Fuzzy-interval approach and sensitivity analysis to assess dimensional tolerances of parallel manipulators

Abstract

This paper proposes an alternative approach based on a fuzzy-interval analysis to assess how dimensional tolerances affect the kinematics of parallel manipulators. The novel contribution of the proposed approach consists of modeling and analyzing the dimensional tolerances as fuzzy-intervals based on fuzzy theory and interval sensitivity methods. The dimensional tolerances are modeled as intervals weighted by the membership function for the proposed fuzzy-interval approach. The proposed fuzzy-interval analysis is used to analyze the positioning error, sensitivity, and design analysis. The proposed fuzzy-interval method was applied using numerical simulations to assess the effect of dimensional tolerances and sensitivity of parallel manipulators. The proposed approach showed as a complementary procedure that can be used in the design phase of parallel manipulators.

Introduction

Several methods have been proposed to evaluate the uncertainties and dimensional tolerances on the parallel manipulator’s kinematics to enhance the design procedure. These computational methods aim to evaluate the contribution of different uncertain input factors to a mathematical model’s output to examine the error analysis, sensitivity, and optimization.

Concerning the error analysis, several procedures have been studied to evaluate the consequences of errors and joint clearances on positioning accuracy based on kinematics. The 3-UPU parallel mechanism sensitivity showed high sensitivity of kinematic outputs to specific clearances in the universal joint. ¹ The positioning error was obtained based on a kinematic approach rather than kinetostatics methods, which considers the loads applied to the mechanism. ² Voglewede and Ebert-Uphoff ³ applies generation methods in the workspace to define the unconstrained the end-effector motion of parallel manipulators. Venanzi and Parenti-Catelli ² include the joint clearance error as small-displacements screws that permit the maximum end-effector error. Meng et al. ⁴ analyzes the parallel manipulators accuracy subjected to joint clearances. A study of the error analysis applied on parallel manipulators has been carried out based on Venanzi and Parenti-Catelli ² by using standard convex optimization problem. ⁵

On the other hand, the sensitivity analysis of mechanism dynamics has captured the attention of researchers for analysis and design proposes. ⁶ Several approaches have been proposed to analyze the sensitivity of mechanism and robotic manipulators: response surface method,^7,8 local derivative-based method,^9–11 and interval linearization method. ¹² The kinematics and dynamics of the parallel manipulators were assessed by applying the sensitivity analysis. The sensitivity has been used mainly to evaluate the relative influence of every geometric errors on the end-effector positioning. Consequently, the sensitivity analysis allows determining the critical parameters to positioning accuracy in the early design phase.

The robust optimization of kinematic performance maximizes the kinematic performance and simultaneously minimizes the errors produced by the dimensional tolerances of the geometric parameters. ¹³ The design criteria sensitivity to geometric parameters tolerances has been used for the optimal design of manipulators ¹⁴ ; moreover, the sensitivity analysis was used to minimize the robustness against uncertainties. ¹⁵

Several research studies have suggested that complementary methodologies used for the optimization of robotic manipulators are necessary to enhance the results obtained from the optimal design.^16,17 In this way, the sensitivity analysis has presented potential benefits in the optimal design of manipulators. ¹⁸ Moreover, the development of novel numerical methods to quantify the effects of dimensional tolerances, manufacturing errors and varying parameters is necessary. In this direction, theoretical developments to obtain novel performance indices have been formulated based on the error analysis, ¹⁹ uncertainty quantification, ²⁰ and reliability. ²¹ Nevertheless, few studies have focused on the sensitivity of parallel robots.

