Complexity

Volume 2018, Article ID 4180275, 12 pages

https://doi.org/10.1155/2018/4180275

## Coordinated Depth Control of Multiple Autonomous Underwater Vehicles by Using Theory of Adaptive Sliding Mode

^{1}School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an 710072, China^{2}National Key Laboratory of Underwater Information Processing and Control, Xian, Shaanxi 710072, China

Correspondence should be addressed to Yintao Wang; nc.ude.upwn@oatniygnaw

Received 16 July 2018; Revised 5 October 2018; Accepted 8 November 2018; Published 2 December 2018

Guest Editor: Zhaojie Ju

Copyright © 2018 Yintao Wang and Yani Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper addresses the coordinated depth control problem of multiple autonomous underwater vehicles, which means to maneuver a group of underwater vehicles which move at the same depth synchronously. Firstly, a coordinated error of depth between vehicles and the common desired depth is defined by using extended graph theory in a distributed manner; then a deep-pitch double loop control algorithm based on sliding mode is designed for each vehicle, by which each vehicle is driven to and move at the common depth coordinately. In particular, a pitch reference command is firstly calculated by the predefined coordinated depth error, which can be regarded as the outer loop control, and then, the input rudder angle for each vehicle is derived according to the pitch reference command being as the inner loop control. Considering the uncertainties of the model hydrodynamic parameters, an online parameter adaptive algorithm is introduced to improve the performance of the sliding mode control algorithm proposed. Simulations were performed to verify the theoretical results proposed.

#### 1. Introduction

With the development of artificial intelligence, robots play more and more roles in our daily lives. The high demand of industrial applications and automation devices always needs the robot operate in a complex environment which is posing requirements for the system stability, safety, and strong challenges for the controller design [1–3]. As a typical underwater robot, Autonomous Underwater Vehicle (AUV) plays great roles in ocean exploring missions and even more in the field of militaries. In order to fulfill these applications, it is usually necessary to maneuver an AUV cruise at a fixed and expected depth; this task is always called the vertical plane control of AUV, which is a typical motion form in the field of marine crafts [4]. Many works have been done to develop strategies capable of depth control of a single AUV, mainly include fuzzy control [5], neural networks [6], and sliding mode [7, 8]. Although significant progress has been made in the area, however, much work remains to be done to develop strategies for multiple AUVs. Considering the fact that current missions are getting more complex and AUV technology matures, it is common to require multiple AUVs to work cooperatively to solve these types of tasks with low cost, high adaptively, and easy maintenance.

Consensus theory and algorithms have proven to be effective tools to perform the network-wide distributed computation tasks such as computing aggregate quantities and functions over networks. There have been much works on consensus problems of first-order agents, such as consensus under time-varying topology [9], finite-time consensus [10], consensus over random networks [11], and asynchronous consensus [12]. Taking into account the fact that many vehicles such as AUVs considered in this paper and mobile robots are always controlled directly by their accelerations rather than by their velocities, hence it is also necessary to investigate consensus problems of second-order agents. In [13, 14], the authors studied conditions on the interaction graph and the control gains for two different consensus algorithms to ensure agreement on both positions and velocities. Despite significant progress has been made in aforementioned works, much work remains to be done for the AUV systems due to the fact that the dynamics of AUV are often complex and cannot be simply ignored or drastically simplified for control design purpose. In the field of marine vehicles, authors in [15] proposed a synchronized path following controller for fully actuated surface ships by using passivity theory. A decentralized formation controller was derived in [16], which deals with the cooperative problem of fully actuated surface vehicles with considering the influencing of the sea currents and model uncertainty. Even all the results proposed in the above woks are decentralized because each member of the group only needs to exchange the necessary information under local interactions, but all of them assumed that there is a global reference speed for the whole group which should be known to all the team members. In this sense, the controller proposed is not distributed because there is a global vector which should be known to all of them.

