2.1. Structuring and presenting tolerancing

Different process models exist in industrial practice (Stockinger Reference Stockinger2010; Schleich Reference Schleich2017) for organizing the diverse tolerancing activities. For example, in the automotive industry, tolerancing processes are described with seven to nine predefined steps (Bohn & Hetsch Reference Bohn and Hetsch2013; Mölzer & Strobelt Reference Mölzer and Strobelt2013), which are transferred into a standardized process through the German Association of the Automotive Industry (VDA) (Verband der Automobilindustrie 2013). However, the focus is on a high organizational level and does not address specific aspects of implementing tolerance analysis, for example.

In contrast, several research projects have been carried out to develop a structured tolerancing process on a more specific level. Some focus on embedding tolerancing activities in early stages of the product development process (Roy et al. Reference Roy, Liu and Woo1991; Islam Reference Islam2009), e.g., by a structured translation of functional requirements into geometrical definitions in the concept design (Roy et al. Reference Roy, Pramanik, Sudarsan, Sriram and Lyons2001; Dantan, Anwer & Mathieu Reference Dantan, Anwer and Mathieu2003), an approach enabling the earliest possible definition of geometry (Ballu et al. Reference Ballu, Falgarone, Chevassus and Mathieu2006; Costadoat et al. Reference Costadoat, Mathieu, Falgarone and Fricero2009; Costadoat, Mathieu & Falgarone Reference Costadoat, Mathieu, Falgarone and Bernard2011) or a combination of tolerancing and Robust Design activities (Hu & Peng Reference Hu and Peng2007; Goetz, Schleich & Wartzack Reference Goetz, Schleich and Wartzack2020). In addition to intensive frontloading, the aim is to achieve parallelization of design and tolerancing as well as consistency (Desrochers & Laperrière Reference Desrochers, Laperrière, Bourdet and Mathieu2003; Dufaure & Tessandier Reference Dufaure and Tessandier2008). Other approaches concentrate on the structure and detailed description of tolerancing activities as a process, highlighting the interdisciplinarity and variety of tolerancing approaches, e.g., by providing a SysML-based representation of a variation management process (Horber et al. Reference Horber, Goetz, Schleich and Wartzack2022), a computer-aided toolchain enabling the geometry assurance from concept phase to verification and production (Söderberg, Lindkvist & Carlson Reference Söderberg, Lindkvist and Carlson2006; Söderberg et al. Reference Söderberg, Lindkvist, Wärmefjord and Carlson2016), a framework to consider perceived quality in tolerancing (Stylidis, Wickman & Söderberg Reference Stylidis, Wickman and Söderberg2020), a tolerance analysis and synthesis approach integrating non-geometrical functional relations (Germer & Franke Reference Germer and Franke2003) and a description of computer-aided tolerancing from tolerance specification phase to product usage (Wartzack et al. Reference Wartzack, Meerkamm, Stockinger, Stoll, Stuppy, Voß, Walter and Wittmann2011). The partly addressed iterative character is explicitly described in the closed-loop tolerance engineering model based on four basic recurring steps (Krogstie & Martinsen Reference Krogstie and Martinsen2012; Schleich & Wartzack Reference Schleich and Wartzack2014). These iterative and cross-domain, cross-phase, and cross-activity processes lead to different levels of detail of tolerance information. To improve consistency and traceability, various unifying information models for the combination of these processes exist (Bradley & Maropoulosa Reference Bradley and Maropoulosa1998; Feng & Song Reference Feng and Song2000; Johannesson & Söderberg Reference Johannesson and Söderberg2000; Sudarsan et al. Reference Sudarsan, Roy, Narahari, Sriram, Lyons and Pramanik2000; Roy et al. Reference Roy, Pramanik, Sudarsan, Sriram and Lyons2001; Desrochers & Laperrière Reference Desrochers, Laperrière, Bourdet and Mathieu2003; Ballu, Dufaure & Teissandier Reference Ballu, Dufaure, Teissandier, Cunha and Maropoulos2007; Mathieu & Ballu Reference Mathieu, Ballu and Davidson2007; Shen, Shah & Davidson Reference Shen, Shah and Davidson2008; Dantan, Ballu & Mathieu Reference Dantan, Ballu and Mathieu2008a; Zhang, Jiang & Ning Reference Zhang, Jiang and Ning2009; Rhahli et al. Reference Rhahli, Bosch-Mauchand, Anselmetti and Eynard2010). Linking to this, the ISO GPS system can be used for consistent communication of tolerances between the various actors along the product life cycle, explicitly addressing the link between specification and verification of tolerances (ISO/TC 213 2011, 2014).

While these approaches support engineers in the proper definition and communication of tolerance information or tolerancing procedures, they do not support the consistent description of tolerancing, especially when considering the various fields of application, stages and stakeholders along the product life cycle (Maltauro, Meneghello & Concheri Reference Maltauro, Meneghello and Concheri2024b), see also Figure 1. This also applies to the few approaches that assign tolerancing activities to individual product development process steps, for example, in the VDI 2221 (Leidich, Ebermann & Gröger Reference Leidich, Ebermann, Gröger and Stelzer2014; Goetz Reference Goetz2022), with the aim of structuring tolerancing in product design. This is due to the fact that they focus only on certain aspects of tolerancing, such as early tolerance management (Goetz Reference Goetz2022). The heterogeneity of these individual approaches already emphasizes that there can be no generally valid and yet concrete tolerancing process, thus hindering the exchange in research and industry (Sigurdarson et al. Reference Sigurdarson, Eifler and Ebro2018). This is further amplified by different views on tolerancing between distinct companies, departments, or researchers (Krogstie, Martinsen & Andersen Reference Krogstie, Martinsen and Andersen2014) and the interdisciplinary nature of tolerancing since similar methods, e.g., for tolerance analysis, are used in early Robust Design (Goetz, Schleich & Wartzack Reference Goetz, Schleich and Wartzack2018a), final tolerance optimization (Hallmann, Schleich & Wartzack Reference Hallmann, Schleich and Wartzack2021b), and tolerancing-driven process design (Heling et al. Reference Heling, Oberleiter, Rohrmoser, Kiener, Schleich, Hagenah, Merklein, Willner and Wartzack2019).

