Abstract
The fast finitetime output bipartite tracking of networked heterogeneous robotic systems with matrixweighted digraphs, parametric uncertainties and external disturbances is studied in this article. Besides, solving the fast finitetime bipartite tracking problem in this paper implies that the system states are forced to reach the employed nonsingular finitetime sliding surface in a predefined time, which thus called fast finitetime control. To address the aforementioned issues, a fast finite time hierarchical control algorithm utilizing estimator methodologies are proposed. The Lyapunov stability theory is used to derive some sufficient requirements for performing output bipartite tracking in a fast finite time manner. To verify that the theoretical results are valid, numerical simulation examples are given.
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1 Introduction
A growing number of robotics and automation researchers are interested in distributed cooperative control of Euler–Lagrange systems (ELSs). Robots may now use decentralized control algorithms to make local decisions using information from adjacent neighbors, which reduces the need for centralized control and increases system robustness and efficiency [11, 28]. As a result, many academics are inspired to research different control methods to achieve different control targets, such as consensus [14, 20, 21], bipartite consensus [7, 16, 31], formation control [5, 24, 25], containment control [27, 29] and optimal tracking [2, 12]. Among these, bipartite tracking is crucial in multiagent systems, where all agents converge to a same leader state but with opposite sign.
It is worth mentioning that the results discussed above mostly pertain to integratortype and linear dynamic systems. However, nonlinear systems, particularly ELSs, have garnered considerable interest. Because it may be expanded to meet complicated problems with constraints, it is necessary for a variety of realworld engineering issues, such as those involving aeronautical systems [18], mobile robots [13, 38] and autonomous vehicles [3]. As the studies mentioned above that concentrated on joint space coordination (i.e., statespace) are less suitable for modeling actual engineering systems, it is required to take more appropriate endeffectors control (i.e., outspace) into account. Operators no longer need to describe specific joint movements since taskspace control makes it possible to intuitively regulate a robot’s location and orientation in realworld space. It, therefore, promotes further investigation into output space control strategies of ELSs [10, 15, 26]. In comparison to identical mechanical structures, the heterogeneous agents are more flexible in how they execute their task. In [15], a controllerestimator method was presented to achieve multitarget tracking of networked heterogeneous robotic systems (NHRSs). A unique hierarchical finitetime control protocol is employed in [30] to explore bipartite tracking for NHRSs.
On the other hand, it should be noted that conventional consensus primarily concentrates on the collaborative relationships among agents in multiagent systems. Nonetheless, in specific scenarios, agents may interact in a cooperative or competitive manner. For example, in the ecosystem, species engage in both collaboration, such as mutualistic relationships between pollinators and plants, and competition for essential resources, such as food and territory. In the realm of technology, two companies may collaborate on open source projects or industry standards while competing to develop proprietary technologies. With regard to this issue, bipartite consensus was initially proposed in [1] with signed graphs. Bipartite tracking has been developed and studied as a result of Altafini’s pioneering research [1]. The majority of recent work relies on scalarweighted graphs, nevertheless in scientific and engineering scenarios, matrixweights should be employed to depict the information exchange between highdimensional interactions. Matrixweighted consensus problem was investigated in [23]. The bipartite consensus over an unbalanced graph was studied in [22] to evaluate the effect of matrix weights.
Besides, convergence rate is a crucial factor in assessing how well the proposed control protocol works, which implies that bipartite tracking, achieved only in infinite time, can not satisfy realistic needs. It has sparked a significant lot of attention among finitetime control [8, 9, 35], but its settling time is primarily influenced by the initial states for each agent. The fixedtime approach has been established in [6, 33, 37] to loosen this constraint. The results mentioned above depend on the NHRSs control parameters, which are not explicit in practice. To overcome the deficiencies, there has been considerable exploration into the predefinedtime method [17, 19, 36, 39], which entails setting the convergence time in advance. Inspired by existing finitetime control and predefinedtime control, we aim to exploit a new control algorithm to combine the both two control technologies for achieving a newly designed fast finitetime control by simultaneously considering the issues of uncertainty and nonlinear models.
