3D Modelling Of Customized Lasts Based On Anthropometric Data Acquired From 3D Foot Scanning-One Study Case


 Designing and manufacturing personalized lasts are the first steps in obtaining the right fitted footwear for various users, especially for sport or/and medical purposes. The accurate dimensional relationship between foot and last represents the key element for this activity. The critical shape of the last should always be determined by the shape of the foot and the cumulative relationship between lengths, widths, heights and girths, whatever method is used, Some corrections and constraints must always be considered because the shoe-last is not identical to the foot. The foot anthropometric measurements are modified based on biomechanical constraints and technological limitations and they are interactively transformed into last’s dimensions by using 3D modelling. The present study brings together the modern scanning technique with the new methodology for modifying a reference last, and it is aimed to explore the philosophy of re-designing functional lasts. It also tests and highlights the limits of the actual methodology for shoe-last virtual prototyping based on anthropometric data acquired from one commercially available 3D foot scanning system.

The subject's foot is scanned by using a 3D foot scanning system; respectively the INFOOT USB Standard Model IFU-S-01, provided with eight progressive ¼' CCD cameras and four laser instruments, class 1M. INFOOT scans a foot and positions the anatomical landmarks, which are used to measure automatically/calculate up to 20 measuring items. It scans the 3D foot form and the anatomical points in about 10 seconds per foot, and the dimensions and angles are automatically calculated and viewed in a few seconds. The subject stands with one barefoot inside the scanner and one foot outside the scanner, and the entire mass of the subject is equally distributed on both feet.
The scanned foot data can be used for foot morphological analysis, footwear/last selection, and also for designing new lasts or re-designing existing ones.
The scanned data have the points cloud format, wireframe or solid format, and they are saved as FBD binary data that gives both the 3D foot shape and the position of the anatomical points. The binary le can also be exported by a speci c INFOOT software module (for example File Converter) as *.csv, *.dxf, *.vrml or *.stl formats. These exported formats could be imported into different modelling or designing software. For this study, the OrthoLast modelling software from Delcam Crispin has been used.

Positioning of the anatomical points on foot
Accurate positioning of the anatomical points in uences the value of anthropometric parameters. For the hereby-presented study case, the anatomical points mapping (Fig. 1) suggested by the scanner's producer -INFOOT used (*** INFOOT website). The landmarks are automatically given by software in few seconds. Because several problems and structural modi cations against normal foot have been identi ed for this case, each anatomical point is checked, and it is moved (if necessary) in its right position. Also, each transversal section is checked and corrected in case shape distortions occurred during scanning. Even if the scanning process takes several seconds, the correction process can take a long time. The commercially available scanning systems recognize the anatomical points for normal feet; in the case of feet having anomalies, this standard facility is less useful. Therefore, the accurateness in measurements taken for customized footwear can be affected by introducing huge errors regarding positioning the anatomical points (Sarghie et al., 2013).

Measurement/calculus of the main anthropometric measurements
The applied anthropometry into footwear industry aims to measure the foot. The foot measurements are assessed through precisely de ned points that are called anatomical points. The anatomical points are some protuberances of the foot skeleton or its joints, and they are becoming well-shaped limits of the soft tissues. Several basic measurements are mentioned (Xiong S., et.al. 2008) for characterizing the foot dimensions and, therefore, its anthropometric. The longitudinal measurements (lengths of the foot) represent the distances from the heel extreme point (landing point or nearby it) to a series of precise anatomical points (for example, 1st or 2nd toe, instep point, 1st metatarsal head and 5th metatarsal head, etc.). These distances are measured up along the longitudinal axis of the foot. There are different opinions among specialists regarding the right position for this axis (Reel S., et.al. 2010, Chantelau E., and Gede A., 2002, Nikolaidou M.E., and Boudolos K.D. 2006). To keep the same reference as for the longitudinal axis of the last, the longitudinal axis of the foot is given in this study by the line that joins the heel centre with the head of 2nd metatarsal bone (Pastina et al., 2012). The transversal measurements are represented by widths and girths. The widths are measured on the outline of the foot or the footprint, perpendicularly on the foot's longitudinal axis or in line with ball direction. The girths are circumferences of foot measured up according to with previous de ned sectional planes on metatarsal heads, instep, heel, ankle, etc. The heights represent the vertical distance measured up from the footing surface. These dimensions are measured through precisely de ned points that are called anatomical points. The anatomical points are some protuberances of the skeleton or the joints, and they are becoming well-shaped limits of the soft tissues. Due to lack of generalized, universal accepted rules for taking measurements on foot/last, as well due to the need of standardized models for transforming the anthropometric data into dimensional parameters of last, the foot anthropometry applied to designing well-tted footwear has been found quite di cult (Bertram H., 2011).

