This paper presents a natural extension of the results obtained by Feintuch and Francis in (2012a,b) concerning the so-called robot rendezvous problem. In particular, we revisit a known necessary and sufficient condition for convergence of the solution in terms of Cesaro convergence of the translates S(k)x(0), k >= 0, of the sequence x(0) of initial positions under the right-shift operator S, thus shedding new light on questions left open in Feintuch and Francis (2012a,b). We then present a new proof showing that a certain stronger ergodic condition on x(0) ensures that the corresponding solution converges to its limit at the optimal rate 0(t(-1/2)) as t -> infinity. After considering a natural two-sided variant of the robot rendezvous problem already studied in Feintuch and Francis (2012a) and in particular proving a new quantified result in this case, we conclude by relating the robot rendezvous problem to a more realistic model of vehicle platoons. (C) 2017 Elsevier Ltd. All rights reserved.