Milling force estimation based on harmonic force model and acceleration feedback

Abstract Monitoring milling forces is crucial as they directly reflect the milling process. Direct force measurement using a dynamometer is challenging for industrial applications. However, indirect force measurement can be achieved with efficient Kalman filter estimation algorithms and low-cost accelerometers that can be conveniently installed in machine tools. Existing research establishes the force-acceleration relationship through modal analysis of the frequency response function (FRF). High-order system models for acceleration response not only increase computational effort but also introduce tuning issues for the estimator. This paper introduces a novel and simple Kalman filter system model. In this approach, the cutting force coefficients are treated as unknown state variables, resulting in a $$4 \times 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo>×</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> identity state transition matrix. The acceleration measurement is related to the estimated coefficient states by modeling the milling forces as a sum of tooth passing frequency harmonics. First, an analytical derivation of the force harmonics in the angular domain is presented. Then, utilizing the FRF gains and phases at the tooth passing frequency harmonics, the acceleration is expressed in terms of coefficient state variables and integrated into the output matrix of the Kalman filter. By updating the output matrix with the measured angular position of the tool, a linear time-varying system model of the cutting process is achieved. The proposed method is validated through real cutting tests conducted under various process parameters. Comparison with dynamometer measurements shows that the milling force can be estimated with an average RMS error of around 5%