In this paper, we consider N agents and the directed topology graph among the agents, which can be described as G = (V, E). V = {1, …, N} represents the set of nodes. E ⊆ V × V represents the set of edges. An edge can be described as e _ji = (j, i) ∈ E, which expresses that agent i can get the information from agent j. Meanwhile, agent j is described as the neighbor of agent i. Then, the neighbor set of agent i is represented by N _i = {j|(j, i) ∈ E}. Furthermore, the adjacency matrix is defined as A = [a _ij ] ∈ R ^N×N . The element a _ij > 0 if e _ji = (j, i) ∈ E; otherwise, a _ij = 0. Generally, self-edge (i, i) is not allowed, which means that the diagonal elements of A are all zeros, i.e., a _ii = 0. Next, we define an in-degree matrix D = diag{d ₁, d ₂, …, d _N } ∈ R ^N×N as a diagonal matrix, and its diagonal elements are d i = ∑ j = 1 N a i j for agent i. Hence, the Laplacian matrix of digraph G can be expressed as L = D − A.

The augmented graph G ̄ can be described as the corresponding topology between the leader and the followers generally when the leader is considered. Therefore, G ̄ = ( V ̄ , E ̄ ) represents the augmented graph. V ̄ represents the node sets, including the leader and the followers. E ̄ represents the edge sets, displaying the communication relationship between the leader and the followers. The diagonal elements b _i > 0 of B = diag{b _i } ∈ R ^N×N denote that there is a weight between agent i and the leader, which is b _i = 0 otherwise. If a directed graph has a directed path from the root to every other node, it is said to have a spanning tree.

We consider a nonlinear flexible-joint robotic MAS with a leader and N followers. The dynamics of agent i in Figure 1 are given as follows: J 1 q ̈ i , 1 + F 1 q ̇ i , 1 + K q i , 1 − q i , 2 N + M g d ⁡ cos q i , 1 = 0 , J 2 q ̈ i , 2 + F 2 q ̇ i , 2 − K N q i , 1 − q i , 2 N = K t I i , L I i ̇ + R I i + K b q ̇ i , 2 = u i , (1) where q _i,1, q ̇ i , 1 , and q ̈ i , 1 represent the link position, velocity, and acceleration, respectively. q _i,2, q ̇ i , 2 , and q ̈ i , 2 denote the motor shaft position, velocity, and acceleration, respectively. I _i denotes the armature current. J ₁ and J ₂ represent the link inertias. F ₁ and F ₂ are the viscous friction constants. K, K _t , and K _b denote the spring constant, torque constant, and back-emf constant, respectively. M, g, d, N, L, and R are the link mass, acceleration of gravity, position of the link center of gravity, gear ratio, armature inductance, and armature resistance, respectively. u _i is the armature voltage.

For the convenience of studying system (1), we define x _i,1 = q _i,1, x i , 2 = q ̇ i , 1 , x _i,3 = q _i,2, x i , 4 = q ̇ i , 2 , and x _i,5 = I _i . Then, system (1) can be transformed as follows: x ̇ i , 1 = x i , 2 , x ̇ i , 2 = δ i , 2 x i , 1 , x i , 2 , x i , 3 + x i , 3 , x ̇ i , 3 = x i , 4 , x ̇ i , 4 = δ i , 4 x i , 1 , x i , 2 , x i , 3 , x i , 4 , x i , 5 + x i , 5 , x ̇ i , 5 = δ i , 5 x i , 1 , x i , 2 , x i , 3 , x i , 4 , x i , 5 + 1 L u i , y i = x i , 1 , (2) where δ i , 2 ( x i , 1 , x i , 2 , x i , 3 ) = J 1 − 1 ( − M g d ⁡ cos x i , 1 − F 1 x i , 2 − K ( x i , 1 − x i , 3 N ) − x i , 3 ) xi,3), δ i , 4 ( x i , 1 , x i , 2 , x i , 3 , x i , 4 , x i , 5 ) = J 2 − 1 ( K N ( x i , 1 − x i , 3 N ) − F 2 x i , 4 + K t x i , 5 − x i , 5 ) , δ _i,5(x _i,1, x _i,2, x _i,3, x _i,4, x _i,5) = L ⁻¹(−Rx _i,5 − K _b x _i,4), and x i = x i , 1 , ⋯ x i , 5 T ∈ R 5 are the whole states of agent i. y _i and u _i represent the output and control input of agent i, respectively. To simplify the design procedure of the controller, the parameters of the functions will be omitted in the subsequent design procedure. For example, the function δ _i,2(x _i,1, x _i,2, x _i,3) is expressed as δ _i,2. After that, the dynamic model of the leader is expressed as x ̇ d = f d x d , t , y d = x d , (3) where y _d ∈ R means the output of the leader. f _d (x _d , t) is a piecewise continuous function, which meets the local Lipschitz condition about x _d for t ≥ 0.

