Abstract

Conventional inclination measurement systems such as precision vials and capacitance measuring systems can measure inclination in only one direction at a time. We present what is believed to be a new optoelectronic system that can measure inclination angles along two orthogonal directions simultaneously by using a simple pendulum, two mirrors, a 2D position-sensing detector (PSD), and a laser diode. The light ray from the laser is projected onto a mirror that is fixed to a pendulum whose relative angle modifies in response to inclination changes of the inclinometer's housing. The light ray reflected by the mirror is sensed by the PSD, after which the signal can be interpreted by a PSD signal processor, recorded, or output to a computer. This study uses skew-ray tracing methodology to obtain implicit nonlinear system equations to model the relations of the relative inclination angles of the various components, PSD position, and world frame. A first-order Taylor series expansion is used to obtain a linear form of the system equations. To validate the proposed methodology, an actual prototype system is built and experimental results show that the performance of this system is excellent.

© 2006 Optical Society of America

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References

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  1. Wyler AG, Inclination Measurement with Compendium Yesterday Today Tomorrow (Wyler 2001).
  2. Tilt Measurement Limited, "ELH103 High Precision Submersible Sensing Head range +/-0.5 degrees," http://www.tilt-measurement.com (2004).
  3. GEMAC mbH, "2D Inclination Sensors with CAN Bus Interface," http://www.gemac-chemnitz.de (2005).
  4. F. Donatini, J. Monin, and G. Noyel, "Investigation of an inclination sensor using magnetic fluid for angle measurement up to 90 degrees," Meas. Sci. Technol. 6, 1-3 (1995).
    [CrossRef]
  5. O. Baltag, D. Costandache, and A. Salceanu, "Tilt measurement sensor," Sens. Actuators A 81, 226-339 (2000).
    [CrossRef]
  6. R. Olaru and D. D. Dragoi, "Inductive tilt sensor with magnets and magnetic fluid," Sens. Actuators A 120, 424-428 (2005).
    [CrossRef]
  7. P. S. Huang, Q. J. Hu, and F. P. Chiang, "Double three-step phase-shifting algorithm," Appl. Opt. 41, 4503-4509 (2002).
    [CrossRef] [PubMed]
  8. G. Pedrini, I. Alexeenko, W. Osten, and U. Schnars, "On-line surveillance of a dynamic process by a moving system based on pulsed digital holographic interferometry," Appl. Opt. 45, 935-943 (2006).
    [CrossRef] [PubMed]
  9. T.-T. Liao and P. D. Lin, "Analysis of optical elements with flat boundary surfaces," Appl. Opt. 42, 1191-1202 (2003).
    [CrossRef] [PubMed]

2006

2005

R. Olaru and D. D. Dragoi, "Inductive tilt sensor with magnets and magnetic fluid," Sens. Actuators A 120, 424-428 (2005).
[CrossRef]

2003

2002

2000

O. Baltag, D. Costandache, and A. Salceanu, "Tilt measurement sensor," Sens. Actuators A 81, 226-339 (2000).
[CrossRef]

1995

F. Donatini, J. Monin, and G. Noyel, "Investigation of an inclination sensor using magnetic fluid for angle measurement up to 90 degrees," Meas. Sci. Technol. 6, 1-3 (1995).
[CrossRef]

AG, Wyler

Wyler AG, Inclination Measurement with Compendium Yesterday Today Tomorrow (Wyler 2001).

Alexeenko, I.

Baltag, O.

O. Baltag, D. Costandache, and A. Salceanu, "Tilt measurement sensor," Sens. Actuators A 81, 226-339 (2000).
[CrossRef]

Chiang, F. P.

Costandache, D.

O. Baltag, D. Costandache, and A. Salceanu, "Tilt measurement sensor," Sens. Actuators A 81, 226-339 (2000).
[CrossRef]

Donatini, F.

F. Donatini, J. Monin, and G. Noyel, "Investigation of an inclination sensor using magnetic fluid for angle measurement up to 90 degrees," Meas. Sci. Technol. 6, 1-3 (1995).
[CrossRef]

Dragoi, D. D.

R. Olaru and D. D. Dragoi, "Inductive tilt sensor with magnets and magnetic fluid," Sens. Actuators A 120, 424-428 (2005).
[CrossRef]

Hu, Q. J.

Huang, P. S.

