Extension of the Stream-of-Variation Model for General-Purpose Workholding Devices: Vices and Three-Jaw Chucks

Nowadays, advanced manufacturing models, such as the stream-of-variation (SoV) model, have been successfully applied to derive the complex relationships between fixturing, manufacturing, and datum errors throughout a multistage machining process. However, the current development of the SoV model is still based on 3-2-1 fixturing schemes, and although some improvements have been done, e.g., N-2-1 fixtures, the effect of general workholding systems, such as bench vices or three-jaw chucks, has not yet been included into the model. This article presents the extension of the SoV model to include fixture and datum errors considering both bench vices and three-jaw chucks as fixturing devices in multistage machining processes. The model includes different workholding configurations, and it is shown how to include the workholding accuracy to estimate part quality. The extended SoV model is validated in a three-stage machining process by both machining experimentation and CAD simulations. Note to Practitioners—Part quality estimation in multistage machining systems is a challenging issue. The stream-of-variation (SoV) model is a straightforward model that can be used for this purpose. However, the current model is limited to fixture based on punctual locators, and common shop-floor devices are not considered yet. To overcome this limitation, this article extends the current SoV model to include vices and three-jaw chucks as workholding devices. The proposed methodology lets practitioners estimate the manufacturing capability of a process considering the technical specifications of these devices (e.g., parallelism and perpendicularity of vice surfaces and total indicator runout of chucks), or it can be used for diagnosing workholding issues. The model assumes that the workpiece acts as a rigid part and errors as deformations during clamping are assumed to be negligible in comparison with fixture- and datum-induced errors.

M ANUFACTURING processes have to be environmental friendly and safe and deliver high-quality products rapidly adapted to customer requirements at a minimum cost. One of the most important challenges in modern industry is the implementation of manufacturing systems capable of generating products with zero defects. A recent roadmap promoted by the European Commission has presented the state of art and the research priorities in the field of zero-defect manufacturing processes [1]. According to the roadmap, a research priority for the development of zero-defect manufacturing processes is related to the "integration of machine, fixture, tool, and workpiece models for quality and resource deterioration prediction" in multistage manufacturing processes (MMPs).
MMPs are manufacturing processes that consist of a sequence of stages where manufacturing operations, such as assembly or machining operations, are sequentially conducted to manufacture a part or product. Typical examples of MMPs are automobile body assembly processes and multijob machining processes where a part moves from one stage to another until a semifinished or finished product is obtained. Due to the sequential nature of these manufacturing operations, the error generated at the first stages may be propagated downstream to other stages, which produces additional manufacturing errors. These complex error interactions make it difficult to control product quality, and tasks such as predictive maintenance, process control, quality assurance, and fault diagnosis are challenging.
In order to illustrate the error propagation in an MMP, consider an MMP composed of three stages in a machining line, as shown in Fig. 1. As it can be observed, stage 1 presents a deviation of the cutting-tool trajectory, producing a machininginduced error. The resulting part moves to stage 2, where there are no additional errors. However, since the previous machined surface is used as a datum surface, the square shoulder machining operation is deviated with respect to the top surface, producing a datum-induced error. Finally, the part is moved to stage 3, where a locator has deviated from its nominal position, which produces a deviation of part location, and thus, the drill is misplaced producing a fixtureinduced error. As it can be seen, in MMPs where machining operations are conducted, three main sources of errors arise: machining-induced errors, datum-induced errors, and fixtureinduced errors. Note that similar reasoning can be conducted in assembly lines where welding operations are performed instead of machining operations.
Despite being very common manufacturing systems in the industry, the MMPs are usually too complex to be mathematically modeled, and the development of tools and strategies for effective quality assurance and fault diagnosis is currently a challenging task that hinders the deployment of zero-defect manufacturing processes. In the literature, some approaches have been proposed to model the error propagation within these types of manufacturing systems. One of these approaches is the so-called stream-of-variation (SoV) approach, which was successfully developed in the late 90s for multistage assembly processes by Jin and Shi [2]. The SoV model is based on the state-space model from control theory to define mathematically the relationships between fixture and machining errors on machined surfaces, and the datum errors are introduced to link the errors between the stages. The SoV model was expanded to include multistage machining processes in [3], and later, the model was highly improved by Zhou et al. [4] with the introduction of differential motion vectors (DMVs) to model the small displacements of each geometrical feature as it is used in the field of robotics [5]. This model can be considered as the SoV reference model within multistage machining processes, where the methodology to derive the model is explained in detail under the limitation of fixture devices based on 3-2-1 punctual locators. The model was expanded by Abellán-Nebot et al. [6] to include specific machining errors, such as tool wear errors, deflection errors, and kinematic errors from the tool axis. In regards to assembly processes, the SoV model was first developed for rigid sheet metal parts in [2], but it was later extended to deal with compliant sheet metal parts by Camelio et al. [7]. In [8], the SoV model was expanded to deal with compliant parts using N-2-1 locating schemes based on punctual locators, and later, the mathematical derivation to consider 3-D rigid assemblies instead of sheet metal parts was presented in [9]. More recently, the model was also extended to deal with composites in multistage assembly processes for the aeronautic industry considering compliant parts with anisotropic properties [10].
