Paper deep dive
Adaptive stabilization of a leaderless bearing-constrained formation with disturbances
Minh Hoang Trinh, Chuong Van Nguyen, Quoc Van Tran, Tuynh Van Pham
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 95%
Last extracted: 9/9/2026, 3:58:07 AM
Summary
This paper addresses the problem of regulating and maintaining a target formation characterized by bidirectional bearing constraints in a leaderless multi-agent system subject to bounded, time-varying, and unknown disturbances. The agents are modeled as single integrators. The authors propose adaptive sliding mode control laws to achieve uniform global asymptotic stability of the moving target formation and reject matched disturbances. To mitigate chattering inherent in sliding mode control, smooth adaptive control laws are also designed to ensure uniform global boundedness. The study considers both relative position and bearing-only sensing scenarios.
Entities (8)
Relation Signals (7)
Single Integrators → models → Agents
confidence 97% · The agents in the formation are modeled by single integrators
Bearing Constraints → characterizes → Target Formation
confidence 96% · target formation characterized by a set of bidirectional bearing constraints
Adaptive Sliding Mode Control → achieves → Uniform Global Asymptotic Stability
confidence 95% · adaptive sliding mode control laws to uniformly globally asymptotically stabilizes the moving target formation
Smooth Adaptive Control → alleviates → Chattering
confidence 95% · to alleviate chattering phenomena from sliding mode control, smooth adaptive control laws are then designed
Smooth Adaptive Control → guarantees → Uniform Global Boundedness
confidence 95% · smooth adaptive control laws are then designed to guarantee uniform global boundedness of the desired formation.
Adaptive Sliding Mode Control → usedfor → Disturbance Rejection
confidence 95% · We propose adaptive sliding mode control laws to ... reject the matched disturbances.
Disturbances → causes → Time-Varying Target Formation
confidence 93% · Due to the time-varying disturbances, the target formation is time-varying.
Cypher Suggestions (0)
No Cypher suggestions yet.
Abstract
Abstract:In this paper, we consider the problem of regulating and maintaining a target formation characterized by a set of bidirectional bearing constraints under disturbances. The agents in the formation are modeled by single integrators with bounded continuous disturbances of which the upper bound is unavailable for the control design. Due to the time-varying disturbances, the target formation is time-varying. We propose adaptive sliding mode control laws to uniformly globally asymptotically stabilizes the moving target formation and reject the matched disturbances. In addition, to alleviate chattering phenomena from sliding mode control, smooth adaptive control laws are then designed to guarantee uniform global boundedness of the desired formation. Finally, simulation results are given to support the analysis.
Tags
Links
- Source: https://arxiv.org/abs/2609.06355v1
- Canonical: https://arxiv.org/abs/2609.06355v1
Trouble viewing inline? Open PDF directly →
Full Text
89,757 characters extracted from source content.
Expand or collapse full text
Adaptive stabilization of a leaderless bearing-constrained formation with disturbances Minh Hoang Trinh* Chuong Nguyen Quoc Van Tran Tuynh Van Pham †thanks: *Corresponding author.†thanks: M. H. Trinh is with East Asia University of Technology, Hanoi, Vietnam (e-mail: minhtrinh@ieee.org). C. Nguyen is with Viterbi School of Engineering – Department of Aerospace and Mechanical Engineering, University of Southern California, CA, USA and Ghost Robotics, Philadelphia, PA, USA (e-mail: vanchuong@usc.edu). Q. V. Tran is with Department of Mechatronics, School of Mechanical Engineering, Hanoi University of Science and Technology, Hanoi, Vietnam (e-mail: quoc.tranvan@hust.edu.vn). T. V. Pham is with Department of Automation Engineering, School of Electrical and Electronic Engineering, Hanoi University of Science and Technology, Hanoi, Vietnam (e-mail: tuynh.phamvan@hust.edu.vn ) Abstract In this paper, we consider the problem of regulating and maintaining a target formation characterized by a set of bidirectional bearing constraints under disturbances. The agents in the formation are modeled by single integrators with bounded continuous disturbances of which the upper bound is unavailable for the control design. Due to the time-varying disturbances, the target formation is time-varying. We propose adaptive sliding mode control laws to uniformly globally asymptotically stabilizes the moving target formation and reject the matched disturbances. In addition, to alleviate chattering phenomena from sliding mode control, smooth adaptive control laws are then designed to guarantee uniform global boundedness of the desired formation. Finally, simulation results are given to support the analysis. Index Terms: multi-agent systems, vision-based control, sliding-mode control, chattering alleviation I Introduction Over the past decade, formation control research has attracted significant interest from the robotics and control systems communities [1, 2]. Formations of unmanned robots have been developed for both civilian and military applications, including truck platooning, drone squads for highway monitoring, error checking in solar panel fields, precision agriculture, search and rescue, unmanned underwater vehicles for seabed mapping, and satellite formations for positioning and remote sensing. Moreover, research progresses in formation control are often applicable to its dual problem—the sensor network localization problem [3, 4]. Let a group of autonomous agents (AGVs, UAVs, UUVs, satellites,…) be arbitrarily distributed as an initial configuration in the space. Formation control focuses on designing control algorithms to move a formation from the initial configuration along a continuous trajectory to a target configuration, which is specified by a set of geometric constraints. Each agent in the group can be fully controlled by its onboard controller, and the formation acquisition task is usually performed in an ambient space that is significantly larger in comparison to agents’ size. As a result, in a canonical formulation [2], the agents can be modeled as single-integrator dynamics, with the control input being its velocity. The agent’s internal controller then determines actual actuator signals to realize the commanded velocity. An important requirement in formation control is that the control law must be decentralized or distributed. To meet this requirement, each agent is considered as an independent system capable of measuring and/or communicating (via wireless channels) certain geometric variables related to neighboring agents. These variables may include global or relative positions, inter-agent distances, offset angles, and direction vectors. Based on the information accessible to each agent, formation control strategies are categorized into position-, displacement-, distance-, bearing-, and angle-based methods [2]. Generally, the complexity of the formation control problem is inversely proportional to the amount of information each agent can access, measure, or exchange. Recently, formation control algorithms based on bearing vectors have gained increasing attention [5]. The concept of control and localization using bearing measurements is inspired by the vision-based navigation observed in animal behavior [6, 7]. Bearing vectors can