3.1. Previous work

Engelhardt et al. (Reference Engelhardt, Birkhofer, Kloberdanz and Mathias2009) proposed an holistic framework, the uncertainty mode and effects analysis (UMEA), for the analysis of uncertainty in load-carrying structures and their evaluation based on the caused effect on the functional performance of a considered system (cf. Engelhardt et al. Reference Engelhardt, Birkhofer, Kloberdanz and Mathias2009; Engelhardt Reference Engelhardt2012). The structure and examples for suitable models and methods supporting each step of the UMEA are shown in Figure 2. The UMEA was utilised by Würtenberger to manage the effects of uncertainty in product models for development purposes but also made the transfer to mathematical product models (cf. Würtenberger et al. Reference Würtenberger, Lotz, Freund and Kirchner2017; Würtenberger Reference Würtenberger2018). Moreover, the UMEA was carried over to the field of sensor uncertainty by Vorwerk-Handing et al. and linked to some examples of SME (cf. Vorwerk-Handing, Welzbacher & Kirchner Reference Vorwerk-Handing, Welzbacher and Kirchner2020; Vorwerk-Handing Reference Vorwerk-Handing2021). Eifler analysed the effect of uncertainty along the product life cycle and evaluated the usability of the UMEA from a different perspective (cf. Eifler Reference Eifler2015). Freund focused on mechanical robust design approaches using the variation management framework (VMF) proposed by Howard et al. (Reference Howard, Ebro, Eifler, Pedersen, Göhler, Christiansen and Rafn2014) to differentiate the effect of robust design measures (cf. Freund et al. Reference Freund, Würtenberger, Lotz, Rommel and Kirchner2017; Freund Reference Freund2018). Lotz finally analysed the effects of uncertainty arising on product families due to similarity (cf. Lotz Reference Lotz2018). For this purpose, Lotz (Reference Lotz2018) applied a nondimensional calculus to understand size dependencies and to identify contradicting growth laws. The effects analysed by Lotz (Reference Lotz2018) need to be considered when assessing functional limits of, for example, sensory rolling bearings, in which the electric capacitance of the lubrication film is used as a sensory effect. When analysing the effects of loaded and unloaded rolling contacts in a rolling bearing, as described by Schirra et al. (Reference Schirra, Martin, Puchtler and Kirchner2021), based on the analysis and modelling of electric damage effects, it turns out that the growth laws describing size dependencies will imply boundary conditions. From this point of view, the use of sensory rolling bearings can be considered as a wildcard when looking at robust design, due to the highly complex, empirical, model necessary to implement the sensory function, cf. Schirra et al. (Reference Schirra, Martin, Puchtler and Kirchner2021).

Figure 2.

Structure of the uncertainty mode and effects analysis (UMEA) (translated from Engelhardt Reference Engelhardt2012).

3.2. Uncertainty: definition, classification and variation management framework

In the ISO-Guide 73 (2009), uncertainty is defined as ‘state, even partial, of deficiency of information related to understanding or knowledge of an event, its consequence, or likelihood’ (ISO-Guide 73 2009). This definition is used in the following.

There are different approaches in engineering to categorise uncertainty, for example, based on its nature, level or manifestation in the system model (cf. Walker et al. Reference Walker, Harremoes, Rotmans, van der Sluijs and van Asselt2003; Olausson & Berggren Reference Olausson and Berggren2010; Kreye, Goh & Newnes Reference Kreye, Goh and Newnes2011). Uncertainty can be classified as depicted in Figure 3, combining the classification of uncertainty according to its nature – based on the type of relationship between information and uncertainty – and level – based on the amount of available and reliable information. Further details about these two classification approaches are provided by Walker et al. (Reference Walker, Harremoes, Rotmans, van der Sluijs and van Asselt2003) and Olausson & Berggren (Reference Olausson and Berggren2010).

Figure 3.

Classification scheme for uncertainty (following Lotz Reference Lotz2018).

