adaptive control law is developed for the AUV to track the desired trajectory. This desired state-dependent regressor matrix-based controller provides consistent results under hydrodynamic parametric uncertainties. Stability of the developed controller is verified using the Lyapunov’s direct method. Numerical simulations are carried out to study the efficacy of the proposed adaptive controller.

## Keywords

Autonomous underwater vehicle (AUV) adaptive control law regressor matrix Lyapunov’s stability path following## 1 Introduction

Over the last two decades, research on control of autonomous underwater vehicles (AUVs) has become an important topic due to their wide applications in security patrols, search and rescue in hazardous environments, etc.[1, 2, 3]. In military missions, a group of AUVs are required to maintain a specific formation for an area coverage and reconnaissance. Remotely operated vehicles (ROVs) and AUVs are directly involved in exploitation of resources located at deep oceanic environment[4]. AUVs are also used in risky and hazardous operations such as bathymetric surveys, oceanographic observations, recovery of lost man-made objects and ocean floor analysis.

The trajectory planning and tracking of an AUV is an important research topic. During traversing in a computed path, an AUV provides real-time data to be compared with the designed model[5]. The path is planned by considering two points such as start point and a destination point with a velocity field, where the path is to be traversed within a minimum time[6, 7]. The optimum path is efficiently generated using path parameterization, cost function techniques and minimum expenditure of energy[8]. The coordinated path following problem of marine craft was proposed in [9], where the path following situation was discussed based on the convergence of geometric errors at the origin of each vehicle. The problem of path planning was solved based on fast marching algorithm in [10], where the path tracking algorithm was designed by assuming the AUV to maneuver on a fixed depth. The AUVs were given a task of mine counter measure (MCM) based on synthetic aperture sonar (SAS) data collected at sea[11]. The path following problem was solved by getting inspiration from biological agents in [12]. This bio-inspired model in horizontal plane was capable of generating real time smooth forward and angular velocities. The path following and trajectory tracking problems were solved by a backstepping approach using a special type of kinematics which was developed by Lyapunov’s direct method, and then it was extended to solve the dynamic problem in [13]. The problem of trajectory planning and tracking control was solved by using error dynamics of AUV in the horizontal plane in [14], in which the closed loop tracking controller was developed using backstepping techniques. An adaptive controller was also developed for controlling the nonholonomic mobile robot in [15, 16], where the learning method on neutral network is used to design the robust adaptive controller. In [17], a fuzzy adaptive linearized controller with backstepping-like feedback was used to control the wheeled mobile robot for tracking of the desired trajectories. An adaptive sliding mode controller was used to control the uncertainties accompanied by the camera mounted on the nonholonomic dynamic mobile robot in [18]. A nonlinear iterative sliding mode incremental feedback controller that was developed to track an AUV along the desired trajectories in the horizontal plane is described in [19]. An adaptive neural network controller based on dynamic surface control (DSC) and minimal learning parameters (MLP) was proposed in [20]. The problem of trajectory tracking of AUV was solved in [21] by using sliding mode controller combined with line of sight (LOS) and cross track error methods. The adaptive controller along with radial basis function neural network (RBF-NN) was employed for controlling AUVs to follow the desired trajectories in [22]. A static output feedback controller was proposed in [23] for controlling an AUV to track along the desired trajectory. A virtual vehicle concept that is used to develop the backstepping controller for controlling unmanned underwater vehicle (UUV) to track along the desired trajectory was presented in [24]. A feedback controller based on LOS in the presence of oceanic current disturbances was developed in [25] to control the AUVs for tracking the desired trajectories. A region boundary adaptive controller developed for trajectory tracking of AUV was presented in [26], where the entire boundary was united by the use of multiplicative potential energy functions. In [27], a backstepping controller with a LOS guidance system was used to control an AUV to track the desired trajectory in the presence of ocean current, in which the unknown parameters were estimated by using parameter rejection techniques. A new frame work for object detection and tracking method of an AUV was presented in [28]. This controller was designed for a vision-based AUV which carries a forward looking sonar. In [29], a backstepping adapting controller was developed for an AUV to track the desired path. The stability of the developed controller was proved by using a Lyapunov’s criterion.