Motivated by the above observations, we design a novel fast finitetime hierarchical control algorithm (HCA) for the NHRSs and devote to achieve the output bipartite tracking in the fast finite time. The designed control algorithm includes the distributed predefinedtime estimator layer and the local control layer. In detail, the states of the leader are estimated in a predefined time at the distributed estimator layer. Then, based on a nonsingular finitetime sliding surface and the controlled system is forced to reach the sliding surface in a predefined time. This can be further derived that the output bipartite tracking problem is solved in a fast finite time. The main contributions lie on threefold.

A novel fast finitetime HCA that follows the property of the matrixweighted Laplacian has been proposed to effectively solve the output bipartite tracking problem of NHRSs.

Different from undirected matrixweighted graphs studied in [22, 23], this paper concentrates on bipartite tracking problem of NHRSs under matrixweighted digraphs.

Different from the researches on asymptotic stability [7, 16, 22, 31], the fast finitetime stability can improve convergence rate of the system. Different from the finitetime sliding mode control [8, 21, 30], in which the reaching time can not be determined easily. The reaching time of the proposed algorithm can be designed in advance by the user.
What follows is an outline of the rest of this article. In Sect. 2, preliminaries are provided. Bipartite tracking for NHRSs is discussed through the application of a fast finitetime control method in Sect. 3. Section 4 presents the simulations while Sect. 5 suggests conclusions.
Notations: Let \(\mathbb {R}^k (\mathbb {R}_+)\) represent the k(1)dimensional real vectors, \(1_k=[1,\ldots ,1] \in \mathbb {R}^k \), \(\mathbb {R}^{k\times k}\) refers to the sets of \(k \times k\) real matrix and \(I_k\) denotes the korder identity matrix. \(\lambda _{min}(\cdot )\) and \(\lambda _{max}(\cdot )\) are the minimum and maximum eigenvalue, respectively. \(sign (\cdot )\) is the sign function. \(\iota \in \mathbb {R}\), then a symbol is indicated as \(w^{[\iota ]}=[\textrm{sgn}(w_{1}){w_{1} ^{\iota }},\ldots , \textrm{sgn}(w_{r}){w_{k} ^{\iota }}]^{T}\). \(\Vert w\Vert \) symbolizes the Euclidean norm. \(\otimes \) represents the Kronecker product. \([\hspace{1.49994pt}[ y ]\hspace{1.49994pt}]\) equals to \(\sqrt{\frac{1}{2}{y^T}y}\), it thus can be obtained that \({ [\hspace{1.49994pt}[ y ]\hspace{1.49994pt}] ^\alpha } = {(\frac{1}{2}{y^T}y)^{\frac{\alpha }{2}}}\), where \(y \in {R^n}\).
2 Preliminaries
2.1 Graph Theory
A matrixweighted digraph \(\mathcal {G} = \left( {\mathcal {V},\mathcal {E},\mathcal {W}} \right) \) is used to represent connection between n agents, where \(\mathcal {V} = \left\{ {{v_1},{v_2},\ldots ,{v_n}} \right\} \), \(\mathcal {E} \subseteq \mathcal {V} \times \mathcal {V}\) and \(\mathcal {W} = [{W_{ij}}] \in \mathbb {R}^{np\times np}\) represent node set, the edge set and the adjacency matrix associated with matrixweighted edge, respectively. Besides, \(W_{ij}>0\) and \(W_{ij}<0\) elaborate on the cooperative and antagonistic interaction. It is supposed that \(\mathcal {G}\) is no selfloops, i.e., \({W_{ii}} = 0_{p \times p}\). Besides, \(\mathcal{B}\)=blkdiag\(\left( \mathcal{B}_ 1, \mathcal{B}_ 2,\ldots ,\mathcal{B}_np\right) \) is the block matrix, where \(\mathcal{B}_ i>0\) implies that the ith robot can receive information from the leader and \(\mathcal{B}_ i=\textbf{0} \) otherwise. \(\mathcal {G}\) is referred to as strongly connected If any pair of nodes has at least one directed path. Define the Laplacian matrix \(\bar{L}=[\bar{L}_{ij}] \), where \(\bar{L}_{ii}=\sum \limits _{ j \in \mathcal {N}_{i}}{W_{ij} }\) and \(\bar{L}_{ij}=W_{ij}\) for \(i\ne j \in \mathcal {V} \). A matrixweighted digraph \(\mathcal {G}\) is considered to be structurally balanced if there exists a bipartition \(\left\{ \mathcal {V}_1,\mathcal {V}_2\right\} \) of the vertices, where \(\mathcal {V}_1\cup \mathcal {V}_2=\mathcal {V} \) and \(\mathcal {V}_1\cap \mathcal {V}_2=\emptyset \), such that \(W_{ij}\) is positivedefinite for \(\forall i,j \in \mathcal {V}_{n_1}(n_1\in \{1,2\})\), and \(W_{ij}<0\) is negativedefinite for \(\forall i \in \mathcal {V}_{n_1}\), \(\forall j \in \mathcal {V}_{n_2}\), \(n_1\ne n_2, (n_1, n_2 \in \{1,2\})\).
