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Research Papers

# Minimal Spatial Accelerometer Configurations

[+] Author and Article Information
Thomas R. Williams

Postdoctoral Researcher
e-mail: thomasw@ualberta.ca

Donald W. Raboud

Associate Professor
e-mail: don.raboud@ualberta.ca

Ken R. Fyfe

e-mail: ken.fyfe@ualberta.ca
Department of Mechanical Engineering,
University of Alberta,

The specific force is the difference between the inertial acceleration and the local gravity vectors.

Note that these two $3×1$ matrices are not the component matrices of vectors $ωij$ and $ωii$; they are simply convenient groupings of quadratic combinations of the elements of the component matrix of the angular velocity vector, $[ω]$.

A material unit vector is related to a regular unit vector of E in the same way as a material particle is related to a point of the affine space.

The strict inequalities are used because if $θC=0$ or $π/2$, then the binary rotation pair of accelerometers would be parallel.

Contributed by the Dynamic Systems Division of ASME for publication in the Journal of Dynamic Systems, Measurement, and Control. Manuscript received May 16, 2008; final manuscript received September 7, 2012; published online February 21, 2013. Editor: J. Karl Hedrick.

J. Dyn. Sys., Meas., Control 135(2), 021016 (Feb 21, 2013) (9 pages) Paper No: DS-08-1149; doi: 10.1115/1.4023058 History: Received May 16, 2008; Revised September 07, 2012

## Abstract

It is well established that it is necessary to use a minimum of six accelerometers to determine the general motion of a rigid body. Using this minimum number of accelerometers generally requires that a nonlinear differential equation be solved for the angular velocity and that the estimate of angular velocity that is obtained from the solution of this equation be used in the calculation of the specific force at a point. This paper serves two main purposes. First it discusses, for the first time, the geometric conditions that must be satisfied by an arrangement of six accelerometers so that it is possible, in principle, to determine the motion of the body to which they are attached. Second, a special class of minimal accelerometer configurations that yields angular acceleration as a linear combination of accelerometer measurements is identified, and a design methodology for this special class is presented.

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Topics: Accelerometers

## Figures

Fig. 1

Schuler's minimal configuration, consisting of three parallel pairs of accelerometers, with orthogonal directions

Fig. 2

Chen's minimal configuration (solid arrows), with Tan's proposed addition of a triaxial accelerometer (dashed arrows), and Ding's addition of three accelerometers (dotted arrows)

Fig. 3

A hyperbolic paraboloid defined by the lines of three accelerometers. If a fourth accelerometer is located so that its line is a ruling (of the same family) on this surface, the resulting configuration will be minimally dependent.

Fig. 4

A sampling of accelerometers in the 23 plane of F that are free of [ωij]

Fig. 5

The types of accelerometer placement that are free of [ωii]

Fig. 6

The numbering scheme of the accelerometer placements used in the first special minimal configuration design

Fig. 9

A binary rotation pair of accelerometers

Fig. 7

A special minimal configuration obtained using one of each type of axial (A) and planar (P) placement

Fig. 8

The numbering of the accelerometer placements used in the second special minimal configuration design

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