Figure 1. Tolerance-related activities along the product life cycle. Inspired by Taguchi et al. (Reference Taguchi, Chowdhury, Wu, Taguchi and Yano2005), Schleich (Reference Schleich2017) and Roth (Reference Roth2024)).

Similar problems are faced in the Robust Design domain. For this reason, a matrix-based classification framework differentiating between “System Design,” “Parameter Design,” and “Tolerance Design” on the one hand and between “Principles and Practices,” “Methods and Tools,” “Application Cases,” and “Implementation” on the other hand is proposed (Eifler & Schleich Reference Eifler and Schleich2021). Although this allows the mapping of Robust Design research and thus the identification of remaining research potentials, the classification process itself remains challenging since, for example, a clear distinction between contributions proposing a new method or its application is difficult (Eifler & Schleich Reference Eifler and Schleich2021). This indicates that a simple classification with two dimensions inevitably leads to unambiguity, as most contributions address multiple aspects.

2.2. Fundamental tolerancing activities

Considering the various referenced works, Figure 1 provides an overview of the tolerance-related activities along several product design and realization stages. Product realization is to be understood as the product life cycle phases that convert the ideas from design into the real physical product, so it includes pre-production and production steps.

Following the definition in Schleich (Reference Schleich2017) and focusing on the product design perspective, tolerancing can be traced back to the activities of tolerancing requirements definition, tolerance specification, tolerance allocation, tolerance analysis, and tolerance synthesis. These partly overlap with the activities of the late Robust Design, namely the tolerance design according to Taguchi et al. (Reference Taguchi, Chowdhury, Wu, Taguchi and Yano2005). However, since most Robust Design activities, especially in the early system and parameter design, are assigned to the product design process itself and only have overlaps with tolerance activities, only the five key tolerancing activities, described in the following, are used as the basis for the proposed description scheme.

• • Tolerancing requirements definition: The globally defined product requirements (from user demand, etc.) are broken down into a set of quantitative tolerancing requirements that are directly related to geometry assurance and tolerancing (Schleich Reference Schleich2017). Hence, this step is essential at the beginning of tolerancing to define the targets to be achieved by all the subsequent tolerancing activities.

• • Tolerance specification: A clear and unambiguous specification of tolerances in compliance with international GPS (Geometrical Product Specification) standards is used to describe how the physical proportions, i.e., features, of the individual part geometries are allowed to deviate from their ideal (Kumar et al. Reference Kumar, Soundararajan and Alagumurthi2009; Armillotta Reference Armillotta2013; American Society of Mechanical Engineers 2018). It includes choosing dimensional and geometrical tolerance types for all functionally relevant features and datum reference frames (Armillotta Reference Armillotta2013).

• • Tolerance allocation: In this activity, one specific value for each specified tolerance callout is allocated to define the amount of acceptable variation, corresponding to the size of the tolerance intervals and zones (Armillotta Reference Armillotta2013). This step is mainly based on experience, standards, textbooks, heuristics, rules of thumb, or more sophisticated approaches (Singh et al. Reference Singh, Jain and Jain2009b) and is directly linked to the necessary effort in product realization.

• • Tolerance analysis: The aim is to check if the previously defined part tolerances can assure product quality (Shen et al. Reference Shen, Ameta, Shah and Davidson2005). It uses variation simulation and sensitivity analysis techniques to study the product under variations and assess the probabilistic response with the aid of quality metrics (Morse et al. Reference Morse, Dantan, Anwer, Söderberg, Moroni, Qureshi, Jiang and Mathieu2018).

• • Tolerance synthesis: The boundaries between tolerance allocation and synthesis are blurred as their definitions are often similar in literature (Roth Reference Roth2024). Nonetheless, it is reasonable to see tolerance synthesis as an additional activity following tolerance analysis aiming to refine either the pre-defined allocation or specification scheme to meet the tolerancing requirements (Schleich Reference Schleich2017), in the best case in an optimal way in contrast to the initial allocation, see Figure 1.

3.1. Product design-driven tolerancing

To ensure the applicability of the description scheme, an illustrative visualization principle supporting communication and collaboration (Gebhardt, Beckmann, & Krause Reference Gebhardt, Beckmann and Krause2014; Gebhardt & Krause Reference Gebhardt and Krause2015; Kernbach & Svetina Nabergoj Reference Kernbach and Svetina Nabergoj2018) is utilized, following examples from product design with two-dimensional figurative flowcharts, e.g., the scenario funnel (Gausemeier, Fink & Schlake Reference Gausemeier, Fink and Schlake1998) or the V-model for systems development (Clark Reference Clark2009), and three-dimensional geometrical primitives, e.g., cubes for systems engineering (Hick, Bajzek & Faustmann Reference Hick, Bajzek and Faustmann2019) or safety-oriented product design (Rajabalinejad Reference Rajabalinejad2018).

Considering the five common elements as different views, visualizing the scheme as a pyramid is an obvious choice. The lateral faces represent the activities, methods, tools, and models, while the fundamental connecting all four elements is data, information, and knowledge, representing the base of the pyramid, see Figure 3.

Figure 3. General structure of the proposed description scheme, visualized as a pyramid with its five mutual elements and interrelations.

