Parameter sensitivity analysis of a 5-DoF parallel manipulator

Robotics and Computer–Integrated Manufacturing 46 (2017) 1–14

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Parameter sensitivity analysis of a 5-DoF parallel manipulator ⁎

Binbin Lian, Tao Sun , Yimin Song

crossmark

Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, Tianjin University, Tianjin 300072, China

A R T I C L E I N F O

A BS T RAC T

Keywords: Parallel manipulator Parameter sensitivity Response surface method Performance reliability

With the capability of determining main/subordinate parameters, parameter sensitivity analysis plays an important role in eliminating unimportant parameters and simplifying performance analysis and optimization model. Taking a 5 degree-of-freedom parallel manipulator (T5 PM) as an example, the effects of joint stiffness/ compliance coefficients and parameters of cross section to the mass and stiffness performance are investigated through parameter sensitivity analysis based on response surface method and performance reliability. By selecting experimental strategy and implementing accuracy assessment, the response surface method is adopted to establish the mapping model of parameters and performance with high efficiency and accuracy. In the light of reliability sensitivity, the performance reliability to parameter mean value and variance are simultaneously considered by the global parameter sensitivity index, which is the principle for determining the impact extent of parameters. Moreover, how the parameters affect the targeted performance can be evaluated through RSAV, RSPC and RSNC defined by the performance reliability to the parameter mean value. After verifying the sensitivity analysis approach by SolidWorks simulation, the parameter discussion of T5 PM is carried out. 15 parameters are selected from the original 39 parameters and effects of these parameters are clearly demonstrated, which provide reference for the future optimization process.

1. Introduction In the parallel manipulator (PM) research community, the analysis and optimization of parameters with respect to manipulator performance (for instance, kinematic, stiffness, dynamic or accuracy performance), is the investible process from certain topology structure to the PMs satisfying specific engineering requirements [1–5]. One of the difficulties for this process is the sophisticated calculation due to the large numbers of parameters coming from the complex composition and irregular structures of PMs. Especially for the performance optimization, the substantial parameters make the optimal scheme become complicated, sometimes might lead to fail execution. In addition, the lack of clear understanding between parameters and performance lower the efficiency and accuracy of the optimal calculation dramatically. What's more, the modification of manipulator prototype according to the optimized parameters is difficult even if the large quantity of parameters is obtained. In order to find out the scientific method to simplify the performance analysis or optimization model, the parameter sensitivity analysis becomes necessary as it focuses on the relationship between concerned parameters and targeted performance [6–8]. In general, the parameter sensitivity analysis is the technique to understand how the change of variables will impact the output [9]. As

⁎

far as the parameter sensitivity of PMs is taken into account, it is found that literatures available at hand are mainly about the kinematic/ accuracy sensitivity, which can be roughly divided into two categories: analytical approach and probabilistic approach. Noted that the differentiation of closed-loop equations represents the position/orientation errors of PMs with respect to component geometric errors, analytical approach is to compute the kinematic/ accuracy sensitivity by solving classic linear or nonlinear equations. Caro [10,11] obtained the sensitivity coefficients of 3-RPR PM and Orthoglide PM through direct numerical calculation or numerical iteration of the linear equations. Herein, R and P denote revolute joint and actuated prismatic joint. Focusing on the uncertainties of geometric errors and joint clearance, Wu and Rao [12] proposed the fuzzy error analysis for the non-linear equations of PMs and employed Newton-Raphson based interval calculation combining bisection algorithm to get accurate sensitivity coefficients. It is found that analytical approach deals with the analytical equations thus is able to carry out the deterministic analysis and obtains reliable solutions with direct and clear physical meaning [13]. However, this approach is only suitable for simple PMs since the calculation procedure becomes difficult when large numbers of parameters are involved. In addition, it cannot take the strong coupling, highly nonlinear and uncontrollable error source into account [14].

Corresponding author. E-mail address: [email protected] (T. Sun).

http://dx.doi.org/10.1016/j.rcim.2016.11.001 Received 28 December 2015; Received in revised form 21 October 2016; Accepted 10 November 2016 0736-5845/ © 2016 Elsevier Ltd. All rights reserved.

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Fig. 1. Experimental Strategies for response surface method (blue – experimental point, red – central point, green – factorial point) (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.).

parameters influence parameters. Moreover, in order to further ensure the correctness of the parameter sensitivity analysis, the performance reliability is introduced to study the probability of the PMs achieving expected performance under certain parameters [28–31]. In this way, the statistic characteristics in terms of mean value and variance are taken into account in the parameter sensitivity, which is more accurate and practical. Composing by redundant substructure I and overconstrained substructure II articulated by interlinked R joints, a 5-DoF PM (named as T5 PM) was proposed for the purpose of high-precision machining of aircraft components with large scale, thin wall and complex surface [32]. The stiffness optimization is the key concern for the design of T5 PM. As mentioned above, the parameter sensitivity analysis for a complicated system as T5 PM is of vital importance. Therefore, taking T5 PM as an example, the parameter sensitivity analysis is illustrated by establishing performance functions in light of response surface method and carrying out parameter reliability sensitivity of performance on the basis of stiffness performance reliability. The reminder of this paper is organized as follow: Section 2 introduces the response surface method. In Section 3, a global sensitivity index is proposed considering both parameter mean value and variance, and three criteria are defined for evaluating the effects of parameters based on performance reliability. As a case study, Section 4 carries out parameter sensitivity analysis of T5 PM with respect to the stiffness performance. After verifying the proposed approach by SolidWorks simulation, parameter discussion is carried out in Section 5. Finally, conclusions are drawn in Section 6.

As the mapping operator between joint space and operated space, the mathematic features of Jacobian matrix have been widely adopted in the kinematic/accuracy analysis of PMs [15,16]. Based on the sensitivity indices defined by Jacobian matrix, the geometric errors and joint clearances are regarded as stochastic phenomenon and corresponding statistic regularities of parameter sensitivity are analyzed in probabilistic approach. Xu [17] carried out the Monte-Carlo simulation for kinematic error analysis of spatial linkage by probability density function of joint clearance. Based on the uncompensatable pose error, Huang [18], Sun [19] and Chen [20] formulated the probabilistic error models with normal distribution, and then accomplished the accuracy sensitivity of 2DoF overconstrained PM, 3-PRS PM and SCARA mechanism, respectively. Herein, S denotes the spherical joint. Besides the probabilistic distribution, Sultan [21] utilized a stochastic-based technique to identify the parameters. Chaker [22] investigated the stochastic results of moving platform errors caused by manufacturing errors for a class of spherical PMs. On the whole, the probabilistic approach is able to build a pseudoerror sensitivity model that has a competitive computational time while being simple and general [23]. Inspired by kinematic/accuracy sensitivity analysis, it is concluded that the probabilistic approach is more suitable for simplifying parameter analysis or optimization of complicated PMs, as its capability of considering large numbers and different types of parameters in highly efficient manner with acceptable accuracy. In order to compute parameter sensitivity probabilistic distribution, Monte-Carlo simulation and response surface method are the most common and effective methods adopted so far. By describing or establishing the probabilistic distribution, Monte-Carlo simulation uses repeated sampling to obtain numerical results, which is believed to be precise in terms of simulating the estimated value [23–25]. However, it is found in the kinematic/ accuracy sensitivity analysis that Monte-Carlo simulation analyzes the impact of each parameter separately without considering the coupling effect of these parameters. In addition, Monte-Carlo simulation can only reveal the impact extend of parameters but fail to provide the information of how the parameter affect the targeted performance. Proposed by Box and Wilson in 1951, the response surface method is to employ a sequence of deterministic experiments to establish a polynomial surface function for mapping the relationships between input variables and one or more output response [26–28]. With the explicit mapping model of concerned parameters to the targeted parameters, the response surface method takes the nonlinear feature and coupling effect of parameters into account, thus it is a promising solution for analyzing the effect degree of parameters and understanding how the

