Adaptive estimation and nonlinear variable gain compensation of the contouring error for precise parametric curve following

Contour following is one of the most important issues faced by many computer-numerical-control （CNC） machine tools to achieve high machining precision. This paper presents a new real-time error compensation method aiming at reducing the contouring error caused by facts such as servo lag and dynamics mismatch in parametric curved contour-following tasks. Due to the lack of high-precision contouring-error estimation method for free-form parametric curved toolpath, the error can hardly be compensated effectively. Therefore, an adaptive accurate contouring-error estimation algorithm is proposed first, where a tangential-error backstepping method based on Taylor＇s expansion is developed to rapidly find the closest point on the parametric curve to the actual motion position. On this foundation, the contouring error is compensated using a proposed nonlinear variable-gain compensation method, where the compensation gain is obtained according to not only the contouring-error magnitude but also its direction variation. The stability of the system after compensation is analyzed afterwards according to the Jury stability criterion. By design of the compensator in accordance with the presented contouring-error compensation method as well as the stability analyzation result, the balance between the response speed and the contour control stability can be effectively made. Experimental tests demonstrate the feasibility of the presented methods in both contouring-error estimation and contour-accuracy improvement. Contributions of this research are significant for enhancing the contour-following performance of the CNC machine tools.