The application of the SoV model has been widely studied in the last two decades, and promising results have been presented in different fields, such as fault diagnosis and quality control [11]- [16], process planning [17], [18], and manufacturing tolerance allocation and predictive maintenance [19], [20]. However, despite the efforts made by many researchers, the SoV model still presents some drawbacks for its application in MMPs. One of the major criticisms refers to fixture error modeling, which is focused on punctual locators based on 3-2-1 schemes or N-2-1 schemes when compliant parts are considered, but positioning cases with plane/plane contact or cylinder/cylinder floating contact are not considered [21]. Under 3-2-1 schemes, the touching points between the locating surface and the fixture device are known, and the mathematical model that relates the error of each locator and the deviation of workpiece location can be determined. However, other common fixture devices, such as vices or chucks, do not follow this behavior, and the touching points of the locating surface and the fixture device may depend on previous errors. In this situation, the mathematical model between fixture errors and workpiece location errors cannot be determined in advance, and it will depend on the workpiece errors at the moment of clamping. This problem was tackled by Abellán-Nebot et al. [22], where a generic procedure for modeling fixtures based on surfaces instead of punctual locators was presented. Although the methodology deals with different configurations that depend on previous errors, the research does not deal with specific fixtures, such as vices, and omits other types of fixtures, such as chucks. More recently, the inclusion of the bench vice errors into the SoV model has been introduced in [23]. However, the proposed methodology only showed the result for a specific vice without deriving a generic approach based on differential and homogeneous transformation matrices among fixture/workpiece features. Therefore, more general vices or alternative ones (i.e., rotatory or universal vices) cannot be modeled. Furthermore, the applicability of the methodology is limited since there is no clear use of common workholding specifications in the modeling approach, which prevents their use in industry.
This article shows a methodology to model the effect of fixturing errors from two common fixtures in MMPs: threejaw chucks and bench vices. The mathematical development of these models follows the structure of the SoV model proposed in the aforementioned studies, providing compatibility with the general SoV approach. The model includes different workholding configurations, and it is also shown how to include the workholding accuracy to estimate part quality. The mathematical derivation of the models is validated through both CAD simulations and machining experimentation proving the high accuracy of the model despite linearization errors. Please note that, despite the low accuracy of these workholding devices in comparison with dedicated fixtures, their level of clamping and locating accuracy can be enough for low volume production systems where manufacturing tolerances of tenths of a millimeter are allowed [24], and thus, the inclusion of these devices into the SoV model may be of interest.
This article is organized as follows. Section II provides the general methodology of the SoV model in order to identify the parts of the model that have to be extended. Section III shows the mathematical derivation of the fixture-and datuminduced errors for bench vices, whereas Section IV presents the mathematical derivation for three-jaw chucks. Section V shows a case study where an MMP with both types of fixtures is applied and the model is validated through CAD simulations and machining experiments. Finally, Section VI shows the conclusions of this article.
In this article, terms ω R 0 L x , ω R 0 L y and ω R 0 L z are expressed as proper Euler angles between RCS and 0 LCS in a Z-Y'-Z" order (this means a rotation of RCS around its Z -axis, followed by a rotation around the new Y -axis, and finally, a rotation around the new Z -axis).  MMP of N-stages can be defined as [4] x where A k−1 · x k−1 represents the variations transmitted by datum features generated at upstream stages; B f k ·u f k represents the fixture-induced variations within stage k, where u f k denotes the fixture errors; B m k · u m k represents the machining-induced variations within stage k, where the cutting-tool path deviation is denoted as u m k ; and w k is the unmodeled system noise and linearization errors. The derivation of this model is detailed in [4], where it is presented the procedure to obtain matrices A k−1 , B f k , and B m k at each stage, according to given product and process information (part geometry and fixture layouts). Fig. 3 shows the auxiliary matrices to build the SoV main matrices according to the methodology detailed in Zhou's et al. research work [4]. Following their methodology, the matrices A k−1 , B f k , and B m k are defined as where A 1 k is the relocating matrix, A 2 k is the datum-induced variation matrix, A 3 k is the fixture-induced variation matrix, A 4 k is the feature generation matrix, and A 5 k is the selector matrix. Matrices A 2 k and A 3 k are currently derived for 3-2-1 punctual schemes [4], the N-2-1 extension [7], general punctual fixture configurations [8], and surface-based fixtures [22].
