A Liquid-Surface-Based Three-Axis Inclination Sensor for ... - MDPI

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A Liquid-Surface-Based Three-Axis Inclination Sensor for Measurement of Stage Tilt Motions Yuki Shimizu *, Satoshi Kataoka, Tatsuya Ishikawa, Yuan-Liu Chen Hiraku Matsukuma ID and Wei Gao

ID

, Xiuguo Chen

ID

,

Department of Finemechanics, Tohoku University, Sendai 980-8579, Japan; [email protected] (S.K.); [email protected] (T.I.); [email protected] (Y.-L.C.); [email protected] (X.C.); [email protected] (H.M.); [email protected] (W.G.) * Correspondence: [email protected]; Tel.: +81-22-795-6950 Received: 20 December 2017; Accepted: 29 January 2018; Published: 30 January 2018

Abstract: In this paper a new concept of a liquid-surface-based three-axis inclination sensor for evaluation of angular error motion of a precision linear slide, which is often used in the field of precision engineering such as ultra-precision machine tools, coordinate measuring machines (CMMs) and so on, is proposed. In the liquid-surface-based three-axis inclination sensor, a reference float mounting a line scale grating having periodic line grating structures is made to float over a liquid surface, while its three-axis angular motion is measured by using an optical sensor head based on the three-axis laser autocollimation capable of measuring three-axis angular motion of the scale grating. As the first step of research, in this paper, theoretical analysis on the angular motion of the reference float about each axis has been carried out based on simplified kinematic models to evaluate the possibility of realizing the proposed concept of a three-axis inclination sensor. In addition, based on the theoretical analyses results, a prototype three-axis inclination sensor has been designed and developed. Through some basic experiments with the prototype, the possibility of simultaneous three-axis inclination measurement by the proposed concept has been verified. Keywords: three-axis measurement; inclination; linear slide; optical angle sensor; laser autocollimation

1. Introduction A precision linear slide is one of the most important components in precision machine tools and precision measuring instruments [1]. With the enhancement of precision positioning sensors such as laser interferometers or linear encoders, a precision linear slide can realize highly accurate and precise positioning [2]. Meanwhile, in recent years, it has become more important to verify the angular error motion of a linear slide quantitatively to achieve further higher positioning accuracy [3]. For evaluation of the tilt angles of a precision linear slide, autocollimators and electronic levels have often been employed so far as traditional optical measuring instruments [4,5]. By employing two-degree-of-freedom position detectors such as CCDs [6] or multi-cell photodiodes [7], simultaneous two-degree-of-freedom measurement of tilt angles of a linear slide can be carried out in a simple manner just by placing a target reflector on a slide table and adjusting the optical axis of an autocollimator with respect to the target. When a two-axis autocollimator is placed in such a way that its optical axis coincides with a primary motion axis of a linear slide, both pitch and yaw angles of the linear slide can be measured simultaneously. Meanwhile, a roll angle, which could significantly affect the positioning accuracy of a linear slide when the measurement axis of a displacement sensor is placed with an offset from the motion axis of the linear slide [2], cannot be measured. For measurement of the roll angle, an additional autocollimator should therefore be employed. However, this also induces another problem;

Sensors 2018, 18, 398; doi:10.3390/s18020398

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for measurement of the roll angle of a linear slide, a reference mirror with the length covering the long travel range of the linear slide should be employed. Although a laser interferometer system can be employed instead of an autocollimator to evaluate the roll angle of a linear slide having a long travel range [8,9], the measured information will be lost and the evaluation has to be re-started from the beginning if the optical path of the laser interferometer is blocked; this could be a fatal problem in the case of production process of the slide, where the measurement will repeatedly be carried out together with a hand scraping operation for correction of the form errors of slide guideways [10]. Therefore, angle sensors/inclination sensors that can be mounted on a slide table and can carry out highly-accurate simultaneous tilt angle measurement about the X-, Y- and Z-axes are desired to be realized. Gyroscopes such as ring laser gyroscopes [11], fibre optic gyroscopes [12] and vibrating structure gyroscopes [13] could also be one of the candidate instruments satisfying the requirements mentioned above [14]. By utilizing a gyro for each measurement axis, simultaneous measurement of three-axis tilt angles can be realized. Especially, ring laser gyroscopes [11] have a high enough measurement resolution to be applied for the evaluation of tilt angles of a precision linear slide. However, due to the principle of gyroscope in which the tilt angle will be acquired by integrating detected angular velocity, even a high precision ring laser gyroscope could have a bias instability on the order of 0.01◦ /h (which roughly corresponds to 1 arc-second/100 s) [15]; which cannot be neglected for the evaluation of the angular error motion of a precision linear slide, which is typically on the order of several arc-seconds. In addition, a high cost of the ring laser gyro could also be an obstacle to employ it for the evaluation of linear slides in a machine shop. The authors have developed an optical angle sensor referred to as the three-axis laser autocollimator [16] that can carry out simultaneous measurement of three-axis tilt angles by using a single laser beam. By employing a line scale grating having periodic line grating structures as a reflector, instead of a flat mirror reflector employed in traditional one-axis or two-axis autocollimators based on the laser autocollimation [17], simultaneous measurement of three-axis tilt angles with a resolution better than one arc second has been realized [16]. In the three-axis laser autocollimator, both the zeroth-order and a first-order diffracted beam from the scale grating are required to be detected by an optical sensor head. Meanwhile, the orientation of the first-order diffracted beam from the scale grating is different from the axis of the measurement beam from the laser autocollimator. The working distance of the three-axis laser autocollimator is therefore limited by the allowable size of the optical sensor head, resulting in the difficulty of evaluating a linear slide having a long stroke. In responding to the background described above, the authors have also developed a high-resolution inclination sensor [18] based on the fluid-based type commercial electronic level [19], which does not require a reflector because its angle reference is the level of liquid enclosed inside of the sensor body [20,21]. One of the main features of the high-resolution inclination sensor is its compact size and simple procedure for the measurement of the roll angle of a slide table in a long-stroke linear slide; this is a great advantage from the viewpoint of its usage in the linear slide production line. It should be noted that the concept of using liquids as reference for flatness and a constant level tilt has long been used for measurement of the pitch or roll angles of a linear slide [22]. Two-axis electronic levels based on this concept that can make simultaneous two-axis measurement of pitch and roll angles are also commercially available [23]. However, a conventional electronic level cannot measure the yaw angle of a linear slide in principle, which is a major drawback compared with the autocollimators and laser interferometers. Although the three-axis tilt angle measurement of a linear slide can be realized by the combination of an electronic level and an autocollimator, this will increase the cost and complicity of the measurement system. Meanwhile, in the case of measurement of a linear slide having long travel range, a long optical path of autocollimator could degrade the measurement accuracy of yaw measurement due to the external disturbances such as temperature deviation in a machine shop. A new type of inclination sensor capable of carrying out simultaneous three-axis tilt angles, not only the pitch and roll angles, but also the yaw angle, is therefore desired for measurement of the tilt angles of a precision linear slide.

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inclination sensor capable of carrying out simultaneous three-axis tilt angles, not only the pitch and roll angles, but also the yaw angle, is therefore desired for measurement of the tilt angles of a Sensors 2018,linear 18, 398 slide. 3 of 22 precision In this paper, to achieve simultaneous measurement of three-axis tilt angles of a precision linear slide, a new concept of three-axis inclination sensor is proposed. In the concept, a principle of In this paper, to achieve simultaneous measurement of three-axis tilt angles of a precision linear fluid-based type clinometer is extended by integrating the principle of three-axis laser slide, a new concept of three-axis inclination sensor is proposed. In the concept, a principle of autocollimator. Simultaneous measurement of three-axis tilt angles can be achieved in such a way fluid-based type clinometer is extended by integrating the principle of three-axis laser autocollimator. that a reference float mounting a scale grating will be made to float over a liquid surface in a casing, Simultaneous measurement of three-axis tilt angles can be achieved in such a way that a reference float while monitoring the tilt angles of the reference float by using an optical sensor head of the mounting a scale grating will be made to float over a liquid surface in a casing, while monitoring the three-axis laser autocollimator, which is kinematically connected to the casing. As the first step of tilt angles of the reference float by using an optical sensor head of the three-axis laser autocollimator, research, in this paper, a theoretical analysis on the angular motion of the reference float about each which is kinematically connected to the casing. As the first step of research, in this paper, a theoretical axis based on a simplified kinematic model is at first carried out to search for the possibility of analysis on the angular motion of the reference float about each axis based on a simplified kinematic realizing the proposed concept of the three-axis inclination sensor. In addition, based on the results model is at first carried out to search for the possibility of realizing the proposed concept of the of the theoretical analyses, a prototype three-axis inclination sensor is designed and developed to three-axis inclination sensor. In addition, based on the results of the theoretical analyses, a prototype verify the possibility of simultaneous measurement of three-axis tilt angles by the proposed three-axis inclination sensor is designed and developed to verify the possibility of simultaneous concept. measurement of three-axis tilt angles by the proposed concept. 2. Principle of the Proposed Three-Axis Inclination Sensor 2. Principle of the Proposed Three-Axis Inclination Sensor 2.1. A A Concept Concept of of the the Proposed Proposed Three-Axis Three-Axis Inclination Inclination Sensor Sensor 2.1. A schematic schematic of of the the proposed proposed concept concept of of the the three-axis three-axis inclination inclination sensor A sensor is is shown shown in in Figure Figure 1. 1. The sensor is composed of an optical sensor head and a reference float that floats on a liquid surface The sensor is composed of an optical sensor head and a reference float that floats on a liquid surface in aa casing. casing. The The optical optical sensor sensor head head is is kinematically kinematically connected connected to to the the casing. casing. The The reference reference float float in consists of a scale grating with line pattern structures and a float on which the scale grating is consists of a scale grating with line pattern structures and a float on which the scale grating is mounted. mounted. Although similar concepts using liquids as reference for flatness and a constant level tilt Although similar concepts using liquids as reference for flatness and a constant level tilt can be found can beliterature found in [22], the literature [22], three-axis the proposed three-axis inclination sensor expected to realize in the the proposed inclination sensor is expected to is realize simultaneous simultaneous measurement of rotational motion about the three-axes with the enhancement of measurement of rotational motion about the three-axes with the enhancement of three-axis laser three-axis laser autocollimation [16]. The center point in the bottom surface of float is supported by autocollimation [16]. The center point in the bottom surface of float is supported by the tip of a the tip of a tiny needle, which is rigidly fixed to of thethe bottom ofso the casing, so that the reference float tiny needle, which is rigidly fixed to the bottom casing, that the reference float can be held can be held stationary along the X-, Yand Z-directions in the sensor system, while allowing stationary along the X-, Y- and Z-directions in the sensor system, while allowing relative angular relativeof angular motionfloat of the reference with respect to each the casing each of the three-axes. motion the reference with respectfloat to the casing about of theabout three-axes.

