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Design and analysis of a compact flexure-based precision pure rotation stage without actuator redundancy
Mechanism and Machine Theory 105 (2016) 129–144
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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Design and analysis of a compact flexure-based precision pure rotation stage without actuator redundancy Leon Clark a, * , Bijan Shirinzadeh a , Yongmin Zhong b , Yanling Tian c , Dawei Zhang d a
Robotics and Mechatronics Research Laboratory, Department of Mechanical and Aerospace Engineering, Monash University, Clayton, VIC 3800, Australia School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, VIC 3083, Australia c School of Engineering, University of Warwick, Coventry CV4 7AL, UK d Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, Tianjin University, Tianjin 30072, China b
A R T I C L E
I N F O
Article history: Received 26 January 2016 Received in revised form 20 June 2016 Accepted 20 June 2016 Available online xxxx Keywords: Micro-nano positioning Rotation stage Compliant mechanism design Additive manufacturing
A B S T R A C T This paper presents the mechanical design, optimisation, and computational and experimental analyses of a flexure-based single degree of freedom rotation stage. The mechanism possesses a rotationally symmetric configuration, whilst only employing a single piezoelectric actuator, which increases the mechanism’s ability to reject cross-coupled drift of the rotation centre. This layout is facilitated by a novel multi-level structure, which exploits emerging additive manufacturing techniques for its construction, and is compact, with little unused volume. Computational analysis has been employed for both the optimisation of the mechanism, to increase its workspace whilst maintaining a small physical footprint, and subsequently to predict its performance. The cross-coupled drift, particularly its variation with respect to assembly and manufacturing errors, is explored in depth. A prototype has been manufactured, which fits within a 128mm×153mm×30mm bounding box, and its working range has been experimentally determined to be 2.540mrad, with a first natural frequency of 175.3Hz. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Compliant mechanisms have emerged as fundamental components within many emerging devices in nanotechnology, including inside scanning probe microscopes (SPM), high-density data storage, and manipulators for micro-manufacturing, assembly and biohandling applications [1–5]. Their applicability for these tasks is drawn from their ability to produce smooth, continuous motions, free of backlash and other non-linear effects which inhibit the use of conventional joints. Various designs have been proposed to facilitate the production of ultrahigh precision motions, particularly with regard to translational positioning. Whilst scanners have been developed which produce linear motions with one, two and three degrees of freedom (DOF), the majority of existing research has focused on two DOF scanners, due to their applications within SPM and nano-imprint lithography [6–10]. The demands of high-dexterity manipulation tasks and optical alignment have led to the
* Corresponding author. E-mail addresses: [email protected] (L. Clark), [email protected] (B. Shirinzadeh), [email protected] (Y. Zhong), [email protected] (Y. Tian), [email protected] (D. Zhang).
http://dx.doi.org/10.1016/j.mechmachtheory.2016.06.017 0094-114X/© 2016 Elsevier Ltd. All rights reserved.
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requirement for positioners also possessing angular axes of motion. Recent research efforts have led to the development of positioners with two to six DOFs, all including at least one angular motion axis [11–15]. Many different modes of actuation have been employed to provide the necessary input to these mechanisms. These include voice coil motors, electrostatic drives, thermal actuators and magnetic actuators. Within existing literature, the piezoelectric actuator (PEA) is most commonly favoured, due to its high resolution and compact size. However, PEAs exhibit a rate-dependent hysteresis effect, with its magnitude being a significant proportion of the full stroke at high frequencies, as well as drift in their response. For this reason, almost all PEA-driven nanopositioners employ a combination of feedforward and feedback control in practice. Typically, feedforward strategies employ inversions of the modelled PEA hysteresis [16–18], before the feedback component works to eliminate residual errors and maintain robustness to instability [19–22]. The objective of the study presented in this paper is to produce a compliant pure rotational scanner with the largest possible angular range, which minimises the cross-axis coupled drift of the rotation centre, whilst only being driven by a single actuator. The stiff structure of PEA-driven compliant mechanisms allows them to exhibit high motion bandwidths, and they are capable of large output forces [23, 24]. However, due to the PEA’s limited stroke, the mechanism must be optimised to maximise the output range, subject to constraints on the maximum mechanism size. Such a mechanism would have applications within optical steering, calibration and workpiece alignment tasks [25–27]. Cross-coupled drift of the rotation centre during motion would detract from its performance. An integral strategy for minimising drift is the incorporation of rotational symmetry within the design of the mechanism and its actuation. In particular, if the mechanism were designed with two-fold rotational symmetry about its centre (invariance to a rotation by 180◦ ), then any translation within an output motion must be invariant to this rotation. Therefore any cross-coupled translation is necessarily eliminated within these ideal conditions. A key objective within the design of the mechanism is the elimination of actuator redundancy — the use of only one actuator to actuate the single motion axis. Manufacturing processes such as wire electrical discharge machining (WEDM) typically limit positioner designs to lie within a single plane, and subsequently such rotational scanners must have two or more actuators to allow for symmetry in the design. In this case, particularly if PEAs are employed, the reduction of cross-coupling cannot be guaranteed. As a result of the unknown and possibly unequal level of hysteresis, uneven preloads, mounting defects and manufacturing imperfections, there will be an inevitable difference between the two displacement inputs, breaking the rotational symmetry and leading to a coupled translation in the output. Frontier additive manufacturing technologies, such as fused deposition modelling (FDM) and stereolithography, are increasingly providing higher resolution fabrication capabilities from a broadening set of materials. In this paper, such techniques are exploited to free the mechanism from being constrained to a single plane. This further allows the use of a single actuator, whilst maintaining the requisite symmetry. A pure rotation scanner has previously been designed by Lee et al. [28]. However, this mechanism employed two PEAs, and also did not possess rotational symmetry in its design; making it susceptible to drift of the rotation centre. Cathie and Janky also proposed a flexure-based rotation stage, however, a method of actuation was not provided [29]. Planar XY h positioners could also be utilised for this purpose, however this would require added consideration of actuation, sensing and feedback schemes for the linear axes to compensate for drift. The remainder of the paper is structured as follows: in Section 2 the design of the mechanism, and its subsequent optimisation, are provided. Section 3 details a computational examination of the optimised design, with a focus on the working range and undesirable cross-coupled motions, and their variation as manufacturing errors are introduced. Results from experimentation performed on a manufactured prototype are presented in Section 4. Finally, considerations for practical implementations are discussed in Section 5.
2. Mechanism design Fig. 1a shows a rigid body schematic of the underlying amplification structure for the proposed rotational scanner. The principal design element is the four bar crank-rocker mechanism, from which the output rotation is taken to be at the crankbase joint. An input lever provides amplification to the vertical input, whilst also ensuring that the input is horizontally aligned with the output hinge. The rigid body equivalent to the compliant mechanism is immobile due to this lever. Rotational symmetry of the design is achieved by the addition of a duplicate of the underlying structure, which replaces the output hinge. The duplicate is rotated by 180◦ about the output point, as illustrated in Fig. 1b. With a symmetric input, the output rotation is then centred on the crank linkage. Fig. 2 shows the compliant design basis of the proposed rotational mechanism. The mechanism has been separated into two levels: the upper level provides the functionality of the schematic in Fig. 1, whilst the lower level provides the symmetric inputs to the upper level. The input bridge amplifier stage of the lower level transforms the expansion of a PEA into transverse motions to be transferred to the upper stage. In particular, the two double parallelogram linkages of the input stage ensure that the PEA is not exposed to shearing forces. Due to the duplication of the structure to achieve the symmetry of the design and the removal of the flexure hinge, the output link has an added (translational) degree of freedom when the input is fixed. As a consequence, undesirable vibrational modes exist at low frequencies, which are discussed within the computational analysis in Section 3. To remedy this, cantilevers have
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Fig. 1. Rigid body schematic of the rotation stage: (a) underlying structure, (b) rotationally symmetric version with virtual rotation centre.
