REVIEW OF SCIENTIFIC INSTRUMENTS 85, 046102 (2014)

Note: A piezo tip/tilt platform: Structure, kinematics, and experiments Z. Du,a) Y. Su,a) W. Yang, and W. Dongb) State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150080, China

(Received 6 December 2013; accepted 20 March 2014; published online 3 April 2014) A Piezo Tip/Tilt Platform (PT2 P) is presented with its structure, kinematics, and preliminary experiments. Two essential models of the presented PT2 P, an equivalent hinge of the flexure hinge and a simplified model of the transmission mechanism, are discussed with the analysis on the structure of the PT2 P. Based on these models, the inverse kinematics of the PT2 P is derived. Two experiments are conducted on a prototype of the PT2 P. The kinematic model is verified with experimental results, which also indicate that the resolution and the repeatability of the PT2 P is, respectively, better than 0.50 μrad and 0.25 μrad. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4870062] Piezo Tip/Tilt Platforms (PT2 Ps) have been widely applied in many areas to provide tip/tilt motions of very small angles.1, 2 In recent studies, there are mainly two types of PT2 Ps: the three-piezoelectric-stack-driven PT2 Ps,3–5 which are capable to provide necessary actuations for tip/tilt motions with lower costs, and the four-piezoelectric-stack-driven PT2 Ps,6–8 which have better stabilities by applying differential designs but need sophisticated control methods. In this note, we present a four-piezoelectric-stack-driven PT2 P with its structure, kinematics, and preliminary experiments. The structure of the PT2 P, shown in Figures 1(a) and 1(b), consists of three parts: the base, the actuators, and the cover. The base contains four slots for mounting the actuators. On the bottom of the slots, set-screws are adopted to adjust the pre-load forces between the actuators and the cover. The cover is an integration of the platform and the flexure hinge. The flexure hinge is manufactured by cutting two stagger rows of slits on the cover. This structure can be modelled as a continuous beam supported by laterally and rotationally fixed supports. With this model, the bend deformation is mainly generated on the single flexure,9 which enables the platform to provide the piston and the tip/tilt motions when applying the even and the uneven forces, respectively (an example of the platform rotating when applying uneven forces is shown in Figure 2(b)). Therefore, the flexure hinge is simplified as an equivalent hinge of three Degree-of-Freedoms (DoFs). With the presented structure, a actuator pushes the platform with its steel-ball. This transmission mechanism is described by the point-contact-without-friction model,10 with which only the component of the velocity normal to the contact plane is transmitted from the steel-ball to the platform. But since the motion of the platform is very small, the contact constrain is simplified by letting the location of the contact point be constant and letting the magnitude of the velocity transmitted to the platform equal to that of the steel-ball as shown in Figure 1(c).

a) Z. Du and Y. Su contributed equally to this work. b) Author to whom correspondence should be addressed. Electronic mail:

[email protected] 0034-6748/2014/85(4)/046102/3/$30.00

Based on the above analysis, we can build the kinematic model of the PT2 P with the coordinate systems illustrated in Figure 1(b). The world frame {N} is an inertial frame whose origin is conveniently chosen at the geometry center of the flexure hinge. The platform frame {P} is fixed on the platform with its origin locating at the origin of the world frame. Let the vector ci ∈ R3 represent the location of the ith contact point (i.e., the point where the steel-ball of the ith actuator contacts the platform with i indexing the actuators). With the simplified contact constrain, it is constant and can be estimated by measuring the structural design model of the PT2 P. Let νp = [vpT ωpT ]T ∈ R6 denote the twist of the platform consisting of the translational velocity vp ∈ R3 and the angular velocity ωp ∈ R3 . It is constrained by the flexure hinge with ˙ νp = Jh p, Jh = [03×2 I3×3 03×1 ]T ∈ R3×6 ,

(1)

where p˙ = [˙z r˙x r˙y ]T denotes the velocity of the flexure hinge. Therefore, the platform actually has three DoFs and, one can adopt the motion of the flexure hinge to describe the motion of the platform. Define the vector vc = [vc,1 vc,2 vc,3 vc,4 ] ∈ R4 with vc,i ∈ R denoting the component of the velocity normal to the platform at the ith contact point (i.e., the component velocity along z axis in frame {P}). It is related to the twist of the platform by vc = Jp νp , ⎛ 0 0 ⎜ Jp = ⎝ 0 0

1 ···

c12

−c11

0

1

c42

−c41

0



(2)

⎟ 4×6 ⎠∈R ,

where ci1 and ci2 are the components of the vector ci denoted by ci = [ci1 ci2 ci3 ]T . Let q˙ = [q˙1 q˙2 q˙3 q˙4 ]T represent the magnitudes of the expansion velocities of the actuators, where q˙i ∈ R denotes the magnitudes of the expansion velocity of the ith actuator. With the simplified contact constrain, it is related to the vector

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Rev. Sci. Instrum. 85, 046102 (2014)

FIG. 1. The component of the presented PT2 P is shown in (a). As whose detailed structure shown in (b), the PT2 P consists of three parts: the base (coloured in green), four piezoelectric stack actuators (red), and the cover (grey). The kinematic model of the PT2 P is shown in (c). The flexure hinge is simplified as a hinge of three DoFs and, the transmission mechanism between the steel-balls of the actuators and the platform is described using a simplified point-contact-withoutfriction model, with which the velocity transmitted to the platform (vc,i ) equals to the expansion velocity of the actuator (q˙i ) as shown in (c).

vc by ˙ vc = q.

