This article studies the distributed optimal formation problems for multiagent systems subject to nonlinear inequality constraints. This problem can be formulated into a nonlinear mixed-integer programming (NMIP) problem. We first decompose the NMIP problem into a formation optimal matching problem, which is actually an integer linear programming (ILP) problem, and an optimal formation reference center problem described as a constrained quadratic optimization problem. Subsequently, we develop a discrete-time perturbation-based distributed dual consensus ADMM (PDC-ADMM) algorithm, which achieves an optimal integer solution to the ILP problem and eliminates the unmatched phenomenon. In addition, we propose a distributed optimistic gradient descent ascent (D-OGDA) algorithm with a constant step size, which guarantees exact convergence to the optimal formation reference center. Finally, three simulation examples are carried out to demonstrate the effectiveness of the developed algorithms.

This article focuses on the constrained optimization problem for second-order multiagent systems that experience heterogeneous communication delays. Specifically, the involved agents work together to find the optimal solution of a global payoff function, which is summed by multiple strongly convex local payoff functions, with each function being exclusively owned by an individual agent. However, the feasible solutions must satisfy a coupled equality constraint, formulated by individual parameters assigned to each agent. Initially, a basic anti-delay distributed protocol is developed, which leverages a scattering transformation to enhance the generation of received information. Using the Lyapunov framework, we demonstrate that the agents coordinated by this anti-delay distributed protocol can effectively reach a consensus on the expected optimal solution, despite the presence of communication delays. In addition, we present two results that extend the basic anti-delay distributed protocol. First, we consider the scenario of lacking velocity and develop a velocity-free anti-delay distributed protocol to achieve the concerned constrained optimization objective. Next, we augment the system order and develop an anti-delay distributed optimization protocol for integrator chain multiagent systems. Finally, we confirm the anti-delay performance of the developed distributed protocols through simulations.

In this work, we propose a solution that leverages geospatial data to initialize the monocular visual-inertial navigation system. For Visual-Inertial Navigation Systems (VINS) operating on UAVs, the ability to perform initialization and relocalization in mid-air is essential. However, degenerate motion can cause VINS to lose scale, making traditional initialization algorithms less reliable. To address this issue, we fuse geographic information in the initialization process, and utilize a learning-based feature matching algorithm to associate the information with inertial states. The proposed approach demonstrates adaptability to the degenerate motions of UAVs and significantly surpasses the estimation accuracy of conventional VINS initialization algorithms. Compared to methods that assist initialization by using a laser-range-finder (LRF), the proposed method solely relies on low-cost satellite imagery and elevation information. We evaluate the proposed approach on a large-scale UAV dataset, and compare with existing methods. The results demonstrate the superior effectiveness of the proposed method.

This article considers the distributed Nash equilibrium seeking problem of a multicluster aggregative game subject to local set constraints, consensus constraints in the same cluster, and coupled linear equality and nonlinear inequality constraints among all clusters. In the considered game, each cluster is composed of a group of players formulated by uncertain Euler-Lagrange (EL) dynamics, and its objective is to minimize its own cost function, which is the sum of the local functions of all players in the cluster. The local cost function of each player depends on its own decision and an aggregate of the decisions of all the players. An adaptive continuous-time distributed strategy is developed for uncertain EL systems to reach the generalized Nash equilibrium (GNE) of multicluster aggregative game. In particular, a new auxiliary system is synthesized using a projection operator, gradient descent, and dynamic average consensus to estimate the GNE. Based on the outputs of the auxiliary system, an adaptive tracking algorithm is developed for an EL system with uncertain parameters. Using the Lyapunov stability theory, it is shown that the developed distributed strategy achieves accurate convergence to the GNE. Finally, a numerical example is presented to demonstrate the theoretical results.

