Full length ArticleOptimal design, modeling and control of a long stroke 3-PRR compliant parallel manipulator with variable thickness flexure pivots
Introduction
Compliant mechanisms that apply flexure hinges to serve as rotational joints have been widely adopted in precision engineering [1], [2], [3], [4], since it provides an effective way to overcome backlash, friction, and wear in the conventional rigid mechanical systems [5], [6], [7]. Flexure hinges transmit motion via elastic deformation of their structure, according to the deformation principle, the frequently used flexure hinges can be divided into two types: the lumped-compliance flexure hinges e.g., notched flexure hinges, and the distributed-compliance flexure hinges e.g., flexure pivots.
The motion ranges of notched flexure hinges based compliant mechanisms are limited within micro scale because of stress concentration [8], which cannot meet the increasing demand for workspace in biomedical science and precision manipulation [9], such as drug discovery, optical alignment and microscopy. In these areas, precision manipulators of long-range translation motion over several centimeters with micron accuracy and long-range rotational motion with micro-degree accuracy are urgently needed [3], [4], [9]. While the flexure pivots are good candidates to build long stroke compliant mechanisms, which can undertake deformation all along the length direction of the leaves [10]. A side-effect of this deformation principle is that the parasitic rotational center shift of a flexure pivot is much larger than that of a notched flexure hinge [11], [12], which will deteriorate the motion accuracy of the compliant mechanism, especially when the deformation angle of a flexure pivot is relatively large.
Many new configurations of distributed-compliance flexure hinges are proposed to decrease parasitic rotation errors, such as the butterfly flexure pivot proposed by Henein [13], the X2 pivot proposed by Martin [14], the Q-LITF flexure pivot designed by Yu [15]. Those flexure pivots are designed by constant thickness spring leaves in series configurations, the parasitic motion of the whole structure can be decreased by decreasing the deformation on each spring leaf. However, the series configuration also decreases the support stiffness and stability of the pivot. Inspired by the structure of the notched flexure hinge, a new kind of flexure pivot, VTFP, is proposed in our previous research [16]. The VTFP is constructed by spring leaves with variable thickness which has better motion accuracy and better ability to resist axial force compared with the conventional flexure pivot since the deformation of the spring leaves are more concentrated on the designed rotational center.
The 3-PRR planar parallel manipulators have been successfully applied to industrial automation due to their simple structure and easy to control [17], [18]. The motion performance of the manipulator can be further improved if the rigid rotational joins are replaced by flexure hinges to eliminate clearance and backlash in the rotational joints [19], [20]. In this paper, a long stoke 3-PRR CPM is designed by employing the VTFPs as rotational joints. Furthermore, to synthesize a CPM which can achieve backlash-free motion over centimeter's translation range and 10 degrees’ rotation range with desirable performance, both the geometric parameters of the VTFP and the CPM are optimized.
The inverse kinematic model of a conventional rigid 3-PRR parallel mechanism has been investigated extensively in the last decade [21], [22], [23]. However, it cannot predict the kinematics of the long stroke CPM with enough precision, because of the flexure pivots’ rotation center shift is not considered in the model, even the transmission error of a VTFP decreased to almost half of a classical flexure pivot [16]. Therefore, an inverse kinematic model which considering the rotation center shift of the flexure pivot is required for tracking control. In this paper, AIKM is established by combining the deformation equation of the VTFPs, the geometric closed chain equation and the static equilibrium equation of the manipulator under the deformed configuration.
Except the center shift of the VTFPs, there are also several other factors which affect the prediction accuracy of the CPM, such as manufacturing defects and assembling errors [24]. These factors are highly nonlinear and difficult to formulate. Neural network is an effective approach to compensate nonlinear coupled errors of mechanical systems, and the RBFNN has widely used in practical systems for its simple formulation and easy to train [25], [26]. Thus, an adaptive online learning RBFNN is adopted to compensate the unmodeled factors of the CPM in this paper. By adding the RBFNN compensator with the established AIKM, an enhanced inverse kinematic model (EIKM) is proposed to predict the inverse kinematic behavior of the CPM. In addition, to improve the motion accuracy and robustness of the system, a close-loop control scheme which applies the EIKM is established through a DOB strategy.
