Product-of-exponential formulas for precision enhancement of five-axis machine tools via geometric error modeling and compensation

This paper proposes the precision enhancement of five-axis machine tools using product-of-exponential (POE) formulas through geometric error modeling and compensation. Firstly, two issues of geometric error Denavit-Hartenberg (D-H) models for five-axis machine tools are discussed for further improvement: models of squareness errors and position of rotary axes. Three D-H models of squareness errors and three expressions of position of rotary axes are represented and analyzed. Secondly, POE formulas are established to solve the two problems and to realize the modeling and compensation. The rotation twists and rotation POE formulas of rotary axes are established with the help of the clear geometric meaning of twists to describe their positions and motions. Then, corresponding POE formulas of squareness errors are established by analyzing their geometric definition. All motion POE formulas, rotation POE formulas, and error POE formulas are multiplied in certain order to obtain the final geometric error POE formula of five-axis machine tools. In addition, Jacobian of twists is calculated easily and reasonably with the twists of each axis for the geometric error compensation. Thirdly, experiments are conducted on one SmartCNC500 five-axis machine tool in order to verify the effectiveness of error POE formula and corresponding compensation.