This article studies the leader-following formation tracking control problem of multiple vertical takeoff and landing (VTOL) unmanned aerial vehicles (UAVs) subject to uncertain parameters, in which the target formation configuration is defined by using the interneighbors' bearing vectors. An adaptive formation control algorithm with bearing-only measurements is proposed under a hierarchical control framework. More specifically, a saturated adaptive distributed control force is developed in the position loop with bearing-only measurements. Since the bearing vectors with respect to the neighbors can be directly measured by low-cost onboard cameras, the proposed formation algorithm can be implemented without interagent communication. Subsequently, in the attitude loop, an adaptive hybrid control scheme via two modified Rodrigues parameters (MRPs) sets is proposed, which achieves the command attitude tracking globally and also avoids the unwinding problem of MRPs. Based on the Lyapunov stability analysis and hybrid theory, we prove that the overall closed-loop error system is globally asymptotically stable. Finally, we provide a numerical example to demonstrate the effectiveness of the proposed control algorithm.

This paper presents an efficient and accurate range-aided visual-inertial odometry (RVIO) system for the global positioning system denied environment. In particular, the ultra-wideband (UWB) measurements are integrated to reduce the long-term drift of the visual-inertial odometry (VIO) system. Our approach starts with a filter-based scheme to localize the unknown UWB anchor in the local world frame. In particular, a novel surface-based particle filter is proposed to localize the UWB anchors efficiently. When the initialization is complete, the UWB location information is utilized to support the subsequent long-term robot positioning. An observability-constrained optimization approach is developed to combine the visual, inertial, and UWB range measurements. Such a framework takes advantage of both VIO and UWB measurements and is feasible even when the number of observed UWB anchors is below four. Experiments on both simulated and real-world scenes demonstrate the validity and superiority of the proposed system.

This article studies the distributed constrained optimization problems for the discrete-time second-order multiagent systems (MASs), in which each agent privately owns local cost function and nonidentical convex set constraints. To solve this problem, a projection-based distributed event-triggered algorithm is developed via the constant step-sizes, which achieves an ergodic convergence rate$O(1/k)$for the general convex functions. By applying the event-triggered mechanism, the proposed algorithm can avoid unnecessary communication among the agents. Moreover, it is shown that the introduced event-triggered component does not sacrifice the convergence rate. Finally, a simulation example is carried out to demonstrate the theoretical results.

Bundle adjustment (BA) is a well-studied fundamental problem in the robotics and vision community. In man-made environments, coplanar points and lines are ubiquitous. However, the number of works on bundle adjustment with coplanar points and lines is relatively small. This paper focuses on this special BA problem, referred to as$\pi-\mathbf{BA}$. For a point or a line on a plane, we derive a new constraint to describe the relationship among two poses and the plane, called$\pi$-constraint. We distribute$\pi$-constraints into different groups. Each group is called a$\pi$-factor. We prove that, with some simple preprocessing, the computational complexity associated with a$\pi$-factor in the Levenberg-Marquardt (LM) algorithm is$O(1)$, independent of the number of$\pi$-constraints packed into the$\pi$-factor. In$\pi-\mathbf{BA}, \pi$-factors replace original reprojection errors. One problem is how to divide$\pi$-constraints into$\pi$-factors. Different strategies may result in different numbers of$\pi$-factors, which in turn affects the efficiency. It is difficult to get the optimal division. We present a greedy algorithm to overcome this problem. Experimental results verify that our algorithm can significantly accelerate the computation.

The optimization problem of second-order discrete-time multiagent systems with set constraints is studied in this article. In particular, the involved agents cooperatively search an optimal solution of a global objective function summed by multiple local ones within the intersection of multiple constrained sets. We also consider that each pair of local objective function and constrained set is exclusively accessible to the respective agent, and each agent just interacts with its local neighbors. By borrowing from the consensus idea, a projection-based distributed optimization algorithm resorting to an auxiliary dynamics is first proposed without interacting the gradient information of local objective functions. Next, by considering the local objective functions being strongly convex, selection criteria of step size and algorithm parameter are built such that the unique solution to the concerned optimization problem is obtained. Moreover, by fixing a unit step size, it is also shown that the optimization result can be relaxed to the case with just convex local objective functions given a properly chosen algorithm parameter. Finally, practical and numerical examples are taken to verify the proposed optimization results.