The single elements of tolerancing or pyramid’s faces intend to serve as general containers to collect individual entities to describe a given approach. For the main five activities, it makes sense to, e.g., arrange them according to their usual sequence, starting with basic activities essential for a complete tolerance scheme, including the definition of global tolerancing requirements, specification of part tolerances, and allocation of values, and continuing with optional activities of tolerance analysis and synthesis for making individual adjustments to tolerance specifications and their values, see Figure 4 (left).

Figure 4. Allocation of the five fundamental tolerancing activities (left) and exemplary tolerancing methods (right) in the proposed pyramid description scheme.

In general, there is no absolute position where to place the entities on the respective triangular-shaped faces (near the bottom or the top). This basically differs, e.g., from the popular visualization of ISO GPS standards in a pyramid to illustrate their hierarchy (ISO/TC 213 2011; Charpentier Reference Charpentier2014). Nevertheless, it can be helpful to place fundamental entities near the base of the pyramid and the more specific ones near the top – even though the placement is just qualitative and up to the user’s preferences. In contrast to the five activities, the entities of the other faces are not predefined. The diversity of methods, models, and tools and the non-standardized use of names severely complicate the definition of a complete list with fixed terms and definitions, even though it would foster clarity in the description of the approaches. Nonetheless, referring to exemplary entities in the following is sufficient for explaining the logical structure of the novel description scheme.

Figure 4 (right) shows the allocation of exemplary tolerancing methods, which range from basic problem structuring, e.g., using the Variation Management Framework (Howard et al. Reference Howard, Eifler, Pedersen, Göhler, Boorla and Christensen2017), to specific problem-solving, e.g., tolerance-cost-optimization (Hallmann et al. Reference Hallmann, Schleich and Wartzack2020a). Focusing on the tools facet, assistance by computer and most likely specific computer-aided tolerancing software is common as it simplifies the application of tolerancing methods, alone or in combination, see Figure 5 (left). Examples include interactive tools, such as GD&T advisors ensuring tolerance specification consistency (Gaunet Reference Gaunet, Bourdet and Mathieu2003; Anselmetti et al. Reference Anselmetti, Chavanne, Yang and Anwer2010), commercial variation simulation software supporting tolerance analysis and automating the often challenging identification of tolerance chains (Glancy, Stoddard & Marvin Reference Glancy, Stoddard, Marvin and Drake1999), as well as tools with a high level of automation, for example, to directly propose a tolerance specification based on machine learning (ML) (Cui et al. Reference Cui, Sun, Cao, Zhao, Zeng and Guo2021). Tolerancing models range from tolerance representation and analysis models to knowledge models, with differentiation between mathematical or explicit, for instance, given in vectorial tolerance models (Cao et al. Reference Cao, Liu and Yang2018), and numerical or implicit definitions, such as when using quantifiers (Dantan et al. Reference Dantan, Mathieu, Ballu and Martin2005) or Finite Element models (Morse et al. Reference Morse, Dantan, Anwer, Söderberg, Moroni, Qureshi, Jiang and Mathieu2018), see Figure 5 (right).

Figure 5. Allocation of exemplary tolerancing tools (left) and exemplary tolerancing models (right) in the proposed pyramid description scheme.

Regarding the bottom face of the pyramid, it is generally difficult to make an absolute distinction between data, information, and knowledge since it is relative, depending on how data is used (Ahmed et al. Reference Ahmed, Blessing and Wallace1999). Despite critical opinions on its merit (Hubka & Eder Reference Hubka and Eder1996; Gericke et al. Reference Gericke, Eckert and Stacey2017), a differentiation can help to provide a first association with the type of the respective entity needed to feed the models, tools, and methods. Hence, the pyramid’s base comprises raw data, such as measurement point cloud data of physical part entities (Wärmefjord et al. Reference Wärmefjord, Söderberg, Lindkvist, Lindau and Carlson2017), information, such as resulting part tolerance probability distributions for variation simulation (Hallmann et al. Reference Hallmann, Schleich and Wartzack2021b), and contextualized, respectively formalized knowledge, such as tolerancing rules for tolerance specification (Goetz & Schleich Reference Goetz and Schleich2020), see Figure 6.

Figure 6. Allocation of exemplary tolerancing data, information, and knowledge in the proposed pyramid description.

Reviewing tolerancing approaches presented in literature shows that most approaches do not use just one single method, model, or tool. It is more of a set of entities to reach the specific goal of the respective activity. For example, sampling-based tolerance analysis, as it is called in Hallmann et al. (Reference Hallmann, Schleich and Wartzack2021b), includes the use of a sampling method, such as Monte Carlo (MCS) or Latin Hypercube Sampling (LHS) (Hallmann et al. Reference Hallmann, Schleich and Wartzack2021b). From a more global viewpoint, this results in a large number of possible combinations or combination spaces of various methods, tools, and models, as well as the underlying data, information, and knowledge for the fundamental tolerancing activities – although the reason for the choice of a specific combination is often not further discussed. The interrelations and constraints, for example, given when using sampling and sensitivity analysis methods in combination (Saltelli Reference Saltelli2008), are often only known by a few tolerancing experts. Formalizing this knowledge is the key to giving users clear guidance in finding an appropriate set of entities from all five faces. Starting from any pyramid face, it is then possible to systematically narrow down the solution space and find a suitable combination for a given context of use, see Figure 3 (right).

