Reconstructing Free-Flight Angular Velocity from a Miniaturized Wireless Accelerometer

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.