This article investigates the noncooperative game of multiple clusters composed by double-integrator agents subject to set constraints. In particular, by decomposing each cluster cost function into multiple individual ones and allocating them to respective agents, the generalized Nash equilibrium (GNE) associated with the upper cluster layer is sought by the lower agent layer in terms of a hierarchical structure. A particular agent in each cluster is appointed as the messenger that is specialized in interacting with adjacent messengers from other clusters. It is first shown that the concerned seeking objective is achieved provided that the agents from the same cluster reach a consensus on their corresponding component of the GNE. Motivated by this consensus idea, a distributed seeking strategy through the intracluster interaction is first developed, which is applicable to the case where each agent has access to the information of the messengers from other clusters. Based on this result, we next consider a relaxed case where the messenger information is absent for each agent and the inter-cluster information interaction is only allowed via their messengers. Another distributed seeking strategy resorting to a distributed observer is developed via both the inter- and intra-cluster interactions. It is shown that both the developed distributed seeking strategies are capable of seeking out the concerned GNE. As extensions to the double-integrator model, we further consider two evolved models, i.e., Euler–Lagrange model and multi-integrator model. Two modified distributed seeking strategies are developed for achieving the concerned NE seeking objective. Finally, the effectiveness of the developed distributed seeking strategies is validated by an application.

Visual Inertial Odometry (VIO) is a widely studied localization technique in robotics. State-of-the-art VIO algorithms are composed of two parts: a frontend which performs visual perception and inertial measurement pre-processing, and a backend which fuses vision and inertial measurements to estimate the robot's pose. Both image processing in the frontend and sensor fusion in the backend are computationally expensive, making it very challenging to run the VIO algorithm, especially the optimization-based VIO algorithm in real time on embedded platforms with limited power budget. In this paper, a real-time optimization-based monocular VIO algorithm is proposed based on algorithm-and-hardware co-design and successfully implemented on an embedded platform with only 2.6W processor power consumption. In particular, the time-consuming Harris corner detection (HCD) is accelerated on Field Programmable Gate Array (FPGA), achieving an average 16 × processing time reduction compared with the ARM implementation. Compared with the state-of-the-art HCD accelerator provided by Xilinx, the hardware resource required of our accelerator is largely reduced without any compromise in speed, thanks to the proposed dedicated pruning and paral-lelization techniques. Finally, experiment on the public dataset demonstrates that the proposed real-time VIO algorithm on the FPGA-based platform has comparable accuracy with respect to the existing state-of-the-art VIO algorithm on the desktop, and 3 × faster frontend processing speed over the ARM-based implementation.

The time-varying economic dispatch over a network is considered where both the cost function and the equality constraint are time-varying. A distributed algorithm is first designed combing the prediction-correction framework and the consensus + innovations approach. In the prediction steps, the sensitivity analysis method is introduced to calculate the shift between the estimated and the exact optimization solution of the correction steps. Afterwards, the overall optimal errors in two steps are calculated by the sub-optimal analysis method. The convergence analysis shows that the dispatch errors are bounded and the convergence process is Q-linear. A dynamic prediction-correction algorithm is then presented to improve the accuracy at every time instant. Finally, numerical examples using the IEEE 118-bus system are presented to illustrate the validity of the algorithms.

This article studies a fully distributed optimal coordinated control problem with the global cost function for networked Euler-Lagrange (EL) systems subject to unknown model parameters. In particular, the global cost function is the sum of all the local cost functions assigned to each agent and only available to itself. The objective is to minimize the global cost function in a distributed manner while achieving a consensus on its optimal solution. Since the model parameters of the considered EL systems are not available, a new auxiliary system is introduced as a reference model, and its outputs exponentially converge the optimal solution of the global cost function. A fully distributed optimal control algorithm without requiring global information is first proposed. Then, an alternative distributed optimal algorithm via the event-triggered mechanism is proposed to reduce the communication cost. In particular, by combining an edge-based adaptive gain method, the proposed event-triggered optimal algorithm is also fully distributed. Finally, numerical simulation is carried out to validate the theoretical results.