3.2. Linking product design-driven tolerancing to tolerancing in product realization

In product realization, the focus shifts from the virtual to the physical realm (Schleich Reference Schleich2017). For this reason, there is also a shift in responsibility. Manufacturing- and inspection-driven tolerancing activities become relevant to assure and verify the design-intended part and assembly quality. From the manufacturing perspective, they comprise activities to design the single- and multistage part and assembly manufacturing processes as well as their planning and organization. Inspection-driven tolerancing addresses all activities necessary to provide insights on deviations and variations (The terms “deviation” and “variation” are often used as synonyms. To be more precise, “[t]he term geometrical deviation refers to the deviation from the nominal shape for one part or the mean-value deviation from the nominal shape for a set of parts. […] Geometrical variation is the variation of the deviation for different parts produced in the same process.” (Wärmefjord, Hansen, & Söderberg Reference Wärmefjord, Hansen and Söderberg2023)), and achievable tolerances on both part and assembly levels. Despite the change of perspective, manufacturing- and inspection-driven tolerancing have strong overlaps in the entities of the five main elements of product-driven tolerancing. Figure 7 highlights examples of entities to illustrate the link between the different activities. Sampling-based variation simulation and contributor analysis, for instance, are general methods used for design, manufacturing, and inspection purposes (Söderberg et al. Reference Söderberg, Lindkvist, Wärmefjord and Carlson2016; Morse et al. Reference Morse, Dantan, Anwer, Söderberg, Moroni, Qureshi, Jiang and Mathieu2018). Thus, the same CAT tools, aiming for broad applicability, such as RD&T, can be applied for different activities (Söderberg et al. Reference Söderberg, Lindkvist, Wärmefjord and Carlson2016). Besides, Skin Model Shapes (SMS) are one example of a general concept for the representation of variations that can be used along the product life cycle (Schleich et al. Reference Schleich, Anwer, Mathieu and Wartzack2016; Hofmann, Gröger & Anwer Reference Hofmann, Gröger and Anwer2022), whereas depending on the approach, a mapping between different representations may further be required, e.g., to enhance applicability by linking with domain-standard representations (Hallmann, Goetz, & Schleich Reference Hallmann, Goetz and Schleich2019). Tolerancing data, information, and knowledge play an essential role since they directly connect all the individual activities (Schleich & Anwer Reference Schleich and Anwer2021) defining their main in- and outputs. This applies not only to the use of product and GD&T information from the product design phase in all downstream activities but also to upstream feedback of, for instance, measurement data (Madrid et al. Reference Madrid, Vallhagen, Söderberg and Wärmefjord2016), measurement uncertainty information (Kaufmann, Effenberger & Huber Reference Kaufmann, Effenberger and Huber2021), process capability information (Robles & Roy Reference Robles and Roy2004), or knowledge about achievable manufacturing tolerances (Sauer et al. Reference Sauer, Heling, Schmutzler and Schleich2019) to design- and manufacturing-driven activities.

Figure 7. The five elements of the description scheme serve as links between product-, manufacturing- and inspection-driven tolerancing, illustrated by examples.

The result is a dense network of interrelations in which the individual entities connect the different tolerancing approaches. While this is supported, e.g., by the ISO GPS standards, the inherent principle of duality between specification and verification (ISO/TC 213 2011) corresponds to the differentiation of tolerance-related activities. Nevertheless, the meaning of interchangeability and interoperability of tolerancing data, information, and knowledge is undisputed, also emphasizing the role of neutral exchange formats to ensure the digital thread (Schleich et al. Reference Schleich, Wärmefjord, Söderberg and Wartzack2018; Schleich & Anwer Reference Schleich and Anwer2021). Interoperability is, however, just as decisive for methods, tools, and models to exploit synergy effects when individual entities are (re-)used across activities over the entire product life cycle.

In addition, there is a trend to shift individual activities from later stages to the design stage. Keeping the general vision of Concurrent Engineering alive (Smith Reference Smith1997), integrated approaches frontload further activities to design-driven tolerancing, for example, machine and process selection or fixture layout design into tolerance synthesis (Rezaei Aderiani et al. Reference Rezaei Aderiani, Hallmann, Wärmefjord, Schleich, Söderberg and Wartzack2021; Hallmann, Schleich & Wartzack Reference Hallmann, Schleich and Wartzack2021a). This makes particular sense for products where additional joining processes, such as welding or riveting, are used for assembly since they influence the final product design and quality far more than the individual part feature accuracies (Ding et al. Reference Ding, Jin, Ceglarek and Shi2005). Shifting of activities, however, has its limits since it can lead to conflicts, for example, when tolerance experts from product design largely define the production process and thus limit the scope of action for detailing the manufacturing process. Hence, the links through the elements and their entities are instrumental in bringing the disciplines closer together.

3.3. Linking tolerancing to further domains

Although tolerancing is a vast discipline of its own, it is crucial to keep in mind that it is just a small part of engineering, where “knowledge of the mathematical or physical sciences […] is applied” (Dugger Reference Dugger1993) (referring to the original definition of the Accreditation Board for Engineering Technology (ABET)). Hence, tolerancing is mainly based on general mathematics, statistics, and informatics principles, and it applies general methods, tools, and models. Consequently, all tolerancing approaches fall back on general entities and combine them in the context of use for tolerancing, forming tolerancing-specific models, methods, and tools. Metaphorically speaking, the three tolerancing pyramids from Figure 7 are surrounded by one or multiple further layers, depending on the level of abstraction of the observation. An explicit assignment of the entities to specific domains is not necessarily unambiguous. Figure 8 illustrates an outer pyramid as a totality of all elements not specific to tolerancing and shows exemplary interrelations. Tolerancing methods are, for example, often based on traditional methods from statistics, such as advanced sampling methods for tolerance analysis (Bacharoudis, Popov & Ratchev Reference Bacharoudis, Popov, Ratchev and Ratchev2021), analysis of variance (ANOVA) to perform tolerance synthesis (Kusiak & Feng Reference Kusiak and Feng1995), and fuzzy logic for estimation of tolerancing costs in tolerance synthesis (Dupinet, Balazinski & Czogala Reference Dupinet, Balazinski and Czogala1996), or machine learning and deep learning methods developed in computer science, such as for the generation of shapes with systematic and random variations (Qie, Schleich & Anwer Reference Qie, Schleich and Anwer2023). In some cases, this requires a thorough adaptation of these traditional methods or the adaptation and mapping of tolerance-related data and information, such as distribution functions. A large number of tolerancing models are also based on general representations, such as Finite Element models (Franz, Schleich & Wartzack Reference Franz, Schleich and Wartzack2021), ontologies (Qin et al. Reference Qin, Qi, Lu, Liu, Scott and Jiang2018), or voxel models (Moroni, Petrò & Polini Reference Moroni, Petrò and Polini2017). Speaking of tools, it becomes apparent that the presented approaches often use general software, such as spreadsheet software (Gerth Reference Gerth1994) or graph editors (Goetz et al. Reference Goetz, Kirchner, Schleich and Wartzack2021b), as well as engineering tools, such as CAD (Goetz et al. Reference Goetz, Schleich and Wartzack2018a) and Finite Element Analysis (FEA) software for structural analysis (Corrado & Polini Reference Corrado and Polini2018), as aids to achieve the goals in the respective activities. All these examples are just a few of many to illustrate the basic concept that tolerancing may benefit from engineering approaches, e.g., by expanding the area of application or increasing efficiency.