(3)

Therefore, the inverse kinematics is gained by substituting (1) and (2) into (3), ˙ q˙ = Jp Jh p.

(4)

Since (4) is gained using the screw theory, the results are also applicable for the infinitesimal motions, δq = Jp Jh δp,

(5)

where δq = [δq1 δq2 δq3 δq4 ]T and δp = [δz δrx δry ]T , respectively, denote the infinitesimal expansions of the actuators and the infinitesimal displacements of the flexure hinge. Therefore, given the respective initial values p0 and q0 , the required expansions of actuators for moving the flexure hinge to a desired configuration is accounted by solving (5) subject to δq = q − q0 and δp = p − p0 . We adopted four piezoelectric stack actuators (i.e., PSt 150/7/40 VS12) in the prototype of the PT2 P. The expansion range of these actuators is 0–38 μm, which decreases to 0– 35 μm under the pre-load condition. Each actuator contains a resistance strain gauge sensor to output expansion of the actuator to a servo driver. Two servo drivers (i.e., XMT XE-517) are adopted. Each driver controls two actuators. The driver takes the input of desired expansions of the actuators and uses

FIG. 2. The design of the flexure hinge is shown in (a). A finite element analysis result on force-deformation of the flexure hinge is shown in (b), where the flexure hinge rotates 678 μrad when applied 200 N force (coloured in red).

the Proportional-Integral-Derivative (PID) control methods to follow the input. Two experiments were conducted to verify the kinematic model and investigate the resolution and the repeatability of the PT2 P. The setup of the experiments is shown in Figures 3(a) and 3(b). A calibration block was mounted on the platform with a one channel capacitance gauge (i.e., MicroEpsilon RS6500) measuring the translation of the measurement point on the calibration block. The measuring range and the resolution of the capacitance gauge are, respectively, 200 μm and 0.6 nm. We measured the repeatability of the capacitance gauge with the standard deviation of 5 min measurements taken at the initial point (p0 = (20 μm, 0 μrad,

7 and the capacFIG. 3. The experiment setup is shown in (a). The PT2 P  6 were mounted on a air floated platform  8 . The PTZ itance gauge probe  4 and  5 were controlled by a computer  3 . The capacitance gauge  1 drivers  2 for reading the measurements. Panel (b) dewas connected to a computer  tails the capacitance gauge probe and the calibration block. Panel (c) details the calibration block and the nominal location of the measurement point.

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Rev. Sci. Instrum. 85, 046102 (2014) TABLE III. Repeatability experiment results.

TABLE I. Parameters of the kinematic model. Symbol c1 c2 c3 c4 t

Nominal value

Calibrated value

Unit

Test points

(15.0a , 0.0b , . . . c ) (0.0b , −15.0a , . . . c ) (−15.0a , 0.0b , . . . c ) (0.0b , 15.0a , . . . c ) (−31.8a , 31.8a , . . . c )

(15.1, 0.0, . . . ) (0.0, −15.0, . . . ) (− 15.0, 0.0, . . . ) (0.0, 15.1, . . . ) (−32, 1, 30.2, . . . )

mm mm mm mm mm

std(δm)a

a

Values are parameters identified before experiments. Values are constants as they are easily guaranteed in manufacturing. c Values are arbitrary as they do not affect the kinematic model. b

0 μrad)). We measured twenty times, the result was averagely 4 nm ranging from 2 nm to 7 nm. The location of the measurement point is shown in Figure 3(c). The measurement of the capacitance gauge is related to the motion of the platform by δm = Jt Jh δp,