In Sections III and IV we present the mathematical derivation of the corresponding matrices A 2 k and A 3 k when the workholding device is a bench vice or a three-jaw chuck. Please note that, in this article, it is assumed that the workpiece acts as a rigid part and errors as deformations during clamping are assumed to be negligible in comparison with fixtureand datum-induced errors. For the sake of simplicity, form errors are assumed to be negligible, but their inclusion can be straightforward when using small jaws or locators by treating form tolerances as an independent fixture error on each locator/jaw, as it is explained in [25]. If surfaces are used for locating, e.g., vice jaws, it can be considered that form errors have little or no effect on the result, as a feature's form tolerance is always smaller than its location/orientation tolerance. Form errors will be only significant if the surfaces in contact present a specific shape, and the high points of both surfaces are aligned and touch each other, which is very remote [26, Ch. 20].

III. BENCH VICES
Bench vices are common devices for holding workpieces on a milling machine table. Among bench vices, plain vices are probably the most widely used fixture device on the shop floor. A plain vice has two jaws, one fixed and one movable, and the workpiece is held by the force exerted from the movable jaw to the fixed one with a pull-down action. Although the jaws are usually plain, special jaws with irregular shapes are sometimes used to hold nonprismatic parts. Fig. 4(a) shows a typical bench vice for milling where the components of the vice (jaws, supports, and pins), and the fixture coordinate system (FCS) are identified; Fig. 4(b) shows the workpiece held in the vice and the datum CS: A-CS, B-CS, and C-CS.
Following the methodology proposed in [4], the relationship between the errors of vice surfaces (fixture-induced errors) and the deviation of the FCS is defined by the matrix A 3 k , and the relationship between the errors of datum surfaces and the position of the FCS (datum-induced errors) is defined by matrix A 2 k . However, it should be noted that the final assembly fixture-workpiece depends on the position and orientation deviation of the fixed jaw, the support surface, and primary and secondary datums, and thus, the superposition of fixture and datum errors as presented in previous studies cannot be straightforward applied. In other words, the values of both matrices A 2 k and A 3 k should be expressed as a function of the interaction between the current position and the orientation of datum and locating surfaces.

A. Fixture-Induced Errors
To understand the FCS deviation due to fixture errors, let us explain the clamping process in a plain vice. First, the workpiece is placed over the support, which makes the support surface and primary datum to be coplanar. Then, the workpiece is moved over the support to first touch the primary datum with the fixed jaw and then to touch the tertiary datum with the pin of the vice. Finally, the movable jaw moves until it clamps the part, exerting a force toward the fixed jaw and the support due to its pull-down action. Due to this clamping force, the primary datum will move to be coplanar with the fixed jaw, which may cause that the secondary datum lifts from the support. Therefore, the fixed jaw blocks three degrees of freedom (DOFs): two rotations and one translation; the support blocks the other two DOF: one rotation and one translation; and the pin blocks the remaining DOF.
The fixturing errors in vice fixtures can be defined as where z 1 , α 1 , and β 1 refer to the Z -axis deviation and orientation deviations around X-and Y -axes of the fixed jaw (1-CS), respectively; z 2 , α 2 , and β 2 refer to similar deviations but from the support (2-CS); and z 3 refers to the deviation of the locating pin of the vice fixture (3-CS). Therefore, the resulting deviation of the FCS due to fixture errors is defined as where matrix A 3 k can be estimated as follows. As stated above, in a vice workholding system, there is a plane-to-plane contact between the fixed jaw and the primary datum. Thus, three DOFs are blocked by the fixed jaw, and the errors z 1 , α 1 , and β 1 are directly propagated to the FCS in these DOF. Therefore where In (7), the HTM H 1 F is equal to 0 H 1 F since there is a plane-to-plane contact, and the movements along the X-axis and rotations around the Y -and Z -axes of FCS are blocked.