2. Response surface method Based on design of experiment, the core idea of the response surface method is the match of mathematical models to experimental results and the verification of the model by statistical techniques [33]. Targeted for the structural parameters to stiffness or dynamic performance, the approximate model formulated by the response surface method can be implemented as: 1) Determine parameters and performance response, 2) Select experimental design strategy and execute designed experiments, 3) Obtain response model and assess its accuracy. In the presented process, selection of experimental strategy and assessment of the obtained model are the key steps. For the experimental design strategy, several methods shown in Fig. 1 have been successfully applied to the manufacturing: Full or fractional factorial design (FFD), Central composite design (CCD), Box2

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3. Parameter sensitivity indices

Behnken design (BBD) and Latin hypercube design (LHD). Among them, FFD studies all the factors at all p levels, and it is capable of evaluating interaction effects clearly by calculating all possible combinations of variables. But FFD becomes computational intensive when solving second or higher order polynomial model. Hence, the fractional factorial design, including the famous orthogonal array assuming parameters independence is applied to the design of experiments. Correspondingly, CCD has as much information as the three-level FFD with fewer experiments. But for the two-level experiment, CCD contains 2k FFD, 2k axial points and one center point, which would significantly increase numbers of experiments when large numbers of analyzed parameters are involved. Without sequential experiments, BBD requires fewer designed points than CCD and is more suitable for parameters with non-linear features. However, its drawback lies in that BBD can only be applied to quadratic model. Comparatively, LHD allows random combination of experimental sampling and the historical design experiments that can be inherited in later iterations. As a result, the limited number of design experiments is required even for high-dimensional problems. But most previous points will keep to the next iteration without adding new points if the cutting is too conservative. According to the features of the analyzed parameters and concerned performance, designers can choose any of the above-mentioned experimental strategy. There has been software that provides simple and clear use of these methods, for instance, Design Expert (Stat – Ease Inc.), Minitab (Minitab Inc.), Matlab (MathWorks), Isight (Dassault Systeme). After determining the experimental strategy, the response surface model can be built with N experiments. Mathematically, the linear and quadratic model can be formulate as t

f (X ) = a +

t

∑ bi xi ,

f (X ) = a +

i =1

t

t

For the sake of comprehensively evaluating parameter sensitivity of PMs, the targeted performance to the parameter mean value and variance is considered, which can be obtained by the performance reliability that describes the probability of the PMs achieving desired performance. The definition of sensitivity indicates that parameter sensitivity can be expressed as the differentiation of the function to the variables, based on which, the reliability sensitivity is able to calculate by the differentiation of performance reliability to the parameter mean value and variance. To achieve that, the formulation of performance reliability is brought in as

R=

i =1

g (X ) = fU − f (X ) or g (X ) = f (X ) − fL

t

i =1 i < j

(1)

f (X ) = a +

t

∑ bi xi + ∑ i =1

i =1

t

ci xi2

+

t

E (X ) = [ μ1 μ2 ⋯ μn ]T = μ, Cov (X ) = diag ( σ12 σ22 ⋯ σn2 )

i =1 i < j

g (X ) = g (μ ) +

ei xi3

m

RMAE =

∑i =1 (yi − yˆi )2 m ∑i =1

(yi −

y )2

∑i =1 yi − yˆi m ∑i =1

yi − y

max{ yi − yˆ1 , ⋯, yi − yˆm } , RMSE = m ∑i =1 yi − y / m

X=μ

where

∂g (X ) ∂XT

X=μ

⎡ ∂g (X ) = ⎢ ∂x ⎣ 1

H (g (X )) 2!

(X − μ ) T (X − μ ) X=μ

∂g (X ) ∂x 2

⎤ ⋯ ∂g (X ) ⎥ , and H (g (X )) is the ∂xn ⎦ X=μ

Hessian matrix. Then the mean value and variance of the state function g (X ) is calculated as

μg = Eg (X ) = g (μ) +

m

, RAAE =

(X − μ ) +

(7)

Before determining the suitable response surface model, the error analysis is required to evaluate the accuracy of the obtained models. Thus, additional m sampling points are selected for the accuracy assessments. According to [34], four metrics, i.e. R Square (RS), Relative Average Absolute Error (RAAE), Relative Maximum Absolute Error (RMAE), Root Mean Square Error (RMSE), are applied for this purpose. Let y = f (X ), the evaluating metrics are expressed as

RS = 1 −

∂g (X ) ∂X T

(2)

i =1

n

1 2

∑ σ j2 j =1

∂ 2g (X ) ∂xj2

, n

m

∑i =1 (yi − y )2 m

(6)

Based on Taylor expansion, the state function at the point X = μ is expressed as

t

∑ ∑ dij xi xj + ∑

(5)

where fU , fL denote the upper and lower performance limits from the engineering requirements. If both upper and lower limits are required, two state functions shown in Eq. (5) would be formulated simultaneously with respect to the same performance function. When g (X ) ≤ 0 , the PM system is at failure state, When , it is at safe state. As shown in [35], the reliability sensitivity can be achieved by the differentiation of composite function formulated by the mean value and variance of g (X ). Assume μj , σ j2 are the mean value and variance of xj (xj ∈ X , j = 1, 2, ⋯, n ), the mean value vector and covariance matrix of X can be computed.

where X = ( x1 x2 ⋯ xt )T is the design variables. The coefficients a , bi , ci , dij are obtained from estimated regression resorting to the least square method. xi xj denotes two-parameter interactions and xi2 represents the second order nonlinearity of . Similarly, by adding cubic term to the quadratic model, the third order response surface model is expressed as t

(4)

herein, R is the performance reliability, g (X ) denotes performance state function, p (X ) represents the probabilistic density function of X . The state function shows the safe or failure state of the PM system, which can be defined by the performance function obtained by the response surface method.

∑ bi xi + ∑ ci xi2 + ∑ ∑ dij xi xj i =1

∫g (X ) > 0 p (X ) dX

σg2 = E (g (X ))2 − μg2 =

⎧ ⎪⎡

⎤2

∑ σ j2 ⎨ ⎢ ∂g (μ) ⎥ j =1

(3)

⎩ ⎣ ∂xj ⎦ ⎪

(8)

X=μ

− g (μ )

⎤2

where yˆi denotes the result from response surface model, yi is the exact value of ith sampling point and y represents the corresponding mean value. It is known from Eq. (3) that smaller RAAE, RMAE, RMSE and larger RS indicate higher accuracy of the response surface model. Among them, RS, RAAE, RMSE evaluate the overall performance in the design space while RMAE focuses on the maximum error in one region even the other metrics show good global behavior. With the consideration of both global performance and worst cases from the four metrics, the appropriate surrogate response surface function can be achieved.

−

where

⎡ n ∂ 2g (μ) 1⎢ ∑ σ j2 2 ⎥⎥ ⎢ 4 ⎣ j =1 ∂xj ⎦

⎫ ∂ 2g (μ) ⎪ ⎬ 2 ⎪ ∂xj ⎭

∂g (μ) ∂xj

=

∂g (X ) ∂xj

, X=μ

∂ 2g (μ) ∂x j2

(9)

=

∂ 2g (X ) ∂x j2

. X=μ

To further simplify the calculation of reliability sensitivity, let β = μg / σg denote the reliability index, the equivalent random variable Y = [g (X ) − μz ]/ σz is introduced to the reliability equation. Y is a random variable subject to standard normal distribution, and Y ~N (0, 1). Hence, Eq. (4) can be rewritten as 3

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1 2π

R = P {g (X ) > 0} = 1 − P {Y ≤ −β} = Φ (β ) =

β

t2

∫−∞ e− 2 dt

3) Reliability Sensitivity Negative Correlation (RSNC). The minus sign of κj stands for the state function reduces successively as the increasing of the RSAV. If upper limit is provided from the engineering, the aimed performance increases as the increment of parameter with RSNC. If lower limit is considered, the aimed performance decreases when the parameter with RSNC increases.

(10)

Therefore, the reliability sensitivity with respect to parameter mean value is derived by the differentiation of reliability index, then mean value of the state function and mean value of parameter.