been added to the output platform which resist this translation whilst permitting the rotational output. To reduce the impedance of this added stiffness on the output, the cantilevers were made to terminate as close to the rotation centre as possible. The lower section was chosen to be 15 mm thick, whilst the upper section was chosen to be 12 mm thick. A gap of 3 mm separated the two levels. The two-level flexure design employs four different types of compliant elements. All hinges within the crank-rocker sections employ circular flexure hinges, whilst the remaining hinges are filleted leaf hinges. Similarly, the cantilevers have filleted radii at their attachment points. The input bridge elements are reinforced to reduce axial elongation, concentrating rotation to the thinner sections of the bridge structures. The two-level design can be contrasted with a complementary design, using a single actuator and the same underlying structure, but constructed within a single plane, as shown in Fig. 3. To achieve a comparable output range the layout is dominated by input amplification levers, which constrains the size of the output platform, and increases the mechanism size and amount of void space. Due to the lack of design symmetry, careful optimisation of the geometric parameters would be required to increase rotational precision, and the incidence of any cross-coupled drift would be much more sensitive to manufacturing defects. 2.1. Design optimisation The goal of the mechanism design process was to produce a prototype with the largest possible angular range, whilst retaining a small footprint. The design was optimised employing the response surface optimisation technique, using computational analysis to study the dependence of the mechanism’s behaviour on multiple design parameters. From Fig. 1a, it can be seen that there are three sources of amplification involved during the transformation of the input to the output. These have been named the input amplification, the transfer amplification, and the crank-rocker amplification for the remainder of this paper. The input amplification depends on the geometry of the link possessing the hinges labelled a, b and c in Fig. 1a, whilst the transfer amplification depends on the hinges labelled c, d and e. The instantaneous amplifications at the rest position, with the lengths defined in Figs. 2b and c, are given by: aIA =
LCO LCI
(1a)
aT =
LUR LUT
(1b)
aCR =
LUR . LUC
(1c)
These amplifications, combined with the bridge amplifier thickness t and angle 0, also defined in Fig. 2b, were the design parameters considered during the optimisation. As the PEA is constrained to be at the centre of the lower stage, the lengths defined in Fig. 2 are further subjected to the constraint: LUC − LUR + LUT + LCO − LCI = 0.
(2)
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Fig. 2. Flexure design of rotation stage: (a) 3D view with upper section cutout, (b) cross-section of lower section, (c) cross-section of upper section.
Considering the size requirements of the optical measurement equipment to be used in experimental work, the output platform width was fixed to be 25 mm (LUC = 12.5 mm). Hence, the remaining dimensions of the upper stage are fully determined by Eqs. (1a), (1b), (1c) and (2). The optimisation employed a static structural finite element analysis to estimate the angular output at the full driving voltage. Computational simulations were performed within the ANSYS software package. A proximity based meshing method was utilised, which ensured that the compliant elements of the mechanism had a more refined mesh. The meshed model, as shown in Fig. 4, was re-meshed for each set of design parameters, however all meshes contained in excess of 200,000 nodes and 115,000 elements. Displacement inputs of 5.8 lm were applied to the two input surfaces on the lower level, representing a full stroke of 11.6 lm, matching the nominal capabilities of the NEC/Tokin AE0505D16F PEA used for experimentation. The average displacement of nodes on the surface of a cylindrical pilot hole through the centre of the output stage was used to determine the resultant mechanism output. The nodes on the side boundaries of the mechanism were fixed within these analyses. Material properties of Polyjet RGD720, which was used in the 3D printing of the experimental prototype were assumed in the analysis. The material has a density of approximately 1.19 g/cm3 , an elastic modulus of 2.5 GPa and 50 MPa tensile strength.
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Fig. 3. Design of flexure mechanism if constrained to planar manufacturing.
Two analyses were employed to estimate the output rotation. The first of these, employing the constraints above was used to find the raw output in response to the 11.6 lm input. However, due to the input stiffness of the mechanism kin , the actuator’s range is reduced according to: g=
kPEA . kPEA + kin
(3)
The second analysis estimated the input stiffness by providing 1 N forces to each of the input surfaces. The average displacement of one surface was then used to determine the stiffness: kin =
Fin . 2xin
Fig. 4. Cross-sections of meshed model used for computational design optimisation and validation: (a) lower section, (b) upper section.
(4)
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L. Clark, et al. / Mechanism and Machine Theory 105 (2016) 129–144 Table 1 Design parameter ranges considered for optimisation. Parameter Input thickness (mm) Input angle Crank-rocker amplification Transfer amplification Input amplification
t 0 a CR aT a IA
Min.
Max.
Opt. value
1.0 8.0◦ 3.0 1.5 1.5
2.0 25.0◦ 6.0 5.0 5.0
1.000 11.61◦ 6.000 2.167 2.358
Hence, the optimisation sought to maximise the stiffness-reduced output: hpred = ghFEA .
(5)
The optimisation proceeded in three phases to minimise the total number of design points to be simulated. Initially, it was assumed that the amplification of the input stage would be decoupled from the kinematics of the upper level. Hence, optimisations of the input stage (t and 0) were carried out separately to the upper level (a CR , a T , and a IA ). Table 1 shows the ranges considered for each design parameter. In particular, the maximum value of a CR was chosen based on the maximum allowable dimensions of the mechanism. Other candidate variables considered for optimisation were the separation of the central cantilevers on the output stage and their lengths. It was observed that the output stiffness with respect to rotation was minimised by reducing this separation (thus increasing the output range), and it was subsequently set to its minimum value. Lengthening the upper level cantilevers led the reduction of the output stiffness, but it also reduced the stiffness with respect to undesirable coupled motion in the X direction, which caused the corresponding natural mode (shown later in Fig. 6b) to have a lower frequency. The cantilever length was chosen such that the rotational mode remained as the fundamental mode of the mechanism (as discussed in the next section). The first two optimisation operations, carried out sequentially, resulted in the generation of 228 and 437 feasible datapoints, respectively. Parameters were chosen by finding the design variables resulting in the maximum value predicted by a surface of best fit to the generated points. Surfaces were fitted using the Krige interpolation method, whilst a genetic algorithm was employed to find the maximising parameters [30]. After performing the two optimisations, the assumption that the two phases were decoupled was found to be inaccurate. Consequently, optimisation over all parameters was required. However, as shown in Fig. 5c, the angular range increased with
Fig. 5. Optimality of final design — cross-sections of response surface: variation with (a) input angle (0), (b) input thickness (t), (c) crank-rocker amplification (a CR ), (d) transfer amplification (a T ), (e) input amplification (a IA ).
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Fig. 6. First four natural modes of the proposed angular stage: (a) h motion (217 Hz), (b) X motion (219 Hz), (c)–(d) out-of-plane motions (379 Hz, 477 Hz).
the crank-rocker amplification, whereupon it was set to its maximum value. A final optimisation over the remaining four parameters, employing 442 feasible simulations, was used to determine the final design parameters, as given in Table 1. Fig. 5 shows cross-sections of the response surface with respect to each of the design parameters considered, demonstrating the optimality of the design.
3. Computational verification The static structural analysis was repeated to verify the performance of the optimised design candidate. The model was replaced with a finer mesh, consisting of at least 475,000 nodes and 290,000 elements. The input stiffness was instead evaluated as the slope of the line of best fit to responses to 1 N, 2 N and 3 N force inputs. All simulations were repeated sixteen times, re-meshed each time, to improve the numerical stability of the solutions generated. Table 2 shows the results of this static analysis, including the translational motion (drift) of the rotation centre. For comparison, the performance is also provided for the case where the central cantilevers were omitted from the design. It can be seen that the increase in output stiffness due to the introduction of the cantilevers reduces the working range. As the standard error in the predicted drift is comparable in magnitude to the mean drift, it is likely that the computed drift for each simulation is due to asymmetry in the meshed models, rather than a systematic error in the design. A modal analysis was also performed on the meshed model of the mechanism. In order to avoid excessive computation time, the number of nodes was reduced to 172,000, with 101,000 elements. A linear spring was added to the model between the two input surfaces to simulate the presence of the PEA. Fig. 6 shows the first four modes of the mechanism predicted by the computational analysis. In particular, it can be seen that the first natural mode corresponds to the desired angular motion. As was mentioned in the previous section, the second mode is a motion in the X direction. Its natural frequency could be further increased above the h mode by increasing the stiffnesses of the central cantilevers. Fig. 7 shows the first four modes of the mechanism if the central cantilevers were omitted. The rotational mode is preceded by two translational modes, caused by the crank-rocker section being underconstrained. Without the cantilevers, these translational motions are likely to be excited and coupled at lower frequencies, thus decreasing the mechanism’s operational bandwidth. 3.1. Drift of rotation centre The small amount of drift and out-of-plane rotation predicted in Table 2 stems from the rotational symmetry of the design. However, in practice this symmetry cannot be realised perfectly, due in part to errors in assembly, manufacturing and apparatus alignment. In this section, the impact from two of these sources, improper actuator mounting and manufacturing errors, on the drift of the output stage are studied through computational simulation. Table 2 Computational predictions of maximum mechanism output. Final design
h kin xdrift ydrift zdrift Roll Pitch
Cantilevers omitted
Mean
Std. err.