Figure 8. Models, methods, and tools link tolerancing to other engineering domains and disciplines, illustrated by examples.

Since product design-driven tolerancing is a part of product design, there are particularly strong links to design methods, models, and tools used in other activities. Manufacturing-driven tolerancing, in contrast, tends to rely on entities used to design the process, such as manufacturing simulation software (Schleich & Wartzack Reference Schleich and Wartzack2013). Characterizing the tolerancing approaches with the proposed elements and drawing interrelations between the entities on the tolerancing domain level, but also on the product design or higher levels, establishes a direct relationship to similar basic elements, e.g., sampling strategies or ontological models. In this way, similarities between tolerancing approaches, which at first glance appear very different, can be revealed. This helps less-experienced engineers get started and facilitates embedding the approaches in established product development processes. Moreover, it expands the view to identify overlaps with approaches outside the tolerancing domain.

3.4. Expected benefits of the proposed description scheme

The previous sections focused on the general structure of the description scheme, its implementation for product design-driven tolerancing, and its link to manufacturing- and inspection-driven tolerancing, see Section 3.2, as well as to other domains, see Section 3.3. While some expected benefits of the novel description scheme have already been mentioned in the individual passages, this section aims to consolidate the expected benefits that have driven the scheme’s development.

The authors expect the main merit of the novel description scheme to serve as a practical instrument…

1. (1) … to classify and more clearly describe existing tolerancing approaches and procedures, either presented in scientific articles or applied in industry in a holistic way,

2. (2) … to facilitate the communication and collaboration between the different stakeholders involved in the variation management process from different domains with different backgrounds and wordings.

As a consequence, it aims to help…

1. (3) … to support the transfer of research methods into practice,

2. (4) … to facilitate the introduction and implementation of a thorough tolerancing process in practical applications,

3. (5) … to analyze existing tolerancing procedures implemented in the industry aiming to identify current shortcomings and potentials for improvement,

4. (6) … to identify current gaps and trends in literature, providing directions for future research works.

4.1. Mapping of tolerancing research work

Aiming to overcome the problems associated with an unclear classification of the various tolerancing research approaches, see Section 1, the research works from the biennial CIRP Conference on Computer Aided Tolerancing are exemplarily mapped using the proposed description scheme. This conference represents the activities within the international tolerancing research community from different views and levels (Garaizar et al. Reference Garaizar, Anwer, Mathieu and Qiao2014; Morse Reference Morse2020), including the design, manufacturing, and inspection perspectives, see Figure 7.

The proceedings from the years 2020 and 2022 are the basis for this mapping. A total of 78 research papers were published at both conferences and initially classified according to the five fundamental tolerancing activities from a product design perspective, whereby papers solely addressing manufacturing-driven and inspection-driven tolerancing were not assigned to these design-driven tolerancing activities. While a complete presentation of all papers and their classification can be found in the table in the provided supplementary material, Figure 9 summarizes the classification according to the tolerancing activities. Thereby, multiple assignments are considered when one contribution addresses multiple activities, e.g., by providing an approach for both tolerance analysis and tolerance specification (Goetz, Lechner & Schleich Reference Goetz, Lechner and Schleich2022) or by unifying the design-driven tolerance synthesis with manufacturing- and inspection-driven activities (Dantan et al. Reference Dantan, Etienne, Mohammadi, Khezri, Homri, Tavakkoli-Moghaddam and Siadat2022), thus demonstrating the strong linking along the product life cycle described in Section 3.2.

Figure 9. Illustration of the results from the classification of the research papers.

This classification indicates that, from a design perspective, the focus of the research presented at the conference was primarily on tolerance analysis, while the definition of tolerance requirements and tolerance allocation were barely considered. First, this reflects the scope of the conference. Second, it indicates that there is currently a lack of research in these areas. This may be because no acute need for research is apparent here since concrete, practical procedures, including realistic tolerance values, are of primary interest instead, for example, for the initial tolerance allocation. In addition, the close linking of tolerancing to other engineering domains, as discussed in Section 3.3, becomes evident, for example, for the tolerancing requirements definition, which adopts models, methods, and tools from requirements engineering and quality management, where extensive research driven by the emerging large language models takes place. Thus, taking into account a longer period of time, this classification fosters the identification of current research trends and focuses, as well as possible shifts to other areas.

