A mixed-copula-based integral method for reliability analysis of a novel multi-functional rescue end-effector

To improve the performance of a conventional rescue equipment, we designed a novel multi-functional rescue end-effector comprising a separation mechanism and a grasping mechanism. The failure causes of the separation mechanism are analyzed, and three main failure modes are determined. Meanwhile, their performance functions are established as well. To solve the complex correlations among multiple failure modes, a novel mixed copula function that can capture the upper and lower tail dependencies simultaneously is proposed, which comprises the Gumbel copula and the Clayton copula. Further, a novel procedure that adopts the non-parametric kernel density estimation (NKDE) technique to estimate the probability density function (PDF) and the cumulative distribution function (CDF) for each performance function of the separation mechanism is proposed. The parameters of the mixed copula function are estimated using the method of maximum likelihood estimation. Subsequently, to further study the separation mechanism’s reliability during working, a time-variant parameter estimation method is introduced to build the time-variant mixed copula function. Finally, both the reliability and time-variant reliability of the separation mechanism are analyzed via the proposed integral methods. Comparisons with other analysis schemes show that the proposed integral method is not only effective but also efficient in analyzing the reliability of our target mechanism.


Introduction
Conventionally, the rescue end-effector is installed at the end of rescue equipment to perform rescue tasks. 1,2 The end-effector not only is a core device but also plays a deterministic role to the overall performance of rescue equipment. To modify the performance of traditional rescue equipment, we designed a novel multifunctional rescue end-effector with tonging, shearing, and grasping capabilities, as illustrated in Figure 1. 3 This new invented rescue end-effector comprises a separation, a grasping, and some additional supporting mechanisms, and is easy to be manipulated. Therefore, it may save the time required for replacing individual end-effector so as to improve rescue efficiency. Despite the above merits of our new invented rescue end-effector, further studies on its major switch function as well as the reliability of each mechanism is essential. It is 1 School of Mechanical and Electric Engineering, Sanming University, Sanming, Fujian Province, China known that the reliability of each mechanism significantly affects the overall reliability of the end-effector directly. Besides, among all parts of a rescue end-effector, the separation mechanism plays the most important role since it determines the expansion range of grasping and the switching functions between the tonging-shearing and grasping forms. Furthermore, it is noticeable that when the end-effector performs the tonging-shearing task, the separation mechanism is subjected to a very large force. This may result in crack of material and even the failure of whole rescue equipment. Therefore, this paper aims to analyze the reliability of the separation mechanism for our new invented rescue end-effector.
The motion mechanism is generally a mechanical system comprising multiple component mechanisms that accomplish various functions by transmitting either the motion or load. [3][4][5] The reliability of a motion mechanism depends on two crucial factors which are the structure as well as the motion function reliability of individual component. The structure reliability of each component mainly refers to the reliability in term of the strength or stiffness of the component, and the component reliability is a prerequisite to ensure the motion function reliability of the mechanism. 6,7 The reliability of the motion function is defined as the ability of the mechanism to complete specified functions accurately in a timely and coordinated manner under specified conditions of usage and within specified periods. [8][9][10] The failure of the mechanism's motion function is defined as the failure form of the mechanism caused by mismatching of the geometric or mechanical relationship between components, yet the component mechanism does not really fail. Therefore, the reliability of a mechanism usually involves multiple failure modes and complex failure mechanisms. Furthermore, in engineering practice, the output motion of a mechanism is usually time-variant because that its applying load is time-variant and stochastic, material properties are time-variant, manufacturing errors are dynamic, and motion is time-variant. [11][12][13][14] Hence, it is necessary to analyze the time-variant reliability of the mechanism. In addition, the reliability and time-variant reliability model of a system with k failure modes can be written as follows, respectively.
where G i X, t ð Þ is the performance function of the ith failure mode; X is the random variable vector.
