Because the circular weld in the pipeline lacks point and straight-line features, research to automatically locate the welds in the pipeline using a monocular vision nondestructive testing robot is still challenging. The localization of positive circular welds under the influence of the robot’s left and right swings is investigated in this paper. The governing equation of the special simultaneous localization and mapping (SLAM) problem for locating the distance between the positive circular weld and the inspection robot is established using the geometric imaging model of the circular weld in the inspection robot camera and the robot’s motion speed. The problem of solving the governing equation is transformed into a loss function minimization optimization problem. As the optimization problem is highly nonlinear, this paper employs a genetic algorithm to solve it and achieves the positive annular weld location while the robot swings from side to side. Using the synthetic video of the robot detecting the circular weld in the cylindrical pipe and the video captured during the actual operation, experimental results show that the proposed method can achieve good results.