Abstract

Remote measurement of object orientation is used in various scientific fields, such as robotics, optics, and biology (e.g., optical tweezers). Roll angle is one of the three angles that describe the orientation of an object in space. A common method to measure the roll angle is based on analyzing the polarization of the backreflection of a beam. The accuracy of the measurement is degraded by low signal-to-noise ratio (SNR). The low SNR is the result of the large distance between the measurement device and the object, or due to the small backreflection cross section. We perform a laboratory experiment and derive a mathematical model for the probability density function of the measured roll angle and its expectation value. This model makes it possible to calculate the accuracy of the roll angle measurement at low SNRs. Experiments and theoretical analysis using our model were performed and good agreement between the two approaches has been found.

© 2014 Optical Society of America

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References

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  1. S. Arnon, “Lunar optical wireless communication and navigation network for robotic and human exploration,” Proc. SPIE 8162, 816202 (2011).
    [CrossRef]
  2. K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods 5, 491–505 (2008).
    [CrossRef]
  3. I. Rahneberg, H.-J. Büchner, and G. Jäger, “Optical system for the simultaneous measurement of two-dimensional straightness errors and the roll angle,” Proc. SPIE 7356, 73560S (2009).
    [CrossRef]
  4. G. C. Gilbreath, N. G. Creamer, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, J. A. Vasquez, R. Mahon, E. Oh, P. G. Goetz, and S. Mozersky, “Modulating retro-reflectors for space, tracking, acquisition and ranging using multiple quantum well technology,” Proc. SPIE 4821, 494 (2002).
    [CrossRef]
  5. Y. Zhai, Q. Feng, and B. Zhang, “A simple roll measurement method based on a rectangular-prism,” Opt. Laser Technol. 44, 839–843 (2012).
    [CrossRef]
  6. E. Shi, J. Guo, Z. Jia, and Y. Huang, “Theoretic study on new method for roll angle measurement of machines,” in IEEE International Conference on Automation and Logistics (IEEE, 2008), pp. 2722–2726.
  7. E. Plosker, D. Bykhovsky, and S. Arnon, “Evaluation of the estimation accuracy of polarization-based roll angle measurement,” Appl. Opt. 52, 5158–5164 (2013).
    [CrossRef]
  8. D. V. Hinkley, “On the ratio of two correlated normal random variables,” Biometrika 56, 635–639 (1969).
    [CrossRef]
  9. G. Marsaglia, “Ratios of normal variables,” J. Stat. Software 16, 1–10 (2006).
  10. T. Pham-Gia, N. Turkkan, and E. Marchand, “Density of the ratio of two normal random variables and applications,” Commun. Stat., Theory Methods 35, 1569–1591 (2006).
    [CrossRef]
  11. S. G. Lambert and W. L. Casey, Laser Communications in Space (Artech House, 1995).
  12. E. Hecht, Optics, 4th ed. (Addison-Wesley, 2001).
  13. N. Blaunstein, S. Arnon, N. Kopeika, and A. Zilberman, Applied Aspects of Optical Communication and LIDAR (CRC Press, 2010).
  14. S. Arnon, J. Barry, G. Karagiannidis, R. Schober, and M. Uysal, eds., Advanced Optical Wireless Communication Systems (Cambridge University, 2012).

2013

2012

Y. Zhai, Q. Feng, and B. Zhang, “A simple roll measurement method based on a rectangular-prism,” Opt. Laser Technol. 44, 839–843 (2012).
[CrossRef]

2011

S. Arnon, “Lunar optical wireless communication and navigation network for robotic and human exploration,” Proc. SPIE 8162, 816202 (2011).
[CrossRef]

2009

I. Rahneberg, H.-J. Büchner, and G. Jäger, “Optical system for the simultaneous measurement of two-dimensional straightness errors and the roll angle,” Proc. SPIE 7356, 73560S (2009).
[CrossRef]

2008

K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods 5, 491–505 (2008).
[CrossRef]

2006

G. Marsaglia, “Ratios of normal variables,” J. Stat. Software 16, 1–10 (2006).

T. Pham-Gia, N. Turkkan, and E. Marchand, “Density of the ratio of two normal random variables and applications,” Commun. Stat., Theory Methods 35, 1569–1591 (2006).
[CrossRef]

2002

G. C. Gilbreath, N. G. Creamer, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, J. A. Vasquez, R. Mahon, E. Oh, P. G. Goetz, and S. Mozersky, “Modulating retro-reflectors for space, tracking, acquisition and ranging using multiple quantum well technology,” Proc. SPIE 4821, 494 (2002).
[CrossRef]

1969

D. V. Hinkley, “On the ratio of two correlated normal random variables,” Biometrika 56, 635–639 (1969).
[CrossRef]

Arnon, S.