be acquired from onboard cameras, which provide data on the relative direction between agents. Compared to other approaches, bearing-based control reduces the number of sensors required per agent and minimizes dependence on a global reference frame [8]. Furthermore, since cameras are passive sensors, this approach is well-suited for military applications where signal emission must be limited [9]. Bearing rigidity theory and bearing-only stabilization of leaderless formations of single-integrators have been studied in [10, 11, 12]. While formation acquisition is a fundamental task, it is necessary to consider disturbances and uncertainties in any practical formation [13]. For formations with leaders, [14, 15] showed that formation control with disturbance and formation tracking are actually the same problem. Accordingly, different robust bearing-only control strategies for formation acquisition or tracking were proposed [16, 17, 18, 19, 20]. However, existing works either assumed availability of a-priori information on the disturbance’s upper bound and/or consider formations with several leaders to fix the scale of the target formation. For example, the authors in [16, 21, 17] considered the formation tracking problem with constant leaders’ velocity. Bearing-only formation under disturbances was considered in [20], where the upper bound of the disturbances are known. Bearing-only formation disturbance rejection and tracking with unknown disturbance’s upper bound using adaptive sliding-mode control were proposed in [15]. The work [19] proposed a finite-time bearing-only formation tracking strategy with unknown leader velocities based on the inverse of the minimum eigenvalue of local orthogonal projection matrices. However, the method can only be used with acyclic leader-follower graph topologies and under assumption that immediate leaders never take up collinearity positions. Bearing only leaderless formation stabilization was considered in [22, 23], where a robust adaptive gradient control strategy was employed to suppress the effects of the unknown disturbance and maintain the bearing error within a certain upper bound. However, the upper bound of the bearing error level in [22, 23] is state-dependent and cannot be computed. The issue of actuator faults and time-delay were considered in [24]. It is worth mentioning recent works based on sign-elevation angle rigidity and angle rigidity theory on stabilizing a desired formation using only local bearing measurements [25, 26, 27, 28, 29]. The control of leaderless formation with mismatched disturbances, however, has not been considered in these works. This paper studies a distributed control strategies to achieve a desired bearing-based formation in a leaderless multi-agent system of single-integrator dynamics, under the presence of deterministic, unknown but bounded disturbances. Two sensing scenarios are considered for each agent: (i) relative position measurements with respect to neighboring agents (bearing-based control), and (i) bearing vector measurements only (bearing-only control). To address these challenges, we develop adaptive sliding-mode control strategies that guarantee fast convergence and robustness against disturbances [30]. The main idea is to construct an adaptive mechanism that increases the control gain in response to the formation error, thereby progressively estimating the unknown upper bound of the disturbance [31, 32]. In particular, each bearing constraint specifying the desired formation is associated with an individual adaptive gain updating according to the error between the actual and desired bearing. By this way, by only seeing each others, the agents eventually move to the set of target formation. Since the proposed formation control law cannot reject mismatched disturbances, the target configuration is time-varying and disturbance-dependent. We characterize the time-varying target formation and establish uniform asymptotic stability of the target formation under the proposed control scheme. Furthermore, smooth adaptive control laws are developed to mitigate the chattering phenomenon and guarantee that the target formation remains uniformly ultimately bounded. The rest of this paper is organized as follows. Section I presents theoretical background on bearing rigidity theory and formulates the problems. Sections I and IV propose and analyse the formation control laws based on displacements and bearing vectors, respectively. Section V provides numerical simulations. Lastly, section VI concludes the paper. I Problem formulation I-A Bearing rigidity theory Consider a set of n non-collocated points in the d-dimensional space (n≥2n≥ 2, d≥2d≥ 2) i∈ℝdp_i ^d, with i≠j,∀i≠j,i,j≤np_i _j,~∀ i≠ j,~i,j≤ n. A framework (or a formation) in the d-dimensional space (,G,p) is defined by an undirected graph =(,ℰ)G=(V,E) and a configuration =[1⊤,…,n⊤]⊤∈ℝdnp=[p_1 ,…,p_n ] ^dn. Specifically, =1,…,nV=\1,…,n\ is the set of n vertices and ℰ⊂×E ×V is the set of m edges. Each edge (i,j)(i,j) in E joins two distinct vertices i,j∈Vi,j∈ V, and we do not distinguish between (i,j)(i,j) and (j,i)(j,i) as they denote a same edge. The neighbor set of a vertex i, denoted by iN_i, consists of all vertices j that are directly joined to i by edges (i,j)∈E(i,j)∈ E. We will also use e1,…,eme_1,…,e_m to denote the edges of ℰE indexing in a certain order. With ek=(i,j)∈ℰe_k=(i,j) , k=1,…,mk=1,…,m, the bearing vector from ip_i to jp_j is defined by ij=ij‖ij‖g_ij= z_ij\|z_ij\|, where ij=j−iz_ij=p_j-p_i is the displacement vector between ip_i and jp_j. Choosing an arbitrarily orientation and indexing of the edges in ℰE, we can define a corresponding incidence matrix =[hki]∈ℝm×nH=[h_ki] ^m× n of the graph as follows hki=1,ek=(i,j),−1,ek=(j,i),0,otherwise. h_ki= \ array[]rl1,&e_k=(i,j),\\ -1,&e_k=(j,i),\\ 0,&otherwise. array . Using the incidence matrix, we can write the displacement vector as =[…,ij⊤,…]⊤=[1⊤,…,m⊤]⊤=¯z=[…,z_ij ,…] =[z_1 ,…,z_m ] = Hp. For each vector ijg_ij, a corresponding orthogonal projection matrix ij=d−ijij⊤P_g_ij=I_d-g_ijg_ij can be defined. The matrix ijP_g_ij is symmetric positive semidefinite, idempotent ij2=ijP_g_ij^2=P_g_ij, and has spectrum 0,1,…,1\0,1,…,1\. For any vector ∈ℝdy ^d, ij=dP_g_ijy=0_d if and only if two vectors y and ijg_ij are parallel. Two formations (,)(G,p) and (,′)(G,p ) are bearing equivalent if and only if ij(j′−i′)=d,∀(i,j)∈ℰP_g_ij(p_j -p_i )=0_d,~∀(i,j) . The formations (,)(G,p) and (,′)(G,p ) are bearing congruent if and only if ij(j′−i′)=d,∀i,j∈,i≠jP_g_ij(p_j -p_i )=0_d,~∀ i,j ,i≠ j. A formation (,G,p) is globally bearing rigid if any formation having the same bearing constraints with (,)(G,p) is also bearing congruent with (,)(G,p). Let =[1⊤,…,m⊤]⊤∈ℝdmg=[g_1 ,…,g_m ] ^dm, the bearing rigidity matrix is defined by b=b()=∂≜blkdiag(k‖k‖)¯∈ℝdm×dn, _ b=R_ b(p)= ( P_g_k\|z_k\| ) H ^dm× dn, (4) where ¯=⊗d H=H _d. A formation is infinitesimally bearing rigid in ℝdR^d if and only if rank(b)=dn−d−1rank(R_ b)=dn-d-1, this means ker(b)=im([n⊗d,−n⊗¯])ker(R_ b)=im([1_n _d,p-1_n p]), where ¯=1n(n⊤⊗d) p= 1n(1 _n _d)p denotes the formation’s centroid. The augmented bearing rigidity matrix ~b=blkdiag(‖k‖⊗d)b=blkdiag(k)¯ R_ b=blkdiag(\|z_k\| _d)R_ b=blkdiag(P_g_k) H has the same rank and the same null space as bR_ b but does not contain information of the relative distances between the agents ‖k‖\|z_k\|. The bearing Laplacian b=~b⊤~bL_ b= R_ b R_ b is symmetric positive semidefinitea and is a special type of matrix-weighted Laplacian [33][Chapter 