Mean
Std. err
2.7962 mrad 1.6802 × 106 Nm −1 −0.1141 nm 0.1077 nm 0.3622 lm −25.14 nrad 2.760 nrad
0.161 lrad 75.8 Nm −1 1.16 nm 0.179 nm 0.300 nm 22.5 nrad 22.1 nrad
3.7886 mrad 1.4823 × 106 Nm −1 −3.430 nm 3.395 nm 3.332 lm −57.44 nrad 88.03 nrad
0.210 lrad 80.9 Nm −1 7.84 nm 6.44 nm 3.64 nm 135 nrad 21.9 nrad
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Fig. 7. Natural modes of the mechanism with the central cantilever omitted: (a)–(b) X–Y motions (80 Hz, 107 Hz), (c) h motion (188 Hz), (d) out-of-plane motion (225 Hz).
3.1.1. Deviation in input position Incorrect mounting of the actuator during the mechanism assembly could cause the symmetry of the input to be lost, thereby causing motion along undesirable axes in the output response. To simulate this effect, the CAD model of the mechanism was altered so that the input would be localised to two surfaces. These surfaces were the size of the faces of the PEA used for experimentation. The positions of these two surfaces could be altered in the Y and Z directions from the centre location, as shown in Fig. 8. The computational analysis was then repeated for this modified model, where a force along the axis of the line separating the surfaces was applied as the two inputs. The force was computed from the results of Table 2, such that the predicted h value was produced when there was no deviation in input position. The simulation was repeated for 801 combinations of input positions, with the Y and Z positions on both faces uniformly randomly selected from the interval [−2, 2]. Fig. 9 shows the results of the analysis. The drift is plotted against the variable which had the greatest Pearson correlation coefficient when using a linear regression. The raw Y and Z deviations of the inputs, as well as the offset and difference, illustrated in Fig. 8, were considered. The offset is the deviation of the mean position of the input surfaces, whilst the difference is the projection of the vector from the left (the surface with negative X coordinates) to the right input surface onto the Y–Z plane. The drift of the stage in the X–Y plane was confined within 22 nm. The drift being relatively small is likely to be due to the redundancy and number of compliant elements between the input and output stages. Whilst the correlation is weak, it was expected that the strongest dependence of the drift would be on the Y offset, as the offset in the Z direction does not contribute to a loss in symmetry. Similarly, symmetry would be maintained with a difference in Y position without an offset. The Y difference can be seen to have a strong influence on the angular output, which also appears to be inversely related to the Z drift of the stage. The coupled pitch and roll of the stage were also computed, which both remained beneath 280 nrad for all simulated combinations. No strong correlations were observed between these rotations and the input positions. 3.1.2. Manufacturing noise The manufacturer of the 3D printer, used for manufacturing of the prototype detailed in Section 4, specifies a build resolution of approximately 42 lm in the X and Y directions, and 16 lm in the Z direction, with an overall accuracy below 85 lm for parts smaller than 50 mm, or 200 lm for the largest printable object. Consequently, inaccuracies in manufacturing or surface finishing could lead to asymmetry in the fabricated mechanism.
Fig. 8. Setup of CAD geometry to simulate deviation in the actuator positioning.
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Fig. 9. Drift of the rotation centre due to variation in the actuator positioning: (a) drift in the X and Y directions, (b) variation in the output rotation, and drift in the Z direction.
These effects have been simulated by adding noise to the surface of the compliant elements of the mechanism, which is illustrated in Fig. 10. This was achieved by subdividing the faces of these elements, before applying Gaussian noise along the surface normal to a triangulated model of the mechanism. The subdivided facets had a height of 1.5 mm, and a width of 0.6 mm for the circular hinges, 1.0 mm for leaf hinges, 2.0 mm for cantilevers, and 1.8 mm for the bridge amplifier. In total, the locations of 2412 vertices were randomly displaced during this procedure. Noise with standard deviations from 0 lm to 100 lm in 20 lm increments was considered. Due to the procedure used to triangulate, distort and reimport the geometry, for the 0 lm case, 0.1 lm noise was employed to preserve the facets of the geometry and allow a direct comparison with the other cases. Due to the complexity of the deformed surfaces, a different meshing method had to be employed to accommodate the changed geometry. Meshes with approximately 730,000 nodes and 500,000 elements were used for most models, however at higher noise levels a mesh with lower complexity was sometimes required due to the difficulty in meshing these geometries. Each analysis was repeated ten times to reduce the dependence on the meshing, and all six cases were repeated five times with differing random number generator seeds. A displacement input was utilised, reduced from the maximum PEA stroke to reflect the input stiffness, using the results of Table 2. The predicted drift in the X–Y plane of the rotation centre, as well as pitching and rolling motions of the output stage, are shown in Fig. 11. It has been assumed that the manufacturer’s specified surface accuracy represents the 95% confidence interval, hence two standard deviations have been used to represent the surface noise. The translational drift, shown together with error bars indicating the standard error, have been computed using: r=
x¯ 2 + y¯ 2
10 2 2
1 xi (xi − x¯ ) yi (yi − y¯ ) dr = + 10 r r
(6)
(7)
i=1
where x¯ =
1 10
10 i=1
xi is the mean drift in the X direction. Whilst positive and negative deviations were observed, the figure shows
their absolute values.
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Fig. 10. Modified CAD geometry showing subdivided facets and simulated surface noise added to compliant elements: (a) upper level flexure hinge, (b) lower level cantilever and bridge element. The upper edge of the surfaces which have been deformed for the elements pictured has been indicated. The standard deviation of the noise added to the surfaces of this model was 100 lm.
