Collective Dynamics and Control for Multiple Unmanned Surface Vessels
Abstract
A multiunmanned surface vessel (USV) formation control system is established on a novel platform composed of three 1.2 meterlong hydraulic jet propulsion surface vessels, a differential GPS reference station, and intervessel Zigbee communication modules. The system is also equipped with an upper level collective multiUSV protocol and a lower level vessel dynamics controller. The system is capable of chasing and surrounding a target vessel. The results are supported by rigorous theoretical analysis in terms of asymptotical surrounding behavior and trajectory regulation. Extensive experiments are conducted to demonstrate the effectiveness and efficiency of the proposed hardware and software architectures.
I Introduction
Unmanned surface vessels (USVs) have extensive applications in marine resource exploration, water pollution clearance, disaster searching and rescue, marine patrol and prospection, for their lowcost, high efficiency, agility and flexibility. Most existing research on USVs focuses on a single vessel. As representative works, recurrent neural networkbased predictive controllers were designed in [1, 2] to address the nonlinearity of the USV dynamics. Trajectory tracking controllers were proposed for path planning of USVs subject to input saturation, system uncertainties, and wind/wave disturbances in [3, 4, 5, 6, 7].
With the tremendous development over the past years, multiUSV systems have become indispensable tools for developing marine economic, protecting marine environment, and preserving marine rights. In particular, a single USV is far less capable than a multiUSV formation, especially in fulfilling complex tasks of patrol, rescue, smuggle seizing, water pollution clearance, and material delivery. For example, in harsh marine environments with severe external disturbances, a single USV is more vulnerable than a multiUSV setting where one malfunctioned USV can be replaced and/or rescued by another.
In the field of formation control of multiple unmanned vehicle/robot/vessel, called a multiagent system (MAS) in general, these years have witnessed many research outcomes, including lattice flocking in [8], a secondorder CuckerSmale model in [9] and its prediction version in [10, 11], homogeneous and heterogeneous collective circular motion control protocols in [12, 13, 14], an arbitrary collective closed envelope motion control scheme in [15], and formation control protocols for EulerLagrangian systems in [16]. More results can be referred to in the survey papers, e.g., [17, 18].
Especially on formation control of multiUSVs, the representative works are discussed as follows. A slidingmode formation control scheme was designed in [19] for USVs to form arbitrary formations. A coordinative control protocol governing a multiUSV system was developed in [20] to a desirable stationary formation with identical orientations. Formation control of USVs in the presence of uncertainties and ocean disturbances was studied in [21]. Based on a fuzzy estimator, a distributed constrained control law was proposed in [22] for multiple USVs guided by a virtual leader moving along a parameterized path. A smooth timevarying distributed control law was proposed in [23] that assures that a multiple USV can globally exponentially converge to a desirable geometric formation. The objective of this paper is to drive a team of vessels to surround a target vessel within their convex hull, which is different from the aforementioned formation control. A relevant theoretical work can found in [24] where the vehicles are initially placed within a circle and/or using a predefined standoff distance between the vehicles and the target. A novel kinematic control scheme is proposed in this paper that does not require such an initial setup.
Most of the aforementioned works focus on formation control protocols of kinematic models, but not taking complicated surface vessel dynamics into consideration. It is of great theoretical challenge to consider the complicated interaction of an upper level collective multiUSV protocol and a lower level vessel dynamics controller. Specifically, this paper answers how to achieve the upper level collective behavior subject to the regulation error from the lower level controller, as well as how to drive the regulation error to zero exponentially for a specified trajectory from the upper level.
Also, theoretical research has rarely been tested in real environment due to the challenging practical issues in establishment of a real experimental platform. Rare relevant results can be found in [25, 26, 27, 28] where experiments were conducted on real water surfaces including rivers, lakes and seas. These works however focus on a single USV. In this paper, we aim to test the design in a real lakebased multiUSV formation control platform that is composed of three 1.2 meterlong jetpropelling vessels equipped with onboard differential GPS receivers and imaging processing modules, located at Songshan Lake, Guangdong, China.
Ii Modeling
Consider a multiUSV system consisting of vessels. Let . Denote the complete position distribution of the system as , where , , represents the Cartesian coordinates of the th vessel. Denote be the convex hull of , , , that is,
Also, let
(1) 
be the distance between a point and . Obviously, if and only if .
The kinematics model for each vessel is given as follow
(2) 
for a rotation matrix
(3) 
In the model, , and represent the forward (surge) velocity, the orientational (yaw) angle and the transverse (sway) velocity, respectively, as illustrated in Fig. 1. Denote the orientational (yaw) angular velocity .
