  The continuum deformation leader-follower cooperative control strategy models
vehicles in a multi-agent system as particles of a deformable body. A desired
continuum deformation is defined based on leaders' trajectories and acquired by
followers in real-time through local communication. The existing continuum
deformation theory requires followers to be placed inside the convex simplex
defined by leaders. This constraint is relaxed in this paper. We prove that
under suitable assumptions any (n+1)(n=1,2,3) vehicles forming an n-D simplex
can be selected as leaders while followers, arbitrarily positioned inside or
outside the leading simplex, can acquire a desired continuum deformation in a
decentralized fashion. The paper's second contribution is to assign a
one-to-one mapping between leaders' smooth trajectories and homogeneous
deformation features obtained by continuum deformation eigen-decomposition.
Therefore, a safe and smooth continuum deformation coordination can be planned
either by shaping homogeneous transformation features or by choosing
appropriate leader trajectories. This is beneficial to efficiently plan and
guarantee inter-agent collision avoidance in a large-scale group. A simulation
case study is reported in which a virtual convex simplex contains a quadcopter
vehicle team at any time t; A* search is applied to optimize quadcopter team
continuum deformation in an obstacle-laden environment.
