Bipartite Formation Research Articles

Hierarchical bipartite formation tracking of networked perturbed robotic systems with nonlinear state‐to‐output mappings

AbstractThis article investigates the hierarchical bipartite formation tracking (HBFT) problem of the networked perturbed robotic system (NPRS) with dynamical uncertainties, actuator faults and nonlinear state‐to‐output mappings. Different from the traditional single‐layer formation tracking problems based on only cooperative interconnections, solving the HBFT problem implies that (a) the robots of the NPRS are divided into two layers of subnetworks, where the ones in subnetwork 1 can directly interconnect with the leader, those in subnetwork 2 can only interconnect with each other or subnetwork 1, and the ones in the different subnetworks form hierarchical formations; (b) each formation is further grouped into two complementary sets according to the cooperative and antagonistic interconnections, where the robots of each formation forms two opposite subformations; (c) for the same subnetwork, the geometric centers of the two subformations track the positive and negative states of the leader respectively. The positive state equals to the leader state, and the negative state has the elements with the same absolute value and opposite sign of the corresponding one in the leader state. Then, a uniform cooperative observer‐based fault‐tolerant control (COFTC) algorithm is designed to solve these challenging problem. By employing the Lyapunov theory, the sufficient conditions are derived for guaranteeing the convergence of the proposed COFTC algorithm. Eventually, simulation results are performed on NPRS to verify the effectiveness of the theoretical results.

Observer‐based bipartite Mittag‐Leffler time‐varying formation for fractional multi‐agent systems with discontinuous inherent dynamics

SummaryIn this article, the bipartite Mittag‐Leffler time‐varying formation issue of fractional multi‐agent systems is discussed, where the dynamical nodes of followers are modeled to be discontinuous and nondecreasing. A observer‐based adaptive distributed control scheme, which is dependent on output information, is designed to realized the bipartite formation control goal. Meanwhile, the efficient algorithm with respect to the control protocol also is presented. Under fractional Filippov differential inclusion framework, by constructing Lur'e Postnikov‐type Lyapunov functional and applying Clarke's non‐smooth analysis technique, the bipartite Mittag‐Leffler time‐varying formation conditions are derived in terms of linear matrix inequalities. At last, a simulation example is carried out to verify the effectiveness of the proposed scheme.

Fully distributed practical bipartite formation tracking of networked Lagrangian systems with actuator faults and bounded inputs

This paper investigates the bipartite formation tracking (BFT) problem of networked Lagrangian systems (NLSs) with signed directed graphs. Each system suffers from actuator fault with bounded input. Firstly, the unknown input of the leader is estimated for all the followers based on an observer in the prescribed time, which is constructed by employing a time base generator (TBG). Then, a hierarchical fully distributed controller is proposed to achieve BFT without using the global information with respect to the communication graphs. Then, the related sufficient condition for the system stability is derived by means of the Lyapunov theory. Finally, a simulation example is proposed to demonstrate the performance of the hierarchical fully distributed controller for achieving BFT.

Predictor‐based bipartite time‐varying formation control of nonlinear multi‐agents systems via disturbance observer

AbstractFormation control is one of critical control issues in multi‐agent systems (MASs). In this article, we extend the traditional formation to a bipartite one in a structurally balanced signed graph and propose a bipartite formation control strategy for an MAS with unknown external disturbances and nonsymmetric input saturation constraints. Neural networks (NNs) are developed to approximate followers' dynamics, and their transient performance are improved by prediction errors. On this basis, an NN‐based nonlinear disturbance observer (NDO) is constructed to estimate generalized disturbances including external disturbances and NNs approximation errors. In order to reduce the number of learning parameters, a normalization method is introduced for NNs weights and applied to the proposed NDO. An auxiliary system is constructed to compensate for asymmetric input saturation, and it is not required to access unknown control gains. The stability analysis is presented, and it is proven that all signals in the MAS are bounded. A numerical simulation and a team of quadrotors examples corroborate the effectiveness and feasibility of the strategy.

