Experimental and fuzzy modelling analysis on dynamic cutting force in micro milling

Abstract

Prediction of cutting forces is very important for the design of cutting tools and for process planning. This paper presents a fuzzy modelling method of cutting forces based on subtractive clustering. The subtractive clustering combined with the least-square algorithm identifies the fuzzy prediction model directly from the information obtained from the sensors. In the micro-milling experimental case study, four sets of cutting force data are used to generate the learning systems. The systems are tested against each other to choose the best model. The obtained results prove that the proposed solution has the capability to model the cutting force in spite of uncertainties in the micromilling process.

Appendix

For the first-order model presented in this paper, the consequent functions are linear. In the method of Sugeno and Kang (1988), least-square estimation is used to identify the consequent parameters of the TSK model, where the premise structure, premise parameters, consequent structure and consequent parameters were identified and adjusted recursively.

In a TSK FLS, rule premises are represented by an exponential membership function. The optimal consequent parameters p ^k₀ , p ^k₁ , p ^k₂ ,…, p ^k _n (coefficients of the polynomial function) for a given set of clusters are obtained using the least-square estimation method.

When certain input values x ⁰₁ , x ⁰₂ ,…, x ⁰ _n are given to the input variables x ₁, x ₂,…,x _n , the conclusion from the kth rule in a TSK model is a crisp value w ^k:

$$ w^{k} = p_{0}^{k} + p_{1}^{k} x_{1}^{0} + p_{2}^{k} x_{2}^{0} + \cdots + p_{n}^{k} x_{n}^{0} $$

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with a certain rule firing strength (weight) defined as

$$ \alpha^{k} = \mu_{1}^{k} (x_{1}^{0} ) \cap \mu_{2}^{k} (x_{2}^{0} ) \cap \cdots \cap \mu_{n}^{k} (x_{n}^{0} ) $$

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where μ ^k₁ (x ⁰₁ ), μ ^k₂ (x ⁰₂ ),…, μ ^k _n (x ⁰ _n ) are membership grades for fuzzy sets Q ^k₁ , Q ^k₂ ,…, and Q ^k _n in the kth rule. The symbol ∩ is a conjunction operator, which is a T-norm (the minimum operator ∧ or the product operator *).

Moreover, the output of the model is computed (using weighted average aggregation) as

$$ w = \frac{{\sum\nolimits_{k = 1}^{m} {\alpha^{k} w^{k} } }}{{\sum\nolimits_{k = 1}^{m} {\alpha^{k} } }} $$

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Suppose

$$ \beta^{k} = \frac{{\alpha^{k} }}{{\sum\nolimits_{k = 1}^{m} {\alpha^{k} } }} $$

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Then, (9) can be converted into a linear least-square estimation problem, as

$$ w = \sum\limits_{k = 1}^{m} {\beta^{k} w^{k} } $$

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For a group of $\lambda$ data vectors, the equations can be obtained as

$$ \begin{gathered} w^{1} = \beta_{1}^{1} (p_{0}^{1} + p_{1}^{1} x_{1} + p_{2}^{1} x_{2} + \cdots + p_{n}^{1} x_{n} ) + \beta_{1}^{2} (p_{0}^{2} + p_{1}^{2} x_{1} + p_{2}^{2} x_{2} + \cdots + p_{n}^{2} x_{n} ) + \cdots \hfill \\ + \beta_{1}^{m} (p_{0}^{m} + p_{1}^{m} x_{1} + p_{2}^{m} x_{2} + \cdots + p_{n}^{m} x_{n} )) \hfill \\ \end{gathered} $$

$$ \begin{gathered} w^{2} = \beta_{2}^{1} (p_{0}^{1} + p_{1}^{1} x_{1} + p_{2}^{1} x_{2} + \cdots + p_{n}^{1} x_{n} ) + \beta_{2}^{2} (p_{0}^{2} + p_{1}^{2} x_{1} + p_{2}^{2} x_{2} + \cdots + p_{n}^{2} x_{n} ) + \cdots \hfill \\ + \beta_{2}^{m} (p_{0}^{m} + p_{1}^{m} x_{1} + p_{2}^{m} x_{2} + \cdots + p_{n}^{m} x_{n} )) \hfill \\ \end{gathered} $$

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$$ \begin{gathered} w^{\lambda } = \beta_{\lambda }^{1} (p_{0}^{1} + p_{1}^{1} x_{1} + p_{2}^{1} x_{2} + \cdots + p_{n}^{1} x_{n} ) + \beta_{\lambda }^{2} (p_{0}^{2} + p_{1}^{2} x_{1} + p_{2}^{2} x_{2} + \cdots + p_{n}^{2} x_{n} ) + \cdots \hfill \\ + \beta_{\lambda }^{m} (p_{0}^{m} + p_{1}^{m} x_{1} + p_{2}^{m} x_{2} + \cdots + p_{n}^{m} x_{n} )) \hfill \\ \end{gathered} $$

These equations can be represented as

$$ \left[ {\begin{array}{*{20}c} {\beta_{1}^{1} x_{1} } & {\beta_{1}^{1} x_{2} \cdots \beta_{1}^{1} x{}_{n}} & {\beta_{1}^{1} } & \cdot & \cdot & \cdot & {\beta_{1}^{m} x_{1} } & {\beta_{1}^{m} x_{2} \cdots \beta_{1}^{m} x_{n} } & {\beta_{1}^{m} } \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ {\beta_{\lambda }^{1} x_{1} } & {\beta_{\lambda }^{1} x_{2} \cdots \beta_{\lambda }^{1} x_{n} } & {\beta_{\lambda }^{1} } & \cdot & \cdot & \cdot & {\beta_{\lambda }^{m} x_{1} } & {\beta_{\lambda }^{m} x_{2} \cdot \cdot \cdot \beta_{\lambda }^{m} x_{n} } & {\beta_{\lambda }^{m} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {p_{1}^{1} } \\ \begin{gathered} p_{2}^{1} \hfill \\ : \hfill \\ \end{gathered} \\ \begin{gathered} p_{n}^{1} \hfill \\ p_{0}^{1} \hfill \\ \end{gathered} \\ \cdot \\ \cdot \\ {p_{1}^{m} } \\ \begin{gathered} p_{2}^{m} \hfill \\ : \hfill \\ \end{gathered} \\ \begin{gathered} p_{n}^{m} \hfill \\ p_{0}^{m} \hfill \\ \end{gathered} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {w^{1} } \\ {w^{2} } \\ \cdot \\ \cdot \\ {w^{3} } \\ \end{array} } \right] $$

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Using the standard notation AP = W, this becomes a least square estimation problem where A is a constant matrix (known), W is a matrix of output values (known) and P is a matrix of parameters to be estimated. The well-known pseudo-inverse solution that minimizes $\left\| {AP - W} \right\|^{2}$ is given by

$$ P = (A^{\text{T}} A)^{ - 1} A^{\text{T}} W $$

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