Abstract: The trajectory planning in the Simultaneous Localization And Mapping(SLAM) problem is investigated.A novel objective function is proposed,which simultaneously considers the effect of the robot motion on the coverage area of the map,uncertainty of the map,uncertainty of the robot location,and the cost of navigation.The concepts of single step optimal trajectory planning and multi-step optimal trajectory planning are put forward,the planning algorithms under the two optimal criterions and the approximated computation methods for these algorithms are designed.Finally,contrastive simulation experiments validate the feasibility of the methods,and the future research topics are provided.