In this classification scheme, uncertainty is generally distinguished from determinacy, which describes a condition where exact models and complete information about all relevant parameters exist. It has to be noted that determinacy is an ideal theoretical condition, whereas real technical systems are always subject to uncertainty. Uncertainty itself can range from stochastic uncertainty, where its variation and its probability are known, to nescience, where neither the disturbing influencing parameter nor the disturbance’s cause or a model for describing the interrelations between influencing parameters and the system are sufficiently known. Nescience comprises cases, where uncertainty is suspected but its effects remain unknown. Between these extremes, gradations of uncertainty were defined, based on the amount of available and reliable information. If the probability density function of the output parameter is not known exactly, as for stochastic uncertainty, an estimated probability density function can be applied. The estimated function is categorised under estimated uncertainty and describes the occurrence of a variation of influencing parameters based on known intervals of input–output relationships. In the category of estimated uncertainty, the unknown variability of the output depending on the input is also included. A special case is ignored uncertainty. In this category, the variation is known or estimated but is neglected on purpose, like, for example, in terms of simplifications in the modelling process (cf. Lotz Reference Lotz2018; Pelz et al. Reference Pelz, Groche, Pfetsch and Schaeffner2021).

The impact of uncertainty on a technical system – and consequently sensing technology – as well as its sensitivity to parameter variations becomes notable within the VMF proposed by Howard et al. (Reference Howard, Ebro, Eifler, Pedersen, Göhler, Christiansen and Rafn2014). It visualises how variations, in Figure 4 process properties, propagate via transfer functions through different domains to a measure of interest, in case of Figure 4 the customers’ satisfaction. The final result is influenced by the three linked, sequential steps of production, development and customer perception as depicted. Each transfer function describes the relation of process properties to design parameters during the production, the design parameters towards the system behaviour by the development and finally the system behaviour and the customer’s satisfaction by the customers’ perception (cf. Howard et al. Reference Howard, Ebro, Eifler, Pedersen, Göhler, Christiansen and Rafn2014, Reference Howard, Eifler, Pedersen, Göhler, Boorla and Christiansen2017). How exactly a scattering impacts the measure of interest, in this case, the costumers perception, can be understood as uncertainty.

Figure 4.

Variation management framework (translated from Freund Reference Freund2018, based on Howard et al. Reference Howard, Eifler, Pedersen, Göhler, Boorla and Christiansen2017).

For the design of and with sensing technology, in particular SME, the consideration of uncertainty is of great importance, since occurring uncertainty can limit their functionality and thus have a significant impact on the benefit of their application. Consequently, uncertainty needs to be systematically identified, mathematically described, to be able to quantify its effects, and considered when designing sensing technology or technical systems with integrated sensing technology, to ensure their functionality. Therefore, for example, the UMEA proposed by Engelhardt et al. (Reference Engelhardt, Birkhofer, Kloberdanz and Mathias2009) is applicable. Applying the VMF in the framework of the UMEA clarifies, at which step uncertainty arises within the development of and with sensing technology and thus how it can be eliminated or at least reduced by applying robust design. During the design of and with sensing technology, uncertainty can occur, for example, in the following forms (cf. Vorwerk-Handing et al. Reference Vorwerk-Handing, Welzbacher and Kirchner2020; Vorwerk-Handing Reference Vorwerk-Handing2021):

1. (i). manufacturing uncertainty, tolerances and uncertainty due to calibration;

2. (ii). modelling uncertainty of the relation between the input and output of the sensing technology;

3. (iii). uncertainty caused by disturbances of/in the usage phase of the system as further described in Taguchi, Chowdhury & Wu (Reference Taguchi, Chowdhury and Wu2004) and Pelz et al. (Reference Pelz, Groche, Pfetsch and Schaeffner2021).

Since SME are based on standardised conventional machine elements that can normally be manufactured within narrow tolerances, in terms of, for example, geometry or material properties, manufacturing uncertainty of SME mainly arises from the additional nonstandardised components required for the realisation of integrated measuring functions. Furthermore, the sensory function of SME, thus the relation between the input – the process or state variable to be measured – and the actually recorded and subsequently interpreted output has to be modelled based on the used physical effects and their corresponding individual laws. For this purpose, physical effect catalogues, for example, the one proposed by Vorwerk-Handing (Reference Vorwerk-Handing2021), can be used. The resulting model finally underlies uncertainty due to, for example, simplifications, made consciously or unconsciously, unknown disturbances or the temporal change of included design parameters, for example, due to corrosion or wear. These aspects are related to the SME itself and can in general be determined for each SME. In addition, when designing with SME, uncertainty arises due to the integration of the SME into a technical system, which needs to be considered, too. The uncertainty results, for example, from a change in the environmental conditions or a different assembly situation compared with standard testing conditions. These system aspects have to be investigated for each technical system using SME and taken into account (cf. Kreye et al. Reference Kreye, Goh and Newnes2011; Vorwerk-Handing et al. Reference Vorwerk-Handing, Gwosch, Schork, Kirchner and Matthiesen2020; Hausmann et al. Reference Hausmann, Koch and Kirchner2021).

