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The low absolute positioning accuracy of industrial robots is one of the bottlenecks preventing industrial robots from precision applications. Kinematic calibration is the main way to improve the absolute positioning accuracy of industrial robots, which greatly relies on three-dimensional (3D) measurement instruments, including laser trackers and pull rope mechanisms. These instruments are costly, and their required intervisibility space is large. In this paper, a precision 3D measurement instrument integrating multiple laser range sensors is designed, which fuses the information of multiple redundant laser range sensors to obtain the coordinates of a 3D position. An identification model of laser beam position and orientation parameters based on redundant distance information and standard spherical constraint is then developed to reduce the requirement for the assembly accuracy of laser range sensors. A hybrid identification algorithm of PSO-LM (particle swarm optimization Levenberg Marquardt) is designed to solve the high-order nonlinear problem of the identification model, where PSO is used for initial value identification, and LM is used for final value identification. Experiments of identification of position and orientation, verifications of the measuring accuracy, and the calibration of industrial robots are conducted, which show the effectiveness of the proposed 3D measurement instrument and identification methods. Moreover, the proposed instrument is small in size and can be used in narrow industrial sites.