⎧ ⎡ n dR ∂β ∂μg ∂R 1 dR ⎪ ∂g (μ) 1 ∂ ⎢ ∂ 2g (X ) ⎨ = = + ∑ σ j2 dβ ∂μg ∂μ σg dβ ⎪ ∂μ1 ∂μ 2 j =1 ∂μ1 ⎢ ∂xj2 ⎣ ⎩ ⎡ 2 ⎤ n ∂g (μ) 1 ∂ ⎢ ∂ g (X ) ⎥, ⋯, , + ∑ σ j2 ⎥ ∂μ2 2 j =1 ∂μ2 ⎢ ∂xj2 ⎣ X=μ ⎦ ∂g (μ) 1 + ∂μn 2

n

∑ σ j2 j =1

⎡ ∂ ⎢ ∂ 2g (X ) ∂μn ⎢ ∂xj2 ⎣

⎤ ⎥ ⎥ X=μ ⎦

It is found that through the criteria defined above, not only the main/ subordinate parameters can be distinguished by RSAV, but also how the parameters would affect the performance can be assessed by RSPC or RSNC of the mean value. Note that the existing PM sensitivity analysis approaches merely determine the extent of impact from independent or non-coupled parameters to performance, the proposed method based on performance reliability enables designers to thoroughly and explicitly understand the relationship between parameters and performance, which would provide reference for the optimization process.

⎤ ⎫T ⎥⎪ ⎬ ⎥⎪ X=μ ⎦ ⎭

β2

≈

T ∂g (μ) ⎤ e− 2 ⎡ ∂g (μ) ∂g (μ) ⎢ ⎥ , , ⋯, ∂μ2 ∂μn ⎦ σg 2π ⎣ ∂μ1

(11)

4. Parameter sensitivity of T5 PM

Similarly, the reliability sensitivity to parameter variance matrix is computed by the differentiation of reliability index, variance of the state function and variance of parameter.

On the basis of the response surface model and reliability sensitivity, T5 PM is taken as an example to demonstrate the parameter sensitivity analysis with respect to the static performance, which can be implemented as: 1) Determine static performance indices as objectives and relating parameters as variables, 2) Establish response surface models and evaluate the accuracy of the obtained models with R2 , RAAE, RMAE and RMSE, 3) Obtain parameter sensitivity by computing global sensitivity index, RSAV, RSPC or RSNC of parameter mean value.

⎡⎡ 2 μg dR ∂σg2 ∂g (μ) ⎤ ∂ 2g (μ) dR ∂β ∂R diag ⎢ ⎢ = =− 3 ⎥ − g (μ ) 2 ⎢⎣ ⎣ ∂x1 ⎦ dβ ∂σg ∂Cov (X ) ∂Cov (X ) 2σg dβ ∂x12 −

1 ∂ 2g (μ) 2 ∂x12

1 ∂ 2g (μ) − 2 ∂x 22

n

⎤2

2

⎡

∂xj

⎣ ∂x2 ⎦

∑ σ j2 ∂ g (2μ) ,⎢ ∂g (μ) ⎥ j =1 n

2

j =1

∂xj

− g (μ )

∑ σ j2 ∂ g (2μ) , ⋯,

⎡ ∂g (μ) ⎤2 ∂ 2g (μ) 1 ∂ 2g (μ) − ⎥ − g (μ ) ⎢ 2 ⎣ ∂xn ⎦ 2 ∂xn2 ∂xn =−

μg 2σg3

β2 e− 2

2π

∂ 2g (μ) ∂x 22

⎡ ∂σ 2 ⎤ g diag ⎢ 2 ⎥ ⎢⎣ ∂σ j ⎥⎦

n

∑ j =1

σ j2

4.1. Determination of objectives and variables

⎤ ∂ 2g (μ) ⎥ ∂xj2 ⎥⎦

Note that the instantaneous energy [36], which is converted from the virtual work calculated by the product of instantaneous payload and the corresponding instantaneous deformations, has unified physical unit thus can deal with both linear and angular stiffness. And it is able to evaluate overall and worst-case static performance within prescribed workspace. Hence, the stiffness index based on instantaneous energy index is adopted for assessing stiffness performance of T5 PM. Moreover, considering the horizontal layout of T5 PM, the mass of the execution part should be taken into account. T5 PM is composed of redundant substructure I and over-constrained substructure II connected by interlinked R joints. Analyzing the semi-analytical stiffness model of T5 PM [37], it is found that the compliance of end reference point can be achieved by the superposition of the compliance models of the two substructures after evaluating the impacts of interlinked R joints to them. Therefore, the parameter sensitivity analysis relating to static performance of T5 PM can be evaluated from two subsystems, subsystem I for redundant substructure I with the consideration of interlinked R joints, subsystem II for over-constrained substructure II including the effect of interlinked R joints. In conclusion, the static performance of T5 PM is determined as the mass (M ), instantaneous linear stiffness performance (ηlx , ηly , ηlz ), overall instantaneous stiffness performance (η ) of subsystem I and II, in which, the mass and overall instantaneous stiffness performance are to check the global static performance of subsystem I and II whereas the instantaneous linear stiffness performance is taken the worst-case optimization into account. As mentioned in [37], subsystem I consists of one fixed base, five UPS limbs, one UP limb, two interlinked R joints (IR), and platform I. See Fig. 2, the 1st and 2nd UPS limbs connect to platform I through IR1 joint, while the 3rd and 4th UPS limbs link to platform I by IR2 joint. The 5th UPS limb and UP limb attach to platform I directly. Thus, parameters of subsystem I can be divided into four groups: the 1st to 4th UPS limbs, IR joints, the 5th UPS limb, UP limb. According to the

(12)

The reliability sensitivity to parameter mean reveals how the parameter value impacts the performance reliability while the reliability sensitivity to parameter variance matrix indicates the effect of the change of parameter value. Considering both parameter mean value and variance, a global sensitivity index is defined in this paper for the convenience of eliminating non-significant parameters from optimization model by means of sensitivity assessment as

εj =

⎛ κj ⎞2 ⎛ δj ⎞2 ⎜ ⎟ +⎜ ⎟ , j = 1, 2, ⋯, n ⎝ κmax ⎠ ⎝ δ max ⎠

(13)

where κj = ∂R /∂μj , δj = ∂R /∂Cov (xj ), j = 1, 2, ⋯, n .κmax , δ max denote the maximum of absolute values parameter sensitivity in terms of mean value and variance. Moreover, three criteria can be proposed for evaluating the influence of parameter by the reliability sensitivity to parameter mean value as 1) Reliability Sensitivity Absolute Value (RSAV). The absolute values of κj indicates the effect degree of mean value of xi to the performance reliability. The higher absolute value, the higher sensitivity of the studied parameters to the concerned performance. 2) Reliability Sensitivity Positive Correlation (RSPC). The plus sign of κj indicates the state function increases progressively with the increment of the RSAV. If the engineering requirement provides upper limit, the targeted performance decreases as the increasing of parameter with RSPC. Correspondingly, when lower limit is offered, the targeted performance shows the same change tendency as parameter with RSPC.

4

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Fig. 2. Composition of T5 PM.

