Determining rigid body motion from accelerometer data through the square-root of a negative semi-definite tensor, with applications in mild traumatic brain injury

Mild Traumatic Brain Injuries (mTBI) are caused by violent head motions or impacts. Most mTBI prevention strategies explicitly or implicitly rely on a "brain injury criterion". A brain injury criterion takes some descriptors of the head's motion as input and yields a prediction for that motion's potential for causing mTBI as the output. The inputs are descriptors of the head's motion that are usually synthesized from accelerometer and gyroscope data. In the context of brain injury criterion, the head is modeled as a rigid body. We present an algorithm for determining the complete motion of the head using data from only four head mounted tri-axial accelerometers. In contrast to inertial measurement unit based algorithms for determining rigid body motion, the presented algorithm does not depend on data from gyroscopes, which consume much more power than accelerometers. Several algorithms that also make use of data from only accelerometers already exist. However, those algorithms, except for the recently presented AO (accelerometer-only) algorithm [Rahaman MM, Fang W, Fawzi AL, Wan Y, Kesari H (2020): J Mech Phys Solids 104014], give the rigid body's acceleration field in terms of the body frame, which in general is unknown. Compared to the AO-algorithm the presented algorithm is much more insensitive to bias type errors, such as those that arise from inaccurate measurement of sensor positions and orientations. (c) 2021 Elsevier B.V. All rights reserved.