Whilst there is a large variation in the drift even at a single noise level, it is clear that the maximum drift increases with noise. The maximum translational drift was observed to lie within an exponential envelope, whilst the largest rotation varied linearly with increasing noise. 4. Experimental prototype A prototype of the rotational scanner has been fabricated using an Objet Eden 260 V 3D printer. Photographs of the prototype showing the upper and lower levels are provided in Fig. 12. The bounding box of this prototype has dimensions 128 mm × 153 mm × 30 mm. A detailed photograph of the surface finish near the output stage is shown in Fig. 12c, where the horizontal layering from the printing process is evident. To arrive at this prototype, the original flexure design shown in Fig. 2 was modified to remove material from the base which would not experience large deformations. This has the effect of reducing the prototype’s mass, and consequently lowered the printing time and material cost. In addition, as shown in Fig. 12b, four slots were introduced to provide access to the 3 mm gap separating the two levels, primarily to assist the removal of the support material required during fabrication. One of the slots would also provide a passage for the input cables to the PEA. Two grooves were manufactured within the output platform, which allow the installation of a press-fit reflector mount. The mechanism was mounted within the apparatus shown in Fig. 13. Actuation was provided by a NEC/Tokin AE0505D16F PEA capable of a full stroke of 11.6±1.3 lm at a 100 V input voltage. The PEA was driven by a voltage amplifier (Thorlabs MDT693B), which received input from the output of a 16-bit digital-to-analogue converter (DAC) from a control computer. Measurement of the angular position was provided by two sensors, a digital autocollimator (Microradian T30D) and a laser interferometer (Zygo HSPMI 2001-02). Compared to the interferometer, the autocollimator allowed for angular measurement over a much larger range, albeit at reduced resolution and sampling rate. The steady state RMS noise was measured to be 86.7 nrad using the interferometer, and 1.56 lrad for the autocollimator. The preload to the actuator was provided by unscrewing a steel M6 hex-head screw/nut pair placed between the actuator and an input surface. Small props were employed within the PEA mountings to aid in their location, and to reduce any deviation in the actuator positioning. The added mass of the reflectors, in addition to the mechanism being fixed at the two diagonally-opposite screw hole locations shown in Fig. 12b, rather than over the entirety of the side boundaries, will lead to a reduction in performance compared to that predicted by the computational model in the previous section. These analyses have therefore been repeated, with the removed material, PEA mountings, reflectors and their mounting represented within the model. Nodes were fixed at only the surfaces of the two cylindrical screw holes. Employing this revised model, the angular range was predicted to be reduced to 2.4288 mrad, whilst the natural frequency of the rotational mode was predicted to reduce to 185.7 Hz. 4.1. Experimental validation To determine the output range of the angular scanner, a smoothed triangle wave open-loop input voltage between 0 V and 100 V was applied to the PEA. The measured position during this operation is shown in Fig. 14. The angular range of the mechanism was 2.540 mrad, 4.6% larger than the computational prediction. However, this discrepancy is within the 11.5% uncertainty
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Fig. 11. Introduction of output drift due to simulated surface noise: (a) X–Y plane, (b) out-of-plane rotation. The red coloured markers in (a) indicate simulations which employed a mesh with fewer than 700,000 nodes. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
in PEA stroke stated by the manufacturer. Other errors may also be present within the computational model, such as due to the unmodelled contact stiffness condition at the input, and inaccuracies in the material properties assumed within the analysis. The first natural frequency of the rotational scanner has been determined. A sinusoidal sweep signal with its frequency increasing exponentially from 0.05 Hz to 400 Hz has been applied to the PEA input, with a peak-to-peak actuation magnitude of 1.0 V. As the sampling rate of the autocollimator was limited, data for this experiment was gathered using the interferometer, at a control frequency of 800 Hz. The frequency response to this input is shown in Fig. 15. The first resonant frequency was encountered at 175.3 Hz. The discrepancy between the computational prediction (185.7 Hz) is likely to be due to other modelling errors, such as an error in the estimated PEA stiffness. The resolution of the angular positioning was tested by measuring the feedback control response to nanoradian scale steps. A proportional-integral controller was employed, with a feedforward component approximating the result of Fig. 14 as a proportional input. Feedback was provided by the interferometer, with the control computations taking place at 800 Hz. Fig. 16 shows the response to 250 nm steps at 1.33 Hz. The RMS tracking error for the time shown was 90.6 nrad, with a maximum error of 371.8 nrad. At this scale, the tracking error was limited by the noise within interferometer measurement, as well as the resolution of the computer’s DAC, where the quantisation step was approximately 38.8 nrad. 4.2. Influence of 3D printed fabrication Previous research, including the results shown in this paper, have demonstrated the capability for 3D printed photopolymer mechanisms to exhibit positioning with precision at the sensor noise level [31, 32]. However, it was observed that the output of the mechanism would drift over time, most likely due to the creep of the mechanism under the stress of the preload. This drift is shown for the 24 hours immediately after the application of the preload in Fig. 17. It should be noted that the environmental climate control was disabled overnight between hours 8 and 22, which further shows the thermal dependence of the drift. Due to this drift, there are several factors that must be taken into consideration when employing such mechanisms. Firstly, as these mechanisms drift, their working range and kinematics will change. Consequently, this may require periodic adjustment of
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Fig. 12. Close-up photographs of the fabricated rotational scanner prototype: (a) upper level, (b) lower level, (c) surface finish near output stage.
the preload, and the use of adaptive control schemes to compensate for the varying mechanical characteristics. In addition, it has also been observed that mechanisms fabricated from photopolymer exhibit hysteresis in their kinematics, which in combination with PEA hysteresis, need to be factored into controller design. It is expected that these issues could be mitigated through the use of a 3D printed metal for fabrication, which is also likely to increase the bandwidth of achievable motions. 5. Discussion Manufacturing error, in the form of surface noise, was shown to potentially lead to a linearly increasing amount of drift. As shown in Fig. 12c, the actual surface profile was quite smooth, apart from variations resulting from, and of similar size to, the (16 lm) layering of the printing process. It is important to note that, at the 80 lm noise level (assumed to be comparable to the specified build accuracy), the maximum predicted pitch and roll drift motions were less than 0.3% of the full range output
Fig. 13. Photograph of the experimental apparatus.
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Fig. 14. Experimentally determined working range of the rotational scanner.
rotation. However, this analysis was limited to variations fixed to the chosen grid on the compliant elements, and did not consider material anisotropy, which could further increase the amount of drift. The drift due to variation in the input position was observed to be negligible in comparison to the drift due to manufacturing error. In particular, the drift observed in the X and Y axes, as well as pitch and roll motions, shown in Fig. 9, was of a comparable magnitude to the standard deviation in the sample, showing that the mechanism has a strong ability to reject variations in the actuator positioning. It is also worth noting that the 2 mm variation in all directions considered is likely to be an exaggeration, and tighter assembly tolerances would be encountered in practice. Whilst the setup of the computational simulation, in particular the use of a force input, may not be a completely accurate depiction of the true behaviour in practice, the analysis has shown that these errors lead to a small amount of drift, which should not change dramatically if the computational model were altered. It should also be noted that in a practical experimental environment, the performance would be affected by the setup of the mechanism and its accompanying instruments. In particular, an offset between the true X axis and a sensor’s median axis will give rise to Abbe errors within measurement, as illustrated in Fig. 18. As a consequence of this offset, a spurious X displacement would be measured as: xAbbe = d tan h ≈ dh
(8)
where d is the offset between the true and measured axes. This error can form a significant component of the measured output; at the maximum measured 2.540 mrad, an offset of only 0.4 mm could cause a 1 lm error. As this error increases linearly with the output angle, it is indistinguishable from the drift of the rotation centre. Consequently, calibration and correction of Abbe
Fig. 15. Frequency response of the rotational scanner.
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Fig. 16. Controlled response to a nanoradian scale step input.
errors would necessarily also compensate for any drift. It is for this reason that experimental measurement of the drift has not been performed.
6. Conclusion This paper has presented the design of a compliant pure rotation stage which employs a single actuator, whilst possessing rotational symmetry in its design to minimise undesirable drift motions. The mechanism’s geometric parameters were optimised to maximise the working range, whilst retaining a small physical size. Computational analysis was performed to determine the capabilities of the mechanism, and in particular the cross-coupled motions were analysed with respect to simulated manufacturing and assembly errors, which cause a loss of design symmetry. The drift due to errors in the mounting of the input was found to be negligible when considered against those resulting from manufacturing errors. Nevertheless, pitch and roll drift motions were found to be insignificant in comparison to the principal yaw motion. Furthermore, the drift in the X–Y plane will be compensated simultaneously with calibration for Abbe errors, which are likely to be larger in magnitude. Experimental results demonstrated the ability to produce an angular output over a 2.540 mrad range, with a positioning resolution comparable to the sensor noise level, and a first resonant frequency of 175.3 Hz, which compared favourably with computational predictions. However, these results for the fabricated prototype were reduced from the initial ideal model, and it is expected that performance can be improved by reducing the load mass and employing more rigid fixings at the base. The output exhibited a long term drift, due to the creep effect of the printed photopolymer material, which is likely to be mitigated if a metal were alternatively used for fabrication.
Fig. 17. Drift of mechanism after application of preload.
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Fig. 18. Abbe error resulting from experimental offset.
Acknowledgements This work was supported by ARC LIEF (Grants LE0347024 and LE0775692), and ARC Discovery Projects (Grant DP140104019). The writing of this manuscript was supported by the Monash Postgraduate Publications Award. Fabrication of the compliant mechanism was performed at the Melbourne Centre for Nanofabrication (MCN) in the Victorian Node of the Australian National Fabrication Facility (ANFF). The authors would like to thank Mr. Soon Hock Ng from the Melbourne Centre for Nanofabrication (MCN) for his assistance in the fabrication of the compliant mechanism.
