When a malfunction is observed in a manufacturing process, it must be corrected to maintain control over the process. It is in this context that Statistical Process Control (SPC) (cf. Pillet [1]) is currently being implemented in several companies to best prevent incidents from occurring during production. Its main objective in industrial production is to maintain the quality characteristics of the manufactured product as close as possible to their target values defined in the design. To this end, the aim is to detect a malfunction as early as possible in order to trigger appropriate corrective action. There are several methods and tools in the literature for detecting out-of-control processes.

Walter A. Shewhart [2] did remarkable work by first proposing the one-dimensional control chart for monitoring production processes. He extended his idea to introduce the X ¯ control chart to monitor the means and standard deviation of a process. Later, this same Shewhart [3] showed that the statistical approach based on the detection of out-of-control situations is well-suited for solving manufacturing process problems.

E. Del Castillo [4] tested the X ¯ average control chart when the standard deviation is unknown, for small batches. He demonstrated that it is preferable to use the X ¯ chart with a standard deviation estimated from individual samples, rather than using it with an average standard deviation, which generates false alarms.

Other types of control charts, such as CUSUM (Cumulative Sum) or EWMA (Exponentially Weighted Moving Average), have been used by authors (Box [5], Yi-Dai [6], Huwang [7]) for monitoring manufacturing processes. Some authors have compared the effectiveness of different control charts, following the example of G. Brunschwig [8] or Boudaoud [9]. G. Brunschwig showed that the Shewhart control chart is more suitable for detecting sudden and significant drifts, while the CUSUM and EWMA charts are more effective in detecting slow drifts. N. Boudaoud and C. Zohra [9] proposed a drift detection approach using a multimodal control chart based on the statistical approach of Hotelling T² [10]. He made a comparative study of the CUSUM and EWMA charts and the multimodal T² chart. He showed that the ARL (Average Run Length) value of the T² chart is much higher than that of the first two charts, but that the CUCUM chart is better in performance in the case of detecting slow drifts.

The author Douglas C. Montgomery [11] demonstrated the joint use of SPC (Statistical Process Control) and EPC (Engineering Process Control) for process monitoring through the Shewhart control chart, CUSUM, and EWMA. He indicated that the role of SPC is to correct the process when it is deemed out of control, while the role of EPC is to continually adjust the process to avoid drift.

Frank E. Grubbs [12], in turn, proposed a statistical approach to detect outliers in individual samples based on a test criterion (Tn) which corresponds to the square root of the statistical distance. This criterion is associated with a risk of rejecting good observations. He shows that if the risk is between 1% and 5%, the test gives a more reliable detection result, but that beyond 5% of the risk, the result is not guaranteed.

Researchers (Lafaye [13], Epprecht [14]) have worked on multidimensional process monitoring using multivariate control charts to track multiple process parameters simultaneously.

Robert L. Mason and Young [15] have done excellent work on the statistical monitoring of multivariate processes. He used Hotelling’s statistical distance (T²) to design the T² chart to detect out-of-control situations using an elliptical representation of correlated variables. He shows that for a bivariate normal function, there is up to a 95% probability of finding all points with the same statistical distance within the elliptical contour. Points outside the elliptical contour correspond to out-of-control situations.

Del Castillo et al. [16] have developed a sequential process adjustment based on a stochastic approximation to approximate the average value of the target. They used the rule of Grubbs [12], which is to adjust the machine by a successive approach. Kibe et al. [17] proposed to adjust the initial position of tools using an in-situ measurement because the initial setting of the tools is very approximate. The authors (Bourdet [18], Anselmetti et al. [19]) showed that if we can act on tool offsets in the machine to reduce the deviation between the theoretical part and the machined part, it can also be reduced by acting on the optimum dimensions chosen intelligently. Goldschmidt [20] and Pairel (Pairel et al. [21], Pairel et al. [22]) have developed the methodology Copilot Pro^®, which allows adjusting all the tools for roughing and finishing by measuring the manufacturing dimensions established by the Office of the methods. Works on an inertial approach for setting machines were conducted by Pillet et al. [23] in the case of inertial tolerancing.

The authors (Pillet [24], Del Castillo et al. [16]) worked on the statistical approach to process adjustment in the one-dimensional case. Pillet [25] showed that using the classic Shewhart map allows for early process adjustment because it generates false alarms. He then introduced a tuning rule that takes into account short-term dispersion and the measured deviation to calculate corrections. In this article, we exploit the statistical distance T² to detect cutting tool drift at an opportune time without generating false alarms.

This approach complements the work carried out by M. Pillet in the context of inertial control. It provides a significant improvement to inertial control for effectively adjusting processes in a multivariate control framework. The results of 10⁶ simulations generated using data from the Mersenne Twister algorithm, performed on three types of maps, provide relevant results that validate the approach. A comparative result based on ARLs of these charts is presented in the article. We illustrate our remarks using an example of a simple part whose sole purpose is to show the influence of three correlated characteristics on the setting of the tool that produces them.

In the case of one-dimensional charting, in the literature, there are many methods and tools for the detection of a non-controlled process. Walter A. Shewhart [2][3] accomplished a tremendous work with his one-dimensional control chart for process surveillance. Later, he extended his idea with the introduction of the X ¯ control chart for monitoring the mean values and the standard deviation of a process. E. Del Castillo [4] tested this type of chart when the standard deviation is unknown, in the case of small-sized batches. He demonstrated that it is better to use this chart with an estimated standard deviation from individual samples than with a mean standard deviation in terms of false alarm rate. Other types of control charts, such as CUSUM (Cumulative Sum) or EWMA (Exponentially Weighted Moving Average) charts, have been exploited by authors (Box [5], Dai [6], Huwang [7] for monitoring of the manufacturing process.