Path planning of different autonomous agents such as marine crafts, mobile robots and AUVs has been reported in literature. Tracking of an AUV along a desired trajectory is an important task. It is difficult to develop a controller for an AUV in the dynamic environment where hydrodynamic uncertainties exist. Under these difficulties, the focus of this paper is to develop a trajectory tracking control law of the AUV. The contribution of this paper lies in the development of an adaptive controller such that the AUV can track the desired trajectory in the dynamic environment in the presence of uncertainties owing to hydrodynamic parameters.

In this work, we assume that while moving in a horizontal plane, the dynamic behavior of an AUV is similar to that of an under-actuated surface vessel. Also the orientation of the AUV is similar to that of the surface vessel. It is intended to design a control law such that the AUV can track a desired trajectory accordingly.

In the development of the controller for an under-actuated AUV, we consider four degrees of freedom (DOF) for simplicity. We propose an adaptive controller using a regressor matrix consisting of unknown parameters. The stability of the developed control law is verified using Lyapunov’s stability criterion. From the numerical simulations presented, it can be seen that the proposed controller is efficient in providing appropriate adaptive control action to force the AUV to follow the desired trajectory in the presence of hydrodynamic parameter uncertainties.

Some challenges involved with this approach, as compared to some existing approaches[14, 19], are as follows. The dynamics of the AUV is nonlinear and uncertain, and the effects of Coriolis, gravitational and damping factors in six degrees of freedom increase the mathematical complexity. Thus, designing an adaptive control law and its implementation are challenging. To ensure the stability of the proposed controller by choosing the Lyapunovs function is also difficult.

The rest of the paper is organized as follows. Section 2 presents the problem statement. AUV kinematics and dynamics are discussed in Section 3. The development of a new adaptive control law based on regressor approach for the AUV to achieve tracking of a desired trajectory is developed in Section 4. In this section, the stability of the developed adaptive control law is verified by using Lyapunov’s stability criterion. To verify the efficiency of the proposed control law, simulation studies are made and results analyses are provided in Section 5. Conclusions are presented in Section 6.

## 2 Problem statement

*η*= [

*x*,

*y*,

*z*,

*φ*,

*θ*,

*ψ*]

^{T}is the position and orientation vector in the inertial frame;

*x*,

*y*,

*z*are the coordinates of position, and

*φ*,

*θ*,

*ψ*are orientations in the longitudinal, transversal and vertical axes, respectively.

*η*

_{ d }= [

*x*

_{ d },

*y*

_{ d },

*z*

_{ d },

*φ*

_{ d },

*θ*

_{ d },

*ψ*

_{ d }]

^{T}is the position and orientation vector of the desired trajectory;

*x*

_{ d },

*y*

_{ d },

*z*

_{ d }are the coordinates of the desired position, and

*φd*,

*θ*

_{ d },

*ψ*

_{ d }are orientations in the desired longitudinal, transversal and vertical axes, respectively.

## 3 AUV kinematics and dynamics

*ν*is the velocity vector in the body-fixed frame,

*u, v*and

*w*denote linear velocities;

*p, q, r*are angular velocities;

*X, Y, Z*are forces;

*K, M, N*denote moments.

*τ*is the vector of forces and moments acting on the AUV in the body-fixed frame.

*M*(

*η*) is the inertia matrix including added mass, \(C(\eta, \dot \eta )\) is the matrix of Coriolis and centripetal terms including the added mass. \(D(\eta, \dot \eta )\) denotes the hydrodynamic damping and lift matrix,

*g*(

*η*) is the vector of gravitational forces and moments, and

*J*(

*η*) is the velocity transformation matrix between AUV and earth fixed frames.

*J*(

*η*) can be obtained as

**Assumption 1**. Due to asymmetric body structure of the AUV, deriving a control law is very difficult. For sake of convenience, the following assumptions are made.

- 1)
Center of mass (CM) and center of buoyancy (CB) coincide with each other.

- 2)
Mass distribution all over the body is homogeneous.

- 3)
The hydrodynamic terms of higher order as well as pitch and role motions are negligible.