Figure 1. 1. A A schematic schematic of of the the liquid-surface-based liquid-surface-based three-axis three-axis inclination inclination sensor. sensor. Figure

Three-axis inclination measurements are carried out by mounting the inclination sensor on a Three-axis inclination measurements are carried out by mounting the inclination sensor on a measurement target such as the slide table in a precision linear slide. Since the normal direction of measurement target such as the slide table in a precision linear slide. Since the normal direction of the the liquid surface in the casing always coincides with the orientation of gravity, the reference float liquid surface in the casing always coincides with the orientation of gravity, the reference float being being floated on the liquid surface can be treated as a datum for tilt angle measurement about the floated on the liquid surface can be treated as a datum for tilt angle measurement about the X- and Y-axes. By detecting relative tilt angles of the optical sensor head with respect to the reference float, tilt angles of the measurement target about the X- and Y-axes can be measured.

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The reference float can also be employed as a datum for tilt angle measurement about the Z-axis. Since the reference float is being floated on a liquid surface, an external torque to be applied to the reference float will be generated by shearing force between the float surface and the liquid, as well as by the friction force between the bottom surface of the reference float and a supporting needle introduced to support the reference float. On the assumption that the frictional force due to the supporting needle is small enough to be neglected, the angular motion of the float about the Z-axis can be suppressed by employing liquid with low viscosity resistance, and therefore the reference float is expected to be used as a datum for measurement of inclination angle about the Z-axis since the angular position of the reference float about the Z-axis with respect to a fixed table, where the measurement target is mounted, is expected to be kept stationary with the enhancement of the inertial force of the reference float. It should be noted that, for the evaluation of stage tilt, employing another reference three-axis inclination sensor mounted on a table where the stage is mounted realizes relative angular position measurement with respect to the angular position of the stage at its initial position. In the following section, to verify the feasibility of proposed three-axis inclination sensor, a theoretical analysis on the angular motion of the reference float is carried out based on simplified kinematic models of the float on the liquid surface. 2.2. Modeling of the Yaw Motion of Inclination Angle Reference Float In the proposed three-axis inclination sensor, drift motion of the reference float about the Z-axis (yaw drift), which could occur due to the liquid flow around the reference float induced by the angular motion of the casing about the Z-axis, degrades the accuracy of yaw angle measurement. The reference float is therefore required to be designed while paying attention to its sensitivity against the liquid flow about the Z-axis. The equation of motion of the reference float about the Z-axis can be expressed as follows by the second-order differential equation: Iz

d2 θYaw (t) = MExt (t) dt2

(1)

where θ Yaw and IZ are the angular displacement and moment of inertia of the reference float about the Z-axis, MExt is external torque applied to the reference float about the Z-axis, and t is time. Since both θ Yaw and MExt are functions of t, now we discretize the equation over time interval ∆t as follows: Iz

∆θYaw (ti ) ∆t

∆θYaw (ti−1 ) ∆t

− ∆t

= MExt (ti )

(2)

where ti (i = 1, 2, . . . , N) is discrete time, and ∆θYaw (ti ) = θYaw (ti ) − θYaw (ti −1 ). By solving this equation about ∆θYaw (ti ), the following equation can be obtained: ∆θYaw (ti ) = ∆θYaw (ti−1 ) +

MExt (ti ) 2 ∆t Iz

(3)

Equation (3) shows that large moment of inertia IZ of the reference float contributes to reduce the yaw drift, as well as the decrease of the external torque MExt . In this paper, the external torque to be applied to the reference float is modeled as a sum of shear torque MBottom (t) generated by the liquid flow between the bottom surface of the casing and that of the float, and shear torque MSide (t) generated by the liquid flow between the side wall of the casing and that of the reference float. At first, MBottom (t) is estimated based on a simple model shown in Figure 2. Now we assume that the liquid in the casing can be treated as the Newtonian fluid, and it flows only in the circumferential direction about the Z-axis in the figure. In this case, the shear stress τ in the liquid can be expressed as follows [24]: dv τ=µ (4) dz

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 

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dv dz

(4)

where µ is the dv/dz derivative of of thethe velocity v parallel to the where the shear shearviscosity viscosityofofthe theliquid, liquid,and and dv/dzisisthe the derivative velocity v parallel to direction of shear relative to the Z-directional displacement z. the direction of shear relative to the Z-directional displacement z. Z

Casing

MBottom(t)

Z

Float Liquid surface

z=h

v

f(t)

(r,,z)

r

z=h

z=0

z=0 r=0

 c(t)

r vc=rc(t)

v

Bottom of the casing

r=a Bottom of casing

Float

vf=rf(t)

r

Distribution of v

Figure 2. Modeling of the shear torque in between the bottom surface of casing and that of reference Figure 2. Modeling of the shear torque in between the bottom surface of casing and that of reference float. float.

Denotingthe the angular angular velocities and thethe casing as ωasf (t)ωand ω c (t), the fluid Denoting velocitiesofofthe thefloat float and casing f(t) and ωcrespectively, (t), respectively, the velocity at the bottom surface of float v and that at the bottom surface of casing v can be expressed as c fluid velocity at the bottom surface off float vf and that at the bottom surface of casing vc can be v = rω (t) and v = rω (t), respectively. The shear stress dτ to be applied to the area element dS can f f expressed as vf =c rωf(t)c and vc = rωc(t), respectively. The shear stress dτ to be applied to the area therefore be expressed asbe follows: element dS can therefore expressed as follows: ω(t(t))− ωff((tt)) v −  v v vf µr r c c d µ c c f = dτ = h h h h

(5) (5)

Now coordinate system system can can be beexpressed expressedas asdS dS==rdrdθ, rdrdθ,where wherer Now the the area element dS in the polar coordinate rand andθθare area aradial radialposition position and angular position, respectively. shear torque to applied be applied to and anan angular position, respectively. TheThe shear torque to be to the the area element dS can therefore be derived as follows: area element dS can therefore be derived as follows:

 (t)   (t) t) dM Bottom t) rdτdS r 33 ωcc (t) −f ωf (drd   rddS= µr dMBottom (t) (= drdθ hh

(6) (6)

The (t) to be applied to the bottom surface of the float can be acquired The total total shear shear torque torque M MBottom Bottom (t) to be applied to the bottom surface of the float can be acquired by by integrating integrating Equation Equation (6) (6) over over the the whole bottom surface as follows: 2 a

2π a a 4 3  c (t )   f (t ) drd   πµa4(c (t)  f (t)) M Bottom (tZ) Z   3rω ω ( t ) c (t) − f h MBottom (t) = drdθ = 2h (ωc (t) − ωf (t)) 0 µr 0 h 2h

(7) (7)

0 0

where a is the radius of the float. Asa ais next step of modeling of the yaw motion of the reference float, the shear torque where the radius of the float. MSide(t) estimated. Figure 3 shows a yaw schematic modelingfloat, of shear torque between the As is a next step of the modeling of the motionofofthe the reference the shear torque MSide (t) is sidewall ofFigure casing3and thata schematic of float. Inofthis a cylindrical coordinate system is employed to estimated. shows the model, modeling of shear torque between the sidewall of casing describe motion of reference Denoting the radial and circumferential fluid and that the of float. In this model, afloat. cylindrical coordinate system is employed tocomponents describe theof motion velocity as vfloat. r and Denoting vθ, respectively, shear τrθ to be generated by of the liquid viscosity of reference the radial andstress circumferential components fluid velocity as vr can and be vθ , expressed as follows [24]:τrθ to be generated by the liquid viscosity can be expressed as follows [24]: respectively, shear stress    ∂  v θv  11∂v vrr   r     τrθ r= µ r +      ∂rr r r  rr ∂θ

(8) (8)

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Z

Casing Liquid surface

Mside(t)

Float

T

f(t) Sside

v

Casing

z=h

Float v (r) r

z=0 r=0

r r=a

c(t)

r=R

r

f(t) c



Z r

Distribution of v

Figure 3. 3. Modeling of of thethe shear torque between the sidewalls ofof casing and float. Figure Modeling shear torque between the sidewalls casing and float.

