Finite-time coordinated path-following control of leader-following multi-agent systems

This paper presents applications of the continuous feedback method to achieve path-following and a formation moving along the desired orbits within a finite time. It is assumed that the topology for the virtual leader and followers is directed. An additional condition of the so-called barrier function is designed to make all agents move within a limited area. A novel continuous finite-time path-following control law is first designed based on the barrier function and backstepping. Then a novel continuous finite-time formation algorithm is designed by regarding the path-following errors as disturbances. The settling-time properties of the resulting system are studied in detail and simulations are presented to validate the proposed strategies.


Introduction
Currently, the theory of the formation control problem has emerged as a hot topic and attracted great attention from researchers. To achieve better measurements of biological variables across a range of spatial and temporal scales in the applications in oceanic and planetary explorations (Bertozzi et al., 2005;Fiorelli et al., 2006), unmanned systems are required to simultaneously follow a set of given orbits with a desired formation, which is a special formation control problem called the coordinated pathfollowing control problem. In the area of coordinated path-following control, many scholars focused on the asymptotic stability of the resulting multi-agent systems. In Cao et al. (2009), a discrete-time consensus-based algorithm was developed to force each follower to track a leader with the desired dynamics, which is also called the consensus tracking control problem. The continuoustime consensus tracking control laws were given in cases of time-invariant formation in Cao and Ren (2012), the time-varying formation in Yu et al. (2018), and the containment motion in Zhang FX and Chen (2022). In Ghabcheloo (2007), a coordinated path-following control law was designed by parameterizing the desired trajectories while synchronizing the orbital parameters. This idea was used in the case of uncertain dynamics in Peng et al. (2013). Noting the geometry of the orbit, a novel geometry extension method was proposed and then integrated into the consensus of the generalized arc-lengths (the smooth functions) to achieve the coordinated pathfollowing task in Zhang FM and Leonard (2007) and Chen and Tian (2015). The geometry extension method was also used to solve the asymptotic coordinated path-following problem with time-varying flows in Chen et al. (2021aChen et al. ( , 2021b. However, the coordinated path-following control problem within a finite settling time is still unsolved. Recently, finite-time control laws in multi-agent systems concentrate on the consensus (or consensus tracking) problems. In Xiao et al. (2009), a finitetime consensus tracking law was designed for a structure that consists of one leader and bidirectional connected followers based on the sliding-mode method. The sliding-mode method was used in the case of directed topologies in Cao et al. (2010) and Wang L and Xiao (2010), in the case of uncertainties in Khoo et al. (2009), and in under-actuated systems in Li TS et al. (2018). The finite-time properties of a sliding-mode-based consensus tracking system can be analyzed using the degree of homogeneity; details can be found in Guan et al. (2012) and Dou et al. (2019). Note that the above control laws are nonsmooth and thus sometimes cannot be directly used in actual continuous systems (Qian and Lin, 2001). There is a trend toward designing a continuous finitetime controller for the coordinated control problem. In Li SH et al. (2011), a continuous finite-time consensus law was designed for second-order multi-agent systems under one leader and bidirectional connected followers. A similar idea was designed in Du et al. (2013) using dynamic output feedback. In Huang et al. (2015), an adaptive finite-time consensus algorithm was designed for uncertain nonlinear mechanical systems. The continuous finite-time consensus method was developed to deal with high-order nonholonomic mobile robots with bidirectional topologies in Du et al. (2017) and surface vehicles under the assumption that all followers can access to the leader in Wang N and Li (2020). Note that the objectives of the coordinated path-following control problem include path-following and formation, which are different from those of the consensus problem. It is essential to give a finite-time method to the coordinated path-following problem.
This paper gives a continuous solution to the finite-time control problem of coordinated pathfollowing under directed topologies. To solve the trajectory restriction problem, we present a new bar-rier function definition that is integrated into backstepping to design a novel continuous finite-time path-following control input projected on the normal vector on the orbit. Another continuous finitetime formation control input projected on the tangential vector on the orbit is designed by regarding the path-following errors as disturbances. Note that the proposed method in this paper is different from our previous adaptive method in Chen et al. (2021b) concerning two conditions: (1) directed networked second-order agents are under consideration and the first-order systems are replaced with bidirectional topologies; (2) a continuous finite-time design method is used to replace the adaptive methods.