3]. For an infinitesimally rigid framework, ~b L_ b has exactly d+1d+1 zero eigenvalues and ker(~b)=( L_ b)=ker(~b)( R_ b). I-B Problem formulation We consider an n-agent system in the d-dimensional space. Assume that each agent i maintains a local coordinate system Σi^i , which is attached to each agent. The axes’ of n local coordinate systems are aligned, but the origins of these coordinate systems differ from each other. The agent’s motions are modeled by the equation ˙i=i+i,i=1,…,n, p_i=u_i+d_i,\,i=1,…,n, (5) where ip_i and id_i correspondingly denote the position of the agent and the disturbance, both written in a global coordinate system. The disturbance vector =[1⊤,…,n⊤]⊤d=[d_1 ,…,d_n ] is a bounded, time-varying, uniformly continuous functions. The upper bound of the disturbance is denoted as supt≥0‖∞=β>0 _t≥ 0\|d\|_∞=β>0, and this information is unknown to each agent. Let ∗=[(1∗)⊤,…,(n∗)⊤]⊤∈ℝdnp^*=[(p_1^*) ,…,(p^*_n) ] ^dn denote a desired formation (or target formation), which does not have any collocated points. The desired formation induces a set of desired bearing vectors Γ=ij∗=j∗−i∗‖j∗−i∗‖(i,j)∈ℰ =\g_ij^*= p_j^*-p_i^*\|p_j^*-p_i^*\|\_(i,j) . It is assumed that ∗p^* is infinitesimally bearing rigid in ℝdR^d. We define the set of formations which are bearing congruent to ∗p^* as =∈ℝdn|~b∗=dm,i≠j∀i,j∈,i≠jD=\p ^dn|~ R_ b^*p=0_dm,p_i _j∀ i,j ,i≠ j\, where ~b∗≜~b(∗)=∗¯ R_ b^* R_ b(p^*)=P_g^* H and ∗≜blkdiag(k∗)P_g^* (P_g_k^*). To control the local bearing constraints to match with those defined from the target formation, the agents need to sense some geometric variables with regard to their neighbors. Two types of relative sensing variables, namely, the displacements ij=j−i,z_ij=p_j-p_i, and the bearing vectors ij=j−i‖j−i‖,∀j∈ig_ij= p_j-p_i\|p_j-p_i\|,∀ j _i will be considered in this paper. Problem I.1 Suppose that each agent can sense the displacements with regard to its neighbors. Design control law for each agent using the available information so that p asymptotically converges to the set of formations which is bearing congruent to ∗p^*. Problem I.2 Suppose that each agent can sense the bearing vectors with regard to its neighbors. Design control law for each agent using the available information so that p converges to a formation which is bearing congruent to ∗p^*. I Adaptive bearing-based formation control with disturbance rejection In this section, we first propose a control law for Problem I.1. Second, we analyse the effect of disturbance on the formation acquisition process, describe the time-varying target formation via the set of desired bearing constraints and the mismatched disturbance. Convergence of the moving target formation is then established via Barbalat’s lemma. Finally, we propose a smooth adaptive formation control law that ensures global uniform ultimate boundedness of the desired moving formation. I-A Proposed control law We propose the following bearing-based control law to solve Problem I.1: i _i =−∑j∈iγijij∗sgn(ij),i=1,…,n, =- _j _i _ijP_g^*_ijsgn(q_ij),\,i=1,…,n, (6a) ij _ij =ij∗(i−j), =P_g^*_ij(p_i-p_j), (6b) γ˙ij γ_ij =kγ||ij||1,∀(i,j)∈ℰ, =k_γ | |q_ij | |_1,\,∀(i,j) , (6c) where ij∗=d−ij∗(ij∗)⊤P_g^*_ij=I_d-g^*_ij(g^*_ij) can be determined from the desired bearing vector ij∗g^*_ij, sgn(⋅)sgn(·) is the signum function defined element-wise for each element of a vector, γij _ij are adaptive gains satisfying γij(0)>0 _ij(0)>0, and kγ>0k_γ>0 is a positive constant. Let =[1⊤,…,n⊤]⊤u=[u_1 ,…,u_n ] , =[…,γij,…]⊤=[γ1,…,γm]⊤ γ=[…, _ij,…] =[ _1,…, _m] , =diag() =diag( γ), and ¯=⊗d = _d. The multiagent system under the proposed control law (6) can be expressed in the following form: ˙ p =−(~b∗)⊤¯sgn(~b∗)+, =-( R_ b^*) sgn ( R_ b^*p )+d, (7a) ˙ γ =kγ[‖1‖1,…,‖m‖1]⊤, =k_γ [\|q_1\|_1,…,\|q_m\|_1 ] , (7b) It is noted that the right-hand-side of Eq. (7a) is discontinuous and we understand the solution of (7) in Filippov sense [34]. Furthermore, (7b) implies that the adaptive gains γij _ij are non-decreasing in time. I-B Effects of the matched and mismatched disturbances Let ¯(t)=1n∑i=1ni=1n(n⊤⊗d) p(t)= 1n _i=1^np_i= 1n(1_n _d)p be the formation’s centroid at time t. Let (t)=1+2+3d(t)=d_1+d_2+d_3, where 1(t)∈im(~b(∗))d_1(t) ( R_ b(p^*)) is a matched disturbance which can be handled by the control input u, 2(t)∈im(n⊗d)d_2(t) (1_n _d) is the disturbance part corresponding to a translation of the whole formation, and 3(t)∈ker()∖im(n⊗d)d_3(t) (L) (1_n _d) is the disturbance part corresponding to a formation scaling about the formation’s centroid. It is noticed that dim(ker()∖im(n⊗d))=1dim(ker(L) (1_n _d))=1. Under the presence of the mismatched disturbances, the sliding-mode control law (6) has no effect outside im(~b)im( R_ b). The formation’s uncontrollable space is driven by the disturbance parts 2d_2 and 3d_3, respectively. Defining ()≜=−n⊗¯, (p) =p-1_n p, (8) and let dn≠∗∈ker()∖im(n⊗d)0_dn ^* (L) (1_n _d) be a nonzero constant vector. Without loss of generality, ∗r^* is selected to be ∗=∗−n⊗¯∗r^*=p^*-1_n p^*, where ¯∗=1n(n⊤⊗d)∗ p^*= 1n(1_n _d)p^*. Then, three components of the disturbance can be expressed as follows 1 _1 =(~b∗)⊤1(t), =( R_ b^*) f_1(t), (9a) 2 _2 =n⊗2(t), =1_n _2(t), (9b) 3 _3 =f3(t)^∗, =f_3(t) r^*, (9c) where 1(t)=[11⊤,…,1m⊤]⊤∈ℝdmf_1(t)=[f_11 ,…,f_1m ] ^dm, 1k∈ℝd,∀k=1,…,mf_1k ^d,∀ k=1,…,m, 2(t)∈ℝdf_2(t) ^d, ^∗≜∗‖∗‖ r^* r^*\|r^*\| and f3(t)∈ℝf_3(t) . It is not hard to verify that 1,2,3d_1,d_2,d_3 are mutually orthogonal to each other. The following additional assumption is imposed on the disturbance. Assumption I.1 The function f3(t)f_3(t) satisfies ∫0t|f3(t)|t<|⊤(0)^∗| _0^t|f_3(t)|dt<|r (0) r^*|, for all t≥0t≥ 0. Due to the assumption on the boundedness of d, the disturbance terms if_i are bounded and uniformly continuous. We can thus assume that there exist βi>0 _i>0 such that supt≥0‖i‖∞<βi _t≥ 0\|f_i\|_∞< _i, ∀i=1,2,3∀ i=1,2,3. The restricted effect of f3(t)f_3(t) in Assumption I.1 prevents n agents to shink into a point due to the scaling motion caused by disturbance. We can now prove the following lemma on the effects of the disturbance to the formation. Lemma I.1 Consider Problem I.1 and suppose that Assumption I.1 holds. Under the control law (6), the following claims hold. i. The formation’s centroid moves with the velocity 2(t)f_2(t). i. The formation’s scale s(t)=⊤(t)^∗s(t)=r (t) r^* satisfies |s(t)|>0|s(t)|>0, ∀t≥0∀ t≥ 0. Proof: i. For the first claim, we have ¯˙ p =1n(n⊤⊗d)˙ = 1n(1_n _d) p =−1n(n⊤⊗d)(~b∗)⊤(¯sgn(~b∗)−1) =- 1n(1_n _d)( R_ b^*) ( sgn ( R_ b^*p )-f_1 ) +1n(n⊤n⊗d)2+1n(n⊤⊗d)f3(t)∗‖∗‖ + 1n(1_n 1_n _d)f_2+ 1n(1_n _d)f_3(t) r^*\|r^*\| =2. =f_2. (10) since im(n⊤n⊗d)∈ker(~b∗) im(1_n 1_n _d)∈ ker( R_ b^*) and ∗⟂im(n⊗d)r^* im(1_n _d). i. Clearly, s˙=˙⊤∗=f3(t) s= r r^*=f_3(t). Thus, ||s(t)|−|s(0)||≤|s(t)−s(0)|≤∫0t|f3(τ)|τ<|s(0)|||s(t)|-|s(0)||≤|s(t)-s(0)|≤ _0^t|f_3(τ)|dτ<|s(0)| and it follows that |s(t)|>0,∀t≥0|s(t)|>0,∀ t≥ 0. ∎ It is noted that |s(t)|>0|s(t)|>0 is only necessary but sufficient for avoiding collisions between agents in the formation. We will consider collision avoidance among n agents after showing convergence of p to the set of target formation D. I-C Stability analysis From the analysis in the previous section, we can describe a time-varying target formation d(t)p^d(t) as follows • Centroid: ¯d(t)=¯(t) p^d(t)= p(t), • Scale: sd=s(t)s^d=s(t), • Bearing constraints: ij∗(id−jd)=dP_g_ij^*(p_i^d-p_j^d)=0_d, ∀(i,j)∈ℰ∀(i,j) . Thus, d≜n⊗¯(t)+nsd(t)∗‖∗‖ ^d 1_n p(t)+ ns^d(t) r^*\|r^*\| (11) is the time-varying target formation satisfying three