4.2. Response surface models Based on the virtual prototype built by kinematic optimization and engineering experience, the initial sectional parameters and joint stiffness coefficients of T5 PM are obtained, from which the range of variables can be settled. Let the sectional parameters change between ± 10% and stiffness/compliance coefficients vary between ± 30%. Considering the large amount of parameters and uncertain orders of approximate models, it is found that LHD is the most appropriate experimental strategy for T5 PKM as its strong potential for computation intensive problem with vast variables and its capability of generating saturated design with limited experiments. By means of Matlab software, the numerical model between structural parameters and static performance of T5 PM in the prescribed workspace can be established, which will be treated as the ‘experiments’ to achieve the response values with respect to the variables designed by experimental strategy. Therefore, with determined range of parameters, experimental strategy and experiments, response surface models are derived resorting to Isight software. The sampling points and error analyzing points change along with the number of parameters and selected orders. For subsystem I with 20 parameters, the sampling points for the first, second and third orders of response surface models are 42, 462, 502, and the error analyzing points are 21, 231, and 251, respectively. Corresponding accuracy assessments are shown in Table 1. It is found that accuracy of the response surface model for M1 increases as the developing of the orders, and the cubic response model is determined as the most appropriate model. As for ηlx,1, ηly, 1 and ηlz,1, the RS of linear and cubic response surface models cannot meet the requirements, and the RMAE of cubic models are far beyond acceptance. Thus, quadratic models are chosen. None of accuracy evaluation for linear response surface model of η1 is acceptable, and RS and RMAE of cubic model exceed accepted level. Overall, quadratic model is the most suitable response surface for η1. Similarly, for subsystem II with 19 parameters, the sampling points for linear, quadratic response surface models are 40, 420, 458, and the error analyzing points are 20, 210, and 229, respectively. The accuracy assessments of these models are shown in Table 2. As the cubic response models are with the highest accuracy for all the static performance, they are selected as the analyzing models.

Fig. 3. Structure and parameters of subsystem I.

structure of the four groups in subsystem I, all possible sectional parameters and related joint stiffness coefficient are shown in Fig. 3. For the 1st to 4th UPS limbs, ku denotes the stiffness coefficient of U joint along the direction of P joint and ds represents the screw diameter. Dop and dop are the external and internal diameters of the outer pipe, while Dip and dip are the external and internal diameters of the inner pipe. ks denotes the stiffness coefficient of S joint along the direction of P joint. For the IR joints with structure of ‘T’ letter, Dir1 and dir1 represent the external and internal diameters of the horizontal part of IR joints whereas dir1 stands for the diameter of the vertical part. For the 5th UPS limb, the concerned parameters are with the same meaning as that of the 1st to 4th UPS limbs, distinguished by extra subscript ‘5’. For UP limb, kU , kV denote the stiffness coefficients of central U joint. Dct and dct are the external and internal diameters of central tube. Altogether subsystem I has 20 parameters. As stated in [37], subsystem II is composed of parallelogram-based closed-loop I and closed-loop II. By regarding platform I from subsystem I as the fixed base, IR1 and IR2 joints are treated as the actuated joints of subsystem II. Closed-loop I consists of the 1st bracket, the 1st and 2nd rods, and the moving platform whereas closed-loop II is made up of the 2nd bracket, the 3rd and 4th rods, and plate II. The two closed-loops are articulated by one R joint. According to the structure of the closed-loop I and II in subsystem II, all possible sectional parameters and related joint compliance coefficient are shown in Fig. 4. The main features of components in closed-loop I and II are extracted as the beam elements drawn by the black lines. aij (i = 1~5 when j = 1; i = 1~3 when j = 2 ) denotes the shorter edges of cross section, and bhj (h = 1~3 when j = 1; h = 1, 2 when j = 2 ) represents the longer edges of the cross section. c11, c21, c31, c41, c51 are the compliance coefficients of the U joints. Overall, subsystem II contains 19 parameters.

4.3. Reliability sensitivity After obtaining the performance function by the response surface model, upper limits of the instantaneous energy and overall mass are required for the state function. Herein, the maximum values in the prescribed workspace are selected as the upper limits, and the state function can be formulated by Eq. (4). From the sampling points of the determined response model, the mean value and standard variance of the parameters in subsystem I and II are listed in Tables 3 and 4. Moreover, the mean value and variance of the state functions, as well as the reliability index can be 5

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Fig. 4. Structure and parameters of subsystem II.

parameters change with the same value ± 5mm . Moreover, the measurement planning as shown in Fig. 5 is carried out for each set of parameters since global behavior of the mass is concerned. The upper, medial and lower surface (z = 1200mm , z = 1350mm and z = 1500mm ) of the cylindrical workspace are chosen, on which evenly distributed measuring points are located on the circle R = 0 , R = 200mm , R = 400mm (6 points on each circle). The mean values of these measuring points are regarded as the simulation results. Therefore, the SolidWorks simulation procedure is concluded as:

computed by Eqs. (8) and (9), which are shown in Table 5. Resorting to Eqs. (11) and (12), T5 PM parameter sensitivity of static performance reliability to the mean value and variance can be computed as shown in Appendix A and B. And the global parameter sensitivity considering both mean value and variance are obtained from Eq. (13), which is illustrated in Table 6 and Table 7. 5. Parameter sensitivity verification and discussion 5.1. Parameter sensitivity verification

(1) Set the material property of each component according to the calculation settings. (2) Establish fixed frame and end reference point. (3) Drive the prototype going through all measuring points and obtain the mean value of the mass of subsystem I. (4) Change the value of the parameter and repeat step (3). (5) Select another parameter and repeat step (3) and (4).

In order to validate the parameter sensitivity analysis based on the response surface model and performance reliability, the mass of subsystem I is selected as the example to examine the results in Section 4. Note that SolidWorks is capable of measuring prototype mass, the verification process can firstly compare simulation from SolidWorks and calculation from the response surface model, and then discuss the impact of changed parameters to the mass of subsystem I for checking reliability sensitivity. Five parameters (Dct , dct , Dip , dop , ku ) with maximum, median and minimum global reliability sensitivity, as well as different RSAV, RSPC and RSNC are selected. Regard the initial parameters of the virtual prototype as the baseline, let the five

Among the five parameters, it is found form SolidWorks software that ku has no effect on the mass of subsystem I, which is accordance with the corresponding zero value in response model and reliability sensitivity analysis. For the four remaining parameters, the compar-

Table 1 Accuracy assessment of response models for subsystem I. Error (Accepted level)

Order

M1

ηlx,1

ηly,1

ηlz,1

η1

RS ( > 0.9)

linear quadratic cubic linear quadratic

0.99868 1 1 0.00774

0.55946 0.95688 0.01265 0.15036 0.01767

0.86912 0.96161 0.0354 0.09107 0.0195

0.83863 0.98759 0.08955 0.1008 0.01472

0 0.91945 0.0101 1.10848 0.02048

cubic

2.4765×10−8 0.01497

0.00853

0.00858

0.00913

0.00836

0.30800 0.26604

0.22248 0.18656

0.23559 0.12557

2.7669 0.29081

0.98117

0.98648

0.97309

0.98843

4.6011×10−8

0.17077 0.02621

0.10615 0.02704

0.11757 0.02034

1.35566 0.03094

2.80174×10−8

0.06776

0.06813

0.06726

0.06824

RAAE ( < 0.2)

RMAE ( < 0.3)

linear quadratic cubic

RMSE ( < 0.2)

linear quadratic cubic

3.5024×10−8

1.1228×10−7 8.20041×10−8 0.00919

6

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Table 2 Accuracy assessment of response models for subsystem II. Error (Accepted level)

Order

M2

ηlx,2

ηly,2

ηlz,2

η2

RS ( > 0.9)

linear quadratic cubic linear quadratic

0.99844 1 1 0.00834

0.96876 0.99556 0.99990 0.04259 0.01056

0.96893 0.99557 0.99990 0.04250 0.01052

0.9661 0.99564 0.99990 0.03797 0.00998

0.98119 0.99636 0.99987 0.03513 0.00935

0.00168

0.00170

0.00181

0.00193

0.09129 0.03575

0.09054 0.03577

0.09290 0.0329

0.08687 0.04388

0.00876

0.00875

0.00837

0.00675

0.04767 0.1309

0.04408 0.01219

0.04216 0.01194

0.00254

0.00247

0.00252

RAAE ( < 0.2)

1.58548×10−5

cubic RMAE ( < 0.3)

5.28461×10−6 0.02043

linear quadratic

6.94572×10−5

cubic RMSE ( < 0.2)

1.42586×10−5 0.01035

linear quadratic

2.11149×10−5

0.04783 0.01314

cubic

6.46371×10−6

0.00252

response surface model and SolidWorks simulation are 1.5292 kg and 1.2832 kg, which are within acceptable accuracy. It proves that the response surface model takes multiple parameters into account at the same time rather than analyzing single parameter in sequence. Therefore, based on the response surface model, the reliability sensitivity is capable of comprehensively analyzing the statistical regularity of the impact of parameters to the performance reliability.