Graph theory and barrier functions
The network topology of the coordinated pathfollowing system can be described by a digraph G = {V, E}, where nodes V = {V 0 , V 1 , · · · , V n } are associated with a virtual leader labeled V 0 and n vehicles labeled V 1 , V 2 , · · · , V n , and E ⊆ V × V is a set of network links. A directed path from node V i to node V j is a sequence of edges in the network topology with distinct nodes V i k , k = 1, 2, . . . , l. A digraph is called a directed tree if there exists a node, called the root, that has directed paths to all the other nodes in the digraph. Let, for i, j = 0, 1, . . . , n, a ii = 0 and a ij = 1 if (V i , V j ) ∈ E, and a ij = 0 otherwise. In addition, define the Laplacian matrix L [l ij ] n i,j=0 with l ii = n j=1 a ij and l ij = −a ij , for any i = j, i, j = 0, 1, . . . , n. Assumption 1 The digraph consisting of a virtual leader and n vehicles contains a directed spanning tree with root V 0 .
For the considered coordinated path-following system, the Laplacian matrix L can be written as where l 0 = [l 10 , l 20 , · · · , l n0 ] T ∈ R n×1 and L 1 ∈ R n×n . Suppose that Assumption 1 holds. L 1 is a nonsingular M-matrix and all eigenvalues of L 1 have positive real parts (Zhang Y and Tian, 2009). ρ L1 denotes the smallest eigenvalue of L 1 .
To keep each agent's trajectory staying in a restricted area when applying the geometry extension method, a new definition of the barrier function Ψ i is given:

Remark 1
The barrier function in Definition 1 is different from those in traditional definitions, because condition (C4) is added and used to yield the finite-time convergence of the resulting system with the state constraint It is noted that the additional condition (C4) is not difficult to satisfy in traditional barrier functions. For example, where c 1 and c 2 are positive constants. In this case, one can select parameter c 2 from 0.3 to 0.8 to yield condition (C4), as shown in Fig. 1.

Some lemmas
Lemma 1 (Chen and Tian, 2015) Consider any simple, closed, and regular orbit satisfying the following conditions: which is a tubular neighborhood of the orbit, and a smooth function which is called the orbit function (its value is called the orbit value), such that the following conditions hold: C ic is a level line of the orbit function λ i (p i ), and the orbit value associated with orbit C i0 is zero. Definition 2 (Chen et al., 2021a) The generalized arc-lengths ξ i are C 1 functions of the arc-lengths s i and ∂ξi ∂si ≥ c ξ > 0. Lemma 2 (Chen et al., 2021a) The C 1 invertible mappings ξ i : R → R define a change of coordinates, which allows formulation of the coordinated pathfollowing problem with state variables s i into the consensus problem described by with state variables ξ i (s i ). To form the desired formation, the desired arc-length s * i of the i th follower is determined by the arc-length of the leader s 0 via s * i = g sis0 (s 0 ), where g sis0 : R → R is an invertible mapping explicitly defined by the desired formation. By Eq.
(2), ξ 0 and ξ i are such that , which properly defines ξ i for any given ξ 0 , since g sis0 is known. Here, "•" is the symbol for composition of functions; that is, ξ i is the composition of ξ 0 and g −1 sis0 .

Problem formulation
In this study, we first consider the cases of static virtual leader and dynamic virtual leader. In a fixed inertial reference frame, the model of the static virtual leader is the first-order dynamics such thaṫ T ∈ R 2 is its position.
For i = 1, 2, · · · , n, the dynamic equation for the i th follower satisfying the second-order dynamics is given by where denote the position and velocity, respectively.
Suppose that the desired orbit associated with each agent is a simple, closed, and regular curve with nonzero curvature. According to Lemma 1, this curve can be extended geometrically to a set of level curves, which can be defined by a smooth function (the orbit function) λ i : The path-following error can be described by the value of the orbit function and the path-following task is achieved if Let the arc-lengths be given by where φ * i is the parameter associated with the starting point of the arc of s i . The generalized arclengths ξ i : R → R of s i are used to describe the formation along the curves. ∂ξ i /∂s i is a constant and and satisfies c ξ ≤ |∂ξ i /∂s i | ≤ cξ with two positive constants c ξ and cξ. From Lemma 2, the task of achieving coordinate formation in finite time T can be described as follows: The finite-time coordinated path-following control problem is as follows: For i = 1, 2, · · · , n, consider system (3) and the initial position p i (0) ∈ Ω i . Suppose that Assumption 1 holds. Design a finitetime coordinated path-following control u i such that the closed-loop system satisfies Eqs. (4) and (7). Remark 2 The discontinuous laws based on sgn(·) (Khoo et al., 2009;Xiao et al., 2009;Cao et al., 2010;Wang L and Xiao, 2010;Guan et al., 2012;Li TS et al., 2018;Dou et al., 2019) might cause signal chattering in the closed-loop system. In practice, it is difficult to accomplish these discontinuous laws.

Remark 3
This paper is devoted to designing a continuous finite-time control law for directed networking second-order agents for the coordinated path-following problem. However, Chen et al. (2021b) dealt with the adaptive design for first-order agents with unknown time-varying parameters and bidirectional topologies.