aforementioned constraints. It follows from Lemma I.1 that d∈∖im(n),∀t≥0p^d im(1_n),∀ t≥ 0. Lemma I.2 Suppose that Assumption I.1 holds, the formation error ≜−d δ -p^d satisfies (t)⟂ker(b∗) δ(t) ker(L_ b^*), for all time t≥0t≥ 0. Proof: We have ˙ δ =−(~b∗)⊤¯sgn(~b∗)+1 =-( R_ b^*) sgn ( R_ b^*p )+d_1 =−(~b∗)⊤(¯sgn(~b∗)+1). =-( R_ b^*) ( sgn ( R_ b^*p )+f_1 ). (12) Thus, ˙⟂ker(~b∗)=ker(b∗) δ ker( R_ b^*)= ker(L_ b^*), which implies that [n⊗d,^∗]⊤(t)=[n⊗d,^∗]⊤(0)=d+1[1_n _d, r^*] δ(t)=[1_n _d, r^*] δ(0)=0_d+1. It follows that (t)⟂ker(b∗) δ(t) ker(L_b^*), for all t≥0t≥ 0. ∎ Using the fact that ∗¯=∗¯−∗¯d=∗P_g^* H δ=P_g^* Hp-P_g^* Hp^d=P_g^*z, equation (12) can be rewritten as a δ-dynamics follows ˙ δ =−(~b∗)⊤(¯sgn(~b∗)+1). =-( R_ b^*) ( sgn ( R_ b^* δ )+f_1 ). (13) The main result of this section is stated in the following theorem. Theorem I.1 (Stability of the desired formation) Consider Problem I.1 and suppose that Assumption I.1 is satisfied. Then, the following claims hold. i. If there exists T≥0T≥ 0 such that γij(T)>β1∀(i,j)∈ℰ _ij(T)> _1∀(i,j) , the system (13) is globally finite-time stable. i. In general case, (t)→dn δ(t) 0_dn as t→+∞t→+∞. In both cases, there exists a constant vector ∗∈ℝm γ^* ^m such that →∗ γ→ γ^*, as t→+∞t→+∞. Proof: i. For the first claim, consider the Lyapunov function V=12‖2V= 12\| δ\|^2, which is continuously differentiable, positive definite, radially unbounded, and being bounded by two class ∞K_∞ functions h12‖2 h_12\| δ\|^2 and h22‖2 h_22\| δ\|^2, for any 0<h1<1<h20<h_1<1<h_2. Then, we can compute the generalized gradient of V [34] for t≥Tt≥ T as ∂V=∂ V=\ δ\. The time derivative of V is then V˙∈a.eV~˙=⋃∈∂V⊤K[˙], V∈^a.e V= _ ξ∈∂ V ξ K[ δ], where K[sgn](⋅) K[ sgn](·) denotes the differential inclusion, which is defined as K[sgn](x)=1,x>0,−1,x<0,[−1,1],x=0, K[ sgn](x)= \ array[]rl1,&x>0,\\ -1,&x<0,\\ [-1,1],&x=0, array . for a real number x. Further, K[sgn](⋅) K[ sgn](·) is defined element-wise for a real vector. It follows that V˙ V =⊤(~b∗)⊤(−¯K[sgn](∗)+1) = δ ( R_ b^*) (- K[sgn] (P_g^*z )+f_1 ) =⊤∗(−¯K[sgn](∗)+1) =z P_g^* (- K[sgn] (P_g^*z )+f_1 ) =−∑(i,j)∈ℰ(γij∥ij∗ij∥1−ij⊤ij∗1k) =- _(i,j) ( _ij\|P_g_ij^*z_ij\|_1-z_ij P_g_ij^*f_1k ) ≤−∑(i,j)∈ℰ(γij−∥1k∥∞)∥ij∗ij∥1. ≤- _(i,j) ( _ij-\|f_1k\|_∞ )\|P_g_ij^*z_ij\|_1. Let ζ=min(i,j)∈ℰ(γij(T)−β1)=mink=1,…,m(γk(T)−β1)>0ζ= _(i,j) ( _ij(T)- _1)= _k=1,…,m( _k(T)- _1)>0, we have V˙ V ≤−ζ∑k=1m∥k∗k∥1≤−ζ∥∗∥1≤−ζ∥~b∗∥1 ≤-ζ _k=1^m\|P_g_k^*z_k\|_1≤-ζ\|P_g^*z\|_1≤-ζ\| R_ b^*p\|_1 ≤−ζ‖~b∗‖≤−ζ(⊤b∗)1/2. ≤-ζ\| R_ b^*p\|≤-ζ ( δ L_ b^* δ )^1/2. (17) Now, using the fact that (t)⟂ker(b∗) δ(t) ker(L_ b^*), we have V˙ V ≤−ζ(λd+1(b∗)⊤)1/2 ≤-ζ ( _d+1(L_ b^*) δ δ )^1/2 ≤−ζλd+1(b∗)‖ ≤-ζ _d+1(L_ b^*)\| δ\| ≤−ζ2λd+1(b∗)⏟≜ρV1/2. ≤- ζ 2 _d+1(L_ b^*)_ ρV^1/2. (18) Consider the differential equation y˙=−ρ′y1/2 y=-ρ y^1/2, y(t)=V(t)≥0y(t)=V(t)≥ 0, t≥Tt≥ T. Separating the variables gives dy=−ρdt dy y=-ρ dt, and thus, the solution can be found as y(t)=y(T)−12εt y(t)= y(T)- 12 t for T≤t≤T0T≤ t≤ T_0 and y(t)=0y(t)=0 for t≥T0≜2y(0)εt≥ T_0 2 y(0) . Based on comparison lemma [35, Section 3.4], V(t)≤y(t),∀t≥TV(t)≤ y(t),∀ t≥ T. Thus, V(t)=0V(t)=0 and =dn δ=0_dn for t≥T0t≥ T_0. Since =dn δ=0_dn implies that ij=dq_ij=0_d, γ˙ij= γ_ij=0 and thus γij=γij(T0) _ij= _ij(T_0), ∀t≥T0∀ t≥ T_0. i. Consider the Lyapunov function V=12‖2+12kγ‖−γ∗n‖2V= 12\| δ\|^2+ 12k_γ\| γ-γ^*1_n\|^2, for some γ∗>β1γ^*> _1. Then, V is continuously differentiable, positive definite and radially unbounded with regard to =[⊤,(−γ∗n)⊤]⊤ η=[ δ ,( γ-γ^*1_n) ] . Similar to the proof of Lemma I.1, we can compute V˙ V =−∑k=1m(γk∥k∗k∥1−k⊤k∗k) =- _k=1^m ( _k\|P_g_k^*z_k\|_1-z_k P_g_k^*f_k ) +∑k=1m(γk−γ∗)∥k∗k∥1 + _k=1^m( _k-γ^*)\|P_g_k^*z_k\|_1 ≤−∑k=1m(γ∗−β1)∥k∗k∥1 ≤- _k=1^m(γ^*- _1)\|P_g_k^*z_k\|_1 ≤−(γ∗−β1)∑k=1m∥k∗k∥ ≤-(γ^*- _1) _k=1^m\|P_g_k^*z_k\| ≤−(γ∗−β1)(⊤b∗)1/2 ≤-(γ^*- _1) ( δ L_ b^* δ )^1/2 ≤−(γ∗−β1)2λd+1(b∗)‖≤0, ≤-(γ^*- _1) 2 _d+1(L_ b^*)\| δ\|≤ 0, (19) which implies that δ, −∗ γ- γ^* are uniformly bounded, and so is γ. Moreover, there exists limt→+∞V(t)≥0 _t→+∞V(t)≥ 0. Since V˙ V is uniformly continuous, it follows from the Barbalat’s lemma [35] that V˙→0 V→ 0, as t→+∞t→+∞. Because ⟂ker(b∗) δ (L_ b^*), it follows that →dn δ 0_dn, as t→+∞t→+∞. Finally, as γijγ_ij are bounded and non-decreasing, there must exists finite constants γij∗ _ij^* such that γij→γij∗ _ij→ _ij^*, ∀(i,j)∈ℰ∀(i,j) . ∎ From Theorem I.1, we conclude that (t)→p(t) as desired. Now, we impose further requirements for collision avoidance. Corollary I.1 (Collision avoidance 1) Suppose that Assumption I.1 holds, and inft≥0‖id−jd‖−2nV(0)>0,∀i,j∈,i≠j, _t≥ 0\|p_i^d-p_j^d\|- 2nV(0)>0,\;∀ i,j ,\;i≠ j, (20) there will be no collision between n agents for all time t≥0t≥ 0. Proof: Consider the inequality ‖id−jd‖ \|p_i^d-p_j^d\| =‖(id−i)−(jd−j)+(i−j)‖ =\|(p_i^d-p_i)-(p_j^d-p_j)+(p_i-p_j)\| ≤‖i−id‖+‖j−jd‖+‖i−j‖, ≤\|p_i-p_i^d\|+\|p_j-p_j^d\|+\|p_i-p_j\|, ≤n∑i=1n‖i−id‖2+‖i−j‖ ≤ n _i=1^n\|p_i-p^d_i\|^2+\|p_i-p_j\| =n‖+‖i−j‖, = n\| δ\|+\|p_i-p_j\|, From the analysis in Theorem I.1, there holds ‖(t)‖≤2V(t)≤2V(0)\| δ(t)\|≤ 2V(t)≤ 2V(0). By combining ‖i−j‖≥‖id−jd‖−2nV(0)≥inft≥0‖ijd‖−2nV(0)\|p_i-p_j\|≥\|p_i^d-p_j^d\|- 2nV(0)≥ _t≥ 0\|z_ij^d\|- 2nV(0) with the inequality (20), the distance between any pair of agents i,j∈,i≠ji,j ,i≠ j maintains positive. ∎ Remark I.1 Although being the formation control law (6a) is expressed in the same form as a matrix-weighted consensus protocol [33][Chapter 9], it can be implemented using only bearing measurements since sgn(ij∗(i−j))=sgn(ij∗ij) sgn(P_g_ij^*(p_i-p_j))= sgn(P_g_ij^*g_ij). Thus, if the magnitude of the disturbance is known, (6a) provides a bearing-only stabilization control law. However, the adaptive mechanism (6c) uses displacements (in ‖ij‖\|q_ij\|) for adjusting γij _ij. We observe from simulations that a modification of adaptive law (6c) using only bearing measurements also work, however, did not found a proof for this fact. I-D A smooth bearing-based control law A main issue in implementing sliding-mode control laws is chattering, which is due to the discontinuities of the signum function. An effective strategy for avoiding chattering is compromising the control performance, for example, relaxing the requirement for perfect disturbance rejection. To this end, the following adaptive bearing-based control law is proposed for each agent i _i =−kp∑j∈ij−∑j∈iγijij‖ij‖+ε, =-k_p _j _iq_ij- _j _i _ij q_ij\|q_ij\|+ , (21a) γ˙ij γ_ij =kγ(‖ij‖2‖ij‖+ε−αγij),∀j∈i, =k_γ ( \|q_ij\|^2\|q_ij\|+ -α _ij ),\;∀ j _i, (21b) where ij=ij∗(i−j)q_ij=P_g_ij^*(p_i-p_j), ε>0 >0, and α>0α>0 are small positive constants, and kp>0k_p>0 is a positive control gain. Similar to the previous subsection, we can prove that if Assumption I.1 holds, under the smooth control law (21), both claims of Lemma I.1 are still valid. Therefore, we can also define the time-varying target formation dp^d as in Eq. (11). Let =−d δ=p-p^d, and ~g∗ϵ≜blkdiag(gk∗‖k‖+ε) P_ g^*^ε blkdiag ( P_ g^*_k\|q_k\|+ ), the δ-dynamics can be written as follows ˙ δ =−kpb∗−¯⊤~g∗(¯~g∗ϵ¯+1). =-k_pL_ b^* δ- H P_ g^* ( P_ g^*^ε H δ+f_1 ). (22) The main result of this subsection is stated in the following theorem. Theorem I.2 (Globally uniform ultimate boundedness of the desired formation) Consider Problem I.1 and suppose that Assumption I.1 holds. Under the control law (40), δ and γ are globally uniformly ultimately bounded, and the ultimate bound is adjustable by the control law’s parameters kp,kγ,ε,k_p,k_γ, , and α. Proof: The proof can be found in Appendix A-A. ∎ IV Adaptive bearing-only formation