E. Plosker, D. Bykhovsky, and S. Arnon, “Evaluation of the estimation accuracy of polarization-based roll angle measurement,” Appl. Opt. 52, 5158–5164 (2013).
[CrossRef]

S. Arnon, “Lunar optical wireless communication and navigation network for robotic and human exploration,” Proc. SPIE 8162, 816202 (2011).
[CrossRef]

N. Blaunstein, S. Arnon, N. Kopeika, and A. Zilberman, Applied Aspects of Optical Communication and LIDAR (CRC Press, 2010).

Blaunstein, N.

N. Blaunstein, S. Arnon, N. Kopeika, and A. Zilberman, Applied Aspects of Optical Communication and LIDAR (CRC Press, 2010).

Büchner, H.-J.

I. Rahneberg, H.-J. Büchner, and G. Jäger, “Optical system for the simultaneous measurement of two-dimensional straightness errors and the roll angle,” Proc. SPIE 7356, 73560S (2009).
[CrossRef]

Bykhovsky, D.

Casey, W. L.

S. G. Lambert and W. L. Casey, Laser Communications in Space (Artech House, 1995).

Creamer, N. G.

G. C. Gilbreath, N. G. Creamer, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, J. A. Vasquez, R. Mahon, E. Oh, P. G. Goetz, and S. Mozersky, “Modulating retro-reflectors for space, tracking, acquisition and ranging using multiple quantum well technology,” Proc. SPIE 4821, 494 (2002).
[CrossRef]

Feng, Q.

Y. Zhai, Q. Feng, and B. Zhang, “A simple roll measurement method based on a rectangular-prism,” Opt. Laser Technol. 44, 839–843 (2012).
[CrossRef]

Gilbreath, G. C.

G. C. Gilbreath, N. G. Creamer, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, J. A. Vasquez, R. Mahon, E. Oh, P. G. Goetz, and S. Mozersky, “Modulating retro-reflectors for space, tracking, acquisition and ranging using multiple quantum well technology,” Proc. SPIE 4821, 494 (2002).
[CrossRef]

Goetz, P. G.

G. C. Gilbreath, N. G. Creamer, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, J. A. Vasquez, R. Mahon, E. Oh, P. G. Goetz, and S. Mozersky, “Modulating retro-reflectors for space, tracking, acquisition and ranging using multiple quantum well technology,” Proc. SPIE 4821, 494 (2002).
[CrossRef]

Guo, J.

E. Shi, J. Guo, Z. Jia, and Y. Huang, “Theoretic study on new method for roll angle measurement of machines,” in IEEE International Conference on Automation and Logistics (IEEE, 2008), pp. 2722–2726.

Hecht, E.

E. Hecht, Optics, 4th ed. (Addison-Wesley, 2001).

Hinkley, D. V.

D. V. Hinkley, “On the ratio of two correlated normal random variables,” Biometrika 56, 635–639 (1969).
[CrossRef]

Huang, Y.

E. Shi, J. Guo, Z. Jia, and Y. Huang, “Theoretic study on new method for roll angle measurement of machines,” in IEEE International Conference on Automation and Logistics (IEEE, 2008), pp. 2722–2726.

Jäger, G.

I. Rahneberg, H.-J. Büchner, and G. Jäger, “Optical system for the simultaneous measurement of two-dimensional straightness errors and the roll angle,” Proc. SPIE 7356, 73560S (2009).
[CrossRef]

Jia, Z.

E. Shi, J. Guo, Z. Jia, and Y. Huang, “Theoretic study on new method for roll angle measurement of machines,” in IEEE International Conference on Automation and Logistics (IEEE, 2008), pp. 2722–2726.

Kopeika, N.

N. Blaunstein, S. Arnon, N. Kopeika, and A. Zilberman, Applied Aspects of Optical Communication and LIDAR (CRC Press, 2010).

Lambert, S. G.

S. G. Lambert and W. L. Casey, Laser Communications in Space (Artech House, 1995).

Mahon, R.

