Daniel Zelazo

Edge Agreement: Graph-Theoretic Performance Bounds and Passivity Analysis

This work explores the properties of the edge variant of the graph Laplacian in the context of the edge agreement problem. We show that the edge Laplacian, and its corresponding agreement protocol, provides a useful perspective on the well-known node agreement, or the consensus algorithm. Specifically, the dynamics induced by the edge Laplacian facilitates a better understanding of the role of certain subgraphs, e.g., cycles and spanning trees, in the original agreement problem. Using the edge Laplacian, we proceed to examine graph-theoretic characterizations of the H2 and H∞ performance for the agreement protocol. These results are subsequently applied in the contexts of optimal sensor placement for consensus-based applications. Finally, the edge Laplacian is employed to provide new insights into the nonlinear extension of linear agreement to agents with passive dynamics.

Performance and design of cycles in consensus networks

Abstract This work explores the role of cycles in consensus seeking networks for analysis and synthesis purposes. Cycles are critical for many reasons including improving the convergence rate of the system, resilience to link failures, and the overall performance of the system. The focus of this work examines how cycles impact the H 2 performance of consensus networks. A first contribution shows that the addition of cycles always improves the performance of the system. We provide an analytic characterization of how the addition of edges improves the performance, and show that it is related to the inverse of the cycle lengths and the number of shared edges between independent cycles. These results are then used to consider the design of consensus networks. In this direction we present an l 1 -relaxation method that leads to a convex program for adding a fixed number of edges to a consensus networks. We also demonstrate how this relaxation can be used to embed additional performance criteria, such as maximization of the algebraic connectivity of the graph.

Duality and network theory in passivity-based cooperative control ☆

This paper presents a class of passivity-based cooperative control problems that have an explicit connection to convex network optimization problems. The new notion of maximal equilibrium independent passivity is introduced and it is shown that networks of systems possessing this property asymptotically approach the solutions of a dual pair of network optimization problems, namely an optimal potential and an optimal flow problem. This connection leads to an interpretation of the dynamic variables, such as system inputs and outputs, to variables in a network optimization framework, such as divergences and potentials, and reveals that several duality relations known in convex network optimization theory translate directly to passivity-based cooperative control problems. The presented results establish a strong and explicit connection between passivity-based cooperative control theory on the one side and network optimization theory on the other, and they provide a unifying framework for network analysis and optimal design. The results are illustrated on a nonlinear traffic dynamics model that is shown to be asymptotically clustering.

Rigidity maintenance control for multi-robot systems

Rigidity of formations in multi-robot systems is important for formation control, localization, and sensor fusion. This work proposes a rigidity maintenance gradient controller for a multi-agent robot team. To develop such a controller, we first provide an alternative characterization of the rigidity matrix and use that to introduce the novel concept of the rigidity eigenvalue. We provide a necessary and sufficient condition relating the positivity of the rigidity eigenvalue to the rigidity of the formation. The rigidity maintenance controller is based on the gradient of the rigidity eigenvalue with respect to each robot position. This gradient has a naturally distributed structure, and is thus amenable to a distributed implementation. Additional requirements such as obstacle and inter-agent collision avoidance, as well as typical constraints such as limited sensing/communication ranges and line-of-sight occlusions, are also explicitly considered. Finally, we present a simulation with a group of seven quadrotor UAVs to demonstrate and validate the theoretical results.

Bearing Rigidity and Almost Global Bearing-Only Formation Stabilization

A fundamental problem that the bearing rigidity theory studies is to determine when a framework can be uniquely determined up to a translation and a scaling factor by its inter-neighbor bearings. While many previous works focused on the bearing rigidity of two-dimensional frameworks, a first contribution of this paper is to extend these results to arbitrary dimensions. It is shown that a framework in an arbitrary dimension can be uniquely determined up to a translation and a scaling factor by the bearings if and only if the framework is infinitesimally bearing rigid. In this paper, the proposed bearing rigidity theory is further applied to the bearing-only formation stabilization problem where the target formation is defined by inter-neighbor bearings and the feedback control uses only bearing measurements. Nonlinear distributed bearing-only formation control laws are proposed for the cases with and without a global orientation. It is proved that the control laws can almost globally stabilize infinitesimally bearing rigid formations. Numerical simulations are provided to support the analysis.