2.2 System Formulation
Consider the following differential equations of ELSs with n agents.
where \({q_i}, \dot{q}\) and \(\ddot{q} \in \mathbb {R}^{n_i}\) symbolize vectors of the generalizedjoint position, velocity and acceleration in joint space, \(x_{i}\), \(\dot{x}_{i} \in \mathbb {R}^{p} \) denote the vectors of position and velocity in task space, \(M_{i}(q_{i})=\overline{M}_{i}(q_i)+ eM_i(q_i) \in \mathbb {R}^{n_i \times n_i}\) denotes the inertia matrix, \(C(q_i,\dot{q}_i)=\overline{C}_i(q_{i}, \dot{q}_i)+eC_i(q_i,\dot{q}_i) \in \mathbb {R}^{n_i \times n_i}\) symbolizes the Corioliscentrifugal matrix, \(g_i(q_i)=\overline{g}(q_i)+ eg_i(q_i)\in \mathbb {R}^{n_i }\) represents the gravity vector, \(\tau _i \in \mathbb {R}^{n_i }\) denotes the torque input, \(d_i(t)\) is the external disturbance implementing that \(\Vert d_i(t)\Vert . \leqslant \overline{\sigma }_{i,d}\). \(\zeta _i(q_i) \in \mathbb {R}^{p}\) and \(J_{i}(q_{i})=\partial \zeta _{i}(q_{i})/\partial q_{i} \in \mathbb {R}^{p \times n_i}\) are the forward kinematics and the nonsingular Jacobian matrix, respectively. Furthermore, \(\overline{M}_i(q_i)\), \(\overline{C}_i(q_{i})\) and \(\overline{g}(q_i)\) are the nominal terms that can be involved in the control design, \(eM(q_i)\), \(eC(q_i)\) and \(eg(q_i)\) refer to the uncertain terms. This allows (1) to be rewritten as follows:
where \(\gamma _i(t)=eM_i(q_i)\ddot{q}_ieC_i(q_i,\dot{q}_i)\dot{q}_ie g_i(q_i)+ d_i(t)\). Without loss of generally, following [34], \(\gamma _i(t)\) is bounded, i.e., \( \Vert \gamma _i(t)\Vert \le \overline{\ell }_{i,1} +\overline{\ell }_{i,2} \Vert q_i\Vert +\overline{\ell }_{i,3} \Vert \dot{q}_i \Vert ^2 + \overline{\ell }_{i,d},\) where \(\overline{\ell }_{i,d}\), \(\overline{\ell }_{i,1},\) \(\overline{\ell }_{i,2}\) and \(\overline{\ell }_{i,3}\) are positive constants. Furthermore, the virtual leader’s trajectory is defined as
where \(x_{l}(t)\), \(v_{l}(t) \in \mathbb {R}^{n }\) are the vectors of the virtual of position and velocity, respectively.
Assumption 1
With a spanning tree, the matrixweighted signed digraph G is structurally balanced.
Assumption 2
The acceleration and velocity of all leaders are bounded.