G. C. Gilbreath, N. G. Creamer, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, J. A. Vasquez, R. Mahon, E. Oh, P. G. Goetz, and S. Mozersky, “Modulating retro-reflectors for space, tracking, acquisition and ranging using multiple quantum well technology,” Proc. SPIE 4821, 494 (2002).
[CrossRef]

Marchand, E.

T. Pham-Gia, N. Turkkan, and E. Marchand, “Density of the ratio of two normal random variables and applications,” Commun. Stat., Theory Methods 35, 1569–1591 (2006).
[CrossRef]

Marsaglia, G.

G. Marsaglia, “Ratios of normal variables,” J. Stat. Software 16, 1–10 (2006).

Meehan, T. J.

G. C. Gilbreath, N. G. Creamer, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, J. A. Vasquez, R. Mahon, E. Oh, P. G. Goetz, and S. Mozersky, “Modulating retro-reflectors for space, tracking, acquisition and ranging using multiple quantum well technology,” Proc. SPIE 4821, 494 (2002).
[CrossRef]

Mozersky, S.

G. C. Gilbreath, N. G. Creamer, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, J. A. Vasquez, R. Mahon, E. Oh, P. G. Goetz, and S. Mozersky, “Modulating retro-reflectors for space, tracking, acquisition and ranging using multiple quantum well technology,” Proc. SPIE 4821, 494 (2002).
[CrossRef]

Nagy, A.

K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods 5, 491–505 (2008).
[CrossRef]

Neuman, K. C.

K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods 5, 491–505 (2008).
[CrossRef]

Oh, E.

G. C. Gilbreath, N. G. Creamer, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, J. A. Vasquez, R. Mahon, E. Oh, P. G. Goetz, and S. Mozersky, “Modulating retro-reflectors for space, tracking, acquisition and ranging using multiple quantum well technology,” Proc. SPIE 4821, 494 (2002).
[CrossRef]

Pham-Gia, T.

T. Pham-Gia, N. Turkkan, and E. Marchand, “Density of the ratio of two normal random variables and applications,” Commun. Stat., Theory Methods 35, 1569–1591 (2006).
[CrossRef]

Plosker, E.

Rabinovich, W. S.

G. C. Gilbreath, N. G. Creamer, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, J. A. Vasquez, R. Mahon, E. Oh, P. G. Goetz, and S. Mozersky, “Modulating retro-reflectors for space, tracking, acquisition and ranging using multiple quantum well technology,” Proc. SPIE 4821, 494 (2002).
[CrossRef]

Rahneberg, I.

I. Rahneberg, H.-J. Büchner, and G. Jäger, “Optical system for the simultaneous measurement of two-dimensional straightness errors and the roll angle,” Proc. SPIE 7356, 73560S (2009).
[CrossRef]

Shi, E.

E. Shi, J. Guo, Z. Jia, and Y. Huang, “Theoretic study on new method for roll angle measurement of machines,” in IEEE International Conference on Automation and Logistics (IEEE, 2008), pp. 2722–2726.

Turkkan, N.

T. Pham-Gia, N. Turkkan, and E. Marchand, “Density of the ratio of two normal random variables and applications,” Commun. Stat., Theory Methods 35, 1569–1591 (2006).
[CrossRef]

Vasquez, J. A.

G. C. Gilbreath, N. G. Creamer, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, J. A. Vasquez, R. Mahon, E. Oh, P. G. Goetz, and S. Mozersky, “Modulating retro-reflectors for space, tracking, acquisition and ranging using multiple quantum well technology,” Proc. SPIE 4821, 494 (2002).
[CrossRef]

Vilcheck, M. J.

G. C. Gilbreath, N. G. Creamer, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, J. A. Vasquez, R. Mahon, E. Oh, P. G. Goetz, and S. Mozersky, “Modulating retro-reflectors for space, tracking, acquisition and ranging using multiple quantum well technology,” Proc. SPIE 4821, 494 (2002).
[CrossRef]

Zhai, Y.

Y. Zhai, Q. Feng, and B. Zhang, “A simple roll measurement method based on a rectangular-prism,” Opt. Laser Technol. 44, 839–843 (2012).
[CrossRef]

Zhang, B.

Y. Zhai, Q. Feng, and B. Zhang, “A simple roll measurement method based on a rectangular-prism,” Opt. Laser Technol. 44, 839–843 (2012).
[CrossRef]

Zilberman, A.

N. Blaunstein, S. Arnon, N. Kopeika, and A. Zilberman, Applied Aspects of Optical Communication and LIDAR (CRC Press, 2010).

