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Enhancing the multi-encoder-based cutting force estimation along the stationary axis of a machine tool with multiple inertia dynamics

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Abstract

Wideband cutting force sensing is a key technology for process monitoring. Sensorless cutting force estimation using the internal servo information of a machine tool with ball-screw-driven stages has been studied owing to its high maintainability and ease of introduction. In the motor current-based method, the cutting force estimation along the stationary axis is challenging, and the estimation bandwidth is significantly limited owing to the low sensitivity of the motor current in the high-frequency range. The dual-inertia model-based load-side disturbance observer (LDOB) can estimate the cutting force along the stationary axis using the relative position obtained from the rotary and linear encoder. The linear encoder is installed relatively near the cutting point and has a high sensitivity in the high-frequency range. However, this approach is not applicable to machine tools with complicated structural dynamics. To address this challenge, we propose a cutting force estimation method along the stationary axis using the Kalman filter (KF) based on a multiple inertia model derived solely from the relative position signal. The dynamics, depending on the stage position of the feed drive, were modeled using linear interpolation. Through end milling tests, we confirmed that the cutting force estimation accuracy along the stationary axis of a machine tool with multiple inertia dynamics was significantly improved by the proposed method compared to the current and LDOB-based methods. Additionally, the wideband cutting force could be estimated using the proposed method for bandwidths up to 1000 Hz.

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Abbreviations

\({a}_{1},{a}_{2},{a}_{3}\) :

Coefficients of the quadratic function model of stiffness

\({b}_{1},{b}_{2},{b}_{3}\) :

Coefficients of the quadratic function model of modal parameters

\({C}_{t}\) :

Damping coefficient of the transitional element

\({D}_{m}\) :

Damping coefficient of the rotational element

\({F}_{cut}\) :

Cutting force

\({F}_{h}\) :

Impulse force

\(i\) :

Mode index

\({I}_{a}\) :

Motor current

\({J}_{m}\) :

Total inertia of the motor, coupling, and ball-screw

\({K}_{r}\) :

Axial stiffness of the ball-screw

\({K}_{t}\) :

Torque coefficient of the servo motor

\(l\) :

Pitch length of the ball screw

\({M}_{t}\) :

Total movable mass

\({m}_{i},{c}_{i},{k}_{i}\) :

Modal mass, damping coefficient, and stiffness of the \(i\) th eigenmode

\(N\) :

Total number of modes

\(p, q\) :

Interpolation coefficients

\({r}^{2}\) :

Coefficient of determination

\(R\) :

Transformation coefficient from rotational to translational motion (= ℓ/2π)

\({T}_{s}\) :

Discrete sampling time

\(v\) :

System noise

\(w\) :

Measurement noise

\({x}_{i}\) :

Displacement of the \(i\) th eigenmode

\({x}_{m}\) :

Equivalent value of \({\theta }_{m}\) in transitional motion \((=R{\theta }_{m})\)

\({x}_{r}\) :

Relative position between the motor and stage

\({x}_{t}\) :

Stage position

\(y\) :

Measurement value

\(\alpha ,\zeta ,\omega\) :

Residue, damping ratio, and resonance angular frequency

\({\alpha }_{i},{\zeta }_{i},{\omega }_{i}\) :

Residue, damping ratio, and resonance angular frequency of \(i\) th mode considering stage position dependency

\({\theta }_{m}\) :

Motor angle

\({\varvec{A}}, {\varvec{B}}, {\varvec{C}}\) :

Coefficient matrices of the continuous-time state-space model

\({{\varvec{C}}}_{0}\) :

Conversion matrix from state vector to cutting force

\({{\varvec{H}}}_{{\varvec{m}}{\varvec{e}}{\varvec{a}}{\varvec{s}}}\) :

Complex column vector of the measured FRF dataset

\({{\varvec{H}}}_{{\varvec{m}}{\varvec{o}}{\varvec{d}}{\varvec{e}}{\varvec{l}}}\) :

Complex column vector of the modeled FRF dataset

\({\varvec{K}}\) :

Continuous Kalman gain matrix

\({\varvec{P}}\) :

Covariance matrix of the state estimation error

\({\varvec{Q}},\boldsymbol{ }{\varvec{R}}\) :

Covariance matrix of the system and measurement noise

\({\varvec{x}}\) :

State vector

\({\left(\right)}_{a}, {\left(\right)}_{c}, {\left(\right)}_{m}\) :

Anti-motor side, center point, and motor side of the feed drive

\({\left(\right)}_{CUR}\) :

Current-based estimated force

\({\left(\right)}_{i}\) :

\(i\) Th mode

\({\left(\right)}_{LDOB}\) :

LDOB-based estimated force

\({\left(\right)}_{y}\) :

Y-axis

\({\left(\right)}_{\alpha },{\left(\right)}_{\zeta },{\left(\right)}_{\omega }\) :

Residue, damping ratio, and resonance angular frequency

\({\left(\right)}^{ref}\) :

Reference value

\(\left(\widehat{}\right)\) :

Estimated value

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Funding

This study was partially supported by JSPS KAKENHI Grant Number JP19H00735.

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Authors and Affiliations

  1. Department of System Design Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa, 223-8522, Japan

    Keisuke Yamamoto, Aya Kamba & Yasuhiro Kakinuma

  2. Research and Development Center, Shibaura Machine Co., Ltd, 4-29-1 Hibarigaoka, Zama, Kanagawa, 252-0003, Japan

    Kazuhiro Takeuchi, Jun Fujita & Yusuke Fujimagari

Authors
  1. Keisuke Yamamoto
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Correspondence to Keisuke Yamamoto.

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Yamamoto, K., Kamba, A., Takeuchi, K. et al. Enhancing the multi-encoder-based cutting force estimation along the stationary axis of a machine tool with multiple inertia dynamics. Int J Adv Manuf Technol 123, 1215–1229 (2022). https://doi.org/10.1007/s00170-022-10245-9

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