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Action recognition is an important component to improve autonomy of physical rehabilitation devices, such as wearable robotic exoskeletons. Existing human action recognition algorithms focus on adult applications rather than pediatric ones. In this paper, we introduce BabyNet, a light-weight (in terms of trainable parameters) network structure to recognize infant reaching action from off-body stationary cameras. We develop an annotated dataset that includes diverse reaches performed while in a sitting posture by different infants in unconstrained environments (e.g., in home settings, etc.). Our approach uses the spatial and temporal connection of annotated bounding boxes to interpret onset and offset of reaching, and to detect a complete reaching action. We evaluate the efficiency of our proposed approach and compare its performance against other learning-based network structures in terms of capability of capturing temporal inter-dependencies and accuracy of detection of reaching onset and offset. Results indicate our BabyNet can attain solid performance in terms of (average) testing accuracy that exceeds that of other larger networks, and can hence serve as a light-weight data-driven framework for video-based infant reaching action recognition.