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A handshake is an embodied interaction to display closeness using physical contact. In the case of a handshake between a human and a robot, robots can now smoothly communicate and coexist with humans without eliciting feelings of aversion in humans. To enable such a handshake, we proposed a model that simulates a handshake approach motion by analyzing the human-human handshake motion. With this model, a robot generates a handshake motion when a handshake is requested by a human. However, embodied interaction between a human and a robot can be promoted if, instead, a robot requests a handshake from a human. Therefore, in this paper, we propose a handshake request motion model with which a robot requests humans for a handshake. In this model, a robot stretches its hand out to a human to request a handshake. Furthermore, a gaze presentation is generated based on the analysis of the handshake between humans. A handshake robot system with the proposed model is developed, and the effectiveness of the model is experimentally demonstrated.