isons of response surface model and SolidWorks simulation is as shown in Fig. 6, from which conclusion can be drawn as: (1) The response surface model is with high accuracy since the deviation between the two sets of results are within 1.5 kg (the maximum difference is 1.3179 kg at dct = 149mm , which is resulted from the stiffener inside the central tube). The simulation results are slightly lower due to the irregular features of components in the virtual prototype, such as chamfer or fillet. (2) When the parameters change with the same value ( ± 5mm ), the impact of the changed parameter to the mass in subsystem I are ranked as: Dct , dct ,,Dip , dop , which agree with rank of their global reliability sensitivity value (1.4142, 10368, 0.9655, 0.6280) and the RSAV (2.8311, 2.5173, 1.8786, 1.7654). (3) The mass in subsystem I shows the same tendency with the change of Dct , Dip . Take response surface model as an example, when Dct and Dip reduce to 165mm and 149mm, the mass decreases by 14.7688 kg, 9.53653 kg, respectively. While Dct and Dip increase to 175mm and 159mm, the mass raises 15.2096 kg and 10.5404 kg. It confirms the RSNC of Dct , Dip to the mass reliability (−2.8311,−1.8786 ) since upper limit is provided in this case. (4) The mass in subsystem I has the opposite change to the variation of dct , dop . For instance, in the SolidWorks simulation, when dct and dop drop to 45 mm and 55mm, corresponding mass increases by 12.9243 kg, 9.4308 kg. While dct and dop change to 55 mm and 65 mm, the mass cuts down by 13.7151 kg, 9.6900 kg. The change of dct , dop to the mass in subsystem I shows the same changing rules indicated by RSPCs of dct , dop (+2.5173, +1.7654).

5.2. Parameter discussion After verifying the proposed approach by SolidWorks simulation, the parameter discussion based on the reliability sensitivity can be applied to determine main/subordinate parameters for the performance optimization. The global sensitivity considering both mean value and variance of the parameters are the primary principle. And the reliability sensitivity to parameter means are then taken into account to evaluate how the changed parameters affect the performance. 5.2.1. Parameters in Subsystem I Fig. 8 shows the percentages of parameter global sensitivity in subsystem I, in which the red numbers are the performance reliability to parameter mean value. It can be concluded that: (1) For the mass of subsystem I (M1), the 1st to the 4th UPS limbs and the UP limb have greater influence while the 5th UPS limb and IR joints show smaller impact. Overall, stiffness coefficients (ku , ks , kU , kV , ks5) exert little change to the mass. In the 1st to the 4th UPS limbs and the UP limb, the effects of external diameters (Dop , Dip , Dct ) are bigger than that of corresponding internal diameters (dop , dip , dct ). Top four RSAV are Dct , dct , Dop , Dip , among which Dct , Dop , Dip are with RSNC and dct is with RSPC, indicating that decreasing

In order to further validate the effectiveness of the response surface model, two parameters are changed simultaneously and the comparison with SolidWorks simulation is as shown in Fig. 7. Deviations of the Table 3 Mean value and standard variance of the parameters in subsystem I.

quadratic cubic

quadratic cubic

*

μ σ μ σ μ σ μ σ

ku

Dop

dop

ds

Dip

dip

ks

Dir1

dir1

dir2

247.5341 42.6825 222.3225 28.4260 kU 233.5503 40.6658 232.3747 40.40494

72.0284 4.1394 71.9564 4.0974 kV 210.034 36.5113 211.0164 36.49294

59.8445 3.4063 59.9695 3.4713 Dct

16.0045 0.9251 16.0250 0.9295 dct 152.6217 8.5584 152.6461 8.5934

49.9984 2.9199 50.0279 2.9209 Dop5

34.9470 2.0621 35.0020 2.0294 dop5 60.0299 3.5263 60.0614 3.4577

7.7962 1.3543 7.8556 1.3546 ds5 15.9864 0.9342 16.0047 0.9363

63.1298 3.6582 62.9694 3.6367 Dip5

37.9465 2.2167 37.9746 2.2183 dip5 34.964 2.0373 34.9616 2.0524

29.9658 1.7411 30.0129 1.7231 ks5 7.9915 1.3952 7.9500 1.3880

171.7191 9.2283 171.7925 9.1594

71.9315 4.1613 72.0776 4.1920

Unit for parameters of cross section is mm, unit for stiffness coefficient is N/μm .

7

50.0672 2.9109 49.9462 2.8904

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Table 4 Mean value and standard variance of the parameters in subsystem II.

*

cubic

μ σ

cubic

μ σ

dir2

a11

b11

a21

a31

a41

b 21

a51

b31

c11

29.9841 1.7359 c21 0.0150 0.0026

14.0046 0.8118 c31 0.0008 0.0001

79.9720 4.652041 c41 6.0055 1.0433

22.9914 1.3354 c51 10.1475 1.0705

35.9908 2.0826 a12 14.9969 0.8728

46.9972 2.7328 b12 32.99053 1.9136

60.0048 3.4796 a22 25.0024 1.4545

18.0067 1.0439 a32 17.9985 1.0410

25.0004 1.4464 b 22 34.9988 2.0258

0.0034 0.0006

Unit for parameters of cross section is mm, unit for compliance coefficient is μm/N or rad/(N⋅m) .

Table 5 Mean value, variance and reliability index of the state functions.

M1

ηlx,1

ηly,1

ηlz,1

η1

M2

ηlx,2

ηly,2

ηlz,2

η2 1.0405

μg

86.5801

4.8141

8.5959

0.2614

98.8652

1.9984

2.4286

2.4303

1.8836

σg2

1579.1105

7.0501

21.6865

0.0475

1257.9305

0.3325

1.8855

1.8824

0.8322

0.2728

β

2.1788

1.8131

1.8459

1.1994

2.7875

3.4655

1.7686

1.7714

2.0648

1.9923

Table 6 Global parameter sensitivity of performance reliability in subsystem I.

ku

Dop

dop

ds

Dip

dip

ks

Dir1

dir1

dir2

ε M1

0.0000

1.0477

0.6280

0.0154

0.9655

0.4847

0.0000

0.1272

0.0799

0.0191

εηlx,1

0.0006

0.0056

0.0126

0.1824

0.0525

0.0627

0.2176

0.0559

0.0500

0.3225

εηly,1

0.0016

0.0326

0.0479

0.2169

0.0193

0.0795

1.4142

0.0301

0.0126

0.0781

εηlz,1

0.0014

0.0269

0.0353

0.1465

0.0186

0.0646

1.2048

0.2352

0.0668

0.7934

εη1

0.0026

0.1086

0.0219

1.1074

0.0115

0.1927

0.5806

0.0989

0.3254

1.0115

kV 0.0000

Dct 1.4142

dct 1.0368

Dop5 0.2094

dop5 0.1618

ds5 0.0038

Dip5

ε M1

kU 0.0000

0.1958

dip5 0.1352

ks5 0.0000

εηlx,1

0.0174

0.0029

0.0197

0.0157

0.0155

0.0474

0.2377

0.0265

0.1123

1.4142

εηly,1

0.0002

0.0009

0.0117

0.0040

0.0210

0.0502

0.2510

0.0110

0.1046

0.0509

εηlz,1

0.0083

0.0037

0.0121

0.0039

0.0265

0.0527

0.1915

0.0137

0.0961

1.4142

εη1

0.0120

0.0043

0.1881

0.1636

0.1239

0.2179

0.8177

0.1754

0.4198

0.2546

Table 7 Global parameter sensitivity of performance reliability in subsystem II.

