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This paper describes a comprehensive solution to the problem of reconstructing the multijoint movement trajectories of the human body from diverse motion capture data. The problem is formulated in a probabilistic framework so as to handle multiple and unavoidable sources of uncertainty: sensor noise, soft tissue deformation and marker slip, inaccurate marker placement and limb measurement, and missing data due to occlusions. All unknown quantities are treated as state variables even though some of them are constant. In this way, state estimation and system identification can be performed simultaneously, obtaining not only the most likely values but also the confidence intervals of the joint angles, skeletal parameters, and marker positions and orientations relative to the limb segments. The inference method is a Gauss-Newton generalization of the extended Kalman filter. It is adapted to the kinematic domain by expressing spatial rotations via quaternions and computing the sensor residuals and their Jacobians analytically. The ultimate goal of this project is to provide a reliable data analysis tool used in practice. The software implementation is available online.