Fixed-time Leader-Following Formation Control of AUVs Without Velocity Measurements

This paper is concerned with formation control of autonomous underwater vehicles (AUVs), focus-ing on improving system convergence speed and over-coming velocity measurement limitation. By employ-ing the ﬁxed-time control theory and command ﬁltering technique, a full state feedback formation algorithm is proposed, which makes the follower AUV track the leader in a given time with all signals in the system globally practically stabilized in ﬁxed time. To avoid degraded control performance due to inaccurate velocity measurement, a ﬁxed-time convergent observer is designed to estimate the velocity of AUVs. We then give an observer-based ﬁxed-time control method, with which acceptable formation tracking performance can be achieved in ﬁxed time without velocity measurement. The eﬀectiveness and performance of the proposed method is demonstrated by numerical simulations.


Introduction
Autonomous underwater vehicles (AUVs) have played important roles in various underwater applications, e.g., ocean sampling, oceanographic survey and inspection, seafloor mapping and modeling, submarine mine neutralization, offshore oil and gas exploration [1,9,10,26,28]. To improve efficiency and to extend task capacity, many tasks in these applications involve collaboration of multiple AUVs in a certain formation [2,3,4,5,6,7,11,19]. Formation control is therefore very important and there are various methods widely in use, among which the leader-follower scheme is the most popular due to simplicity and scalability. Some outstanding works on leader-following control can be seen in the literature [11,16].
What are the challenging issues in formation control of AUVs? Some well-known issues include energy limitation [2], communication constraint [11], uncertainties (modeling error, parameter variation, and external disturbance) [20], sensor and actuator constraints [29], etc. The first issue concerned in this paper is the low convergence speed of the formation tracking control system, which is partly due to actuator constraints. Most AUVs formation control methods in existence can only guarantee asymptotic convergence of the tracking errors. Namely, the convergence speed is at best exponential, which implies that the tracking errors will converge to zero with infinite settling time [11,20]. Clearly, it would be a great advance if AUVs can track a given formation in finite time. Actually, some research attention has been paid to finite-time formation control of various vehicles (e.g., mobile robots, surface vessels and aircrafts) in recent years [12,21]. However, no similar work on formation control of underwater vehicles has been seen yet, although a finite-time trajectory tracking problem was studied for a single AUV in [23]. Even for this much simpler problem, the settling time of the method in [23] depends on the initial states of the vehicle and cannot be guaranteed since the initial conditions are usually unknown. Fortunately, some recent advances showed that it is possible to realize group coordination control with guaranteed settling time regardless of the initial conditions [13,27,30,32]. These research works are based on the theory of fixed-time stability [27]. Although the idea of fixed-time control is interesting, research works are mainly focused on consensus of multi-agent systems. Here, we aim to find a fixed-time solution to AUVs formation control problem.
The second challenging issue relates to the velocity measurement limitation of AUVs. Most of the existing results on AUVs formation control assume precise and full position and velocity feedback [11,29]. Typically, the position of AUVs can be measured with certain accuracy using advanced positioning technologies such as acoustic ultrashort baseline system (USBL) and the global positioning system (GPS, possibly with the help of surface ships). However, accurate velocity measurement of AUVs is more difficult, due to technology limitations and environment disturbances. In order to eliminate the need of velocity measurement and maintain high control accuracy, observers are used in many methods [23,20,33]. In [20], a dynamic surface control-based formation tracking scheme was proposed, in which the velocity of the leader AUV is estimated using a linear observer. In [33], an adaptive trajectory tracking control method was present based on a linear velocity observer. In [23], a global finite-time observer was proposed to reconstruct the vehicle velocity. However, neither the linear observer in [20,33] nor the finite-time observer in [23] can ensure that the observed velocity can accurately track the true value within a given time.
This paper gives a fixed-time leader-follower formation tracking control method for AUVs using a fixedtime velocity observer. Firstly, a fixed-time formation tracking control law based on full position and velocity information is presented. Then, to deal with the difficulty of velocity measurement, we give a state observer that can converge in a fixed time with uncertainties due to model dynamics and environmental disturbances. An adaptive formation control method based on the fixedtime velocity observer and the position measurement is then presented, which can track the given formation in a fixed time. To the best of our knowledge, this is the first study of its type on formation tracking control of AUVs.
The main contributions of this paper are summarized as follows: (1). A unified novel framework for AUVs formation control with both full state feedback and velocity estimation is developed. The proposed control algorithm can ensure not only a given settling time regardless of the initial conditions of the system, but also a better tracking performance and faster convergence speed of the AUVs formation system than the existing methods [11,20].
(2). The velocity observer itself is also fixed-time convergent, which can estimate the actual AUV velocity with zero error in a designated fixed time regardless of the initial states of the system. In this sense, it is superior to both the linear observer in [20,33] and the finite-time observer in [23].
This paper is organized as follows. Section II gives some preliminaries and the problem formulation. Section III contains the fixed-time formation control method with full velocity and position feedback. Section IV gives the fixed-time velocity observer and the formation control scheme with velocity observer. Simulation studies are given in Section V, which is followed by the conclusions and future works in Section VI.
Notations: Throughout this paper, R n and R n×n denote the n-Euclidean space and the set of all n × n real matrices, respectively; · stands for either the Euclidean vector norm or the spectral norm of a matrix; λ max (X) (λ min (X)) represents the maximum (minimum) eigenvalue of matrix X; X > 0 (X < 0) means that matrix X is positive (negative) definite; For two positive definite diagonal matrices A and B with the same dimension, A > B (A < B) means each diagonal element satisfies a ii > b ii (a ii < b ii ), A B means aii bii ; 1 denotes the diagonal matrix with all elements equal to one.