This is primarily facilitated by the structured description scheme with predefined activities and elements. In contrast to titles or keywords, which can be arbitrarily defined by authors, this ensures a uniform description of tolerancing research works and thus helps to identify similarities. Within this scheme, the papers “An integrated resource allocation and tolerance allocation optimization: A statistical-based dimensional tolerancing” (Khezri et al. Reference Khezri, Homri, Etienne and Dantan2022) and “How to Consider Over-constrained Assemblies with Gaps in Tolerance-Cost Optimization?” (Hallmann, Schleich & Wartzack Reference Hallmann, Schleich and Wartzack2020b) similarly contribute to the tolerance synthesis activity, while the titles suggest otherwise due to the different methods, models, and tools as well as data, information, and knowledge used in areas with different backgrounds, see the table in the supplementary material. Thus, the proposed description scheme might facilitate communication between these areas.

Moreover, it supports bringing together different points of view and levels of detail, e.g., when proposing tolerance specification approaches utilizing knowledge from the ISO GPS system. While one publication addresses a method for the implementation of the GPS system in the practical tolerance specification process (Schuldt & Gröger Reference Schuldt and Gröger2022) from an organization or training perspective, the other proposes a method for the specification of highly interrupted collected features, such as textured surfaces (Fischer & Morse Reference Fischer and Morse2020). Although both pursue the goal of a standard-compliant tolerance specification, the two approaches are intended for different target groups, namely product developers and tolerance experts. While associated approaches are usually completely separate from each other, the description via the five common elements enables the identification of intersections. It thus promotes possible cooperation across different levels in this case.

This also applies to approaches on similar levels, which, however, use different models for tolerance analysis, such as Skin Model Shapes (Shao, Liu & Ding Reference Shao, Liu and Ding2020), polytope models (Teissandier, Delos & García Reference Teissandier, Delos and García2020), or vector models (Schaechtl et al. Reference Schaechtl, Hallmann, Schleich and Wartzack2020). If these approaches are assigned to similar tools, this fosters the combination of different models into a hybrid tolerance analysis model that unites the advantages of several models (Heling, Hallmann & Wartzack Reference Heling, Hallmann and Wartzack2017). Besides, the proposed description scheme also facilitates the differentiation and potential integration on a detailed level. Thus, for example, the proposed open-source tolerance analysis tool based on Skin Model Shapes (Garcia et al. Reference Garcia, Teissandier, Anwer, Delos, Ledoux and Pierre2022) could be extended by a Skin Model Shape generation based on generative adversarial networks (Qie, Balaghi & Anwer Reference Qie, Balaghi and Anwer2022). Due to the usually inconsistent titles and keywords, this intersection also only becomes clear via the description scheme.

The application of the proposed scheme to the research work from the selected conferences highlights the need for a differentiated classification going beyond one purely based on just one category, e.g., the activity or just the title, which is not useful. This fosters an unambiguous description and classification of research work and enables easy structuring and clustering. Thus, it provides an overview of existing approaches with the corresponding methods, models, and tools as well as data, information, and knowledge for the different activities and facilitates differentiation and collaborations, ensuring scientific novelty.

4.2. Development of an electrified cross skate

While the previous section focused on the academic perspective, the proposed description scheme is subsequently applied in a practical tolerancing process scenario for a newly developed electrified cross skate. The scheme is used to describe the multiple, sometimes alternative, activities for ensuring and optimizing product quality in a structured manner. This is intended to simplify the communication between tolerance experts from different backgrounds and thus establish a procedure for targeted quality improvement that can be integrated into the development process.

The Technical Product Cross skates are roller skates with two wheels suitable for off-road use and enable skating like in cross-country skiing. During skating, upper and lower body muscles can be trained (Ross Reference Ross, Werd and Knight2010). Hence, they offer a health-promoting micro-mobility solution to the so-called “last mile problem.” The Institute of Engineering Design (KTfmk) of the Friedrich-Alexander-Universität Erlangen-Nürnberg took up this general idea and has developed an innovative, electrified version of a cross skate, called E-Cross Skate in the following (Kramer et al. Reference Kramer, Wirsching, Wartzack and Miehling2023), see Figure 10. It is attached to the rider’s foot by a binding, enabling its use with regular street shoes. The binding is mounted on four strain gauge load cells (two each at the front and rear), allowing driving power regulation of the electrically driven rear wheel. The front wheel is designed to be steerable to improve cornering and driveability, see the cross-section of the front wheel assembly in Figure 10 (right). For this purpose, a wheel hub steering system is suitable to meet the actuation and design space requirements. Hub-center steering is characterized by the steering pivot points located inside rather than outside or above the wheel hub as in conventional arrangements of steering mechanisms. Two servomotors redundantly actuate the steering system via steering rods depending on the inclination about the roll axis (x-axis in Figure 10), which is measured by an inclination sensor (gyroscope) on the controller board of each skate. The E-Cross Skate provides two driving modes. In the all-electric mode, the skates are aligned in parallel, and driving power is provided solely by the electric motors. In addition to purely electric driving, the E-Cross Skate features a new, patented hybrid driving mode (Rathert et al. Reference Rathert, Miehling, Wartzack, Biehler, Kramer, Duschl and Kuhn2022) in which the user moves in the skating technique as in cross-country skiing, supported by electric motors. The amount of auxiliary torque is determined based on the instantaneous distribution of the driver’s weight between the two E-Cross Skates and among the individual load cells.

Figure 10. Overview of the CAD model of the E-Cross Skate (left) and its front wheel assembly (right) serving as the case study to emphasize the benefits of a tolerancing-supported product design.

Tolerancing for E-Cross Skate The E-Cross Skate in Figure 10 shows the status of the prototype developed by the product design team. After verifying its functionality in physical tests, it had to be leveled up for series production by assuring product quality and robustness considering geometrical variations. It was the responsibility of the tolerancing team, formed by the dimensional management group of the institute of the authors, to improve the current status using different tolerancing approaches developed in research and established in industry but strongly differing in their focus. Hence, the following section does not aim to propose a chronological procedure with fixed activities. Instead, it illustrates the diversity of different activities with multiple people involved. At this time, the interdisciplinary team consisted of eight tolerancing experts (#1–#8), who are all part of the author team, with an experience level in tolerancing between 1.5 and 7.5 years (on average 4.1 years) and experience from industry in this field between 0 and 1.5 years (on average 0.3 years). Figures 11–12 and the following paragraphs provide a global overview of the different activities applied. More details on the activities can be found in their respective descriptions given in the supplementary material.