2.3 Some Definitions and Lemmas
Lemma 1
[4] Under Assumption 1, let \( \iota =[\iota _{1}, \iota _{2}, \ldots ,\iota _{np}]^{T}=H_{s}^{1}\textbf{1}_{np}\), \(\nu =[\nu _{1},\nu _{2},\ldots ,\nu _{np}]^{T}=H_{s}^{T}\textbf{1}_{np}\), \(U=diag(u_i)=diag(\nu _i/\iota _i)\), and \(\Theta =U H_s +H_s^{T} U\), where \(H_{s}=E H E\), E=blkdiag\((e_1,e_2,\ldots ,e_n) \in \mathbb {R}^{np \times np}\), \(e_i \in \left\{ I_p,I_p \right\} \) and \(H=\mathcal {L}+\mathcal {B}\). Then, \(U>0\) and \(\Theta >0\).
Lemma 2
[32] Given \({z _{i}} \ge 0, i={1,\ldots ,n}\), \(\mathrm{{0}} \le p \le 1\) and \(q\ge 1\), there holds
Lemma 3
[36] Consider the following differential system
where \(x\in \mathbb {R}^{n}\)and \(f:\mathbb {R}\times \mathbb {R}^{n}\rightarrow \mathbb {R}^{n}\) is continuous. Suppose that there exits a continuous positive definite function \(V(X):\mathbb {R}^{n}\rightarrow \mathbb {R}\) such that \(\dot{V}\left( x(t)\right) \le \frac{\chi \left( z\right) }{T_{z}}\left( \omega _1 V^{c_1}\big (x(t)\big )+\omega _2 V^{c_2}\big (x(t)\big )\right) , \) where \(x(t)\in \mathbb {R}^{n}\backslash \{0\},z=[\omega _1,\omega _2,c_1,c_2]^{T}\in \mathbb {R}^{4},T_{z}, \omega _1,\omega _2>0, 0< c_1<1, c_2>1,\) and \(\chi (z)=\frac{\Gamma (\frac{1 c_1}{c_2c_1})\Gamma (\frac{c_21}{c_2c_1})}{\omega _1(c_2c_1)}(\frac{\omega _1}{\omega _2})^{\frac{1 c_1}{c_2c_1}}.\) Then the origin of \(\dot{x}(t)=f(t,x)\) is global predefinedtime stable and the settling time \(T_z\) is not affected by the initial state.
Definition 1
The fast finitetime output bipartite tracking problem is solved if there exists a \(T_z >0\) such that
where \(\dot{x}_{i} \buildrel \Delta \over = v_{i}\) is the velocity vector of the ith agent in the output space.
3 Main Results
3.1 The Fast FiniteTime HCA
In this section, the fast finitetime HCA is developed to address the the output bipartite tracking problem for NHRSs. Before moving on, introducing the following sliding mode:
where \(w_i=x_i\hat{x}_i\), \(\dot{w}_i=\dot{x}_i\hat{v}_i\), \(\hat{x}_i\) and \(\hat{v}_i\) denote the estimates of \(x_i\) and \(v_i\), respectively. Moreover, the constants satisfy \(l_{1}>0\) and \(1<\upsilon <2\).
The following is the proposed HCA when external perturbations and parametric uncertainties are presented.
with
where \(l_4 \in \mathbb {R}^{p \times p}\), \(0<a<1<c\), \( l_2, l_3, \xi , \eta ,T_{p1}, T_{p2}, T_L, \rho >0,\) \(0<h_1, h_2<1\) and \(g_1, g_2>1\). The vector \(\zeta _i\) satisfies \(J_i\zeta _i=0.\) If the ith agent is nonredundant, \(\zeta _i=0\), otherwise \(\zeta _i\) is developed to accomplish the subtasks for redundant agent. \( \Psi _1, \Psi _2\) will be designed later. Consider that the proposed HCA is composed of the distributed estimator layer and the local control layer shown in Fig. 1.
Let \( \tilde{\delta }_{x,i}=\hat{x}_{i}e_ix_{l,i}\), \(\tilde{\delta }_{v,i}=\hat{v}_{i}e_iv_{l,i}\). Substituting the proposed HCA (8) into (2) obtains the cascade closedloop system as follows.