The secondary datum blocks other two DOF: the movement of the part along the Z -axis of FCS and the rotation of the part around the X-axis of FCS. Furthermore, since the planeto-plane contact is given at the primary datum, the secondary datum and the support touch each other at least in two points. Denoting the two contact points as p D and p E , wherẽ p = [p T , 1] T , we havẽ and the same expression forp F E holds. From (10), we know = 0 since the contact points define the location of the part in the Z -direction of the FCS, and the orientation deviation along the X-axis is the same as the orientation deviation of the support, which blocks this DOF. Thus, (10) can be solved to relate x F 0 F (3) and x F 0 F (4) with the fixed jaw errors together with the support errors.
Finally, the locating pin blocks the movement of the part along the Y -axis; thus, following the same procedure, we havẽ wherep G is the contact point defined by the locating pin of the fixture andp F G (2) = 0. Equation (11) can be solved to relate x F 0 F (2) with the locating pin errors together with support and fixed jaw errors.
Following the steps shown above, the DMV x F 0 F can be expressed as a function of fixture errors through matrix A 3 k . For the workpiece and vice shown in Fig. 4, the numerical solution of this matrix is where t F 1y and t F 1z refer to the location of the fixed jaw CS with respect to the FCS, t F 2x and t F 2y refer to the location of the support CS with respect to the FCS, and t F 3x and t F 3z refer to the location of the locating pin CS with respect to the FCS. Parameter a depends on the fixture and datum assembly and resulting contact points p D and p E . As shown in Fig. 5, the contact points between part and fixture depend on orientation deviations of support and fixed jaw and orientation deviation of primary and secondary datums. For the example given in Fig. 4, a has the following values: where L s is the length of the contact between support and workpiece, F = 90 • +β 1 −α 2 , and P = 90 Bx . For practical purposes, it is of interest to relate the fixture errors with the technical specifications of the vice. From common technical specifications, we may remark accuracy in clamping repeatability and parallelism and perpendicularity specification of vice surfaces. These geometrical specifications can be translated to DMV limits in the u F k parameters, as shown in [25], and thus, the estimation of part quality variability for a given bench vice can be conducted. Therefore, considering L v and H v as the length and height of the fixed jaw, respectively, W s as the contact width between support and part, and c , pa , and pe as clamping accuracy and parallelism and perpendicularity of vice surfaces, we have |α 2 | ≤ pa /W s , |β 2 | ≤ pa /L s , |z 1 | ≤ c /2, |α 1 | ≤ pe /H v , and |β 1 | ≤ pe /L v ; and their relationships are defined as In addition, we may add the alignment error of the vice in the machine tool as alig and the position error of the vice on the machine-tool table during the setup process (e.g., touch probe inaccuracy) as stp . Therefore, we have |z 2 | ≤ stp and |z 1 | ≤ c /2+ stp , and β 1 , which is previously defined, will add the alignment error alig .
Another common vice configuration is presented when the location of the workpiece in the parallel direction of the jaws is undefined, so the pin locator is removed from the workholding device. Under this configuration, the possible deviation of machined features along this direction is undetermined, and t F 3x and t F 3z from matrices A 2 k and A 3 k are replaced by U , which refers to an undetermined component. To operate with U , the following properties apply:

B. Datum-Induced Errors
Considering the primary datum as the RCS of the workpiece, the deviation of the FCS with respect to RCS is modeled by the DMV x R F , and it can be defined as [4] x where x R 2 and x R 3 are the DMVs that define the deviations of the secondary and tertiary datums of the workpiece that corresponds with the workpiece surfaces that touch the support and the locating pin, and A 2 k = [T 1 T 2 ]. Following the procedure presented in [4], [23], the matrices T 1 and T 2 for the vice and workpiece shown in Fig. 4 are defined as where t B Fx and t B Fy are the X and Y coordinates of FCS with respect to B-CS, respectively, t C Fx and t C Fy are the same but with respect to the C-CS, t F 3x and t F 3z refer to the position of the locating pin of the vice, and the parameter a depends on fixture and workpiece assembly (see Fig. 5) and presents the values shown in (13). Please refer to Appendix I for the derivation details of T 1 and T 2 .