Liao, T.-T.

Lin, P. D.

Monin, J.

F. Donatini, J. Monin, and G. Noyel, "Investigation of an inclination sensor using magnetic fluid for angle measurement up to 90 degrees," Meas. Sci. Technol. 6, 1-3 (1995).
[CrossRef]

Noyel, G.

F. Donatini, J. Monin, and G. Noyel, "Investigation of an inclination sensor using magnetic fluid for angle measurement up to 90 degrees," Meas. Sci. Technol. 6, 1-3 (1995).
[CrossRef]

Olaru, R.

R. Olaru and D. D. Dragoi, "Inductive tilt sensor with magnets and magnetic fluid," Sens. Actuators A 120, 424-428 (2005).
[CrossRef]

Osten, W.

Pedrini, G.

Salceanu, A.

O. Baltag, D. Costandache, and A. Salceanu, "Tilt measurement sensor," Sens. Actuators A 81, 226-339 (2000).
[CrossRef]

Schnars, U.

Appl. Opt.

Meas. Sci. Technol.

F. Donatini, J. Monin, and G. Noyel, "Investigation of an inclination sensor using magnetic fluid for angle measurement up to 90 degrees," Meas. Sci. Technol. 6, 1-3 (1995).
[CrossRef]

Sens. Actuators

O. Baltag, D. Costandache, and A. Salceanu, "Tilt measurement sensor," Sens. Actuators A 81, 226-339 (2000).
[CrossRef]

R. Olaru and D. D. Dragoi, "Inductive tilt sensor with magnets and magnetic fluid," Sens. Actuators A 120, 424-428 (2005).
[CrossRef]

Other

Wyler AG, Inclination Measurement with Compendium Yesterday Today Tomorrow (Wyler 2001).

Tilt Measurement Limited, "ELH103 High Precision Submersible Sensing Head range +/-0.5 degrees," http://www.tilt-measurement.com (2004).

GEMAC mbH, "2D Inclination Sensors with CAN Bus Interface," http://www.gemac-chemnitz.de (2005).

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Figures (6)

Fig. 1
Fig. 1

Light rays at a flat boundary surface.

Fig. 2
Fig. 2

(Color online) Schematic of the proposed optoelectronic inclinometer.

Fig. 3
Fig. 3

Percentage errors of the inclination angles from the linear form of the system equations.

Fig. 4
Fig. 4

Photograph of the laboratory-built optoelectronic inclinometer.

Fig. 5
Fig. 5

(Color online) Results of system calibration.

Fig. 6
Fig. 6

(Color online) Results of the system stability test.

Tables (3)

Tables Icon

Table 1 Relative Performances of the Inclination Measurement Systems

Tables Icon

Table 2 Components of the Prototype Optoelectronic Inclinometer

Tables Icon

Table 3 Performances of the Prototype Optoelectronic Inclinometer and Wyler Minilevel NT11

Equations (214)

