Introduction

Industrial systems usually have complex structures, and thus, the analysis and optimization of their performance require adequate knowledge of human operators and system information for achieving desired industrial goals. To achieve desired industrial goals, a robotic system is widely being used, and therefore, the importance of the robot reliability and quality has been increased. The study of robot reliability is very complex, since a lot of interlocking variables are involved in evaluation of various reliability levels of these systems (Sharma et al. 2010). The research on robotic systems is reported very large in the literature; however, limited work has been reported on reliability of robotic systems (Leuschen et al. 1998; Carreras et al. 1999; Carlson and Murphy 2003; Sharma et al. 2008; Kumar et al. 2012. Specifically, Khodabandehloo (1996) used fault tree for the safety and reliability analysis of robotic system. To improve the safety and reliability of robot manipulator, (Walker and Cavallero 1996a; Walker and Cavallaro 1996b) used fault tree analysis(FTA) during design phase of the manipulator. Dhillon and Yang (1996) analyze safety and reliability of robotic systems using supplementary variable and Markov techniques, while Carreras et al. (1999) and later on Carreras and Walker (2000) applied interval method for the same purpose. Leuschen et al. (2001) developed a novel fuzzy Markov modeling approach for analyzing fault tolerant of a robot designed for hazardous waste removal. Stancliff et al. (2006) presented a quantitative method for mission reliability estimation of various mobile robots working as teams. The reliability of mechanical robots was improved by Korayem and Iravani (2008) using failure mode analysis and function deployment approach.

From the literature, one can observe that approaches discussed so far for reliability analysis use probabilistic assumptions and crisp historical data which is in general limited and contains vagueness, uncertainty, etc. Practically, for system reliability analysis, constant failure rate model is being utilized, because most of the technical systems show this type of behavior, having some kind of uncertainties (Das 2008; Knezevic and Odoom 2001). Unfortunately, crisp historical data are not sufficient to account for the available uncertainties. Using vague, limited and imprecise data and information about any system such as a robotic system modeling analysis and optimization of the system’s overall performance are very difficult. Thus, the issue of robot reliability is related to uncertainty, and in lack of proper knowledge, it is difficult to analyze and predict its behavior. In addition, a system analyst is unable to suggest any necessary action to optimize the performance of a robotic system by enhancing its reliability. Fuzzy sets as suggested by many researchers are used to account for the uncertainties involved in the available information, i.e., in the extracted data (Chen 1994; Cai 1996; Bai and Asgarpoor 1996). In view of above limitations and the applicability of fuzzy set theory for quantifying involved uncertainties, Sharma et al. (2008) analyzed the performance of a complex robotic system using fuzzy lambda–tau(FLT) methodology. In FLT, traditional lambda–tau method (Mishra 1992) is coupled with fuzzy set theory and \(\alpha\)-cut interval arithmetic operations to analyze fuzzy reliability of any repairable system, and has been employed to analyze the fuzzy reliability of various industrial systems including butter–oil, paper, and fertilizer manufacturing plants, waste clean up manipulator etc (Komal et al. 2009, 2010; Kumar et al. 2012). It is noticed that FLT gives wide spreads for computed fuzzy reliability parameters of any complex repairable system with numerous components.

Considering the influence of growing event of fuzziness, it may be possible that the decision-maker may suggest some impressive corrections which may improve the system performance. However, it may also be possible that after incorporating suggested corrections, system performance may not improve up to the desired level, because suggestions are inappropriate due to the wide ranges of prediction. To reduce the range of prediction and to make effective decisions, soft computing-based techniques were developed for analyzing the fuzzy reliability of complex repairable systems (Komal et al. 2009, 2010).

The aim of the present work is to analyze the reliability of a robotic system up to a desired degree of accuracy under uncertain environment by reducing the complexity in calculations and utilizing available raw data. The aim is also to reduce the ranges of the prediction for computed reliability indices, so that better decisions may be drawn from the analysis which may help to optimize the performance of the system.

The paper is organized as follows. Section 2 describes the methodology. Robotic system is described together with obtained results in Sect. 3. Sensitivity analysis is also performed in this section followed by concluding remarks in Sect. 4.

Methodology

The main aim of the paper is to evaluate the performance of a robotic system by utilizing vague, imprecise, and uncertain data. For the reliability analysis of any repairable system, the following assumptions are being used in the literature:

  • the failures and repair rates of the components are statistically independent and obey exponential distribution function;

  • the product of the failure rate and repair time is small (less than 0.1);

  • after repairs, the repaired component is considered as good as new;

  • system structure is precisely known.

