Complexity

Volume 2018, Article ID 4975187, 8 pages

https://doi.org/10.1155/2018/4975187

## Consensus Path-following Control of Multiple Underactuated Unmanned Underwater Vehicles

^{1}School of Marine Science and Technology, Northwestern Polytechnical University, Xian 710072, China^{2}National Key Laboratory of Underwater Information Processing and Control, Xian, Shaanxi 710072, China^{3}Jiangsu Automation Research Institute, Lianyungang, China

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

Received 12 June 2018; Accepted 9 September 2018; Published 5 November 2018

Guest Editor: Junpei Zhong

Copyright © 2018 Yintao Wang and Yao Yao. 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 work mainly studies the problem of how to steer a group of underactuated unmanned underwater vehicles to specified paths coordinately. The algorithm proposed consists of a single path-following strategy and a path parameter consensus tracking strategy. In the context of single path following, we describe the path to be followed by an arbitrary scalar, then by using Lyapunov and backstepping theories, a single path-following strategy was derived to drive each UUV move to the predefined path asymptotically. In the coordinated level, we focus on the coordination for the scalar parameters. In particular, we show that all the path parameters can track with a virtual reference leader who is a neighbor of only a subset of following UUVs with local interactions. The stability of the closed system was proved and analyzed theoretically. The validity of the algorithm proposed is supported by simulation results.

#### 1. Introduction

With the development of artificial intelligence, robots have been widely used for complex tasks such as handling searching and rescuing in a dangerous environment. In [1], the teleoperated robots working in dynamic and unstructured environments to perform complex tasks were considered. In this paper, neural networks methods were used to deal with the uncertainties of the telerobot model to ensure a guaranteed performance; moreover, extreme learning machine-based control scheme and combined radial basis function with neural networks techniques were developed to compensate for unknown nonlinearity in the manipulator dynamics and communication delays in [2, 3], respectively. Compared to telerobots considered in the aforementioned works, unmanned underwater vehicle (UUV) works in a more complex environment and with more uncertainties. In order to fulfill these applications, we always need to maneuver an UUV along a desired geometric path. Typically, there are two ways to define the expected geometric path for an UUV should move to [4]. These two methods are called trajectory tracking and path following, respectively. In the context of trajectory tracking, it always requires the UUV to converge to the expected path determined by specific times. However, we usually need one UUV to be driven to spatial positions as the main goal rather than dynamic behaviors described by time, and this is the main characteristic of the concept of path following.

The key idea of path following is how to define the path to be followed and assign the according reference position for each vehicle. In [5], the reference point is selected as the closest point to the current position of the vehicle. This method only needs to control the error in a normal direction relative to the path and consequently allowing a quick convergence to the path; however, this method demands that the initial position of the vehicle must be located in a circle with a radius of curvature centered at this closest point. To relax this conservative and necessary condition, Lapierre and Jouvencel [6] took an arbitrary point as the reference point and describe the path by the curvilinear abscissa of this point; the algorithm proposed in this paper took this point as a virtual leader and derived an updating law for the curvilinear abscissa of this point which enables following errors to converge to zero asymptotically. References [7, 8] considered the single path-following problem of the underactuated autonomous underwater vehicle in 3D case, respectively. In [7], external disturbances and vehicle’s model uncertainty are both considered using Lyapunov direct method, backstepping, and parameter projection techniques. The results show that path-following errors can be made arbitrarily small. In [8], the problem was decoupled to a 3D guidance law and a heuristic fuzzy control law; both internal and external uncertainties are considered in this work.