Comparing the foot against the reference last
The reference last, which is imported from an existing database, is subject to an interactively comparing process against the scanned foot. Following simpli ed hypotheses were considered for this study: the selected reference last has appropriate size towards subject's foot length, it has low heel height, and it has rounded toe. On these lines, by using the Compare module of OrthoLast-Delcam Crispin software, the two 3D shapes (foot and last) were brought together on the same screen. The reference last and the foot are successively moved and rotated to align them in the same plane (Fig. 3).
When the two forms (last and foot) are correctly positioned, the face centre line of the foot should match with the face centre line of the last and the centre back line of the foot should match with the back centre line of the last. This process is somehow time-consuming, and it requires from designer to have strong visual abilities for correct perspectives on 3D forms moving into 2D space available on at computer screens. At this point, any further technological developments on similar software could be very useful regarding making this process in an automatic manner, just with several corrections at the end.  Figure 4 illustrates the methodology for measuring the herebymentioned parameters.
The technique of transforming an initial 3D structure into a new one, namely Free-From Deformation of Solid Geometric Models, represents one of the graphic procedures that allow for modifying a 3D structure by moving the basic points/nodes of its grid (Sederberg, T.W. 1986, Mochimaru M. and Kouchi M. 2011; Dumitras C., Cozminca I., 2008). Mochimaru M. et.al (2000) used this method for building new deformed grids suitable for grading the lasts. In our study, one structure (the last) represents the grid that will be interactively modi ed by moving precise points, and the other structure (the foot) represents the comparing form. The last and the foot are being compared until they are overlapping in as many points as possible. The two 3D shapes have different appearances: draft solid for foot and gridded frame for last. By overlapping, it can be seen the differences between the foot and the last. Therefore, the last will be modi ed in precisely selected areas (Fig. 5).
The last is modi ed acting on nine typical dimensional parameters, namely interactively modi ed parameters: SL, LW, BG, IUG, HW, HH, HC, TS, and TT. These parameters have been selected based on an initial analysis of the need for modi cation according to with the subject's foot. The other dimensional parameters that also describe the modi ed shape of the last represent the outcome-modi ed parameters, and they are BL, BUG, BWC, BWL, IWC, IWL, SHC, EW, TL, AC, AW. On each step, one single parameter from the rst category is interactively modi ed. The modi cation upon one parameter is affecting all studied parameters that allow for collecting series of data to be statistically analyzed.