Main results
In this study, we first consider the cases of static virtual leader and dynamic virtual leader. The design precedence is as follows: (1) decouple the whole system as a path-following subsystem and a formation subsystem; (2) regard v Ni as a virtual controllerv Ni and design u Ni by backstepping to achieve finite-time path-following along the given orbits (Theorem 1); (3) regard v Ti as a virtual controllerv Ti and design u Ti by backstepping to achieve finite-time formation along the given orbits (Theorem 2); (4) according to Theorems 1 and 2, give Theorem 3 to show the finite-time convergence of the coordinated path-following control system. Then a corollary is given to show the case of the dynamic virtual leader.
Section 3.1 gives the open-loop system (i.e., the error equations of the coordinated path-following control system), which is used to design the pathfollowing control law in Section 3.2 and the formation control law in Section 3.3.

Open-loop system
By differentiating λ i , the path-following dynamics of agent i is obtained as follows: where v Ni = N T i v i denotes the velocity projected on vector N i which is normal to the level orbit of the current position of agent i and N i = ∇λi ∇λi . Differentiating both sides of v Ni yieldṡ where u Ni = N T i u i denotes the control input projected on the normal vector Let T i denote the vector which is tangent to the level orbit of the current position of agent i and Then the dynamics of ξ i is given bẏ where where u Ti = T T i u i denotes the control input projected on the tangent vector N i and Δ errors. The dynamics of ς i is described bẏ As a result, the equations of the formation tracking control system are given by Eqs. (8), (9), (11), and (12).

Path-following controller design
Let us first consider the path-following subsystem consisting of Eqs. (8) and (9) and let the virtual controlv Ni bev where 1 ≤ α = p1 p2 , p 1 and p 2 are positive odd integers, and the control gain k 1 will be selected later. Consider the path-following candidate Lyapunov function as where The proof of inequality (15) is provided in the supplementary materials.
Suppose that V P (t) = 0. Substituting Eq. (19) into inequality (18) yieldṡ . By Eq. (19), the closed-loop equation associated with the path-following subsystem for the i th follower is

Remark 4
It is obvious that the closed-loop system (22) for path-following is not homogeneous. Therefore, the finite-time stability analysis methods given in Guan et al. (2012) and Dou et al. (2019) cannot be applied in this study.
Note that 0 < (1 + 1 α )/2 < 1. To apply Theorem 4.2 in Bhat and Bernstein (2000) (which is provided in the supplementary materials), we will show that g P has a lower bound. From condition (C4), we have where β P4 is positive and bounded. According to Theorem 4.2 in Bhat and Bernstein (2000), we establish the following theorem:

Theorem 1
Suppose that the initial positions of vehicles are such that p i (0) ∈ Ω i . Assume moreover that Assumption 1 holds. Then the path-following objective (Eq. (4)) can be achieved by the finite-time control u Ni given in Eq. (19), for i = 1, 2, · · · , n.
Proof From inequality (21), we conclude that the function V P is bounded all the time, which implies that the objective (Eq. (5)) is satisfied according to conditions (C1) and (C2). From inequalities (21) and (24), we conclude that λ i = 0 and thatv Ni (i = 1, 2, · · · , n) are the finite-time stable equilibria of the closed-loop path-following subsystem (22).

Coordinated formation controller design
In the following, we will consider the formation subsystem consisting of Eqs. (11) and (12). Let the virtual controlv Ti bê where k 3 is a positive control gain and will be selected later. Consider the coordinated formation candidate Lyapunov function as . In Eq. (26), the first term on the right-hand side contributes to achieving the formation objective, i.e., Eq. (7), and the second term contributes to guaranteeing the convergence of where Note that where γ 3 = max ∀i n j=0 a ij and γ 4 = max ∀i,j {a ij }. From Lemmas A.1 and A.2 in Qian and Lin (2001) where where c σ1 = γ 3 cξ + (n + 1)γ 4 cξ and c vT1 = γ 3 cξc F1 + nγ 4 cξc F1 . Note that where α . The proof of inequality (37) is provided in the supplementary materials.
From inequalities (34) and (39), we have Substituting the above inequality and inequality (33) into Eq. (27) yieldṡ which makes the choices such that where the control gain k 4 will be set later. As a result, the closed-loop formation subsystem for the i th follower is in which control gains are chosen as follows: where β F1 is an arbitrary positive constant. As a result, we havė Let l F = max 1, 2 1− 1 . Then we have which yields V (44) can be rewritten aṡ where β F2 = β F1 l (1+ 1 α )/2 F and g F3 = g F1 + g F22 . Due to 0 < (1 + 1 α )/2 < 1, β F2 has a lower bound. g F3 approaches zero as lim t→T λ i (t) = 0 and lim t→TvNi (t) = 0, as proved in Theorem 1. We give the following result directly: Theorem 2 Suppose that the initial positions of vehicles are such that p i (0) ∈ Ω i . Assume moreover that Assumption 1 holds. Then the formation objective (Eq. (7)) can be achieved by the finite-time control u Ti given in Eq. (41), for i = 1, 2, · · · , n. Proof The proof follows the same argument as the proof of Theorem 5.3 in Bhat and Bernstein (2000) (which is provided in the supplementary materials). Hence, it is omitted.
Theorems 1 and 2 yield the following result: Theorem 3 Suppose that the initial positions of vehicles are such that p i (0) ∈ Ω i . Assume moreover that Assumption 1 holds. For i = 1, 2, · · · , n, the finite-time coordinated path-following control problem is solved by the coordinated path-following control: where u Ni and u Ti are as given in Eqs. (19) and (41), respectively.

