Human motion analysis using inertial measurement units (IMUs) has recently been shown to provide accuracy similar to the gold standard, marker-based optical motion capture, but at much lower costs and while being less restrictive and time-consuming. However, IMU-based motion analysis requires precise knowledge of the orientation in which the sensor is attached to the body segments. This knowledge is commonly obtained via an anatomical calibration procedure based on precisely defined poses or motions, which is time-consuming and error-prone. In the present work, we propose a self-calibrating approach for magnetometer-free joint angle tracking that is suitable for joints with two degrees of freedom (DoF), such as the elbow, ankle, and metacarpophalangeal finger joints. The proposed methods exploit kinematic constraints to simultaneously identify the joint axes and the heading offset. The experimental evaluation shows that the proposed methods are able to estimate plausible and consistent joint axes from just ten seconds of arbitrary elbow joint motion. Comparison with optical motion capture shows that the proposed methods yield joint angles with similar accuracy as a conventional IMU-based method while being much less restrictive. Therefore, the proposed methods improve the practical usability of IMU-based motion tracking in many clinical and biomedical applications.