A novel NURBS surface approach to statistically monitor manufacturing processes with point cloud data

Abstract

As sensor and measurement technologies advance, there is a continual need to adapt and develop new Statistical Process Control (SPC) techniques to effectively and efficiently take advantage of these new datasets. Currently high-density noncontact measurement technologies, such as 3D laser scanners, are being implemented in industry to rapidly collect point clouds consisting of millions of data points to represent a manufactured parts' surface. For their potential to be realized, SPC methods capable of handling these datasets need to be developed. This paper presents an approach for performing SPC using high-density point clouds. The proposed approach is based on transforming the high-dimensional point clouds into Non-Uniform Rational Basis Spline (NURBS) surfaces. The control parameters for these NURBS surfaces are then monitored using a surface monitoring technique. In this paper point clouds are simulated to determine the performance of the proposed approach under varying fault scenarios.

Appendix A

This appendix covers the derivation of the estimated control point matrix given in Eq. (6) and the NURBS model residuals matrix given in Eq. (13).

Estimated control point matrix

This section of the appendix covers the derivation from Eqs. (5) to (6), which are identified in the appendix as Eqs. (A1) and (A11), respectively. The NURBS model used in this paper can be expressed in matrix notation as

$$ {\mathbf{B}}_{u} {\mathbf{F}}_{u} = {\mathbf{S}} + {\mathbf{E}} . $$

(A1)

Here ${\mathbf{B}}_{u}$ is the matrix of B-Spline basis functions for the $u$ direction,

$$ {\mathbf{B}}_{u} = \left[ {\begin{array}{*{20}c} {B_{u,1} \left( {\bar{u}_{1} } \right)} & \cdots & {B_{{u,n_{u} }} \left( {\bar{u}_{1} } \right)} \\ \vdots & \ddots & \vdots \\ {B_{u,1} \left( {\bar{u}_{l} } \right)} & \cdots & {B_{{u,n_{u} }} \left( {\bar{u}_{l} } \right)} \\ \end{array} } \right],\varvec{ } $$

(A2)

${\mathbf{F}}_{u}$ is the matrix of control points for the isoparametric curves,

$$ {\mathbf{F}}_{u} = \left[ {\begin{array}{*{20}c} {f_{1,1} } & \cdots & {f_{1,h} } \\ \vdots & \ddots & \vdots \\ {f_{{n_{u} ,1}} } & \cdots & {f_{{n_{u} ,h}} } \\ \end{array} } \right], $$

(A3)

${\mathbf{S}}$ is the matrix of measured points,

$$ {\mathbf{S}} = \left[ {\begin{array}{*{20}c} {{\mathbf{s}}\left( {\bar{u}_{1} ,\bar{v}_{1} } \right)} & \cdots & {{\mathbf{s}}\left( {\bar{u}_{1} ,\bar{v}_{h} } \right)} \\ \vdots & \ddots & \vdots \\ {{\mathbf{s}}\left( {\bar{u}_{l} ,\bar{v}_{1} } \right)} & \cdots & {{\mathbf{s}}\left( {\bar{u}_{l} ,\bar{v}_{h} } \right)} \\ \end{array} } \right], $$

(A4)

and ${\mathbf{E}}$ is the matrix of errors,

$$ {\mathbf{E}} = \left[ {\begin{array}{*{20}c} {{\mathbf{e}}\left( {\bar{u}_{1} ,\bar{v}_{1} } \right)} & \cdots & {{\mathbf{e}}\left( {\bar{u}_{1} ,\bar{v}_{h} } \right)} \\ \vdots & \ddots & \vdots \\ {{\mathbf{e}}\left( {\bar{u}_{l} ,\bar{v}_{1} } \right)} & \cdots & {{\mathbf{e}}\left( {\bar{u}_{l} ,\bar{v}_{h} } \right)} \\ \end{array} } \right]. $$

(A5)

From Eq. (A1) the estimate for ${\mathbf{F}}_{u}$ can be calculated as

$$ {\hat{\mathbf{F}}}_{u} = \left( {{\mathbf{B}}_{u}^{\text{T}} {\mathbf{B}}_{u} } \right)^{ - 1} {\mathbf{B}}_{u}^{\text{T}} {\mathbf{S}}. $$

(A6)

Applying this estimate, Eq. (3) can be written in matrix form as

$$ {\mathbf{B}}_{v} {\mathbf{P}}^{\text{T}} = {\hat{\mathbf{F}}}_{u}^{\text{T}} + {\mathbf{G}}, $$