3.3. Robust design

Uncertainty in different areas of the mechanical engineering domain can be addressed by applying robust design during the product development process to achieve insensitivity, that is, robustness, of a technical system against this uncertainty. A technical system is defined as ‘robust’ when it is able to ‘[…] fulfill its predefined functions [not only] at the design point, but also in the surrounding neighborhood […]’ (Pelz et al. Reference Pelz, Groche, Pfetsch and Schaeffner2021). This definition is similar to the understanding of robustness by Taguchi et al. (Reference Taguchi, Chowdhury and Wu2004), which is often used in this context, who defines a system or process as robust ‘[…] when it has limited or reduced functional variation, even in the presence of noise’.

To achieve robustness of a technical system, Mathias et al. (Reference Mathias, Kloberdanz, Engelhardt and Birkhofer2010) defined three strategies, so-called robust design strategies, for handling disturbances causing uncertainty. These can already be applied in the development phase of the system. They differ, as shown in Figure 5, with regard to their individual starting point in the influencing chain of the disturbances:

1. (i). Elimination of the disturbance: prevent the disturbance from coming up at all.

2. (ii). Suppression of the influence of the disturbance: prevent the caused influence of the disturbance from acting on the (sub)system.

3. (iii). Reduction of the impact of the disturbance: prevent the impact of the disturbance from influencing the behaviour of the system considered and consequently causing problematic effects.

Figure 5.

Robust design strategies (based on Mathias et al. Reference Mathias, Kloberdanz, Engelhardt and Birkhofer2010).

To implement these strategies in the development phase of a technical system, Ebro and Howard proposed different robust design principles (cf. Ebro & Howard Reference Ebro and Howard2016; Howard et al. Reference Howard, Eifler, Pedersen, Göhler, Boorla and Christiansen2017). These principles are categorised according to their influence, to be applicable for handling variations of the design parameters, influencing the sensitivity of the system towards the variation of design parameters and for handling the variation of the system’s behaviour. All principles contribute to the reduction of the variation of behaviour as depicted in Figure 6.

Figure 6.

Robust design in the development phase using principles for influencing the transfer function between design parameters and the product behaviour following Ebro (translated from Freund Reference Freund2018, based on Ebro & Howard Reference Ebro and Howard2016; Howard et al. Reference Howard, Eifler, Pedersen, Göhler, Boorla and Christiansen2017).

4.1. Quantitative description of uncertainty

As mentioned in Section 3.2, the classification scheme for uncertainty distinguishes between the categories ‘determinacy’, ‘stochastic uncertainty’, ‘estimated uncertainty’, ‘ignored uncertainty’ and ‘nescience’, cf. Figure 3, depending on how much information about uncertainty and the caused effect is known and thus how precise it can be described in terms of modelling. This classification is not limited to mechanical systems, but is also applicable to signal transfer analysis. The mathematical expressions introduced in the following are summarised in Figure 8. They allow a formal description of the uncertainty-effect relation and to identify suitable calculation methods, for instance Gaussian error propagation, to quantify the final effect caused by uncertainty.

Figure 8.

Classification of uncertainty (utilising basic definitions from Knetsch Reference Knetsch2004; Engelhardt Reference Engelhardt2012).

The ideal case would be determinacy, which means complete knowledge about the system and influences on the systems would exist. Consequently, the effects of uncertainty on the signal emitted by the SME and its probability could be explicitly determined. The caused effects on the system could be described mathematically without a lack of model parameters. Depending on the type of disturbances identified, different possibilities would exist to describe the effect of the disturbance on the system. If the disturbance is continuous, it can be mathematically described following Eq. (1), whereby the derivation of the function indicates the sensitivity of the system to this disturbance.