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Contents lists available at ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Design and analysis of a compact flexure-based precision pure rotation stage without actuator redundancy Leon Clark a, * , Bijan Shirinzadeh a , Yongmin Zhong b , Yanling Tian c , Dawei Zhang d a
Robotics and Mechatronics Research Laboratory, Department of Mechanical and Aerospace Engineering, Monash University, Clayton, VIC 3800, Australia School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, VIC 3083, Australia c School of Engineering, University of Warwick, Coventry CV4 7AL, UK d Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, Tianjin University, Tianjin 30072, China b
A R T I C L E
I N F O
Article history: Received 26 January 2016 Received in revised form 20 June 2016 Accepted 20 June 2016 Available online xxxx Keywords: Micro-nano positioning Rotation stage Compliant mechanism design Additive manufacturing
A B S T R A C T This paper presents the mechanical design, optimisation, and computational and experimental analyses of a flexure-based single degree of freedom rotation stage. The mechanism possesses a rotationally symmetric configuration, whilst only employing a single piezoelectric actuator, which increases the mechanism’s ability to reject cross-coupled drift of the rotation centre. This layout is facilitated by a novel multi-level structure, which exploits emerging additive manufacturing techniques for its construction, and is compact, with little unused volume. Computational analysis has been employed for both the optimisation of the mechanism, to increase its workspace whilst maintaining a small physical footprint, and subsequently to predict its performance. The cross-coupled drift, particularly its variation with respect to assembly and manufacturing errors, is explored in depth. A prototype has been manufactured, which fits within a 128mm×153mm×30mm bounding box, and its working range has been experimentally determined to be 2.540mrad, with a first natural frequency of 175.3Hz. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Compliant mechanisms have emerged as fundamental components within many emerging devices in nanotechnology, including inside scanning probe microscopes (SPM), high-density data storage, and manipulators for micro-manufacturing, assembly and biohandling applications [1–5]. Their applicability for these tasks is drawn from their ability to produce smooth, continuous motions, free of backlash and other non-linear effects which inhibit the use of conventional joints. Various designs have been proposed to facilitate the production of ultrahigh precision motions, particularly with regard to translational positioning. Whilst scanners have been developed which produce linear motions with one, two and three degrees of freedom (DOF), the majority of existing research has focused on two DOF scanners, due to their applications within SPM and nano-imprint lithography [6–10]. The demands of high-dexterity manipulation tasks and optical alignment have led to the
* Corresponding author. E-mail addresses: [email protected] (L. Clark), [email protected] (B. Shirinzadeh), [email protected] (Y. Zhong), [email protected] (Y. Tian), [email protected] (D. Zhang).
http://dx.doi.org/10.1016/j.mechmachtheory.2016.06.017 0094-114X/© 2016 Elsevier Ltd. All rights reserved.
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requirement for positioners also possessing angular axes of motion. Recent research efforts have led to the development of positioners with two to six DOFs, all including at least one angular motion axis [11–15]. Many different modes of actuation have been employed to provide the necessary input to these mechanisms. These include voice coil motors, electrostatic drives, thermal actuators and magnetic actuators. Within existing literature, the piezoelectric actuator (PEA) is most commonly favoured, due to its high resolution and compact size. However, PEAs exhibit a rate-dependent hysteresis effect, with its magnitude being a significant proportion of the full stroke at high frequencies, as well as drift in their response. For this reason, almost all PEA-driven nanopositioners employ a combination of feedforward and feedback control in practice. Typically, feedforward strategies employ inversions of the modelled PEA hysteresis [16–18], before the feedback component works to eliminate residual errors and maintain robustness to instability [19–22]. The objective of the study presented in this paper is to produce a compliant pure rotational scanner with the largest possible angular range, which minimises the cross-axis coupled drift of the rotation centre, whilst only being driven by a single actuator. The stiff structure of PEA-driven compliant mechanisms allows them to exhibit high motion bandwidths, and they are capable of large output forces [23, 24]. However, due to the PEA’s limited stroke, the mechanism must be optimised to maximise the output range, subject to constraints on the maximum mechanism size. Such a mechanism would have applications within optical steering, calibration and workpiece alignment tasks [25–27]. Cross-coupled drift of the rotation centre during motion would detract from its performance. An integral strategy for minimising drift is the incorporation of rotational symmetry within the design of the mechanism and its actuation. In particular, if the mechanism were designed with two-fold rotational symmetry about its centre (invariance to a rotation by 180◦ ), then any translation within an output motion must be invariant to this rotation. Therefore any cross-coupled translation is necessarily eliminated within these ideal conditions. A key objective within the design of the mechanism is the elimination of actuator redundancy — the use of only one actuator to actuate the single motion axis. Manufacturing processes such as wire electrical discharge machining (WEDM) typically limit positioner designs to lie within a single plane, and subsequently such rotational scanners must have two or more actuators to allow for symmetry in the design. In this case, particularly if PEAs are employed, the reduction of cross-coupling cannot be guaranteed. As a result of the unknown and possibly unequal level of hysteresis, uneven preloads, mounting defects and manufacturing imperfections, there will be an inevitable difference between the two displacement inputs, breaking the rotational symmetry and leading to a coupled translation in the output. Frontier additive manufacturing technologies, such as fused deposition modelling (FDM) and stereolithography, are increasingly providing higher resolution fabrication capabilities from a broadening set of materials. In this paper, such techniques are exploited to free the mechanism from being constrained to a single plane. This further allows the use of a single actuator, whilst maintaining the requisite symmetry. A pure rotation scanner has previously been designed by Lee et al. [28]. However, this mechanism employed two PEAs, and also did not possess rotational symmetry in its design; making it susceptible to drift of the rotation centre. Cathie and Janky also proposed a flexure-based rotation stage, however, a method of actuation was not provided [29]. Planar XY h positioners could also be utilised for this purpose, however this would require added consideration of actuation, sensing and feedback schemes for the linear axes to compensate for drift. The remainder of the paper is structured as follows: in Section 2 the design of the mechanism, and its subsequent optimisation, are provided. Section 3 details a computational examination of the optimised design, with a focus on the working range and undesirable cross-coupled motions, and their variation as manufacturing errors are introduced. Results from experimentation performed on a manufactured prototype are presented in Section 4. Finally, considerations for practical implementations are discussed in Section 5.
2. Mechanism design Fig. 1a shows a rigid body schematic of the underlying amplification structure for the proposed rotational scanner. The principal design element is the four bar crank-rocker mechanism, from which the output rotation is taken to be at the crankbase joint. An input lever provides amplification to the vertical input, whilst also ensuring that the input is horizontally aligned with the output hinge. The rigid body equivalent to the compliant mechanism is immobile due to this lever. Rotational symmetry of the design is achieved by the addition of a duplicate of the underlying structure, which replaces the output hinge. The duplicate is rotated by 180◦ about the output point, as illustrated in Fig. 1b. With a symmetric input, the output rotation is then centred on the crank linkage. Fig. 2 shows the compliant design basis of the proposed rotational mechanism. The mechanism has been separated into two levels: the upper level provides the functionality of the schematic in Fig. 1, whilst the lower level provides the symmetric inputs to the upper level. The input bridge amplifier stage of the lower level transforms the expansion of a PEA into transverse motions to be transferred to the upper stage. In particular, the two double parallelogram linkages of the input stage ensure that the PEA is not exposed to shearing forces. Due to the duplication of the structure to achieve the symmetry of the design and the removal of the flexure hinge, the output link has an added (translational) degree of freedom when the input is fixed. As a consequence, undesirable vibrational modes exist at low frequencies, which are discussed within the computational analysis in Section 3. To remedy this, cantilevers have
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Fig. 1. Rigid body schematic of the rotation stage: (a) underlying structure, (b) rotationally symmetric version with virtual rotation centre.
