For many micro and nano manipulation tasks, the need for small-sized and ultra-high-precision positioning and manipulation devices is essential [1
]. Among them, piezo-actuated stick–slip micro-drives (PASSMDs) have received much attention given their main advantages of a very simple design consisting of only few parts (see Figure 1
a) and noticeable working properties. PASSMDs are guided and driven by piezo actuators supplied by saw-tooth driving voltages (see Figure 1
d). The motion of these devices consists of sequences of stick and slip-phases. In the stick phase (see Figure 1
b), the mobile part, called the runner, is slowly displaced along with slow expansion of the actuators due to the static friction force in the guiding contacts. This phase is followed by the slip-phase with a rapid contraction of the actuators in the opposite direction (see Figure 1
c). The runner cannot fully follow this displacement and thus slides relatively with respect to the actuators. As a result, after a small back-step, a step of the runner is generated. This motion principle has an advantage for movements of the runner with not only very high resolution but also theoretically with unlimited range, although the displacement of the actuators is limited.
The modeling and control of PASSMDs are important research topics. However, although numerous prototypes and systems based on the stick–slip principle have been introduced (see [2
] and chapter 2 of [4
]), the proof-of-concept of individual drives is the main focus. There have been only few works concerning the physical modeling of micro-drives intensively [4
]. All of these works have tried to model frictional behavior in the guiding contacts with or without considering dynamic effects of the piezo actuators.
There is an important phenomenon of PASSMDs experimentally observed. If the actuation amplitude of the piezo actuators is smaller than a critical value, no measurable step of the runner is obtained. This means the runner only oscillates around its initial position without step generation. In this paper, we call the actuation amplitude at this critical value the “critical amplitude, ”. Explanations for this phenomenon are classified into two groups:
The work of van der Wulp [5
] is classified into the first group. In this work, the author explained that the critical amplitude is caused by the presliding, a maximal deformation of the guiding contacts before a full slip occurs, and the dynamic effects of the piezo actuators. The guiding contacts are described by the classical tangential contact theory. The piezo actuators are modeled as a mass–spring–damper system. However, only simple analytic formulations for critical amplitude were generated. Dynamic simulations were not implemented. Therefore, further investigation should be performed.
In the second group [4
], the authors explained the critical amplitude by only the presliding. Dynamic effects of the piezo actuators do not influence this phenomenon. Therefore, the friction model for the guiding contacts was concentrated on.
Breguet used the LuGre model, invented by Canudas de Vit et al. [14
], for modeling a linear stick–slip micro-drive [7
]. This model is capable of predicting basic friction properties. However, the presliding cannot be covered, as the deformation of the contacts is considered as plastic. The presliding is simply stated as “contact deformation”. Peng and Chen [8
] exploited the elastoplastic model (EPM) [15
], an extension of the LuGre model, for their investigations. Because the EPM takes the elastic property into account, it allows one to predict the presliding, but it is only qualitative. Consequently, in [9
], Edeler et al. made an adaption for the EPM by adding six empirical parameters to establish the CEIM-model (Christoph Edeler - Ingo Meyer Model). This model shows a higher simulation quality.
It is known that all of the above-mentioned models are extended from Dahl’s single-state friction law [16
], which is based on the interaction of surface asperities of contacting bodies. These models are used in such a way that all important parameters are empirically chosen or fitted. The real physical properties of frictional contacts are neglected. Therefore, a new approach was first proposed in [11
] and further studied in [4
]. In this approach, the authors used the standard tangential contact theory and applied the method of dimensionality reduction (MDR) (see [17
]) to describe the friction behavior of the guiding contacts. The model of this approach, namely, the TC-MDR (tangential contact with MDR), permits a much higher modeling quality, as all important influencing parameters of the contacts are taken into account and thus none of the empirical or fitted parameters are used. However, the relation between the critical amplitude and the presliding in the case of the multi-contact model is still qualitatively explained.
While all above-mentioned models allow for an increasing simulation quality, we believe that the explanation for the critical amplitude phenomenon has not been fully satisfied yet, as its relation to the dynamic effects of the piezo actuators is not considered. Therefore, in this work, we extend the TC-MDR model by taking the dynamics of the piezo actuators into account to establish a new model. This model will allow us to explain clearly the causation of the critical amplitude phenomenon. Additionally, the remaining deviation between experiments and simulations can be analyzed and narrowed.
The optimization of the driving waveform is significant for the higher performance of PASSMDs. An optimized waveform may allow us to increase the step size and reduce back-step and micro-vibration, and thus it results in a faster operational velocity, a higher repeatability and less chaos for the drives. There have been both hardware- and model-based approaches. For the hardware-based approach, in [19
], Špiller and Hurák introduced a hybrid charge amplifier. The device allows us to provide a high-voltage asymmetric saw-tooth-like signal while maximizing the slew rate and compensating for the hysteretic behavior of the piezo actuator. As a result, it exhibits the minimum back-step and thus induces less micro-vibration. The model-based approach has often been used [4
]. In [20
], Bergander and Breguet used a method called “signal shaping” to optimize the driving waveform for a linear stick–slip micro-stage. The method permits reducing vibrations of the piezo actuators efficiently and thus heightens the operational velocity as well as the repeatability. Nevertheless, this method can only be applied for drives having actuators with a relatively low natural frequency. Recently, Hunstig et al. [21
] proposed a novel driving method for a macroscopic stick–slip drive with the actuators’ stroke of 27
m. It is known that the optimized signal allows the drive to operate with very high velocities in the slip–slip regime, while the reachable maximum velocity in stick–slip operation is principally limited. Neither the works of Bergander et al. nor Hunstig et al. consider the dynamic behavior of the guiding contacts influencing the drivers’ performance. This is particularly important for micro-drives. Consequently, a new driving waveform was first introduced in [23
] and was further studied in [4
]. On the basis of a simplified model of a micro-drive using the MDR, the motion of the drive in one period is analyzed, in which the dynamic behavior of the guiding contacts is involved. This leads to an optimized waveform allowing us to reduce micro-vibrations as well as the back-step. The experimental validation results show that the performance of the drive is substantially improved with possible driving frequencies up to 13 kHz.