A lot of past studies on multiple failure modes of motion mechanism were focusing on how to accurately solve the related linear correlation coefficients with appropriate methods, such as the widely used narrow boundary method. [15][16][17][18][19][20][21] However, the linear correlation coefficient cannot reflect the complex nonlinear relationship that exists among different multiple failure modes. In addition, some scholars used the method of weakest link theory to improve the computational efficiency by simplifying the characteristic of correlation. 22,23 The main feature of this theory is that the failure probability of the weakest failure mode is regarded as the failure probability of the system, which then leads to large errors in calculation. Generally speaking, the well-known Monte Carlo (MC) method is so far the most versatile and accurate method, and the results from MC method are often regarded as the base for comparison with other methods. [24][25][26] However, in MC, large numbers of sample points must be used in order to estimate a convergent failure probability for small probability events in engineering. In a short comment, to overcome the appeared difficulties of the aforementioned proposed methods, it is beneficial to adopt other auxiliary methods such as ADAMS and ANSYS in the reliability analysis. 27,28 To be sure, it will take more simulation time.
Compared to the time-invariant reliability analysis, the time-variant reliability analysis is much more complicated. Jiang et al. 29 proposed an outcrossing rate model to analyze time-variant reliability of the system, and the effectiveness of the method was proved. Hu and Mahadevan 30 proposed a method to characterize the system as a random field with two dimensions for time-varying reliability analysis. Zhou et al. 31 analyzed the time-variant reliability of system based on moment method and probability density evolution method. Yu et al. 11,12 proposed a method which combined the extreme moment method and the improved maximum entropy method for time-varying reliability analysis. However, the above methods are based on approximate analytical solutions, which may have low accuracy in dealing with highly nonlinear problems.
On the other hand, the copula function (CF) method proposed by Sklar 32 provided an effective tool to describe the correlations among failure modes and additional random variables. The CF method is outstanding in analyzing the joint distribution construction problem with multidimensional random variables. The CF can be used to estimate the marginal probability distributions combined with a dependent structure among various marginal distributions. By this way, the joint probability distribution can then be constructed more simply and flexibly as a result. The CF methods were widely used in the field of reliability analysis for multiple failure modes. 7,33,34 It is noticeable that different CFs may have different characteristics. Fang et al. 35 and Tang et al. 36 used different copula functions to analyze the reliability of their systems and compared the obtained results with others' works. In addition, since it is difficult to accurately describe the complex correlation among multiple failure modes using a single join model, several researchers thus attempted to construct mixed copula functions to provide a better description. [37][38][39] Wang et al. 38 applied the Clayton copula, Gumbel copula, and Frank copula to establish a reliability model for a rolling bearing. Jiang et al. 39 established a mixed copula model using the linear weighting method in the structural reliability analysis. In short, the above investigations indicate that the mixed copula approach is suitable for establishing a reliability model of the system with multiple failure modes.
In this paper, the failure causes and failure mechanisms for our new invented separation mechanism are analyzed, and its three main failure modes are identified as well. Particularly, both Clayton copula and Gumbel copula functions are adopted to construct a mixed copula function that can capture two tail correlations simultaneously for use in the reliability analysis of separation mechanism. It is noticeable that no matter the single or the mixed copula function mentioned above is merely a static copula function.
It is known that, for a motion mechanism, it is inadequate to use a static copula function to describe the correlation change among multiple failure modes during working. Therefore, we propose a time-variant parameter estimation method to construct a timevariant mixed copula function for proper use in the motion mechanism. The reliability of the separation mechanism can then be calculated based on this timevariant mixed copula model. The implementation process of this study is as follows. First, we introduce the working principle and analyze the applying forces of the separation mechanism. Second, we analyze the failure causes and failure mechanisms of the separation mechanism. Third, a reliability analysis method based on the time-variant mixed copula function is proposed. Fourth, we analyze the reliability and the time-variant reliability of the separation mechanism. Finally, the time-variant reliability of the separation mechanism is modeled using a polynomial equation with least square method.
The novelty of this study includes three points. First, we design a novel multi-functional rescue end-effector comprising a separation mechanism and a grasping mechanism. Second, we propose a novel mixed copula function that can capture the upper and lower tail dependencies simultaneously, which comprises the Gumbel copula and the Clayton copula. Third, a nonparametric kernel density estimation (NKDE) technique is then used to estimate the probability density function (PDF) and the cumulative distribution function (CDF) for each performance function of the proposed separation mechanism.

