- Original Article
- Open Access
Closed-Form Modeling and Analysis of an XY Flexure-Based Nano-Manipulator
© The Author(s) 2018
- Received: 27 April 2016
- Accepted: 14 January 2018
- Published: 28 February 2018
Flexure-based mechanisms are widely utilized in nano manipulations. The closed-form statics and dynamics modeling is difficult due to the complex topologies, the inevitable compliance of levers, the Hertzian contact interface, etc. This paper presents the closed-form modeling of an XY nano-manipulator consisting of statically indeterminate symmetric (SIS) structures using leaf and circular flexure hinges. Theoretical analysis reveals that the lever’s compliance, the contact stiffness, and the load mass have significant influence on the static and dynamic performances of the system. Experiments are conducted to verify the effectiveness of the established models. If no piezoelectric actuator (PEA) is installed, the influence of the contact stiffness can be eliminated. Experimental results show that the estimation error on the output stiffness and first natural frequency can reach 2% and 1.7%, respectively. If PEAs are installed, the contact stiffness shows up in the models. As no effective method is currently available to measure or estimate the contact stiffness, it is impossible to precisely estimate the performance of the overall system. In this case, the established closed-form models can be utilized to calculate the bounds of the performance. The established closed-form models are widely applicable in the design and optimization of planar flexure-based mechanisms.
- Flexure-based mechanism
- Statically indeterminate structure
- Lever mechanism
- Piezoelectric actuator
The integrations of piezoelectric actuators (PEAs) and flexure-based mechanisms have been widely utilized in nano-positioning and manipulations [1–5]. On the one hand, the shape of a PEA changes if charge or voltage is exerted, and thus generating sub-nanometer resolution actuation. However, PEAs suffer from the inherent hysteresis and creep nonlinearities [6–8]. Many feedforward and feedback methodologies have been proposed to compensate for the hysteresis and creep nonlinearities of PEAs [9, 10]. On the other hand, flexure-based mechanisms are capable of transmitting high-precision motions via the elastic deformations of the flexure hinges, making it ideal in building the transmission chains for PEAs [11, 12]. Widely utilized flexure hinge profiles include circular [13–16] and leaf [17, 18].
A single flexure hinge can be treated as a revolute joint during micro- and nano-scale motions. In literature, many analytical and empirical models have been established for the compliance/stiffness of a single flexure hinge [19–21]. In order to improve the performance, multiple flexure hinges are generally combined in various configurations, such as the parallelograms [22–24] and the statically indeterminate symmetric (SIS) structures . In these structures, it is common to treat the flexure hinges as flexible, and all the other components as rigid. Considering the widely-utilized lever mechanism as an example, the lever is frequently assumed to be rigid [26, 27] so as to facilitate the design and modeling processes. However, this assumption may increase the estimation error of the analytical model, especially when the lever is long or the compliance of the lever is not negligible.
A PEA is brittle and very weak when subjected to large lateral forces or torques. As a result, a PEA is not allowed to be firmly fixed to the mechanism during the installation. Many commercial PEAs use ball tips to eliminate the bending torques. In this case, a Hertzian contact interface forms between the tip and the mechanism. One significant drawback of Hertzian contact is its low contact stiffness that consumes large portion of the PEA’s displacement. The contact stiffness is highly dependent on the material properties and the contact status. Currently, there is no effective and reliable model to estimate the contact stiffness. Thus, the contact stiffness is frequently identified from the measured data .
As a flexure-based mechanism is generally light and compact, its performance is likely to be affected by the load mass, including the sensors, end-effectors, fixtures, and other accessories installed on the mechanism. The load mass increases the effective mass and moment of inertia of the system, leading to a slow response. Thus, the influence of the load mass should be taken into consideration in the design and modeling of flexure-based mechanisms.
This paper presents the closed-form modeling of an XY flexure-based nano-manipulator developed in our previous work . In this nano-manipulator, the flexure hinges are arranged in SIS configurations to transmit linear or angular motions. Analytical modeling reveals that the lever’s compliance significantly increases the estimation error. Thus, a threshold is proposed to determine whether the lever’s compliance can be neglected or not. Subsequently, a systematic modeling methodology is established to investigate the behavior of the nano-manipulator during linear and angular motions. Experimental results show that the modeling accuracy is significantly improved if the influence of the lever’s compliance, the contact stiffness, and the load mass is taken into consideration.
3.1 In-plane Compliance of a Single Flexure Hinge
For hinges with a rectangular cross section, κ = 5/6.
In this paper, P1‒P3 are only dependent on the geometric parameter of the hinge, and thus they are named as fundamental integrations.
3.2 Stiffness Modeling of the SIS Structures
In-plane stiffness of the SIS structures
4.350 × 10−2
5.182 × 10−2
3.170 × 103
3.3 Stress Concentration of the SIS Structure
Eqs. (16) and (17) are utilized to obtain the maximum allowable deflections of the SIS structure. For Structure I and II, Eq. (16) shows that the maximum stress locates at both ends of the hinge. For Structure III, the location of the maximum stress can be obtained by differentiating Eq. (17) to x1. Taking the yield strength of the material into consideration, the maximum allowable deflections of Structure I-III are calculated to be 1.46 mm, 1.30 mm, and 5.349 mrad, respectively.
The first three modes of the nano-manipulator are the linear motions in the x- and y-axes and the angular motion about the z-axis. The linear motions in each axis are the primary motions, whereas the angular motion about the z-axis is an unexpected motion degrading the motion accuracy. In this section, the dynamics models in both linear and angular motions are established.
