Conjugate Unscented Transformation: Applications to Estimation and Control

[+] Author and Article Information
Nagavenkat Adurthi

Department of Mechanical and Aerospace
University at Buffalo,
Buffalo, NY 14260-4400
e-mail: nagavenk@buffalo.edu

Puneet Singla

Associate Professor
Department of Mechanical and Aerospace
University at Buffalo,
The State University of New York,
Buffalo, NY 14260-4400
e-mail: psingla@buffalo.edu

Tarunraj Singh

Fellow ASME
Department of Mechanical and Aerospace
University at Buffalo,
The State University of New York,
Buffalo, NY 14260-4400
e-mail: tsingh@buffalo.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received February 15, 2017; final manuscript received June 26, 2017; published online November 8, 2017. Assoc. Editor: Prashant Mehta.

J. Dyn. Sys., Meas., Control 140(3), 030907 (Nov 08, 2017) (22 pages) Paper No: DS-17-1095; doi: 10.1115/1.4037783 History: Received February 15, 2017; Revised June 26, 2017

This paper presents a computationally efficient approach to evaluate multidimensional expectation integrals. Specifically, certain nonproduct cubature points are constructed that exploit the symmetric structure of the Gaussian and uniform density functions. The proposed cubature points can be used as an efficient alternative to the Gauss–Hermite (GH) and Gauss–Legendre quadrature rules, but with significantly fewer number of points while maintaining the same order of accuracy when integrating polynomial functions in a multidimensional space. The advantage of the newly developed points is made evident through few benchmark problems in uncertainty propagation, nonlinear filtering, and control applications.

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Grahic Jump Location
Fig. 1

Unscented transformation points in 2D: (a) points with corresponding weights and (b) effect of κ on weight

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Fig. 2

Symmetric set of points and axes 2D and 3D space: (a) 2D axes, (b) 3D first quadrant view 1, and (c) 3D first quadrant view 2

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Fig. 3

Comparison of number of points for Gaussian PDF: (a) fifth-order, (b) seventh-order, and (c) ninth-order

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Fig. 4

Comparison of number of points for uniform PDF: (a) fifth-order, (b) seventh-order, (c) seventh-order (CUT versus Sparse GLgn4), and (d) ninth-order

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Fig. 5

Simulation results for polar to Cartesian transformation: (a) μr=1 m, μθ=90 deg, σr2=0.022 m2, σθ2=152 deg2 and (b) μr=50 m, μθ=0 deg, σr2=0.022 m2, σθ2=302 deg2

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Fig. 6

Nonpolynomial function-uniform PDF: % relative error: (a) Gauss–Legendre, (b) sparse grid Gauss–Legendre, and (c) 10% relative error GLgn versus CUT-U

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Fig. 7

Air traffic scenario: (a) typical aeroplane trajectory and (b) estimated aeroplane trajectory

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Fig. 8

Iterative block schematic of the filtering algorithm

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Fig. 9

State estimation error versus the measurement time interval, T for the air traffic problem: (a) position (all filters), (b) velocity (all filters), (c) angular rate (all filters), (d) position (PF, CUT4, CUT6, CUT8), (e) velocity (PF, CUT4, CUT6, CUT8), and (f) angular rate (PF, CUT4, CUT6, CUT8)

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Fig. 11

Comparison of filters for Lorenz model with varying measurement time intervals T: (a) 2-norm of RMSE and (b) max of RMSE

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Fig. 12

Example 2: three mass system: uncertain spring and damping constants

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Fig. 13

Distribution of cost function J under parametric uncertainty: (a) up to second moment) cost, (b) up to fourth moment) cost, (c) up to sixth moment) cost, and (d) min–max control cost



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