Abstract

Simulation results for an optical realization of a factorized extended Kalman filter algorithm are presented, minimum word lengths required for accurate tracking are empirically determined, and computation times for an optical realization are quantified.

© 1988 Optical Society of America

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References

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  1. J. L. Fisher, D. P. Casasent, C. P. Neuman, “Factorized Extended Kalman Filter for Optical Processing,” Appl. Opt. 25, 1615 (1986).
    [CrossRef] [PubMed]
  2. R. A. Singer, “Estimating Optimal Tracking Filter Performance for Manned Maneuvering Targets,” IEEE Trans. Aerosp. Electron. Syst. AES-6, 473 (1970).
    [CrossRef]
  3. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984), Chap. 10.
  4. G. F. Franklin, J. D. Powell, Digital Control of Dynamic Systems (Addison-Wesley, Reading, MA, 1980).
  5. J. J. D’Azzo, C. H. Houpis, Linear Control System Analysis and Design (McGraw-Hill, New York, 1981).
  6. T. Kailath, Linear Systems (Prentice-Hall, Englewood Cliffs, NJ, 1980).
  7. D. Casasent, B. K. Taylor, “Banded-Matrix High-Performance Algorithm and Architecture,” Appl. Opt. 24, 1476 (1985).
    [CrossRef] [PubMed]
  8. H. J. Whitehouse, J. M. Speiser, “Linear Signal Processing Architectures,” in Aspects of Signal Processing with Emphasis on Underwater Acoustics, Part 2, G. Tacconi, Ed. (Reidel, Boston, 1976), pp. 669–702.
  9. D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate Numerical Computation by Optical Convolution,” Proc. Soc. Photo-Opt. Instrum. Eng.151 (1980).
  10. D. Casasent, A. Ghosh, “LU and Cholesky Decomposition on an Optical Systolic Array Processor,” Opt. Commun. 46, 270 (1983).
    [CrossRef]
  11. G. J. Bierman, Factorization Methods for Discrete Sequential Estimation (Academic, New York, 1977).
  12. D. P. Casasent, S. E. Riedl, “Time and Space Integrating Optical Laboratory Matrix–Vector Analog Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 698, 151 (Aug.1986).
  13. G. J. Bierman, “A Comparison of Discrete Linear Filtering Algorithms,” IEEE Trans. Aerosp. Electron. Syst. AES-9, 28 (1973).
    [CrossRef]
  14. G. J. Bierman, C. L. Thornton, “Numerical Comparison of Kalman Filter Algorithms: Orbit Determination Case Study,” Automatica 13, 23 (Jan.1977).
    [CrossRef]
  15. C. L. Thornton, G. J. Bierman, “UDUT Covariance Factorization for Kalman Filtering,” in Control and Dynamic Systems, C. T. Leondes, Ed. (Academic, New York, 1980), Vol. 16, pp. 117–248.
    [CrossRef]

1986 (2)

J. L. Fisher, D. P. Casasent, C. P. Neuman, “Factorized Extended Kalman Filter for Optical Processing,” Appl. Opt. 25, 1615 (1986).
[CrossRef] [PubMed]

D. P. Casasent, S. E. Riedl, “Time and Space Integrating Optical Laboratory Matrix–Vector Analog Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 698, 151 (Aug.1986).

1985 (1)

1983 (1)

D. Casasent, A. Ghosh, “LU and Cholesky Decomposition on an Optical Systolic Array Processor,” Opt. Commun. 46, 270 (1983).
[CrossRef]

1980 (1)

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate Numerical Computation by Optical Convolution,” Proc. Soc. Photo-Opt. Instrum. Eng.151 (1980).

1977 (1)

G. J. Bierman, C. L. Thornton, “Numerical Comparison of Kalman Filter Algorithms: Orbit Determination Case Study,” Automatica 13, 23 (Jan.1977).
[CrossRef]

1973 (1)

G. J. Bierman, “A Comparison of Discrete Linear Filtering Algorithms,” IEEE Trans. Aerosp. Electron. Syst. AES-9, 28 (1973).
[CrossRef]

1970 (1)

R. A. Singer, “Estimating Optimal Tracking Filter Performance for Manned Maneuvering Targets,” IEEE Trans. Aerosp. Electron. Syst. AES-6, 473 (1970).
[CrossRef]

Bierman, G. J.

G. J. Bierman, C. L. Thornton, “Numerical Comparison of Kalman Filter Algorithms: Orbit Determination Case Study,” Automatica 13, 23 (Jan.1977).
[CrossRef]

G. J. Bierman, “A Comparison of Discrete Linear Filtering Algorithms,” IEEE Trans. Aerosp. Electron. Syst. AES-9, 28 (1973).
[CrossRef]

C. L. Thornton, G. J. Bierman, “UDUT Covariance Factorization for Kalman Filtering,” in Control and Dynamic Systems, C. T. Leondes, Ed. (Academic, New York, 1980), Vol. 16, pp. 117–248.
[CrossRef]

G. J. Bierman, Factorization Methods for Discrete Sequential Estimation (Academic, New York, 1977).

Carlotto, M.