Existing Fuzzy Lambda-Tau (FLT) technique

Fuzzy lambda–tau technique developed by Knezevic and Odoom (2001) is a traditional method for analyzing fuzzy reliability of the system. In this methodology, Petri-nets(PN) is used for the qualitative modeling and lambda–tau method of solution derived in Mishra (1992) is utilized for quantitative modeling. In lambda–tau method, mathematical expressions for failure rate (\(\lambda _s\)) and repair time (\(\tau _s\)) of the system are formulated in terms of its constituting component’s failure rates (\(\lambda _i\)’s) and repair times (\(\tau _i\)’s). The basic expressions of lambda–tau method are expressed in Table 1. The various reliability parameters can be evaluated according to Table 2 using fuzzy arithmetic operations given in (Table 3).

This approach is suitable for small-scale simplified systems. If FLT is implemented for analyzing fuzzy reliability of any complex system with numerous components, then computed fuzzy reliability indices have wide spreads. The reason is the growing event of fuzziness due to the use of \(\alpha\)-cut interval arithmetic operations in the computations (Chen 1994). Thus, using highly uncertain data and the obtained results, decision-maker cannot suggest any specific and influential managerial strategy to prevent unexpected failures and consequently to improve the industrial system performance. To overcome this problem, a hybridized technique is used in this paper and described in the next subsection.

Table 1 Basic expressions of lambda–tau methodology
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Table 2 Some reliability indices for repairable system with constant repair rate model
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Table 3 Basic operations on fuzzy numbers
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Hybridized technique

This technique utilizes fuzzy set theory to quantify uncertainties, fault tree to model the system, and lambda–tau method to formulate the mathematical expressions of system’s failure/repair rates and the genetic algorithm is utilized to solve established nonlinear programming problems. The expression of the various reliability parameters of the system is evaluated in terms of component’s failure rate and repair time of the system using Tables 1 and  2. The evaluated reliability parameters so obtained are nonlinear as the system has complex structure. The failure and repair data of system’s components are exponentially distributed and, therefore, are not precisely known. Quantification of involved uncertainties is done through data fuzzification using triangular fuzzy numbers. The optimization problem (1) is used for finding system fuzzy reliability parameters.

Minimize/maximize:

$$\begin{aligned} \tilde{P}{(\lambda _1,\lambda _2,\ldots ,\lambda _n,\tau _1,\tau _2,\ldots ,\tau _n)} \quad {\text{or}} \quad \tilde{P}{(t/\lambda _1,\lambda _2,\ldots ,\lambda _n,\tau _1,\tau _2,\ldots ,\tau _n)} \end{aligned}$$
(1)
$$\begin{aligned} \text {Subject\;to} :&=\nu _{\lambda _i}(x)\ge \alpha , \\&\nu _{\tau _i}(x)\ge \alpha , \\&0\le \alpha \le 1,\\&i=1,2,\ldots ,n. \end{aligned}$$

Here, \(\tilde{P}{(\lambda _1,\lambda _2,\ldots ,\lambda _n,\tau _1,\tau _2,\ldots ,\tau _n)}\) and \(\tilde{P}{(t/\lambda _1,\lambda _2,\ldots ,\lambda _n,\tau _1,\tau _2,\ldots ,\tau _n)}\) are time-independent (system failure rate, repair time, and MTBF) and time-dependent (system reliability, availability, and ENOF) fuzzy reliability parameters, respectively. The obtained minimum and maximum values of \(\tilde{P}\) are denoted by \(P_{\text {min}}\) and \(P_{\text {max}}\), respectively. The membership function values of \(\tilde{P}\) at \(P_{\text {min}}\) and \(P_{\text {max}}\) are both \(\alpha\), that is

$$\begin{aligned} \nu _{\tilde{P}}(P_{\text {min}})=\nu _{\tilde{P}}(P_{\text {max}})=\alpha . \end{aligned}$$
(2)

The optimization problem so obtained is highly nonlinear. Various methods and algorithms have been developed for optimization and have been utilized in various research areas (Tillman et al. 1980; Ravi et al. 1997, 2000; Konak et al. 2006). Genetic algorithms (GAs) is one of the most effective evolutionary algorithms. Genetic algorithms have been applied to the optimization of various reliability problems (Juang et al. 2008; Lapa et al. 2006; Martorell et al. 2005; Konak et al. 2006; Komal et al. 2009, 2010).