Assumption 1

In the augmented graph G ̄ , the leader is the root node of a directed spanning tree. Additionally, each agent i can only receive state information from its neighbors.

Assumption 2

There is a continuous function f(⋅) and a positive constant X _d , which makes the inequalities f d ( x d , t ) ≤ f ( x d ) and x d ( t ) ≤ X d hold for all t ≥ t ₀.

Our goal is to present an adaptive consensus tracking control protocol for the flexible-joint robotic MASs (2) to make sure that the tracking errors converge to a small neighborhood of origin within a finite time and that all signals in the closed-loop systems remain bounded. Therefore, the following knowledge is needed in the design process of the controller:

Lemma 1

(See [43]). For ∀ψ ∈ R, the following inequality is true: 0 ≤ ψ − ψ ⁡ tanh ψ t ≤ ρ t , ∀ t ≥ 0 , (4) where ρ = 0.2785.

Lemma 2

(See [44]). For any variable, ι, γ, one has ι κ 1 γ κ 2 ≤ κ 1 κ 1 + κ 2 κ 3 ι κ 1 + κ 2 + κ 2 κ 1 + κ 2 κ 3 − κ 1 κ 2 γ κ 1 + κ 2 , (5) where κ ₁, κ ₂, and κ ₃ are arbitrary positive constants.

Lemma 3

(See [45]). For s _i ∈ R, i = 1, 2, …, n, and 0 < j ≤ 1, it holds that ∑ i = 1 n s i j ≤ ∑ i = 1 n s i j ≤ n 1 − j ∑ i = 1 n s i j . (6)

Definition 1

(See [41]). For any incipient condition ζ(0) ∈ ζ ₀, if there is a constant ɛ > 0 and a settling time T(ɛ, ζ ₀) < ∞ such that ζ t < ε , ∀ t > T , (7) the solution, which belongs to the nonlinear system ζ ̇ = f ( ζ ( t ) ) , is practical finite-time stable.

Lemma 4

(See [46]). The solution of ζ ̇ = f ( ζ ( t ) ) is practical finite-time stable if there is a positive-definite function that meets the following formula: V ̇ ζ ≤ − α V ζ − β V p ζ + Γ , (8) where the design constants α > 0, β > 0, 0 < p < 1, and 0 < Γ < ∞.

RBF NNs [47]: In this paper, RBF NN technology is utilized to approximate unknown continuous functions. For the unknown continuous nonlinear function h(Z): R ^S → R defined over a compact set Ω _Z ∈ R ^s and the given precision ɛ* > 0, h(Z) can be approximated by RBF NNs as follows: h Z = θ * T φ Z + ε Z , (9) where Z ∈ Ω _Z ⊂ R ^s is the input vector. ɛ(Z) denotes the approximation error with ε ( Z ) ≤ ε * . φ ( Z ) = [ φ 1 ( Z ) , … , φ l ( Z ) ] T represents the basis function vector, and l > 1 represents a positive integer. Generally, the basis function φ _i (Z) can been chosen as the following Gaussian function: φ i Z = exp − Z − μ i T Z − μ i κ i 2 , i = 1 , … , l , (10) where μ i = [ μ i 1 , … , μ i l ] T and κ _i are the center and the width of the Gaussian function, respectively. In addition, θ * = [ θ 1 , … , θ l ] T ∈ R l represents an ideal weight vector defined as θ * = arg min θ ∈ R l sup Z ∈ Ω Z h Z − θ T φ Z , (11) where θ is the weight vector.