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate Numerical Computation by Optical Convolution,” Proc. Soc. Photo-Opt. Instrum. Eng.151 (1980).

Casasent, D.

D. Casasent, B. K. Taylor, “Banded-Matrix High-Performance Algorithm and Architecture,” Appl. Opt. 24, 1476 (1985).
[CrossRef] [PubMed]

D. Casasent, A. Ghosh, “LU and Cholesky Decomposition on an Optical Systolic Array Processor,” Opt. Commun. 46, 270 (1983).
[CrossRef]

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate Numerical Computation by Optical Convolution,” Proc. Soc. Photo-Opt. Instrum. Eng.151 (1980).

Casasent, D. P.

J. L. Fisher, D. P. Casasent, C. P. Neuman, “Factorized Extended Kalman Filter for Optical Processing,” Appl. Opt. 25, 1615 (1986).
[CrossRef] [PubMed]

D. P. Casasent, S. E. Riedl, “Time and Space Integrating Optical Laboratory Matrix–Vector Analog Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 698, 151 (Aug.1986).

D’Azzo, J. J.

J. J. D’Azzo, C. H. Houpis, Linear Control System Analysis and Design (McGraw-Hill, New York, 1981).

Fisher, J. L.

Franklin, G. F.

G. F. Franklin, J. D. Powell, Digital Control of Dynamic Systems (Addison-Wesley, Reading, MA, 1980).

Ghosh, A.

D. Casasent, A. Ghosh, “LU and Cholesky Decomposition on an Optical Systolic Array Processor,” Opt. Commun. 46, 270 (1983).
[CrossRef]

Houpis, C. H.

J. J. D’Azzo, C. H. Houpis, Linear Control System Analysis and Design (McGraw-Hill, New York, 1981).

Kailath, T.

T. Kailath, Linear Systems (Prentice-Hall, Englewood Cliffs, NJ, 1980).

Neft, D.

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate Numerical Computation by Optical Convolution,” Proc. Soc. Photo-Opt. Instrum. Eng.151 (1980).

Neuman, C. P.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984), Chap. 10.

Powell, J. D.

G. F. Franklin, J. D. Powell, Digital Control of Dynamic Systems (Addison-Wesley, Reading, MA, 1980).

Psaltis, D.

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate Numerical Computation by Optical Convolution,” Proc. Soc. Photo-Opt. Instrum. Eng.151 (1980).

Riedl, S. E.

D. P. Casasent, S. E. Riedl, “Time and Space Integrating Optical Laboratory Matrix–Vector Analog Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 698, 151 (Aug.1986).

Singer, R. A.

R. A. Singer, “Estimating Optimal Tracking Filter Performance for Manned Maneuvering Targets,” IEEE Trans. Aerosp. Electron. Syst. AES-6, 473 (1970).
[CrossRef]

Speiser, J. M.

H. J. Whitehouse, J. M. Speiser, “Linear Signal Processing Architectures,” in Aspects of Signal Processing with Emphasis on Underwater Acoustics, Part 2, G. Tacconi, Ed. (Reidel, Boston, 1976), pp. 669–702.

Taylor, B. K.

Thornton, C. L.

G. J. Bierman, C. L. Thornton, “Numerical Comparison of Kalman Filter Algorithms: Orbit Determination Case Study,” Automatica 13, 23 (Jan.1977).
[CrossRef]

C. L. Thornton, G. J. Bierman, “UDUT Covariance Factorization for Kalman Filtering,” in Control and Dynamic Systems, C. T. Leondes, Ed. (Academic, New York, 1980), Vol. 16, pp. 117–248.
[CrossRef]

Whitehouse, H. J.

H. J. Whitehouse, J. M. Speiser, “Linear Signal Processing Architectures,” in Aspects of Signal Processing with Emphasis on Underwater Acoustics, Part 2, G. Tacconi, Ed. (Reidel, Boston, 1976), pp. 669–702.

Appl. Opt. (2)

Automatica (1)

G. J. Bierman, C. L. Thornton, “Numerical Comparison of Kalman Filter Algorithms: Orbit Determination Case Study,” Automatica 13, 23 (Jan.1977).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst. (2)

G. J. Bierman, “A Comparison of Discrete Linear Filtering Algorithms,” IEEE Trans. Aerosp. Electron. Syst. AES-9, 28 (1973).
[CrossRef]

R. A. Singer, “Estimating Optimal Tracking Filter Performance for Manned Maneuvering Targets,” IEEE Trans. Aerosp. Electron. Syst. AES-6, 473 (1970).
[CrossRef]

Opt. Commun. (1)

D. Casasent, A. Ghosh, “LU and Cholesky Decomposition on an Optical Systolic Array Processor,” Opt. Commun. 46, 270 (1983).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate Numerical Computation by Optical Convolution,” Proc. Soc. Photo-Opt. Instrum. Eng.151 (1980).

