Distributed stabilization control of rigid formations with prescribed orientation

Most rigid formation controllers reported in the literature aim to only stabilize a rigid formation shape, while the formation orientation is not controlled. This paper studies the problem of controlling rigid formations with prescribed orientations in both 2-D and 3-D spaces. The proposed controllers involve the commonly-used gradient descent control for shape stabilization, and an additional term to control the directions of certain relative position vectors associated with certain chosen agents. In this control framework, we show the minimal number of agents which should have knowledge of a global coordinate system (2 agents for a 2-D rigid formation and 3 agents for a 3-D rigid formation), while all other agents do not require any global coordinate knowledge or any coordinate frame alignment to implement the proposed control. The exponential convergence to the desired rigid shape and formation orientation is also proved. Typical simulation examples are shown to support the analysis and performance of the proposed formation controllers.Comment: This paper was submitted to Automatica for publication. Compared to the submitted version, this arXiv version contains complete proofs, examples and remarks (some of them are removed in the submitted version due to space limit.

Formation Control of Rigid Graphs with a Flex Node Addition

This paper examines stability properties of distance-based formation control when the underlying topology consists of a rigid graph and a flex node addition. It is shown that the desired equilibrium set is locally asymptotically stable but there exist undesired equilibria. Specifically, we further consider two cases where the rigid graph is a triangle in 2-D and a tetrahedral in 3-D, and prove that any undesired equilibrium point in these cases is unstable. Thus in these cases, the desired formations are almost globally asymptotically stable.Comment: The full version of this paper with general extensions has been submitted to a journal for publicatio

Controlling rigid formations of mobile agents under inconsistent measurements

Despite the great success of using gradient-based controllers to stabilize rigid formations of autonomous agents in the past years, surprising yet intriguing undesirable collective motions have been reported recently when inconsistent measurements are used in the agents' local controllers. To make the existing gradient control robust against such measurement inconsistency, we exploit local estimators following the well known internal model principle for robust output regulation control. The new estimator-based gradient control is still distributed in nature and can be constructed systematically even when the number of agents in a rigid formation grows. We prove rigorously that the proposed control is able to guarantee exponential convergence and then demonstrate through robotic experiments and computer simulations that the reported inconsistency-induced orbits of collective movements are effectively eliminated.Comment: 10 page

Identification of Hessian matrix in distributed gradient-based multi-agent coordination control systems

Multi-agent coordination control usually involves a potential function that encodes information of a global control task, while the control input for individual agents is often designed by a gradient-based control law. The property of Hessian matrix associated with a potential function plays an important role in the stability analysis of equilibrium points in gradient-based coordination control systems. Therefore, the identification of Hessian matrix in gradient-based multi-agent coordination systems becomes a key step in multi-agent equilibrium analysis. However, very often the identification of Hessian matrix via the entry-wise calculation is a very tedious task and can easily introduce calculation errors. In this paper we present some general and fast approaches for the identification of Hessian matrix based on matrix differentials and calculus rules, which can easily derive a compact form of Hessian matrix for multi-agent coordination systems. We also present several examples on Hessian identification for certain typical potential functions involving edge-tension distance functions and triangular-area functions, and illustrate their applications in the context of distributed coordination and formation control

Taming mismatches in inter-agent distances for the formation-motion control of second-order agents

This paper presents the analysis on the influence of distance mismatches on the standard gradient-based rigid formation control for second-order agents. It is shown that, similar to the first-order case as recently discussed in the literature, these mismatches introduce two undesired group behaviors: a distorted final shape and a steady-state motion of the group formation. We show that such undesired behaviors can be eliminated by combining the standard formation control law with distributed estimators. Finally, we show how the mismatches can be effectively employed as design parameters in order to control a combined translational and rotational motion of the formation.Comment: 14 pages, conditionally accepted in Automatic Control, IEEE Transactions o

Consensus problems in networks of agents with switching topology and time-delays

In this paper, we discuss consensus problems for networks of dynamic agents with fixed and switching topologies. We analyze three cases: 1) directed networks with fixed topology; 2) directed networks with switching topology; and 3) undirected networks with communication time-delays and fixed topology. We introduce two consensus protocols for networks with and without time-delays and provide a convergence analysis in all three cases. We establish a direct connection between the algebraic connectivity (or Fiedler eigenvalue) of the network and the performance (or negotiation speed) of a linear consensus protocol. This required the generalization of the notion of algebraic connectivity of undirected graphs to digraphs. It turns out that balanced digraphs play a key role in addressing average-consensus problems. We introduce disagreement functions for convergence analysis of consensus protocols. A disagreement function is a Lyapunov function for the disagreement network dynamics. We proposed a simple disagreement function that is a common Lyapunov function for the disagreement dynamics of a directed network with switching topology. A distinctive feature of this work is to address consensus problems for networks with directed information flow. We provide analytical tools that rely on algebraic graph theory, matrix theory, and control theory. Simulations are provided that demonstrate the effectiveness of our theoretical results