Lemma 5

(See [31]). Suppose h ( x ̄ q ) = [ h 1 ( x ̄ q ) , … , h l ( x ̄ q ) ] T is the basis function vector of RBF NNs, where x ̄ q = [ x 1 , … , x q ] T . Then, the following inequality holds: h x ̄ q 2 ≤ h x ̄ L 2 , (12) where the arbitrary positive integer L satisfies L ≤ q.

In this section, we design an adaptive command-filtered finite-time consensus tracking control scheme for MASs (2). The consensus tracking error of agent i is defined as z i , 1 = ∑ j = 1 N a i j y i − y j + b i y i − y d , (13) where a _ij and b _i are defined in the graph theory.

Remark 1

It is worth noting that (13) includes a _ij and b _i . Therefore, the consensus tracking error z _i,1 is influenced by the topology structure of the augmented graph G ̄ . Furthermore, b _i + d _i > 0 is met for all the agents.

The coordinate transformation is designed as follows: z i , k = x i , k − α ̄ i , k , (14) where k = 2, …, 5. α ̄ i , k represents the output of the following command filter: τ i , k α ̄ ̇ i , k + α ̄ i , k = α i , k , α ̄ i , k 0 = α i , k 0 , (15) where τ _i,k > 0 is a design constant. α _i,k is both the input of the command filter and the virtual controller, which will be presented later. Then, considering the impact of the error brought by the command filter (15), we define the following compensating signals: η ̇ i , 1 = − c i , 1 + 1 η i , 1 + d i + b i η i , 2 + d i + b i α ̄ i , 2 − α i , 2 − λ i , 1 sgn η i , 1 , η ̇ i , 2 = − c i , 2 + 1 η i , 2 + η i , 3 + α ̄ i , 3 − α i , 3 − d i + b i η i , 1 − λ i , 2 sgn η i , 2 , η ̇ i , 3 = − c i , 3 + 1 2 η i , 3 + η i , 4 + α ̄ i , 4 − α i , 4 − η i , 2 − λ i , 3 sgn η i , 3 , η ̇ i , 4 = − c i , 4 + 1 η i , 4 + η i , 5 + α ̄ i , 5 − α i , 5 − η i , 3 − λ i , 4 sgn η i , 4 , η ̇ i , 5 = − c i , 5 + 1 η i , 5 − η i , 4 − λ i , 5 sgn η i , 5 , (16) where c _i,k > 0, λ _i,k > 0, and η _i,k (0) = 0 for k = 1, 2, 3, 4, 5. Next, we define the compensated tracking error v _i,k = z _i,k − η _i,k for k = 1, 2, 3, 4, 5.

Then, the virtual controllers are designed as follows: α i , 2 = 1 d i + b i − c i , 1 + 1 z i , 1 − e i , 1 v i , 1 2 p − 1 − 1 2 a i , 1 2 v i , 1 θ ̂ i , 1 φ i , 1 T φ i , 1 , α i , 3 = − c i , 2 + 1 z i , 2 − e i , 2 v i , 2 2 p − 1 − 1 2 a i , 2 2 v i , 2 θ ̂ i , 2 φ i , 2 T φ i , 2 − d i + b i z i , 1 + α ̄ ̇ i , 2 , α i , 4 = − c i , 3 + 1 2 z i , 3 − e i , 3 v i , 3 2 p − 1 − z i , 2 + α ̄ ̇ i , 3 , α i , 5 = − c i , 4 + 1 z i , 4 − e i , 4 v i , 4 2 p − 1 − 1 2 a i , 4 2 v i , 4 θ ̂ i , 4 φ i , 4 T φ i , 4 − z i , 3 + α ̄ ̇ i , 4 , (17) where 1/2 < p < 1, p = ϖ ₁/ϖ ₂ and ϖ ₁, ϖ ₂ are odd integers. θ ̂ i , k denotes the estimation of θ _i,k , and the estimation error is θ ̃ i , k = θ i , k − θ ̂ i , k . e _i,k and a _i,k are positive design parameters.

