Asymptotic Tracking and Robustness of MAS Transitions Under a New Communication Topology

We have recently applied the principles of continuum mechanics to develop a new leader–follower model for the collective motion of a multiagent system (MAS). Agents are modeled as particles of a continuum body that can deform in <inline-formula> <tex-math notation="LaTeX">$\mathbb {R}^{n}$ </tex-math></inline-formula> (<inline-formula> <tex-math notation="LaTeX">$n=1,2,3$ </tex-math></inline-formula>) under a specific class of mappings, called the homogeneous transformation. This paper shows how a desired homogeneous deformation is uniquely specified based on the trajectories chosen by <inline-formula> <tex-math notation="LaTeX">$p\,+1$ </tex-math></inline-formula> (<inline-formula> <tex-math notation="LaTeX">$p\leq n$ </tex-math></inline-formula>) leaders, and it is acquired by the remaining agents, called followers, through local communication. Under this setup, every follower interacts with <inline-formula> <tex-math notation="LaTeX">$p+1$ </tex-math></inline-formula> local agents with fixed communication weights that are uniquely determined based on the initial positions of the agents. Although asymptotic convergence of the agents’ transient positions to the desired final positions (prescribed by a homogeneous transformation) can be assured by applying the proposed paradigm, follower agents deviate from the desired positions during evolution. The main objective of this paper is to assure that the transient error, the difference between the actual and desired positions of each follower, converges to zero during evolution. For this purpose, each leader chooses a time-dependent polynomial vector of order (<inline-formula> <tex-math notation="LaTeX">$h-1$ </tex-math></inline-formula>) (<inline-formula> <tex-math notation="LaTeX">$h\in { {\mathbb N}}$ </tex-math></inline-formula>) for its trajectory connecting two consecutive way points, and each follower applies continuous time or discrete time linear time invariant dynamics to update its state based on the states of <inline-formula> <tex-math notation="LaTeX">$p+1$ </tex-math></inline-formula> local agents. The second objective of this paper is to develop a paradigm for the homogeneous deformation of an MAS that is robust to communication failure. For this purpose, we will show how followers can acquire desired positions prescribed by a homogeneous mapping to preserve volumetric ratios under either fixed or switching communication topologies, where there is no restriction on the number of the agents, if every follower communicates with <inline-formula> <tex-math notation="LaTeX">$m_{i}\geq p+1$ </tex-math></inline-formula> local agents. In addition, agents’ collective motion can be stably continued even if some followers give up communication with other agents at some time during evolution.</p><p><italic>Note to Practitioners</italic>—A multiagent system evolution as continuum deformation is a novel idea for avoiding interagent collision and collisions of agents with obstacles, while a capability to manage large deformations (expiation or contraction) is provided. This idea becomes more interesting if deviations of agents from desired positions defined by continuum deformation vanish during evolution when agents only access neighboring agent state information.