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P i x i + P i y j + P i z k
P j i = [ P i x P i y P i z 1 ] T
P j i
( x y z ) j
P j i
P k i
P k i = A k j P j i
A k j
4 × 4
( x y z ) j
( x y z ) k
l i j = [ l i x l i y l i z 0 ] T
( x y z ) 1
r i i
[ β i 0 0 1 ] T ( β i 0 )
y i
r i i = Rot ( y i , α i ) [ β i 0 0 1 ] = [ β i C α i 0 β i S α i 1 ] ,
Rot
β i
α i
n i i
n i i = s i ( ( r i i ) α i ( r i i ) β i ) / | ( r i i ) α i ( r i i ) β i | = s i [ 0 0 1 0 ] T ,
s i
+ 1
1
C θ i > 0
r i i
n i i
( x y z ) 1
( x y z ) 1
( x y z ) i
A 1 i = [ I i x J i x K i x t i x I i y J i y K i y t i y I i z J i z K i z t i z 0 0 0 1 ] .
n i
( x y z ) 1
n i = [ n i x n i y n i z 0 ] T = A i 1 n i i = s i [ I i y J i y K i y 0 ] T .
P i 1 = [ P i 1 x P i 1 y P i 1 z 1 ] T
l i 1 = [ l i 1 x l i 1 y l i 1 z 0 ] T
r i
Q i
Q i = [ P i 1 x + l i 1 x λ P i 1 y + l i 1 y λ P i 1 z + l i 1 z λ 1 ] T ,
λ 0
P i 1
Q i
λ = λ i
P i
P i = [ P i x P i y P i z 1 ] T = [ P i 1 x + l i 1 x λ i P i 1 y + l i 1 y λ i P i 1 z + l i 1 z λ i 1 ] T
r i i
Q i i
[ Q i i = A 1 i Q i
Q i
( x y z ) i
λ i = ( I i y P i 1 x + J i y P i 1 y + K i y P i 1 z + t i y ) I i y l i 1 x + J i y l i 1 y + K i y l i 1 z = B i G i .
α i
β i
l i
θ i
C θ i = l i 1               T n i = s i ( I i y l i 1 x + J iy l i 1 y + K i y l i 1 z ) .
m i
n i
l i 1
m i = [ m i x m i y m i z 0 ] T = n i × l i 1 / S θ i .
l i
n i
m i
θ i
l i = [ m i x       2 ( 1 C θ i ) + C θ i m i y m i x ( 1 C θ i ) m i z S θ i m iz m ix ( 1 C θ i ) + m i y S θ i 0 m i x m i y ( 1 C θ i ) + m i z S θ i m iy 2 ( 1 C θ i ) + C θ i m i z m i y ( 1 C θ i ) m ix S θ i 0 m i x m i z ( 1 C θ i ) m i y S θ i m i y m i z ( 1 C θ i ) + m i x S θ i m iz 2 ( 1 C θ i ) + C θ i 0 0 0 0 0 ] [ n i x n i y n i z 0 ] .
S θ i m i × n i = ( n i × l i 1 ) × n i = l i 1 ( n i T l i 1 ) n i = l i 1 + n i θ i
l i = [ l i x l i y l i z 0 ] = [ l i 1 x 2 I i y ( I i y l i 1 x + J i y l i 1 y + K i y l i 1 z ) l i 1 y 2 J i y ( I i y l i 1 x + J i y l i 1 y + K i y l i 1 z ) l i 1 z 2 K i y ( I i y l i 1 x + J i y l i 1 y + K i y l i 1 z ) 0 ] .
r i
P i
l i
l 1
l 2
n 2
l 1
l 2
P 4
( x y z ) 1
l 1
x 1 z 1
x 1
( x y z ) i
( i = 2 , 3 , 4 )
( x y z ) 1
( x y z ) 3
( x y z ) 4
4 × 4