dir2

a11

b11

a21

a31

a41

b 21

a51

b31

c11

ε M2

0.0000

1.0000

0.6372

0.8247

0.5000

0.4267

0.3342

0.4963

0.3575

0.6471

εηlx,2

0.0012

0.0003

0.0002

0.0000

0.0000

0.0000

0.0000

0.0001

0.0000

1.4142

εηly,2

0.0011

0.0003

0.0002

0.0000

0.0000

0.0000

0.0000

0.0001

0.0000

1.4142

εηlz,2

0.0011

0.0002

0.0001

0.0008

0.0000

0.0000

0.0000

0.0002

0.0001

1.0013

εη2

0.0059

0.0100

0.0007

0.0008

0.0001

0.0001

0.0001

0.0213

0.0053

0.0960

ε M2

c21 0.0717

c31 1.0027

c41 0.0000

c51 0.0000

a12 0.5038

b12 0.7333

a22 0.6654

a32 0.6834

b 22 0.3515

εηlx,2

0.0099

0.6471

0.0000

0.0000

0.0001

0.0002

0.0007

0.0000

0.0001

εηly,2

0.0101

0.6315

0.0000

0.0000

0.0001

0.0002

0.0007

0.0000

0.0001

εηlz,2

0.1190

1.3722

0.0000

0.0000

0.0004

0.0006

0.0001

0.0000

0.0000

εη2

0.5170

1.4142

0.0001

0.0005

0.0008

0.0012

0.0002

0.0000

0.0000

(ηly,1), the impact from the 1st to the 4th UPS limbs takes up almost 75% of the total influence, followed by the 5th UPS limb, then the IR joints and UP limb. Similar to ηlx,1, S joint stiffness coefficient (ks , ks5) and diameters of ball screws (ds , ds5) show greater influence. And ks is the dominant factor due to the percentage of 58%, which can also be seen from its larger RSAV. The rank of top four parameters are: ks , ds5, ds , dip5, in which ks and dip5 are with RSPC, ds5 and ds are with RSNC. Therefore, improving ks , dip5 and lowering ds5, ds would reduce the instantaneous linear stiffness index along y subsystem I. (4) For the instantaneous linear stiffness index along z of subsystem I (ηlz,1), the effect from the 5th UPS limb is slightly bigger and the

Dct , Dop , Dip and increasing dct would help reduce the mass of subsystem I. (2) For the instantaneous linear stiffness index along x of subsystem I (ηlx,1), the 5th UPS limb and IR joints impose bigger effects than the 1st to the 4th UPS limbs and the UP limb. Particularly, S joint stiffness coefficient (ks , ks5) have the greatest influence, then the diameter of IR joints (dtr2 ), the next are the diameters of the ball screws (ds , ds5). Among them, ks5 accounts for nearly half of the contribution to ηlx,1. Top four RSAV are ks5, dtr2 , ds5, ks , and they are all with RSPC. Thus, increasing these values would reduce the instantaneous linear stiffness index along x of subsystem I. (3) For the instantaneous linear stiffness index along y of subsystem I 8

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Fig. 5. Measurement planning for SolidWorks simulation.

Fig. 7. Comparisons for two changing parameters.

IR joints (Dir1, dir2 ). The rank of RSAV of the four parameters are ks5, ks , dir2 , Dir1. They are all with RSPC, revealing that increasing these values would decrease the instantaneous linear stiffness index along z of subsystem I. (5) For the overall stiffness index of subsystem I (η1), the 1st to the 4th UPS limbs, IR joints and the 5th UPS limb are almost equally occupy the same proportion whereas the UP limb has smaller influence. In the UPS limbs, S joint stiffness coefficient (ks , ks5) and diameters of ball screw (ds , ds5) account for more effects. The top four parameters are ds , dir2 , ds5, ks . The RSAV of dir2 indicates that it is the dominant element in the overall stiffness performance of subsystem I. All the four parameters are with RSPC, thus increasing the parameters helps reduce the overall stiffness index of subsystem I.

Fig. 6. Comparison of subsystem I mass from response surface model and SolidWorks simulation (Baseline parameters: Dct = 170mm , dct = 154mm , Dct = 50mm , dop = 60mm ).

In conclusion, parameters of the 1st to the 4th UPS limbs are the main factors for M1, ηly,1. The diameters of outer and inner pipe (Dop , dop , Dip , dip ) contribute more to the mass whereas S joint stiffness coefficient (ks ) and diameters of ball screw (ds ) have greater influence on the

effect of the 1st to the 4th UPS limbs and IR joints are nearly the same, while UP limb show little impact. Overall, S joint stiffness coefficient (ks , ks5) contribute the most to ηlz,1, then the diameters of

Fig. 8. Proportion of parameter impacts to the performance reliability of subsystem I (green: 1st to 4th UPS limbs, yellow: IR joints, blue: UP limb, pink: 5th UPS limb) (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.).

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Fig. 9. Proportion of parameter impacts to the performance reliability of subsystem II (green: cross section parameters in closed-loop I, pink: closed-loop II, yellow: joint stiffness coefficients) (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.).

6. Conclusion

stiffness performance of subsystem I. The 5th UPS limb impose much effect to ηlx,1 and ηlz,1, which are mainly from ks5 and ds5. IR joints, especially the diameter of vertical part of the ‘T’ structure (dir2 ), plays an important role in ηlx,1,ηlz,1, η1. Parameters of UP limb (mainly the Dct , dct ) account for nearly half of M1. They show little impact on the linear stiffness performance of subsystem I but their proportion become bigger in the overall stiffness performance. Taking all the performances into account, the main parameters of subsystem I are determined as Dop , ds , Dip , ks , Dct , dct dir2 , ds5,ks5.

Aiming at determining main/subordinate parameters for performance optimization of T5 PM, this paper carries out parameter sensitivity analysis with respect to the mass and stiffness performance on the basis of response surface method and performance reliability, which is verified by SolidWorks simulation. The conclusions are drawn as follow: (1) Based on design of experiment, the response surface method is to establish the approximate model between any parameters and the corresponding performance, in which experimental strategy and accuracy assessment are the key steps. (2) By introducing the performance reliability, a global parameter sensitivity index is proposed to consider both performance reliability to parameter mean value and variance, which is the principle for determine the effect degree of the parameters. Moreover, RSAV, RSPC and RSNC are defined to evaluate how the parameters affect the performance reliability. (3) T5 PM is divided into subsystem I and subsystem II considering its unique structure. Joint stiffness/compliance coefficients and parameters of cross sections are the concerned parameters whereas mass, linear instantaneous stiffness indices and overall instantaneous stiffness index are the targeted performance. After verifying the effectiveness of the approach by SolidWorks simulation, parameter discussion is carried out based on global sensitivity index, RSAV, RSPC and RSNC. 15 parameters are selected from 39 initial parameters and the effects of the parameters are clearly demonstrated, which would provide reference for the optimization process.

5.2.2. Parameters in subsystem II The proportions of parameters global sensitivity to the performance in subsystem II is as shown in Fig. 9, where red numbers are the corresponding reliability sensitivity to parameter mean values. It is concluded that: (1) For the mass in subsystem II (M2 ), cross sections parameters have greater impact than the U joint compliance coefficients. Among them, parameters of 1st and 2nd brackets impose more influence. Top four parameters are a11, c31, a21, b12 . Note that c31 is included even with smaller RSAV, which is resulted from the large value of M2 to its variance. The four parameters are with RSNC, stating that reducing these values would reduce the mass of subsystem II. (2) For the stiffness performance of subsystem II (ηlx,2 , ηly,2 , ηlz,2 , η2 ), compliance coefficients have much bigger effect than the cross section parameters, especially linear compliance coefficients (c11, c21, c31). Except for c31 in ηlx,2 (with RSPC), all compliance coefficients mean are with RSNC, indicating that decreasing these values, i. e. increasing the stiffness of U joints, would reduce ηlx,2 , ηly,2 , ηlz,2 , η2 . Overall, parameters of cross sections account more for the mass of subsystem II whereas U joint compliance coefficients impose greatest influence on the stiffness performance of subsystem II. Taking both mass and stiffness performance into account, the main parameters of subsystem II are determined as a11, a21, b12 , c11, c21, c31.