control with disturbance rejection This section focuses on Problem I.2, i.e., designing bearing-only formation control for n agents with disturbances. The complexity of the problem is significantly increased, as the agents need to simultaneously achieve all the desired bearing constraints and suppress the disturbances at the same time relying on only the measured bearing vectors. We will firstly propose an adaptive sliding-mode based bearing-only control law, and then provide a corresponding stability analysis. Then, we provide an analysis on a smooth adaptive bearing-only control law, which guarantees uniform ultimate boundedness of the desired formation. IV-A Proposed control law Consider the system of single-integrator agents with uncertainty (5). The proposed bearing-only control law for each agent i=1,…,ni=1,…,n is given as follows i _i =−∑j∈iγijijsgn(ijij∗), =- _j _i _ijP_g_ijsgn (P_g_ijg_ij^* ), (23a) γ˙ij γ_ij =kγ‖ijij∗‖1,γij(0)>0,∀j∈i. =k_γ\|P_g_ijg_ij^*\|_1,\; _ij(0)>0,\;∀ j _i. (23b) Since the right-hand-side of Eq. (24) is discontinuous, the solution of (24) is understood in Filippov sense. Clearly, γij(t)≥γij(0)>0 _ij(t)≥ _ij(0)>0 for all t≥0t≥ 0 and (i,j)∈ℰ(i,j) . IV-B Stability analysis In this subsection, the stability analysis of the n-agent system under the bearing-only control law (23) will be considered. We will first consider the following assumption on the disturbance in the stability analysis, as the target formation in this case is well-defined. Assumption IV.1 The disturbance d is uniformly continuous and bounded. Moreover ⟂d for all time t≥0t≥ 0. Under Assumption IV.1, the disturbance can be expressed as =~b⊤1+n⊗2d= R_ b f_1+1_n _2, where the upper bounds supt≥0‖1‖∞=β5>0 _t≥ 0\|f_1\|_∞= _5>0 and supt≥0‖2‖∞=β2>0 _t≥ 0\|f_2\|_∞= _2>0 are finite. By denoting =blkdiag(k)P_g= blkdiag(P_g_k), we can express the n-agent system under the control law (23a) in the matrix form as follows: ˙ p =~b⊤(¯sgn(∗)+1)+n⊗2 = R_ b ( sgn (P_gg^* )+f_1 )+1_n _2 =~b⊤(¯sgn(blkdiag(‖kd‖k)∗)+1)+n⊗2 = R_ b ( sgn (blkdiag(\|z_k^d\|P_g_k)g^* )+f_1 )+1_n _2 =~b⊤(¯sgn(d)+1)+n⊗2 = R_ b ( sgn (P_gz^d )+f_1 )+1_n _2 =−~b⊤(¯sgn((−d))−1)+n⊗2 =- R_ b ( sgn (P_g(z-z^d) )-f_1 )+1_n _2 =−~b⊤(¯sgn(¯(−d))−1)+n⊗2. =- R_ b ( sgn (P_g H(p-p^d) )-f_1 )+1_n _2. (24) Similarly, the vector form of (23b) is given as ˙=kγ[‖11∗‖1,…,‖mm∗‖1]⊤, γ=k_γ[\|P_g_1g_1^*\|_1,…,\|P_g_mg_m^*\|_1] , (25) where =[…,γij,…]⊤=[γ1,…,γm]⊤ γ=[…, _ij,…] =[ _1,…, _m] . We have the following lemmas on (24). Lemma IV.1 Consider the Problem I.2 and suppose that Assumption IV.1 holds. Under the control law (23), the following claims hold. i. The formation’s centroid ¯=1n(n⊤⊗d) p= 1n(1_n _d)p moves at velocity 2f_2 and the formation’s scale s(t)≜1n‖(t)‖s(t) 1 n\|r(t)\| is time-invariant. i. ‖ij‖=‖i−j‖≤2sn−1,∀i,j=1,…,n\|z_ij\|=\|p_i-p_j\|≤ 2s n-1,~∀ i,j=1,…,n, ∀t≥0∀ t≥ 0. i. Let d,u∈ℝdnp^d,p^u ^dn be two configurations that satisfy d ^d =n⊗¯+‖(0)‖∗‖∗‖=n⊗¯+ns∗‖∗‖, =1_n p+\|r(0)\| r^*\|r^*\|=1_n p+ ns r^*\|r^*\|, u ^u =n⊗¯−‖(0)‖∗‖∗‖=n⊗¯−ns∗‖∗‖. =1_n p-\|r(0)\| r^*\|r^*\|=1_n p- ns r^*\|r^*\|. Then, we have (a) Formation’s centroid: ¯d=¯u=¯(t)=2 p^d= p^u= p(t)=f_2, (b) Formation’s scale: s=sd=1n‖d‖=1n‖(0)‖=s(0)=1n|u|=sus=s^d= 1 n\|r^d\|= 1 n\|r(0)\|=s(0)= 1 n\|r^u\|=s^u, where d=(d)r^d=r(p^d) and u=(u)r^u=r(p^u) (c) Bearing vectors: jd−id‖jd−id‖=ij∗ p_j^d-p_i^d\|p_j^d-p_i^d\|=g_ij^*, and ju−iu‖ju−iu‖=−ij∗ p_j^u-p_i^u\|p_j^u-p_i^u\|=-g_ij^*, ∀i,j=1,…,n∀ i,j=1,…,n, i≠ji≠ j. Proof: i. Consider the system (24), we have ˙ p =1n(n⊤⊗d)~b(¯sgn(¯(−d))−1) = 1n(1_n _d) R_ b ( sgn (P_g H(p-p^d) )-f_1 ) +1n(n⊤⊗d)(n⊗2)=2, + 1n(1_n _d)(1_n _2)=f_2, s˙ s =⊤(˙−n⊗¯˙)n‖=0 = r ( p-1_n p ) n\|r\|=0 because ⟂im(~b)r im( R_ b) and ⟂im(n⊗d)r im(1_n _d). i. This proof is similar to [10][Corollary 2]. We have ‖i−¯‖2=‖∑j=1,j≠in(j−¯)‖2≤(n−1)∑j=1,j≠in‖j−¯‖2\|p_i- p\|^2=\| _j=1,j≠ i^n(p_j- p)\|^2≤(n-1) _j=1,j≠ i^n\|p_j- p\|^2, which implies that ‖i−¯‖2≤n−1n‖−n⊗¯‖2\|p_i- p\|^2≤ n-1n\|p-1_n p\|^2. Thus, ‖i−¯‖≤n−1s\|p_i- p\|≤ n-1s. Since ‖i−j‖≤‖i−¯‖+‖j−¯‖\|p_i-p_j\|≤\|p_i- p\|+\|p_j- p\|, we have ‖i−j‖≤‖i−¯‖+‖j−¯‖≤2n−1s\|p_i-p_j\|≤\|p_i- p\|+\|p_j- p\|≤ 2 n-1s. i. It is not hard to verify that ¯d p^d =1n(n⊤⊗d)d=¯(t), = 1n(1_n _d)p^d= p(t), d ^d =d−n⊗¯d=‖(0)‖∗‖∗‖, =p^d-1_n p^d=\|r(0)\| r^*\|r^*\|, sd s^d =‖d‖=||‖(0)‖∗‖∗‖||=|(0)|=s(0), =\|r^d\|= | |\|r(0)\| r^*\|r^*\| | |=\|r(0)\|=s(0), and a similar formulas can be derived for ¯u p^u and sus^u. ∎ We have the following lemma: Lemma IV.2 Consider Problem I.2 and suppose that Assumption IV.1 holds. Under the control law (23), if γij(0)>β5,∀(i,j)∈ℰ _ij(0)> _5,∀(i,j) , the desired configuration =dp=p^d is finite time stable. Proof: Let =−d=−d δ=p-p^d=r-r^d, it follows from (24) that ˙∈−~b⊤(¯K[sgn](¯)−1). δ∈- R_b ( K[sgn] (P_g H δ )-f_1 ). (27) The Lyapunov function V=12‖2V= 12\| δ\|^2 is continuously differentiable, positive definite and radially unbounded, and has V˙∈a.eV~˙ V∈^a.e V =−⊤¯⊤(~K[sgn](¯)−1) =- δ H P_g ( K[sgn] (P_g H δ )-f_1 ) =−∑k=1m(γk∥kd∥1−(kd)⊤k1k) =- _k=1^m ( _k\|P_g_kz^d\|_1-(z^d_k) P_g_kf_1k ) ≤−∑k=1m(γk(0)−∥1k∥∞)∥kkd∥1. ≤- _k=1^m( _k(0)-\|f_1k\|_∞)\|P_g_kz_k^d\|_1. (28) Let ζ=mink∈1,…,m(γk(0)−β5)>0ζ= _k∈\1,…,m\( _k(0)- _5)>0, then we have V˙≤−ζ∑k=1m∥kdk∥1≤−ζ∥d∥1≤−ζ∥d∥2. V≤-ζ _k=1^m\|P_g_kz^d_k\|_1≤-ζ\|P_gz^d\|_1≤-ζ\|P_gz^d\|_2. (29) By noting that kd=‖kd‖k∗z^d_k=\|z^d_k\|g_k^*, we can write ‖d‖22 \|P_gz^d\|_2^2 =(d)⊤d =(z^d) P_gz^d =∑k=1m(kd)⊤kkd = _k=1^m(z^d_k) P_g_kz^d_k =∑k=1m‖kd‖2(k∗)⊤kk∗. = _k=1^m\|z^d_k\|^2(g^*_k) P_g_kg^*_k. (30) Moreover, it follows from OPEN(ij∗)⊤ijij∗=(ij∗)⊤(d−ijij⊤)ij∗=1−(ij⊤)ij∗)2=(ij)⊤(d−ij∗ij∗⊤)ij=(ij)⊤ij∗ij(g^*_ij) P_g_ijg^*_ij=(g^*_ij) (I_d-g_ijg_ij )g^*_ij=1-(g_ij )g^*_ij)^2=(g_ij) (I_d-g_ij^*g_ij^* )g_ij=(g_ij) P_g_ij^*g_ij, equation (29) and inequality (30) that V˙ V ≤−ζ(∑k=1m‖kd‖2‖k‖2k⊤k∗k)12 ≤-ζ ( _k=1^m \|z_k^d\|^2\|z_k\|^2z_k P_g_k^*z_k ) 12 ≤−ζ(mink‖kd‖24(n−1)s2)12⏟≜ς(∑k=1mk⊤k∗k)12 ≤- ζ ( _k\|z_k^d\|^24(n-1)s^2 ) 12_ ( _k=1^mz_k P_g_k^*z_k ) 12 ≤−ς(⊤∗)12 ≤- (z P_g^*z ) 12 ≤−ς(⊤¯⊤∗¯)12 ≤- ( δ H P_g^* H δ ) 12 ≤−ς(⊤b∗)12≤0. ≤- ( δ L_ b^* δ ) 12≤ 0. (31) Furthermore, V˙=0 V=0 if and only if ∈ker(b(∗)) δ (L_b(p^*)). It follows from Lemma IV.1 that =dp=p^d or =up=p^u. To establish instability of up^u under the proposed control law (23), we consider the Lyapunov function W=12‖′‖2W= 12\| δ \|^2, where ′≜−u δ -p^u and the ′ δ -dynamics. Since sgn(¯′)=−sgn(¯u)=−sgn(u)=sgn(d)=sgn(∗) sgn(P_g H δ )=- sgn(P_g Hp^u)=- sgn(P_gz^u)= sgn(P_gz^d)= sgn(P_gg^*), we can write W˙ W =−(′)⊤¯⊤(−~K[sgn](¯′)−1) =-( δ ) H P_g (- K[sgn] (P_g H δ )-f_1 ) =∑k=1m(γij‖kku‖1−1k⊤kku) = _k=1^m( _ij\|P_g_kz^u_k\|_1-f_1k P_g_kz^u_k) ≥∑k=1m(γij−β5)‖kku‖1≥0. ≥ _k=1^m( _ij- _5)\|P_g_kz^u_k\|_1≥ 0. (32) For any open ball centered at dn0_dn which does not contain d0^d and ′=d−u δ =p^d-p^u, W˙>0 W>0. Based on Chetaev’s instability theorem [35], we conclude that up^u is unstable. Due to the existence of disturbance, the system cannot stay identically at up^u. Thus, there exists a finite time t1≥0t_1≥ 0 such that (t1)≠up(t_1) ^u. Thus, for t≥t1t≥ t_1, V˙ V is negative definite, which implies that (t)→dp(t) ^d asymptotically. Decomposing =||+⟂ δ= δ^||+ δ , where ||∈ker(b∗) δ^|| (L_ b^*), and ⟂∈im(b∗) δ (L_ b^*) as illustrated in Fig. 1. Since ‖+d‖=‖d‖=ns(0)\| δ+r^d\|=\|r^d\|= ns(0) is time-invariant, let α(t)∈[0,π2)α(t)∈[0, π2) denote the angle between (t) δ(t) and −∗-r^*, for t≥t1t≥ t_1, it follows that ⊤b∗ δ L_b^* δ =(⟂)⊤b∗(⟂) =( δ ) L_ b^*( δ ) ≥λd+2(b∗)‖⟂‖2 ≥ _d+2(L_ b^*)\| δ \|^2 ≥λd+2(b∗)sin2α‖2. ≥ _d+2(L_ b^*) ^2α\| δ\|^2. As (t1)≠up(t_1) ^u, and p asymptotically converges to dp^d, for t≥t1t≥ t_1, we have α is non-decreasing. Thus, V˙≤−ςsin(α(t1))2λd+2(b∗)⏟≜ϱ‖2≤−ϱV12, V≤- (α(t_1)) 2 _d+2(L_ b^*)_ \| δ\| 2≤- V 12, (33) which implies that V(t)→0V(t)→ 0 in finite time by a similar argument as in the proof of Theorem I.1. ∎ The main result of this section is given in the following theorem. Theorem IV.1 Consider the Problem I.2 and suppose that Assumption IV.1 holds. Under the adaptive bearing-only control law (23), the following claims hold. i. (t)→dp(t) ^d as t→+∞t→+∞, i. There exists a constant vector ∗ γ^* such that →∗ γ→ γ^*, as t→+∞t→+∞, i. Additionally, if γi∗>β5,∀i _i^*> _5,~∀ i, and there exists a finite time T such that |γi(T)−γi∗|<mini=1,…,m|γi(T)−β5|,∀i=1,…,n,| _i(T)- _i^*|< _i=1,…,m| _i(T)- _5|,~∀ i=1,…,n, then (t)→dp(t) ^d in finite time. Proof: i. Consider the Lyapunov function V=12‖2+12k∑k=1m‖kd‖‖γk−β6n‖2V= 12\| δ\|^2+ 12k _k=1^m\|z_k^d\|\| _k- _61_n\|^2, for some β6>β5 _6> _5. Similar to Lemma IV.2, we have V˙∈a.eV~˙ V∈^a.e V =−⊤¯⊤(~K[sgn](¯)−1) =- δ H P_g ( K[sgn] (P_g H δ )-f_1 ) +∑k=1m∥kd∥(γk−β5)∥k∗∥1 + _k=1^m\|z^d_k\|( _k- _5)\|P_g_kg_k^*\|_1 =−∑k=1m(γk∥kkd∥1−(kd)⊤k1k) =- _k=1^m ( _k\|P_g_kz^d_k\|_1-(z^d_k) P_g_kf_1k ) +∑k=1m(γk−β6)∥kkd∥1 + _k=1^m( _k- _6)\|P_g_kz_k^d\|_1 (34) ≤−∑k=1m(β6−∥1k∥∞)∥kkd∥1. ≤- _k=1^m ( _6-\|f_1k\|_∞ )\|P_g_kz_k^d\|_1. For β6>β5≥supt≥0‖1‖∞ _6> _5≥ _t≥ 0\|f_1\|_∞, and thus ζ=β6−β5>0ζ= _6- _5>0. We obtain V˙≤−ζ‖kkd‖1 V≤-ζ\|P_g_kz_k^d\|_1. Similar to the proof of Theorem IV.1, we can eventually prove that V˙≤−φ(⊤b∗)12≤0 V≤- ( δ L_ b^* δ ) 12≤ 0, for some φ>0 >0. It follows that δ and γ are uniformly bounded and there exists finite limt→+∞V(t)≤V(0) _t→+∞V(t)≤ V(0). Due to the uniformly continuity of f, it follows from (34) that V˙ V is uniformly continuous. As assumptions of the Barbalat’s lemma is satisfied, we conclude that V˙→0 V→ 0. However, V˙=0 V=0 if and only if =dp=p^d or =up=p^u. The instability of up^u can be similarly proved as in Theorem IV.1, using the function W=12‖′‖2+12kγ∑k=1m‖kd‖‖γk−β6n‖2W= 12\| δ \|^2+ 12k_γ _k=1^m\|z_k^d\|\| _k- _61_n\|^2. Therefore, →dp ^d as t→+∞t→+∞. i. This claim follows from the fact that γk,k=1,…,m, _k,k=1,…,m, are non-decreasing and upper bounded. i. If the assumptions of this claim hold, then γ(T)≥β5γ(T)≥ _5. The claim follows from Theorem IV.1. ∎ The collision avoidance condition in Corollary I.1 still holds, and we can prove the following corollary. Corollary IV.1 (Collision avoidance 2) Under the control law (23), if inft≥0‖id−jd‖−2nV(0)>0,∀i,j∈,i≠j, _t≥ 0\|p_i^d-p_j^d\|- 2nV(0)>0,\;∀ i,j ,\;i≠ j, (35) no collision happens between n agents for all time t≥0t≥ 0. Fig. 1: The diagram describes the system (38). If the scaling disturbance component 3d_3 is zero, then the system is constrained on ‖+d‖=‖d‖=‖d(0)‖\| δ+r^d\|=\|r^d\|=\|r^d(0)\|. If the scaling disturbance 3d_3 is nonzero, then ‖\| δ\| may increase or decrease. Remark IV.1 (Effects of scaling disturbance) We consider the disturbance d to consist of an additional term 3=f3(t)‖d_3=f_3(t) r\|r\| corresponding to a scaling of the whole formation. In this case, ˙=−~b⊤(sgn(~b)−1)+n⊗2+f3‖. p=- R_ b ( sgn( R_ b δ)-f_1)+1_n _2+f_3 r\|r\|. (36) With s=1n‖s= 1 n\|r\|, we have s˙ s =⊤n‖(~b⊤(sgn(~b)−1)+n⊗2+f3‖) = r n\|r\| ( R_ b ( sgn( R_ b δ)-f_1)+1_n _2+f_3 r\|r\| ) =f3⊤n‖2=f3n. = f_3r r n\|r\|^2= f_3 n. The desired formation is defined as d=n⊗¯+‖∗‖∗‖=n⊗¯+ns∗‖∗‖. ^d=1_n p+\|r\| r^*\|r^*\|=1_n p+ ns r^*\|r^*\|. (37) Let =−d=ns(‖−∗‖∗‖)=−‖∗‖∗‖=−d δ=p-p^d= ns ( r\|r\|- r^*\|r^*\| )=r-\|r\| r^*\|r^*\|=r-r^d, we have ˙ δ =−~b⊤(sgn(~b)−1)+f3(‖−∗‖∗‖) =- R_ b ( sgn( R_ b δ)-f_1 )+f_3 ( r\|r\|- r^*\|r^*\| ) =−~b⊤(sgn(~b)−1)+f3‖ =- R_ b ( sgn( R_ b δ)-f_1 )+ f_3\|r\| δ (38) For simplicity, suppose that γij _ij are sufficient large to suppress the disturbance term 1d_1, we consider the Lyapunov function V=12‖2V= 12\| δ\|^2, we have V˙ V OPEN=⊤K[˙]=⊤(−~b⊤(K[sgn](~b)−1))+f3‖) = δ K[ δ]= δ (- R_ b ( K[ sgn]( R_ b δ)-f_1) )+ f_3\|r\| δ ) ≤−ζmink‖kd‖2n−1s⊤b∗+1nsf3‖2, ≤- ζ _k\|z_k^d\|2 n-1s δ L_ b^* δ+ 1 nsf_3\| δ\|^2, (39) where the derivation of immediate inequalities are similar to those in Lemma IV.2 and Lemma IV.1 i.b. As depicted in Fig. 1, the 3d_3 changes the magnitude of d=s^∗r^d=s r^*, which in turn may increase or decrease ‖\| δ\| depending on the sign of f3(t)f_3(t). As a result, the angle α(t)α(t) may not monotonically increasing as in the analysis of Lemma IV.2. V˙≤−(ζmink‖kd‖2n−1sλd+2(∗)sin2α−1nssupt≥0|f3|)‖2 V≤- ( ζ _k\|z_k^d\|2 n-1s _d+2(L^*) ^2α- 1 ns _t≥ 0|f_3| )\| δ\|^2 However, we can still assume that f3f_3 is sufficiently small, so that the formation scale are bounded sM>|s|>sm>0s_M>|s|>s_m>0, ∀t≥0∀ t≥ 0. In this case, suppose further that α(0)α(0) is sufficiently large, we have V˙≤0 V≤ 0. Fig. 1 illustrates a scenario where the scaling disturbance perturbs δ to ~=+3 δ= δ+d_3, with ‖~‖>‖\| δ\|>\| δ\|. Due to the control input, δ is actually controlled to ′ δ , with ‖′‖<‖\| δ \|<\| δ\| and α′>α >α. As α increases, the inequality of V˙<0 V<0 holds with ‖≠0\| δ\|≠ 0, and thus, →dn δ 0_dn. IV-C A smooth adaptive bearing-only control law A main issue in implementing sliding-mode control laws is the chattering phenomenon, which arises from the discontinuity of the signum function. To mitigate this effect, we modify the bearing-only control law (23) for each agent as follows i _i =−kp∑j∈ijij∗−∑j∈iγijijij∗‖ijij∗‖+ε, =-k_p _j _iP_g_ijg_ij^*- _j _i _ij P_g_ijg_ij^*\|P_g_ijg_ij^*\|+ , (40a) γ˙ij γ_ij =kγ(‖ijij∗‖2‖ijij∗‖+ε−αγij),∀j∈i, =k_γ ( \|P_g_ijg_ij^*\|^2\|P_g_ijg_ij^*\|+ -α _ij ),\;∀ j _i, (40b) where ε,α ,α are small positive constants, kpk_p, kγk_γ are positive control gain and adaptive rate, respectively. We rewrite equation (40) governing the motion of n-agent system in matrix-form as follows ˙ p =kp~b⊤∗+¯⊤¯ε∗+ =k_p R_ b g^*+ H P_g g^*+d =kp~b⊤∗+¯⊤¯ε∗+~b⊤1+n⊗2 =k_p R_ b g^*+ H P_g P_g g^*+ R_ b f_1+1_n _2 =kp~b⊤∗+~b⊤(¯ε∗+1)+n⊗2, =k_p R_ b g^*+ R_ b ( P_g g^*+f_1 )+1_n _2, (41) where ε≜blkdiag(k‖kk∗‖+ε)P_g blkdiag ( P_g_k\|P_g_kg_k^*\|+ ), and we have used the fact that ε=ε=εP_gP_g =P_g P_g=P_g . It is not hard to verify that ˙(t)⟂ker(~b) p(t) ker( R_ b), and thus all claims of Lemma IV.1 hold with the system (41), (40b). Let =−d δ=p-p^d, the main result of this subsection is provided in the following theorem. Theorem IV.2 (Uniform ultimate boundedness of the desired formation) Suppose that the assumptions of Problem I.2 hold. Under the control law (40), δ and γ are globally uniformly ultimately bounded, and the ultimate bound is adjustable by the control law’s parameters kp,kγ,ε,k_p,k_γ, , and α. Proof: The proof is provided in Appendix A-B. ∎ Finally, we remark that a similar collision avoidance condition in Corrolary IV.1 holds for the system (41), (40b). V Simulation results In this section, we provide simulation examples to demonstrate the effectiveness of the proposed formation control laws proposed in Sections I and IV. In each simulation, we consider a system of 6 agents in the two-dimensional space with the interaction graph G as depicted in Fig. 2. The desired formation is specified by a set of 12 desired bearing vectors, corresponding to a six-regular polygon configuration ∗p^* with 1∗=[32,12]⊤p_1^*=[ 32, 12] and i+1∗=(π3)p_i+1^*=R ( π3 ), for i=2,…,6i=2,…,6 and (α)=[cosα−sinαsinαcosα]R(α)= bmatrix α&- α\\ α& α bmatrix. The initial values of agents’ positions and adaptive gains in all simulations are chosen as in Table I. TABLE I: The initial positions k(0)p_k(0), k=1,…,6,k=1,…,6, and adaptive