Fully Distributed Event-Triggered Bipartite Formation Tracking Control for Heterogeneous Multi-Agent Systems on Signed Digraph

Considering that communication between neighboring agents may be cooperative or antagonistic, this study is concerned with the bipartite time-varying output formation tracking (BTVOFT) problem for heterogeneous multi-agent systems (MASs) with multiple leaders on signed digraph. A fully distributed event-triggered output feedback formation tracking (FDETOFFT) control scheme is constructed to address this problem. It is noteworthy that the control scheme does not involve any global information, which enables it to be implemented in a fully distributed fashion. The biggest contribution of this study is to extend event-triggered techniques to the formation tracking control scheme. Compared with previous related works, continuous communication between neighboring agents is avoided. Moreover, it is further proved that no agents exhibit the Zeno behavior.

Fully distributed event-triggered bipartite formation tracking for multi-agent systems with multiple leaders and matched uncertainties

This paper investigates the bipartite time-varying output formation tracking issue for general linear multi-agent systems with multiple leaders and matched uncertainties. The outputs of followers are required to form two antagonistic prescribed time-varying subformations, one of which tracks the convex combination of the outputs of multiple leaders and the other tracks the symmetric convex combination. An independent asynchronous fully distributed event-triggered control protocol is formulated by using relative information between neighboring agents. It is shown that, with the formulated protocol, the bipartite time-varying output formation tracking can be achieved if the feasible formation condition is satisfied. The Zeno behavior is proved to be excluded. Then, we further construct a novel self-triggered control protocol to avoid continuous monitoring of estimate error. Since the protocol incorporates both event-triggered control and adaptive control, it efficiently avoids continuous communication among agents and can be implemented in a fully distributed manner. Moreover, it is noteworthy that in the case where relative information is available while absolute information is not, the protocol is applicable. Finally, the feasibility of the constructed protocols is verified by a numerical example.

Observer-Based Bipartite Formation Control for MASs With External Disturbances Under Event-Triggered Scheme

This brief investigates the bipartite time-varying formation tracking control for multi-agent systems with cooperative-antagonistic interactions and external disturbances under event-triggered scheme. To deal with the external disturbances, an extended state observer is employed to illustrate the control protocol. Then, a time-varying formation control protocol is developed under event-triggered scheme to eliminate continuous communication, further reducing unnecessary computational cost successfully. Besides, Zeno behavior is verified to be excluded and convergence analysis of proposed protocol is demonstrated by the lights of Lyapunov stability theory. Finally, the effectiveness of the proposed protocol is indicated by a numerical simulation, where the agents realize two time-varying antagonistic sub-formations with disturbance rejection under event-triggered scheme.

Event-Triggered Control for Guaranteed-Cost Bipartite Formation of Multi-Agent Systems

This paper is concerned with event-triggered guaranteed-cost bipartite formation control of multi-agent systems with antagonistic interactions. In order to save the limited network communication bandwidth of multi-agent systems, event-triggered sampled-data transmission strategy is adopted. Event conditions are designed for both leader and followers to reduce the frequency of state transmission more effectively. According to the event-triggered sampled-data of the leader and followers, bipartite formation controllers are designed for followers. By using the event conditions and the Lyapunov method, the sufficient conditions for the realization of bipartite formation and guaranteed-cost bipartite formation are obtained in terms of linear matrix inequality, respectively. Finally, the effectiveness of the theoretical results is demonstrated by numerical examples.

Bipartite Tracking Formation Control of Nonlinear Multi-Agent Systems Using Adaptive Output–Feedback Protocols

This paper proposed distributed adaptive formation control for leader-follower bipartite time-varying formation (BTVF) of a nonlinear multi-agent system (MAS). The proposed nonlinear MAS satisfies the one-sided Lipschitz-type condition. In the topological graphs with directed spanning trees, the design of adaptive protocols does not depend on the known communication topology, which can avoid the use of global information. Given limited information, the proposed MAS can achieve desired formation tracking by utilizing an observer protocol and converting the bipartite formation control problem into a stability problem of system errors. The analysis results of the systematic errors by using Lyapunov candidate functions, indicate that the MAS can be globally stable with a certain convergence rate during operation. Finally, numerical simulations are presented to confirm the validity of the proposed approach.