IV. THREE-JAW SELF-CENTERING CHUCKS
A three-jaw self-centering chuck is a workholding device used in turning and milling processes to hold regular-shaped parts, such as cylinders. This type of chuck consists of a cylindrical base with three slots carved from the center to the exterior, separated 120 • from each other. There is a jaw in each slot, and all three jaws slide simultaneously by the same amount if one of the three pinions is rotated. A three-jaw chuck can present different configurations depending on the main locating surfaces used. Fig. 6 shows three configurations analyzed in this article. In the first configuration, the main locating surface is the outer diameter of the workpiece, and thus, the jaws block four DOFs, whereas a pin locator blocks the Z movement of the part. The second configuration is similar to the previous one, but no locating pin is used, so the position of the workpiece in the Z -direction is undetermined. The third configuration uses the end flat surface of the workpiece as the main locating surface that blocks three DOFs (the Z movement and two rotations) due to the contact with the jaws. In this configuration, the clamping process locates the part in X-and Y -directions. In all cases, rotation around the Z -axis is limited by the friction of the workpiece and the jaws.

A. Fixture-Induced Errors
Some studies have studied the errors of three-jaw chucks and the methods to improve chuck accuracy [24], [27], [28]. The main identified errors in three-jaw chucks are the radial displacement error of individual jaws due to internal wear or backlash; taper in jaw alignment; nonsymmetric deformation of jaw-workpiece; and kinematic redundancy. In this article, we consider that the results of those fixture errors are reflected in the deviation of the jaws from their nominal position and orientation. Therefore, we consider as fixture errors the position deviation of each jaw in the radial direction, the position error of the locating pin or the jaw to place the end face of the workpiece, and the orientation error due to jaw alignment.
For any of the chuck configurations defined above, the position of the workpiece in the X-and Y -axes is defined by the position deviation of each jaw in the radial direction. Considering that the jaws are placed perpendicular to the slots and the chuck base, the top-down view of a chuck holding a perfect cylinder (XY view) can be defined, as shown in Fig. 7. Points P, Q, and R (the contact point of the jaws) are separated a distance G from the center plus a jaw error δ, expressed outward the center. Thus, the deviation of the jaws from nominal positions is denoted as δ P , δ Q , and δ R . The deviation of the center of the workpiece clamped with respect to the center of the chuck is estimated to be 2/3 of the jaw deviation along the direction of jaw deviation, as shown in Appendix II.
Furthermore, the workpiece may be deviated from the Z -axis if the chuck constrains this direction as in configurations 1 and 3. In configuration 1, the main locating surface is the cylindrical surface of the workpiece, and an orientation deviation of the jaws will produce a Z deviation of the part when the end face and the locating pin of the chuck touch each other. In configuration 3, the main locating surface is the end face of the workpiece, and the Z deviation will depend on the position and orientation deviation of the plane defined by the three jaws that contact the end face. In any case, we represent the Z deviation as δ z and the orientation deviation of the chuck as δ α and δ β , which reproduces the same inclination of the FCS from its nominal position.
Therefore, the fixture errors in a three-jaw chuck can be defined as and the corresponding matrix A 3 k can be defined as Please note that, if configuration 2 applies, there is no control about the Z position of the workpiece, and thus, z L is replaced by U , an undetermined component. From previous equations, the angle has been included to take into account that the position of the jaws may be rotated from the FCS on the machine-tool table, so jaw P maybe not in the +Ydirection. Furthermore, note that, if δ P = δ Q = δ R , then the center of the part is the same as the center of the chuck, and thus, x As it was presented in the vice, it is of interest to obtain the relation between the technical specifications about the accuracy of the three-jaw chuck and the identified fixture errors. Common technical specifications in chucks refer to maximum total indicator runout (TIR) values in the radial and axial directions, as shown in Fig. 8. As it has been shown, the deviation of the jaws will define the centering error, which, in turn, produces a constant radial run-out defect when rotating a cylindrical part. For the configuration 3 [see Fig. 8(a)], the radial TIR alongside the jaws, denoted as TIR r , can be defined as two times the centering offset, and thus, this accuracy term of the chuck can be represented as the deviation of the jaws, δ P , δ Q , and δ R , in a range of [0, (3/4) · TIR r ]. On the other hand, the axial TIR is related to the orientation deviation of the chuck defined by δ α and δ β and the diameter of the tested part. Denoting TIR a as the axial TIR and D t the diameter of the part tested for the axial TIR, we have |δ α | ≤ TIR a /D t and |δ β | ≤ TIR a /D t , and the following relationship holds: For the configuration 1 [see Fig. 8(b)], the radial TIR is measured at the length L t of the tested part. Similar to the configuration 3, the deviation of jaws are limited to a range of [0, (3/4) · TIR r ], but now, due to the effect of orientation deviations δ α and δ β at the L t position of the dial indicator, we have |δ α | ≤ TIR r /(2L t ) and |δ β | ≤ TIR r /(2L t ), and in addition, the following relationship holds: (24) where c * = cos(30 • ).