Preliminaries
Lemma 1 [22,27] For systemẋ(t) = f (x(t)) along a trajectory x(t), if there exists a positive definite func- for some α, β, p, q > 0, with 0 < p < 1 and q > 1; then, the origin of the system is globally fixed-time stable with ϑ = 0, and the settling time T can be estimated by .
For ϑ ∈ (0, ∞), the system trajectory x(t) is practically fixed-time stable, and the residual set of the solution of the system can be given by where θ is a scalar and satisfies 0 < θ < 1. The time needed to reach the residual set is bounded as Now, consider a command filter [14] as described below: where α 1 is the input signal, z 1 and z 2 are the filter output, f > 0 and ζ ∈ (0, 1] are the filter gains to be determined, which denote the natural frequency and the damping ratio of the command filter, respectively.
Choose appropriate parameters f and ζ, the following lemma holds.
Remark 1 Since the signal passing through the filter will have a phase offset, that is, there is a phase difference between the filtered signal and the original signal, which leads to the existence of filtering error |z 1 − α 1 |.

Lemma 4 [30] Consider a nonlinear systeṁ
where x = [x 1 , x 2 , · · · , x n ] T ∈ R n is the state, f (x) : R n → R n is a nonlinear function. If f (x) is a homogenous vector function in the bi-limit with associated triples (r 0 , k 0 , f 0 ) and (r ∞ , k ∞ , f ∞ ), moreover, if the original systemẋ = f (x) and the approximating systemṡ x = f 0 (x),ẋ = f ∞ (x) are globally asymptotically stable, then we have the following results: (1) The origin of (6) is fixed-time stable when condition k ∞ > 0 > k 0 holds; (2) Let d V0 and d V∞ be real numbers such that d V0 > max 1≤i≤n r 0,i and d V∞ > max 1≤i≤n r ∞,i . There exists a continuous and positive definite function V : R n → R + such that the function ∂V ∂xi is homogeneous in the bi-limit with associated triples r 0 , d V0 − r 0,i , ∂V0 ∂xi and r 0 , d V∞ − r 0,i , ∂V∞ ∂xi , and the function ∂V ∂x f (x) is negative definite, and satisfies where k υ is positive constant and function Γ :