In this article, a class of distributed nonlinear placement problems is considered for a multicluster system. The task is to determine the positions of the agents in each cluster subject to the constraints on agent positions and the network topology. In particular, the agents in each cluster are placed to form the desired shape and minimize the sum of squares of the Euclidean lengths of the links amongst the center of each cluster and its corresponding cluster members. The problem is converted into a time-varying noncooperative game and then a distributed Nash equilibrium-seeking algorithm is designed based on a distributed observer method. A new iterative approach is employed to prove the convergence with the aid of the Lyapunov stability theorem. The effectiveness of the distributed algorithm is validated by numerical examples.

This article investigates the targeted bipartite consensus problem of opinion dynamics in cooperative-antagonistic networks. Each agent in the network is assigned with a convergence set to represent a credibility interval, in which its opinion is trustworthy. The network topology is characterized by a signed switching digraph. The objective is to achieve the bipartite consensus targeted within these credibility intervals. A gradient term is introduced in the opinion dynamics besides the consensus term. Under the assumption that the underlying graph is uniformly jointly strongly connected and structurally balanced, it is first shown that the considered opinion dynamics reaches the targeted bipartite consensus within the intersections of the convergence sets associated with two antagonistic groups. Next, by relaxing the connectivity condition to uniform joint quasistrong connectivity, the targeted bipartite consensus result is also proven with an additional convergence set assumption. Numerical examples are provided to validate the proposed theoretical results.

In this article, the distributed nonlinear placement problem for a class of multicluster Euler–Lagrange systems is considered. The problem is first converted into a time-varying noncooperative game. A distributed Nash equilibrium seeking algorithm composed of an auxiliary double-integrator system and a coordinated-tracking observer is designed to solve the problem. The convergence results are established by an iterative approach and the small gain theorem. The effectiveness of the algorithm is demonstrated via simulations.

In this article, we consider a convex optimization problem which minimizes the sum of local agents' cost functions subject to certain local constraints. Besides, both the local cost function and local constraints are only known by the local agent itself. To solve this problem, a new accelerated distributed gradient-based algorithm is proposed, which is inspired by the “momentum” phenomena in nature and aims to accelerate the convergence speed of conventional distributed gradient algorithms. Sufficient conditions for the stepsizes and the acceleration gains are derived to ensure the convergence of the proposed algorithm. Furthermore, based on this proposed fast distributed algorithm, a new decentralized approach is proposed to solve economic dispatch problem, especially for a large-scale power system. Based on the idea of virtual agent, it is proved that this decentralized algorithm is equivalent to the original fast distributed gradient method. Several case studies implemented on IEEE 30-bus, IEEE 118-bus power systems, and a large-scale power system consisting of 1000 generators are conducted to validate the proposed method.

In this article, a class of distributed quadratic games is considered over an undirected graph. The issue on the communication topology restriction is introduced and the players’ dynamics are with nonlinear dynamics and unknown time-varying perturbation. A distributed Nash equilibrium seeking algorithm is proposed based on a high-gain observer method, and the convergence is analyzed by the Lyapunov stability theory. It is shown that each player estimates the rival players’ states, and the errors between the players’ states and the Nash equilibrium are ultimately bounded by a small bound. Moreover, the presented algorithm is free of chattering phenomena because it is designed using the hyperbolic tangent function instead of the signum function to dominant the perturbation. The effectiveness of the proposed algorithm is validated via a simulation of the oligopoly game in which five firms produce the same products in a duopoly market structure.

In signed networks, the cooperation and antagonism cause great difficulties for their behavior analysis, especially, when they are subject to time-varying topologies. This is different from unsigned networks involving only cooperations, of which behavior analysis can be feasibly achieved based on the nonnegative matrix theory. With these facts, this article first bridges a relation between signed and unsigned networks and then takes advantage of the relation for the behavior analysis of signed networks under directed switching topologies. In particular, a solution is provided for the connection of signed and unsigned networks via solving a class of linear dynamic systems, which is obtained by separating antagonisms from cooperations. The solution makes it possible to employ the convergence results for unsigned networks to address the convergence issues for signed networks. If the joint spanning tree condition is met, then switching signed networks can achieve the quasi-interval bipartite consensus. Moreover, the established results can be applied to signed networks with both continuous-time and discrete-time dynamics.