B. Datum-Induced Errors
Datum-induced errors in three-jaw chuck mainly depend on the chuck configuration. A chuck with configuration 2 presents only a primary datum, the cylindrical feature, and there is no secondary datum since there is no constraint over the Z position of the workpiece. Therefore, no datum-induced errors apply. Similarly, a chuck with configuration 3 presents the end face of the workpiece as the primary datum, and the cylindrical feature is defined as the secondary datum. Since the chuck is a self-centering chuck and form errors are not considered, the center of the workpiece would be only defined by the jaw errors even though the cylindrical feature would present an orientation deviation, so no datum-induced errors apply. However, when the chuck presents configuration 1, where a locating pin blocks the Z movement of the part, the primary datum is the cylindrical feature, and a datum-induced error may arise, as shown in Fig. 9. As it can be seen, only the position along the nominal Z -axis is modified due to datum errors since the cylindrical surface is oriented according to the three-jaw orientation, which is considered perfect when only datum-induced errors are analyzed. Then, following the methodology presented in [4], the deviation of the nominal FCS with respect to RCS, x R F , can be obtained as where x R 2 is the deviation of the secondary datum with respect to the part reference CS. The solving steps are detailed in Appendix III. Once solved, the deviation of the FCS with respect to RCS in the Z -axis direction is defined as In the matrix form, the final matrix A 2 k from Zhou's methodology can be expressed as Note that p F L is the position of the locating point with respect to FCS, and it depends on the distance from the center of the chuck, r loc , and the angle with respect to the X-axis of FCS, φ loc , as shown in Fig. 9. Then, the X and Y coordinates with respect to FCS are p F Ly = r loc · sin(φ loc ).
(29) Fig. 10. Three-stage machining process evaluated. Fig. 11. Part specifications to be inspected. Other dimensions are omitted for simplicity.

V. CASE STUDY
In order to validate the extension of the SoV model, a three-stage machining process is analyzed where both bench vices and three-jaw chucks are used. As shown in Fig. 10, the manufacturing process consists of a face milling and endmilling operation at the first stage holding the part on a bench vice, the second stage where similar machining operations are conducted but using as fixturing device a three-jaw chuck with a centered locator, and the third stage where the part is held on the same three-jaw chuck in order to conduct a circular pocketing operation. As shown in Fig. 11, the inspected part specifications are the distance between both squareshoulder features, the position of the cylinder with respect to datums A, B, and C, and the coaxiality of the circular pocketing with respect to its datum D. The position and orientation vectors for the main features are shown in Table I. The raw material is an aluminum block with dimensions 100 × 100 × 100 mm whose surfaces have been premachined, so flatness and square errors between surfaces can be assumed negligible.
The validation is conducted in two ways: 1) by using the SoV model to predict the deviations of the three geometrical specifications of the part and comparing these results with the resulting deviations obtained using a CAD software and 2) by machining the part and comparing the results with those expected by the SoV and CAD model. The CAD software used is SolidWorks, and we basically model the fixtureworkpiece assembly with the surface errors introduced in the tested cases to check the final part deviation. This is a tedious and time-consuming procedure that can be used for checking the effect of few errors at the same time assuming the rest negligible, and it cannot be used for checking the manufacturing process capability. For the machining experimentation, the machining center used for the experimentation is a Deckel Maho DMC 70V machining center, and the features are inspected in a Brown & Sharpe Mistral 775 coordinate measuring machine. The vice used is a Fresmak Arnold Twin with 0.02 mm of parallelism and perpendicularity and 0.01-mm clamping accuracy, and the three-jaw chuck is an Optimum K11-125 chuck model mounted according to configuration 1 with an inspected TIR r of 0.11 mm for a length of L t = 50 mm. The setup process of the vice and threejaw chuck in the machine-tool table is conducted with a Renishaw touch probe, and the alignment and positioning errors are assumed to be alig = 0.020 mm/100 mm and stp = ±0.015 mm. The first part is machined to calibrate the process (e.g., tool dimensions and offsets due to clamping deformation). Fig. 12 illustrates the experimental setup.