been added to the output platform which resist this translation whilst permitting the rotational output. To reduce the impedance of this added stiffness on the output, the cantilevers were made to terminate as close to the rotation centre as possible. The lower section was chosen to be 15 mm thick, whilst the upper section was chosen to be 12 mm thick. A gap of 3 mm separated the two levels. The two-level flexure design employs four different types of compliant elements. All hinges within the crank-rocker sections employ circular flexure hinges, whilst the remaining hinges are filleted leaf hinges. Similarly, the cantilevers have filleted radii at their attachment points. The input bridge elements are reinforced to reduce axial elongation, concentrating rotation to the thinner sections of the bridge structures. The two-level design can be contrasted with a complementary design, using a single actuator and the same underlying structure, but constructed within a single plane, as shown in Fig. 3. To achieve a comparable output range the layout is dominated by input amplification levers, which constrains the size of the output platform, and increases the mechanism size and amount of void space. Due to the lack of design symmetry, careful optimisation of the geometric parameters would be required to increase rotational precision, and the incidence of any cross-coupled drift would be much more sensitive to manufacturing defects. 2.1. Design optimisation The goal of the mechanism design process was to produce a prototype with the largest possible angular range, whilst retaining a small footprint. The design was optimised employing the response surface optimisation technique, using computational analysis to study the dependence of the mechanism’s behaviour on multiple design parameters. From Fig. 1a, it can be seen that there are three sources of amplification involved during the transformation of the input to the output. These have been named the input amplification, the transfer amplification, and the crank-rocker amplification for the remainder of this paper. The input amplification depends on the geometry of the link possessing the hinges labelled a, b and c in Fig. 1a, whilst the transfer amplification depends on the hinges labelled c, d and e. The instantaneous amplifications at the rest position, with the lengths defined in Figs. 2b and c, are given by: aIA =
LCO LCI
(1a)
aT =
LUR LUT
(1b)
aCR =
LUR . LUC
(1c)
These amplifications, combined with the bridge amplifier thickness t and angle 0, also defined in Fig. 2b, were the design parameters considered during the optimisation. As the PEA is constrained to be at the centre of the lower stage, the lengths defined in Fig. 2 are further subjected to the constraint: LUC − LUR + LUT + LCO − LCI = 0.
(2)
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Fig. 2. Flexure design of rotation stage: (a) 3D view with upper section cutout, (b) cross-section of lower section, (c) cross-section of upper section.
Considering the size requirements of the optical measurement equipment to be used in experimental work, the output platform width was fixed to be 25 mm (LUC = 12.5 mm). Hence, the remaining dimensions of the upper stage are fully determined by Eqs. (1a), (1b), (1c) and (2). The optimisation employed a static structural finite element analysis to estimate the angular output at the full driving voltage. Computational simulations were performed within the ANSYS software package. A proximity based meshing method was utilised, which ensured that the compliant elements of the mechanism had a more refined mesh. The meshed model, as shown in Fig. 4, was re-meshed for each set of design parameters, however all meshes contained in excess of 200,000 nodes and 115,000 elements. Displacement inputs of 5.8 lm were applied to the two input surfaces on the lower level, representing a full stroke of 11.6 lm, matching the nominal capabilities of the NEC/Tokin AE0505D16F PEA used for experimentation. The average displacement of nodes on the surface of a cylindrical pilot hole through the centre of the output stage was used to determine the resultant mechanism output. The nodes on the side boundaries of the mechanism were fixed within these analyses. Material properties of Polyjet RGD720, which was used in the 3D printing of the experimental prototype were assumed in the analysis. The material has a density of approximately 1.19 g/cm3 , an elastic modulus of 2.5 GPa and 50 MPa tensile strength.
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Fig. 3. Design of flexure mechanism if constrained to planar manufacturing.
Two analyses were employed to estimate the output rotation. The first of these, employing the constraints above was used to find the raw output in response to the 11.6 lm input. However, due to the input stiffness of the mechanism kin , the actuator’s range is reduced according to: g=
kPEA . kPEA + kin
(3)
The second analysis estimated the input stiffness by providing 1 N forces to each of the input surfaces. The average displacement of one surface was then used to determine the stiffness: kin =
Fin . 2xin
Fig. 4. Cross-sections of meshed model used for computational design optimisation and validation: (a) lower section, (b) upper section.
(4)
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L. Clark, et al. / Mechanism and Machine Theory 105 (2016) 129–144 Table 1 Design parameter ranges considered for optimisation. Parameter Input thickness (mm) Input angle Crank-rocker amplification Transfer amplification Input amplification
t 0 a CR aT a IA
Min.
Max.
Opt. value
1.0 8.0◦ 3.0 1.5 1.5
2.0 25.0◦ 6.0 5.0 5.0
1.000 11.61◦ 6.000 2.167 2.358
Hence, the optimisation sought to maximise the stiffness-reduced output: hpred = ghFEA .
(5)
The optimisation proceeded in three phases to minimise the total number of design points to be simulated. Initially, it was assumed that the amplification of the input stage would be decoupled from the kinematics of the upper level. Hence, optimisations of the input stage (t and 0) were carried out separately to the upper level (a CR , a T , and a IA ). Table 1 shows the ranges considered for each design parameter. In particular, the maximum value of a CR was chosen based on the maximum allowable dimensions of the mechanism. Other candidate variables considered for optimisation were the separation of the central cantilevers on the output stage and their lengths. It was observed that the output stiffness with respect to rotation was minimised by reducing this separation (thus increasing the output range), and it was subsequently set to its minimum value. Lengthening the upper level cantilevers led the reduction of the output stiffness, but it also reduced the stiffness with respect to undesirable coupled motion in the X direction, which caused the corresponding natural mode (shown later in Fig. 6b) to have a lower frequency. The cantilever length was chosen such that the rotational mode remained as the fundamental mode of the mechanism (as discussed in the next section). The first two optimisation operations, carried out sequentially, resulted in the generation of 228 and 437 feasible datapoints, respectively. Parameters were chosen by finding the design variables resulting in the maximum value predicted by a surface of best fit to the generated points. Surfaces were fitted using the Krige interpolation method, whilst a genetic algorithm was employed to find the maximising parameters [30]. After performing the two optimisations, the assumption that the two phases were decoupled was found to be inaccurate. Consequently, optimisation over all parameters was required. However, as shown in Fig. 5c, the angular range increased with
Fig. 5. Optimality of final design — cross-sections of response surface: variation with (a) input angle (0), (b) input thickness (t), (c) crank-rocker amplification (a CR ), (d) transfer amplification (a T ), (e) input amplification (a IA ).
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Fig. 6. First four natural modes of the proposed angular stage: (a) h motion (217 Hz), (b) X motion (219 Hz), (c)–(d) out-of-plane motions (379 Hz, 477 Hz).
the crank-rocker amplification, whereupon it was set to its maximum value. A final optimisation over the remaining four parameters, employing 442 feasible simulations, was used to determine the final design parameters, as given in Table 1. Fig. 5 shows cross-sections of the response surface with respect to each of the design parameters considered, demonstrating the optimality of the design.
3. Computational verification The static structural analysis was repeated to verify the performance of the optimised design candidate. The model was replaced with a finer mesh, consisting of at least 475,000 nodes and 290,000 elements. The input stiffness was instead evaluated as the slope of the line of best fit to responses to 1 N, 2 N and 3 N force inputs. All simulations were repeated sixteen times, re-meshed each time, to improve the numerical stability of the solutions generated. Table 2 shows the results of this static analysis, including the translational motion (drift) of the rotation centre. For comparison, the performance is also provided for the case where the central cantilevers were omitted from the design. It can be seen that the increase in output stiffness due to the introduction of the cantilevers reduces the working range. As the standard error in the predicted drift is comparable in magnitude to the mean drift, it is likely that the computed drift for each simulation is due to asymmetry in the meshed models, rather than a systematic error in the design. A modal analysis was also performed on the meshed model of the mechanism. In order to avoid excessive computation time, the number of nodes was reduced to 172,000, with 101,000 elements. A linear spring was added to the model between the two input surfaces to simulate the presence of the PEA. Fig. 6 shows the first four modes of the mechanism predicted by the computational analysis. In particular, it can be seen that the first natural mode corresponds to the desired angular motion. As was mentioned in the previous section, the second mode is a motion in the X direction. Its natural frequency could be further increased above the h mode by increasing the stiffnesses of the central cantilevers. Fig. 7 shows the first four modes of the mechanism if the central cantilevers were omitted. The rotational mode is preceded by two translational modes, caused by the crank-rocker section being underconstrained. Without the cantilevers, these translational motions are likely to be excited and coupled at lower frequencies, thus decreasing the mechanism’s operational bandwidth. 3.1. Drift of rotation centre The small amount of drift and out-of-plane rotation predicted in Table 2 stems from the rotational symmetry of the design. However, in practice this symmetry cannot be realised perfectly, due in part to errors in assembly, manufacturing and apparatus alignment. In this section, the impact from two of these sources, improper actuator mounting and manufacturing errors, on the drift of the output stage are studied through computational simulation. Table 2 Computational predictions of maximum mechanism output. Final design
h kin xdrift ydrift zdrift Roll Pitch
Cantilevers omitted
Mean
Std. err.