None of published works known to the authors exploit the micro-vibration as a source to increase the step size and thus the operational velocity for the drives. In this work, we propose a new ideal for the driving signal. The optimized waveform is obtained by analyzing the movement of the runner and its micro-vibration on the basis of the dynamic model of the guiding contacts using MDR. This new ideal will allow us to significantly increase the step size of the runner and thus the speed.
Consequently, the contribution of this paper is twofold. On the one hand, an extended model is derived to describe the whole macroscopic movement of the drive and the microscopic dynamic behavior of the guiding contacts. A validation of the model indicating its accuracy is provided. On the other hand, a new driving waveform is proposed, which allows us to increase the operational velocity for the drives.
2. Investigated Drive
The drive investigated in this work was first introduced by Edeler [10
] and was adapted by the first author of this paper [12
]. Experimental results for the characterization of the device are presented in detail in [4
]. In order to validate our further improvements for the drive, we use these already published experimental results. For the sake of brevity, in the following, we shortly describe the construction and working principle of the device.
The test stand for the investigated drive, presented in [12
] by the first author of this paper, is shown in Figure 2
a. The linear drive consists of a runner, which is guided and driven by six ruby hemispheres. These ruby hemispheres are glued onto six piezo actuators. The actuators are arranged on both sides of the runner with an angle of 90° to each other (two on the right and four on the left; cf. Figure 2
b,c) and are mounted onto the actuator holders. This arrangement allows for horizontally translational forward and backward movements of the runner. The actuator holder on the right is fixed, whereas that on the left is guided by a very low friction linear ball bearing. This will make sure that the preload applied on the runner-hemisphere contacts is only impacted by the preload spring. The level of the preload is measured by a force sensor (M17 by ATI Industrial Automation, Pinnacle Park Apex, NC, USA). The position of the runner is recorded by the high-resolution laser interferometer (SP-120 by SIOS Meßtechnik GmbH, Ilmenau, Germany). Measurements of the forces generated by the drive are implemented with a miniature load cell (M31 by Sensing and Control Automation and Control Solutions Honeywell, Golden Valley, MN, USA).
The design of the piezo actuators is schematically shown in Figure 2
d. The ruby hemisphere is glued on two segments of the structured actuator, which is fabricated out of piezoceramic plates (PIC-151 by PI Ceramic GmbH, Lederhose, Germany). Two electrodes are coated on the top surface of the piezo segments. When two saw-tooth-like voltages with opposite signs are supplied on these electrodes, the piezo segments will be deformed, leading to the saw-tooth-like rotation of the ruby hemisphere. The synchronous saw-tooth-like rotation of all six ruby hemispheres results in the translational motion of the runner according to the stick–slip principle. The applied voltages on the piezo have a maximal amplitude of 300 V
and a typical slew-rate of 150 V/
s. The maximal displacement of the contact center of each hemisphere depends on its radius and the dimension of the piezo segments, as well as the amplitude of the driving voltages. This displacement is denoted as the actuation amplitude. All important influencing parameters, including the radius of the frictional contacts, preloads, and the control amplitude of the driving signal, can be independently controlled and investigated.
6. Conclusions and Outlook
In this paper, the modeling and the waveform optimization of stick–slip micro-drives have theoretically been studied. We have introduced an extended model for the drive, which can be used to investigate both the macroscopic behavior, including the dynamics of the actuators and the runner, and the microscopic properties of the frictional contacts. Two important characteristics of the drive, consisting of the critical amplitude phenomenon and the force generation, have systematically been analyzed. The achieved results confirm that the critical amplitude depends on the presliding of the guiding contacts, the dynamic properties of the actuators and the interaction among them. A more general determination for the critical amplitude is derived. The causation for the discrepancy in levels of the generated forces between the experimental and simulation results of the previously published work has been presented. The use of a more-realistic friction coefficient and the combination of the dynamics of the actuators and the runner in our introduced model allow for much better results. The simulation of the force generation fits excellently with the experiment.
We have introduced a novel driving waveform for the investigated drive that is based on exploiting the micro-vibration and considering the contact status. The optimized waveform is received under an assumption that the actuators are ideally rigid. This waveform allows for a maximal higher operational velocity for the drive in comparison with the standard saw-tooth waveform. Influences of parameters such as the preload, the radius of hemispheres and Young’s modulus on the runner’s velocity have been investigated. The preload and the radius of the hemispheres have been found to have significant influences, while the influence of Young’s modulus can be negligible.
It is noted that the results obtained in Section 5
are only from the simulations. Therefore, for future works, we will study influences of the actuator dynamics on the efficiency of the new waveform. Moreover, experimental validation for this waveform is of importance and should therefore be given.