Any nonlinear optimization problem without checking the convexity and differentiability of objective functions can be solved effectively using Genetic algorithms (Goldberg 1989; Konak et al. 2006). Therefore, we have selected GAs as a tool to solve the optimization problem (1). In the literature, the variety of GAs are available such as binary-coded GA, real-coded GA, NSGA, etc. Among the many GAs available in the literature, the present study utilizes binary-coded GA to solve above formulated nonlinear optimization problems. In the beginning of the solution process, failure rates (\(\lambda _i\)’s) and repair times (\(\tau _i\)’s) of system’s components are encoded as the strings of selected bit length l, a parameter of GA, that finally constitute a chromosome. For the maximization problem (1), the objective function is considered as the fitness function, while reciprocal of the objective function is considered as the objective function for the minimization problem (1). Roulette wheel selection process is employed for the selecting potentially useful solutions for the recombination. Crossover and mutation are another two important basic operators used in any GA. There are many types of crossover and mutation operators depending upon the type of encoding and also on a problem. In the present study, one-point crossover and random-point mutation operators are employed. Maximum number of generations and change in population fitness value are used to stop the optimization process. The coding of the above developed GA has been done in MATLAB 7.1 environment. Different sets of values of all the parameters of GA are tested and the best set of values is selected for analysis which gives better optimum solution of the problem. After solving above formulated nonlinear optimization problems (1) for each cut-level \(\alpha\) using developed binary-coded GA, we have fuzzy reliability indices with reduced spread at each cut-level \(\alpha\).

System description and results

In the present study, various reliability parameters of a robotic system with a conveyer unit are evaluated. The system consists of two robots and one conveyor unit between them. This system may be utilized as a conveyor system in industries. There are three joints in each robot. Each joint has one motor (\(M_i,1\le i\le 6\)) and one sensor (\(S_i,1\le i\le 6\)). The conveyor unit has a bearing (\(\text {Br}\)) and a roller (\(\text {Rl}\)) as its components. The fault tree model of the robot is depicted in Fig. 1. Minimal cut sets of the system are {\(M_i,1\le i\le 6\)}; {\(S_i,1\le i\le 6\)}; {\(\text {Br}\)}; and {\(\text {Rl}\)} which are obtained using matrix method (Knezevic and Odoom 2001).

Using Table 1, the expression for the system failure rate (\(\lambda _s\)) and repair time (\(\tau _s\)) takes the following forms:

$$\begin{aligned} \lambda _s= & {} \sum _{i=1}^{14}\lambda _i\end{aligned}$$
(3)
$$\begin{aligned} \tau _s= & {} \frac{\sum _{i=1}^{14}\lambda _i\tau _i}{\lambda _s}. \end{aligned}$$
(4)

Utilizing these expressions along with expressions given in Table 2, the system reliability parameters are obtained. The failure rate and repair time of the main components of the robotic system following exponential distribution and their values are given in Table 4 (Sharma et al. 2010). Quantification of involved uncertainties in crisp input data is done through data fuzzification using triangular fuzzy numbers with \(\pm 15, \pm 25\) and \(\pm 50\%\) spreads. For an example, the input for \(\pm 15\%\) (failure rate and repair time) for the motor is shown in Fig. 2. Using fuzzy input and expressions for reliability parameters, fuzzy reliability parameters of the system have been evaluated using hybridized technique for mission time \(t=100\)(h) to analyze the behavior of the system. For this technique, a nonlinear optimization problem (1) has been formulated. To solve the established optimization problem (1), the parameters of GA are taken as follows:

  • Population size = 120

  • Probability of crossover (\(P_\text {c}\)) = 0.85

  • Probability of mutation (\(P_\text {m}\)) = 0.005

  • Number of iterations = 500

  • Number of runs = 20.