## 4 Path following control law

*η*

_{ d }= [

*x*

_{ d },

*y*

_{ d },

*z*

_{ d },

*φ*

_{ d },

*θ*

_{ d },

*ψ*

_{ d }]

^{T}is the position and orientation vector of the desired trajectory in the inertial frame. Consider the control law in the following form:

*Y*\(\hat \alpha \) is the feed forward term,

*K*

_{ D }

*s*is a simple propositional differential (PD) term.

*K*

_{ D }is a positive definite gain matrix and the error vector

*s*is defined as

It will be shortly proved that, the proposed parameter adaptation law given in (11) is able to drive the AUV in the desired trajectory and the stability of the AUV system is ensured.

*V*(

*t*) satisfies (12) and is given by (13)

We will prove that the system is asymptotically stable in the sense of Lyapunov if (12) is satisfied.

**Proof (Stability of the proposed control law)**. Let

*V*(t) be a Lyapunov candidate function and be given by

*Ms*and \({V_2}(t) = {1 \over 2}{{\tilde \alpha }^{\rm{T}}}{\Gamma ^{ - 1}}\tilde \alpha \).

Analysis of second term of (14):

Equation (26) satisfies the Lyapunov’s stability criterion for AUV dynamics with a stable controller. Hence the proposed adaptive control law (9) yields a stable closed system. □

*τ*= [

*τ*

_{ X },

*τ*

_{ y },

*τ*

_{ Z },

*τ*

_{ R }]

^{T}.

*α*is defined as

Using the regressor matrix \(Y({\nu _d},{{\dot \nu }_d})\), the AUV model (27) can be expressed in a linear parametric form. The derivation of the regressor matrix of a high-DOF is very tedious. Hence, only four DOFs are considered for simplicity. For real-time realization, the regressor matrix is to be computed in each control cycle; also the computation of each parameter of the matrix is performed in each iteration[32]. The regressor matrix is a state dependent matrix. The elements of this matrix contain velocity and acceleration terms which are computed in each iteration of the control cycle.

## 5 Results and discussions

To verify the efficacy of the proposed control law, numerical simulations are carried out. Consider various desired trajectories and conditions as described next.

**Case 1**. Desired circular path. We consider a desired circular path as

Parameters of AUV used for simulation

Mass (kg) | Damping coefficients (kg/s) | Other parameters |
---|---|---|

| | |

| | Λ=diag{50, 10, 20, 20} |

| | Γ = diag{200, 10, 20, |

| | 0, 0, 0, 10, 10} |

Next, we present the results obtained as follows.

The comparison of the desired and actual positions as well as orientations of the AUV in the earth fixed inertial frame of reference is shown in Fig. 4(b). From this figure, it is observed that the actual and desired positions matched with each other in 10 s.

*x*,

*y*,

*z*,

*ψ*directions. From this result, it is observed that the position errors converged to zero. Thus the proposed controller steered the AUV to track the desired path accurately.

**Case 2**. Desired spiral path. Next we consider a spiral path in the space as

As in the case of circular path, for the spiral path the derivatives up to the second order, i.e., \({{\dot x}_d},{{\dot y}_d},{{\dot z}_d},{{\dot \psi }_d},{{\ddot x}_d},{{\ddot y}_d},{{\ddot z}_d},{{\ddot \psi }_d}\) are also necessary. The other parameters such as *m* _{11}, *m* _{22}, *m* _{33}, *m* _{44}, *d* _{11}, *d* _{22}, *d* _{33}, *d* _{44}, *K* _{ D }, Λ and Γ are given in Table 1.

## 6 Conclusions

In this paper, a new adaptive control law for trajectory tracking of AUV moving in space is addressed. The adaptive control law is developed with estimation of uncertain parameters associated with the hydrodynamic damping coefficients. This law is used to generate appropriate control for the AUV to track the desired trajectories. The stability of the control law is verified using Lyapunov’s direct method. By comparing the desired positions and velocities with the actual positions and velocities, respectively, it is found that all the errors are converge to zero quickly. From the numerical simulation results, the efficacy and accuracy of the developed control law are verified.

## Notes

### Acknowledgements

The authors wish to thank the editor and reviewers for their suggestions to improve the quality of the paper.

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