Results
When wearing shoes, the foot is constrained to modify its shape and dimensions among certain admissible limits of tightening. The constructive dimensional parameters of last provide limits of tightening the foot by footwear. As a result, a lower level of tightening the foot by footwear that will reduce the risk of high pressures on concrete foot surfaces is one functional requirement in this study case ( Table 1 shows the obtained data from 3D interactive modi cations on dimensional parameters by following up nine successive steps. Each step is based on the results obtained in the previous step. When one parameter is modi ed, the entire range of the studied parameters is collected. While the modelling process advances, there can be determined paired relationships among sets of parameters that characterize the shape of last at each step of modi cation. Also, the set of dimensional parameters for the resulting last could be compared against the initial one. The dimensional parameters of the last, both the interactively modi ed parameters and the outcome parameters have different values on successive steps of modi cation. For comparing on a common basis the range of variations (increase or decrease) for each parameter, following calculus has been considered: Where: i -step of modi cation, i = 1, 2, 3, 4, 5, 6, 7, 8, 9 Di -increase or decrease of modi ed parameter on step i,, in % Pi -value of modi ed parameter on step i, in mm Pi-1 -value of modi ed parameter on step i-1 (previous step), in mm Di variations are calculated against the previously obtained values; thus, the dependencies between two successive modi ed lasts can be progressively noticed and quanti ed. Additionally, after the nal iteration is performed (corresponding to the 9th step), the variation is calculated with the same relation (Eq. 1), by comparing nal obtained parameters with parameters of the reference last.
Because the order of modifying the last was empirically established, a mathematical method is proposed by the authors using a DSM matrix. The right method will be the one in which the last's parameters are closer to the scanned foot.
DSM matrix design was rstly described by Yassine in 1999, being developed and applied later in many elds (Mihai A. et al. 2009). In this type of analysis, three steps are followed: 1. The product is decomposed in elements and the relationships and connections between them are established; 2. The identi ed elements are written in the same order in a matrix. The connection between elements is marked inside the matrix; 3. The matrix is transformed using special algorithms in a low triangular shape by arranging rows such that the marked points are located close to the diagonal of the matrix. A specialized software facilitates and simpli es the procedure.
In our case, the algorithm used is the one A. Kusiak et al., developed in 1994 and available at: http://css.engineering.uiowa.edu/~ankusiak/processmodel.html was used.
Based on the obtained results, the initial last will be modi ed respecting the order given by the DSM matrix. The results can be seen in the table below: The results of the two methods, empirically one and DSM method are presented in the table below. Comparing the results obtained in both cases, the empirical method and the DSM method, lower deviation from the initial last are obtained in the user de ned order: SL > LW > BG > IUG > HH > HW > HC > TS > TT. Research is recommended to continue, for testing another order for the parameters.
All successive modi cations on the empirical method are made considering the foot's anthropometric measurements and the dimensional parameters of the reference last (Table 5). The result is an adapted last to the subject's foot (Fig. 8). Table 5 Correspondences among the foot, the initial last and the modi ed last needed to assess entirely customized lasts and customized footwear. The lack of information in this area still determines the designers to select the last just based on empirical approaches gained from previous experiences (Wang Z. et al, 2009).
The described method, as well as the developed methodology for analyzing the nal results of the modelling process, could be replicated and applied for any new study case. In order to obtain the modi ed last, ve working stages have been followed up: 1) scanning the foot; 2) positioning the anatomical points on foot; 3) measuring/calculating the main anthropometric measurements; 4) comparing the foot against the reference last; 5) modifying the last towards the foot shape.
The studied case refers to a subject (47-year-old, female, diagnosed with arthritis and incipient stages of Hallux-Valgus) having visibly identi ed foot problems that ask for a careful interpretation of the design features based on anthropometric data, biomechanics and orthopaedic requirements. Several initial stages of structural modi cation related with arthritic feet have been identi ed on this subject, especially in the forefoot area.
The obtained data from 3D interactive modi cations on dimensional parameters follows up nine successive steps. Each step is based on the results obtained in the previous step. When one parameter is modi ed, the entire range of the studied parameters is collected. While the modelling process advances, there are determined paired relationships among sets of parameters that characterize the shape of last at each step of modi cation.
The dimensional parameters of the last, both the interactively modi ed parameters and the outcome parameters have different values on successive steps of modi cation.
Each modi cation on a parameter brings changes to the other parameters. The designer has to follow each change, so that at the end, the last will be proper to the subject's foot. This research has indicated that modifying last's parameters are not enough for designing a customized last, the relationship between parameter has to be taken into account. The limitation of this paper given by the fact that some foot area could be tightened while wearing a footwear product, and to know if these are acceptable, the subject has to wear a shoe designed on these speci c lasts.

Conclusions
This research follows up an extended and integrated application on combining CAD techniques for 3D shoe-last modelling with 3D foot scanning and measuring procedures. Both 3D scanning and 3D virtual modelling in the practice of designing fully customized lasts for special purposes are analyzed and presented as grounded related results for further innovation and development on software applications in the footwear industry. The present study brings together the modern scanning technique with the new methodology for modifying a reference last through interactive 3D modelling. Thus, it aims to go deeply into re-designing process of functional lasts based on anthropometric data obtained from 3D foot scanning. This research emphasizes the necessity of reuniting various modelling methods and scanning techniques into a new common standardized approach that should make the data transforming process more easily and precisely.

Declarations
Ethics approval Not applicable.
Consent to participate All authors have approved to participate.
Consent for publication The manuscript is approved by all authors for publication.
Con ict of interest The authors declare that they have no con ict of interest. Interactively aligning the foot to the reference last Illustrated methodology for measuring dimensional parameters of shoe-last Figure 5 The nine-step methodology for modelling the last.

Figure 6
Establishing the matrix elements Figure 7 Generating the matrix Figure 8 Foot (a), initial last (b) and modi ed last (c)