Determining the acceleration field of a rigid body using three accelerometers and one gyroscope, with applications in mild traumatic brain injury

Acceleration–deceleration mild traumatic brain injury (mTBI) arises when the head experiences rapid motions, as in sports and military training. The risk of mTBI strongly correlates with the head's kinematics, motivating the development of wearable devices that can reconstruct those. Accelerometer-only (AO) algorithms (such as Rahaman et al. (J. Mech. Phys. Solids, 2020) and Padgaonkar, Krieger, and King (J. Appl. Mech., 1975)) are attractive for they enable the wearables based on them to comprehensively reconstruct the kinematics while at the same time being stable, and not requiring the numerical differentiation of angular velocities. However, the AO algorithms require their accelerometers not to lie in a plane. This greatly limits the scope and adoptability of their wearables, since they can no longer have a headband-style design. In this work, we present a new algorithm that employs three accelerometers and a gyroscope, which we term the A3G1 algorithm. It has all the aforementioned attractive features of the AO algorithms. Additionally, it allows its sensors to lie in a plane. We provide a mathematical proof that the only constraint on the A3G1's sensors placement is that its three accelerometers do not lie in a line. We experimentally demonstrate the algorithm's utility and robustness by applying it in a mock soccer-header exercise.