D. P. Casasent, S. E. Riedl, “Time and Space Integrating Optical Laboratory Matrix–Vector Analog Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 698, 151 (Aug.1986).

Other (7)

C. L. Thornton, G. J. Bierman, “UDUT Covariance Factorization for Kalman Filtering,” in Control and Dynamic Systems, C. T. Leondes, Ed. (Academic, New York, 1980), Vol. 16, pp. 117–248.
[CrossRef]

G. J. Bierman, Factorization Methods for Discrete Sequential Estimation (Academic, New York, 1977).

H. J. Whitehouse, J. M. Speiser, “Linear Signal Processing Architectures,” in Aspects of Signal Processing with Emphasis on Underwater Acoustics, Part 2, G. Tacconi, Ed. (Reidel, Boston, 1976), pp. 669–702.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984), Chap. 10.

G. F. Franklin, J. D. Powell, Digital Control of Dynamic Systems (Addison-Wesley, Reading, MA, 1980).

J. J. D’Azzo, C. H. Houpis, Linear Control System Analysis and Design (McGraw-Hill, New York, 1981).

T. Kailath, Linear Systems (Prentice-Hall, Englewood Cliffs, NJ, 1980).

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Figures (6)

Fig. 1
Fig. 1

Simplified schematic of a high-accuracy vector inner product processor (N-bit accuracy, M-element vector).

Fig. 2
Fig. 2

Root-mean-square acceleration error for the benign trajectory case study.

Fig. 3
Fig. 3

Comparison of the true x-acceleration and the x-acceleration estimated by our factorized EKF for the maneuvering target case study.

Fig. 4
Fig. 4

Root-mean-square acceleration error in the estimate obtained from our factorized EKF for the maneuvering target case study. (Note that both curves are identical for the vertical axis shown.)

Fig. 5
Fig. 5

Root-mean-square velocity error in the estimate obtained from our factorized EKF for the maneuvering target case study.

Fig. 6
Fig. 6

Root-mean-square position error in the estimate obtained from our factorized EKF for the maneuvering target case study.

Tables (9)

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Table I Equations Defining the Nonlinear Measurement Vector h[xk]

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Table II Numerical Constants and Initial Values used in our Case Study

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Table III System and Measurement Models and Extended Kalman Filter Algorithm

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Table IV Definition of Symbols used in Discrete-Time Linear and Extended Kalman Filter Algorithms

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Table V Count of Linear Algebra Operations for Basic Matrix–Vector Operations In Units of VIPs (the Time Unit T2 for our Processor)

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Table VI Steps Required in the Computation of Pk and the Number of Basic Linear Algebra Function that each Step Requires

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Table VII Steps Required in the Computation of Mk+1 and the Number of Basic Linear Algebra Functions that each Step Requires

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Table VIII Execution Times for the Calculations in one Cycle of the EKF Equations in Terms of the Number of States n and the Number of Measurements r

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Table IX Target Accelerations used in the Maneuvering Case Study

Equations (21)

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

x = [ z ¨ , y ¨ , x ¨ , z ˙ , y ˙ , x ˙ , z , y , x ] T
z = [ r ˙ , r , θ , ϕ ] T
z = h ( x ) + v ,
H ( x k ) = h [ x k ] x k = H i j = h i x j ,
E { x ¨ ( t ) x ¨ ( t + τ ) } = σ 2 exp ( β | τ | ) , β 0 ,
A = [ β I 0 0 I 0 0 0 I 0 ] ,
x ˙ ( t ) = A x ( t ) + B u ( t ) + w c ( t ) ,
Q c ( t ) = E { w c ( t ) w c T ( s ) } = diag [ 2 β σ 2 δ ( t ) , 2 β σ 2 δ ( t ) , 2 β σ 2 δ ( t ) , 0 , 0 , 0 , 0 , 0 , 0 ] ,
x k + 1 = Φ x k + Γ u k + w k ,
( 1 P 0 + 2 P max 2 A max )
σ 2 = A max 2 3 ( 1 + 4 P max P 0 ) .
[ ( x ¨ x ¨ ) 2 + ( y ¨ y ¨ ) 2 + ( z ¨ z ¨ ) 2 ] 1 / 2
x k = 0 = [ 0 , 0 , 0 , 0 , 0 , 300 , 3000 , 2850 , 3000 ] T .
x ¯ k = 0 = [ 0 , 0 , 10 , 0 , 0 , 300 , 3000 , 2850 , 3000 ] T ,
x k = 1 = Φ x k + Γ u k + w k
z k = h [ x k ] + v k
P k 1 = M k 1 + H T [ x ¯ k ] R k 1 H [ x ¯ k ]
K k = P k H T [ x ¯ k ] R k 1
x ˆ k = x ¯ k + K k { z k h [ x ¯ k ] }
M k + 1 = Φ P k Φ T + Q k
x ¯ k + 1 = Φ x ˆ k + Γ u k

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