The AUV dynamics
Consider N AUVs with a global leader, labeled as l, and N − 1 followers, labeled as f 1 to f N −1 . Each follower is equipped with a sensor to measure its own position and receive that of the leader in the global coordinate frame {E}. In addition, each follower can receive velocity vector of the leader in the body coordinate frame {B}. Assume that each AUV i, i = 1, · · · , N , has fixed attitudes and the translational dynamics is given by [19]: where denotes the generalized attitude in Euler angles of roll φ i , pitch θ i , and yaw ψ i , in frame {E}, J i (Θ i ) denotes the kinematic transformation matrix from frame {B} to {E}. For short, throughout the paper, we will denote J(Θ) as J.
notes the environmental disturbance forces and moments due to waves, wind and ocean current, respectively. M i , D i (υ i ) and g i (Θ i ) are the inertia matrix, the damping matrix, and the restoring force vector, respectively, with For simplicity, the dynamic part (i.e., the second equation in (8)) can be rewritten as: Let ϕ i = [ϕ ui , ϕ vi , ϕ ωi ] T , where each element is a timevarying function satisfying the following assumption.
Assumption 1 [31] For each element in ϕ i , there is a known and positive constant * satisfying (13) where * = u i , v i , ω i and let i = diag( ui , vi , ωi ).

The objective
In this paper, we consider an interesting scenario in which the N −1 follower vehicles can each independently follow the leader. Namely, the entire formation can be decomposed into N −1 subformations, each of which comprises a follower AUV and the leader, as shown in Fig. 1. In each subformation, the follower AUV tries to maintain a desired distance relative to the global leader. Without loss of generality, the objective here can be described as designing a control law τ f i for follower AUV f i , so that AUV f i and the leader can achieve a formation given in {E}. Namely, the distance between the follower and the leader reaches a desired value in finite time, i.e., where δ 0i is an arbitrary small positive constant and 0 ≤ T < ∞ is the given convergence time.