Working principle
The separation mechanism mainly comprises a frame, two translational hydraulic actuators, two guide rails, and four sliding pairs, as illustrated in Figure 2. Because the separation mechanism is symmetrical, only one side of the schematic diagram is shown for clear observation.
When the end-effector carries out the tongingshearing work, the translational hydraulic actuator retracts the piston to the right limited position and then locks, as shown in Figure 1(a). In contrast, when the end-effector carries out the grasping work, the piston of translational hydraulic actuator protrudes as far as possible to the right limited position and then is locked in place, as shown in Figure 1(b). Obviously, the separation mechanism is a key component in the transformation of work types for a rescue end-effector.

Force analysis of separation mechanism during transformation
Extension of a separation mechanism may enlarge the grasping space of the end-effector. The main driving force of the separation mechanism is supplied by a translational hydraulic actuator. We now analyze all possible applying forces on the separation mechanism. As shown in Figure 3, the driving force of the translational hydraulic actuator is F t , the length of the sliding pair is d, the distance between the two guide rails is D, and the weight of the motion mechanism is G. Because the geometric shape of the separation mechanism is symmetrical, only a half of the motion mechanism is needed to be shown for clear illustration.
The distance between the application point O 5 of the translational hydraulic actuator and the point O in the Z-axis direction is h 1 . The arm length of the separation mechanism is h 1 . This arm will cause the mechanism a clockwise tilt around the Z-axis on the XOY plane during motions. In this situation, the two sliding pairs on the separation mechanism are subjected to two pairs of forces, F 1 and F 2 , F 11 and F 22 , from the guide rail on the YOZ plane. In addition, we denote the distance between O 5 and O 6 (the midpoint between two guides in the X-axis direction) as dp. The separation mechanism will have a clockwise tilt around the X-axis on the XOY plane during motions because of the effect for the arm of the separation mechanism with distance dp. In this situation, the two sliding pairs are subjected to the forces F 3 , F 4 , F 33 , and F 44 from the guide rail on the XOZ plane. Then, according to the static force equilibrium conditions for our separation mechanism system, the following equations can be established: where u is the friction coefficient between the sliding pair and the guide rail, F fi represents the corresponding friction forces of F i , and F fi = uF i , where i = 1, 2, 3, 4, 11, 22, 33, 44.  To improve the rigidity of the separation mechanism, we design two parallel guide rails as additional supports. However, these rails give redundant constraints and result in a statically indeterminate state of the whole structure. Therefore, according to the deformation compatibility condition, we may obtain: Using equations (2) and (3), F i can be calculated. The opening and closing motions of the separation mechanism are controlled by the translational hydraulic actuator, which enable the end-effector to transform between the tonging-shearing and grasping forms. The premise of a nature separation mechanism is that it can be normally opened and closed, that is, the translational hydraulic actuator must be able to overcome the friction forces of the sliding pair so as to control the tonging and grasping motions of the end-effector. Therefore, F t must be restricted as follows: When the mechanical structure parameters h 1 , D, d, and dp satisfy equation (4), the driving force required by the translational hydraulic actuator, F t0 , should meet the condition that F t0 .F t , and thus the seizure motion may be avoided.