Assuming vrv= 0=for thethe sake of of simplicity, thethe above equation can bebe simplified asas follows: Assuming 0 for sake simplicity, above equation can simplified follows: r  v   r  r ∂  vθ  τrθ = µr r  r  ∂r r

(9)(9)

bebeobtained The Thecircumferential circumferentialliquid liquidvelocity velocitycomponent componentvθvcan obtainedbybyderiving derivingNavier–Stokes Navier–Stokes θ can equations.Denoting Denoting Z-directional liquid velocity component body forcesalong alongthe the equations. thethe Z-directional liquid velocity component asas vZvand body forces Z and radial, circumferential and the Z-directions as F , F and F , respectively, incompressible momentum radial, circumferential and the Z-directions ras θFr, Fθ Z and FZ, respectively, incompressible Navier-Stokes equations inequations the cylindrical coordinate system can be expressed as follows [25]: as momentum Navier-Stokes in the cylindrical coordinate system can be expressed follows [25]:

2

vθ ∂vr vθ ∂vr r + vr ∂v ∂r + r ∂θ − r 2 + vz ∂z  ∂2 vr 2 ∂vθ v v ∂2vvr v1 ∂vr vrv 1 ∂2 vr vr= − 1∂p  vrρ ∂rr +  ν ∂r2r + r ∂r v−z r2 r+ r2 ∂θ 2 − r2 ∂θ + ∂z2 + Fr r  r t r z 2 ∂v ∂v v v v ∂vθ 1v p θ +  θ vθ −1 rvθ + vv ∂vθ 1  2vr 2 v  2vr  ∂t +   Fr   2 22  2  r ∂r   r ∂θ22r  r r  zr2 ∂z ∂p r + ν r∂ v2θ +r 1∂v r θ −r v2θ + r + ∂2zv22θ  + Fθ r 12∂ v2θ +r 22 ∂v = − 1ρr∂θ r ∂r ∂θ ∂r r r ∂θ r ∂z ∂vr ∂t

(10)

(10) (11)

vθ ∂vz ∂vz ∂vz ∂vz rvr∂θ v ∂tv + vvr ∂rv+  v2  + vz ∂z v  (12)  vr  1 ∂p  ∂ vz  v z1 ∂vz ∂2 v z 1 ∂2 v z = − + + Fz + ν t r r2 ρr ∂z r  ∂rz + r2 ∂θ 2 + ∂z2 ∂r (11)  2v 1and v dynamic  2v vrliquid  2vin the casing. Since these v 1 p a pressure 1 viscosity 2 of where ρ, p and ν areadensity,        F  2 2  2   r 2 we r rthe r apply  ranalytically,  equations cannot be solved boundary zlisted r 2  2 rconditions   as follows to convert these equations into linear differential equations so that the circumferential velocity component of the flow vθ can be obtained [25]: vz v v v v  vr z   z  v z z t r z r  (a) The flow is not a function of time (steady flow: ∂/∂t = 0). (12) 2   v 1 p 1 vz 1  2v z  2v z  (b) The flow is incompressible   (ρ =const.)  2   F   2z  z 2  r r r r   z velocity z 2  are zero (vr = 0, vz = 0) (c) The radial and the Z-directional components of the flow 

weνconsider the equation of and continuity in viscosity the cylindrical coordinate system expressed are a density, a pressure dynamic of liquid in the casing. Since these where Here, ρ, p and as follows: equations cannot be solved analytically, we apply the boundary conditions listed as follows to 1 ∂(vr r ) 1 ∂vθ ∂vz convert these equations into linear differential so0 that the circumferential velocity + equations + = (13) r ∂r r ∂θ ∂z component of the flow vθ can be obtained [25]: θ Applying the boundary condition (c) to the above equation results in ∂v ∂θ = 0, which means (a)thatThe flow is not a function of time (steady flow: ∂/∂t = 0). the circumferential velocity component of the flow is constant with respect to θ. Regarding this, (b)now The incompressible (ρ = const.) weflow alsoisassume the following boundary conditions: (c) The radial and the Z-directional velocity components of the flow are zero (vr = 0, vz = 0) θ (d) The circumferential velocity component of the flow is uniform along the Z-direction ( ∂v ∂z = 0) Here, we consider the equation of continuity in the cylindrical coordinate ∂p system expressed as (e) The pressure distribution is uniform along the circumferential direction ( ∂θ = 0) follows:

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By applying these boundary conditions to Equations (10) to (12), the following linear differential equations can be obtained: vθ 2 1 ∂p =− (14) r ρ ∂r 0=

v ∂2 v θ 1 ∂vθ − 2θ + r ∂r ∂r2 r 0=−

(15)

1 ∂p ρ ∂z

(16)

By deriving Equation (14), the following equation can be obtained: vθ = C1 r + C2

1 r

(17)

where C1 and C2 are constants. Now the liquid velocity component vθ at the sidewall of float can be expressed as aω f (t), where a is the radius of float. In the same manner, the fluid velocity component vθ at the sidewall of casing can be expressed as Rω f (t), where R is the radius of casing. The constants C1 and C2 can therefore be derived as: C1 =

  1 2 2 t − R ω t a ω ( ) ( ) c f a2 + R2

(18)

 a2 R2  t ω t − ω ( ) ( ) c f a2 + R2

(19)

C2 =

Applying Equations (18) and (19) to Equation (17) gives the following equation:    1 1 2 2 2 2 vθ = 2 a ω f ( t ) − R ωc ( t ) r + a R − ω f ( t ) + ωc ( t ) r a − R2

(20)

By applying the above equation to Equation (9), the shear stress τ rθ to be generated by the fluid viscosity can be expressed as follows: τrθ

 i   a2 ω f (t) − R2 ωc (t) + a2 R2 −ω f (t) + ωc (t) r12 2µa2 R2 (ωc (t)−ω f (t))

d = µr dr

=



1

h

a2 − R2

(21)

( R2 − a2 )r 2

The shear stress on the sidewall of float can therefore be obtained from Equation (21) as follows:

τrθ |r= a =

  2µR2 ωc (t) − ω f (t)

(22)

R2 − a2

Since the area of the sidewall of float beneath the liquid surface Sside = 2πaT (T: draft of the float), the shear torque MSide (t) to be applied to the sidewalls of the float can be obtained as follows: Mside (t)

= τrθ · Sside · a 4πµa2 R2 T (ωc (t)−ω f (t)) = R2 − a2

(23)

Finally, By applying the derived torque MBottom (t) and Mside (t) to Equation (1), the following equation can be obtained: Iz

d2 θYaw (t) dt2



a4 2h



4R2 a2 T R2 − a2

ωc ( t ) − ω f ( t )  = πµK ( a, h, R, T ) ωc (t) − ω f (t)

= πµ

+

 (24)

equation can be obtained: Iz

2 2 d 2Yaw t   4    a  4R2 a T2  c t    f t  2 dt  2h R  a   K a, h, R, T  c t    f t 







(24)



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Now we include a factor K, which is a function containing design parameters of both the reference float and the casing as its arguments. For the reduction of yaw motion of the reference Now we include a factor K, which is aoffunction containing parameters of both the float induced by the angular motion the casing, as can design be seen in the equation, the reference factor K is float and the casing as its arguments. For the reduction of yaw motion of the reference float induced by to preferred to be reduced as much as possible. A design study is therefore carried out in this paper theinvestigate angular motion of the casing, as can be seen inK. theItequation, thenoted factorthat K is the preferred to the be reduced how the design parameters affect should be draft of float T is as fixed muchto asbe possible. A regarding design study therefore carried out sensor in this paper to investigate how thesection design of 8.8 mm, theisdeveloped prototype described in the following parameters affect K. It should be noted that the draft of the float T is fixed to be 8.8 mm, regarding the this paper. developed prototype described theafollowing section of thisofpaper. Figure 4 showssensor the variation of in K as function of the radius reference float a in the case of R Figure shows the variation of KofasKacalculated function of the aradius of reference floatbottom a in thesurfaces case of of = 140 mm.4 In the figure, variations with distance between the R= 140 and mm.that In the figure, hvariations of K calculated with a distance between the bottom surfaces float of casing ranging from 1 mm to 128 mm are plotted. It should be noted that the of ordinate float andinthat casing ranging from 1 mm toAs 128can mmbeare plotted. shouldK be noted that thethe theofgraph is hscaled logarithmically. seen in theItfigure, decreases with ordinate in of thea,graph As can be the seendecrease in the figure, K decreases decrease while is itsscaled slope logarithmically. continually increases with of a from 135 mm with to 10the mm; decrease of a,means while its continually withtothe decrease a from 135 of mm to 10 mm; this result thatslope the decrease of increases a contributes reduce the of yaw motion reference float this result means the decrease contributes to reduce theKyaw motion reference float induced induced by thethat angular motion of ofathe casing. Meanwhile, is found to of dramatically increase with bythe theincrease angular of motion the mm; casing. K is that found dramatically increasea with the increase a overof135 thisMeanwhile, result implies thetoincrease of a reduces clearance between of side a over 135 mm; implies that theresulting increase of reduces a clearance betweentorque side surfaces of surfaces of this the result float and the casing, in athe increase of transmitting in between thethem floatdue andto the casing, increasethe of transmitting between them duetotoreduce the the liquidresulting viscosity.inInthe addition, increase of h torque is also in found to contribute liquid K. viscosity. In addition, the increase of h is also found to contribute to reduce K. 0

10

h mm

-1

K(a, h ,R,T)

10

-2

10

-3

10

-4

10

32 64 128

2016

4 8

1 2

-5

10

-6

10

20 40 60 80 100 120 140 Radius of the float: a mm

Figure 4. 4. Variation ofof the factor KK asasa afunction Figure Variation the factor functionofofthe thefloat floatradius radiusa aininthe thecase caseofofRR==140 140mm mm and and T T == 8.8 8.8 mm. mm.

The theoretical analyses described above have revealed the following designs are expected The theoretical analyses described above have revealed thatthat the following designs are expected to to reduce the drift yaw drift the reference reduce the yaw of theofreference float:float: (1) (2) (3)

Set the radius of casing R as large as possible with respect to the radius of reference float a. Design the radius of float a as small as possible. Set the distance h between the bottom surface of the float and that of the casing as large as possible.