(6)

where δm = m − m0 ∈ R denotes the incremental measurement caused by the incremental motion with m0 and m, respectively, denoting the measurement taken at the initial point and the measurement after the movement, the matrix Jt maps the incremental motion of the platform to the incremental measurement Jt = −[0 0 1 t2 − t1 0], where t = [t1 t2 t3 ]T ∈ R3 denotes the location of the measurement point. In order to identify the parameters of (5) and (6), seven points (i.e., displacements of the platform) randomly generated to evenly cover the workspace were substituted into (5) with the nominal parameters, listed in Table I, to account for the control commands (i.e., expansions of actuators). Three trials were taken at each point. The measurements and the commands were recorded to identify the parameters using the least square method. The identified parameters, listed in Table I, were used to account for commands and predict measurements in the following experiments. To investigate the resolution, five test points were selected to cover the workspace. Five trials were carried out at each point. In each trial, the PT2 P was assigned to conduct a small differential movement after reaching the test point from the initial point. We selected a differential rotation of 0.5 μrad because the expected incremental measurements of it (15 nm), approximately being double to the maximal repeatability of the capacitance gauge (7 nm), was minimal and reliable. The results are shown in Table II. The averages of the actual measurements taken at the test points m1 and that after TABLE II. Resolution experiment results. Test points m1 ˆ1 m m2 ˆ2 m δm ˆ δm

0,0

600,0

900,0

0,600

0,900

Unit

0.000 0.000 −0.015 −0.015

−18.023 −18.180 −18.039 −18.195

−27.163 −27.270 −27.177 −27.285

19.347 19.260 19.362 19.276

29.063 28.890 29.079 28.906

μm nm μm nm

−16 −15

−14 −15

15 16

16 16

nm nm

−15 −15

0,0

600,0

900,0

0,600

0,900

Unit

4/0.12

6/0.19

8/0.25

4/0.13

7/0.23

nm/μrad

a

The standard deviation of the incremental measurements (with unit nm) is translated to the rotation of the platform (with unit μrad) with (6).

the incremental movements m2 are used to calculate the incremental measurements δm = m2 − m1 , which should equal ˆ calculated with (6). The paired-sample to the expectations δ m t-test indicates that the incremental measurements equals to the expectations at the 5% significance level, which suggests the PT2 P is capable to reliably provide small motion of 0.50 ˆ 1 and μrad. To verify the kinematic model, its predictions, m ˆ 2 , are accounted and shown in Table II. The paired-sample tm test indicates that the predictions equal to the actual measureˆ 1 and m2 = m ˆ 2 ) at the 5% significance ments (i.e., m1 = m level. The same test points were adopted in the experiment to investigate the repeatability. For each test point, the PT2 P was assigned to conduct reciprocate incremental motions of 50 μrad at the test point for five times. The standard deviations of the differential measurements are accounted to indicate the repeatability. As shown in Table III, the maximum value is 7 nm (while average repeatability of the capacitance gauge is 4 nm), which is 0.25 μrad rotation of the platform. In conclusion, we have developed a PT2 P and derived its kinematic model. The experiment results have verified the kinematic model and shown that the resolution and the repeatability of the PT2 P is, respectively, better than 0.50 μrad and 0.25 μrad. It is worth to point out that limited by the vibration of the experiment environment, the repeatability of the capacitance gauge (maximal 7 nm) was too large to measure higher resolution of the PT2 P. In future work, we plan to investigate resolution and dynamic characteristics of the PT2 P with vibration isolator. This research is funded by the Chinese Advanced Armament Research Project of 12th Five-year Plan (Grant No. 51320030201), the Fundamental Research Funds for Central Universities (HIT.NSRIF.2014061), the HIT Disruptive Innovation Program, and the HIT Overseas Talents Introduction Program. 1 C. Mio, T. Gong, A. Terray, and D. W. M. Marr, Rev. Sci. Instrum. 71, 2196

(2000). Du, R. Shi, and W. Dong, IEEE Trans. Robotics 30, 131 (2014). 3 J.-C. Tsai, L.-C. Lu, W.-C. Hsu, C.-W. Sun, and M. C. Wu, J. Micromech. Microeng. 18, 015015 (2008). 4 B. Shao, L. Chen, W. Rong, C. Ru, and M. Xu, Smart Mater. Struct. 18, 035009 (2009). 5 W. Dong, M. Gauthier, C. Lenders, and P. Lambert, J. Micromech. Microeng. 22, 057001 (2012). 6 Y. Zhu, W. Liu, K. Jia, W. Liao, and H. Xie, Sens. Actuators A 167, 495 (2011). 7 L. Liu, Y. Bai, D. Zhang, and Z. Wu, Sensors 13, 9070 (2013). 8 W. Liao, W. Liu, Y. Zhu, Y. Tang, B. Wang, and H. Xie, IEEE Sens. J. 13, 2873 (2013). 9 H. A. Eschenauer, Applied Structural Mechanics: Fundamentals of Elasticity, Load-Bearing Structures, Structural Optimization (Springer, 1997). 10 D. Prattichizzo and J. C. Trinkle, in Springer Handbook of Robotics (Springer, 2008), pp. 671–700. 2 Z.

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Tilt Platform: structure, kinematics, and experiments.

A Piezo Tip/Tilt Platform (PT(2)P) is presented with its structure, kinematics, and preliminary experiments. Two essential models of the presented PT(...
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