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# Angle Estimation of Simultaneous Orthogonal Rotations from 3D Gyroscope Measurements

Faculty of Electrical Engineering, University of Ljubljana, Ljubljana 1000, Slovenia

* Author to whom correspondence should be addressed.

Received: 11 August 2011 / Revised: 29 August 2011 / Accepted: 30 August 2011 / Published: 2 September 2011

(This article belongs to the Section Physical Sensors)

# Abstract

A 3D gyroscope provides measurements of angular velocities around its three intrinsic orthogonal axes, enabling angular orientation estimation. Because the measured angular velocities represent simultaneous rotations, it is not appropriate to consider them sequentially. Rotations in general are not commutative, and each possible rotation sequence has a different resulting angular orientation. None of these angular orientations is the correct simultaneous rotation result. However, every angular orientation can be represented by a single rotation. This paper presents an analytic derivation of the axis and angle of the single rotation equivalent to three simultaneous rotations around orthogonal axes when the measured angular velocities or their proportions are approximately constant. Based on the resulting expressions, a vector called the simultaneous orthogonal rotations angle (SORA) is defined, with components equal to the angles of three simultaneous rotations around coordinate system axes. The orientation and magnitude of this vector are equal to the equivalent single rotation axis and angle, respectively. As long as the orientation of the actual rotation axis is constant, given the SORA, the angular orientation of a rigid body can be calculated in a single step, thus making it possible to avoid computing the iterative infinitesimal rotation approximation. The performed test measurements confirm the validity of the SORA concept. SORA is simple and well-suited for use in the real-time calculation of angular orientation based on angular velocity measurements derived using a gyroscope. Moreover, because of its demonstrated simplicity, SORA can also be used in general angular orientation notation.*Keywords:*gyroscope; angular velocity; angular orientation; simultaneous rotations; rotation axis; rotation angle; spatial angle

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