(1)$$ S\hskip1.5pt =\hskip1.5pt f(x)\iff \Delta S\hskip1.5pt =\hskip1.5pt {\left.\frac{\mathrm{d}f}{\mathrm{d}x}\right|}_x\cdot \Delta x. $$

The evaluation of the derivative at the nominal design point $ x $ in Eq. (1) assumes, as mentioned earlier, that the model $ f $ is sufficiently smooth and continuous. Provided that the primary functions remain unaffected by the influence of the disturbance $ \Delta x $. If the sensory function shows any kind of discontinuity, the determinacy condition for Eq. (1) is violated. If stochastic uncertainty occurs, it has a known effect on the system but cannot be fully quantified. However, the character and extent of the impact are known to have a certain probability density function and can be considered during the system development. This leads to a direct dependency of the signal variation $ \Delta S $ on the uncertainty $ \Delta x $, which is described through Eq. (2). Therefore, the continuity of the uncertainty is assumed, too.

(2)$$ S\hskip1.5pt =\hskip1.5pt c+k\cdot {x}^m\iff \Delta S\hskip1.5pt =\hskip1.5pt k\cdot m\cdot {x}^{m-1}\cdot \Delta x. $$

Estimated uncertainty is present if partial information about the value of influencing parameters, their impact and/or the probability of occurrence are not known. It is distinguished between an uncertain variability and an unknown variability. For uncertain variability, the probability density functions or assignment functions are estimated and one of the model parameters $ k $ and $ m $ remains undetermined, cf. Eq. (3). This leads to a loss of prediction accuracy. The effect of an uncertainty $ \Delta x $ can only be expressed in a vicinity of the current design state.

(3)$$ S\hskip1.5pt \propto \hskip1.5pt {x}^m\iff \Delta S\hskip1.5pt \propto \hskip1.5pt {x}^{m-1}\cdot \Delta x. $$

The combination of Eqs. (1)–(3) leads to the conclusion that effects of uncertainty can be described almost completely for the first two cases, whereas dependencies can only be stated for the third case. For unknown variability, the probability density functions or assignment functions are known or partly unknown and a tendency statement can be given but no precise quantity can be calculated. The leading power $ m $ in Eq. (3) is often known but the coefficient is not. Consequently, only rough guesses can be stated for unknown variability, which are rather tendency estimates, as described with Eq. (4).

(4)$$ \Delta S\propto \pm \Delta x. $$

Disregarded uncertainty is characterised by the fact that the effects of disturbances are deliberately neglected, mostly based on some intuitive estimates to limit the complexity of the description. The effects of uncertainty itself can be of any type described, they might be understood but excluded.

Unknown uncertainty, analogue to nescience in the classification scheme from Figure 3, is characterised by the absence of applicable engineering intuition. Therefore, the system behaviour is not predictable by the engineer. Estimates of the occurring uncertainty cannot be made explicit due to, for example, contradicting dependencies. None of the dependencies described in Eqs. (1) through (4) can be applied to estimate the effects of uncertainty.

4.2. Consideration of uncertainty in the transfer path

For the consideration of uncertainty, the signal chain shown in Figure 7 can be used as a guideline to determine the sensitivity of a signal with respect to occurring uncertainty, which affects the signal on its way from the measurand $ \Phi $ to the process control unit, where a signal $ {S}^{\ast } $ is received.

Under the assumption that determinacy is present and the relation between the measured $ \Phi $ and the emitted signal $ S $ is completely known, the sensitivity of the sensing technology can be calculated as described in Eq. (5) with regard to variations in both the measurand $ \Phi $ and the uncertainty affected quantity $ x $.

(5)$$ S\hskip1.5pt =\hskip1.5pt f\left(\Phi, x\right)\iff \Delta S\hskip1.5pt =\hskip1.5pt {\left.\frac{\partial f}{\mathrm{\partial \Phi }}\right|}_{\Phi, x}\cdot \Delta \Phi +{\left.\frac{\partial f}{\partial x}\right|}_{\Phi, x}\cdot \Delta x. $$

In case the relation $ f $ between the measurand $ \Phi $ and the emitted signal $ S $ cannot be completely described including all relevant and consequently required parameters, the derivatives in (1) cannot be determined, leaving in best case a tendency statement in form of Eq. (2). Hence, if this relation cannot be fully described, there is at least estimated uncertainty present, which allows only trend predictions.

