Real-time motion tracking of kinematic chains is a key prerequisite in the control of, e.g., robotic actuators and autonomous vehicles and also has numerous biomechanical applications. In recent years, it has been shown that, by placing inertial sensors on segments that are connected by rotational joints, the motion of that kinematic chain can be tracked accurately. These methods specifically avoid using magnetometer measurements, which are known to be unreliable since the magnetic field at the different sensor locations is typically different. They rely on the assumption that the motion of the kinematic chain is sufficiently rich to assure observability of the relative pose. However, a formal investigation of this crucial requirement has not yet been presented, and no specific conditions for observability have so far been given. In this work, we present an observability analysis and show that the relative pose of the body segments is indeed observable under a very mild condition on the motion. We support our results by simulation studies, in which we employ a state estimator that neither uses magnetometer measurements nor additional sensors and does not impose assumptions on the accelerometer to measure only the direction of gravity, nor on the range of motion or degrees of freedom of the joints. We investigate the effect of the amount of excitation and of stationary periods in the data on the accuracy of the estimates. We then use experimental data from two mechanical joints as well as from a human gait experiment to validate the observability criterion in practice and to show that small excitation levels are sufficient for obtaining accurate estimates even in the presence of time periods during which the motion is not observable.

Inertial measurement units are commonly used in a growing number of application fields to track or capture motions of kinematic chains, such as human limbs, exoskeletons or robotic actuators. A major challenge is the presence of magnetic disturbances that results in unreliable magnetometer readings. Recent research revealed that this problem can be overcome by exploitation of kinematic constraints. While typically each segment of the kinematic chain is equipped with an IMU, a novel approach called sparse inertial motion tracking aims at inferring the complete motion states from measurements of a reduced set of sensors. In the present contribution, we combine the magnetometer-free and the sparse approach for real-time motion tracking of double-hinge joint systems with non-parallel joint axes. Analyzing the observability of the system, we find a condition which assures that the relative orientations between all segments are uniquely determined by a kinematic constraint, which contains only the gyroscope readings. Furthermore, we propose a moving-horizon estimator and validate it in a simulation study of four movements, which differ by their degrees of excitation. The results of this study confirm the theoretical conjectures and demonstrate that magnetometer-free sparse inertial real-time motion tracking is feasible under precise and simple excitation conditions.