The paper deals with the development and application of a new adaptive calibration method that extends the geometrical calibration of mechanisms from calibration of only dimensions and kinematical joint positions into the calibration of kinematic joint shape imperfections. Originally unknown nonlinear properties of the kinematical pairs are adaptively included into the kinematical models that are used during the calibration calculation. The method uses description by local linear models and validity functions in order to identify and describe nonlinear properties of the kinematical pairs. During each calibration step, various models with growing complexity are considered before the best model variant is selected to improve calibration results. The method is mainly devoted to the structures with many loops like parallel and fiber-driven parallel mechanisms. The method is applied to parallel mechanism Sliding Star and parallel fiber-driven mechanism Quadrosphere.