The topicality of the theme is related to the development of the theory and constructions of high-speed mechatronic positioning systems, which allow to perform automated biological tests and analyses, which increase the productivity of technological process several times (about 10 times) and replace exhausting manual labour in 3D motion. In the dissertation is developed principle new high-speed 2D (two dimensional) and 3D (three dimensional) robotic executive mechanisms of biochip systems (actuators) and improve the theory for the optimal synthesis of machines and mechanisms in the field of the positioning theory. The novelty of the research comprises the studies in the field of the synthesis of robotic executive mechanisms (in positioning). The work deals with the optimisation of positioning systems, where the optimisations of limited force, limited phase coordinates and mixed-type control systems have been analysed. The practical application of the research comprises the creation of real robotic executive mechanisms and their experimental testing. The number of positioning robots has been sold in USA and other countries as well.