gains γk(0),k=1,…,12 _k(0),k=1,…,12. k 1 2 3 4 5 6 kp_k [1.96−3.24] bmatrix1.96\\ -3.24 bmatrix [0.52.35] bmatrix0.5\\ 2.35 bmatrix [−3.39−0.2] bmatrix-3.39\\ -0.2 bmatrix [−5.6−0.3] bmatrix-5.6\\ -0.3 bmatrix [−2.0−5.3] bmatrix-2.0\\ -5.3 bmatrix [3.8−1.2] bmatrix3.8\\ -1.2 bmatrix γk _k 0.13 0.51 0.18 0.79 0.85 0.5 γ6+k _6+k 0.85 0.08 0.51 0.07 0.43 0.1 123456 Fig. 2: A regular graph G of six-vertices in simulation. V-A Simulation 1: Bearing-based control First, we simulate the six-agent system under the control law (6), with kγ=0.1k_γ=0.1. The disturbance vector d consists of three terms as described in (9), with parameters 1 _1 =14b∗(2⊗[1,2,3,4,5,6,0,0.5,1,1.5,2,2.5]⊤), = 14L_ b^*(1_2 [1,2,3,4,5,6,0,0.5,1,1.5,2,2.5] ), 2 _2 =[3+cos(0.125πt),1.5sin(πt/6)]⊤, = [3+ (0.125π t),1.5 (π t/6 ) ] , f3 f_3 =0.125(8e−0.1t+sin(πt/10)). =0.125 (8e^-0.1t+ (π t/10 ) ). (a) (b) (c) Fig. 3: Simulation of six-agent system under the adaptive sliding-mode bearing-based control law (6). (a) (b) (c) Fig. 4: Simulation of 6-agent system under the adaptive bearing-based control law (21). Simulation results for t∈[0,50]t∈[0,50] (second) are given in Fig. 3. The matched disturbances are compensated along with the attainment of the set of desired bearing vectors. Bearing errors ‖−∗‖\|g-g^*\| converges to 0 and γk _k converge to some constant values for all k=1,…,12k=1,…,12 after about 5 seconds. The formation is driven by two disturbance terms 2d_2 (translation velocity 2f_2) and 3d_3 (formation rescales at rate f3(t)f_3(t)) in the set of desired formations. Second, we consider the system under the smooth adaptive bearing-based control law (21), with kγ=0.2k_γ=0.2, ε=0.05 =0.05 and α=0.05α=0.05. The mismatched disturbance terms are changed correspondingly to 2=[5cos(0.04πt),2sin(0.04πt)]⊤f_2=[5 (0.04π t),2 (0.04π t)] and f3=0.25(3e−0.1t+sin(πt10))f_3=0.25 (3e^-0.1t+ ( π t10 ) ). Simulation results for t∈[0,60]t∈[0,60] (seconds) are depicted in Fig. 4. After 5 seconds, the bearing errors ‖(t)−∗‖\|g(t)-g^*\| are kept within the interval [0,5⋅10−3][0,~5· 10^-3]. The adaptive gains do not converge to fixed values, and also vary within a positive, and bounded interval. The formation moves with velocity 2f_2 and changes the scale due to the mismatched disturbances 2d_2 and 3d_3, respectively. V-B Simulation 2: Bearing-only control In this subsection, we first consider the six-agent system under the adaptive sliding-mode bearing-only control law (23), with kγ=0.25k_γ=0.25. The functions 1,2,f3f_1,f_2,f_3 are selected the same as in Simulation 1, but the disturbances’ components are 1=b⊤1d_1=R_ b f_1 and 3=f3^d_3=f_3 r. From the simulation results displayed in Fig. 5, the adaptive gains converge to constant values, the desired bearing constraints are almost achieved after 10 seconds, and the bearing error is kept smaller than 5⋅10−45· 10^-4 for t≥10t≥ 10. The motions of the formation consists of a translation with velocity 2f_2 and a scaling motion f3^f_3 r. . (a) (b) (c) Fig. 5: Simulation of six-agent system under the adaptive sliding-mode bearing-only control law (23). Second, we consider the six-agent system under the smooth adaptive bearing-only control law (40). The disturbance components are changed to 2=[2.5cos(0.015πt),sin(0.015πt)]⊤f_2=[2.5 (0.015π t), (0.015π t)] and f3=0.75e−0.1t+sin(0.015πt)f_3=0.75e^-0.1t+ (0.015π t). (a) (b) (c) Fig. 6: Simulation of six-agent system under the smooth bearing-only control law (40). With the control parameters selected as kp=4k_p=4, kγ=0.5k_γ=0.5, ε=0.03 =0.03 and α=0.02α=0.02, the simulation results of the system for 0≤t≤1200≤ t≤ 120 are given in Fig. 6. Observe that the bearing error is maintained smaller than 0.05 for t≥20t≥ 20 (second). The adaptive gains do not converge, and vary within some positive, small intervals. Thus, we conclude that simulation results are consistent with the analysis. VI Conclusions This paper investigates bearing-constrained formation control under unknown bounded disturbances using both displacement-based and bearing-only measurements. The disturbances were decomposed into matched and mismatched components, which introduce deterministic time-varying perturbations to the target formation. Adaptive sliding-mode controllers were proposed to uniformly stabilize the set of desired formations and completely reject the matched disturbance. Furthermore, simple smooth adaptive control laws were proposed to ensure uniform ultimate boundedness of the target formation and eliminate chattering phenomenon. While this paper confines the focus on bearing-constrained formation, the adaptive control design strategies in this paper are applicable to other categories of leaderless formation control schemes. References [1] B. D. O. Anderson, C. Yu, B. Fidan, and J. M. Hendrickx, “Rigid graph control architectures for autonomous formations,” IEEE Control Systems Magazine, vol. 28, no. 6, p. 48–63, 2008. [2] K.-K. Oh, M.-C. Park, and H.-S. Ahn, “A survey of multi-agent formation control,” Automatica, vol. 53, p. 424–440, 2015. [3] T. Eren, W. Whiteley, A. S. Morse, P. N. Belhumeur, and B. D. O. Anderson, “Sensor and network topologies of formations with direction, bearing, and angle information between agents,” in Proc. of the 42nd IEEE Conference on Decision and Control, Maui, HI, USA, vol. 3. IEEE, 2003, p. 3064–3069. [4] S. Zhao and D. Zelazo, “Localizability and distributed protocols for bearing-based network localization in arbitrary dimensions,” Automatica, vol. 69, p. 334–341, 2016. [5] H. Su, Z. Yang, S. Zhu, C. Chen, X. Guan, and L. Xie, “Bearing-based multi-agent formation control: A survey and taxonomy,” Annual Reviews in Control, vol. 61, p. 101043, 2026. [6] E. Montijano, E. Cristofalo, D. Zhou, M. Schwager, and C. Sagues, “Vision-based distributed formation control without an external positioning system,” IEEE Transactions on Robotics, vol. 32, no. 2, p. 339–351, 2016. [7] R. Tron, J. Thomas, G. Loianno, K. Daniilidis, and V. Kumar, “A distributed optimization framework for localization and formation control: Applications to vision-based measurements,” Control Systems Magazine, vol. 36, no. 4, p. 22–44, 2016. [8] S. Doodeman, Z. Tang, M. Jacinto, R. Cunha, and C. Silvestre, “Leader-follower formation tracking control of quadrotor uavs using bearing measurements,” in Proc. of the IEEE Conference on Control Technology and Applications (CCTA). IEEE, 2025, p. 970–975. [9] M. Ye, B. D. O. Anderson, and C. Yu, “Bearing-only measurement self-localization, velocity consensus and formation control,” IEEE Transactions on Aerospace and Electronic Systems, vol. 53, no. 2, p. 575–586, 2017. [10] S. Zhao and D. Zelazo, “Bearing rigidity and almost global bearing-only formation stabilization,” IEEE Transactions on Automatic Control, vol. 61, no. 5, p. 1255–1268, 2016. [11] E. Schoof, A. Chapman, and M. Mesbahi, “Bearing-compass formation control: A human-swarm interaction perspective,” in Proc. of the 2014 American Control Conference, 2014, p. 3881–3886. [12] Q. V. Tran, M. H. Trinh, D. Zelazo, D. Mukherjee, and H.