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The paper proposes a novel method for extremely fast inverse kinematics computation suitable for fast-moving manipulators and their path planning and for the animation of anthropomorphic limbs. In a preprocessing phase, the workspace of the robot is decomposed into small cells, and data sets for joint angle vectors (configurations) and hand positions/orientations (postures) are generated randomly in each cell using the forward kinematics. Due to the existence of multiple solutions for a desired posture, data classification is utilized to identify various solutions. The generated and classified data are used to determine the parameters of a simple linear or quadratic model that closely approximates the inverse kinematics within a cell. These parameters are stored in a lookup file. During the online phase, given the desired posture, the index of the appropriate cell is found, the model parameters are retrieved, and the joint angle vectors are computed. The advantages of the proposed method over the existing approaches are discussed. Data resulting from many trial runs are compiled for a manipulator and an anthropomorphic arm to demonstrate the performance of the proposed method.