Inertial motion capture is a promising approach for capturing motion outside the laboratory. However, as one major drawback, most of the current methods require different quantities to be calibrated or computed offline as part of the setup process, such as segment lengths, relative orientations between inertial measurement units (IMUs) and segment coordinate frames (IMU-to-segment calibrations) or the joint positions in the IMU frames. This renders the setup process inconvenient. This work contributes to real-time capable calibration-free inertial tracking of a kinematic chain, i.e. simultaneous recursive Bayesian estimation of global IMU angular kinematics and joint positions in the IMU frames, with a minimal state size. Experimental results on simulated IMU data from a three-link kinematic chain (manipulator study) as well as re-simulated IMU data from healthy humans walking (lower body study) show that the calibration-free and lightweight algorithm provides not only drift-free relative but also drift-free absolute orientation estimates with a global heading reference for only one IMU as well as robust and fast convergence of joint position estimates in the different movement scenarios.