Since uncertainty not only arises within the transformation of the measurand $ \Phi $ into the emitted signal $ S $, but also the signal transfer turning the emitted signal $ S $ into $ {S}^{\ast } $, both transformations must be considered. The argumentation applied in terms of sensing technology can also be applied to the signal transfer. Hence, considering an additional uncertainty $ \Delta \tilde{x} $ arising within the transformation of the emitted signal $ S $ into $ {S}^{\ast } $, described by the function $ g $, the sensitivity of the signal transfer can be calculated as shown in Eq. (6).

(6)$$ {S}^{\ast }=g\left(S,\tilde{x}\right)\iff \Delta {S}^{\ast }={\left.\frac{\partial g}{\partial S}\right|}_{S,\tilde{x}}\cdot \Delta S+{\left.\frac{\partial g}{\partial \tilde{x}}\right|}_{S,\tilde{x}}\cdot \Delta \tilde{x}. $$

Consequently, the accumulated effect $ \Delta {S}^{\ast } $ of the uncertainty portions $ \Delta x $ and $ \Delta \tilde{x} $ on the signal received at the process control unit can be described following (7).

(7)$$ \Delta {S}^{\ast }={\left.\frac{\partial g}{\partial S}\right|}_{S,\tilde{x}}\cdot \left\{{\left.\frac{\partial f}{\mathrm{\partial \Phi }}\right|}_{\Phi, x}\cdot \Delta \Phi +{\left.\frac{\partial f}{\partial x}\right|}_{\Phi, x}\cdot \Delta x\right\}+{\left.\frac{\partial g}{\partial \tilde{x}}\right|}_{S,\tilde{x}}\cdot \Delta \tilde{x}. $$

The challenge is, however, to differentiate a change in the signal received $ \Delta {S}^{\ast } $ by its reason, being caused by either occurring uncertainty or in fact by a change of the measurand $ \Delta \Phi $.

To interpret the reason for a change of the signal received by the process control unit, it is indispensable to understand all differential expressions in Eq. (7). Therefore, all dependencies must be fully determined, which would allow the application of Eq. (1). Oftentimes, engineers try to bypass this problem of limited knowledge by applying additional sensors, for example, a thermocouple to record a drift in temperature. Using this thermocouple, a change in the signal received by the process control unit, which is caused by a thermally induced uncertainty, can be clearly related to the drift in temperature once the temperature sensitivity is known. But, as long as the sensitivity is unknown, not even in the form of Eqs. (2) or (3), one can only guess – but not quantitatively describe – the reason for the change of the received signal $ \Delta {S}^{\ast } $.

This approach always targets at moving individual entries in Eq. (7) up in the hierarchy of managed uncertainty starting from Eqs. (4) to (1). In doing so, estimated uncertainty is gradually reduced to stochastic uncertainty resulting theoretically in determinacy, which would allow the correct interpretation of the different root causes of a signal change $ \Delta {S}^{\ast } $ when uncertainty $ \Delta x $ and $ \Delta \tilde{x} $ are either excluded or measured by individual sensors denoting their change.

The mathematical description of a system or part of it, respectively, provides an opportunity to objectively determine the uncertainty category of the corresponding model. The classification of uncertainty then leads to useful measures how and where to eliminate or gradually reduce disruptions causing uncertainty – in terms of robust design. Furthermore, an already existing system model allows a step by step concretisation, in which the reduction of uncertainty can be tracked by comparing the mathematical description before and after the concretisation and subsequently re-categorising according to the descriptions in Figure 8. The aim of the concretisation is to identify, which information is still required about the system, its behaviour or its interrelations so that the model follows a stochastic uncertainty behaviour in the end. This makes the data acquisition with sensing technology, in particular SME, a robust, and hence an applicable and useful, tool. In the following section, an example is discussed to outline the resulting robust design process.