Force analysis of end-effector during tonging-shearing
For the separation mechanism, the force required to shear a steel bar is much greater than that required for grasping it. Therefore, the shearing operation case should be especially noted and carefully analyzed. Now, supposed that the separation mechanism remains at a closed state when the end-effector is performing the tonging-shearing operation. The force analysis of the end-effector at this state is shown in Figure 4. We can then deduce to obtain that: where F m is the driving force of the main hydraulic actuator, F s is the force that reacts on the cutting edge during tonging-shearing, d s is the cutting edge length, l 5 is the length of linkage, 3 c is the angle between the main hydraulic actuator and the clamp, and c = a tan 0: Þ . The shearing force varies with c. When the piston of main hydraulic actuator protrudes to the furthest position, the clamping state becomes a vertical type and c attains a minimum by now, and the shearing force F s at the cutting edge also reaches a minimum. In this case, F s should meet the design requirements. Further, a force analysis of the cutting edge for end-effector in the vertical clamping state is shown in Figure 5.
Based on the requirement of static equilibrium conditions, the following equations can be established: where f is the angle between F c and the X-axis. By solving equation (6), we may obtain: Force analysis of separation mechanism under selflocking For the concerns of easy and safe manipulations, it is necessary that the separation mechanism must be kept at a closed state when the end-effector is performing the tonging-shearing operation. Specifically, our designed mechanical structure of separation mechanism with well-defined shapes and dimensions may exhibit selflock state automatically without additional supply of any driving force from the translational hydraulic actuator. This self-lock state as well as the force analysis of the whole separation mechanism under the tongshearing operation are shown in Figure 6. Similarly, the geometric design of cutting edge with distance h 2 between point A and point O in the Z-axis direction may cause the separation mechanism to have a clockwise tilt around the X-axis on the XOY plane during motions. This means that the two sliding pairs are subjected to the forces F 5 and F 6 coming from the guide rails on the XOY plane. Then, we can obtain: where F 0 c is the reaction force of the separation mechanism when subjected to the action of the clamping at point A, and F f 5 , F f 6 , and F fn1 are the friction forces due to F 5 , F 6 , and F n1 , respectively.
By solving equation (10), we can obtain: Therefore, the following condition must be met to enable the separation mechanism to achieve a selflocking state: Furthermore, this condition can be rewritten as: Therefore, when the mechanical structure parameters h 1 , h 2 , d, and f satisfy equation (13), the separation mechanism may automatically reach the self-locking state when the end-effector performs the tongingshearing operation.
Failure cause and failure mechanism analysis of separation mechanism As previously stated, when we need to transform the end-effector between the tonging-shearing and grasping forms, the separation mechanisms must perform corresponding actions in advance. By synthesizing the analysis results obtained in Section 2, we propose three possible failure modes for the separation mechanism: self-locking failure, component failure, and motion seizure failure, as described in Table 1.

Influencing factors of self-locking failure
According to the discussions in Section 2.4, when the end-effector is performing the tonging-shearing operation, the separation mechanism must be at the closed state, the static friction force is certainly greater than the reaction force. Therefore, the self-locking failure criterion can be defined as: Furthermore, this criterion can be rewritten as: By combining the working principle of the endeffector with the force analysis of the separating mechanism, we are able to conclude that the main factors influencing the self-locking failure are the friction (parameter: friction coefficient u) and the dimension errors of component shape (parameters: h 1 , h 2 , d, and l 5 ) from manufacturing.

Influencing factors of component failure
According to Section 2.3, as the guide rail is subjected to large forces of F 1 , F 2 , and F n1 , where F 1 and F 2 are equal and act in opposite directions, its maximum normal bending stress can be expressed as: where R is the guide rail radius, L is the distance between the force acting on the guide rail and the two ends of the rails, and I Z is the moment of inertia of the cross-section with respect to the central axis of the guide rail. Therefore, the failure criterion of the guide rail can be defined as: where d p is the allowable normal bending stress of the material. Similar to the manipulation procedure in Section 3.1, the main factors influencing the component failure of the separating mechanism can obtained as the friction coefficient u between the sliding pair and the guide rail, the external force F s , and the dimension errors of component shape (parameters: h 2 , d, ds, and l 5 ) from manufacturing.