Acknowledgement This research work was supported by the National Natural Science Foundation of China (NSFC) under Grant No. 51205278 and 51475321.

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Appendix A. Parameter sensitivity of static performance reliability in subsystem I According to the formula in Section 3.3, the parameter sensitivity of performance reliability can be calculated. For subsystem I, the sensitivity of mass to the mean value of the parameters is calculated as 2 βM

1 ∂R M1 e− 2 ⎡ ∂g M1 = (M1) ⎢ ∂μ 2πσg ⎣ ∂ku

∂g M1

∂g M1 ⎤

⋯

∂Dop

∂ks5

T

⎥ ⎦

= 10−3 [ 0.0000 − 2.1183 1.7654 − 0.0238 − 1.8786 1.3144 0.0000 − 0.3240 0.1954 − 0.0417 0.0000 0.0000 − 2.8331 2.5173 − 0.5305 0.4420 − 0.0059 − 0.4689 0.3282 0.0000 ]T 2 βM 1

⎡ ∂(σ (M1) )2 μgM1 e− 2 ∂R M1 =− diag ⎢ g 2 3 M ( ) 1 ∂Cov (X ) ⎢⎣ ∂σ ku 2 2π (σg )

(M ) 2 ∂(σg 1 )

⋯

2 ∂σ D

op

(A-1)

(M ) 2 ⎤ ∂(σg 1 )

∂σ k2

s

⎥ ⎥⎦

= 10−4diag [ 0.0000 − 2.0141 − 0.2150 − 0.0354 − 1.9258 0.38490.0000 − 0.1529 0.1110 − 0.0335 0.0000 0.0000 − 2.7443 − 1.4662 − 0.2572 0.1174 − 0.0088 − 0.2873 0.1913 0.0000 ]

(A-2)

And the sensitivity of instantaneous linear stiffness performance along x direction in subsystem I to the mean value and variance of parameters are expressed as

∂Rηlx,1

e−

=

∂μ

βη2 lx,1 2

(η ) 2πσg lx,1

⎡ ∂gηlx,1 ⎢ ∂ku ⎣

∂gη

lx,1

∂gη

lx,1

⋯

∂Dop

∂ks5

⎤T ⎥ ⎦

= 10−2 [ 0.0011 − 0.0053 − 0.0076 0.1011 − 0.0922 0.0154 0.3794 0.0975 − 0.0681 0.5395 − 0.0309 0.0051 0.0338 − 0.0244 0.0099 − 0.0678 0.1130 0.0468 0.0907 1.7713]T βη2 lx,1 μηlx,1 e− 2

∂Rηlx,1

=−

∂Cov (X )

⎡ ⎛ (ηlx,1) ⎞2 ⎢ ∂ ⎜⎝σg ⎟⎠ diag (η ) 3 ⎢ ∂σ 2 2 2π (σg lx,1 ) ku ⎣

⎛ ( η ) ⎞2 ∂ ⎜σg lx,1 ⎟ ⎝ ⎠

g

2 ∂σ D

(A-3)

⎛ ( η ) ⎞2 ⎤ ∂ ⎜σg lx,1 ⎟ ⎥ ⎝ ⎠

⋯

∂σ k2

op

s

⎥ ⎦

= 10−2diag [ 0.0000 − 0.0053 − 0.0131 − 0.1912 0.0080 − 0.0686− 0.0424 0.0105 0.0353 − 0.1169 − 0.0001 0.0003 − 0.0054 − 0.0083 − 0.0160 − 0.0309 − 0.2527 − 0.0007 − 0.1103 − 1.1036 ]

(A-4)

The sensitivity of instantaneous linear stiffness performance along y direction in subsystem I to the mean value and variance of parameters are computed as ∂Rηly,1 ∂μ

βη2 ly,1 ⎡ ∂gηly,1 2 (ηly,1) ⎢ ∂k ⎣ u 2πσg −

=

∂gη

e

ly,1

∂Dop

⋯

∂gη

ly,1

∂ks5

⎤T ⎥ ⎦

= 10−2 [ 0.0025 0.0433 −0.0255 0.0012 0.0105 0.0393 1.6189 −0.0354 −0.0008 −0.0396 0.0002 −0.0015 0.01838 −0.0061 −0.0121 −0.0145 −0.0205 −0.0044 0.0512 0.0036 ]T η

∂Rηly,1 ∂Cov (X )

=−

−

μg ly,1 e

βη2 ly,1 2

⎡ ⎛ (ηly,1) ⎞2 ⎢ ∂ ⎜⎝σg ⎟⎠

diag ⎞3 ⎢

⎛ (η ) 2 2π ⎜σg ly,1 ⎟ ⎝ ⎠

⎣

⎛ (η ) ⎞2 ∂ ⎜σg ly,1 ⎟ ⎝ ⎠

∂σ k2

2 ∂σ D

⋯

op

u

(A-5)

⎛ ( η ) ⎞2 ⎤ ∂ ⎜σg ly,1 ⎟ ⎥ ⎝ ⎠ ∂σ k2

s

⎥ ⎦

= 10−2diag [ 0.0001 −0.0188 −0.0459 0.2196 0.0184 −0.0766 −1.0125 0.0209 −0.0127 −0.0751 −0.0001 0.0000 −0.0027 −0.0013 −0.0199 −0.0500 −0.2538 −0.0108 −0.1009 0.0515]

(A-6)

The sensitivity of instantaneous linear stiffness performance along z direction in subsystem I to the mean value and variance of parameters are formulated as ∂Rηlz,1 ∂μ

βη2 lz,1 ⎡ ∂gηlz,1 2 (ηlz,1) ⎢ ∂k ⎣ u 2πσg −

=

e

∂gη

lz,1

∂Dop

⋯

∂gη

lz,1

∂ks5

⎤T ⎥ ⎦

= 10−2 [ 0.0053 0.0845 −0.0596 0.1096 0.0118 0.0768 3.4858 0.8918 −0.2551 2.5004 −0.0324 −0.0145 0.0463 −0.0136 0.0411 −0.1085 −0.0239 0.04206 −0.1699 3.9112 ]T ∂Rηlz,1 ∂Cov (X )

=−

βη2 lz,1 η − μg lz,1 e 2

2

⎡ ⎛ (ηlz,1) ⎞2 ⎢ ∂ ⎜⎝σg ⎟⎠ diag ⎛ (η ) ⎞3 ⎢ ∂σ 2 2π ⎜σg lz,1 ⎟ ku ⎣ ⎝ ⎠

⎛ (η ) ⎞2 ∂ ⎜σg lz,1 ⎟ ⎠ ⎝ 2 ∂σ D

op

⋯

(A-7)

⎛ ( η ) ⎞2 ⎤ ∂ ⎜σg lz,1 ⎟ ⎥ ⎝ ⎠ ∂σ k2

s

⎥ ⎦

= 10−2diag [ 0.0001 −0.0188 −0.0374 0.1688 0.0215 −0.072 −0.9513 −0.0676 −0.0172 −0.5513 −0.0001 0.0000 −0.0030 −0.0022 −0.0285 −0.0526 −0.2247 −0.0099 −0.1006 −1.1735]

(A-8)

11

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The sensitivity of overall instantaneous stiffness performance in subsystem I the mean value and variance of parameters are obtained as ∂Rη1 ∂μ

βη2 1 2 ⎡ ∂gη1 (η1) ⎢ ∂ku 2πσg ⎣ −

=

∂gη

e

1

∂Dop

⎤T

∂gη

1

∂ks5 ⎥ ⎦

⋯

= 10−4 [ 0.0055 −0.2295 0.0124 1.0063 −0.0194 0.1465 0.6617 0.1621 −0.6372 2.1147 0.0253 −0.0087 0.3957 −0.3396 −0.262 −0.4514 0.2262 0.3665 0.5926 0.5309 ]T ∂Rη1

=−

∂Cov (X )

βη2 − 1 μgη1 e 2

⎡ ∂(σ (η1) )2 ⎢ g2

diag (η ) 3 ⎢⎣ 2 2π (σg 1 )

