Three-robot minimax travel-distance optimal formation

Multi-robot formation problem has received increasing attention due to its wide applications such as surveillances and various services. To illustrate a novel framework on optimal multi-robot formation given in our previous work, which aims to answer the long-term ignored fundamental problem of describing the formation and clarifying optimal formation rigorously in a mathematical manner, as a preliminary case study, various cases of the simplest optimal line formation of three robots, i.e. minimax travel-distance line formation problems, where each robot admits to move with the same constant speed along any chosen direction and the three-robot team aims to row on a straight line with the minimum maximal travel distance, are investigated in this note. Such problems look like very easy to resolve, however, to our surprise, mathematical results for these cases established with geometric analysis and inequalities have shown the non-trivialness of the most simple optimal line formation problem. Extensive simulations have also been conducted and briefly reported in this contribution, and these experimental results are found to coincide with the established theoretical results.