Consequently, the adaptive laws are designed as follows: θ ̂ ̇ i , k = r i , k 2 a i , k 2 v i , k 2 φ i , k T φ i , k − σ i , k θ ̂ i , k , (18) where r _i,k and σ _i,k are positive design parameters for k = 1, 2, 4, 5.

Then, we give the detailed design process for the system controllers.

Step 1

Taking the derivative of v _i,1, one has v ̇ i , 1 = z ̇ i , 1 − η ̇ i , 1 = ∑ j = 1 N a i , j y ̇ i − y ̇ j + b i y ̇ i − y ̇ d − η ̇ i , 1 = ∑ j = 1 N a i , j x i , 2 − x j , 2 + b i x i , 2 − f d x d , t − η ̇ i , 1 = d i + b i x i , 2 − b i f d x d , t − ∑ j = 1 N a i , j x j , 2 − η ̇ i , 1 . (19)

Then, we design the following candidate Lyapunov function: V i , 1 = 1 2 v i , 1 2 + 1 2 r i , 1 θ ̃ i , 1 2 . (20)

Next, the derivation of V _i,1 is given as follows: V ̇ i , 1 = v i , 1 v ̇ i , 1 − 1 r i , 1 θ ̃ i , 1 θ ̂ ̇ i , 1 = v i , 1 d i + b i x i , 2 − b i f d x d , t − ∑ j = 1 N a i , j x j , 2 − η ̇ i , 1 − 1 r i , 1 θ ̃ i , 1 θ ̂ ̇ i , 1 = v i , 1 d i + b i v i , 2 + d i + b i η i , 2 + d i + b i α ̄ i , 2 − b i f d x d , t − ∑ j = 1 N a i , j x j , 2 − η ̇ i , 1 − 1 r i , 1 θ ̃ i , 1 θ ̂ ̇ i , 1 . (21)

According to Assumption 2 and Lemma 1, it is easy to get − b i v i , 1 f d x d , t ≤ b i v i , 1 f x d ≤ b i v i , 1 f x d tanh v i , 1 t i , 1 + ρ t i , 1 . (22)

Substituting (22) into (21) yields V ̇ i , 1 ≤ v i , 1 d i + b i v i , 2 + d i + b i η i , 2 + d i + b i α ̄ i , 2 + g ̄ i , 1 − η ̇ i , 1 + ρ t i , 1 − 1 r i , 1 θ ̃ i , 1 θ ̂ ̇ i , 1 , (23) where g ̄ i , 1 = b i f ( x d ) tanh ( v i , 1 t i , 1 ) − ∑ j = 1 N a i , j x j , 2 .

From (9), one can obtain g ̄ i , 1 = θ i , 1 ∗ T φ i , 1 + ε i , 1 , (24) where ε i , 1 ≤ ε i , 1 * , and ε i , 1 * is a positive constant.

By applying Young’s inequality and Lemma 5, one can get v i , 1 g ̄ i , 1 ≤ v i , 1 2 θ i , 1 φ i , 1 T φ i , 1 2 a i , 1 2 + a i , 1 2 2 + v i , 1 2 2 + ε i , 1 * 2 2 , (25) v i , 1 λ i , 1 sgn η i , 1 ≤ 1 2 v i , 1 2 + 1 2 λ i , 1 2 , (26) where θ i , 1 * 2 = θ i , 1 . Then, by substituting (16)–(18) and (24)–(26) into (23), it is derived that V ̇ i , 1 ≤ − c i , 1 v i , 1 2 − e i , 1 v i , 1 2 p + d i + b i v i , 1 v i , 2 + a i , 1 2 2 + ε i , 1 * 2 2 + λ i , 1 2 2 + ρ t i , 1 + σ i , 1 r i , 1 θ ̃ i , 1 θ ̂ i , 1 . (27)

Step 2

Taking the derivative of v _i,2, one can get v ̇ i , 2 = z ̇ i , 2 − η ̇ i , 2 = δ i , 2 + x i , 3 − α ̄ ̇ i , 2 − η ̇ i , 2 . (28)

The candidate Lyapunov function V _i,2 is chosen as follows: V i , 2 = V i , 1 + 1 2 v i , 2 2 + 1 2 r i , 2 θ ̃ i , 2 2 . (29)

Then, the derivation of V _i,2 is given as follows: V ̇ i , 2 = V ̇ i , 1 + v i , 2 δ i , 2 + x i , 3 − α ̄ ̇ i , 2 − η ̇ i , 2 − 1 r i , 2 θ ̃ i , 2 θ ̂ ̇ i , 2 . (30)

From (9), one can obtain δ i , 2 = θ i , 2 ∗ T φ i , 2 + ε i , 2 , (31) where ε i , 2 ≤ ε i , 2 * , and ε i , 2 * is a positive constant.