Mean
Std. err
2.7962 mrad 1.6802 × 106 Nm −1 −0.1141 nm 0.1077 nm 0.3622 lm −25.14 nrad 2.760 nrad
0.161 lrad 75.8 Nm −1 1.16 nm 0.179 nm 0.300 nm 22.5 nrad 22.1 nrad
3.7886 mrad 1.4823 × 106 Nm −1 −3.430 nm 3.395 nm 3.332 lm −57.44 nrad 88.03 nrad
0.210 lrad 80.9 Nm −1 7.84 nm 6.44 nm 3.64 nm 135 nrad 21.9 nrad
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Fig. 7. Natural modes of the mechanism with the central cantilever omitted: (a)–(b) X–Y motions (80 Hz, 107 Hz), (c) h motion (188 Hz), (d) out-of-plane motion (225 Hz).
3.1.1. Deviation in input position Incorrect mounting of the actuator during the mechanism assembly could cause the symmetry of the input to be lost, thereby causing motion along undesirable axes in the output response. To simulate this effect, the CAD model of the mechanism was altered so that the input would be localised to two surfaces. These surfaces were the size of the faces of the PEA used for experimentation. The positions of these two surfaces could be altered in the Y and Z directions from the centre location, as shown in Fig. 8. The computational analysis was then repeated for this modified model, where a force along the axis of the line separating the surfaces was applied as the two inputs. The force was computed from the results of Table 2, such that the predicted h value was produced when there was no deviation in input position. The simulation was repeated for 801 combinations of input positions, with the Y and Z positions on both faces uniformly randomly selected from the interval [−2, 2]. Fig. 9 shows the results of the analysis. The drift is plotted against the variable which had the greatest Pearson correlation coefficient when using a linear regression. The raw Y and Z deviations of the inputs, as well as the offset and difference, illustrated in Fig. 8, were considered. The offset is the deviation of the mean position of the input surfaces, whilst the difference is the projection of the vector from the left (the surface with negative X coordinates) to the right input surface onto the Y–Z plane. The drift of the stage in the X–Y plane was confined within 22 nm. The drift being relatively small is likely to be due to the redundancy and number of compliant elements between the input and output stages. Whilst the correlation is weak, it was expected that the strongest dependence of the drift would be on the Y offset, as the offset in the Z direction does not contribute to a loss in symmetry. Similarly, symmetry would be maintained with a difference in Y position without an offset. The Y difference can be seen to have a strong influence on the angular output, which also appears to be inversely related to the Z drift of the stage. The coupled pitch and roll of the stage were also computed, which both remained beneath 280 nrad for all simulated combinations. No strong correlations were observed between these rotations and the input positions. 3.1.2. Manufacturing noise The manufacturer of the 3D printer, used for manufacturing of the prototype detailed in Section 4, specifies a build resolution of approximately 42 lm in the X and Y directions, and 16 lm in the Z direction, with an overall accuracy below 85 lm for parts smaller than 50 mm, or 200 lm for the largest printable object. Consequently, inaccuracies in manufacturing or surface finishing could lead to asymmetry in the fabricated mechanism.
Fig. 8. Setup of CAD geometry to simulate deviation in the actuator positioning.
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Fig. 9. Drift of the rotation centre due to variation in the actuator positioning: (a) drift in the X and Y directions, (b) variation in the output rotation, and drift in the Z direction.
These effects have been simulated by adding noise to the surface of the compliant elements of the mechanism, which is illustrated in Fig. 10. This was achieved by subdividing the faces of these elements, before applying Gaussian noise along the surface normal to a triangulated model of the mechanism. The subdivided facets had a height of 1.5 mm, and a width of 0.6 mm for the circular hinges, 1.0 mm for leaf hinges, 2.0 mm for cantilevers, and 1.8 mm for the bridge amplifier. In total, the locations of 2412 vertices were randomly displaced during this procedure. Noise with standard deviations from 0 lm to 100 lm in 20 lm increments was considered. Due to the procedure used to triangulate, distort and reimport the geometry, for the 0 lm case, 0.1 lm noise was employed to preserve the facets of the geometry and allow a direct comparison with the other cases. Due to the complexity of the deformed surfaces, a different meshing method had to be employed to accommodate the changed geometry. Meshes with approximately 730,000 nodes and 500,000 elements were used for most models, however at higher noise levels a mesh with lower complexity was sometimes required due to the difficulty in meshing these geometries. Each analysis was repeated ten times to reduce the dependence on the meshing, and all six cases were repeated five times with differing random number generator seeds. A displacement input was utilised, reduced from the maximum PEA stroke to reflect the input stiffness, using the results of Table 2. The predicted drift in the X–Y plane of the rotation centre, as well as pitching and rolling motions of the output stage, are shown in Fig. 11. It has been assumed that the manufacturer’s specified surface accuracy represents the 95% confidence interval, hence two standard deviations have been used to represent the surface noise. The translational drift, shown together with error bars indicating the standard error, have been computed using: r=
x¯ 2 + y¯ 2
10 2 2
1 xi (xi − x¯ ) yi (yi − y¯ ) dr = + 10 r r
(6)
(7)
i=1
where x¯ =
1 10
10 i=1
xi is the mean drift in the X direction. Whilst positive and negative deviations were observed, the figure shows
their absolute values.
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Fig. 10. Modified CAD geometry showing subdivided facets and simulated surface noise added to compliant elements: (a) upper level flexure hinge, (b) lower level cantilever and bridge element. The upper edge of the surfaces which have been deformed for the elements pictured has been indicated. The standard deviation of the noise added to the surfaces of this model was 100 lm.