Meanwhile, in a practical design of the three-axis inclination sensor, the reference float is required to be floated on a liquid surface. The minimum value of float a will therefore be determined while considering not only the buoyancy to be generated by the reference float but also the gravitational force by the masses of scale grating, optical window, and fixing screws. In addition, R and h will be determined by the size allowed for the three-axis inclination sensor. It should be noted that the influence of a on the moment of inertia IZ of the reference float about the Z-axis has not been taken into consideration in this paper for the sake of simplicity, and further detailed investigation is required for the optimization of the design of reference float.

determined while considering not only the buoyancy to be generated by the reference float but also the gravitational force by the masses of scale grating, optical window, and fixing screws. In addition, R and h will be determined by the size allowed for the three-axis inclination sensor. It should be noted that the influence of a on the moment of inertia IZ of the reference float about the Sensors 2018,has 18, 398 of 22 Z-axis not been taken into consideration in this paper for the sake of simplicity, and 9further detailed investigation is required for the optimization of the design of reference float. using design parameters, yaw drift reference float is simulated. Equation (24) ByBy using thethe design parameters, thethe yaw drift of of thethe reference float is simulated. Equation (24) can be modified with respect to the angular velocity of float ω f(t) as follows: can be modified with respect to the angular velocity of float ω (t) as follows: f

df (t)   a4 2 42R2 a2T ( (t)   (t)) πµ a4  4R a T2 dωf (t) I z+ 2h R  a(2ωc (t)c − ωf (f t)) = dt dt Iz 2h R2 − a2

(25) (25)

The angular velocity of reference float ωf(t) can be numerically solved by the Runge–Kutta The angular velocity of reference float ω f (t) can be numerically solved by the Runge–Kutta method [26]. In the numerical simulation, a step input with an angular velocity of casing ωc of 1 method [26]. In the numerical simulation, a step input with an angular velocity of casing ω c of arc-second/s is assumed, while time step is set to be 10 ms. Other parameters for the simulation are 1 arc-second/s is assumed, while time step is set to be 10 ms. Other parameters for the simulation are summarized in Table 1. Figure 5 shows the result of numerical simulation, in which the change in summarized in Table 1. Figure 5 shows the result of numerical simulation, in which the change in the the angular velocity of the float with respect to time t is plotted. By integrating the numerically angular velocity of the float with respect to time t is plotted. By integrating the numerically simulated simulated ωc(t) shown in Figure 5, an angular displacement of the float θc can be acquired as shown ω c (t) shown in Figure 5, an angular displacement of the float θ c can be acquired as shown in Figure 6. in Figure 6. As can be seen in the figure, the amount of yaw drift increases with the increase of time. As can be seen in the figure, the amount of yaw drift increases with the increase of time. However, However, on the other hand, the amount of yaw drift after 100 s is expected to be less than 1 on the other hand, the amount of yaw drift after 100 s is expected to be less than 1 arc-second; which is arc-second; which is comparable or better than the conventional gyros [15]. comparable or better than the conventional gyros [15]. Table 1. Parameters for analytical calculations. Table 1. Parameters for analytical calculations.

Parameters

Float Float

Casing

Parameters Radius

Symbol

Radius Draft a Draft T Moment of inertia about the Z-axis Moment of inertia about Iz the Z-axisRadius

Casing

Water level Radius Water levelType

R h

Type Viscosity Viscosity Density Density

µ ρ

Fluid

Yaw angular velocity of the casing arc-second/s

Fluid

0.06 Casing

1.0

0.04

0.5

0.02 Float

0

20

40 60 Time s

80

0.00 100

Value Value 30 mm 30 mm 8 mm 8 mm -4

2

R h -

9.02 × 10 kg m 4 kg m2 9.02 × 10− 120 mm 80 mm 120 mm Water 80 mm

µ

0.89 Water × 10 Pa s



1.5

0.0

Symbol a T Iz

-3

0.89 × 103−3 Pa s3 kg/m3 1.00×× 10 103 kg/m 1.00

Yaw angular velocity of the float arc-second/s

Component Component

100

2.5

80

2.0

Casing

60

1.5

40

1.0

20 0

Float

0

20

40

60

80

0.5

0.0 100

Yaw drift of the float arc-second

Yaw angle of the casing arc-second

Figure 5. Simulated yaw angular of the the reference reference float float due due to to the the yawing yawing given given to to the the casing. casing. 10 of 22 Sensors 2018, x FOR PEER Figure 5. 18, Simulated yawREVIEW angular velocity velocity of

Time s Figure 6. Simulated yaw drift of the reference float due to the yawing given to the casing. Figure 6. Simulated yaw drift of the reference float due to the yawing given to the casing.

2.3. Modeling of the Pitch and Roll Motions of Inclination Angle Reference Float Following the modeling of yaw motion of the reference float, pitch and roll motions of the reference float are also modeled to further investigate the characteristics of the reference float. Since the reference float in this paper is designed to have a disk-like shape symmetrical about the Z-axis, the same model can be applied to both the roll and pitch motions of the reference float. Therefore,

0

Float

0

20

40

60

80

0.5

0.0 100

Yaw

Yaw a

20

Time s SensorsFigure 2018, 18, 6. 398 Simulated yaw drift of the reference float due to the yawing given to the casing.

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2.3. Modeling of the Pitch and Roll Motions of Inclination Angle Reference Float 2.3. Modeling of the Pitch and Roll Motions of Inclination Angle Reference Float Following the modeling of yaw motion of the reference float, pitch and roll motions of the Following the modeling of yaw motion of the reference float, pitch and roll motions of the reference reference float are also modeled to further investigate the characteristics of the reference float. Since float are also modeled to further investigate the characteristics of the reference float. Since the reference the reference float in this paper is designed to have a disk-like shape symmetrical about the Z-axis, float in this paper is designed to have a disk-like shape symmetrical about the Z-axis, the same model the same model can be applied to both the roll and pitch motions of the reference float. Therefore, can be applied to both the roll and pitch motions of the reference float. Therefore, now we focus on the now we focus on the modeling of the roll motion of the reference float, a schematic of which is modeling of the roll motion of the reference float, a schematic of which is shown in Figure 7. shown in Figure 7.

Roll Buoyancy : Fb Float

Z

Center of gravity:（xg, zg）

Roll

rres g

rb

Air rg

Mf

X

Center of buoyancy:（xf, zf）

Y

Liquid

Volume under the water: Vf

Figure 7. Analytical model for for thethe rollroll motion of the reference float. Figure 7. Analytical model motion of the reference float.

The equation of angular motion of the reference float can be expressed as follows: The equation of angular motion of the reference float can be expressed as follows: d 2 Roll d (26)  c dθRoll  M I X d2 θ Roll IX dt 22 + c dtRoll = M (26) dt dt where IX is the moment of inertia of the reference float about the X-axis at the gravitational center, where IX is the moment of inertia of the reference float about the X-axis at the gravitational center, θ Roll θRoll is the roll angle, c is the viscosity coefficient of the liquid, and M is a torque applied to the is the roll angle, c is the viscosity coefficient of the liquid, and M is a torque applied to the reference reference float. The buoyancy Fb can be expressed as follows: float. The buoyancy Fb can be expressed as follows: t Fb  0 ,  gVf sin  Roll ,  gVf cos  Roll  (27) Fb = t [0, ρgVf sin θ Roll , ρgVf cos θ Roll ] (27) where ρ is the liquid density, g is the gravity, and Vf is the volume of reference float beneath the liquid surface. point of application of buoyancy f, zf)volume coincides with a center of the volume where ρ is theThe liquid density, g is the gravity, and V f is(ythe of reference float beneath the liquid beneath theThe liquid surface, and canof bebuoyancy expressed(yas follows: surface. point of application , z ) coincides with a center of the volume beneath the f f liquid surface, and can be expressed as follows: zdV xdV , z f  R zdV (28) y f   R xdV y f =  RdV , z f = RdV (28) dV dV

 

In this paper, the point of application of buoyancy (yf, zf) is calculated by using a function in a In this paper, the point of application of buoyancy (y , z ) is calculated by using a function in a three-dimensional CAD software. The parameters employedf inf the simulation are summarized in three-dimensional CAD software. The parameters employed in the simulation are summarized in Table 2. Figure 8 shows variations of yf and zf with respect to θRoll ranging from −100 arc-seconds to Table 2. Figure 8 shows variations of yf and zf with respect to θ Roll ranging from −100 arc-seconds 100 arc-seconds in the case of T = 10 mm and a = 30 mm. to 100 arc-seconds in the case of T = 10 mm and a = 30 mm. Table 2. Parameters for simulation. Parameters Moment of inertia of the float about the X-axis Sensitivity coefficient Gravity

Symbol

Value

IX

5.17 × 10−4 kg m2

α g

−0.0281 m/rad 9.80 m/s2

Table 2. Parameters for simulation.

Parameters Moment of inertia of the float about the X-axis

Symbol IX

Sensitivity coefficient



5.17 × 10 kg m −0.0281 m/rad

Gravity

g

9.80 m/s

2

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2

0.010

15 10

yf m

-4

zf

5

0.005 0.000

0 -5

yf

-10 -15 -100

-50

Roll

0 50 arc-second

zf m

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Value

-0.005 -0.010 100

Figure Variation the point application buoyancy. Figure 8. 8.Variation ofof the point ofof application ofof buoyancy.

be seen the figure, and f can be approximated by a linear function and a AsAs cancan be seen in theinfigure, yf and zyf fcan bezapproximated by a linear function and a second-order second-order polynomial functions, respectively, with respect the. change Roll. Since zf is much polynomial functions, respectively, with respect to the change into θ Roll Since zf in is θ much smaller than smaller than y f, now we assume zf to be zero for the sake of simplicity. Under the condition of T = 10 yf , now we assume zf to be zero for the sake of simplicity. Under the condition of T = 10 mm and anda agradient = 30 mm, gradient of line the of approximated yf is evaluated to be Now −0.1091 a =mm 30 mm, of theaapproximated yf is evaluatedline to beof−0.1091 mm/arc-second. a mm/arc-second. Now a positional vector r res pointing toward the gravitational center (yg, zg) of the positional vector rres pointing toward the gravitational center (yg , zg ) of the reference float from the reference float from of buoyancy can be expressed as follows: center of buoyancy canthe becenter expressed as follows: t





rres  rf  rgh 0 , y f  yg , z f  zgi (29) rres = r f − rg = t 0, y f − y g , z f − z g (29) Rotational moment Mf to be applied to the reference float due to the buoyancy can therefore be Rotational moment M f to be applied to the reference float due to the buoyancy can therefore be expressed as follows: expressed as follows: M f  rres  Fb = rhres ×tF0b , y f  yg , z f i zg t 0 , gVf sin Roll , gVf cosRoll  (30) = t 0, ytf(− − zcos × t [0, ρgVf sin θ Roll , ρgVf cos θ Roll ] g  y f yg ,ygz)f gV f Roll  ( z f  zg ) gVf sin  Roll , 0, 0 i h (30) t y ) ρgV cos θ = t (y f− − (z f −), z g0), ρgV g ( y cos f RollRoll f sin θ Roll , 0, 0    gV  z sin 0 f f g Roll h i = t ρgVf (y f cos θ Roll + z g sin θ Roll ), 0, 0 where yg is also assumed to be zero due to the symmetrical shape of the reference float. Furthermore, the to measurement of the developed sensor for Furthermore, both the pitch where yg is also since assumed be zero due ranges to the symmetrical shape inclination of the reference float. and roll angle will be designed to be smaller than ±50 arc-seconds, by approximating cosθ Roll ≈ 1 and since the measurement ranges of the developed inclination sensor for both the pitch and roll angle will sinθ Roll ≈ θRoll, Mf in Equation (30) can be simplified as follows: be designed to be smaller than ±50 arc-seconds, by approximating cosθ ≈ 1 and sinθ ≈θ Mf