Sometimes, we need a group of vehicles to work together to complete complex tasks coordinately, which is called cooperative control [9] and has received more and more attention from the researchers. An important theory and tool to analyze the cooperative behavior is a consensus [10], which means to reach consent on their common states via information communications between the members in the group. Motivated by the idea of consensus, [11] considered the cooperative path-following problem in the field of ground robots. In the field of marine vehicles, authors in [12] proposed a synchronized path-following controller for fully actuated surface ships by using the passivity theory. We also proposed a formation control law for full actuated autonomous underwater vehicles in [13] by using passivity techniques. A decentralized formation controller was derived in [14], which deals with the cooperative problem of fully actuated surface vehicles with consideration of the influence of the sea currents and model uncertainty. Although the aforementioned works have considered about the cooperative path following of marine vehicles, all the vehicles are fully actuated rather than underactuated ones. Considering the fact that underactuated vehicles have fewer actuators than degrees of the freedom, there exist more challenging problems for controller designing because of the nonholonomic constraints. Reference [15] considered the coordinated straight path-following problem for underactuated unmanned surface vehicles. All the vehicles are controlled to converge to the accordingly assigned straight line path by the path-following controller based on line of sight guidance. To move synchronously, the surge speeds of each vehicle are adjusted by the distance along paths. By using nonlinear cascaded systems theory, the stability of the closed system was proved and analyzed. References [16, 17] also executed researches on the coordinated path following of underactuated autonomous vessels. In the coordination level, the time delay between them to exchange the coordination vectors is considered and discussed. Even all the results proposed in the above works are decentralized because each member in the group only needs to exchange the necessary information under local interactions, all of them assumed that there is a global reference speed for the whole group that should be known to all the team members. In this sense, the controller proposed is not distributed because there is a global vector that should be known to all of them.

Motivated by the ideas of consensus tracking and aforementioned works, we consider the truly distributed path-following problem of underactuated unmanned vehicles in this proposal. Firstly, we take an arbitrary variable rather than the curvilinear abscissa like [6–8] to describe the desired path; followed by Lyapunov and backstepping techniques, the path-following errors were driven to zero asymptotically, and this can be regarded as the first contribution of our work. Furthermore, to ensure cooperative path-following behaviors of the vehicles, we proposed a consensus tracking law for the speed of each UUV, by which the reference point of each vehicle was driven coordinately. Compared with [12, 14–17], in our consensus tracking level, we only need the group reference speed to be available to one subset of the following UUVs rather than all of them. So the strategy proposed is truly distributed, and this can be regarded as the second contribution of this work.

The remainder of this paper was organized as: in Section 2, the problems to be studied were introduced and formulated. In Section 3, the kinematic and dynamic model of UUV was presented, and the single path-following controller for each UUV was designed using Lyapunov and backstepping methods; in the coordinated level, a path parameters consensus tracking strategy based was derived for multiple vehicles. To validate the performance of the method proposed, simulation examples are executed and discussed in Section 4. Finally, a brief conclusion is given in Section 5.

#### 2. Problem Formulation

The main objective of path following is to enable the vehicle to converge to and move along the specified path with a dynamic behavior. In this paper, we mainly talked about this problem in planar 2D. The position and velocity of the th vehicle can be denoted by and , where is the index of the vehicle number, and denotes the total numbers of vehicles in the group. The size and orientation of the velocity vector can be characterized by and , respectively.

Let the desired path be represented as , which was parameterized by . Then the consensus path-following problem considered in this paper can be formulated as where (1) and (2) can be regarded as the single path-following objective which requires each individual vehicle to converge to and move along the desired path with a dynamic speed . Equation (3) is the coordinated objective which ensures all the path parameters achieve consensus, then the reference points determined by these parameters can be updated synchronously.

#### 3. Consensus Path-following Controller Design

##### 3.1. Path following for Single UUV

Firstly, we consider path following for one single UUV. Before going on, we give the dynamic model of the UUV in the horizontal plane borrowed from [18]. where denotes the position and heading of UUV in earth-fixed frame ; represents the speed in a body-fixed frame ; means the transformation matrix from frame to ; are mass and hydrodynamic added mass terms, and denote hydrodynamic terms. From (5), we can find that there are two control inputs , , but there are 3 freedoms to determine the specific spatial location of one UUV. So the UUVs considered in this paper are underactuated systems. Before moving on, we make some common assumptions to specify the model of the UUVs considered in this paper: (1)We do not consider the influence of the environmental disturbance, and we also assume that all model parameters are determined without any uncertainty(2)The position and heading states of each UUV can be captured by inertial navigation sensors and compass system, and the velocity can be measured by Doppler sensor(3)All the UUVs are equipped with sonar systems, which enable UUV to have the capabilities to communicate with each other

Let the desired path for the th vehicle be described by a given variable as we have introduced in Section 2, define a path frame with being the origin, specifically, define axis along with the tangent and axis along with normal direction, and name this frame as . Then, the relative position between the vehicle and the reference point can be depicted as shown Figure 1.