A 1 3 = Rot ( z , 90 ° ) Rot ( y , 90 ° ) × Trans ( m 1 m 2 , 0 , 0 ) = [ 0 1 0 0 0 0 1 0 1 0 0 m 1 + m 2 0 0 0 1 ] ,
A 1 4 = Rot ( x , θ 90 ° ) Rot ( z , 90 ° ) Rot ( y , 90 ° ) × Trans ( m 1 m 2 m 3 , 0 , m 4 ) = [ 0 1 0 0 C θ 0 S θ ( m 1 + m 2 + m 3 ) C θ m 4 S θ S θ 0 C θ ( m 1 + m 2 + m 3 ) S θ + m 4 C θ 0 0 0 1 ] ,
( x y z ) 0
( x y z ) 1
A 1 0 = Rot ( z , 90 ° ) Rot ( x , 180 ° ) Trans ( m 1 , 0 , m 5 ) = [ 0 1 0 0 1 0 0 m 1 0 0 1 m 5 0 0 0 1 ] .
0
( x y z ) 0
z 0
( x y z ) 0
( x y z ) 0
( x y z ) 0
( x y z ) 0
ϕ y
y 0
ϕ x
x 0
ϕ x
ϕ y
( x y z ) 0
( x y z ) 0
A 0 0 = Rot ( y , ϕ y ) Rot ( x , ϕ x ) = [ C ϕ y S ϕ y S ϕ x S ϕ y C ϕ x 0 0 C ϕ x S ϕ x 0 S ϕ y C ϕ y S ϕ x C ϕ y C ϕ x 0 0 0 0 1 ] .
( x y z ) 2
x 2 z 2
( x y z ) 0
A 2 0 = Rot ( x , 90 ° ) Trans ( 0 , m 6 , 0 ) = [ 1 0 0 0 0 0 1 0 0 1 0 m 6 0 0 0 1 ] .
( x y z ) 1
( x y z ) 2
A 1 2 = A 0 2 A 0 0 A 1 0 = ( A 2 0 ) 1 A 0 0 A 1 0 = [ S ϕ y S ϕ x C ϕ y S ϕ y C ϕ x m 1 S ϕ y S ϕ x + m 5 S ϕ y C ϕ x C ϕ y S ϕ x S ϕ y C ϕ y C ϕ x m 1 C ϕ y S ϕ x + m 5 C ϕ y C ϕ x m 6 C ϕ x 0 S ϕ x m 1 C ϕ x + m 5 S ϕ x 0 0 0 1 ] .
l 1
P 1 = [ 0 0 0 1 ] T
l 1 = [ C θ 1 0 S θ 1 0 ] T
i = 2
i = 3
i = 4
l 2
l 3
P 2
P 3
P 4
P 2 = [ P 2 x P 2 y P 2 z 1 ] T = [ C θ 1 λ 2 0 S θ 1 λ 2 1 ] T ,
λ 2 = C ϕ y ( m 1 S ϕ x m 5 C ϕ x ) + m 6 C ϕ y ( S ϕ x C θ 1 C ϕ x S θ 1 ) ,
l 2 = [ l 2 x l 2 y l 2 z 0 ] = [ C θ 1 2 C 2 ϕ y S ϕ x ( S ϕ x C θ 1 C ϕ x S θ 1 ) 2 S ϕ y C ϕ y ( S ϕ x C θ 1 C ϕ x S θ 1 ) S θ 1 + 2 C 2 ϕ y C ϕ x ( S ϕ x C θ 1 C ϕ x S θ 1 ) 0 ] ,
P 3 = [ P 3 x P 3 y P 3 z 1 ] = [ P 2 x + l 2 x λ 3 P 2 y + l 2 y λ 3 P 2 z + l 2 z λ 3 1 ] = [ C θ 1 λ 2 + l 2 x λ 3 l 2 y λ 3 0 1 ] ,
λ 3 = λ 2 S θ 1 S θ 1 + 2 C 2 ϕ y C ϕ x ( S ϕ x C θ 1 C ϕ x S θ 1 ) ,
l 3 = [ l 3 x l 3 y l 3 z 0 ] = [ C θ 1 2 C 2 ϕ y S ϕ x ( S ϕ x C θ 1 C ϕ x S θ 1 ) 2 S ϕ y C ϕ y ( S ϕ x C θ 1 C ϕ x S θ 1 ) - S θ 1 - 2 C 2 ϕ y C ϕ x ( S ϕ x C θ 1 C ϕ x S θ 1 ) 0 ] ,
P 4 = [ P 4 x P 4 y P 4 z 1 ] = [ P 3 x + l 3 x λ 4 P 3 y + l 3 y λ 4 P 3 z + l 3 z λ 4 1 ] = [ C θ 1 λ 2 + l 2 x λ 3 + l 3 x λ 4 l 2 y λ 3 + l 3 x λ 4 l 3 z λ 4 1 ] ,
λ 4 = C θ 4 P 3 x + ( m 1 + m 2 + m 3 ) C θ 4 + m 4 S θ 4 C θ 4 l 3 x + S θ 4 l 3 z .
P 4
( x y z ) 1
( x y z ) 4
