Multiple fully actuated mobile robots travel in a plane with upper-limited speeds and are driven by accelerations limited in magnitude. The robots should arrive at a pre-specified distance from an unpredictable speedy target, then maintain this distance and achieve an even self-distribution over the respective moving circle, along with a given angular velocity of rotation about the target. The robots do not communicate with anybody and are anonymous to one another; pre-assignment of different roles to various robots is impossible. Every robot measures only the relative position of the target and companion robots (within a finite range of 'visibility' in the latter case) and has access to the angular velocity of its own pure rotation; access to its own linear velocity may also be needed in some cases. Necessary conditions for the solvability of the mission are disclosed and a distributed control strategy is proposed. Its global convergence and collision avoidance property are rigor- ously proved under slight and partly unavoidable enhancement of the just mentioned necessary conditions. The performance of the control law is illustrated by computer simulations.