A five-axis machining center (5MC) is capable of synchronous control, which makes it a feasible tool for quickly and accurately machining complicated three-dimensional surfaces, such as propellers and hypoid gears. Recently, the necessity of improving both the machined shape accuracy and the machined surface roughness of free-form surfaces is growing. Therefore, in our previous study, we aimed to maintain the feed speed vector at the end-milling point by controlling two linear axes and a rotary axis of the 5MC to improve the quality of the machined surface. Additionally, we developed a method for maintaining the feed speed vector at the end-milling point by controlling the three axes of the 5MC to reduce the shape error of the machined workpieces (referred to as the shape error herein), considering the approach path of the tool determined via calculation. However, a high machining force at the start of the workpiece cutting was observed and the factor contributing to this phenomenon was not determined, although this phenomenon leads to a shape error to a certain degree according to the machining condition. In this study, the main objective is to suggest a method to reduce the machining force at the start of the workpiece cutting and shape error. Hence, we develop a theoretical method to estimate the machining force by using an instantaneous cutting force model, which considers the synchronized motion of two linear axes and a rotary axis of the 5MC. Subsequently, we determine the most suitable approach path of the tool considering the prediction of the machining force. The results of this study indicate that the machining force can be estimated by applying an instantaneous cutting force using the feed per tooth and machining angle, and that both a high machining force at the start of the workpiece cutting and shape error reduction can be realized by using the proposed approach path of the tool.

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