Influencing factors of motion seizure failure
When the end-effector transform between the tongingshearing and grasping forms, the translational hydraulic actuator must provide enough driving forces to assure the opening or closing motions of the separating mechanism. Therefore, the motion seizure failure criterion for the separation mechanism can be expressed as the difference between the maximum driving force provided by the translational hydraulic actuator and the maximum theoretical driving force according to the law of motion, that is: where F t0 is the maximum driving force provided by the translational hydraulic actuator, and F t is the maximum theoretical driving force required according to the law of motion which can be calculated using equation (4). Accordingly, the main factors influencing the motion seizure failure of the separating mechanism are the dimension errors of component shape from manufacturing, including h 1 , D, d, and dp.

Reliability analysis based on mixed copula
Mixed copula model Different copula functions have different tail correlations. Because the correlations in different failure modes for the mechanism system are nonlinear and complex, a single copula function cannot fully describe them. Additionally, the Clayton copula is dependent on the lower tail of variables, but the Gumbel copula is dependent on the upper tail of variables. The expressions for these two copulas are respectively expressed as follows: where u 1 2 0, ' ð Þ and u 2 2 1, ' ½ Þ. We synthesize merits of these two copulas functions to construct a mixed Table 1. Failure modes for separation mechanism.

Failure mode
Factors and criterion of failure Self-locking failure The force of static friction is less than the reaction force during tonging-shearing Component failure The contact stress between the sliding pair and the guide rail during tonging-shearing is too large Motion seizure failure Insufficient driving force (large resistance due to wear and deformation of components) copula to deal with the tail dependence among random variables and the multi-failure modes for the mechanism system. The new proposed mixed copula is: where v is the weight parameter, v 2 0, 1 ð Þ. Accordingly, the reliability of a mechanism system that has k failure modes can be expressed using the above mixed copula model as: where C ð Þ is the mixed copula, and P j is the failure probability of the j-th failure mode.

Parameter estimation in mixed copula
Past studies often adopted empirical theories to calculate the failure probability in copula functions. However, in engineering applications, this approximation may have great deviation due to manufacturing and measurement errors for the dimension parameters of mechanism. Instead of the traditional empiricaltheory approximation, we adopt the non-parametric kernel density estimation (NKDE) algorithm to estimate the failure density function (PDF) and the cumulative distribution function (CDF) of each performance function accurately, which is one of the most wellknown data fitting algorithms. 40 The NDKE algorithm is entirely based on historical data and no necessary to preset any distribution of variables.
Assuming that the concerned separation mechanism has k failure modes, the PDF and the CDF of the i-th failure mode are expressed as f Z i Z i ð Þ and F Z i Z i ð Þ, respectively. The sequence values Z 0 i = Z i1 , Z i2 , Á Á Á , Z in f gof the performance function Z i can then be calculated based on the sequence values x 1j ,x 2j , ÁÁÁ,x mj À Á , j=1,2,ÁÁÁ,n of the random variables X = x 1 ,x 2 , ÁÁÁ,x m ð Þ . The PDF f Z i Z i ð Þ based on the NKDE is then given as follows 40 : where n is the sample size, b i is the bandwidth of the NKDE, K Z i ÀZ ij b i is the selected kernel density func- Á . It can be found from equation (22) that when the number of samples remains constant, the accuracy of f Z î Z i ð Þ is dependent on the kernel density function and the bandwidth. Because the Gaussian function has good smoothness and differentiability, it is therefore selected as the kernel density function as follows: In addition, the bandwidth is optimized using the method proposed in the literature, 41 and the mathematical model can then be expressed as follows: Therefore, the kernel density distribution function F Z i Z i ð Þ of the performance function Z i can be calculated as follows: where b 0 i is the optimal bandwidth. The failure probability of each failure mode can then be estimated using Next, we are going to use the time-variant correlation parameters to describe the changes of correlations in different failure modes for the separation mechanism. A time-variant parameter estimation method is proposed to establish the time-variant mixed copula model to reflect the time-variant reliability of the separation mechanism, as illustrated in Figure 7. The detailed steps of this method are given as follows: Step 1: Determining the total motion cycle N, the initial sample size n, the initial sampling cycle t, and the sampling interval cycle k.
Step 2: Sampling the data of random variables. Then, calculate the sequence values of performance functions.
Step 3: Using NKDE technique to estimate PDFs and CDFs of the performance functions. Calculate the failure probability p fZ i for each failure mode.
Step 5: Performing the maximum likelihood estimation. Assuming that Z it is the performance function i of the t-th cycle and F it (Z it ) and f it (Z it ) are the marginal distribution function and the marginal density function, respectively. Then, the joint distribution function for the variables is c t F 1t z 1t ð Þ, ð F 2t z 2t ð Þ, Á Á Á , F kt z kt ð Þ; u 1 , u 2 , vÞ and the corresponding joint density function can be expressed as: The likelihood function of u 1 , u 2 , v ð Þ can be expressed as: where z 1tj , z 2tj , Á Á Á , z ktj À Á , j = 1, 2, Á Á Á , n. The likelihood function can then be rewritten as: The maximums of the likelihood function can then be obtained as: Step 6: Using the minimum Euclidean distance method (d 2 ) 42 to test the fitting effect. If d 2 ł 0:05, the program then jumps to Step 7; otherwise, let n = n + 100, the program then return to Step 2, and the procedure is repeated until the requirements are met.
Step 7: Establishing a mixed copula model based on the obtained parameters u 1 , u 2 , v ð Þ , and then calculating the reliability of the current motion cycle t using equation (21).
Step 8: Stopping criteria. If t ø N , then the timevariant reliability is fitted via the least-square method with a polynomial function; otherwise, let t = t + k, the program returns to Step 2 and the procedure will be repeated until the above requirements are met.