Leader-Following Bipartite Formation Control of Multiple Nonholonomic Robot Systems Over Signed Graph

This paper studies the bipartite formation control problem of multiple nonholonomic robot systems over signed graph. To overcome nonholonomic and underactuated control challenges, the hand position output is defined to obtain a linear hand position subsystem. A distributed control law is designed, which uses sliding mode control technology to eliminate the influence of dynamic leader, so that the global index of position formation error converges to an arbitrary small neighborhood of the origin. Then, the zero dynamic equation is computed, and based on it and Lyapunov method, the convergence conditions on controller parameters and on leader’s velocities are derived to guarantee the angle error is bounded and convergent. Finally, a simulation example containing eight robots and one leader is given to demonstrate the effectiveness of the proposed control strategy. In contrast to related results, this paper presents the complete stability analysis and corresponding stability conditions, in addition to the control law and stability of position subsystems.

Distributed Prescribed-Time Interval Bipartite Consensus of Multi-Agent Systems on Directed Graphs: Theory and Experiment

This work deals with the analysis and protocol design problems of the prescribed-time interval bipartite consensus of multi-agent systems on signed and directed graphs. A new distributed protocol with hybrid constant and time-varying feedbacks of local signed error is proposed, whose consensus time period is independent of the specific topology among agents and initial states of all agents. By introducing a series of well-structured Lyapunov functions, the technical difficulties arising from the asymmetrical Laplacian matrices of directed graphs are circumvented. The effectiveness of this prescribed-time protocol for multi-agent systems on signed digraphs with a spanning tree is proven both on structurally balanced digraphs and structurally unbalanced ones with a positive root subgraph. An illustrative simulation example and a prescribed-time bipartite formation experiment on a swarm of nano-quadcopters are implemented to show the validity and practicability of these proposed protocols.

Bipartite Consensus Control for a Swarm of Robots

Abstract This paper studies the bipartite consensus problem of a swarm of robots whose dynamics are formulated by Lagrangian equations. Two distributed bipartite consensus control protocols are proposed for a swarm of robots without a leader or with a virtual leader. For the nonleader case, the networked Lagrangian system can reach static bipartite consensus under the control protocol developed, and the final convergent states can be explicitly determined by the specific structure of the Laplacian matrix associated with the cooperative–competitive network topology. For the virtual leader case, all the followers can track the leader's state in a bipartite formation to realize bipartite tracking consensus. Finally, the simulation results are given to verify the theoretical results.

Distributed finite-time bipartite consensus of multi-agent systems on directed graphs: Theory and experiment in nano-quadcopters formation

This work deals with the analysis and protocol design problems of finite-time consensus for multi-agent systems on signed and directed networks. A new distributed protocol with two reciprocal fractional power feedbacks of local consensus error has been proposed to remove the influence of initial states on the consensus time. By introducing a well-structured Lyapunov function, the technical difficulty arising from the asymmetrical Laplacian matrices of directed graphs is circumvented. Furthermore, it is shown that the feasibility of this proposed protocol can be extended to directed graphs with a spanning tree by mathematical induction method. The upper bounds of the finite-time periods are given under the circumstances of directed networks with a spanning tree apart from the strongly-connected ones. This new protocol is then extended to the application of finite-time bipartite formation control of multi-agent systems. Simulations and experiments of bipartite formation of multiple nano-quadcopters are implemented to prove the validity and practicability of this new protocol. In the experiment, a novel fuzzy logic is also introduced to realize the group collision avoidance of all nano-quadcopters.

Bipartite formation problem of second-order nonlinear multi-agent systems with hybrid impulses

In this paper, bipartite formation of second-order nonlinear multi-agent systems (MASs) with hybrid impulses is investigated. At first, state-feedback controllers and impulsive protocols are designed for second-order MASs based on the definition of bipartite formation. Secondly, with the aid of average impulsive interval and average impulsive gain, the design criterion of bipartite formation control protocol is derived such that second-order nonlinear MASs with hybrid impulses can realize bipartite formation. Furthermore, adaptive control is also addressed for second-order MASs with unknown nonlinear terms and the corresponding result is derived. Finally, the validity of the theoretical results is illustrated by two numerical simulations.