Roll

Roll

Roll,

t M f in Equation (30) can be simplified as follows: M f  gVf (  zg )Roll, 0, 0 (31)   t Mf = ρgVf (α +will z g )θbe , 0, 0 (31) As can be seen in the equation, the moment about the X-axis in Figure 7. The Rollgenerated equation of roll motion of the reference float can be obtained as follows by applying Mf in Equation be seen in the equation, the moment will be generated about the X-axis in Figure 7. (31)As to can Equation (26): The equation of roll motion of the reference float can be obtained as follows by applying Mf in d 2 Roll d Equation (31) to Equation (26): (32) IX  c Roll  gVf (  z g ) Roll 2 dt dt 2 d θ Roll dθ Roll =motion ρgVf (α of + zthe (32) It IX equation + cof pitch g ) θ Roll In the same manner, the reference float can be obtained. 2 dt dt should be noted that the natural frequencies of pitch and roll motions of the reference float fp and fr, In the samecan manner, the equation of pitch motion the reference float can be obtained. It should be respectively, be derived from Equation (32) as of follows: noted that the natural frequencies of pitch and roll motions of the reference float f p and f r , respectively, can be derived from Equation (32) as follows:

1 fr = 2π

s

s

−ρgVf α + zg −ρgVf α + zg 1 , fp = IX 2π IY

(33)

Sensors 2018, 18, x FOR PEER REVIEW

fr 

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

 gVf   zg

1 2



IX

,

fp 

1 2



 gVf   zg



IY

12 of (33) 22

Since α < 0, fp can exist over the region of zg < −α. Attention should be paid when designing the Since α < 0, f p can exist over the region of zg < −α. Attention should be paid when designing the reference float so that the frequency bandwidth of the three-axis inclination sensor will not overlap reference float so that the frequency bandwidth of the three-axis inclination sensor will not overlap the the natural frequency fp. natural frequency f p . 3. Design of the Three-Axis Inclination Sensor 3. Design of the Three-Axis Inclination Sensor 3.1.An An Optical Sensor Head 3.1. Optical Sensor Head the proposed three-axis inclinationsensor, sensor, optical sensor head the angle sensor based InIn the proposed three-axis inclination anan optical sensor head ofof the angle sensor based on the three-axis laser autocollimation [16] is employed, while a scale grating having linear pattern on the three-axis laser autocollimation [16] is employed, while a scale grating having linear pattern structureswith witha constant a constant periodp is p is employed a reflector and mounted the float. Figure structures period employed asas a reflector and is is mounted onon the float. Figure 99 shows a schematic of geometric the geometric relationship between the optical sensor head and grating. the scale shows a schematic of the relationship between the optical sensor head and the scale grating. As can be seen in the figure, a measurement laser beam made incident to the scale grating As can be seen in the figure, a measurement laser beam made incident to the scale grating will generate will generate zeroth-order diffracted beam, and first-order beams in positive and zeroth-order diffracted beam, and first-order diffracted beams indiffracted positive and negative directions. negative directions. It should be noted that the negative first-order diffracted beam is omitted in the It should be noted that the negative first-order diffracted beam is omitted in the figure for the sake figure for the sake of clarity. of clarity. X

Roll

QPD 1 (1st order detection)

Yaw Z

dr

Pitch

Yaw

Y r’

Yaw O

Z r 1st +0th order beam

n

Collimator objective (CO1) Collimator objective Diffraction grating (CO0) (pitch : p)

QPD 0 (0th order detection)

Figure 9. 9. Principle ofof the three-axis laser autocollimation. Figure Principle the three-axis laser autocollimation.

Two-dimensional angles scale grating about and/orY-axes Y-axeswith withrespect respecttoto the Two-dimensional tilttilt angles ofof thethe scale grating about thethe X-Xand/or the optical sensor head can be measured based on the two-axis laser autocollimation [17]. By detecting optical sensor head can be measured based on the two-axis laser autocollimation [17]. By detecting displacementsofofaafocused focused laser laser beam beam Δv V0-Vand H0-axes on the quadrant photo displacements ∆v00 and andΔh ∆h0 0along alongthe the 0 - and H0 -axes on the quadrant diode (QPD) for the detection of zeroth-order diffracted beam (QPD0), respectively,the theangular angular photo diode (QPD) for the detection of zeroth-order diffracted beam (QPD0), respectively, displacements of the scale grating Δθ Roll and ΔθPitch about the X- and Y-axes, respectively, can be displacements of the scale grating ∆θ Roll and ∆θ Pitch about the X- and Y-axes, respectively, can be obtained follows [27]: obtained asas follows [27]:

1 v 1 ∆h0h0 Pitch1  arctan ∆v0 0  Roll1  arctan ∆θRoll = arctan , ∆θ = arctan Pitch 2 2 2 f f , 2 f f

(34) (34)

where f is a focal length the collimator objective (CO0). Meanwhile,the thetilttilt angle scale grating where f is a focal length ofof the collimator objective (CO0). Meanwhile, angle ofof scale grating about the Z-axis does not affect the position focused laser beam QPD0; this fact means that the about the Z-axis does not affect the position ofof focused laser beam onon QPD0; this fact means that the change in yaw angle of the scale grating cannot be detected by the QPD0. However, on the other change in yaw angle of the scale grating cannot be detected by the QPD0. However, on the other hand, a QPD positive first-order diffracted beam (QPD1) detect the change in yaw hand, a QPD forfor thethe positive first-order diffracted beam (QPD1) can can detect the change in yaw angleangle of

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the scale grating. Now the diffraction angle of the first-order diffracted beam φ1st with respect to the normal of the scale grating can be expressed as follows [28]: φ1st

  λ = arcsin p

(35)

where λ is the wavelength of laser beam made incident to the scale grating. Now we denote a directional vector of the zeroth-order diffracted beam as n. Also, we denote the directional vectors of the positive first-order diffracted beam with the yaw angle of zero and ∆θ Yaw as r and r’, respectively. According to the geometric relationship, the following equation can be obtained [16]: r0 = r cos ∆θYaw + n(n · r)[1 − cos ∆θYaw ] + (n × r) sin ∆θYaw

(36)

On the assumption that ∆θ Yaw is small, denoting the r’ − r = dr, Equation (36) can be modified as follows: dr = r0 − r ∼ (37) = (n × r)∆θYaw The angle ∆ψZ between the vectors r and r’ can be expressed by using r and dr as follows:

kdrk ∆ψZ ∼ = ∆θYaw sin φ1st = krk

(38)

Since ∆ψZ is the angle of incidence of the first-order diffracted beam with respect to the collimator objective 1 (CO1), according to the principle of the laser autocollimation, the displacement ∆v1_Yawing of focused first-order diffracted beam on the QPD1 along the V 1 -axis due to ∆θ Yaw can be obtained as follows: f λ∆θYaw ∆v1_Yawing = f tan ∆ψZ ∼ (39) = f ∆ψZ = p By modifying the above equation, the change in yaw angle of the scale grating about the Z-axis ∆θ Yaw can be expressed as follows: ∆θYaw =

p ∆v f λ 1_Yawing

(40)

It should be noted that the first-order diffracted beam focused on the QPD1 could be moved along the V 1 -axis not only by ∆θ Yaw but also by ∆θ Pitch . The change in yaw angle of the scale gratings can therefore be obtained from the following equations: ∆θYaw =

 p p  p ∆v1_Yawing = ∆v1 − ∆v1_Pitching = (∆v1 − ∆v0 ) fλ fλ fλ

(41)

Figure 10 shows an optical design of the optical sensor head for the three-axis inclination sensor. The detailed design parameters of the optical sensor head are summarized in Table 3. As the light source for the optical sensor head, a laser diode with a wavelength of 685 nm was employed. As the two-dimensional photodetectors, quadrant photodiodes with insensitive gaps of 5 µm (S6795, Hamamatsu Photonics, Hamamatsu, Japan) and 10 µm (Hamamatsu Photonics S1651-03) were employed to detect the zeroth and first-order diffracted beams, respectively. It should be noted that the widths of insensitive gaps of the QPDs were small enough to detect the laser beam focused by the collimator objectives. All the optical components were assembled in a home-made optical sensor base designed in a size of within 100 mm × 100 mm. It should be noted that the influence of the earth curvature on the evaluation of pitch and roll angles cannot be neglected [29] in the case of evaluating stage systems having a long travel range in the X- and/or Y-directions, and needs to be corrected when further better measurement accuracy is required.

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Figure 10. 10. A A schematic schematic design design of of the the optical optical sensor sensor head. head. Figure Table 3. Parameters of the three-axis angle sensor. Table 3. Parameters of the three-axis angle sensor.