Whilst there is a large variation in the drift even at a single noise level, it is clear that the maximum drift increases with noise. The maximum translational drift was observed to lie within an exponential envelope, whilst the largest rotation varied linearly with increasing noise. 4. Experimental prototype A prototype of the rotational scanner has been fabricated using an Objet Eden 260 V 3D printer. Photographs of the prototype showing the upper and lower levels are provided in Fig. 12. The bounding box of this prototype has dimensions 128 mm × 153 mm × 30 mm. A detailed photograph of the surface finish near the output stage is shown in Fig. 12c, where the horizontal layering from the printing process is evident. To arrive at this prototype, the original flexure design shown in Fig. 2 was modified to remove material from the base which would not experience large deformations. This has the effect of reducing the prototype’s mass, and consequently lowered the printing time and material cost. In addition, as shown in Fig. 12b, four slots were introduced to provide access to the 3 mm gap separating the two levels, primarily to assist the removal of the support material required during fabrication. One of the slots would also provide a passage for the input cables to the PEA. Two grooves were manufactured within the output platform, which allow the installation of a press-fit reflector mount. The mechanism was mounted within the apparatus shown in Fig. 13. Actuation was provided by a NEC/Tokin AE0505D16F PEA capable of a full stroke of 11.6±1.3 lm at a 100 V input voltage. The PEA was driven by a voltage amplifier (Thorlabs MDT693B), which received input from the output of a 16-bit digital-to-analogue converter (DAC) from a control computer. Measurement of the angular position was provided by two sensors, a digital autocollimator (Microradian T30D) and a laser interferometer (Zygo HSPMI 2001-02). Compared to the interferometer, the autocollimator allowed for angular measurement over a much larger range, albeit at reduced resolution and sampling rate. The steady state RMS noise was measured to be 86.7 nrad using the interferometer, and 1.56 lrad for the autocollimator. The preload to the actuator was provided by unscrewing a steel M6 hex-head screw/nut pair placed between the actuator and an input surface. Small props were employed within the PEA mountings to aid in their location, and to reduce any deviation in the actuator positioning. The added mass of the reflectors, in addition to the mechanism being fixed at the two diagonally-opposite screw hole locations shown in Fig. 12b, rather than over the entirety of the side boundaries, will lead to a reduction in performance compared to that predicted by the computational model in the previous section. These analyses have therefore been repeated, with the removed material, PEA mountings, reflectors and their mounting represented within the model. Nodes were fixed at only the surfaces of the two cylindrical screw holes. Employing this revised model, the angular range was predicted to be reduced to 2.4288 mrad, whilst the natural frequency of the rotational mode was predicted to reduce to 185.7 Hz. 4.1. Experimental validation To determine the output range of the angular scanner, a smoothed triangle wave open-loop input voltage between 0 V and 100 V was applied to the PEA. The measured position during this operation is shown in Fig. 14. The angular range of the mechanism was 2.540 mrad, 4.6% larger than the computational prediction. However, this discrepancy is within the 11.5% uncertainty
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Fig. 11. Introduction of output drift due to simulated surface noise: (a) X–Y plane, (b) out-of-plane rotation. The red coloured markers in (a) indicate simulations which employed a mesh with fewer than 700,000 nodes. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
in PEA stroke stated by the manufacturer. Other errors may also be present within the computational model, such as due to the unmodelled contact stiffness condition at the input, and inaccuracies in the material properties assumed within the analysis. The first natural frequency of the rotational scanner has been determined. A sinusoidal sweep signal with its frequency increasing exponentially from 0.05 Hz to 400 Hz has been applied to the PEA input, with a peak-to-peak actuation magnitude of 1.0 V. As the sampling rate of the autocollimator was limited, data for this experiment was gathered using the interferometer, at a control frequency of 800 Hz. The frequency response to this input is shown in Fig. 15. The first resonant frequency was encountered at 175.3 Hz. The discrepancy between the computational prediction (185.7 Hz) is likely to be due to other modelling errors, such as an error in the estimated PEA stiffness. The resolution of the angular positioning was tested by measuring the feedback control response to nanoradian scale steps. A proportional-integral controller was employed, with a feedforward component approximating the result of Fig. 14 as a proportional input. Feedback was provided by the interferometer, with the control computations taking place at 800 Hz. Fig. 16 shows the response to 250 nm steps at 1.33 Hz. The RMS tracking error for the time shown was 90.6 nrad, with a maximum error of 371.8 nrad. At this scale, the tracking error was limited by the noise within interferometer measurement, as well as the resolution of the computer’s DAC, where the quantisation step was approximately 38.8 nrad. 4.2. Influence of 3D printed fabrication Previous research, including the results shown in this paper, have demonstrated the capability for 3D printed photopolymer mechanisms to exhibit positioning with precision at the sensor noise level [31, 32]. However, it was observed that the output of the mechanism would drift over time, most likely due to the creep of the mechanism under the stress of the preload. This drift is shown for the 24 hours immediately after the application of the preload in Fig. 17. It should be noted that the environmental climate control was disabled overnight between hours 8 and 22, which further shows the thermal dependence of the drift. Due to this drift, there are several factors that must be taken into consideration when employing such mechanisms. Firstly, as these mechanisms drift, their working range and kinematics will change. Consequently, this may require periodic adjustment of
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Fig. 12. Close-up photographs of the fabricated rotational scanner prototype: (a) upper level, (b) lower level, (c) surface finish near output stage.
the preload, and the use of adaptive control schemes to compensate for the varying mechanical characteristics. In addition, it has also been observed that mechanisms fabricated from photopolymer exhibit hysteresis in their kinematics, which in combination with PEA hysteresis, need to be factored into controller design. It is expected that these issues could be mitigated through the use of a 3D printed metal for fabrication, which is also likely to increase the bandwidth of achievable motions. 5. Discussion Manufacturing error, in the form of surface noise, was shown to potentially lead to a linearly increasing amount of drift. As shown in Fig. 12c, the actual surface profile was quite smooth, apart from variations resulting from, and of similar size to, the (16 lm) layering of the printing process. It is important to note that, at the 80 lm noise level (assumed to be comparable to the specified build accuracy), the maximum predicted pitch and roll drift motions were less than 0.3% of the full range output
Fig. 13. Photograph of the experimental apparatus.
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Fig. 14. Experimentally determined working range of the rotational scanner.
rotation. However, this analysis was limited to variations fixed to the chosen grid on the compliant elements, and did not consider material anisotropy, which could further increase the amount of drift. The drift due to variation in the input position was observed to be negligible in comparison to the drift due to manufacturing error. In particular, the drift observed in the X and Y axes, as well as pitch and roll motions, shown in Fig. 9, was of a comparable magnitude to the standard deviation in the sample, showing that the mechanism has a strong ability to reject variations in the actuator positioning. It is also worth noting that the 2 mm variation in all directions considered is likely to be an exaggeration, and tighter assembly tolerances would be encountered in practice. Whilst the setup of the computational simulation, in particular the use of a force input, may not be a completely accurate depiction of the true behaviour in practice, the analysis has shown that these errors lead to a small amount of drift, which should not change dramatically if the computational model were altered. It should also be noted that in a practical experimental environment, the performance would be affected by the setup of the mechanism and its accompanying instruments. In particular, an offset between the true X axis and a sensor’s median axis will give rise to Abbe errors within measurement, as illustrated in Fig. 18. As a consequence of this offset, a spurious X displacement would be measured as: xAbbe = d tan h ≈ dh
(8)
where d is the offset between the true and measured axes. This error can form a significant component of the measured output; at the maximum measured 2.540 mrad, an offset of only 0.4 mm could cause a 1 lm error. As this error increases linearly with the output angle, it is indistinguishable from the drift of the rotation centre. Consequently, calibration and correction of Abbe
Fig. 15. Frequency response of the rotational scanner.
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Fig. 16. Controlled response to a nanoradian scale step input.
errors would necessarily also compensate for any drift. It is for this reason that experimental measurement of the drift has not been performed.
6. Conclusion This paper has presented the design of a compliant pure rotation stage which employs a single actuator, whilst possessing rotational symmetry in its design to minimise undesirable drift motions. The mechanism’s geometric parameters were optimised to maximise the working range, whilst retaining a small physical size. Computational analysis was performed to determine the capabilities of the mechanism, and in particular the cross-coupled motions were analysed with respect to simulated manufacturing and assembly errors, which cause a loss of design symmetry. The drift due to errors in the mounting of the input was found to be negligible when considered against those resulting from manufacturing errors. Nevertheless, pitch and roll drift motions were found to be insignificant in comparison to the principal yaw motion. Furthermore, the drift in the X–Y plane will be compensated simultaneously with calibration for Abbe errors, which are likely to be larger in magnitude. Experimental results demonstrated the ability to produce an angular output over a 2.540 mrad range, with a positioning resolution comparable to the sensor noise level, and a first resonant frequency of 175.3 Hz, which compared favourably with computational predictions. However, these results for the fabricated prototype were reduced from the initial ideal model, and it is expected that performance can be improved by reducing the load mass and employing more rigid fixings at the base. The output exhibited a long term drift, due to the creep effect of the printed photopolymer material, which is likely to be mitigated if a metal were alternatively used for fabrication.
Fig. 17. Drift of mechanism after application of preload.
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Fig. 18. Abbe error resulting from experimental offset.
Acknowledgements This work was supported by ARC LIEF (Grants LE0347024 and LE0775692), and ARC Discovery Projects (Grant DP140104019). The writing of this manuscript was supported by the Monash Postgraduate Publications Award. Fabrication of the compliant mechanism was performed at the Melbourne Centre for Nanofabrication (MCN) in the Victorian Node of the Australian National Fabrication Facility (ANFF). The authors would like to thank Mr. Soon Hock Ng from the Melbourne Centre for Nanofabrication (MCN) for his assistance in the fabrication of the compliant mechanism.
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