P 4 4 = A 1 4 P 4 = [ P 4 x 4 0 P 4 z 4 1 ] = [ P 4 x 4 ( ϕ x , ϕ y ) 0 P 4 z 4 ( ϕ x , ϕ y ) 1 ] = [ P 3 y + l 3 y λ 4 0 P 3 x S θ 4 + ( l 3 x S θ 4 l 3 z C θ 4 ) λ 4 ( m 1 + m 2 + m 3 ) S θ 4 + m 4 C θ 4 1 ] .
[ P 4 x 4 P 4 z 4 ] T
ϕ x
ϕ y
ϕ x = 0
ϕ y = 0
[ P ¯ 4 x 4 P ¯ 4 z 4 ] = [ P 4 x 4 ( 0 , 0 ) P 4 z 4 ( 0 , 0 ) ] = [ 0 0 ] .
[ X ( ϕ x , ϕ y ) Z ( ϕ x , ϕ y ) ] = [ P 4 x 4 P 4 z 4 ] [ P ¯ 4 x 4 P ¯ 4 z 4 ] = [ P 3 y + l 3 y λ 4 0 P 3 x S θ 4 + ( l 3 x S θ 4 l 3 z C θ 4 ) λ 4 ( m 1 + m 2 + m 3 ) S θ 4 + m 4 C θ 4 1 ] .
[ P 4 x 4 P 4 z 4 ] T
[ P ¯ 4 x 4 P ¯ 4 z 4 ] T
ϕ x = ϕ y = 0
m i ( i = 1 , 2 , 3 , 4 )
m 1
A 2 0 = Rot ( x , 90 ° )
Trans ( 0 , m 6 , 0 )
Trans ( Δ t 2 x , Δ t 2 y , Δ t 2 z )
Rot ( z , Δ α 2 z )
Rot ( y , Δ α 2 y )
Rot ( x , Δ α 2 x )
Δ t 2 x , Δ t 2 y ,
Δ t 2 z , Δ α 2 x ,
Δ α 2 y
Δ α 2 z
[ X ( ϕ x , ϕ y ) Z ( ϕ x , ϕ y ) ] T
ϕ x
ϕ y
[ X ( ϕ x , ϕ y )
Z ( ϕ x , ϕ y ) ] T
ϕ x
ϕ y
ϕ x = ϕ y = 0
[ X ( ϕ x , ϕ y ) Z ( ϕ x , ϕ y ) ] [ X ( 0 , 0 ) Z ( 0 , 0 ) ] + [ X ϕ x X ϕ y Z ϕ x Z ϕ y ] ϕ x = ϕ y = 0 [ ϕ x ϕ y ] ,
X ( 0 , 0 ) = 0 ,
Z ( 0 , 0 ) = 0 ,
X ( ϕ x , ϕ y ) ϕ x | ϕ x = 0 ϕ y = 0 = 0 ,
X ( ϕ x , ϕ y ) ϕ y | ϕ x = 0 ϕ y = 0 = 2 ( m 5 + m 6 ) 2 S 2 θ 1 ( m 1 + m 2 + m 3 + m 4 2 m 5 + 2 m 6 ) ,
Z ( ϕ x , ϕ y ) ϕ x | ϕ x = 0 ϕ y = 0 = 2 ( m 5 m 6 ) / S θ 1 + 2 S θ 1 ( m 2 + m 3 + m 4 m 5 + m 6 ) ,
Z ( ϕ x , ϕ y ) ϕ y | ϕ x = 0 ϕ y = 0 = 0.
[ ϕ x ϕ y ] T
[ ϕ x ϕ y ] = [ Z ( ϕ x , ϕ y ) / [ 2 ( m 5 m 6 ) / S θ 1 + 2 S θ 1 ( m 2 + m 3 + m 4 m 5 + m 6 ) ] X ( ϕ x , ϕ y ) / [ 2 ( m 5 + m 6 ) 2 S 2 θ 1 ( m 1 + m 2 + m 3 + m 4 2 m 5 + 2 m 6 ) ] ] .
ϕ x
ϕ y
θ = 45 °
m 1 = m 2 = m 3 = m 4 = m 6 = 50   mm
m 5 = 100   mm
ϕ y = 0
± 20 °
5   arc   min
range   of   ± 4000   arc   min
ϕ x
ϕ y
[ P ¯ 4 x 4 P ¯ 4 z 4 ] T
ϕ x = ϕ y = 0
P 4
ϕ x = 68   arc   min
ϕ x = 68   arc   min
ϕ y
10   arc   min
ϕ x
ϕ y
ϕ y = 68   arc   min
ϕ y = 68   arc   min
ϕ x
ϕ y
ϕ x
ϕ y
ϕ x
ϕ y
1 min
± 3   μm
60   s
0 .2   arc   sec
2 .0   arc   sec
Rot ( x , α x ) = [ 1 0 0 0 0 C α x S α x 0 0 S α x C α x 0 0 0 0 1 ]
Rot ( y , α y ) = [ C α y 0 S α y 0 0 1 0 0 S α y 0 C α y 0 0 0 0 1 ]
Rot ( z , α z ) = [ C α z S α z 0 0 S α z C α z 0 0 0 0 1 0 0 0 0 1 ]
Trans ( t x , t y , t z ) = [ 1 0 0 t x 0 1 0 t y 0 0 1 t z 0 0 0 1 ] .

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