(A7)

where ${\mathbf{B}}_{v}$ is the matrix of B-Spline basis functions for the $v$ direction,

$$ {\mathbf{B}}_{v} = \left[ {\begin{array}{*{20}c} {B_{v,1} \left( {\bar{v}_{1} } \right)} & \cdots & {B_{{v,n_{v} }} \left( {\bar{v}_{1} } \right)} \\ \vdots & \ddots & \vdots \\ {B_{v,1} \left( {\bar{v}_{h} } \right)} & \cdots & {B_{{v,n_{v} }} \left( {\bar{v}_{h} } \right)} \\ \end{array} } \right], $$

(A8)

${\mathbf{P}}$ is the matrix of control points,

$$ {\mathbf{P}} = \left[ {\begin{array}{*{20}c} {{\mathbf{p}}_{1,1} } & \cdots & {{\mathbf{p}}_{{i,n_{v} }} } \\ \vdots & \ddots & \vdots \\ {{\mathbf{p}}_{{n_{u} ,1}} } & \cdots & {{\mathbf{p}}_{{n_{u} ,n_{v} }} } \\ \end{array} } \right], $$

(A9)

and ${\mathbf{G}}$ is the matrix of errors from the estimated control points ${\hat{\mathbf{F}}}_{u}$,

$$ {\mathbf{G}} = \left[ {\begin{array}{*{20}c} {{\mathbf{g}}\left( {1,1} \right)} & \cdots & {{\mathbf{g}}\left( {n_{u} ,1} \right)} \\ \vdots & \ddots & \vdots \\ {{\mathbf{g}}\left( {1,h} \right)} & \cdots & {{\mathbf{g}}\left( {n_{u} ,h} \right)} \\ \end{array} } \right]. $$

(A10)

From Eq. (A7) the estimate for ${\mathbf{P}}$ can be calculated as

$$ {\hat{\mathbf{P}}}^{\text{T}} = \left( {{\mathbf{B}}_{v}^{\text{T}} {\mathbf{B}}_{v} } \right)^{ - 1} {\mathbf{B}}_{v}^{\text{T}} {\hat{\mathbf{F}}}_{u}^{\text{T}} . $$

(A11)

NURBS model residuals matrix

This section of the appendix covers the derivation of Eq. (13), which is identified in the appendix as Eq. (A15). Taking the transpose of the estimated control points from Eq. (A11) results in

$$ {\hat{\mathbf{P}}}^{ } = {\hat{\mathbf{F}}}_{u} {\mathbf{B}}_{v} \left[ {\left( {{\mathbf{B}}_{v}^{\text{T}} {\mathbf{B}}_{v} } \right)^{ - 1} } \right]^{\text{T}} . $$

(A12)

Since ${\mathbf{B}}_{v}^{\text{T}} {\mathbf{B}}_{v}$ is an invertible matrix, $\left[ {\left( {{\mathbf{B}}_{v}^{\text{T}} {\mathbf{B}}_{v} } \right)^{ - 1} } \right]^{\text{T}} = \left[ {\left( {{\mathbf{B}}_{v}^{\text{T}} {\mathbf{B}}_{v} } \right)^{\text{T}} } \right]^{ - 1}$ and Eq. (A12) becomes

$$ {\hat{\mathbf{P}}}^{ } = {\hat{\mathbf{F}}}_{u} {\mathbf{B}}_{v} \left[ {\left( {{\mathbf{B}}_{v} {\mathbf{B}}_{v}^{\text{T}} } \right)} \right]^{ - 1} . $$

(A13)

By multiplying both side of Eq. (A13) with ${\mathbf{B}}_{v}^{\text{T}}$ results with

$$ {\hat{\mathbf{P}}}{\mathbf{B}}_{v}^{\text{T}} = {\hat{\mathbf{F}}}_{u} {\mathbf{B}}_{v} \left[ {\left( {{\mathbf{B}}_{v} {\mathbf{B}}_{v}^{\text{T}} } \right)} \right]^{ - 1} {\mathbf{B}}_{v}^{\text{T}} , $$

(A14)

which simplifies to ${\hat{\mathbf{P}}}{\mathbf{B}}_{v}^{\text{T}} = {\hat{\mathbf{F}}}_{u}$. Substituting ${\hat{\mathbf{F}}}_{u} = {\hat{\mathbf{P}}}{\mathbf{B}}_{v}^{\text{T}}$ into Eq. (A1) results in the NURBS model residuals matrix,

$$ {\hat{\mathbf{E}}} = {\mathbf{B}}_{u} {\hat{\mathbf{P}}}{\mathbf{B}}_{v}^{\text{T}} - {\mathbf{S}} $$

(A15)