Motivated by the ideas of consensus tracking and aforementioned works, we consider the truly distributed depth coordinated control problem of AUVs in this proposal. By using extended graph theories, the coordinated depth error between each AUV and the common desired depth was defined firstly, in which we only need the common desired depth be available to one subset of AUVs by local interactions, so the strategy is truly distributed and this can be regarded as our first contribution. Moreover, a double loop based on adaptive sliding mode control algorithm for each AUV was derived with considering the model uncertainties of each AUV, and each AUV can move at the same depth coordinately by using the algorithm proposed. Furthermore, the control parameters used in the algorithm proposed are obtained from the kinematic model of the AUV and the expected dynamic characteristics of each link; except for the initial value of the model parameter vector, all other parameters are irrelevant to the AUV, so the strategy proposed has good application values.

The remainder of this paper is organized as follows: in Section 2, models of the vehicles considered in this work and extended graph theory which will be used throughout this paper are described. We also presented the basic principles of the synchronized depth control by defining a coordinated error vector novelty in this section. Sections 3 and 4 especially present the procedures of designing the sliding mode control for the depth tracking with and without considering the model uncertainties. In Section 5, simulation examples are executed to validate the effectiveness of the strategy proposed. Finally, conclusions and future works are summarized and discussed in Section 6.

#### 2. Problem Formulation

##### 2.1. AUV Model in Vertical Plane

The six degrees of freedom dynamic model of AUV can be decomposed into vertical plane motion and horizontal plane motion independently if some coupling constraints are satisfied. The depth control considered in this paper is a typical freedom in vertical plane, so before going on, we will give the AUV model in vertical plane borrowed from [4]where denotes the position and pitch of AUV in earth-fixed frame and represents the speed in body-fixed frame with being the transformation matrix from frame to . , , , and are the inertial matrix, Coriolis force matrix, damping matrix and gravity, and buoyancy generalized force vector with the following form, respectively:where is the combined moment of inertia with additional mass produced and the rotation inertia of the y-axis and is the rotation torque coefficient produced by rudder angle . Due to the fact that when AUV makes down steering which will produce the negative pitch angular acceleration , therefore, . denotes the control force and torque for AUV which will be designed in following sections to enable each AUV move at the desired depth.

From the above formulations, we can find that the diving equations of AUV motion should include the heave velocity , the angular velocity in pitch , the pitch angle , and the depth . Assume that the forward speed is constant and that the sway and yaw modes can be neglected, this suggests the following simplified equations of AUV motion in vertical plane aswhere denotes the bounded total modeling errors and external disturbance. Based on parameterized linearization techniques, the dynamic equation of pitch angular velocity can be rewritten aswhere

To be noted that, all the AUVs considered will be isomorphic in this paper, so for the sake of convenience, we omitted the index of the AUV number in the presentation of the model, and furthermore, in the implementing of the controller designing, we will make the following assumptions.

*Assumption 1. *The depth can be measured by a pressure meter and the pitch angle can be measured by an inclinometer while the pitch rate requires a rate gyro or a rate sensor. We also assumed that the velocities in the vertical plane can be measured by using a Doppler log for directly obtaining velocity measurements.

*Assumption 2. *The external disturbance is bounded by an unknown upper bound, and is bounded apparently. We assumed that , where to be the unknown upper bound of disturbance [17].

##### 2.2. Extended Communication Graph

Assumed that there are AUVs in the group, it is a conventional way to model the interactions between them using directed graph adjacency matrix and topology graphs [18]. Firstly, we introduce a virtual AUV, denoted as , which specifically represents the desired state. For the AUVs system, the graph contains a set of nodes , and an adjacent matrix , where indicates that the th AUV can obtain state information from th, otherwise . Then we define a degree matrix with the elements . The Laplacian matrix of graph is expressed as and the normalized Laplacian matrix of is , where is the normalization of when ; otherwise [19]. If we add one virtual AUV to the group, then we call the graph as* extended communication graph*. An example of* extended communication graph* is given below; where Figure 1 is a topology graph with four nodes which contains three AUVs and one virtual AUV, its adjacency matrix, degree matrix, normalized adjacency matrix, Laplacian matrix, and normalized Laplacian matrix are, respectively,