(η ) 2 ∂(σg 1 ) 2 ∂σ D op

∂σ k

u

⋯

(A-9)

(η ) 2 ⎤ ∂(σg 1 )

⎥ ⎥⎦

∂σ k2

s

= 10−4diag [ 0.0001 0.0350 −0.1771 −8.3954 0.0583 −1.5066 4.1084 0.5258 1.0319 1.2756 0.0014 0.0093 −0.1627 −0.2630 0.0294 −0.3690 −6.8086 0.2258 −2.6259 0.3556 ]

(A-10)

Appendix B. Parameter sensitivity of static performance reliability in subsystem II Similar to subsystem I, the sensitivity of performance reliability in subsystem II can be calculated resorting to the formula in Section 3.3. The mass of subsystem II to the mean value and variance of parameters are ∂R M2 ∂μ

2 βM 2 2 ⎡ ∂g M2 (M2) ⎢ ∂d 2πσg ⎣ ir 2 −

∂g M2

e

=

∂g M2

⋯

∂a11

∂b 22

⎤T ⎥ ⎦

= 10−4 [ 0.0000 −2.2483 −1.4327 −1.8541 −1.1242 −0.9594 −0.7514 −1.1159 −0.8038 −0.0642 −0.0138 −0.1663 0.0000 0.0000 −1.1328 −1.6487 −1.496 −1.5365 −0.7902 ]T

∂R M2

=−

∂Cov (X )

2 βM 2 − μgM2 e 2

⎡ ∂(σ (M2) )2

diag ⎢ (M ) 3 2 2π (σg 2 ) ⎣

g

(M ) 2 ∂(σg 2 )

∂σ d2 ir 2

⋯

∂σa211

(B-1)

(M ) 2 ⎤ ∂(σg 2 )

∂σ b2

22

⎥ ⎥⎦

= 10−2diag [ 0.0000 −0.0089 −0.0036 −0.0061 −0.0022 −0.0016 −0.0010 −0.0022 −0.0011 1.4141 −0.1562 2.1873 0.0000 0.0000 −0.0023 −0.0048 −0.0040 −0.0042 −0.0011]

(B-2)

The sensitivity of instantaneous linear stiffness performance along x direction in subsystem II to the mean value and variance of parameters are expressed as

∂Rηlx,2

=

∂μ

e−

βη2 lx,2 2

(η ) 2πσg lx,2

⎡ ∂gηlx,2 ⎢ ∂d ⎣ ir 2

∂gη

lx,2

∂a11

⋯

∂gη

lx,2

∂b 22

⎤T ⎥ = 10[ 0.0026 0.0006 0.0004 0.0000 0.0000 0.0000 0.0000 0.00025 0.0000 − 2.281 − 0.0225 0.0201 ⎦

0.0000 0.0000 0.0003 0.0005 0.0017 0.0001 0.0002 ]T ∂Rηlx,2 ∂Cov (X )

=−

μgηlx,2 e− 2 2π

βη2 lx,2 2

(η ) 3 (σg lx,2 )

⎡ ⎛ (ηlx,2) ⎞2 ∂ ⎜σ ⎟ diag ⎢ ⎝ g ⎠ ⎢ ∂σ 2 dir 2 ⎣

⎛ ( η ) ⎞2 ∂ ⎜σg lx,2 ⎟ ⎝ ⎠ ∂σa211

(B-3)

⎛ ( η ) ⎞2 ⎤ ∂ ⎜σg lx,2 ⎟ ⎥ ⎝ ⎠

⋯

∂σ b2

22

⎥ ⎦

= 10 3diag [ 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 − 5.3305 − 0.0016 − 3.4493 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 ]

(B-4)

The sensitivity of instantaneous linear stiffness performance along y direction in subsystem II to the mean value and variance of parameters are calculated as

∂Rηly,2 ∂μ

=

e−

βη2 ly,2 2

(η ) 2πσg ly,2

⎡ ∂gηly,2 ⎢ ∂d ⎣ ir 2

∂gη

ly,2

∂a11

⋯

∂gη

ly,2

∂b 22

⎤T ⎥ = 10[ 0.0026 0.0006 0.0004 0.0000 0.0000 0.0000 0.0000 0.0003 0.0002 − 2.2714 − 0.023 0.0221 ⎦

0.0002 0.0002 0.0003 0.0005 0.0017 0.0001 0.0002 ]T ∂Rηly,2 ∂Cov (X )

=−

βη2 ly,2 η − μg ly,2 e 2

⎡ ⎛ (ηly,2) ⎞2 ⎢ ∂ ⎜⎝σg ⎟⎠

diag ⎞3 ⎢

⎛ (η ) 2 2π ⎜σg ly,2 ⎟ ⎝ ⎠ 10 3diag [

⎣

∂σ d2

ir 2

⎛ ( η ) ⎞2 ∂ ⎜σg ly,2 ⎟ ⎝ ⎠ ∂σa211

⋯

(B-5)

⎛ ( η ) ⎞2 ⎤ ∂ ⎜σg ly,2 ⎟ ⎥ ⎝ ⎠ ∂σ b2

22

⎥ ⎦

= 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 −5.3248 −0.0010 −3.3623 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 ]

(B-6)

The sensitivity of instantaneous linear stiffness performance along z direction in subsystem II to the mean value and variance of parameters are computed as 12

Robotics and Computer–Integrated Manufacturing 46 (2017) 1–14

B. Lian et al.

∂Rηlz,2 ∂μ

=

e−

βη2 lz,2 2

(η ) 2πσg lz,2

⎡ ∂gηlz,2 ⎢ ∂d ⎣ ir 2

∂gη

lz,2

⋯

∂a11

∂gη

lz,2

∂b 22

⎤T ⎥ = 10[ 0.0013 0.0003 0.0001 0.0009 0.0000 0.00010.0000 0.0003 0.0001 − 0.063 − 0.1486 − 1.2489 ⎦

− 0.0001 0.0000 0.0005 0.0007 0.0001 0.0000 0.0000 ]T ∂Rηlz,2 ∂Cov (X )

=−

μgηlz,2 e− 2 2π

βη2 lz,2 2

(η ) 3 (σg lz,2 )

⎡ ⎛ (ηlz,2) ⎞2 ∂ ⎜σ ⎟ diag ⎢ ⎝ g ⎠ ⎢ ∂σ 2 dir 2 ⎣

⎛ ( η ) ⎞2 ∂ ⎜σg lz,2 ⎟ ⎝ ⎠ ∂σa211

⋯

(B-7)

⎛ (η ) ⎞2 ⎤ ∂ ⎜σg lz,2 ⎟ ⎥ ⎝ ⎠ ∂σ b2

22

⎥ ⎦

= 10 2diag [− 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 − 3.3842 − 0.0002 − 3.1787 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 ]

(B-8)

The sensitivity of overall instantaneous stiffness performance in subsystem II the mean value and variance of parameters are obtained as ∂Rη2 ∂μ

βη2 2 2 ⎡ ∂gη2 (η2) ⎢ ∂d 2πσg ⎣ ir 2 −

=

e

∂gη

2

∂a11

⋯

∂gη

2

⎤T ⎥

∂b 22 ⎦

= [ 0.0079 0.0000 0.0009 0.0011 0.0002 0.0002 0.0000 0.0283 0.0070 −0.0322 −0.6867 −1.3283 −0.0001 0.0000 0.0011 0.0016 0.0002 0.0000 0.0000 ]T ∂Rη2 ∂Cov (X )

=−

βη2 2 2

⎡ (η 2 ) 2 diag ⎢ ∂(σg2 ) (η 2 ) 3 ⎢⎣ ∂σ dir 2 2 2π (σg ) −

μgη2 e

(η ) 2 ∂(σg 2 )

∂σa211

⋯

(B-9)

(η ) 2 ⎤ ∂(σg 2 )

∂σ b2

22

⎥ ⎥⎦

= 103diag [ 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 −0.147 0.0068 1.5828 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 ]

(B-10)

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