3.2 Analysis of the Estimator Layer
Theorem 1
Suppose that Assumptions 1–2 hold. Utilizing the distributed estimator algorithm (8b) and (8c), If \(\rho \geqslant a_l^{max}\), then \( \tilde{\delta }_{x}\), \(\tilde{\delta }_{v} \rightarrow \textbf{0}_r \) as \( t \rightarrow T_{p}^{(1)}\), where \(T_{p}^{(1)}= T_{p1}+ T_{p2}, a_l^{max}=max\left\{ a_{l,1}, a_{l,2},\ldots a_{l,np} \right\} \), \(u_{max}=max\left\{ u_1,u_2,\ldots u_{np}\right\} \), \(T_{p1}, T_{p2}\) are the parameters for the predetermined settling time.
Proof
In the first part, convergence of \(\tilde{\delta }_{v}\) within a predefined time is analyzed. Let \(\tilde{\delta }_{x}=[\tilde{\delta }_{x,1}, \tilde{\delta }_{x,2},\ldots , \tilde{\delta }_{x,np}]^{T}\), \(\tilde{\delta }_{v}=[\tilde{\delta }_{v,1}, \tilde{\delta }_{v,2},\ldots , \tilde{\delta }_{v,np}]^{T}\). It follows from (10) that
where H is defined in Lemma 1. Choose \(\bar{\delta }_v=EH \tilde{\delta }_v\). One then derives that
Select the Lyapunov function as
where \(u_i\) is defined in Lemma 1. Taking the derivative of \(V_1\) along the (13) gets
where \(\Theta =UH_s +H_s^{T}U\) is defined in Lemma 1, \(\alpha =\frac{1}{2}\lambda _{min}(\Theta ) min(\xi ,\eta )^2.\) Now, let \(\bar{V}_{1}=\sum \limits _{i=1}^{n}\bar{\delta }_{v,i}^{h_2+1}+\sum \limits _{i=1}^{n}\bar{\delta }_{v,i}^{g_2+1}\). According to Lemma 2, with \(\frac{h_2+g_2}{g_2+1}<1\) and \(2h_2<\frac{h_2+1}{g_2+1}(h_2+g_2)<2g_2\), it can be expected that
One can be deduced from \(\frac{2g_2}{g_2+1}>1\) that
one further has
the above inequalities follow from \(\bar{\delta }_{v,i}^{2g_2\frac{h_2+1}{g_2+1}}\le \bar{\delta }_{v,i}^{2h_2}+\bar{\delta }_{v,i}^{g_2+h_2}\) by applying \(2g_2\frac{h_2+1}{g_2+1} 2h_2>0\) and \(2g_2\frac{h_2+1}{g_2+1}(g_2+h_2)<0\). In view of (14), (15), and (16), it can be acquired that
where \(\varpi _1=\frac{1}{2}\alpha (\frac{h_2+1}{u_{max}})^{\frac{h_2+g _2}{g_2+1}}\), \(\varpi _{2}=\frac{1}{2}\alpha (2n)^{\frac{1g_2}{g_2+1}}(\frac{h_2+1}{u_{max}})^{\frac{2g_2}{g_2+1}}\), \(\Psi _1= \frac{\Gamma (\frac{1 h_2}{g_2h_2})\Gamma (\frac{g_21}{g_2h_2})}{\varpi _1 (g_2h_2)}(\frac{\varpi _1}{\varpi _2})^{\frac{1 h_2}{g_2h_2}}\). Thus from Lemma 3, it can be further concluded that \(\tilde{\delta }_v\) reaches zero within predefined time \(T_{p1}\), that is, \(\tilde{\delta }_{v} \rightarrow 0\) as \(t \rightarrow T_{p1}\) and \(\tilde{\delta }_{v}=0\) when \(t> T_{p1}\).
