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Research Papers

# Angular Velocity Estimation From the Angular Acceleration Matrix

[+] Author and Article Information
Philippe Cardou

Centre for Intelligent Machines, Department of Mechanical Engineering, McGill University, Room 461, Macdonald Engineering Building, 817 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canadapcardou@cim.mcgill.ca

Jorge Angeles

Centre for Intelligent Machines, Department of Mechanical Engineering, McGill University, Room 461, Macdonald Engineering Building, 817 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canada

$CPM(v)$ is defined as $∂(v×x)∕∂x$, for any $x,v∊R3$.

The axial vector $vect(A)$ of matrix $A∊R3×3$ is defined such that $vect(A)×u=(1∕2)(A−AT)u$, for any vector $u∊R3$.

It is common practice to find the null space of a matrix using the singular-value decomposition (SVD). However, algorithms that perform SVD are iterative, which is undesirable for angular-velocity estimation in real time, even for a relatively small matrix.

Note that gyroscopes are also subjected to misalignments, but not to position errors.

J. Appl. Mech 75(2), 021003 (Feb 20, 2008) (8 pages) doi:10.1115/1.2775495 History: Received April 17, 2006; Revised June 14, 2007; Published February 20, 2008

## Abstract

Computing the angular velocity $ω$ from the angular acceleration matrix is a nonlinear problem that arises when one wants to estimate the three-dimensional angular velocity of a rigid-body from point-acceleration measurements. In this paper, two new methods are proposed, which compute estimates of the angular velocity from the symmetric part $WS$ of the angular acceleration matrix. The first method uses a change of coordinate frame of $WS$ prior to performing the square-root operations. The new coordinate frame is an optimal representation of $WS$ with respect to the overall error amplification. In the second method, the eigenvector spanning the null space of $WS$ is estimated. As $ω$ lies in this space, and because its magnitude is proportional to the absolute value of the trace of $WS$, it is a simple matter to obtain $ω$. A simulation shows that, for this example, the proposed methods are more accurate than those existing methods that use only centripetal acceleration measurements. Moreover, their errors are comparable to other existing methods that combine tangential and centripetal acceleration measurements. In addition, errors of 2.15% in the accelerometer measurements result in errors of approximately 3% in the angular-velocity estimates. This shows that accelerometers are competitive with angular-rate sensors for motions of the type of the simulated example, provided that position and orientation errors of the accelerometers are accounted for.

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## Figures

Figure 1

Rotating uω* onto ûω

Figure 2

A brick rotating freely in space

Figure 3

Angular acceleration estimates

Figure 5

Angular-velocity estimates from the existing CA methods

Figure 6

Angular-velocity estimates from the proposed CA methods

Figure 7

Angular-velocity estimates from the existing TCA methods

Figure 8

Errors on the angular-velocity estimates

Figure 9

Errors on the angular-velocity estimates

Figure 10

Errors on the angular-velocity estimates

Figure 11

Norms of the errors on the angular-velocity estimates

Figure 4

Angular-velocity estimates from the existing TA method

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