Three different situations are tested: 1) no errors; 2) relative small fixture errors; and 3) severe fixture errors. The fixture errors that were intentionally added, i.e., cases 2) and 3), are shown in Table II. These errors were physically introduced by adding a feeler gauge between the workholding device and the workpiece or modifying the zero part coordinate system in the CNC matching tool to get the same effect. Furthermore, the SoV model is applied in two ways. The first one considers the errors added into the process and assumes negligible other errors. Conversely, the second one, named "SoV + Monte Carlo," considers the technical specifications related to the accuracy of the workholding systems and simulates additional errors according to these specifications in order to calculate a range of values for each inspected specification. For this purpose, 5000 Monte Carlo simulations were run, and the range that comprises 99.7% of the values was recorded.
The results are shown in Table III. First, case 1) shows the estimation of the range of values for the analyzed part specifications considering the accuracy technical specifications of the vice and the three-jaw chuck used. These ranges are, in fact, the manufacturing process capability according to the workholding specifications and assuming no machininginduced errors exist. For this case study, the high TIR of the three-jaw chuck used is reflected in the high expected coaxiality error, which is indeed confirmed in the experimentation. However, the specification related to the position error of the cylinder can be kept tight despite the bench vice inaccuracies, ensuring a position error less than 0.029 mm under Monte Carlo simulations and experimentally validated with a measured error of 0.035 mm. Second, for both small and severe errors added into the process, the proposed model shows a maximum error of 1% in comparison with the CAD results. The first specification (dimensional deviation between square-shoulder features) gives the same results between CAD and SoV model because, given the errors in Table II, only the deviation in the Z -direction of the three-jaw chuck at stage 2 has an impact, and then, there is no error due to linearizations. However, the position specification shows the effect of linearization errors in the vice due to orientation deviations when comparing with respect to CAD results. According to the results, this error is around 1%.
Finally, the results from the SoV model considering the errors added and the accuracy specifications of the workholding systems are compared with the results obtained after machining and inspecting the parts. As shown in Table III, the inspected specifications are in good agreement with the estimated range of values. The position specifications present slightly higher values than the ones estimated through Monte Carlo, which may be explained by machining-induced errors or deformation variations in the bench vice during clamping. This article has shown how to extend the current SoV model in order to include general-purpose workholding devices, such as bench vices and three-jaw chucks, not considered yet in the literature. In bench vices, the errors included in the model are position and orientation errors of plain jaws, supports, and pins. In three-jaw chucks, the errors included are the position error of jaws in the chuck, the position error of the locating pin to block the Z -direction of the workpiece, and the orientation errors of the jaws. In all cases, the model assumes that the workpiece acts as a rigid part and errors as deformations during clamping are assumed to be negligible in comparison with fixture-and datum-induced errors. The model has been validated on a three-stage machining process through both CAD simulations and machining experimentation. The model performance with respect to CAD simulations showed an error of less than 1% due to linearization, and the machining results validated the capability of the model to estimate 99.7% confidence intervals for different product specifications considering the accuracy of the workholding systems. Unlike previous extensions of the model, the proposed extension lets practitioners apply zero-defect strategies in multistage machining processes where bench vices or chucks are used, and it can also be used for estimating manufacturing process capability under specific workholding devices.

APPENDIX I CALCULUS OF DATUM-INDUCED ERRORS IN VICES
Following the procedure explained in [4], we define the datum points that touch the secondary datum, denoted as p D and p E , which depend on the relationship between the orientation errors of fixture surfaces 1 and 2 and datum surfaces A and B, and the datum point that touches the locating pin of the vice, p G . The nominal coordinates of these three points in FCS are denoted as p F D , p F E , and p F G . Thus, we have Since the contact points between datums and fixture surfaces have a coordinate of 0 in the Z -axis with respect to the each datum coordinate system (note that all datum CS have a Z -axis pointing out to the surface), the coordinate Z of points p B D , p B E , and p C G is equal to 0. Following the steps in [4], previous equations can be simplified to the following expressions in (33) Since the RCS is the primary datum (A-CS), x R F has only three nonzero values in (33). For the fixture and part geometry given in Fig. 4, the resolution of the previous equation is where p F G = [t F 3x , 0, t F 3z ] and the value of p F Ex is the parameter a, which can be 0 or L s depending on the fixture and part assembly.