Parameters Light wavelength Light wavelengthAperture diameter Aperture diameter Grating period Grating period Focal lengthobjective of the collimator objective Focal length of the collimator Parameters

Value Value 685 nm 685 nm 2 mm 2 mm 1.67 µm 1.67 µm 25.4 mm25.4 mm

3.2. Reference Float 3.2. Reference Float Figure 11a shows a schematic of the designed reference float, which is composed of a float, a Figure 11a shows a schematic of the designed reference float, which is composed of a float, a scale scale grating, an optical window, a cover and screws. Regarding the simulation results described in grating, an optical window, a cover and screws. Regarding the simulation results described in the the previous section of this paper, aiming to reduce the yaw drift, the reference float is designed to previous section of this paper, aiming to reduce the yaw drift, the reference float is designed to be be as compact as possible. In addition, attentions have also been paid to suppress the total weight of as compact as possible. In addition, attentions have also been paid to suppress the total weight of the reference float by using materials with light weights for each of the components. The detailed the reference float by using materials with light weights for each of the components. The detailed design parameters of the reference float are summarized in Table 4. The design parameters of the design parameters of the reference float are summarized in Table 4. The design parameters of the reference float were determined mainly by the size of scale grating. From Equation (33), the natural reference float were determined mainly by the size of scale grating. From Equation (33), the natural frequency fr and fp of the reference float were evaluated to be 6.01 Hz. It should be noted that higher frequency fr and fp of the reference float were evaluated to be 6.01 Hz. It should be noted that higher fr and fp can be obtained by decreasing IX and IY with a design modification of the reference float. fr and fp can be obtained by decreasing IX and IY with a design modification of the reference float. Meanwhile, the design modification for decreasing IX and IY could lead to the increase of IZ, Meanwhile, the design modification for decreasing IX and IY could lead to the increase of IZ , resulting resulting in the increase of yaw drift of the reference float. Attentions should therefore be paid to in the increase of yaw drift of the reference float. Attentions should therefore be paid to carry out the carry out the design optimization of the reference float, which will be carried out in future work as design optimization of the reference float, which will be carried out in future work as the next step of the next step of the research. Figure 11b shows a photograph of the developed reference float, on the research. Figure 11b shows a photograph of the developed reference float, on which a scale grating which a scale grating is mounted. is mounted. In the proposed three-axis inclination sensor, the reference float is required to be held stationary along the X-, Y- and Z-directions while allowing its rotational motion about each axis. A supporting needle is therefore introduced in the system in such a way that the needle is fixed at the middle of the bottom surface of the casing, while a jig with dent is placed beneath the reference float as shown in Figure 11c; a tip of the needle is in a point contact with the bottom surface of the reference float through the jig, which makes the reference float possible to be held stationary along the X-, Y- and Z-directions, while allowing the rotational motion about each axis.

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Figure 11. Design of the reference float: (a) Exploded perspective view; (b) A photograph; (c) The floating unit and a supporting needle beneath the liquid surface. Figure 11. Design of the reference float: (a) Exploded perspective view; (b) A photograph; (c) The Table 4. Parameters the reference floating unit and a supporting needle beneath theof liquid surface. float. Parameters

Symbol Table 4. Parameters of the reference float.

Radius a Parameters Thickness Radius Draft T Thickness mass Moment of inertia about the Z-axis Iz Draft Moment of inertia about the mass IX /IY X/Y-axis Material Moment of inertia about the Z-axis Density ρ

Moment of inertia about the X/Y-axis

Symbol a T

Value 30 mm Value 10 mm 30 mm 8.8 mm 10 mm 24.4 × 10−3 kg 9.02 10−4 kg m2 8.8×mm -3

4 kg m2 5.17××1010−kg 24.4

Iz IX/IY

-4

2

Polyoxymethylene 9.02 × 10 kg m 3 1390 -4 kg/m2

5.17 × 10 kg m Material Polyoxymethylene In the proposed three-axis inclination sensor, the reference float is required to be 3held stationary Density  1390 kg/m along the X-, Y- and Z-directions while allowing its rotational motion about each axis. A supporting needle is therefore introduced in the system in such a way that the needle is fixed at the middle of 3.3. System Integration the bottom surface of the casing, while a jig with dent is placed beneath the reference float as shown The developed head is and reference float arethe integrated into the of three-axis inclination in Figure 11c; a tip ofsensor the needle in the a point contact with bottom surface the reference float sensor, athe schematic of which shown in Figure The three-axis composed of through jig, which makes isthe reference float 12. possible to be heldinclination stationarysensor along is the X-, Y- and the referencewhile float allowing and a three-axis angle detection unit,each which consists of the casing with an inner Z-directions, the rotational motion about axis. diameter of 240 mm filled with water, the optical sensor head and its alignment mechanisms. The 3.3. System Integration casing is rigidly connected to the optical sensor head through poles and the alignment mechanisms. The The reference float is madehead to float water float surface, it is point-supported its bottom developed sensor and on thethe reference are while integrated into the three-axisatinclination surface by a needle fixed on the bottom surface of the casing. All the sensor system hasofbeen sensor, a schematic of which is shown in Figure 12. The three-axis inclination sensor is composed the constructed in a diameter of approximately 300 mm and a height of 360 mm. reference float and a three-axis angle detection unit, which consists of the casing with an inner diameter of 240 mm filled with water, the optical sensor head and its alignment mechanisms. The casing is rigidly connected to the optical sensor head through poles and the alignment mechanisms. The reference float is made to float on the water surface, while it is point-supported at its bottom surface by a needle

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fixed on the bottom surface of the casing. All the sensor system has been constructed in a diameter of Sensors 2018, 18, x FOR PEER REVIEW 16 of 22 approximately 300 mm and a height of 360 mm.

Figure schematic the developed three-axis inclination sensor. Figure 12.12. AA schematic ofof the developed three-axis inclination sensor.

Evaluation the Basic Characteristics the Developed Three-Axis Inclination Sensor 4. 4. Evaluation ofof the Basic Characteristics ofof the Developed Three-Axis Inclination Sensor using the developed prototype three-axis inclinationsensor, sensor, the basic characteristics the ByBy using the developed prototype three-axis inclination the basic characteristics ofof the proposed method have been verified in experiments. First, the static characteristics of the proposed method have been verified in experiments. First, the static characteristics of the developed developed sensor head were evaluated. 13 shows photograph of the experimental optical sensoroptical head were evaluated. Figure 13 showsFigure a photograph of athe experimental setup. To avoid setup. To avoid the influence of instability of the reference float being floated on the liquid the influence of instability of the reference float being floated on the liquid surface, the referencesurface, float themounted referenceonfloat was system mounted on on a stage systemsurface fixed on the casing bottominsurface of the casing was a stage fixed the bottom of the which the water wasin which the water tilt wasangles drained. To reference apply tilt float, anglesa to the reference float, a two-axis PZT tilt drained. To apply to the two-axis precision PZT tilt stageprecision was employed stage was employed as the stage system. The tilt angles of the reference float about the and as the stage system. The tilt angles of the reference float about the Y- and Z-axes (the pitchY-and Z-axes (the pitch and yaw angles, respectively) were also measured by using a commercial two-axis yaw angles, respectively) were also measured by using a commercial two-axis laser autocollimator laser autocollimator (ElcomatGermany), 3000, Möller-Wedel, which was employed in theAsetup as a (Elcomat 3000, Möller-Wedel, which was Germany), employed in the setup as a reference. mirror reference. A mirror was mounted on a side face of the PZT tilt stage as a reflector for the was mounted on a side face of the PZT tilt stage as a reflector for the commercial laser autocollimator. commercial laser autocollimator. It should be noted that the commercial autocollimator in the setup It should be noted that the commercial autocollimator in the setup shown in Figure 13 could not shown in could measure roll angle (tilt angle about thebeX-axis). However, this could measure rollFigure angle 13 (tilt anglenot about the X-axis). However, this could measured by placing the be measured by placing the commercial autocollimator in such a way that the measurement axis of commercial autocollimator in such a way that the measurement axis of the commercial autocollimator the commercial coincided with the Y-axis, while re-mounting on the coincided with theautocollimator Y-axis, while re-mounting the reflector on the other sideface ofthe the reflector PZT tilt stage. other sideface of the PZT tilt stage. In the experiments, tilt angles were applied to the reference float In the experiments, tilt angles were applied to the reference float by the PZT tilt stage, while measuring by the PZT tilt stage, while measuring the tilt angles with both the developed optical sensor head the tilt angles with both the developed optical sensor head and the commercial laser autocollimator. and the commercial autocollimator. Figures 14a–c about show the Figure 14a–c show thelaser measured angular displacements the measured X-, Y- andangular Z-axes, displacements respectively. about the X-, Yand Z-axes, respectively. In the figures, the horizontal axes show the tilt angles In the figures, the horizontal axes show the tilt angles measured by the commercial laser autocollimator, measured by the commercial laser autocollimator, the vertical axes show the Linearity tilt angles while the vertical axes show the tilt angles measured by while the developed optical sensor head. measured by shown the developed opticalAs sensor head. errors good are also shown in can the be figures. errors are also in the figures. can be seenLinearity in the figures, correlations foundAs can be seen in the figures, good correlations can be found between the outputs from the developed between the outputs from the developed sensor and those from the commercial laser autocollimator. sensor and those the commercial autocollimator. Theevaluated nonlineartoerrors of the roll, pitch The nonlinear errorsfrom of the roll, pitch and laser yaw measurement were be ±0.93 arc-second, and yaw measurement were evaluated to be ±0.93 arc-second, ±1.05 arc-seconds and ±1.20 ±1.05 arc-seconds and ±1.20 arc-seconds, respectively, over a range of approximately 50 arc-seconds. arc-seconds, respectively, over a range of approximately 50 arc-seconds. These values are These values are comparable to the results shown in [16]. It should be noted that the sensitivity of comparable to the results shown in [16]. It should be noted that the sensitivity of yaw angle measurement was low compared with those of pitch and roll angle measurement; which is mainly due to the different principle of angle detection, as can be seen in Equations (35) and (41). It should also be noted that the calibration and sensitivity verification of the pitch and roll measurement of the developed three-axis inclination sensor can be carried out by the procedure described above,

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yaw angle measurement was low compared with those of pitch and roll angle measurement; which is mainly due to the different principle of angle detection, as can be seen in Equations (35) and (41). It should also beFOR noted that the calibration and sensitivity verification of the pitch and roll measurement Sensors PEER 17 Sensors 2018, 2018, 18, 18, xx FOR PEER REVIEW REVIEW 17 of of 22 22 of the developed three-axis inclination sensor can be carried out by the procedure described above, while yaw measurement cancan be carried out by the three-axis inclination sensor while those for the yaw measurement be out by the inclination while those thosefor forthe the yaw measurement can be carried carried outmounting by mounting mounting the three-axis three-axis inclination onto a precision spindlespindle equipped with precision rotary encoder. sensor onto equipped with rotary sensor onto aa precision precision spindle equipped with precision precision rotary encoder. encoder.