The following \(\tilde{\delta }_x \rightarrow 0\) in predefinedtime is verified. When \(t>T_{p1}\), (11) can be redescribe as
Choose \(\bar{\delta }_x=EH \tilde{\delta }_x\), one can be obtained that
Consider the following Lyapunov function
following a same procedure as in (14)(16) gives that
where \(\varpi _3=\frac{1}{2}\alpha (\frac{h_1+1}{u_{max}})^{\frac{h_1+g_1}{g_1+1}}\), \(\varpi _{4}=\frac{1}{2}\alpha (2n)^{\frac{1g_1}{g_1+1}}(\frac{h_1+1}{u_{max}})^{\frac{2g_1}{g_1+1}}\), \(\Psi _1= \frac{\Gamma (\frac{1 h_1}{g_1h_1})\Gamma (\frac{g_11}{g_1h_1})}{\varpi _3 (g_1h_1)}(\frac{\varpi _3}{\varpi _4})^{\frac{1 h_1}{g_1h_1}}\). By invoking Lemma 3, \(\tilde{\delta }_{x,i}\) convergences to zero in predefined time with \(T_{p}^{(1)}=T_{p1}+T_{p2}\), namely \(\tilde{\delta }_{x,i} \rightarrow 0\) as \(t \rightarrow T_{p}^{(1)}\) and \(\tilde{\delta }_{x,i}=0\) when \(t> T_{p}^{(1)}\). In conclusion, the position and velocity of virtual leaders can be estimated in a predefined time \(T_p^{(1)}\). To sum up, one can conclude that \(\lim \limits _{t\rightarrow T_{p}^{(1)}}\Vert \tilde{\delta }_{v,i}\Vert = 0\), \(\lim \limits _{t\rightarrow T_{p}^{(1)}}\Vert \tilde{\delta }_{x,i}\Vert = 0\) and \(\Vert \tilde{\delta }_{v,i}\Vert =0\), \(\Vert \tilde{\delta }_{x,i}\Vert =0\) for \(t>T_{p}^{(1)}\). \(\square \)
3.3 Analysis of the Fast FiniteTime Bipartite Tracking
The purpose of this subsection is to demonstrate that, given the strength of result in Theorem 1, the output bipartite tracking can be solved using the HCA. Before moving on, define \(\bar{l}_4\) as
Theorem 2
Suppose that Assumptions 1–2 hold. Using the proposed HCA (8) for the system (1), if \(\lambda {min}(\overline{l}_4)\ge 0\), then the fast finitetime output bipartite tracking problem can be settled within a fast finite time \(T_p^{(3)}=T_p^{(1)}+T_p^{(2)}+\frac{\upsilon 2^{\frac{\upsilon 1}{2\upsilon }}}{l_1^{\frac{1}{\upsilon }}(\upsilon 1)}[\hspace{1.49994pt}[ w_{i}(0) ]\hspace{1.49994pt}]^{\frac{1+\upsilon }{\upsilon }}\).
Proof
The first part demonstrates that \(x_i\) and \(v_i\) can respectively tracking the associated estimated terms \(\hat{x}_i\) and \(\hat{v}_i\) in a fast finite time. Then, the positive definite quadratic function is considered by \(V_3=sign(\hat{s}_i)^T\hat{s}_i\), differentiating \(V_3\) along (9) yields that
where \(\Phi =l_1\upsilon \textrm{diag}(\dot{\hat{\epsilon }}_i^{\upsilon 1})\in \mathbb {R}^{p\times p}\), \(\Psi _3= \frac{\Gamma (\frac{1 a}{ca})\Gamma (\frac{c1}{ac})}{\lambda _{min}(\Phi )l_2(ac)}(\frac{l_2}{l_3})^{\frac{1a}{ca}}\). As per Lemma 3, \(\hat{s}_i\) reaches zero in predefined time \(T_{p}^{(2)}=T_{p}^{(1)} +T_L\). It can be concluded that the system (1) will be driven. In addition, once \(\hat{s}_i=0\), note that \( \hat{s}_i=w_{i}+l_{1} sig(\dot{w}_i)^{\upsilon }=0\), by [16] Lemma 3 and [34], \(w_{i} \rightarrow 0\), \(\dot{w}_i \rightarrow 0 \) as \(t \rightarrow T_{p}^{(3)}\), where \(T_{p}^{(3)}=T_{p}^{(2)}+ \frac{\upsilon 2^{\frac{\upsilon 1}{2\upsilon }}}{l_1^{\frac{1}{\upsilon }}(\upsilon 1)}[\hspace{1.49994pt}[ w_{i}(0) ]\hspace{1.49994pt}]^{\frac{1+\upsilon }{\upsilon }}\).
