Optimal thruster configuration for omni-directional underwater vehicles. SamoS: a Pythagorean solution

We show that the design of omni-directional underwater vehicles could gain further insight thanks to the definition of three quality indexes, all related to the notion of isotropy. The first index comes from the concept of manipulability, the second index represents energetic issues and the third index gives exact information on isotropy in force. Basic mathematical relations are first recalled; then details about the derivation of each index are given. A case study is presented, including the description of six different omnidirectional underwater vehicles and a table summarizing index for all vehicles. A "family" of designs obtains good results for all indexes; among this "family" one particular arrangement shows to be the best overall: its arrangement related to the Pythagorean golden ratio leads us to name it SamoS.