4.1 Influence of the lever’s compliance
From the static and dynamic point of view, the PEA is equivalent to an active force generator. In Figure 5(b), the equivalent stiffness, the driving force, and the effective mass of the PEA are denoted as kPEA, FPEA, and mPEA, respectively; and kcon represents the equivalent contact stiffness of the contact interface. Further, a dimensionless parameter, η = kcon/kPEA, is proposed to characterize the contact stiffness. Three generalized coordinates are defined in Figure 5(b), namely, the displacement of the PEA (xPEA), the rotation angle of the lever (θlever), and the linear displacement of the central platform (xeq).
Eq. (23) is the criterion to decide whether the lever can be treated as rigid or not. In this nano-manipulator, klever = 11.7keq and h 3 2 klever = 0.618(h 2 2 kPEA+kR3). As a result, the lever’s compliance must be considered.
4.2 Dynamics of the Nano-manipulator in the x-axis
4.3 Static Analysis of the Nano-manipulator
4.4 Angular Motion of the Nano-manipulator
As illustrated in Figure 8(a), Structure IV rotates about point O3 during the angular motion. If an identical copy of Structure IV is connected to the opposite side, a new structure is obtained, as shown in Figure 8(c). The topology of the new structure is the same as SIS I. The deformation is an angular displacement of θeq about the z-axis.
5.1 Experimental Setups
During the experimental tests, the parameters of the nano-manipulator with and without the PEAs installed are measured individually. In the installations of the PEAs, each PEA is bolt-fixed on the nano-manipulator, and the preload force is manually adjusted. Based on the previous analyses, higher contact stiffness is preferred during the installation. The load mass is not measured in the static test because it has no influence on the static parameters of the nano-manipulator. In the dynamic test, the load mass is measured to be 53.4 g, including the fixtures and two accelerometers.
5.2 Statics of the Nano-manipulator
Output stiffness in the x-axis
In order to investigate the influence of the lever’s compliance, the analytical results with rigid lever assumption (Analytical 2) are also presented in Table 2 for the comparison. These analytical results are obtained by assigning klever a large value according to the criterion defined in Eq. (23). When no PEA is installed, the estimation error with rigid lever assumption is 42%. Such a high overestimation is not acceptable.
5.3 Dynamics of the Nano-manipulator
First natural frequencies in the x-axis
Analytical 1 (Hz)
Analytical 2 (Hz)
In current experimental setup, the third mode (rotations about the z-axis) doesn’t show up in the measured data. Therefore, the computational analysis is employed to evaluate the nano-manipulator’s behavior during the angular motions. Based on the computational results, the third natural frequency with the 53.4 g load mass is found to be 769.2 Hz. The analytical result is 754.1 Hz, corresponding to an estimation error of 2%, and thus validating the analytical model.
An XY flexure-based nano-manipulator is presented in this paper. Two PEAs are employed to generate actuations and the cross-axis couplings are attenuated in the kinematic chains. The flexure hinges, arranged in SIS configurations, function as prismatic and revolute joints. Lever mechanism is utilized to magnify the displacement of the PEA. It is found that the lever’s compliance may significantly affect the estimated parameters of the nano-manipulator, such as the input/output stiffness and the first natural frequency. In this paper, a criterion is proposed to decide whether the lever’s compliance can be neglected or not. The lever’s compliance can be modeled by cascading a linear spring at the end of the lever. Although simple in formulation, this methodology is effective in improving the modeling accuracy, as verified through experimental results.
The dynamics of the nano-manipulator in linear and angular motions is analyzed. The influence of the contact stiffness and the load mass is analytically investigated. Higher contact stiffness results in improved performances, such as larger workspace and higher first natural frequency. The influence of the load mass is also significant as it adds extra inertia to the nano-manipulator.
The nano-manipulator is monolithically fabricated using wire electrical discharge machining technique. During the installation of the PEAs, the preload forces of the PEAs are manually tuned for a high contact stiffness. The analytical results show good modeling accuracy in comparison with the experimental results, and thus verifying the established models. The methodologies proposed in this paper are applicable in the design and optimization of flexure-based mechanisms.
YDQ designed the prototype, carried out the experiments and wrote the paper. XZ participated in the revision of the paper. BS participated in the design of experiments and revision of the paper. YLT and DWZ participated in the mechical design and manufacture of the prototype. All authors read and approved the final manuscript.
Yan-Ding Qin is currently an associate professor at Institute of Robotics and Automatic Information System, Nankai University, China. He received his PhD degree from Tianjin University, China, in 2012. His research interests include micro/nano manipulation and 3D bio-printing.
Xin Zhao is currently a professor at Institute of Robotics and Automatic Information System, Nankai University, China. He received PhD degree from Nankai University, China, in 1997. His research interests include micro operation robotics, MEMS, and biological pattern and tissue formation.
Bijan Shirinzadeh is currently a professor at Department of Mechanical and Aerospace Engineering, Monash University, Australia. He received his PhD degree from The University of Western Australia, Australia, in 1990. His research interests include micro/nano manipulation, systems kinematics and dynamics, haptics and robotic-assisted surgery and microsurgery, and advanced manufacturing.
Yan-Ling Tian is currently a professor at School of Mechanical Engineering, Tianjin University, China. He received his PhD degree from Tianjin University, China, in 2005. His research interests include micro/nano manipulation, mechanical dynamics, surface metrology and characterization
Da-Wei Zhang is currently a professor at School of Mechanical Engineering, Tianjin University, China. He received his PhD degree from Tianjin University, China, in 1995. His research interests include micro/nano positioning techniques, high speed machining methodologies, and dynamic design of machine tools.
Supported by National Natural Science Foundation of China (Grant Nos. 61403214, 61327802, U1613220), and Tianjin Provincial Natural Science Foundation of China (Grant Nos. 14JCZDJC31800, 14JCQNJC04700).
The authors declare that they have no competing interests.
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