APPENDIX II CALCULUS OF SELF-CENTERING ERROR DUE TO SINGLE-JAW DEVIATIONS
Let us consider that the guiding slots of a three-jaw selfcentering chuck can be defined in a 2-D plane as three lines that start from the coordinate origin, separated 120 • to each other. Each jaw can be defined as a point located in each line, as it can be seen in Fig. 13. In this nominal case, jaw P is located in the vertical line, with jaws Q and R named in the clockwise order. Assuming a perfectly working chuck, all jaws are separated a distance G from the origin. Therefore, given a certain G, the position of each jaw in this ideal chuck with respect to the nominal FCS is Fig. 13. Self-centering error y C due to single-jaw deviation δ. Dimensions of δ have been exaggerated to improve comprehension.
A cylinder held by the chuck is represented in the 2-D plane as a circumference tangent to points P, Q, and R. If the jaws are perfectly self-centered, the center of the circumference will be located in the coordinate origin. However, these jaws may present a displacement from its self-centering position. For the sake of simplicity, the displacement, named δ, is applied to the single-jaw P. By clamping the part under this jaw displacement, the following equations hold: Given that the radius of the cylinder is r and that the circumference must be tangent to all three points, the distance from the center of the circumference to the coordinate origin expressed here as (x C , y C ) can be calculated using the following equations: Operating with previous equations, we obtain Solving these equations, we obtain that the deviation of the center x C and y C as Finally, assuming that the displacement δ will be some orders of magnitude smaller than the value of the distance of the jaws to the center G, the previous equation can be approximated to

APPENDIX III CALCULUS OF DATUM-INDUCED ERRORS IN THREE-JAW SELF-CENTERING CHUCKS
In order to calculate matrix A 2 , let p L be the locating pin that touches the secondary datum defined by B-CS in a three-jaw chuck with configuration 1, as shown in Fig. 9. The nominal coordinates are defined as In addition, the following expressions hold: Therefore, substituting (51) and (52) into (50) Neglecting the second-order terms Considering that the three jaws and the locating points are perfect (no fixture errors), the locating point touches the secondary datum, and thus, the Z coordinate ofp B L is zero. .
We can rewrite the left hand of (56) as whereas the right-hand side of (56) is rewritten as (58) Solving (56) and reorganizing the terms, the deviation of the FCS with respect to RCS in the Z -axis direction is defined as

APPENDIX IV IMPLEMENTATION GUIDE TO PRACTITIONERS
In order to facilitate the industrial application of the SoV model based on workholding systems, such as vices and threejaw chucks, we present the following step-by-step implementation guide. Please note that the final purpose of this guide is to estimate part quality according to workholding specifications.
1) Identify the main key characteristics of the workholding systems to be used. For vices, identify parameters such as length and height of jaws (L v , H v ) and contact width between support and workpiece (W s ). In addition, the position of primary, secondary, and tertiary locating features (1-CS, 2-CS, and 3-CS) with respect to FCS should be given. For three-jaw chucks, identify parameters such as jaws position with respect to machine-tool +Y -direction () and locator positions if exist (r loc and θ loc ). 2) Build matrices A 2 k and A 3 k according to (12), (18), and (19) and (21), (22), and (27) for vices and three-jaw chucks, respectively. In vices, these matrices will depend on the relationship between datum and fixture errors (i.e., matrices P and F ).
3) Build the SoV model applying the methodology presented in Zhou et al. [4]. The SoV model requires the matrices previously derived A 2 k and A 3 k . Include these matrices to obtain the SoV model in the form of (1). 4) Identify the technical specifications of the workholding systems provided by vendors. For vices, identify clamping accuracy, parallelism, and perpendicularity of jaws; for three-jaw chucks, identify maximum TIR in radial and axial direction and the dimensions of the part tested (diameter and length D t and L t ) in the calibration sheet. 5) Run M Monte Carlo simulations constrained to previous technical specifications to generate M possible sets of fixture errors for each stage (u F k ). Equations (14), (15), (23), and (24) show some constrains for vices and threejaw chucks according to their technical specifications, so the generated data should be within them. 6) Apply the SoV model using the simulated fixture errors u F k to estimate the deviation of the inspected features from nominal values for the M simulations. An analysis of the deviations of the inspected features for the M simulations will show the capability of the process and the expected quality of the part. Rubén Moliner-Heredia was born in Vila-real,