A complete but complicated nonlinear dynamics model has been proposed in literature based on physical principles with the simplified hydrodynamic effects; see, e.g., Eqs. (46) of [29]. The model was identified with the nominal forward speed up to 20 knots. Therein, several simplified linear variants of the dynamic equations and some control design approaches are also discussed for trajectory tracking including cascaded PD and backstepping control. With the same objective “to obtain a model that is rich enough to enable effective motion planning and control, simple enough for experimental identification, and general enough to describe a variety of vehicles operating over a large range of speeds,” we use the following equations for the dynamics of vessels used in the paper, for ,
(4) 
where the two control variables are the propeller speed and the steering angle . Denote . In particular, this model is given for the vessels working in a medium speed mode (13m/s). In this model, we ignore the highorder nonlinearities except the cross nonlinearity and in the second and fourth equations of (4). This simplification is based on extensive experiments and data matching.
Iii Problem Formulation and Controller Framework
The main technical challenge in multiUSV system control is to propose a decentralized protocol that achieves the specified collaborative behavior through the control to each vessel’s dynamics model. Some preliminary manipulation is first introduced in this section.
Let , , and be the desired signals for , , and , respectively, in the kinematics model (2). Denote
(5) 
Direct calculation shows that
(6) 
with
(7) 
The control design framework of this paper consists of the following two steps.

(Upper level collective protocol) To design the desired , , and , for the kinematics model (6) such that the multiUSV achieves a desired collective behavior, subject to the perturbation approaching zero.

(Lower level vessel dynamics control) To design the actuator input for the dynamics model (4) such that , , and achieve the desired , , and given in (i), in particular, with approaching zero.
The technical objective of this paper is to propose solutions to the two steps. A direct conclusion is as follows, with the two steps in the aforementioned framework solved, the closedloop system composed of (4), (6) and the actuator input , achieves the desired collective behavior specified in step (i). To be more specific, two collective behaviors, i.e., surrounding control and equally surrounding control, are studied in this paper, The rigorous definitions are given below.
Definition 1
A target vessel position is asymptotically surrounded by the vessels of the complete position distribution if
(8) 
Definition 2
A target vessel position is asymptotically equally surrounded by the vessels of the complete position distribution if it is asymptotically surrounded by them with
(9) 
for and .
Remark 1
For every two adjacent vessels, say and , the property (9), together with the geometric constraints, implies .
We propose two approaches in Cartesian coordinate and polar coordinate, respectively, to achieve different collective behaviors with different features.
Approach 1: For any signal to be designed and an arbitrary , let^{1}^{1}1For a vector , let be the angle of the complex number in the complex plan.
(10) 
where is the drift angle, is an integervalued signal such that , i.e., and a continuous implies a continuous in time .
Obviously, the aforementioned step (i) is solvable with an arbitrary and the desired and given by (10) if the following step is solvable.

To design a desired for the kinematics model (11) such that the multiUSV achieves asymptotically surrounding formation, subject to the perturbation approaching zero.
Approach 2: For a specified target vessel position , let
be the polar coordinate of the th vessel where is an integervalued signal such that , i.e., and a continuous implies a continuous in time .
For any signals , let
(12) 
and hence , , and given in Approach 1. Define
i.e.,
Note the following calculation
and
Then, the model (6) becomes
(13) 
Also, it is noted that approaches zero if approaches zero; approaches zero if approaches zero and is asymptotically lower bounded by a positive constant.
Obviously, the aforementioned step (i) is solvable with the desired , , and given by (10) and (12) if the following step is solvable.

To design desired and for the kinematics model (13) such that is asymptotically lower bounded by a positive constant subject to the perturbation approaching zero; and the multiUSV achieves equally asymptotically surrounding formation subject to the perturbation and approaching zero.
Iv Collective Control Design
This section aims to propose a controller for each vessel so that the multiUSV achieves the desired asymptotically surrounding formation in the sense given in (i) or (i).
Iva Asymptotically Surrounding Control
The main objective of this subsection is described in step (i). More specifically, it aims to design the desired for the kinematics model (11) such that a specified target vessel position (may be an enemy vessel) is asymptotically surrounded by the USV team, subject to the perturbation approaching zero.
To give the desired in a distributed manner, we define the set of neighbors of vessel as
with a specified distance . First, assume that is available for all vessels, then the control law for each follower is designed as follows, with and throughout the paper,
(14) 
Now, the main technical result is stated in the following theorem.