Testing of the proposed method
A test example with two failure modes and four factors is performed to test the performance of the proposed algorithm. Here, x 2 is a time-variant factor. The type of random variable and its distribution parameters are listed in Table 2. The performance functions of the failure modes are given as follows: Table 3 and Figure 8 show the comparison results among the methods of Clayton copula, Gumbel copula and mixed copula for the test example, in which they all use the same amount of samples. The results obtained by the MC method are used as the comparison base to calculate the errors. The time interval t of x 2 is set within the range of 1-8.
It is found from Figure 8 that both the correlations between the two failure modes and the failure probability of system change with x 2 . Further, the weight coefficient also changes with x 2 , as indicated in Table 3. The result obtained via the mixed copula is much closer to that obtained by MC than those obtained by other Table 2. Type of random variable and its distribution parameters.

Mean
Standard deviation Distribution type  Figure 9 shows the CDFs of the performance function with the minimum and the maximum sample sizes that happen at the time of t = 2 and t = 5, respectively. It is clear that the theoretical CDF almost overlaps with the NKTD-based CDF. This indicates that the NKDE technique can be used to estimate the distribution of the performance function effectively, and the failure probability of each failure mode can be calculated using the estimated CDF. For further verification, the parameter of Akaike information criterion (AIC) and the minimum Euclidean distance method (d 2 ) 42 are used to test the fitting effects of different copula models. Smaller values of AIC and d 2 indicate better fitting effects for the algorithm. Under the constraint of the same sample size, Table 3 shows that the mixed copula has smaller values for both AIC and d 2 than those of Clayton copula and Gumbel copula. This implies that the mixed copula provides a better fitting effect. More importantly, the results obtained by the mixed copula with a small sample size show a difference of less than 5% when compared with the results obtained via the MC method with a large sample size. This illustrates that the proposed mixed copula method is not only enough accurate but also very effective.