Formation Control with Velocity Measurement
Here, a fixed-time formation control method based on full velocity and position feedback will be given. The controller is designed using the command filter technique and adaptive backstepping technique for each follower AUV to achieve the formation control objective. In what follows, we will drop the subscript i for short.
The controller design process is divided into two steps, i.e., the kinematics control part and the dynamics control part.
Step 1 (Kinematic controller design): Introduce a position tracking error vector e 1 = [e 1x , e 1y , e 1z ] T ∈ R 3 , which is defined by where d lf = [d lf x , d lf y , d lf z ] T ∈ R 3 denotes the desired relative distance vector between the AUVs. Using (8), we can derive the time derivative of (15) aṡ where υ f is the filtered intermediate control vector to be given later.
Note that the controller design procedure is essentially of backstepping type and the use of differential termυ f will be unavoidable in the dynamic controller part (to be given later). Since differential operation is difficult to implement in practice, here a command filter in the form of (5) is used to generate the virtual control signal and its differential term needed. For notation convenience, denote the output of the command filter as υ d f andυ d f . And the input of the filter, called the nominal kinematic controller, is denoted as υ c f . We now choose the nominal kinematic controller υ c f as: where k 1 ∈ R 3×3 , λ 1 ∈ R 3×3 , α ∈ R 3×3 , and β ∈ R 3×3 are positive definite diagonal matrices with k 1 > 2λ 1 , λ 1 > 1, p > 1 and 0 < q < 1 are ratios of two positive odds, and ξ = [ξ u , ξ v , ξ ω ] T denotes the filtered compensating signal to be designed later.
The nominal function υ c f is then passed through a command filter (5) to generate the needed stabilizing control signal υ d f and its derivativeυ d f . The command filter may have errors if not appropriately designed and calibrated, which will in turn affect the performance. Here we introduce a compensating signal to deal with this issue. In doing so, define the filtered error 1 = υ c f − υ d f . Then, the filtered compensating signal ξ is generated by the following system: where ϑ 1 > 0 is a small constant to be chosen, is a positive definite matrix to be chosen such that λ 2 > λ 1 + 1, and ξ(0) = 0.
To show that the designed controller in (17) can guarantee stabilization of the position tracking errors, define the following candidate Lyapunov function: It is obvious that V 1 in (19) is continuously differentiable, positive definite and radially unbounded.
Step 2 (Dynamic controller design): Consider velocity tracking error vector e 2 = [e 2u , e 2v , e 2ω ] T ∈ R 3 defined by whose time derivative is obtained by using (8) as: Here, the ideal form of nonlinear controller where k 2 ∈ R 3×3 is a positive definite diagonal matrix, with k 2 ≥ 1 2 , which is to be determined and ϕ f is given in (12).
In practice, ϕ is very hard to obtain accurately. Hence, for implementation consideration, a radial basis function neural networks (RBFNNs) is employed to approximate ϕ as follows: where Z ∈ R 3×1 and W ∈ R 3l×3 are the input and weight of the NN, l > 0 is the node number and Ω ∈ R 3×1 is a compact set. (Z) is the approximation error having (Z) ≤ * .Since the activation function σ(Z) is bounded, there exists a positive constant σ * ∈ R such that σ(Z) ≤ σ * . So, the actual input can be written as: T is an estimate of the upper bound of external disturbances d f max , whose update law are taken to bė where Γ * ∈ R l×l , Λ * ∈ R l×l are positive-definite design matrices, k 3 * and k 4 * are design constants. Now, we show that the designed controller in (25) can guarantee stabilization of velocity tracking errors. Introduce the following radially unbounded candidate Lyapunov function: (27) is continuously differentiable, positive definite and radially unbounded.
The time derivative of V 2 is given bẏ According to (12), (22) and (25), we can obtain the following inequation: Based on (28) and considering the following facts by completion of squares: we can obtain the following inequation: where From Lemma 1, we know that the velocity tracking errors are practically fixed-time stable. Finally, we proceed to show that the entire tracking errors system composed by (15) and (21) can be stabilized with the designed kinematic and dynamic controllers. Consider the following Lyapunov function for the whole system: It is obvious that V in (33) is continuously differentiable, positive definite and radially unbounded. According to (20), (25), (27), (28), (32), Young's inequality and Lemma 2, we havė 2). Signals e 1 , ξ, e 2 ,W andd f max in the closed-loop formation control system are all practically fixed-time stable.
Proof : The (34) can be rewritten as: where Moreover, assume that there exists an unknown constant ∆ and a compact set D such that Then, we havė where ϑ = ω * =u Again, from Lemma 1, we have that the closed-loop formation tracking system is practically fixed-time stable, and the residual set of the solution of system (32) is calculated as with x = {e 1 , e 2 , ξ,W ,d f max } and the settling time T s satisfies This completes the proof.
Remark 2 According to (39) we can see the maximum convergence time only depends on the controller parameters and parameter θ. Therefore, the presented method allows one to arbitrarily choose the convergence rate of the AUV formation, which makes it feasible for us to meet strict settling time requirements in practical applications. Moreover, the fixed-time algorithm can ensure a fixed settling time regardless of the initial states of AUVs. Table 1 Relationship between parameters and convergence ratẽ Remark 3 By selecting the controller parameters (i.e. α,β, γ 1 , γ 2 ) and θ to satisfy corresponding constraints as mentioned in the above discussions, we can guarantee a bound of the settling time as given in (39), which determines a certain convergence rate. Generally, the relation between the convergence rate and these parameters is shown in Table 1. We can find a set of optimal parameters to yield a minimum settling time by simply introducing a search algorithm like the seeker optimization algorithm and cuckoo search algorithm in [8,17].
Remark 4 From the stability analysis and the definition of convergence time (39), we can see that the values of controller parameters do not affect the stability of the closed-loop system, although they may influence the convergence time. For a practical formation tracking control problem, the desired convergence time cannot be too short, otherwise, the closed-loop system may be instable due to saturation constraint of the thruster. In addition, a feasible convergence time should also take the transient performance into consideration in the design procedure according to the maximum maneuver capability.

Formation Controller without Velocity Measurement
In many practical situations, it is difficult or even impossible to obtain the accurate velocity measurements due to technology limitation or environment disturbances. When the velocity of AUVs is not easily measurable, the state feedback control method in Section III cannot be implemented. Thus, finding a control method based on velocity estimation and position measurements is of great interest. Here we first give a fixed-time observer to estimate the AUV velocity υ f , and then, an observerbased formation control method is presented, see Fig.   2. Fig. 2. Schematic of the AUVs formation control system.

