Zhimin Han

Distributed Formation Control of Multi-Agent Systems Using Complex Laplacian

The paper concentrates on the fundamental coordination problem that requires a network of agents to achieve a specific but arbitrary formation shape. A new technique based on complex Laplacian is introduced to address the problems of which formation shapes specified by inter-agent relative positions can be formed and how they can be achieved with distributed control ensuring global stability. Concerning the first question, we show that all similar formations subject to only shape constraints are those that lie in the null space of a complex Laplacian satisfying certain rank condition and that a formation shape can be realized almost surely if and only if the graph modeling the inter-agent specification of the formation shape is 2-rooted. Concerning the second question, a distributed and linear control law is developed based on the complex Laplacian specifying the target formation shape, and provable existence conditions of stabilizing gains to assign the eigenvalues of the closed-loop system at desired locations are given. Moreover, we show how the formation shape control law is extended to achieve a rigid formation if a subset of knowledgable agents knowing the desired formation size scales the formation while the rest agents do not need to re-design and change their control laws.

Complex Laplacian and pattern formation in multi-agent systems

The paper studies the problem of pattern formation on spatial multi-agent systems. It is shown that complex-valued graph Laplacians in pattern formation are as important as the real one for consensus. First, formation patterns with four degrees of freedom (translation, rotation, and scaling) can be characterized by the null space of complex Laplacians associated with the sensing graph of networked agents. Second, formation patterns can be achieved via simple linear interaction rules related to complex Laplacians and the system can be stabilized by pre-multipling a stabilizing matrix. Several graphical and algebraic conditions are obtained for formation control of spatial multi-agent systems with their interaction topology modeled by an undirected graph.

A Graph Laplacian Approach to Coordinate-Free Formation Stabilization for Directed Networks

This paper concentrates on coordinate-free formation control for directed networks, for which the dynamic motion of each agent is assumed to be governed only by a local control. We develop a graph Laplacian approach to solve the global and exponential formation stabilization problem using merely relative position measurements between neighbors. First, to capture the sensing and control architectures that are needed to maintain the shape of a formation, a necessary and sufficient topological condition is proposed. Second, a Laplacian-based control law is developed for the stabilization problem of a group of mobile agents to a desired formation shape under both fixed and switching topologies due to temporal node failures. Simulation results are provided to demonstrate that our Laplacian-based formation control strategy is inherently fault-tolerant and robust to node failures.

Necessary and Sufficient Graphical Conditions for Affine Formation Control

This paper introduces a new multi-agent control problem, called an affine formation control problem, with the objective of asymptotically reaching a configuration that preserves collinearity and ratios of distances with respect to a target configuration. Suppose each agent updates its own state using a weighted sum of its neighbors relative states with possibly negative weights. Then the affine control problem can be solved for either undirected or directed interaction graphs. It is shown in this paper that an affine formation is stabilizable over an undirected graph if and only if the undirected graph is universally rigid, while an affine formation is stabilizable over a directed graph in the d-dimensional space if and only if the directed graph is (d + 1)-rooted. Rigorous analysis is provided, mainly relying on Laplacian associated with the interaction graph, which contain both positive and negative weights.

Realizability of similar formation and local control of directed multi-agent networks in discrete-time

This paper introduces a new concept called similar formation and studies the realizability condition, concerning with the existence of a protocol that has the ability to drive a directed multi-agent network to a desired formation shape. With the help of complex Laplacian, it is shown that in order to uniquely realize a similar formation in the plane, a necessary and sufficient graphical condition is that the directed multi-agent network is 2-rooted, a new type of connectivity. Moreover, a distributed control law is also derived using local measurements for the purpose of stabilizing a directed multi-agent network to a desired formation shape in discrete time, which ensures globally asymptotic stability of the closed-loop system. Simulation results are provided as well to illustrate our results.

Formation Control With Size Scaling Via a Complex Laplacian-Based Approach

We consider the control of formations of a leader-follower network, where the objective is to steer a team of multiple mobile agents into a formation of variable size. We assume that the shape description of the formation is known to all the agents, which is captured by a complex-valued Laplacian associated with the sensing graph, but the size scaling of the formation is not known or only known to two agents, called the leaders in the network. A distributed linear control strategy is developed in this paper such that the agents converge to the desired formation shape, for which the size of the formation is determined by the two leaders. Moreover, in order to make all agents in a formation move with a common velocity, the distributed control law also incorporates a velocity consensus component, which is implemented with the help of a communication network that may, in general, be of different topology from the sensing graph. Both the setup of single-integrator kinematics and the one of double-integrator dynamics are addressed in the same framework except that the acceleration control in the double-integrator setup has an extra damping term.

Formation control of directed multi-agent networks based on complex Laplacian

Real graph Laplacians are of great importance in consensus of multi-agent systems. This paper introduces complex graph Laplacians as a new tool to study the formation control problem in the plane. It is shown that complex graph Laplacians are of equally great importance for planar formation control like real Laplacians for consensus. First, complex graph Laplacians are used to characterize planar formations under given topology of networked agents. Second, complex graph Laplacians are used to derive local and distributed control strategies for asymptotically achieving formations. This paper explores the relations between graph topology, complex Laplacians, and planar formations, and obtains several graphical and algebraic conditions for realizability of spatial formations. Both simulation and experiment results are provided to illustrate our results.

Double-graph formation control for co-leader vehicle networks

The paper studies the directed formation control problem for co-leader vehicle networks. Double graphs are considered: one for sensing graph and the other for communication graph. Complex-valued Laplacians for graphs whose edges are attributed with complex weights are introduced for the purpose of formation control while real-valued Laplacians are involved for synchronizing the velocities. For both single-integrator kinematics and double-integrator dynamics, necessary and sufficient conditions are obtained for complex-valued Laplacian such that the resulting moving formation will not be deformed. Moreover, since the control law with certain choice of complex weights could not always ensure asymptotic stability, we show the existence of a stabilizing matrix that updates the complex weights, but preserves the equilibrium formation and makes the system asymptotically stable. Simulations are presented to illustrate our results.

A fully distributed approach to formation maneuvering control of multi-agent systems

This paper studies the formation maneuvering control problem for a network of agents with the objective of achieving a desired group formation shape and a constant over-all group maneuvering velocity. A fully distributed approach is developed to solve the problem. That is, a control law is proposed for each agent in the network, with its parameters capable of being designed in a distributed manner, and is implementable locally via relative sensing and communication with neighbors. Necessary and sufficient conditions regarding a critical control parameter are obtained to guarantee the globally asymptotic convergence of the overall system for both the single-integrator kinematics case and the double-integrator dynamics case.

A linear approach to formation control under directed and switching topologies

The paper studies the formation control problem for distributed robot systems. It is assumed each robot only has access to local sensing information (i.e. the relative positions and IDs of its neighbors). Taking into consideration physical sensing constraints (e.g. limited sensing range) and the motion of the robots over time, it may be noted that the sensing graph for the system is directed and time-varying. This presents a challenging situation for formation control. As an initiative attempt to study this challenging situation, we suppose the sensing graph switches among a family of graphs with certain connectivity properties, under which a switching linear control law is then proposed. We show that for arbitrary dwell times or average dwell times, the proposed control law with properly designed control parameters can ensure global convergence to a desired formation shape. The proposed formation control law can be implemented in a distributed manner while the design of certain control parameters requires some global information.