Reliability analysis of separation mechanism
The rescue end-effector was designed by our research group of Wang et al., 3 as shown in Figure 10. The weights of the grasping and tonging-shearing mechanism are 63.3 and 30 kg, respectively. The value of G can therefore be calculated as: According to the design requirements of rescue equipment, the end-effector should be able to cut an HRB400 steel bar with a diameter of 50 mm. The compressive strength of the steel bar ( t ½ ) is 360 MPa, and its shear strength coefficient (l) is 0.8. Therefore, the maximum shear force F s required by the rescue endeffector is: where A is the cross-sectional area of the steel bar. The rescue end-effector was made of 16Mn steel 3 with a yielding stress of d p = 510MPa. When the rescue end-effector is in working, the possible driving force F t0 provided by the translational hydraulic actuator is 2900 N. According to the analysis in Section 3, the main factors influencing the three failure modes are u, D, d, dp, h 1 , h 2 , l 5 , d s , F s , R, and L. Therefore, these parameters can be regarded as random variables in the reliability analysis of the separation mechanism. In addition, the distribution type of each random variable is shown in Table 4. The performance functions for the three failure modes can be rewritten as: Based on the parameters listed in Table 4, the failure probability for each failure mode (P fi MC ), the independent failure probability for each failure mode (P fin ), and the failure probability when considering the correlation among the failure modes (P f MC ) are calculated using the MC method with 10 6 samples. In addition, the failure probabilities P f Clayton , P f Gumbel , and P f mixed are calculated using the Clayton copula, the Gumbel copula, and the proposed method, respectively. In this section, only the original reliability of the separation mechanism is calculated, that is, the total motion cycle N = 0, the initial sampling cycle t = 0, and the sampling interval cycle k = 0. The results are shown in Table 5. The CDFs of the performance functions based on NKDE are shown in Figure 11. These results show that the failure probability in the case without considering the failure correlation is higher than that with considering the failure correlation. This is because the correlation among the failure modes was neglected during analysis and this led to both high failure probability and low reliability. The failure probabilities of three proposed failure modes based on NKDE are very close   to that obtained from MC method. The results also show that the calculations based on the proposed mixed copula model, which adopted a small amount of sample are consistent with those using MC method, which adopted a large amount of samples. Obviously, the proposed mixed copula method has a smaller error level than both the Clayton copula and Gumbel copula methods. The superiority of effectiveness and efficiency of the proposed copula method are thus exhibited. In addition, the component failure mode has a comparatively high failure probability which results in a high failure probability for the separation mechanism. This is because both the guide rail and sliding pair are made of 16Mn steel. For enhancing the structure strength, we adopt 40CrNiMoA steel as the new material of guide rail and sliding pair, which has a yielding stress as high as t ½ = 980Mpa. Now we recalculate the failure probability based on this new material and find that the failure probabilities of the component and separation mechanism are reduced to 2.621e 25 and 7.661e 22 , respectively.
The above analysis indicates that the main failure modes for the improved separation mechanism are selflock failure and motion seizure failure. The PDF and CDF of the mixed copula model for the separation mechanism with new material are shown in Figure  12(a) and (b). These figures show that the Gumbel copula model has a larger weighting coefficient than other models, and the two failure modes both have strong upper as well as lower tail dependence. In addition, P f mixed = 7:572eÀ2, and the error with respect to the MC method is only 21.16%. When we calculate the failure probability of the system with enough accuracy, only few samples are needed in the proposed mixed copula model than those in the other aforementioned methods. Therefore, we will use this model to perform the time-variant reliability analysis in the following section due to its high computational efficiency.

