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Research Papers

# Reconstructing Free-Flight Angular Velocity from a Miniaturized Wireless Accelerometer

[+] Author and Article Information
Ryan McGinnis

Mechanical Engineering,  University of Michigan, Ann Arbor, MI 48109-2125ryanmcg@umich.edu

N. C. Perkins, Kevin King

Mechanical Engineering,  University of Michigan, Ann Arbor, MI 48109-2125

Note that the MEMS accelerometer detects acceleration down to DC and thus it also measures gravity.

For example, refer to [4]. This classical analysis reveals that the “off-axis” components of angular velocity will oscillate with frequency $ωn=Ω(I1-I3)(I2-I1)/I3I2$ or $ωn=Ω(I1-I3)(I3-I2)/I1I2$ for rotations about the major and minor axes, respectively, where $Ω$ is the magnitude of angular velocity component about the major or minor axis.

The error measure is again given by 7 where the angular velocity measured by the angular rate gyros is now used as the “truth” data.

J. Appl. Mech 79(4), 041013 (May 11, 2012) (9 pages) doi:10.1115/1.4006162 History: Received June 25, 2010; Revised February 16, 2012; Posted February 23, 2012; Published May 11, 2012; Online May 11, 2012

## Abstract

The theory governing the torque-free motion of a rigid body is well established, yet direct experimental measurement in the laboratory remains an obvious challenge. This paper addresses this challenge by presenting a novel miniature wireless inertial measurement unit (IMU) that directly measures the motion of a rigid body during free-flight. The IMU incorporates three-axis sensing of acceleration and three-axis sensing of angular velocity with a microcontroller and an RF transceiver for wireless data transmission to a host computer. Experiments consider a rigid body that is spun up by hand and then released into free-flight. The measured rotational dynamics from the IMU are carefully benchmarked against theoretical predictions. This benchmarking reveals that the angular velocity directly measured by the angular rate gyros lies within 6% of that predicted by the (Jacobi elliptic function) solutions to the Euler equations. Moreover, experimentally constructed polhodes elegantly illustrate the expected stable precession for rotations initiated close to the major or minor principal axes and the unstable precession for rotations initiated close to the intermediate axis. We then present a “gyro-free” design that employs a single, triaxial accelerometer to reconstruct the angular velocity during free-flight. A measurement theory is presented and validated experimentally. Results confirm that the angular velocity can be reconstructed with exceedingly small errors (less than 2%) when benchmarked against direct measurements using angular rate gyros. The simpler gyro-free design addresses restrictions imposed by rate gyro cost, size, and measurement range and may enable high-volume commercial applications of this technology in instrumented baseballs, basketballs, golf balls, footballs, soccer balls, softballs, and the like.

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## Figures

Figure 1

Photographs of highly miniaturized, wireless IMU. (a) Analog circuit side with MEMS angular rate gyros and accelerometer. (b) Digital circuit side with microprocessor, wireless transceiver, surface mount antenna, and connectors for battery power and firmware programming.

Figure 2

Photograph of example rigid body employed in experiments

Figure 3

Example time histories of the measured (a) magnitude of the acceleration of point P, (b) magnitude of the angular velocity, (c) the rotational kinetic energy, and (d) magnitude of angular momentum about center of mass. The throw, free-flight, and catch phases are noted. Example trial for rotation initiated nearly about the minor axis.

Figure 4

Measured (solid) and calculated (dashed) angular velocity vector magnitude (black) and components for rotations initiated about the (a) major, (b) intermediate, and (c) minor axes. The blue, green, and red curves correspond to components about the major (ω1), intermediate (ω2), and minor axes (ω3), respectively.

Figure 5

Experimental demonstration of the polhode for rotations initiated close to the (a) major, (b) intermediate, and (c) minor principal axes. The measured angular velocity during the entire free-flight phase (black, scale in deg/s), closely follows the polhode defined by the intersection of the ellipsoids.

Figure 6

Measured (solid) and reconstructed (dashed) angular velocity magnitude (black) and components for rotations initiated nearly about the (a) major, (b) intermediate, and (c) minor axes. The blue, green, and red curves correspond to components about the major (ω1), intermediate (ω2), and minor axes (ω3), respectively.

## Errata

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