In the second step is to prove that the fast finitetime output bipartite tracking problem can be solved. The Triangle Inequality of the Euclidean norm can be employed to get
and
This completes the proof. \(\square \)
Remark 1
Theoretically speaking, we can shorten the upper bound of convergence time by increasing the values of the \(l_2, l_3\) and \(l_4\), and the result is that the amplitude of the input will also increase. To ensure that the robotic system can achieve stable trajectory tracking within a fast finite time, the crucial premise is that the actuators can still produce sufficient actuating torques to drive the system even if there exists the upper amplitude of the actuators. The selection of design parameters needs to take into account the upper bound of the actuator amplitude, so the parameters can be determined by the trialanderror method (Fig. 4).
4 Simulations
The theoretical analysis results presented in this section are validated by the numerical simulations. The considered NHRSs with a virtual leader and eight agents, consisting of six twodegreeoffreedom (i.e., \(i \in \left\{ 2,3,4,5,6,7\right\} \)) and two threedegreeoffreedom (i.e., \(i \in \left\{ 1,8\right\} \) ) agents. The mechanical configurations are shown in Figs. 2 and 3. A matrixweighted sign digraph is presented in Fig. 4. In addition, the physical parameters of the eight manipulators and their estimation values are presented in Tables 1 and 2, respectively. The parameters of the local control layer and estimation layer are shown in Tables 3 and 4, respectively.
Further, the initial values for the state variables and their estimated values are selected randomly from \([1,1]\). Choose the external disturbances \(d_{i}=\left[ 0.1\sin (t),0.1\cos (t)\right] ^{T},i=\left\{ 2,\ldots ,7\right\} \) and \(d_{i}=[0.1\sin (t),0.1\cos (t), 0.1 \sin (t)]^{T}, i=\left\{ 1, 8\right\} \),
To validate the bipartite tracking, the reference trajectory is selected as
Figs. 5, 6, 7, 8 and 9 are presented the simulation results. For more information, one can notice that the evolutions of \(\hat{x}_i\) and \(\hat{v}_i\) which indicate that \(\hat{x}_i \rightarrow e_i x_l, \hat{v}_i \rightarrow e_iv_l\) within a predefined time in Figs. 5c, d and 6c, d. In the same way, Figs. 5a, b and 6a, b show that the fast finitetime convergence \(x_i \rightarrow e_ix_l, v_i \rightarrow e_i v_l\) are completed with the proposed HCA algorithm. Figure 7 illustrates that the convergence performances of \(x_ie_ix_l\) and \(v_ie_iv_l\), where errors converge to zero as \(t \rightarrow \infty \). Figure 8 shows the evolutions of \(\tilde{\delta }_{x,i}\) and \(\tilde{\delta }_{v,i}\), respectively. The effectiveness of the estimation algorithm is demonstrated. Figure 9 shows that the position and velocity error convergence to the zero within the fast finite time. What’s more, It can be inferred from the simulation results depicted in Figs. 5, 6, 7, 8 and 9 that the output bipartite tracking is ensured at approximately T = 3.7 s.
In order to highlight the superiority of the developed fast finitetime control stability, one compared the results with the asymptotic stability in [7, 16, 22, 31] and finitetime stability in [8, 21, 30], which are shown in Table 5.
5 Discussion
We have developed an effective way to solve the output bipartite tracking problem of the NHRSs with matrixweighted digraphs. Unlike the traditional tracking problems with only cooperative interactions, the robots of the NHRSs are divided into two complementary subgroups according to the signed graph, describing the coexistence of cooperative and antagonistic interactions. The developed hierarchical control framework consisting of the distributed predefinedtime estimator layer and the local control layer in this paper can radically reduce the difficulty of the control algorithm design, simplify the corresponding stability analysis, and provide good compatibility. Our results have also demonstrated that the NHRSs described as Euler–Lagrange dynamics is successfully controlled and regulated.