This paper studies the cooperative set aggregation problem for second-order multiagent systems by utilizing the approximate projection algorithm. First, an aggregation law that uses the approximate projection is proposed, where the feasible set of the approximate projection points is established based on an Euclidean distance with respect to the targeted set and a deviated angle with respect to the exact projection point. Under the proposed law, the position vectors of all the agents are shown to reach an agreement in the intersection of their individual targeted sets and the velocity vector of each agent is shown to converge to zero. Then, a time-dependent performance bound is set for the norm of the weighted error of the relative positions among neighboring agents and the absolute velocity of the agent, and a performance-guaranteed aggregation controller is designed to guarantee the prescribed transient performance. During the process of aggregation, the norm of the weighted error is shown to not exceed the performance bound. The convergence conditions of the proposed algorithms with respect to the control strength and the terms induced by the approximate projection are obtained by using the techniques of convex analysis and Lyapunov stability. Simulations are provided to validate the theoretical results.

A distributed control algorithm is proposed in this paper to achieve the leader-follower cooperative attitude tracking of multiple rigid bodies without using absolute and relative angular velocity information. The nonlinear manifold SO(3) is applied to represent the attitude of each rigid body. By considering the case of limited information exchange, i.e., not all the rigid bodies have access to the leader's information, a novel continuous nonlinear distributed estimator satisfying the rotation matrix dynamics is first introduced for each rigid body to estimate the leader's information asymptotically. Then, two auxiliary systems driven by proper inputs are designed such that the issues due to the angular velocity unavailability are effectively solved. Next, a feasible distributed protocol is designed by implementing the outputs from the distributed estimator and the auxiliary systems. It is demonstrated in terms of Lyapunov theory that, the developed distributed control algorithm ensures the asymptotically stable cooperative attitude tracking of multiple rigid bodies under the weak tree connectivity assumption. Finally, illustrative examples are provided to validate and highlight the proposed results.

This paper develops a novel immersion and invariance (I&I)-based adaptive controller for nonlinear underactuated quadrotor systems subject to uncertain inertial parameters. The objective is to drive the quadrotor to accurately track a reference trajectory. The controller synthesis is based on a hierarchical framework. First, the solution to the trajectory tracking problem is transformed to stabilizing its corresponding error system asymptotically. Second, to avoid singularity in the command attitude extraction, a saturated backstepping control strategy with smooth hyperbolic tangent functions is developed to synthesize a command force in the position loop, where an I&I adaptive methodology is introduced to update the mass estimation. The I&I adaption guarantees the asymptotic convergence of the mass estimation to its actual value. Third, in the attitude loop, the projection algorithm is used to update the inertial tensor estimations, which enables the estimations to be maintained within a prescribed range. Then, a rotation matrix-based applied torque with an initial condition constraint is synthesized. It is proven that, with appropriate control parameters, the proposed controller guarantees the asymptotic stability of the error system and then the trajectory tracking of the quadrotor is achieved. Finally, numerical simulations are given to verify and highlight the performance of the proposed controller.

Miniaturization and intellectualization are the major trends of the development of UAV (unmanned aerial vehicle) technology. However, limited by volume, energy consumption and computation capacity, it is still a challenging problem to guarantee stable and accurate object tracking on a nano-scale UAV platform. In this paper, we present a 27 g nano-scale UAV equipped with a mono visual module, with which we firstly realize an artificial target tracking, more specifically, the concentric circle target. Then, hand tracking is realized based on KeF (kernel correlation filter) algorithm and we do not need to manually specify the initial position of hand in the image. The hardware integration, visual tracking and visual servoing control algorithms are explicitly presented. The experimental results validates the effectiveness of the proposed system.