By applying Young’s inequality and Lemma 5, one can get v i , 2 δ i , 2 ≤ v i , 2 2 θ i , 2 φ i , 2 T φ i , 2 2 a i , 2 2 + a i , 2 2 2 + v i , 2 2 2 + ε i , 2 * 2 2 , (32) v i , 2 λ i , 2 sgn η i , 2 ≤ 1 2 v i , 2 2 + 1 2 λ i , 2 2 , (33) where θ i , 2 * 2 = θ i , 2 . Then, by substituting (16)–(18), (27), and (31)–(33) into (30), it is derived that V ̇ i , 2 ≤ − ∑ j = 1 2 c i , j v i , j 2 − ∑ j = 1 2 e i , j v i , j 2 p + v i , 2 v i , 3 + ∑ j = 1 2 a i , j 2 2 + ∑ j = 1 2 ε i , j * 2 2 + ∑ j = 1 2 λ i , j 2 2 + ρ t i , 1 + ∑ j = 1 2 σ i , j r i , j θ ̃ i , j θ ̂ i , j . (34)

Step 3

Taking the derivative of v _i,3, one can obtain v ̇ i , 3 = z ̇ i , 3 − η ̇ i , 3 = v i , 4 + η i , 4 + α ̄ i , 4 − α ̄ ̇ i , 3 − η ̇ i , 3 . (35)

The candidate Lyapunov function V _i,3 is chosen as follows: V i , 3 = V i , 2 + 1 2 v i . 3 2 . (36)

Then, the following equation holds: V ̇ i , 3 = V ̇ i , 2 + v i , 3 v i , 4 + η i , 4 + α ̄ i , 4 − α ̄ ̇ i , 3 − η ̇ i , 3 . (37)

By using Young’s inequality, we get v i , 3 λ i , 3 sgn η i , 3 ≤ 1 2 v i , 3 2 + 1 2 λ i , 3 2 . (38)

By substituting (16)–(18), (34), and (38) into (37), it is obtained that V ̇ i , 3 ≤ − ∑ j = 1 3 c i , j v i , j 2 − ∑ j = 1 3 e i , j v i , j 2 p + v i , 3 v i , 4 + ∑ j = 1 2 a i , j 2 2 + ∑ j = 1 2 ε i , j * 2 2 + ∑ j = 1 3 λ i , j 2 2 + ρ t i , 1 + ∑ j = 1 2 σ i , j r i , j θ ̃ i , j θ ̂ i , j . (39)

Step 4

Taking the derivative of v _i,4, one can get v ̇ i , 4 = z ̇ i , 4 − η ̇ i , 4 = v i , 5 + η i , 5 + α ̄ i , 5 + δ i , 4 − α ̄ ̇ i , 4 − η ̇ i , 4 . (40)

The candidate Lyapunov function V _i,4 is chosen as follows: V i , 4 = V i , 3 + 1 2 v i , 4 2 + 1 2 r i , 4 θ ̃ i , 4 2 . (41)

In addition, the following equation can be obtained: V ̇ i , 4 = V ̇ i , 3 + v i , 3 v i , 5 + η i , 5 + α ̄ i , 5 + δ i , 4 − α ̄ ̇ i , 4 − η ̇ i , 4 − 1 r i , 4 θ ̃ i , 4 θ ̂ ̇ i , 4 . (42)

From (9), we get δ i , 4 = θ i , 4 ∗ T φ i , 4 + ε i , 4 , (43) where ε i , 4 ≤ ε i , 4 * , and ε i , 4 * is a positive constant.