Figure 11. Overview of the different tolerancing activities carried out to assure the product quality of the E-Cross Skate (I/II).

Figure 12. Overview of the different tolerancing activities carried out to assure the product quality of the E-Cross Skate (II/II).

Initially, requirements relevant to tolerancing, such as Key Characteristics (KCs) or functions, were systematically identified and iteratively extended since the initial set of engineering requirements is usually incomplete, especially when product development progresses and new requirements need to be defined (Goetz et al. Reference Goetz, Horber, Schleich and Wartzack2021a). By using the KC-flowdown method, the KCs were derived on different levels, from product KCs, e.g., the lifetime of the E-Cross Skate, over sub-systems, e.g., the steering accuracy of the front wheel mechanism, and part KCs down to process KCs. Natural Language Processing (NLP) was used to extract the information from written product specifications (Goetz et al. Reference Goetz, Horber, Schleich and Wartzack2021a), varying in detail and language. In this way, the tolerance experts #1 and #2 defined the tolerancing requirements from documents and refined them in cooperation with the product design team, see Figure 11.

It is proven that enhancements of the concept and the parameter design in terms of robust design positively impact all subsequent tolerancing activities (Goetz, Roth & Schleich Reference Goetz, Roth and Schleich2021c). Thus, expert #1 first compared two alternative concepts with either two motors with rods or one motor with cable drive for the front wheel steering and studied the overall robustness of the concepts, see Figure 11. Graphs transferred from initial concept sketches were used to automatically detect the KC chains, evaluate their complexity, and perform constraint and mobility analysis using screw theory to apply design concept improvements semi-automatically (Goetz et al. Reference Goetz, Kirchner, Schleich and Wartzack2021b). The steering concept with two motors was identified as less robust than the cable solution. (However, a change of concept typically causes a major redesign of the entire product and influences further requirements. Thus, although the cable drive concept was identified as a more robust solution, it was not chosen for the final design of the E-Cross Skate ready for series production, as the favored concept ensures redundancy.) By using formalized Robust Design principles, a more robust design concept, e.g., through switching from revolute to ball joints, could be found, see Figure 11.

Subsequently, expert #1 searched for a robust combination of the parameters defining the steering mechanism rod lengths using transfer functions and Design of Experiments (DOE). Scatter plots for different configurations of the rod lengths and motor drive angles as inputs and the steering angle as output revealed the most robust parameter design of the mechanism, expressed through a lower sensitivity of the steering angle to changes in the drive angle, see Figure 11.

For subsequent tolerance specification, experts #1 and #3 followed a well-established manual approach. Step by step, they identified the individual contributions of each degree of freedom of the relevant features to the derived KC. Their variations are then limited through size, location, orientation, and form tolerances referring to current and internationally accepted standards on GD&T (ISO and ASME standards). The specification rules are implemented in most common CAD systems as GD&T-advisors, enabling the direct, model-based annotation of the GD&Ts as semantic product and manufacturing information (PMI), see Figure 11. For tolerance allocation, suggestions from handbooks and the ISO 2768 standard were used to complete the tolerance schemes for all parts of the front wheel assembly.

In addition to this manual approach, a knowledge-based approach for automated tolerance specification was used to verify the results. Therefore, the graph representation of the steering mechanism from the previous activities was extended from part to geometry element level, including contact definitions between the parts’ faces and spatial relations among geometry elements within each part. In combination with tolerancing knowledge, formalized by an ontology including manifold Semantic Web Rule Language (SWRL) rules, proposals for tolerance specification and allocation for product functionality were automatically created and served as the basis to refine the manual tolerance allocation results, see Figure 11.

Subsequently, different experts focused on analyzing and improving the defined tolerance schemes. Expert #4 was responsible for tolerance analysis of the steering mechanism, i.e., studying if the proposals made were sufficient to limit the variation of the steering angle, defined as a geometrical, time-variant KC of the system in motion (Walter, Sprügel & Wartzack Reference Walter, Sprügel and Wartzack2013), see Figure 12. The time-variant interrelation between the geometrical tolerances and the steering angle as a function of time was represented by a vectorial tolerance model (Walter et al. Reference Walter, Sprügel and Wartzack2013). Statistical variation simulation based on an LHS was used to create the inputs for density-based sensitivity analysis, to quantify the influence of the individual parts and their tolerances, and to give first hints of the most sensitive areas in the system (Schaechtl et al. Reference Schaechtl, Hallmann, Schleich and Wartzack2020).

Bearing specialist and expert #5 focused on the tolerancing scheme of the bearings and their seats within the front wheel assembly since geometrical variations influence the internal bearing load distributions and thus bearing life (Aschenbrenner et al. Reference Aschenbrenner, Schleich, Tremmel and Wartzack2020). For tolerance analysis of this non-geometrical KC, the geometry of the bearings and seats with their deviations was represented through Skin Model Shapes (Schleich et al. Reference Schleich, Anwer, Mathieu and Wartzack2016), serving as the basis for the analysis of the bearing life (Kramer et al. Reference Kramer, Atalay, Rüth, Wingertszahn, Bartz, Goetz, Schleich, Koch and Wartzack2024). The results indicated that a widening of the bearing seat diameter tolerances from $ \mathrm{g}6 $ (shaft) and $ \mathrm{N}7 $ (housing) to $ \mathrm{h}7 $ and $ \mathrm{N}8 $ does not decrease bearing life while fulfilling the mechanical performance requirements, see Figure 12.