The main advantage of the proposed novel fast finitetime controller is the significantly improved convergence rate. Compared with the existing finitetime algorithms [21, 30], the distributed estimators are employed to estimate the leaders states in finite time, where the upper bound of the settling time dependents on the initial values of the system and thus cannot be arbitrarily predesigned. The designed estimator in this paper can estimator the states of the leader in predefined time, which can be predetermined by the users. Moreover, the predefinedtime reachability of the designed algorithm is guaranteed to reach finitetime sliding surface.
More recent research concentrated on cooperative control of the networked robotic systems being modeled as ELSs [18, 21, 24, 30, 32, 36]. These works can mainly be classified into jointspace control and taskspace control. The second important development is that the developed taskspace algorithms considering kinematics of heterogeneous manipulators are more practical and applicative comparing with jointspace algorithms. On the other hand, multiple manipulators containing both redundant and nonredundant individuals, namely, multiple heterogeneous manipulators, which have a wide application prospect in the areas of military technologies and advanced manufacturing.
In the social network, the individuals reach various agreements through exchanging the data of high dimensionality, while the matrixweighted graph provides a mechanism to model such complicated relationships [22, 23]. Furthermore, the structural balance property of the scalarweighted graphs is no longer holds for bipartite consensus with the signed matrixweighted graphs. We technically have presented several solutions for achieving bipartite tracking with matrixweighted sign digraphs via the newlydesigned hierarchical control framework. Taken together, the fast finite time HCA and their analysis method provide a theoretical guidance for achieving global behaviors of multiagent systems with matricweighted sign digraphs in fast finite time.
6 Conclusion
This paper has studied the fast finitetime output bipartite tracking of the NHRSs in the case of parametric uncertainties and external disturbances under a matrixweighted sign digraph. To deal with such extremely challenging issues, a novel distributed hierarchical control algorithm consists of the estimator algorithm and the terminal sliding mode algorithm have been established. Additionally, sufficient conditions for NHRSs to achieve fast finitetime output bipartite tracking have been acquired. Finally, the results is obtained by a MATLAB R2022a simulation and have effectively validated the theoretical analyses. Besides, it is worth noting that the existence of parametric uncertainties and external disturbances in the NHRSs may lead to actuator bias faults, partial loss of effectiveness actuation faults. Therefore, future works will lie on distributed faulttolerant control of the NHRSs.
7 Appendix
The detailed control process is as follows. To begin with, the states of the leader is estimated in the predefined time \(T_p^{(1)}\) in the distributed estimator layer. Then, there are two involved time in the local control layer, one is that the states of the closedloop system reach to the sliding surface \(\hat{s}_i\) in the predefined time \(T_p^{(1)}+T_L\), the another is that the error states converge to the origin in the finite time \(\frac{\upsilon 2^{\frac{\upsilon 1}{2\upsilon }}}{l_1^{\frac{1}{\upsilon }}(\upsilon 1)}[\hspace{1.49994pt}[ w_{i}(0) ]\hspace{1.49994pt}]^{\frac{1+\upsilon }{\upsilon }}\). Apparently, the whole convergence time is bounded with \(T_p^{(3)}=T_p^{(1)}+T_L+\frac{\upsilon 2^{\frac{\upsilon 1}{2\upsilon }}}{l_1^{\frac{1}{\upsilon }}(\upsilon 1)}[\hspace{1.49994pt}[ w_{i}(0) ]\hspace{1.49994pt}]^{\frac{1+\upsilon }{\upsilon }}\). Furthermore, the Algorithm 1 is introduced to illustrate the design procedure of the proposed controller.
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
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This work was funded by the National Natural Science Foundation of China (62071173) and the Natural Science Foundation of Hubei Province (2022CFB479).
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Gu, R., Han, T., Xiao, B. et al. Fast FiniteTime Output Bipartite Tracking of Networked Heterogeneous Robotic Systems with MatrixWeighted Digraphs. Circuits Syst Signal Process 43, 6132–6154 (2024). https://doi.org/10.1007/s0003402402719w
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DOI: https://doi.org/10.1007/s0003402402719w