Time-variant reliability of separation mechanism
As the number of motion cycles of the separation mechanism increases, accumulation of component damage will lead to reduction of the mechanism's reliability. Analysis of the working principle of the separation mechanism indicates that the damage mainly manifests as wear between the sliding pair and the guide rail, which can then be characterized by the friction coefficient u. In this study, the designed rescue end-effector is used as the experimental platform to analyze the progression of damage to the components of separation mechanism (shown in Figure 10). Furthermore, the relationship between the friction coefficients caused by the opening and closing motions of the separation mechanism is analyzed based on the method of data degradation. The opening and closing motions of the separation mechanism are controlled using a translational hydraulic actuator. The driving force F t of the translational hydraulic actuator is calculated based on the measured pressure. Combing with the results in Section 2.2, the friction coefficient between the moving pair and guide rail can be expressed as: The friction coefficient in any opening and closing cycle can thus be obtained using curve-fitting with the least-square method. The experimental results show that the mean value of the friction coefficient is 0.2 + 5 3 10 2 5n and the standard deviation is 0.02 + 10 2 6 n, where n is the current motion cycle. The MC method and the proposed mixed copula method were both used to calculate the reliability of the separation mechanism at each interval of k cycles. In addition, the total motion cycle N = 8000, the initial sampling cycle t = 0, the sampling interval cycle k = 10, and the distributions of the other parameters are the same as those listed in Table 4. The subsequent fitting of the results to obtain the evolution of the reliability with respect to the motion cycle is shown in Figure 13.
The above analysis indicates that the proposed mixed copula method provides a satisfactory prediction of the reliability of the mechanism for time-variant motion service. Further, it is found that, to reach the required accuracy, it takes about 10 6 samples in MC method and a minimum of 800 samples, a maximum of 1800 samples, and an average of 1370 samples in our proposed mixed copula method. And, the reliability of the separation mechanism initially increases and then decreases with the motion cycle, and the highest reliability of 0.9654 occurs when the motion cycle number is 1600. Apparently, our proposed copula method is more efficient and effective than other methods in analyzing the time-variant reliability of separation mechanism. In addition, we may obtain the relationship between the reliability of these two failure modes and the motion cycle is obtained through calculations, as shown in Figure 14. When the motion cycle number is less than 1600, the self-locking failure mode dominates the motion, which causes the reliability increases rapidly with increasing service time. On the other hand, when the motion cycle number is greater than 1600, the motion seizure failure mode dominates the motion, which causes the reliability decreases slowly with increasing service time.
In addition, the results also show that when the motion cycle number attains 4100, the reliability of the self-locking failure mode is 0.9921 and the reliability of the motion seizure failure mode is 0.8105. When the motion cycle number attains between 1600 and 4100, the reliability of the self-locking failure mode increases to 0.9921, which is very close to 1, while the reliability of the motion seizure failure mode decreases to 0.8105. Combination of these results with data shown in Figure 14 shows that the reliability of the separation mechanism decreases slowly during this period (motion cycle :1600-4100), but when the motion cycle number exceeds 4100, the reliability of the separation mechanism decreases rapidly.
In a whole sense, as the motion cycle exceeds 1600, a drastically increase of resistance as well as the failure probability of motion seizure occurs. At this time, the failure of the separation mechanism is dominated by the motion seizure. Therefore, during this period of service for the rescue end-effector, the guide rail surface should be kept clean, and all related lubrication works should be carefully done. If the guide rail is severely worn or deformed, it should then be replaced as quickly as possible to improve the reliability of the rescue endeffector.

Conclusions
In this paper, initially we analyze the working principle of a rescue end-effector and performed a force analysis of its separation mechanism. Then, we analyze the failure causes and failure mechanisms of the separation mechanism and identified three main failure modes. Specifically, we propose an effective reliability analysis method with multiple failure modes which adopts a novel mixed copula function and develops different failure correlations. In this novel method, the PDF and CDF of each performance function are estimated by NKDE. Furthermore, a time-variant parameter estimation method is proposed so as to establish a timevariant mixed copula function, which may successfully be applied to predict the time-variant reliability of the separation mechanism of multi-functional rescue endeffectors. Verification results demonstrate that our proposed method can not only be used to analyze the reliability for multiple failure modes, but also may acquire satisfying results using few samples, which are very close to the results obtained by the MC method using a large amount of samples.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.