In the case of multidimensional charting, in practice, many industrial processes have their performances tied to a set of interrelated variables. Monitoring such processes is named Multivariate Statistical Process Control (MSPC). Multivariate process control is one of the subjects in the field of statistical process control in perpetual evolution. Many studies have been done about it. Mason and Young [15] and Bersimis [26] wrote a comprehensive review of the literature on multivariate process control chart techniques.

According to Zhang [27], multivariate process measurement benefits from the use of inherent multivariate methods rather than a collection of univariate charting methods applied to the individual components. The development of multivariate control charts originates from the work by Hotelling [10]. In his paper, the author develops a single control chart that regroups the EWMA method and the generalized likelihood ratio (GRL) test. Monitoring the process mean and variability through a mean vector and a covariance matrix.

N. Boudaoud and C. Zohra [9] proposed to detect drifts with a multimodal control chart based upon the Hoteling’s T² statistic ([10]. He conducted a comparative study of CUSUM and EWMA charts and the multimodal T² chart. He has demonstrated that the ARL value (Average Run Length) for the T² chart was superior to the others. But concerning slow drifts, the CUSUM chart remains better.

In this paper, we will show how a specific use of the T² statistic in Inertial Steering can help deal with correlated dimensions by computing T² as a variable for correcting tools.

The objective of any production is to produce parts that conform to the requirements of the geometry established by CAD (Computer Aided Design). This requirement is materialized by a digital target on which is added an acceptable level of variability (tolerances). The Inertial Process Steering approach is a tool that is able to reconcile the real workpiece to its digital model through the measured dimensions of the workpiece. Inertial tolerance is the quality indicator of the part in the Inertial Process Steering approach. The inertial tolerance of a surface is calculated by Equation (1) below.

These results are to be compared with the corresponding ARL following the strategies:

If 1 out of the 3 measurements is outside Shewhart’s limits, the part is discarded.If the one measurement tagged as charted is outside Shewhart’s limits, the part is discarded (The measurements have the same characteristics here, so the choice is irrelevant).

Figure 3. False alarm detection.

The ARL0 (no perturbation) is a false alarm indication. The higher the ARL0 is, the fewer the number of false alarms is. On the contrary, when there is a perturbation, a lower ARL signifies a better detection capability (Figure 3).

According to the graph realized with the simulation, both of the one-dimensional approaches have their advantages and disadvantages. The one-by-one strategy sets fewer false alarms on, whereas the “one chart” strategy has a better detection rate for perturbation.

The T² chart presents the advantages of both. Moreover, with a zoom (Figure 4), the T² detection rate is revealed as even better than the one-by-one chart when the perturbation exceeds about 1.5 σ.

Figure 4. Tool drift detection performance profile using charts measured by ARL.

With a lower false alarm and higher perturbation detection rates, this approach appears as an improvement in the context of Inertial Process Steering filtering.

Today, the common practice in industrial manufacturing is to consider the product characteristics as independent. The Inertial Process Steering method is based on the establishment of a relation between a deviation vector and a correction vector. By nature, it is a multidimensional relation. The Steering Matrix computed from the Incidence matrix addresses both the multidimensional aspect of the problem and the plausible correlation in the data imputed by a problem of tool adjustment.

The use of Hotelling’s T² for CNC machine tools allowed for reduced machine setting time, particularly in companies that manufacture mechanical parts. For example, in watchmaking companies where the watch plate contains no fewer than eighty (80) interdependent dimensions and whose setup takes approximately two (2) weeks, this will significantly reduce setup time, essentially reducing it to the time required to establish the incidence matrix. This approach also reduces scrap rates, improves tool life management, and simultaneously minimizes machine downtime.

This paper has shown, via a simulation, how Hoteling’s T² statistic can help improve detection in Inertial Process Steering. First, by keeping a multidimensional point of view, the gain of information compared to one-dimensional Shewhart charting leads to a better efficiency at filtering the manufactured production. Second, when a process is out of control due to a problem with a tool adjustment, filtering the correction with a chart focusing on the tool instead of the dimensions should be more legitimate. Third, the computation of T² can be easily implemented in the already existing software. Indeed, all the data needed to compute T² are already input (the deviations and the standard deviations).

The performed simulation compared the efficiency between the standard Shewhart and T² charting. A narrower analysis could be realized by comparing this approach with CUSUM and EWMA charts.

This paper deals with the new perspective coming from the study of correlations in IPC, not with the optimal adjustment while running the method. This explains why the χ² statistic, for the purpose of convenience and common use, has been chosen.

One of the major difficulties in the case of vectorial systems is the complexity of the computation, which increases exponentially with the size of the vectors. Hence, through the T² statistic, reducing the multivariate situation to a scalar value while accounting for the values inputted in the deviation vector seems promising.

Using Hoteling’s T² for CNC machine tool control reduces machine setup time, particularly in mechanical parts manufacturing plants. For example, in watchmaking plants where the watch plate contains no fewer than eighty (80) interdependent dimensions and whose setup takes approximately two (2) weeks, this will significantly reduce setup time, essentially reducing it to the time required to establish the incidence matrix. This approach also reduces scrap rates, improves tool life management, and simultaneously minimizes machine downtime.

The study was presented on a specific case of a 1D part, where the method considers the columnar information of the incidence matrix. However, it does not consider both the row and columnar information of the incidence matrix. The approach is limited in its ability to take into account both the correlations between tool offsets and the geometric characteristics of the part. This limits its generalization to more complex parts where several cutting tools act simultaneously on one or more geometric characteristics of the part.