Shaoshuai Mou

Undirected rigid formations are problematic

In a recent paper, a systematic method was proposed for devising gradient control laws for asymptotically stabilizing a large class of rigid, undirected formations in two-dimensional space assuming all agents are described by kinematic point models. The aim of this paper is to explain what happens to such formations if neighboring agents have slightly different understandings of what the desired distance between them is supposed to be. What one would expect would be a gradual distortion of the formation from its target shape as discrepancies in desired distances increase. While this is observed for the gradient laws in question, something else quite unexpected happens at the same time. It is shown for any rigidity-based, undirected formation of this type which is comprised of three or more agents, that if some neighboring agents have slightly different understandings of what the desired distances between them are supposed to be, then almost for certain, the trajectory of the resulting distorted but rigid formation will converge exponentially fast to a closed circular orbit in two-dimensional space which is traversed periodically at a constant angular speed.

Robustness issues with undirected formations

It is shown for any rigidity-based, undirected triangular formation of the type studied in [1], that if neighboring agents in the formation have slightly different understandings of what the desired distance between them is suppose to be, then almost for certain, the trajectory of the resulting distorted but rigid formation will converge exponentially fast to a closed orbit in ℝ2 which is traversed periodically at a single sinusoidal frequency.

Formation movements in minimally rigid formation control with mismatched mutual distances

When a gradient descent control law is employed for stabilizing undirected minimally rigid formations, mismatched desired distances between neighboring agent pairs will cause additional motions of the whole formation. By reviewing and extending the results in [1], [2], we show that in general rotational motions and helical motions will occur for 2-D formations and 3-D formations, respectively. We then consider the problem of how to compute the formulas for the motions caused by constant mismatches. A novel idea based on the angular-momentum concept in rigid body dynamics is proposed for deriving the formation formulas, e.g., angular velocity, rotational radius, etc. in terms of the distance mismatch terms. This has implications on steering and controlling rigid formation motions.

Exponential stability for formation control systems with generalized controllers: A unified approach

This paper discusses generalized controllers for distance-based rigid formation shape stabilization and aims to provide a unified approach for the convergence analysis. We consider two types of formation control systems according to different characterizations of target formations: minimally rigid target formation and non-minimally rigid target formation. For the former case, we firstly prove the local exponential stability for rigid formation systems when using a general form of shape controllers with certain properties. From this viewpoint, different formation controllers proposed in previous literature can be included in a unified framework. We then extend the result to the case that the target formation is non-minimally rigid, and show that exponential stability of the formation system is still guaranteed with generalized controllers.

Finite Time Distance-based Rigid Formation Stabilization and Flocking

Abstract Most of the existing results on distributed distance-based rigid formation control establish asymptotic and often exponentially asymptotic convergence. To further improve the convergence rate, we explain in this paper how to modify existing controllers to obtain finite time stability. For point agents modeled by single integrators, the controllers proposed in this paper drive the whole formation to converge to a desired shape with finite settling time. For agents modeled by double integrators, the proposed controllers allow all agents to both achieve the same velocity and reach a desired shape in finite time. All controllers are totally distributed. Simulations are also provided to validate the proposed control.

Deterministic gossiping with a periodic protocol

A sequence of allowable gossips between pairs of agents in a group is complete if the gossip graph which the sequence generates is a connected spanning subgraph of the graph of all allowable gossip pairs; such a sequence is minimally complete if there is no shorter sequence which is complete. An infinite sequence of gossips is repetitively complete with period T if each successive subsequence of length T within the gossip sequence is complete. Any such sequence converges exponentially fast. A repetitively complete gossip sequence is periodic with period T if each gossip in the sequence is repeated once every T time steps. The rate of convergence of a periodic gossiping process is determined by the Tth root of the second largest eigenvalue in magnitude of the stochastic matrix of the complete gossip subsequence. In the case when the graph of allowable gossips is a tree and the complete gossip subsequence is minimally complete, this eigenvalue is independent of the order in which the gossips occur within the complete gossip subsequences.

Non-robustness of gradient control for 3-D undirected formations with distance mismatch

Gradient control laws can be used for effectively achieving undirected formation shape, by assuming that interagent distances between a certain set of joint agent pairs can be accurately specified and measured. This paper examines the formation behavior in a 3-D space context in the case that the neighboring agent pairs have slightly differing views or estimates about the desired interagent distances they are tasked to maintain. It is shown, by using a tetrahedron formation example, that the final formation shape will be slightly distorted as compared to the desired one. Further, in general each agents motion will be a combination of rotation and translation. Specifically, a helical movement can be observed in the presence of distance mismatch.

Reprint of “A distributed algorithm for efficiently solving linear equations and its applications (Special Issue JCW)”

Abstract A distributed algorithm is proposed for solving a linear algebraic equation A x = b over a multi-agent network, where A ∈ R n × n and the equation has a unique solution x ∗ ∈ R n . Each agent knows only a subset of the rows of [ A b ] , controls a state vector x i ( t ) of size smaller than n and is able to receive information from its nearby neighbors. Neighbor relations are characterized by time-dependent directed graphs. It is shown that for a large class of time-varying networks, the proposed algorithm enables each agent to recursively update its own state by only using its neighbors’ states such that all x i ( t ) converge exponentially fast to a specific part of x ∗ of interest to agent i . Applications of the proposed algorithm include solving the least square solution problem and the network localization problem.

Towards optimal convex combination rules for gossiping

By the distributed averaging problem is meant the problem of computing the average value yavg of a set of numbers possessed by the agents in a distributed network using only communication between neighboring agents. Gossiping is a well-known approach to the problem which seeks to iteratively arrive at a solution by allowing each agent to interchange information with at most one neighbor at each iterative step. In the most widely studied situation, gossiping agents i and j update their current estimates xi(t) and xj(t) of yavg by setting their new estimates xi(t+1) and xj(t+1) equal to the average of xi(t) and xj(t). A more general approach is for gossiping agents i and j to use the convex combination update rules xi(t+1) = wxi(t) + (1 - w)xj(t) and xj(t + 1) = wxj(t) + (1 - w)xi(t) respectively where w is a real number between 0 and 1. While for probabilistic gossiping protocols, a largest convergence rate is attained when w = 0.5, for deterministic gossiping protocols this is not the case. The aim of this paper is to demonstrate by computer experiments and analytically studied examples that for deterministic gossiping protocols which are periodic, the value of w which maximizes convergence rate is not necessarily w = 0.5 and moreover, convergence at the optimal value of w can be significantly faster than convergence at the value w = 0.5. Thus this papers contribution is to provide clear justification for a deeper study of the optimum convergence rate question for gossiping algorithms using convex combination rules.

Convergence analysis for rigid formation control with unrealizable shapes: The 3 agent case

We study the outcome of using a gradient descent control law for a minimally rigid formation consisting of N agents, in which each agent is modeled by a single integrator and the desired interagent distances are specified though they are not realizable. We first formulate the problem for formations of N ≥ 3 agents and derive a condition in terms of the rigidity matrix which the final formation must satisfy. Special attention will be given to the triangular formation for which the desired distances fail to satisfy the triangle inequality. In this case, we show the formation converges to a straight line. Detailed analysis is provided to describe the stability properties in the unrealizable triangle shape control problem.