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In this paper, we present a new method for calibrating a 6 degree-of-freedom (DOF) high-precision flexure parallel robot. The innovative contributions of this work are either in the technological challenge of calibrating a 6 DOF robot with sub-micrometer accuracy either in the way of processing the measurement data in order to correct the robot pose errors. The first part of this work describes the procedure adopted for collecting a set of 6 D (3 translations + 3 rotations) measurement data from the robot. In this procedure, the robot was programmed, using closed-loops with external measurement devices, in order to execute either “pure translational” or “pure rotational” motions. All measurements were carried out on a thermally- stabilized environment. The second part describes the method used to process the acquired data in order to correct the pose errors. We show in which optimal way neural networks (NN) have been used to perform such task. In particular, we show that the use of NN avoids to the robot user the complex task of formulating an analytical geometric model that takes into account many geometric or non-geometric sources of inaccuracy.