In order to describe and compensate for complex hysteresis nonlinearities of piezoelectrically actuated fast tool servo (FTS), a novel
Fast-tool-servo-(FTS-) based single-point diamond turning (SPDT) is considered a very promising technology for the generation of freeform surfaces and complicated micro/nano structures [
Motivated by this, extensive mathematical models for hysteresis have been developed, and they may be classified into two aspects: physics-based [
Fractional order calculus (FOC) theory, which is a generalization of the conventional integration and differentiation to noninteger orders, has found rapidly increasing applications in various fields [
The present paper aims at presenting a linear scheme for describing hysteresis nonlinearities of piezoelectrically actuated FTS. By taking advantage of the nonlocal memory properties of FOC operations, the nonlocal memory-dominant nature of hysteresis nonlinearity is described by a fractional order differential equation (FODE), and a novel
Fractional order calculus is a generalization of the conventional integration and differentiation to noninteger orders with the fundamental operator
Generally, there exist several well-known definitions of FOC operations including the Grunwald-Letnikov (G-L) definition, the Riemann-Liouville (R-L) definition, and the Caputo definition [
As given in (
Both, fractional differentiation and integration are linear operations, which satisfy
For zero initial conditions, the Laplace transform of G-L definition can be written as [
From an electrical circuit point of view, the PEA can be considered as a capacitive component, and the electric-driven circuit can be illustrated in Figure
Equivalent driving circuit of piezoelectric actuator.
Assume that the working frequency of the FTS is much lower than its natural frequency and the dynamic behaviors of the mechanism could be ignored, the relationship between the driving force of the PEA and the displacement of the mechanism can be expressed as
Substituting
Considering hysteresis effects, we modified (
By introducing the fractional order differentiation, the governing equation of the FTS can be expressed as
Taking the Laplace transform of two sides in (
The inverse compensation for hysteresis effects of the piezoelectrically actuated FTS relies on the capacity of hysteresis predictions of the LFDH model. Figure
Schematic of the LFDH model-based inverse compensation strategy.
As for compensation operations, it is essential to obtain the compensated control signal
Generally, to realize the LFDH model-based inverse compensation, there exist three main steps, namely, modeling and identification, compensated command signal calculation, and application. These steps could be formalized as follows.
In this stage, a set of study signals are specified to obtain the characteristics of the FTS system, and then the parameters of the LFDH model are estimated based on certain evolutionary optimization schemes (EOS).
In this stage, a set of desired displacement values
In this stage, the compensated control signal is applied to drive the FTS to achieve the desired displacements.
The authors of this paper carefully designed a short stroke FTS for ultraprecision diamond turning in [
Photographic of the FTS mechanism (
Figure
Schematic of the experimental system (
For the identification process, a hybrid signal with variable amplitudes is employed as the command signal, and the corresponding output of the FTS mechanism is measured via the high precision capacitive sensor. The frequency of the command signal is especially chosen as 1 Hz. The low frequency chosen here can avoid possible separations between the PEA and the tool holder during rapid expansions and retractions of the PEA [
Similar EOSs as shown in [
Accordingly, the identified LFDH model can be obtained as
The displacements generated by the FTS and the identified model are illustrated in Figure
Modeling results. (In (a) and (b), the blue line and the red line denote the displacement generated by the FTS and the LFDH model, resp.).
The displacements generated by the FTS and the model
Hysteresis loop generated by the FTS and the model
The modeling error
According to the identified LFDH model as presented in (
In order to assess the LFDH model-based inverse compensation approach in open-loop operations, a series of experiments are carried out. Three typical signals, which are respectively illustrated in Figures
Responses of the FTS mechanism with inverse compensation. (In (a), the blue line and the red line denote the desired trajectory and the tracking trajectory, resp.).
The desired trajectory and the tracking trajectory
The compensated control signal
The tracking error
Responses of the FTS mechanism with inverse compensation. (In (a), the blue line and the red line denote the desired trajectory and the tracking trajectory, resp.).
The desired trajectory and the tracking trajectory
The compensated control signal
The tracking error
Responses of the FTS mechanism with inverse compensation. (In (a), the blue line and the red line denote the desired trajectory and the tracking trajectory, resp.).
The desired trajectory and the tracking trajectory
The compensated control signal
The tracking error
As we have reported before, the relative error of hysteresis nonlinearity of this FTS mechanism is about 19.5%. But after the proposed compensation strategy is implemented, the relative error caused by hysteresis nonlinearity is strongly reduced to less than ±2%. All of the results indicate that the LFDH model-based inverse compensation approach can significantly suppress the inherent hysteresis of the FTS in open-loop operations.
The main contribution of this paper is to develop a linear mathematical model to describe the complex hysteresis nonlinearity of piezoelectrically actuated FTS. Based on the linear frame, it enables us to implement the well-developed analysis and control theories of linear system into FTS system. The proposed model is constructed based on fractional calculus theory. It can give an analytical description of hysteresis. To verify the effectiveness of the LFDH model and the inverse compensation approach, a series of experiments are conducted. The maximum modeling error is about ±1.75% of the full span range, and the relative positioning error can be strongly reduced to less than ±2% by implementing the inverse compensation approach. The results verify that the proposed LFDH model is efficient for describing the hysteresis behaviors and that the LFDH model-based inverse compensation approach could significantly suppress inherent hysteresis effects in open-loop operations.
The authors are grateful to the financial support from the NSF of China (51175221; 51075041; 50995077), the Ministry of Science and Technology (MoSt) of China (2008AA04Z125), the Ministry of Education (MoE) of China (20070183104), the Department of Science & Technology (DoST) of Jilin Province, China (20090337; 20100359), and the Youth Technology Foundation (YTF) of Jilin University.