Traditional Simultaneous Localization and Mapping (SLAM) passively receives sensor information for mapping and positioning, while active SLAM systems allow robots to acquire environmental information actively. The combination of deep reinforcement learning (DRL) and SLAM allows the robot to directly select actions based on its surrounding environment and reward information, becoming a truly autonomous agent. However, most active SLAM algorithms now use the discrete strategy of DRL, and the robot can only act according to the preset speed and angle. Otherwise, the robot can only use the continuous strategy to explore fewer parts. GRASP-SLAM, an active SLAM framework implemented by building a DRL continuous policy framework that conforms to active SLAM principles. We designed relevant data formats based on the Twin Delayed Deep Deterministic policy gradient (TD3) algorithm to enable the robot to independently select actions and termination times. Reward functions based on robot motion and map exploration rate are also designed to allow the robot to reduce collisions and draw a more accurate map. In a complex environment completely different from the training environment, GRASP-SLAM can still achieve a higher degree of map completion in a shorter time than other methods and can be exploited in a loop closure to reduce its uncertainty.