The surge in autonomous robotic applications across various sectors highlights the crucial need for effective robot battery management to ensure robots perform their tasks successfully with energy constraints. This survey aims to provide a comprehensive overview of the latest advancements in the application Artificial Intelligence in robot battery management, focusing on the integration of task requirements with battery state prediction and management through a task-centric approach. Utilizing the Systematic Literature Review method, this survey analyzes current research findings in the domain of robot battery management, underscoring the significant potential of Deep Learning and Deep Reinforcement Learning techniques in revolutionizing robot battery management. It highlights the effectiveness of various Deep Learning models, such as Feedforward Neural Networks, Extreme Learning Machines, Convolutional Neural Networks, Long Short-Term Memory, Gated Recurrent Units, and Transformers, in accurately predicting battery states, and Deep Reinforcement Learning, including Deep Q-Network, Deep Deterministic Policy Gradient, Twin Delayed DDPG, and Soft Actor- Critic, for optimizing battery usage in response to task requirements and adjusting task plans according to battery health. Building on this survey, we introduce a task-centric neural network framework for robot battery management. This framework is designed to seamlessly integrate robot attributes and task characteristics with real-time battery state and health data, facilitating precise battery management and task planning adjustments. Compared to previous literature focusing on generic battery management systems, this survey provides an analysis of task-centric robot battery management challenges and solutions with Neural Network, setting a new direction for future research in this burgeoning field.