Remark 5
Different from the consensus problem studied in Li SH et al. (2011), this study addresses the coordinated path-following control problem, which includes two subproblems, i.e., pathfollowing and formation control. Moreover, the digraph in this paper, assumed to consist of a virtual leader and n vehicles, contains a directed spanning tree with root V 0 , while in Li SH et al. (2011), each follower was required to access to the leader's states. Now, let us consider a special case: the virtual leader has a velocityξ 0 = η 0 along the responding orbit and its velocity accesses to each follower, where η 0 andη 0 are bounded signals. In this case, the open-loop equations of the path-following subsystem are Eqs. (8) and (9), which are the same as those of the static case. Letξ i = ξ i − ξ 0 . The time derivative ofξ i isξ i = ∂ξi ∂siṽ Ti + Δ ξi , a ij ξ i −ξ j be the formation errors. The dynamics ofς i iṡ which is similar to Eq. (12). As a result, the expressions ofṽ Ni andũ Ni are the same as Eqs. (13) and (19) in the static case, respectively. The expressions ofṽ Ti andũ Ti are also the same as Eqs. (25) and (41), respectively, in the static case by replacing ς i and Δ Ti withς i andΔ Ti , respectively. We now give the following corollary directly: Corollary 1 Consider that a virtual leader has a velocity η 0 along the responding orbit and that the access to each follower has been considered. Suppose that the initial vehicle positions are such that p i (0) ∈ Ω i . For i = 1, 2, · · · , n, the finite-time coordinated path-following control problem is solved by the coordinated path-following control law (47), where u Ni and u Ti are as given in Eqs. (19) and (41), respectively.

Simulations
In this section, we first apply the proposed control laws in Theorem 3 to coordinate the vehicles moving along the elliptic orbits with a triangle pattern in case 1, and then use the control algorithm proposed in Corollary 1 to achieve the in-line formation in case 2. The selected trajectories of the agents are concentric ellipses with a different semi-major axis and semi-minor axis, that is, C l0 : where e l = 1 + 0.5l, a = 3, b = 2, and l = 0, 1, 2, 3, 4.
1. Case 1: static virtual leader The topology for the virtual leader and followers is shown in Fig. 2. The parameters are selected as k 1 = k 3 = 2.7, k 2 = k 4 = 34, and α = 9 7 . The initial generalized arc-length of the virtual leader is ξ 0 (0) = 0. The motion of the agents is illustrated in Fig. 3, where "•", "2", " ", and "+" denote the agents' positions at t = 0, 1, 2, and 7 s, respectively. In this figure, one can see that the four followers converge to the given orbits and achieve the desired formation. The path-following errors λ i and formation errors ξ i − ξ 0 are plotted in Figs. 4 and 5, respec-tively. The above figures show that path-following and formation tracking are achieved.
2. Case 2: dynamic virtual leader We use the coordinated path-following control algorithm given in Corollary 1 to achieve the in-line pattern. The parameters are selected as k 1 = k 2 = k 3 = k 4 = 10 and α = 5 3 . The motion of the agents, the path-following errors λ i , and the formation errors ξ i −ξ 0 are illustrated in Figs. 6, 7, and 8, respectively.

Conclusions
A continuous feedback method to solve the finite-time coordinated path-following control problem is presented, where the topology for the virtual leader and followers is directed. Because the movable ranges of the agents are restricted, a novel barrier function is given. A finite-time coordinated pathfollowing control law in the static virtual leader case t (s) Fig. 8 Formation errors in the dynamic virtual leader case is designed first. Then the control law is obtained in the dynamic virtual leader case, where its velocity can be accessed by each follower. Conditions on the control gains to guarantee that the path-following errors and the formation errors converge to zeros in finite time are presented. In ongoing work, the experiments involving finite-time coordinated pathfollowing problems will be considered.

Contributors
Weibin CHEN designed the research. Yangyang CHEN drafted the paper. Ya ZHANG helped organize the paper.
Weibin CHEN and Yangyang CHEN revised and finalized the paper.

Compliance with ethics guidelines
Weibin CHEN, Yangyang CHEN, and Ya ZHANG declare that they have no conflict of interest.