## 1 Introduction  
The production of parts of non-circular cross-sections in machining is no longer a domain of milling. Such components can be efficiently produced by turning with the use of lathe s fitted with an additional drive. This technology can be based on a rotating tool as in polygon turning [1, 2] or a tool with feed motion operating in one [3, 4, 5] or multiple axes [6]. An example of such a process can be the manufacturing of middle-convex and varying ellipse pistons (MCVEP) widely used in diesel engines, which ensure better strain conditions and guidance quality than circular pistons [7]. Other solutions concern polygonal shafts fabricated without halting the turning process. Driven tools with feed motion referred to as Fast Tool Servos (FTS) are usually controlled by electronic cams that couple the tool linear position with the part angular position following a strict mathematical relation. The technological requirements set for such tools usually exceed the dynamic capabilities of conventional servo drives (e.g., with an electric engine and ball-screw transmission) due to the difficulties with synchronizing the tool's high-frequency reciprocating motion with the rotational speed of the spindle. Linear motion drives are a group that can potentially solve the dynamics issues to some extent. The solutions featuring high dynamics include the Voice Coil Motor (VCM) [8, 9] or piezoelectric actuators. The latter is in the form of stacks and is characterized by high force, rigidity and dynamics. However, they exhibit a limited travel distance which is mostly related to the size of the stack, what can be troublesome in some physical application due to the limited available workspace in the machine. Solid flexures [3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16] reinforcement frames acting as rhomboids that enable achieving a millimeter long travel range but at the cost of deteriorating other performance parameters, especially force and rigidity. An important feature of a piezoelectric drive is its hysteresis which affects the positioning accuracy while operating in an open-loop system. The hysteresis error can amount from several to over a dozen percent of the travel range of a drive unit with a piezoelectric stack. For open-loop systems, predictive methods based on inverse hysteresis models are used to minimize that phenomenon. The commonly used solutions include the Preisach model [17] or its modifications [18; 19; 20], models developed based on a physical analysis or phenomenological approach Jiles-Atherton [21; 22], Bouc-Wen [23; 24], Duhem [25; 26] or NARMAX [27; 28], direct [29] or geometric models [30]. A feedback loop in the form of an external position sensor is often used in FTS drives. Most common solutions include contactless laser or Eddy current transducers, or strain gauges used fitted directly within the piezoelectric stack. The FTS drives can serve numerous functions in manufacturing. Their most important benefits include the high operating frequency, which enables active damping of the vibrations generated during the processing [15] or generating additional precise tool motion [31]. They can also offer a primary cutting motion for surface microtexturing [4]. The high positioning accuracy along with the high resolution of the FTS piezoelectric drive is a good base for designing ultra-precise cutting tools [3; 5; 6; 7; 10; 11; 12; 13; 14; 15; 16]. The FTS drives described in the literature are dedicated devices developed for a specific task, which enables obtaining drive operating at frequencies of 40-600 Hz [3; 7; 11; 16], or even above 2 kHz [10; 12] and positioning accuracy less than 0.3 \(\upmu\)m [3; 10; 16]. The aforementioned studies usually cover mathematical models of the drives, including FEM-based models, and provide primary dynamic specifications for the drive units, i.e., frequency specifications, their step responses, and responses to forced excitation deriving from the actual processing conditions.