Reference Reference angle angle 10 10 arc-second/div. arc-second/div.

-4 -4 -6 -6

22 00 Linearity Linearity error error Reference Reference angle angle 10 10 arc-second/div. arc-second/div.

-2 -2 -4 -4 -6 -6

Sensor Sensor output output

66 44 22 00

Linearity Linearity error error Reference Reference angle angle 10 10 arc-second/div. arc-second/div.

-2 -2 -4 -4 -6 -6

Nonlinearerror errorarc-second arc-second Nonlinear

Linearity Linearity -2 -2 error error

66 44

Sensoryaw yawoutput output Sensor 10arc-second/div. arc-second/div. 10

22 00

Sensor Sensor output output

Nonlinearerror errorarc-second arc-second Nonlinear

66 44

Sensorpitch pitchoutput output Sensor 10arc-second/div. arc-second/div. 10

Sensor Sensor output output

Nonlinearerror errorarc-second arc-second Nonlinear

Sensorroll rolloutput output Sensor 10arc-second/div. arc-second/div. 10

Figure 13. setup. Figure 13. 13. A A schematic schematic of of the the experimental experimental setup. setup. Figure

Figure result of the angle sensor output. Verification result of of the the angle angle sensor sensor output. output. Figure 14. 14. Verification Verification result

After the verification static characteristics of the developed optical sensor head, stability of After the theverification verificationofof ofstatic staticcharacteristics characteristics developed optical sensor head, stability of After ofof thethe developed optical sensor head, stability of the the three-axis inclination sensor was verified in experiments. Figure 15 shows a setup for the the three-axis inclination was verified in experiments. shows setup for the three-axis inclination sensorsensor was verified in experiments. Figure 15Figure shows 15 a setup forathe experiment. experiment. At first, an effect of the point-support by the needle was verified. The reference float experiment. At first, an effect of the point-support by the needle was verified. The reference float At first, an effect of the point-support by the needle was verified. The reference float was made to float was made to float on the water surface in the casing without the support of the needle, and the was made to float on the water surface in the casing without the support of the needle, and the on the water surface in the casing without the support of the needle, and the variations of the three-axis variations of the three-axis outputs from the optical sensor head were evaluated. Figure 16a shows variations of the three-axis outputs optical sensor were the evaluated. shows outputs from the optical sensor headfrom werethe evaluated. Figure head 16a shows results. Figure As can 16a be seen in the results. As can be seen in the figure, it was difficult to continue the experiment due to the the results. As can be seen in the figure, it was difficult to continue the experiment due to the the the figure, it was difficult to continue the experiment due to the instability of the reference float on instability of reference float the water surface. the reference float was point-supported instability of the the reference float on on thewas water surface. Then, Then,by thethe reference float on was water surface. Then, the reference float point-supported needle fixed thepoint-supported bottom surface by the needle fixed on the bottom surface of the float, and the experiment was carried out by the needle fixed on the bottom surface of the float, and the experiment was carried outbeagain. again. of the float, and the experiment was carried out again. Figure 16b shows the result. As can seen Figure 16b shows the result. As can be seen in the figure, stable measurement of the pitch and roll Figure 16b shows the result. As can be seen in the figure, stable measurement of the pitch and in the figure, stable measurement of the pitch and roll angle was realized with the enhancementroll of angle was realized with the enhancement of the point-support by the needle. Furthermore, yaw angle was realized with the enhancement of the point-support by the needle. Furthermore, yaw the point-support by the needle. Furthermore, yaw angle measurement, which could not be carried angle measurement, which could not carried out when the float was freely angle measurement, which not be be carriedfloat out on when the reference reference was made made to tocarried freely out when the reference float could was made to freely the water surface, float was successfully float on the water surface, was successfully carried out with the help of point-support by the needle. float on the water surface, was successfully carried out with the help of point-support by the needle. out with the help of point-support by the needle. It should be noted that the water level in the casing It be that the water level in the was to 28 mm trial and It should should be noted noted that level the casing casing was determined determined to be be to 28adjust mm by by and error error was determined to be 28the mmwater by trial andinerror throughout the experiments thetrial contact force throughout the experiments to adjust the contact force between the reference float and needle for throughout the experiments to adjust the contact force between the reference float and needle for between the reference float and needle for the optimization of sensor stability. The standard deviations the optimization of sensor stability. The standard deviations of the pitch, roll and yaw angle output thethe optimization of sensor stability. The during standard pitch, roll and yaw output of pitch, roll and yaw angle output thedeviations period of of 60the s were evaluated to beangle 3.4, 3.9 and during the period of 60 ss were evaluated to be 3.4, 3.9 and 32.9 arc-seconds, respectively. A during the period of 60 were evaluated to be 3.4, 3.9 and 32.9 arc-seconds, respectively. A 32.9 arc-seconds, respectively. A relatively large deviation of the yaw angle output was considered relatively relatively large large deviation deviation of of the the yaw yaw angle angle output output was was considered considered to to be be mainly mainly due due to to lower lower yaw yaw measurement measurement sensitivity sensitivity and and instability instability induced induced by by standing standing waves waves in in water. water. It It should should be be noted noted that the results acquired by the developed prototype were not as good as expected from that the results acquired by the developed prototype were not as good as expected from the the principle principle of of three-axis three-axis laser laser autocollimation autocollimation [16] [16] due due to to the the instability instability of of the the reference reference float. float. Signal Signal processing processing techniques techniques such such as as low-pass low-pass filtering filtering have have aa possibility possibility of of decreasing decreasing the the resolution resolution to to

water, resulting in the reduction of deviation of yaw angle output. The experiment was extended to address another main concern of the developed three-axis inclination sensor: yaw drift induced by the liquid flow in the casing. Figure 17 shows a schematic of the experimental setup. The three-axis inclination sensor was mounted on a rotary stage that could freely be rotated about the Z-axis. Meanwhile, due to its heavy weight, the rotary stage 18 could Sensors 2018, 18, 398 of 22 not rotate the three-axis inclination sensor with its actuator. Therefore, an external linear slide was employed to give an angular displacement to the inclination sensor about the Z-axis. The external to be mainly dueplaced to lower yaw measurement sensitivity and by standing waves linear slide was beneath the casing, and was made toinstability push a rodinduced connecting the casing and in water. It should be noted that the results acquired by the developed prototype were not as good the optical sensor head. It should be noted that a magnet was embedded to the top plate of the as expected from three-axissensor laser autocollimation [16] due after to thegiving instability of the linear slide so thatthe theprinciple three-axisofinclination could be held stationary the angular reference float. about Signal processing such asAs low-pass filtering a possibility decreasing displacement the Z-axistechniques to the sensor. can be seen have in Figure 17, a ofcommercial the resolution towas the 1employed arc-secondaslevel in the applications low measurement frequency bandwidth of the autocollimator a reference sensor to where verifya the result by three-axis could be accepted. developedinclination three-axis sensor inclination sensor.

Figure on water water surface surface with with the the point-support point-support of of aa needle. needle. Figure 15. 15. The The reference reference float float floating floating on

Since it is difficult to eliminate all the external disturbances which result in the occurrence of the standing waves in water, in this paper, a ring-shaped cover plate was introduced into the three-axis inclination sensor to suppress the influences of standing waves. The cover plate with outer and inner diameters of 90 and 230 mm, respectively, and a thickness of 5 mm was made to float on the water surface. The ring-shaped cover was made to float freely on the liquid surface. The inner and outer diameters of the cover were determined in such a way that the collision between the inner edge of the cover and the reference float could be avoided, while allowing the cover to contact with the casing. Figure 16c shows the measured stability of the three-axis inclination sensor with the cover plate. As can be seen in the figure, deviation of pitch, roll and yaw angle output during the period of 60 s were reduced to be 1.8 arc-seconds, 2.9 arc-seconds and 18.6 arc-seconds, respectively; especially a low-frequency component in the yaw angle output was reduced. The introduced cover plate was considered to work as a damper for reducing the standing waves in water, resulting in the reduction of deviation of yaw angle output. The experiment was extended to address another main concern of the developed three-axis inclination sensor: yaw drift induced by the liquid flow in the casing. Figure 17 shows a schematic of the experimental setup. The three-axis inclination sensor was mounted on a rotary stage that could freely be rotated about the Z-axis. Meanwhile, due to its heavy weight, the rotary stage could not rotate the three-axis inclination sensor with its actuator. Therefore, an external linear slide was employed to give an angular displacement to the inclination sensor about the Z-axis. The external linear slide was placed beneath the casing, and was made to push a rod connecting the casing and the optical sensor head. It should be noted that a magnet was embedded to the top plate of the linear slide so that the three-axis inclination sensor could be held stationary after giving the angular displacement about the Z-axis to the sensor. As can be seen in Figure 17, a commercial autocollimator was employed as a reference sensor to verify the measurement result by the developed three-axis inclination sensor.

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Angular output Angular output 100 arc-second/div. 100 arc-second/div.

(a)

60 60

25.03 25.03 25.02

Temperature ℃ Temperature ℃

Angular output Angular output 100 arc-second/div. 100 arc-second/div.