Theorem 1
Let
that is continuously differentiable and whose derivative is 0 for and
for . Let
whose derivative is, due to the symmetric property of the undirected graph,
Let
Analogously, one has
The derivative of , along the trajectory of (15), is
Denote
Direct calculation gives
As a result, is upper bounded, so is the state .
To prove the moreover part, let , and . Then,
that implies and hence (8). The proof is thus completed.
Next, we will investigate the decentralized scenario that is not available for all the vessels. In such a situation, a decentralized estimator is required for each follower vessel to estimate . Define as the set of vessels that can detect the target (i.e., leaders) and as the set of vessels that cannot (i.e., followers). Let be the set of communication neighbors of vessel , . The estimator is designed as follows:
(16) 
where is the estimate of for vessel . As a result, the controller (14) is modified as follows
(17) 
A similar statement still holds as in the following theorem.
Theorem 2
For the system (11) with exponentially and the controller (16) and (17) with , the states of the closedloop system are bounded if the network determined by , , is connected and the target can be detected by at least one vessel (i.e., ). Moreover, the target vessel position is asymptotically surrounded by the vessels in the sense of (8).
The network (16) is able to achieve , , exponentially, if the network determined by , , is connected and the target can be detected by at least one vessel. Let . One has , exponentially.
IvB Asymptotically Equally Surrounding Control
The main objective of this subsection is described in steps (i). More specifically, it aims to design the desired and for the kinematics model (13) such that a specified target vessel position is asymptotically equally surrounded by the multiUSV, subject to the perturbation and approaching zero. In (13), and are the radius and the moving angle of vessel , respectively, relative to the target . Define where is an integervalued signal.
The main purpose of is to drive all vessels to a circle of a specified radius . The controller for takes the following linear structure
(19) 
To put the desired in a distributed manner, we define the set of neighbors of vessel as
Then, is designed such that the angles of the vessels change along the negative gradient of an energy function to be specified later, that is,
(20) 
Theorem 3
The closedloop system can be rewritten as
(23) 
The proof of (21) is straightforward from the linear system property. To prove the phase distribution property (22), we define a potential function as follows
(24) 
that is continuously differentiable and whose derivative is 0 for and
for . Let
whose derivative is, due to symmetric property of the undirected graph,
Denote
Direct calculation gives
It is noted that is lower bounded and monotonic, so has a finite limit as . Together with the fact that is bounded, it implies and hence , by Barbalat’s lemma [30]. From (20), one has either for or for and . The property (22) thus holds, and the proof is thus completed.
Remark 2
In the controller (14) or (17), the term gives the repulsive velocity between two vessels. In (20), the term gives the repulsive angular velocity between two vessels. In particular, the closer are the two vessels, the larger is the repulsive velocity. It provides a mechanism for collision avoidance among the follower vessels. However, rigorous collision avoidance analysis is an interesting topic for future research.
V Trajectory Regulation
In this section, we will solve the problem formulated in step (ii), that is, to design the actuator input for the dynamics model (4) such that , , and achieve the desired , , and , respectively. Note that (4) is an underactuated system. The states and can be controlled through to achieve the desired and as elaborated in Theorem 4, while cannot be directly controlled. Fortunately, the desired can be arbitrarily selected as explained in Section III as long as . Therefore, we can trivially set that automatically includes regulation of to . In the scenario investigated in this paper, the sway velocity is typically small, which makes hold in general. In practice, if a large occurs in a rare situation, the vessel can be intervened to reduce its sway velocity.
Theorem 4
For sufficiently smooth desired signals and , pick a sufficiently smooth signal
(25) 
Define a lumped reference signal
For the system (4), consider the actuator input as follows
(26) 
for and some positive control parameters and satisfying ; consider the actuator input as follows
(27) 
for
and some positive control parameters and . Then, the states and achieve the desired and , in particular, with in (7) approaching zero exponentially if is bounded.
The dynamics and the controller (26) can be put in the following form, with and ,
(28) 
which is exponentially stable when .
For , one has
Let . Then,
(29) 
where
Direct calculation gives
Noting that
one has
(30) 
From (28), (29), and (30), one has
exponentially. Furthermore, it follows from (25) and (29) that exponentially. It can be verified that
for some between and , noting . Since the norms of the rotation matrix and its derivative, i.e., and , are always bounded, is bounded, and , , , one has , exponentially. The proof is thus completed.
Remark 3
When the vessels work in a low frequency motion scenario with the desired state trajectories and varying slowly and bounded by a constant , one can simplify the controllers by approximately using