-S. Ahn, “Finite-time bearing-only formation control via distributed global orientation estimation,” IEEE Transactions on Control of Network Systems, vol. 6, no. 2, p. 702–712, 2019. [13] S. Aldarmini, A. Vedyakov, and M. Kakanov, “Formation control for multi-agent system with disturbances compensation,” IFAC-PapersOnLine, vol. 55, no. 13, p. 222–227, 2022, proc. of the 9th IFAC Conference on Networked Systems NECSYS. [14] S. Zhao and D. Zelazo, “Translational and scaling formation maneuver control via a bearing-based approach,” IEEE Transactions on Control of Network Systems, vol. 4, no. 3, p. 429–438, 2015. [15] T. T. Nguyen, D. V. Vu, T. V. Pham, and M. H. Trinh, “Bearing-constrained leader-follower formation of single-integrators with disturbance rejection: Adaptive variable-structure approaches,” IEEE Transactions on Cybernetics, vol. 54, no. 11, 2024. [16] J. Zhao, X. Yu, X. Li, and H. Wang, “Bearing-only formation tracking control of multi-agent systems with local reference frames and constant-velocity leaders,” IEEE Control Systems Letters, vol. 5, no. 1, p. 1–6, 2020. [17] M. H. Trinh, Q. V. Tran, D. V. Vu, P. D. Nguyen, and H.-S. Ahn, “Robust tracking control of bearing-constrained leader–follower formation,” Automatica, vol. 131, p. 109733, 2021. [18] Y.-B. Bae, H.-S. Ahn, and Y.-H. Lim, “Leader-follower bearing-based formation system with exogenous disturbance,” in Proc. of the 13th International Symposium on Applied Computational Intelligence and Informatics (SACI). IEEE, 2019, p. 39–44. [19] M. H. Trinh and H.-S. Ahn, “Finite-time bearing-based maneuver of acyclic leader-follower formations,” IEEE Control Systems Letters, vol. 6, p. 1004–1009, 2021. [20] J. J. M., A. Dinesh, and A. K. Mulla, “Bearing-only formation control of multi-agent systems using a signed protocol,” in Proc. of the 2024 European Control Conference (ECC), 2024, p. 2487–2492. [21] S. Li, Q. Wang, E. Wang, and Y. Chen, “Bearing-only adaptive formation control using back-stepping method,” Frontiers in Control Engineering, vol. 2, p. 700053, 2021. [22] Y.-B. Bae, S.-H. Kwon, Y.-H. Lim, and H.-S. Ahn, “Distributed bearing-based formation control and network localization with exogenous disturbances,” International Journal of Robust and Nonlinear Control, vol. 32, no. 11, p. 6556–6573, 2022. [23] Y.-B. Bae, Y.-H. Lim, and H.-S. Ahn, “Distributed robust adaptive gradient controller in distance-based formation control with exogenous disturbance,” IEEE Transactions on Automatic Control, vol. 66, no. 6, p. 2868–2874, 2020. [24] K. Wu, J. Hu, Z. Li, Z. Ding, and F. Arvin, “Distributed collision-free bearing coordination of multi-uav systems with actuator faults and time delays,” IEEE Transactions on Intelligent Transportation Systems, 2024. [25] G. Jing, G. Zhang, H. W. J. Lee, and L. Wang, “Angle-based shape determination theory of planar graphs with application to formation stabilization,” Automatica, vol. 105, p. 117–129, 2019. [26] L. Chen, M. Cao, and C. Li, “Angle rigidity and its usage to stabilize multiagent formations in 2-D,” IEEE Transactions on Automatic Control, vol. 66, no. 8, p. 3667–3681, 2020. [27] L. Chen, Z. Lin, H. G. De Marina, Z. Sun, and M. Feroskhan, “Maneuvering angle rigid formations with global convergence guarantees,” IEEE/CAA Journal of Automatica Sinica, vol. 9, no. 8, p. 1464–1475, 2022. [28] H. L. Cheah and M. Deghat, “Finite-time and fixed-time formation tracking control with unknown disturbance rejection using only local bearing measurements,” Journal of the Franklin Institute, p. 108347, 2025. [29] C. Garanayak and D. Mukherjee, “Sign-elevation angle rigidity for 3-D and almost global bearing-only formation stabilization in agents’ local frames,” IEEE Transactions on Control of Network Systems, p. 1–12, 2025. [30] H. Obeid, L. M. Fridman, S. Laghrouche, and M. Harmouche, “Barrier function-based adaptive sliding mode control,” Automatica, vol. 93, p. 540–544, 2018. [31] T. R. Oliveira, J. P. V. Cunha, and L. Hsu, “Adaptive sliding mode control for disturbances with unknown bounds,” in Proc. of the 14th International Workshop on Variable Structure Systems (VSS). IEEE, 2016, p. 59–64. [32] S. Roy, S. Baldi, and L. M. Fridman, “On adaptive sliding mode control without a priori bounded uncertainty,” Automatica, vol. 111, p. 108650, 2020. [33] M. H. Trinh and H.-S. Ahn, Matrix-weighted graphs: Theory and Applications, ser. Lecture Notes in Control and Information Sciences 498. Switzerland: Springer - Cham, 2026. [34] D. Shevitz and B. Paden, “Lyapunov stability theory of nonsmooth systems,” IEEE Transactions on Automatic Control, vol. 39, no. 9, p. 1910–1914, 1994. [35] H. K. Khalil, Nonlinear systems, 3rd ed. Upper Saddle River, NJ, Prentice Hall, 2002. Appendix A Proofs A-A Proof of Theorem I.2 Using the Lyapunov function V=12‖2+12kγ‖−β4n‖2V= 12\| δ\|^2+ 12k_γ\| γ- _41_n\|^2, for some β4>maxsupt≥0k=1,…,m‖k‖ _4> _k=1,…,m _t≥ 0\|f_k\|, which is continuously differentiable, positive definite and radially unbounded with regard to [⊤,(−γ∗n)⊤]⊤[ δ ,( γ-γ^*1_n) ] , we have V˙ V =−kp⊤b∗+1kγ∑k=1n(γk−β4)γ˙k =-k_p δ L_ b^* δ+ 1k_γ _k=1^n( _k- _4) γ_k −⊤¯⊤¯∗(¯∗ϵ¯+1) - δ H P_g^* ( P_g^*^ε H δ+f_1 ) =−kp⊤b∗−∑k=1m(γk‖k‖2‖k‖+ε−k⊤k) =-k_p δ L_ b^* δ- _k=1^m ( _k \|q_k\|^2\|q_k\|+ -q_k f_k ) +∑k=1m(γk−β4)(‖k‖2‖k‖+ε−αγk) + _k=1^m( _k- _4) ( \|q_k\|^2\|q_k\|+ -α _k ) ≤−kpλd+2(b∗)‖2−∑k=1m(β4‖k‖2‖k‖+ε−‖k‖‖k‖) ≤-k_p _d+2(L_ b^*)\| δ\|^2- _k=1^m ( _4 \|q_k\|^2\|q_k\|+ -\|q_k\|\|f_k\| ) −∑k=1m(α2(2γk2−2β4γk+β42)−α2β42) - _k=1^m ( α2(2 _k^2-2 _4 _k+ _4^2)- α2 _4^2 ) ≤−kpλd+2(b∗)‖2−∑k=1m(α2(γk−β4)2−α2β42) ≤-k_p _d+2(L_ b^*)\| δ\|^2- _k=1^m ( α2( _k- _4)^2- α2 _4^2 ) −∑k=1mβ4∥k∥(‖k‖k‖+ε−1) - _k=1^m _4\|q_k\| ( \|q_k\|\|q_k\|+ -1 ) ≤−kpλd+2(b∗)‖2−α2∑k=1m(γk−β4)2+12mαβ42 ≤-k_p _d+2(L_ b^*)\| δ\|^2- α2 _k=1^m( _k- _4)^2+ 12mα _4^2 (42) Let ζ≜min2kpλd+1(b∗),kγαζ \2k_p _d+1(L_ b^*),k_γα\ and Δ≜12mαβ42 12mα _4^2, we rewrite (42) as V˙≤−ζV+Δ V≤-ζ V+ . By comparison lemma [35], V(t)≤Δζ+exp(−ζt)(V(0)−Δζ)V(t)≤ ζ+ exp(-ζ t) (V(0)- ζ ). This implies global uniform ultimate boundedness of the equilibrium [⊤,(−γ∗n)⊤]⊤=dn+m[ δ ,( γ-γ^*1_n) ] =0_dn+m, with the ultimate bound ζ−1Δ+ϵ1 ζ^-1 + _1, for some small positive ϵ1 _1 [35]. A-B Proof of Theorem IV.2 Consider the Lyapunov function V=12‖2+12kγ∑k=1m‖kd‖(γk−β7)2V= 12\| δ\|^2+ 12k_γ _k=1^m\|z^d_k\|( _k- _7)^2, where β7>maxsupt≥0k‖k‖ _7> _k _t≥ 0\|f_k\|. By denoting k≜kk∗ η_k _g_kg_k^*, with ‖k‖≤1\| η_k\|≤ 1, we have, V˙ V =⊤~b⊤(kp∗+¯ε∗+)+∑k=1m‖kd‖kγ(γk−β7)γ˙k = δ R_ b (k_pg^*+ P_g g^*+f )+ _k=1^m \|z_k^d\|k_γ( _k- _7) γ_k =−kp∑k=1m∥kd∥k⊤k−∑k=1m∥kd∥(γk⊤k‖k‖+ε+k⊤k) =-k_p _k=1^m\|z_k^d\| η_k η_k- _k=1^m\|z_k^d\| ( _k η_k η_k\| η_k\|+ + η_k f_k ) +∑k=1m∥kd∥(γk−β7)(‖k‖2‖k‖+ε−αγk) + _k=1^m\|z_k^d\|( _k- _7) ( \| η_k\|^2\| η_k\|+ -α _k ) ≤−∑k=1m∥kd∥(kp∥k∥2+α2(2γk2−2β7γk+β72)−α2β72) ≤- _k=1^m\|z_k^d\| (k_p\| η_k\|^2+ α2(2 _k^2-2 _7 _k+ _7^2)- α2 _7^2 ) −∑k=1m∥kd∥(β7‖k‖2‖k‖+ε−∥k∥) - _k=1^m\|z_k^d\| ( _7 \| η_k\|^2\| η_k\|+ -\| _k\|\|f_k\| ) ≤−∑k=1m∥kd∥(kp∥k∥2+α2(γk−β7)2−α2β72) ≤- _k=1^m\|z_k^d\| (k_p\| η_k\|^2+ α2( _k- _7)^2- α2 _7^2 ) −∑k=1mβ7∥kd∥k∥(‖k‖k‖+ε−1) - _k=1^m _7\|z_k^d\|\| η_k\| ( \| η_k\|\| η_k\|+ -1 ) (43) In the last term in the right-hand side of Eq. (43), by using ‖k‖k‖+ε<1 \| η_k\|\| η_k\|+ <1, we have V˙ V −kp∑k=1m∥kd∥k∥2−α2∑k=1m∥kd∥(γk−β7)2 -k_p _k=1^m\|z_k^d\|\| η_k\|^2- α2 _k=1^m\|z_k^d\|( _k- _7)^2 +α2β72∑k=1m‖kd‖+εβ7∑k=1m‖kd‖⏟≜Δ. + α2 _7^2 _k=1^m\|z_k^d\|+ _7 _k=1^m\|z_k^d\|_ . (44) With a similar argument as in the proof of Theorem IV.1, we can derive a bound on the first term in the right-hand side of (44) and eventually obtain V˙≤−ζV+Δ, V≤-ζ V+ , (45) where ζ≜minkpλd+2(b∗)mink=1,…,m‖kd‖2(n−1)s2,kαζ \k_p _d+2(L_ b^*) _k=1,…,m\|z_k^d\|2(n-1)s^2,kα\. It follows from comparison lemma [35] that for an arbitrarily small ϵ>0ε>0, there exists a finite time T0≥0T_0≥ 0 such that V(t)≤Δζ+ϵ,∀t≥T0V(t)≤ ζ+ε,\;∀ t≥ T_0. This implies that δ and γ are both uniformly ultimately bounded, with the ultimate bound 2(ζ−1Δ+ϵ) 2(ζ^-1 +ε) independent of the initial condition.