The fuzzy reliability parameters of the system under consideration have also been evaluated using the existing FLT (Sharma et al. 2010). The computed results have been plotted in Fig. 3 for \(\pm 15\%\) uncertainty level. Results clearly indicate that the used hybridized technique provides better fitted results in comparison to FLT technique. Thus, the prediction range for any reliability parameter at each cut-level \(\alpha\) is decreased which will be beneficial for making sound decisions. The crisp and defuzzified values using FLT technique and the hybridized technique with \(\pm 15\), \(\pm 25\), and \(\pm 50\%\) spreads are calculated. The calculated values are tabulated in Table 5. It is obvious from this table that the defuzzified values are changing with the change of spread. Defuzzified values of failure rate, repair time, and ENOF increase as the level of uncertainty increases. In addition, defuzzified values of MTBF, reliability, and availability decrease as the level of uncertainty increases. For example, the defuzzified value of failure rate of the system increases by \(0.3612\%\) and further \(11.2013\%\) for FLT technique, while 0.52 and \(0.8599\%\) for hybrid technique, when spread changes from \(\pm 15\) to \(\pm 25\%\), and \(\pm 25\) to \(\pm 50\%\), respectively. Similarly defuzzified value of MTBF of the system decreases by \(8.7976\%\) and further \(6.6626\%\) for FLT technique, while 1.4408 and \(0.8524\%\) for hybridized technique when spread changes from \(\pm 15\) to \(\pm 25\%\), and \(\pm 25\) to \(\pm 50\%\), respectively.

Fig. 1
figure 1

Fault tree of robotic system

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Fig. 2
figure 2

Fuzzy input for motor

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The effects of various combinations of reliability, failure rate, and availability onto the system MTBF are studied through sensitivity analysis. The effects are shown graphically for both the techniques. The repair time and ENOF are varying along x-axis and y-axis, respectively, whereas MTBF is plotted along z-axis. For the analysis, nine different combinations of reliability, failure rate, and availability are considered. For the both techniques, in all the nine combinations, ranges of repair time and ENOF are fixed and computed by their membership functions (Fig. 4b, d) at cut-level \(\alpha = 0\). The ranges of repair time are taken as 1.2221–3.0264 for FLT technique and 1.6987–2.2236 for hybridized technique, respectively. The ranges of ENOF are taken as 0.0239–0.03248 for FLT technique and 0.0247–0.0320 for hybridized technique, respectively. The effects on MTBF for FLT technique are shown graphically in Fig. 4 and for hybridized technique in Fig. 5, respectively. The ranges of MTBF so obtained are tabulated in Table 6. From Table 6, we can observe that for the first combination, the chosen values of reliability, failure rate, and availability are \(0.9722, 2.825\times 10^{-4},\) \(\text{and}\,0.9994\), respectively. The calculated ranges of MTBF are 3072.808– 4160.350, and 3117.559–4037.021 for FLT technique and hybridized technique, respectively. One can observe that for this combination, the prediction range of MTBF is reduced by approximately \(16\%\) from FLT technique when hybridized technique is utilized. From this observation, we can conclude that if system analyst uses results based on hybridized technique, then he may have less range of prediction, and finally, he will lead to more sound decisions. Similar behavior has also been noticed for rest other combinations. Based on the above observations, the system analyst can analyze the critical behavior of the system and he can prepare a suitable plan for maintenance.

Fig. 3
figure 3

Fuzzy reliability indices plots for robotic system with \(\pm 15\%\) uncertainty

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Fig. 4
figure 4

Behavior analysis plots for robotic system using FLT technique results

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Fig. 5
figure 5

Behavior analysis plots for robotic system using hybridized technique results

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Table 4 Failure rates and repair times data for robotic system
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Table 5 Crisp and defuzzified values of reliability parameters of the system
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Table 6 Change in MTBF for various combinations of reliability indices for the system
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Conclusion

In this paper, a hybridized technique is used for the performance analysis of a robotic system. Various reliability parameters such as system failure rate, repair time, MTBF, ENOF, availability, and reliability are calculated. The calculated reliability parameters are in the form of fuzzy membership functions. Depending upon the confidence level ‘\(\alpha\)’, the analyst can predict the behavior of the system(s). The defuzzified values of reliability indices for different level of uncertainties with their crisp values have been computed and tabulated. It can be concluded that the system analyst should perform the system maintenance on the basis of defuzzified MTBF rather than the crisp value. Similarly, it can be realized that with increasing repair time, the reduced value of reliabillity/availability is more conservative than that of the crisp value. Based on the above observations, the system analyst can analyze the critical behavior of the system and he can prepare a suitable plan for effective maintenance.

We can also conclude that hybridized technique gives reduced range of reliability indices at any cut level. Using these results, the prediction of system behavior can be observed more confidently. This will help the management in reassignment of the resources, taking maintenance decisions effectively, getting more availability of the system, and, therefore, exaggerating the productivity of the system.