Fixed-time velocity observer
Before giving the velocity observer, a coordinate transformation is introduced.

Velocity Observer Design
To estimate the velocity vector χ = [χ u , χ v , χ ω ] T , we introduce the following observer: whereē 1 = p −p,k i ∈ R 3×3 , (i = 1, 2, 3, 4, 5, 6) are positive gain matrices and the parameters ι i ,ῑ i , (i = 1, 2) are given by Theorem 2 The states (p,χ) of the velocity observer in (42) with parameters defined in (43) will globally converge to the real states (p, χ) in a fixed time T 0 if Assumption 1 holds and each element ink i , i = 1, 2, 3, 4, 5, 6 satisfies the following inequalities Namely, the observation error system as defined below is fixed-time stable: Proof : See Appendix

Fixed-time control with velocity observer
Here in this subsection, we present a velocity observerbased formation control method for AUVs. Replacing υ f in (25) by its estimated valueυ f in (42) yields the following control law: with all control parameters are the same with those in Theorem 1. For the velocity observer-based control law, we have the following result. Theorem 3 Consider system (8) under the control law (46) withυ f generated by observer (41). The closed-loop system comprising (16) and (22) under Assumptions 1 and 2 is practically fixed-time stable if the controller parameters k i (i = 1, 2, 3, 4), λ 1 , λ 2 , α, β, p, q, Λ, Γ and the observer parametersk i (i = 1, 2, 3, 4, 5, 6), ι i ,ῑ i (i = 1, 2) are selected as in Theorems 1 and 2, respectively. Furthermore, the convergence time is bounded by T ≤ T 0 + T s .
Proof : It follows from Theorem 2 that there exists a finite time T 0 uniform in initial estimation errors e 1 (0) andē 2 (0) such that υ f (t) =υ f (t) for t ≥ T 0 . As a result, the control law in (46) coincides with state feedback control law (25) for all t ≥ T 0 . Furthermore, if the system trajectory under control law (46) does not escape during interval t ∈ [0, T 0 ], it follows from Theorem 1 that there exists a finite time T s uniform in V (T 0 ) to ensure the fixed-time stability of the formation tracking system. Therefore, the condition that the closed-loop system under control law (46) does not escape in finite time is sufficient to derive the conclusion of Theorem 3. To complete the proof, let us consider the following Lyapunov function: whose derivative along the trajectory of (22) under control law (46) is given bẏ (48) 2 ) for any ς > 0. To prove convergence of the system, two different cases are discussed.
Case 1: Assume that e 2 > J −1ē 2 , which implies sign(e 2 − J −1ē 2 ) = sign(e 2 ). Therefore, one has e T 2ê ς 2 = e 2 e 2 − J −1ē 2 ς for any ς > 0. Then,V 3 in (48) can be rewritten aṡ (49) Taking into account e 2 ≤ J −1ē 2 and the wellknown inequality a − b ς ≤ ( a + b ) ς , inequality (50) satisfieṡ Then, we will show that in both cases there exists a positive constant M such thatV 3 ≤ M at any time. Since Theorem 2 ensures fixed-time convergence ofē 2 , which implies boundness ofē 2 , it follows that there exists a least upper bound of the right hand side of (51). Denote the least upper bound by M = sup{2k 2 J −1ē (49) and (51) thatV 3 ≤ M . Therefore,V 3 as well as the system states e 1 , e 2 cannot escape in any finite time interval. From the above analysis, it can be concluded that the formation tracking system with velocity observerbased control law (46) does not escape in any finite time interval. Following the analysis at the beginning of the proof, one has that the closed-loop system under (17), (18), (42) and (46) is practically fixed-time stable. This completes the proof.
Remark 5 Compared with asymptotic formation control method in [19,11] and finite time trajectory tracking control method in [23], the presented control scheme can achieve fixed-time convergence and higher control precision.

