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A kinematic calibration approach is proposed for a triple-planar 3-PR(4R)R parallel mechanism whose attitude error cannot be corrected by motion. First, the kinematic principle of the 3-PR(4R)R parallel mechanism is presented. Then, an error mapping function model is designed based on the mechanism’s closed-loop vector equations. The model reveals the presence of coupling errors in the mechanism, i.e., the coupling of attitude and position errors, and is used to analyze error sensitivity as an indicator of component machining and assembly accuracy. A kinematic inverse solution is used to build a zero-point calibration and a full calibration method. Calibration is performed by a laser tracker-based calibration system, which generates much lower error values and delivers much improved accuracy. Experiments demonstrate that the proposed approach can potentially be used to calibrate parallel mechanisms with non-compensable error terms.