Decentralized rigidity maintenance control with range measurements for multi-robot systems

This work proposes a fully decentralized strategy for maintaining the formation rigidity of a multi-robot system using only range measurements, while still allowing the graph topology to change freely over time. In this direction, a first contribution of this work is an extension of rigidity theory to weighted frameworks and the rigidity eigenvalue, which when positive ensures the infinitesimal rigidity of the framework. We then propose a distributed algorithm for estimating a common relative position reference frame amongst a team of robots with only range measurements in addition to one agent endowed with the capability of measuring the bearing to two other agents. This first estimation step is embedded into a subsequent distributed algorithm for estimating the rigidity eigenvalue associated with the weighted framework. The estimate of the rigidity eigenvalue is finally used to generate a local control action for each agent that both maintains the rigidity property and enforces additional constraints such as collision avoidance and sensing/communication range limits and occlusions. As an additional feature of our approach, the communication and sensing links among the robots are also left free to change over time while preserving rigidity of the whole framework. The proposed scheme is then experimentally validated with a robotic testbed consisting of six quadrotor unmanned aerial vehicles operating in a cluttered environment.

Agreement via the edge laplacian

This work explores the properties of the edge variant of the graph Laplacian in the context of the edge agreement problem. We show that the edge Laplacian, and its corresponding agreement protocol, provide a useful perspective on the well-known node agreement, or the consensus problem. Specifically, the dynamics induced by the edge Laplacian facilitates a better understanding of the role played by certain subgraphs, e.g., cycles and spanning trees, in the original agreement problem. We also point out a reduced order modeling of the edge agreement as parameterized by the spanning trees of the underlying graph.

On the Robustness of Uncertain Consensus Networks

This work considers the robustness of uncertain consensus networks. The stability properties of consensus networks with negative edge weights are also examined. We show that the network is unstable if either the negative weight edges form a cut in the graph or any single negative edge weight has a magnitude less than the inverse of the effective resistance between the two incident nodes. These results are then used to analyze the robustness of the consensus network with additive but bounded perturbations of the edge weights. It is shown that the small-gain condition is related again to cuts in the graph and effective resistance. For the single edge case, the small-gain condition is also shown to be exact. The results are then extended to consensus networks with nonlinear couplings.

Graph-Theoretic Analysis and Synthesis of Relative Sensing Networks

This work provides a general framework for the analysis and synthesis of a class of relative sensing networks (RSNs) in the context of its <i>H</i>₂ and <i>H</i>_∞ performance. We consider RSNs with homogeneous and heterogeneous agent dynamics. In both cases, explicit graph theoretic expressions and bounds for the <i>H</i>₂ and <i>H</i>_∞ performance are derived. The <i>H</i>₂ performance turns out to be a function of the number of edges in the graph, whereas the <i>H</i>_∞ performance is structure dependent and related to the spectral radius of the graph Laplacian. The analysis results are then used to develop synthesis methods for RSNs. An optimal topology is designed using the Kruskals Algorithm for <i>H</i>₂ performance, and a semi-definite program for the <i>H</i>_∞ performance of uncertain RSNs.

On the observability properties of homogeneous and heterogeneous networked dynamic systems

This work provides a framework for the observability analysis of linear networked dynamic systems (NDS). A distinction is made between NDS that have homogeneous agent dynamics and NDS that have heterogeneous dynamics. In each case, conditions for the observability of such a system are presented; we will also quantify the relative degree of observability of these systems. Moreover, an index of homogeneity and an index of heterogeneity are introduced as the means of quantitatively measuring how homogeneous a particular NDS is.