By using Young’s inequality and Lemma 5, one can get v i , 4 δ i , 4 ≤ v i , 4 2 θ i , 4 φ i , 4 T φ i , 4 2 a i , 4 2 + a i , 4 2 2 + v i , 4 2 2 + ε i , 4 * 2 2 , (44) v i , 4 λ i , 4 sgn η i , 4 ≤ 1 2 v i , 4 2 + 1 2 λ i , 4 2 , (45) where θ i , 4 * 2 = θ i , 4 . Then, by substituting (16)–(18), (39), and (43)–(45) into (42), it is derived that V ̇ i , 4 ≤ − ∑ j = 1 4 c i , j v i , j 2 − ∑ j = 1 4 e i , j v i , j 2 p + v i , 4 v i , 5 + ∑ j = 1,2,4 a i , j 2 2 + ∑ j = 1,2,4 ε i , j * 2 2 + ∑ j = 1 4 λ i , j 2 2 + ρ t i , 1 + ∑ j = 1,2,4 σ i , j r i , j θ ̃ i , j θ ̂ i , j . (46)

Step 5

Taking the derivative of v _i,5, one can get v ̇ i , 5 = z ̇ i , 5 − η ̇ i , 5 = δ i , 5 + 1 L u i − α ̄ ̇ i , 5 − η ̇ i , 5 . (47)

The candidate Lyapunov function V _i,5 is chosen as follows: V i , 5 = V i , 4 + 1 2 v i , 5 2 + 1 2 r i , 5 θ ̃ i , 5 2 . (48)

Next, the following equation can be obtained: V ̇ i , 5 = V ̇ i , 4 + v i , 5 δ i , 5 + 1 L u i − α ̄ ̇ i , 5 − η ̇ i , 5 − 1 r i , 5 θ ̃ i , 5 θ ̂ ̇ i , 5 . (49)

From (9), one has δ i , 5 = θ i , 5 ∗ T φ i , 5 + ε i , 5 , (50) where ε i , 5 ≤ ε i , 5 * , and ε i , 5 * is a positive constant.

By applying Young’s inequality and Lemma 5, one can obtain v i , 5 δ i , 5 ≤ v i , 5 2 θ i , 5 φ i , 5 T φ i , 5 2 a i , 5 2 + a i , 5 2 2 + v i , 5 2 2 + ε i , 5 * 2 2 , (51) v i , 5 λ i , 5 sgn η i , 5 ≤ 1 2 v i , 5 2 + 1 2 λ i , 5 2 , (52) where θ i , 5 * 2 = θ i , 5 .

Furthermore, design the controller u _i is expressed as u i = L − c i , 5 + 1 z i , 5 − e i , 5 v i , 5 2 p − 1 − 1 2 a i , 5 2 v i , 5 θ ̂ i , 5 φ i , 5 T φ i , 5 − z i , 4 + α ̄ ̇ i , 5 . (53)

Finally, by substituting (16)–(18), (46), and (50)–(53) into (49), it is obtained that V ̇ i , 5 ≤ − ∑ j = 1 5 c i , j v i , j 2 − ∑ j = 1 5 e i , j v i , j 2 p + ∑ j = 1,2,4,5 a i , j 2 2 + ∑ j = 1,2,4,5 ε i , j * 2 2 + ∑ j = 1 5 λ i , j 2 2 + ρ t i , 1 + ∑ j = 1,2,4,5 σ i , j r i , j θ ̃ i , j θ ̂ i , j . (54)

Theorem 1

Considering the flexible-joint robotic MASs (1) and (2), the augmented graph G ̄ , the leader (3), the command filter (15), the virtual controllers (17), the adaptive laws (18), and the actual controller (53) satisfying assumptions 1 and 2, and the following conclusions hold:

1) The proposed adaptive command-filtered consensus control scheme can guarantee that the tracking errors converge to a small neighborhood of origin within a finite time

Proof

Based on Young’s inequality, one can obtain ∑ j = 1,2,4,5 σ i , j r i , j θ ̃ i , j θ ̂ i , j ≤ ∑ j = 1,2,4,5 − σ i , j 2 r i , j θ ̃ 2 i , j + ∑ j = 1,2,4,5 σ i , j 2 r i , j θ i , j 2 . (55)