Gyro-less attitude and angular rate estimations are of great importance in small, low-cost spacecraft, where high performance gyroscopes are not available due to multiple limitations. Recent development of accurate, high-bandwidth attitude sensors such as high data-rate star trackers, makes this approach implementable. The gyro-less estimator propagates the estimated states by nonlinear attitude dynamics model, and obtains measurements from nonlinear observer. The state estimation of such nonlinear systems requires approximation of the system equations, as done by the classical extended Kalman filter (EKF), or approximation of the probability densities, as done by Unscented Kalman filter (UKF) including linear regression Kalman filters (LRKFs). In particular, LRKFs utilize a set of deterministically chosen sample points, namely the sigma points, to approximate the state and measurement probability density function. However, the sigma points required in LRKFs grow linearly with the dimension of the state space, which becomes an obstacle for their onboard implementation for overload computational complexity. In this paper, we propose a partial-state sigma-point filter that retains the benefits of LRKFs' ability to deal with nonlinearity and is free from Jacobian matrices derivation. In the proposed filter, the state is decomposed into two parts: the directly observed part of the attitude, and the indirectly observed parts of the angular rate and acceleration. The state and covariance for attitude are approximated based on the geometric simplex transform, which utilize symmetric 4-sigma points to capture the first two moments of a vector in three-dimensional Euclidean space. The remaining states and process noises are treated based on marginalized conditional sigma-point transformation. Therefore, the proposed algorithm requires only four-sigma points to realize a ten-state estimation (attitude, angular rate and angular acceleration). In addition, similar precision to classical UKF is guaranteed and a significantly lower computational complexity is achieved. The performance of the new algorithm is demonstrated by simulations.

The study of distributed remote sensing attracts much attention in the development of space exploration. The key issue towards this field is to study the system property of spacecraft cluster composed of multiple nano/micro modules. In addition, the formation of multiple nano/micro spacecraft is maintained and rebuilt for different application tasks. Due to the fact that micro/nano spacecraft is often lack of fuel, to maintain the stability of distributed remote sensing formation of nano/micro spacecraft for a relatively long period of time under the natural formation is very important. In this paper, a direct parameter method is proposed under the tree communication topology constraints for collaborative observation mode of distributed remote sensing spacecraft cluster. The orbit parameters of the N followers are obtained directly by this method. This paper also establishes a natural evolution model of spacecraft cluster with the advantages of simplicity, intuition and clear physical meaning. According to the assignment of national key research and development plan “distributed reconfigurable remote sensing technology based on nano/micro spacecraft”, we verify the correctness of the proposed method using the pre-given orbit parameters. Also, the numerical simulations of the formation flying are performed through STK and the simulation results show that the proposed approach has the advantages of easy implementation, high precision and robustness.

Controlling directed networks with minimum cost has become an emerging branch in the areas of complex networks and control recently. In this paper, we focus on this minimum cost control problem subject to two types of boundary constraints, namely, trace boundary constraint and orthonormal boundary constraint on the input matrices. First, the minimum cost control problem is formulated as an optimization model for each type of boundary constraint. Next, two iterative algorithms, named as trace-constraint-based projected gradient method and orthonormal-constraint-based projected gradient method, are proposed to solve the optimal problem, respectively. Then, convergence properties of both algorithms are established. Finally, extensive simulation results show the effectiveness of our methods based on detailed comparisons between the two boundary conditions. We believe the results reveal some interesting physical insights for the optimal control of directed networks.

In this paper, we study the coordinated tracking problem of multiple heterogeneous Lagrange systems with a dynamic leader. Only nominal parameters of Lagrange dynamics are assumed to be available. Under the local interaction constraints, i.e., the followers only have access to their neighbors' information and the leader being a neighbor of only a subset of the followers, continuous coordinated tracking algorithms with adaptive coupling gains are proposed. Except for the benefit of the chattering-free control achieved, the proposed algorithm also has the attribute that it does not require the neighbors' generalized coordinate derivatives. Global asymptotic coordinated tracking is guaranteed, and the tracking errors between the followers and the leader are shown to converge to zero. Examples are given to validate the effectiveness of the proposed algorithms.

In this paper, time-domain (Lyapunov theorems) and frequency-domain (the Nyquist stability criterion) approaches are used to study leaderless and leader-following consensus algorithms with communication and input delays under a directed network topology. We consider both the first-order and second-order cases and present stability or boundedness conditions. Several interesting phenomena are analyzed and explained. Simulation results are presented to support the theoretical results.