The rest of this paper is organized as follows. The mechanical design and geometric parameters optimization of the VTFP based 3-PRR CPM are presented in Section 2. Then, an AIKM which considered the rotation center shift of the VTFP is established and verified by FEA in Section 3. A close-loop control scheme is proposed to suppress unmodeled errors and external disturbances of the CPM in Section 4. Experiment test are performed in Section 5 to verify the tracking performance of the proposed control scheme. Finally, the paper is concluded in Section 6.
Section snippets
Geometric parameters optimization of the VTFP
The geometric schematics of a classic flexure pivot and a VTFP are shown in Figs. 1(a) and (b) respectively. Different from the classic flexure pivot where the leaves have a constant cross section, the thicknesses of leaves in VTFP are variable along the length direction. The deformation on the spring leaves will be more concentrated on the region near the rotation center through such a design, just like the notched flexure hinge. Therefore, the VTFP would acquire better transmission
Inverse kinematic modeling of the CPM
For inverse kinematics problem of the CPM, the displacement of the actuators ρi should be solved for a given position and posture of the moving platform (xp, yp, φp). Although the RIKM in Eq. (6) can give a quick and simple solution of the kinematics of the 3-PRR manipulator, the VTFP does not act exactly as the ideal rotational joint during deformation. Parasitic motion will be introduced into the CPM and deteriorate the motion accuracy. To establish an inverse kinematic model for the 3-PRR
Tracking control of the CPM
The above results proved that the proposed AIKM is more accurate than the RIKM to predict the kinematics of the CPM. However, there are still several other factors which affect the accuracy of the CPM, such the manufacturing defects and assemble errors and the external disturbances which are inevitable. Therefore, a close-loop control strategy should be applied to further improve the tracking accuracy of the end-effector. A discrete model that represent the inverse kinematics of the 3-PRR CPM
The experiment setup
A prototype of the proposed 3-PRR CPM is fabricated, as it is depicted in Fig. 13. The VTFPs are made of photosensitive resin and manufactured with stereolithography apparatus. The fixed platform, the moving platform, and passive intermediate links of the CPM are fabricated by Al-7075 alloy. The three kinematic limbs are actuated by three LUSM based linear stages. In each linear stage, the LUSM (Nanomotion HR8) is supported by a linear guide, and a non-contact type linear encoder (Renishaw,
Conclusion
This paper proposed a novel VTFP based long stroke 3-PRR CPM which is capable of achieving high precision planar translation and rotation. The geometric parameters of the VTFP and the manipulator are optimized to take comprehensive consideration of their performance, and a prototype of the optimized 3-PRR CPM is fabricated. The CPM features a regular workspace of centimeters’ translation motion range and 10 degrees’ rotation range, which increases the workspace of the conventional CPM
Acknowledgments
This work was supported by National Natural Science Foundation of China under Grant No. 51475113, Natural Science Foundation of Heilongjiang Province under Grant No. E2015006, the State Key Lab of Self-planned Project under Grant No. SKLRS201701A, and Open Project of Advanced Innovation Center for Intelligent Robots and Systems under Grant No. 2016IRS14.
Conflict of interest
None.
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2021, International Journal of Mechanical SciencesCitation Excerpt :This joint exhibits many advantages of the flexure hinges [15], which are commonly employed in place of classic rigid joints due to their lower maintenance, scalability and monolithic nature [16–19]. In addition, the CAFP offers a more pronounced selective compliant behavior [20], i.e. it provides larger angular strokes and higher resistance to lateral forces and twisting moments, which make it well suited for a wide range of planar design domains [5,21,22]. On the other hand, the CAFP presents some of the typical limitations of distributed compliance joints [23].