Remark 6
The proposed method guarantees that the formation objective can be achieved within an arbitrary time. For practical AUVs subject to actuator saturations, the method is still useful but some amendments are needed and the settling time might be larger. Specifically, one can modify the system model using the auxiliary system technique in [18] or the adaptive approximation method in [22] to address the saturation issue.

Numerical Simulations
In this section, numerical simulations are given to verify the effectiveness of the proposed method. Without loss of generality, consider the following time-varying forces/moment disturbance in frame {E}: (52)
Next, we show that the formation can be achieved within fixed time under the velocity observation-based control law (46). Fig. 8 illustrate the formation tracking results of two AUVs under control law (46), wherein Fig. 8a shows the AUVs trajectories, and Fig. 8b is position tracking errors e 1x , e 1y and e 1z , which all converge to zero after 7 seconds.

Demonstration in the MSS simulator
We further demonstrate the method using the wellknown high fidelity Marine System Simulator (MSS) [15], developed by the Department of Marine Technology, Norwegian University of Science and Technology. The MSS integrates hydrodynamics, structural mechanics, marine machinery, electric power generation and distribution, navigation and automatic control of marine vessels of various types (e.g., surface vessels, hydros, semi-subs). The simulator can better capture the hydrodynamic effects, generalized Coriolis and centripetal forces, nonlinear damping and current forces, and generalized restoring forces. It is composed of powerful modules including the environmental module, the vessel dynamics module, the thruster and shaft module, and the vessel control module.
The simulations are conducted based on the Naval Postgraduate School AUV in the MSS. To make the simulations closer to the practical situation, an ocean disturbance is introduced via the Gaussian Random Process in (52). An input saturation is also considered, which is described by The simulations results are given in Figs. 9-11, which show, respectively, formation tracking control performance without velocity measurement, under input saturation and ocean disturbances. We can clearly see that, the proposed method is effective in all the three cases with the MSS simulator. For the case with saturations, the formation is achieved with a settling time 6 seconds slightly larger than the calculated value 3 seconds for the nominal case without input saturation.

Conclusion
In this paper, the fixed-time formation tracking problem for two AUVs with and without velocity measurements has been studied. In contrast to the existing finite-time control methods, the fixed-time control scheme is independent of the initial conditions and has a more rapid convergence and higher accuracy. Using the fixedtime control theory, a fixed time state feedback controller has been proposed for the formation system. The command filter technique is incorporated to the backstepping scheme, together with an error compensator to eliminate the filtering error and an adaptive NN is introduced to overcome the model uncertainties and external disturbance. To obtain the velocity information for feedback, a coordinate transformation and a global fixed-time convergent state observer has been developed. Then, a fixed-time control scheme with velocity observer has been derived by combining the corresponding state feedback controller and the fixed-time convergent state observer together. Rigorous proof is shown that the formation control can be achieved in a fixed time regardless of the initial states while guaranteeing all tracking errors of the closed-loop formation control system are practically fixed-time stable. Simulation results illustrate the effectiveness of the proposed control scheme. Future work will focus on the extension of the results to formation control of heterogeneous AUVs considering the transmission limitations of underwater acoustic communication systems.

PROOF THE THEOREM 2
We now give the proof, which contains three steps. First, we prove that the closed-loop system (45) with the parameters provided in (43) is globally asymptotically stable, which are followed by the proof that its approximating systems in 0-limit and ∞-limit are also globally asymptotically stable.
Step 1: In this step, we show that the error system (45) is globally asymptotically stable. Consider the following candidate Lyapunov function: It is obvious thatV in (54) is continuously differentiable, positive definite and radially unbounded.
According to (45), the time derivative ofV iṡ On the basis of Assumption 1, we can obtain that From the class Cauchy inequality, one obtains and −k 4ē Also, from (43), and Young's inequality, we have that Then, substituting (56), (57), (58) and (59) into (55), we can havė . With parametersk i selected according to (43), we have that 1 > 0 and 2 > 0. Then it follows from (57) that the error system in (44) is globally asymptotically stable.
Finally, ground on the arguments in Steps 1-3, it can be concluded that the proposed velocity observer in (42) is fixed-time stable under Lemma 4.