Appl. Opt.

Biometrika

D. V. Hinkley, “On the ratio of two correlated normal random variables,” Biometrika 56, 635–639 (1969).
[CrossRef]

Commun. Stat., Theory Methods

T. Pham-Gia, N. Turkkan, and E. Marchand, “Density of the ratio of two normal random variables and applications,” Commun. Stat., Theory Methods 35, 1569–1591 (2006).
[CrossRef]

J. Stat. Software

G. Marsaglia, “Ratios of normal variables,” J. Stat. Software 16, 1–10 (2006).

Nat. Methods

K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods 5, 491–505 (2008).
[CrossRef]

Opt. Laser Technol.

Y. Zhai, Q. Feng, and B. Zhang, “A simple roll measurement method based on a rectangular-prism,” Opt. Laser Technol. 44, 839–843 (2012).
[CrossRef]

Proc. SPIE

I. Rahneberg, H.-J. Büchner, and G. Jäger, “Optical system for the simultaneous measurement of two-dimensional straightness errors and the roll angle,” Proc. SPIE 7356, 73560S (2009).
[CrossRef]

G. C. Gilbreath, N. G. Creamer, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, J. A. Vasquez, R. Mahon, E. Oh, P. G. Goetz, and S. Mozersky, “Modulating retro-reflectors for space, tracking, acquisition and ranging using multiple quantum well technology,” Proc. SPIE 4821, 494 (2002).
[CrossRef]

S. Arnon, “Lunar optical wireless communication and navigation network for robotic and human exploration,” Proc. SPIE 8162, 816202 (2011).
[CrossRef]

Other

E. Shi, J. Guo, Z. Jia, and Y. Huang, “Theoretic study on new method for roll angle measurement of machines,” in IEEE International Conference on Automation and Logistics (IEEE, 2008), pp. 2722–2726.

S. G. Lambert and W. L. Casey, Laser Communications in Space (Artech House, 1995).

E. Hecht, Optics, 4th ed. (Addison-Wesley, 2001).

N. Blaunstein, S. Arnon, N. Kopeika, and A. Zilberman, Applied Aspects of Optical Communication and LIDAR (CRC Press, 2010).

S. Arnon, J. Barry, G. Karagiannidis, R. Schober, and M. Uysal, eds., Advanced Optical Wireless Communication Systems (Cambridge University, 2012).

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Figures (4)

Fig. 1.
Fig. 1.

Physical setup for roll angle measurement.

Fig. 2.
Fig. 2.

Laboratory setup, the dashed line represents the laser beam path. The parts are: 1, laser; 2, LP; 3, P detector head; 4, PBS; 5, S detector head; 6, LP; 7, RR; 8, P detector; 9, S detector; 10, function generator.

Fig. 3.
Fig. 3.

Mean of the roll angle measurement as a function of received power. The line marked with triangles represents the calculated values; the line marked with diamonds represents the measured values.

Fig. 4.
Fig. 4.

Roll angle standard deviation as a function of the received power. The calculated values are marked by triangles and the experimental results are marked with diamonds.

Tables (1)

Tables Icon

Table 1. Equipment Used in the Measurements

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

Eout=[EsEp]T=MPBSMLPMRRMLPEin,
Ein=[E0].
MPBS=[cos(π4)sin(π4)sin(π4)cos(π4)].
MLP=[cos2(θ)cos(θ)sin(θ)cos(θ)sin(θ)sin2(θ)],
MRR=[1001].
ERPR.
Eout=[EsEp][PRcos2(θ)PRcos(θ)sin(θ)],
Is=RPRcos4(θ)+ns(t),
Ip=RPRcos2(θ)sin2(θ)+np(t),
tg2(θ)=Ip+npIs+ns.
pz(z)=b(z)c(z)a3(z)12πσsσp[2Φ(b(z)a(z))1]+1a2(z)πσpσse12(μp2σp2+μs2σs2),
z=tg2(θ),
a(z)=1σx2z2+1σy2,
b(z)=μpσp2z+μsσs2,
c(z)=e12b2(z)a2(z)12(μp2σp2+μs2σs2),
Φ(z)=12πze12u2du,
μs=Is=RPRcos4(θ),
μp=Ip=RPRcos2(θ)sin2(θ),
μθ=g(z)p(z)dz,
σθ2=(zμθ)2g(z)p(z)dz,
g(z)=arctan(z).

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