200 200 150 Pitch angle 150 100 Pitch angle 100 50 050 Roll angle -50 0 Roll angle -50 -100 -100 (Yaw angle could not detect) -150 (Yaw angle could not detect) -150 -200 0 10 20 30 40 50 -200 0 10 20 Time30s 40 50 Time (a) s

Temperature ℃ Temperature ℃

Angular output Angular output 100 arc-second/div. 100 arc-second/div.

Pitch angle 25.02 Pitch angle 25.01 Roll angle 25.01 25.00 Roll angle 25.00 Temperature 24.99 Temperature 24.99 24.98 Yaw angle 24.98 24.97 0 10 20 30 40Yaw angle 50 60 24.97 0 10 20 Time 30s 40 50 60 Time s (b) (b) 26.94 26.94 26.93 Pitch angle 26.93 Pitch angle 26.92 Roll angle 26.92 26.91 Roll angle Temperature 26.91 26.90 Temperature 26.90 26.89 Yaw angle 26.89 26.88 0 10 20 30 40 Yaw50angle60 26.88 0 10 20 Time30s 40 50 60 Time s (c) (c) Figure inclinationsensor sensoroutput: output:(a)(a)Without Without fixing needle; (b) Figure16. 16.Stability Stabilityof of the three-axis inclination thethe fixing needle; (b) With Figure Stability ofwhile the using three-axis inclination sensor output: (a) Without needle; (b) the fixing needle while not the cover the liquid (c) Withthe the fixing needle and With the 16. fixing needle not using the plate coveron plate on thesurface; liquid surface; (c) fixing With the fixing With the fixing needle while not using the cover plate on the liquid surface; (c) With the fixing the cover on the liquid surface. needle and plate the cover plate on the liquid surface. needle and the cover plate on the liquid surface.

Figure 17. A schematic of the experimental setup for evaluation of the yaw drift. Figure 17. 17. A experimental setup setup for for evaluation evaluation of of the the yaw yaw drift. drift. Figure A schematic schematic of of the the experimental

Figure 18a shows the result without the cover plate. In the figure, both the angular output from Figure 18a shows the result the cover plate. In the figure, both the angular from the commercial andwithout the three-axis inclination plotted. As angular canoutput be seen in Figure 18aautocollimator shows the result without the cover plate. sensor In the were figure, both the output the commercial autocollimator and the three-axis inclination sensor were plotted. As can be seen in the figure, the developed three-axis inclination sensor successfully detected the given yaw motion. from the commercial autocollimator and the three-axis inclination sensor were plotted. As can be the figure, to thethe developed sensor successfully detected the given yaw motion. According output three-axis from theinclination commercial autocollimator, an angular displacement of According to the output from the commercial autocollimator, an angular displacement of

2000 1500

Autocollimator output

1000

Yaw drift: 900 arc-seconds

500 0

Angular output arc-second

Angular output arc-second

output from the inclination sensor was approximately 300 arc-seconds; this result implies that the yaw drift due to the angular motion generated by the linear slide was approximately 900 arc-seconds, which was larger than expected from the simulation described in the previous section of this paper. This result implies that the reference float is sensitive to the step input. A possible Sensors 2018, 18, 398 20 of 22 root-cause of these results, which were different from the ones estimated in the analytical calculations, is the frictional force between the reference float and the tip of needle. The change in fluid behavior could also induce the shear effect between the reference float and the casing. seen in the figure, the developed three-axis inclination sensor successfully detected the given yaw Meanwhile, on the other hand, variation of the inclination sensor output after giving the step input motion. According to the output from the commercial autocollimator, an angular displacement of to the float was quite small. In addition, the sensor output was quite stable: this result indicates the approximately 1200 arc-seconds was applied to the three-axis inclination sensor. Meanwhile, the possibility of the application of this sensor for the evaluation of a precision linear slide, which has output from the inclination sensor was approximately 300 arc-seconds; this result implies that the small angular error motion about each axis. Figure 18b shows the result with the cover plate. As can yaw drift due to the angular motion generated by the linear slide was approximately 900 arc-seconds, be seen in the figure, it was verified that the existence of the cover plate did not affect the yaw drift which was larger than expected from the simulation described in the previous section of this paper. of the reference float. This result implies that the reference float is sensitive to the step input. A possible root-cause of these It should be noted that, in this paper, attention has been paid to verifying the possibility of the results, which were different from the ones estimated in the analytical calculations, is the frictional proposed three-axis inclination sensor, and only basic experiments have been carried out to force between the reference float and the tip of needle. The change in fluid behavior could also induce demonstrate the feasibility of the developed prototype sensor. Further detailed investigation on the the shear effect between the reference float and the casing. Meanwhile, on the other hand, variation static and dynamic characteristics of the three-axis inclination sensor with an optimized setup, of the inclination sensor output after giving the step input to the float was quite small. In addition, influence of the frictional force between the supporting needle and the reference float, and the the sensor output was quite stable: this result indicates the possibility of the application of this sensor application to the evaluation of angular error motions of precision linear slides after the for the evaluation of a precision linear slide, which has small angular error motion about each axis. improvement of measurement resolution with the enhancement of signal processing techniques Figure 18b shows the result with the cover plate. As can be seen in the figure, it was verified that the such as low-pass filtering, will be carried out in future work. existence of the cover plate did not affect the yaw drift of the reference float.

Sensor output 0

2

4 6 Time s (a)

8

10

2000 1500

Autocollimator output

1000

Yaw drift: 900 arc-seconds

500 0

Sensor output 0

2

4 6 Time s (b)

8

10

Figure Yawing of the reference measured bydeveloped the developed inclination sensor: (a) Without Figure 18.18. Yawing of the reference floatfloat measured by the inclination sensor: (a) Without the ring-shaped cover cover plate; plate; (b) With ring-shaped cover plate. the ring-shaped (b)the With the ring-shaped cover plate.

5. Conclusions It should be noted that, in this paper, attention has been paid to verifying the possibility of the proposed three-axis inclination sensor, and only basic experiments been carried to In this paper, a new three-axis inclination sensor concept, which have is composed of anout optical demonstrate the feasibility of the developed prototype sensor. Further detailed investigation on the sensor head and a reference float capable of carrying out simultaneous measurement of three-axis static dynamic the three-axis inclination sensor withproposed an optimized setup, influence tilt and angles of a characteristics measurement of target, has been proposed. In the three-axis inclination of sensor, the frictional force between supporting andon theareference float, and application to the the reference float, the which is madeneedle to float water surface, hasthe been employed as a evaluation of angular error motions of precision linear slides after the improvement of measurement reference for the detection of three-axis tilt angles, while the optical sensor head based on the resolution enhancement of techniques as low-pass filtering, will with be three-axiswith laserthe autocollimation hassignal been processing employed to detect its such relative angular displacement carried out future work.float. Meanwhile, stability of the reference float is one of the most important respect toin the reference issues to be addressed for the proposed concept of the three-axis inclination sensor. Therefore, as 5. Conclusions the first step of this research, establishments of simplified kinematic models of the reference float this paper, a new three-axis sensor concept, which is composed of an opticalkinematic sensor forInangular motion about three inclination axes and theoretical analyses based on the established head and ahave reference capable of search carrying out simultaneous measurement three-axis concept tilt angles models been float carried out to for the possibility of realizing theofproposed ofofthe a measurement target, hassensor. been proposed. three-axis inclination the established reference three-axis inclination Results In of the theproposed analytical calculations basedsensor, on the float, which model is madehave to float on athat water hasofbeen employedfloat as adue reference the detection kinematic shown thesurface, yaw drift the reference to the for angular motion of of three-axis tilt angles, while the optical sensor head based on the three-axis laser autocollimation has been employed to detect its relative angular displacement with respect to the reference float. Meanwhile, stability of the reference float is one of the most important issues to be addressed for the proposed concept of the three-axis inclination sensor. Therefore, as the first step of this research, establishments of simplified kinematic models of the reference float for angular motion about three axes and theoretical analyses based on the established kinematic models have been carried out to search

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for the possibility of realizing the proposed concept of the three-axis inclination sensor. Results of the analytical calculations based on the established kinematic model have shown that the yaw drift of the reference float due to the angular motion of the casing in the inclination sensor, can be suppressed to be less than 1 arc-second for 60 s by optimizing the geometrical designs of the reference float and a casing in the three-axis inclination sensor. Furthermore, a prototype of the three-axis inclination sensor has been designed and constructed to search for the possibility of the proposed concept. To carry out measurement by the developed prototype, the reference float has to be placed on the water in a stable state. For stabilizing the reference float on the water surface while not restricting its angular motions about the three axes, a unique point-supporting mechanism with a tiny needle has been introduced to the sensor. Some basic experiments have been carried out by the developed sensor prototype, and an effect of the point-supporting mechanism has successfully been demonstrated. In addition, with the enhancement of a ring-shaped cover plate, which has been integrated into the sensor for the purpose of suppressing standing waves in water, stable simultaneous three-axis angular motion measurement has been realized. It should be noted that this paper has focused on the verification of the possibility of proposed concept of the three-axis inclination sensor. Further detailed investigation on the influence of the change in the moment of inertia Iz of the reference float, verification tests such as evaluation of static and dynamic characteristics of the proposed sensor including analysis on its measurement uncertainty will be carried out in future work, as well as further improvement of the resolution with the enhancement of signal processing techniques such as low-pass filtering. Acknowledgments: This research is supported by Japan Society for the Promotion of Science (JSPS). Author Contributions: Yuki Shimizu; Yuan-Liu Chen and Wei Gao conceived and designed the experiments; Satoshi Kataoka and Tatsuya Ishikawa performed the experiments; Yuki Shimizu, Satoshi Kataoka and Tatsuya Ishikawa analyzed the data; Xiuguo Chen and Hiraku Matsukuma contributed analysis tools; Yuki Shimizu and Wei Gao wrote the paper. Conflicts of Interest: The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.

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