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Research Papers

Angular Rate From the Nine Accelerometer Instrument

[+] Author and Article Information
Jan P. B. Vreeburg

 Wetsat BV, 2314 EX 23 Leiden, The Netherlands

J. Appl. Mech 77(6), 061008 (Aug 20, 2010) (8 pages) doi:10.1115/1.4001992 History: Received August 15, 2008; Revised February 04, 2010; Posted June 16, 2010; Published August 20, 2010; Online August 20, 2010

Instruments composed of arrangements of linear accelerometers are used to determine motion variables. With nine accelerometers, the angular rate of the instrument can be resolved algebraically rather than by the solution of a system of nonlinear differential equations. The algebraic equations consist of three quadratic forms in the angular rate and, in general, lead to multiple solutions. The number and distribution of the multiple roots have been analyzed. Examples illustrate the application of the analyses, for batch and real-time processing. Suitable algorithms have been developed and are detailed. Subjects for further study have been indicated.

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Figures

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Figure 1

Roots u0 (mark o) of cubic form uTA{Bu}Cu=0. Adjoints v0 marked by +, values e/|e| by  ∗.

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Figure 2

Values of f(ti)=s(e)Tω(ti) for eT=[0.74 −0.61 0.28]

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Figure 3

Points with four roots are marked +; the remaining points result in double roots

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Figure 4

Roots from values e(t)/|e(t)| on a great circle through two roots shown in Fig. 1. Foursomes are marked +, pairs o; the foursome from Fig. 1 is marked more prominent.

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Figure 5

As Fig. 4 with three foursome roots; the foursome from Fig. 1 is marked more prominent

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Figure 6

As Fig. 4 for a sequence of values e(t)/|e(t)| on a great circle chosen randomly. A fat mark indicates a root value at the start of the sequence.

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Figure 7

Ballistometer arrangement. Vertices A, B, and C are distance R+d from the origin. Sensors 7, 8, and 9 are midway on the sides of triangle A, B, and C. Sensor 1 is at [R, d, 0]; the locations of sensors 2–6 are permutations of the sensor 1 location coordinates. The sensitive direction is indicated with z.

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Figure 8

Roots obtained from simulated data of a WSM ballistometer configuration subject to angular rate eT=[0.2 cos(0.7t) 0.3 sin(0.3t) −0.12 cos(2t)] for 0≤t≤7. The pole (direction [√3 √3 √3]/3) is at the crossing of two perpendicular great circles, one of which with a marked roots pair e+,−.

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Figure 9

Roots for a single curve arrangement. Either zero or three roots can be solved

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Figure 10

Roots as in Fig. 8, but now recovered from measurements by a ballistometer arrangement with erroneous geometry. The dots show the true data, the circles the recovered values.

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Figure 11

Roots as in Fig. 8, but now recovered from noisy measurements by a ballistometer arrangement with nominal geometry

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Figure 12

Real-time recovery of rate fails at instance 16 and was started again later with different results. True rate components are plotted with drawn lines, iterated solutions are marked +.

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