From rational numbers to algebra: Separable contributions of decimal magnitude and relational understanding of fractions

To understand the development of mathematical cognition and to improve instructional practices, it is critical to identify early predictors of difficulty in learning complex mathematical topics such as algebra. Recent work has shown that performance with fractions on a number line estimation task predicts algebra performance, whereas performance with whole numbers on similar estimation tasks does not. We sought to distinguish more specific precursors to algebra by measuring multiple aspects of knowledge about rational numbers. Because fractions are the first numbers that are relational expressions to which students are exposed, we investigated how understanding the relational bipartite format (a/b) of fractions might connect to later algebra performance. We presented middle school students with a battery of tests designed to measure relational understanding of fractions, procedural knowledge of fractions, and placement of fractions, decimals, and whole numbers onto number lines as well as algebra performance. Multiple regression analyses revealed that the best predictors of algebra performance were measures of relational fraction knowledge and ability to place decimals (not fractions or whole numbers) onto number lines. These findings suggest that at least two specific components of knowledge about rational numbers—relational understanding (best captured by fractions) and grasp of unidimensional magnitude (best captured by decimals)—can be linked to early success with algebraic expressions.

Showing Off

The article examines a group of closing gestures in Mozart's piano concertos, passages of soloistic virtuosity just before the last tuttis of the exposition and recapitulation. Serving as grand cadential clinchers and clothed in conventional figuration, these spots—sometimes called “display episodes”—appear to highlight performative prowess at the expense of compositional invention. As such, beyond an acknowledgment of the appropriateness of stock passagework at this point, analysis can have little to say: the episodes are either self-evidently simple or transparent to theoretical apparatus.I suggest, however, that by using the vantage point of variation, conceived broadly, not just as a technique but as a manner of organizing musical thoughts, new insights can emerge about these passages. This wider conception of variation is sanctioned both by historical precedent (Daube, Koch) and by contemporary usage (Sisman, Agawu); the specific invocation of it in connection with the display episodes is suggested initially by their characteristic bipartite format. Seen through the variation lens, elements such as contrapuntal ingenuity, complex modular arrangements within and between phrases, and an affinity for textural exploration come to the fore. These elements constitute a display of compositional virtuosity that is a counterpart to the more evident performative fireworks.

Skeletal development in Xenopus laevis (Anura: Pipidae).

Postembryonic skeletal development of the pipid frog Xenopus laevis is described from cleared-and-stained whole-mount specimens and sectioned material representing Nieuwkoop and Faber developmental Stages 46-65, plus postmetamorphic individuals up to 6 months old. An assessment of variation of skeletogenesis within a single population of larvae and comparison with earlier studies revealed that the timing, but not the sequence, of skeletal development in X. laevis is more variable than previously reported and poorly correlated with the development of external morphology. Examination of chondrocranial development indicates that the rostral cartilages of X. laevis are homologous with the suprarostral cartilages of non-pipoid anurans, and suggests that the peculiar chondrocranium of this taxon is derived from a more generalized pattern typical of non-pipoid frogs. Derived features of skeletal development not previously reported for X. laevis include 1) bipartite formation of the palatoquadrate; 2) precocious formation of the adult mandible; 3) origin of the angulosplenial from two centers of ossification; 4) complete erosion of the orbital cartilage during the later stages of metamorphosis; 5) development of the sphenethmoid as a membrane, rather than an endochondral bone; and 6) a pattern of timing of ossification that more closely coincides with that of the pelobatid frog Spea than that recorded for neobatrachian species.

Energy transfer between photosystem II units in a connected package model of the photochemical apparatus of photosynthesis

A model of the photochemical apparatus of photosynthesis, presented previously in a tripartite format, is used in a bipartite format to analyze energy transfer between photosystem II units. The model is used to develop analytical expressions for the photochemical properties of chloroplasts that include a term for the probability for energy transfer between photosystem II units. In particular, a normalized plot of the fluorescence of variable yield as a function of the closure of photosystem II reaction centers permits the evaluation of the energy transfer parameter. Data from a study by Melis [Melis, A. (1978) FEBS Lett. 95, 202-206] are used to estimate that the probability for energy transfer between photosystem II units in the alpha component of photosystem II was approximately 0.57 when the photosystem II reaction centers were all open and 0.83 when the centers were closed. This connected package model appears to provide a better description of the system than the matrix model that was used previously.