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Recent research on the coordinated control of biaxial machines for precise contour following has been using various locally defined task coordinate frames (LTCF) “attached” to the desired contour to approximately calculate the contour error for feedback controller designs. Contour error, by definition, is a geometrical quantity depending on the shape of the desired contour only and has nothing to do with the desired motion on the contour. As such, all those moving LTCF-based algorithms have to make the assumptions that the position tracking errors are much smaller than the radius of curvature of the desired contour and the calculated contour error is only an approximation of actual contour error. In contrast, this paper presents an orthogonal global task coordinate frame (GTCF) in which the calculation of contour error is exact to the first-order approximation of the actual contour error, no matter how large the position tracking errors would be. A systematic way to construct curvilinear coordinates of the proposed GTCF using any description of the geometry of the desired contour in a two-dimensional space is also given. Contouring control of a linear motor driven biaxial high-speed industrial gantry is then used as a case study. A simplistic direct adaptive robust controller (ARC) is constructed to deal with the effect of strong coupling of the system dynamics in the task space in addition to modeling uncertainties. The proposed GTCF-based ARC algorithm, along with the traditional LTCF-based ARC ones, are implemented and comparative experimental results are presented. The results validate the effectiveness of the proposed GTCF approach for free-form contouring control with large curvatures and arbitrary position tracking errors and confirm the excellent contouring performance of the proposed approach in general.