These first hints from tolerance analysis bridge to the subsequent tolerance synthesis steps. The tolerance values pre-assigned in tolerance allocation were chosen for function while hoping that they are not cost-intensive. For this reason, expert #3 was responsible for tolerance synthesis, refining the initial tolerance schemes. First, the tolerance analysis software TCVisVA, based on a feature-based representation model, was used for variation simulation and sensitivity analysis. Taking the current tolerances into account, the predicted non-conformance rate of the camber of the E-Cross Skate’s front wheel helped to evaluate if the tolerances are capable of assuring the required product quality, expressed by a maximum non-conformance rate in small parts-per-million (Hallmann et al. Reference Hallmann, Schleich, Heling, Aschenbrenner and Wartzack2018). High-Low-Median (HLM) contributor analysis hints to the relevant tolerances to be manually tightened or widened. This helps to assure product quality and indirectly saves costs (Roth Reference Roth2024). This manual, iterative adaptation and analysis loop was repeated until a satisfactory solution was achieved, see Figure 12.

Second, the cost aspect was included in the described tolerance synthesis loop. Tolerance-cost optimization was applied to minimize the sum of all tolerance-related manufacturing costs, expressed through relative tolerance-cost functions considering the feature type and geometry (Roth Reference Roth2024). The method automates the manual search approach by metaheuristic optimization, where the tolerance values are balanced so they can assure product quality (here at maximum $ 2700\ \mathrm{p}\mathrm{p}\mathrm{m} $ ) but with minimum costs to consider the economic impact of tolerancing already in the design phase (Hallmann et al. Reference Hallmann, Schleich and Wartzack2020a, Reference Hallmann, Schleich and Wartzack2021b), see Figure 12. (The application of tolerance-cost optimization does not necessarily lead to lower costs compared to the initial status. It aims for the best combination of tolerances where the lowest costs can be achieved while ensuring product quality. Though the initial tolerances chosen for comparison in Figure 12 are less cost-intensive than the final solution, they are not capable of meeting the maximum non-conformance rate.)

When switching from product design to product realization, tolerancing activities strongly focus on the manufacturing and inspection process, see Section 3.2. Two exemplary activities from manufacturing-driven tolerancing were applied to the E-Cross Skate. The process design for manufacturing one of the front steering levers was focused on by the experts #4 and #6. They used virtual DOE jointly with a numerical process simulation tool to study the influence of the process on the final part geometry. Covering the relevant process variables for injection molding with their nominal values and variations mapping the real situation helped to virtually find a robust process window where the allocated tolerances can be achieved (Heling et al. Reference Heling, Oberleiter, Rohrmoser, Kiener, Schleich, Hagenah, Merklein, Willner and Wartzack2019; Heling, Schleich & Wartzack Reference Heling, Schleich and Wartzack2020), see Figure 12.

In addition, the two experts, #7 and #8, in tolerancing of lightweight structures studied the potential of weight reduction when substituting the E-Cross Skate frame with a lightweight composite-fiber reinforced plastic (CFRP) part. Variation simulation was used to mimic the manufacturing process, covering ply angle variations on the laminate level and geometrical feature variations on the part level. Since the FEA-based process simulations are computationally expensive and time-consuming (Franz et al. Reference Franz, Schleich and Wartzack2021; Franz & Wartzack Reference Franz and Wartzack2023), surrogate models were applied to be able to first analyze the resulting part deformations and their effect on the final product quality and costs and to second define the process variables in terms of ply angle tolerances in reasonable computing times, see Figure 12.

Evaluation of Using the Proposed Description Scheme In the first meeting, all experts proposed their approaches to be applied to the use case and discussed potential improvements. Although all involved participants were from the same institute, the chosen approaches differed according to the experts’ roles and foci, with inconsistent descriptions and terminology hampering a common understanding. In a subsequent workshop, they used the proposed description scheme to structure their activities and described them in a standardized form using the five main elements and respective entities, see benefit (1) in Section 3.4. The individual feedback confirmed that it fostered communication and collaboration, see benefit (2) in Section 3.4. On the one hand, positive feedback within one activity with multiple stakeholders involved, such as in the tolerancing requirements definition or tolerance allocation and specification, see Figure 11, was reported. On the other hand, an enhanced cross-activity interaction was highlighted, facilitating working on parts shared within different activities, such as the tolerance values to the bearing seats assigned in tolerance analysis and considered within tolerance synthesis, see Figure 12. In addition, it simplified coordination between the tolerancing and the product design team, which was particularly necessary for early tolerancing activities, such as tolerancing requirements definition and robust concept and concept design improvement, see Figure 11. The scheme was perceived as beneficial for working in interdisciplinary teams with people with different backgrounds and levels of experience, as it fosters the accessibility of tolerancing topics for engineers even with little experience in tolerancing.

The structured descriptions presented in the supplementary material emphasize the high variability of the approaches expressed by the different entities chosen. While more qualitative approaches and manual expert knowledge are typically required in the early tolerancing phases, the ratio of simulation-driven methods increases in the later ones. Special applications, such as tolerancing of bearing assemblies or CFRP parts, see Figure 12, often require specific models, tools, and methods. Nevertheless, numerous similarities in the approaches are evident, some of which are not apparent at first glance. For instance, the parameter and process design approaches use DOE to study the influence of nominal parameters while considering their variations. While the first focused on the robustness of the steering mechanism expressed by a transfer function in terms of the varying geometrical characteristics, the latter searched for robust process windows using numerical simulation tools to establish the relationship between varying process parameters and the final part geometry. Synergy effects can be exploited if these similarities are made recognizable via a common description. Hence, the developed scheme helped the tolerancing team to bring the approaches developed in research to practice and to define a continuous, common tolerancing process for a complex but still manageable product, see benefits (3)–(4) in Section 3.4.