The sensitivity analysis has been widely used to quantify how the dimensional tolerances and errors affect the kinematic accuracy. Nevertheless, the error analysis, sensitivity and optimization of manipulators has not assessed the uncertainties as fuzzy variables. The advantage of the proposed approach consists in evaluating the tolerances as intervals that consider a range of variation, and it permits sensitivity assessment. Unlike the previous approaches, the fuzzy-interval approach does not require the definition of the probability distribution function for the tolerances defined as random variables^20,21; as an additional advantage, the tolerance is not evaluated just for a single value.^9–11 The present research study proposes a methodology to analyze parallel manipulators’ kinematic error and sensitivity by considering dimensional tolerances modeled as fuzzy variables. Moreover, the proposed approach quantifies the sensitivity of the kinematic model subject to dimensional tolerance and geometric parameters modeled as intervals derived from the fuzzy variables definition; that is, the sensitivities of the geometric parameters are evaluated aiming at maximizing the workspace size of the manipulators. The novel contribution of the proposed approach consists of modeling and analyzing the dimensional tolerances as fuzzy-intervals based on fuzzy theory and interval sensitivity methods.

Initially, the model of the dimensional tolerances is presented based on the $\alpha -$ level optimization concept. Hereafter, the proposed approach to assess the sensitivity considers the interval analysis that establishes the sensitivity indices by solving an interval problem derived from the interval inputs and considering the interval inputs and outputs ratio. Then, three case studies are considered to evaluate effect of dimensional tolerances on the kinematic characteristics of the parallel manipulators by using the proposed approach. The first case study analyzes the positioning error of a 5R parallel manipulator with active joint errors. The second case study establishes the geometric parameter sensitivity on the positioning accuracy of a Delta manipulator. Moreover, the interval method is compared to the local derivative-based method. The third case study aims to apply interval sensitivity as a complementary procedure for optimizing a Cartesian parallel manipulator. The interval sensitivity approach evaluates the design space to maximize the workspace size.

This paper is organized into four sections. Section 2 presents the proposed interval analysis for the manipulators. Section 3 presents the case studies that analyze the effect of dimensional tolerances modeled as fuzzy-intervals on the parallel manipulators. Finally, Section 4 presents the conclusions.

Fuzzy-interval approach

The approach presented in this contribution for the fuzzy-interval analysis considers non-probabilistic modeling.^22,23

Interval representation of fuzzy variables

Initially, the fuzzy variables of Figure 1(a) are defined based on an interval approach. Fuzzy variables are a simple representation of uncertain parameters that does not require the definition of the probability distribution function of the random variables. Let be ${\lambda }_d$ an element of a universal set $\mathbf{\Lambda }$ . The pertinence of ${\lambda }_d$ to the universal set $\mathbf{\Lambda }$ is defined in a classical way by the discrete membership function ${\mu }_{\lambda }:\mathbf{\Lambda }\to \{0,1\}$ , where ${\mu }_k=1$ indicates that ${\lambda }_d$ belongs to $\mathbf{\Lambda }$ ; otherwise, ${\mu }_k=0$ indicates that ${\lambda }_d$ does not belong to $\mathbf{\Lambda }$ (see Figure 1(a)).

Figure 1.

Fuzzy definitions: (a) fuzzy parameter, (b) interval representation, and (c) fuzzy output.

On the other hand, the fuzzy variable $\tilde{\lambda }$ has its membership function ${\mu }_{\lambda }$ defined as ${\mu }_{\lambda }:\mathbf{\Lambda }\to [0,1]$ where $[0,1]$ is an interval. The fuzzy variable $\tilde{\mathbf{\lambda }}$ is described in equation (1). The membership function, ${\mu }_{\lambda }$ , states the degree of compatibility of $\lambda$ to $\tilde{\mathbf{\lambda }}$ ; that is, when ${\mu }_{\lambda }(\lambda )$ is close to “1” indicates the more belonging of $\lambda$ to $\tilde{\mathbf{\lambda }}$ (see Figure 1(a)).

$$\tilde{\mathbf{\lambda }}=\{(\lambda ,{\mu }_{\lambda }(\lambda ))|\lambda \in \mathbf{\Lambda }\}$$

(1)

where $0\le {\mu }_{\lambda }(\lambda )\le 1$

For computational purposes, the fuzzy variable $\tilde{\mathbf{\lambda }}$ can be modeled as intervals known as $\alpha$ -levels (see Figure 1(b)); the membership function ${\mu }_{\lambda }(\lambda )$ indexes the $\alpha$ -level intervals. Thus, for a considered value of ${\mu }_{\lambda }(\lambda )$ can be obtained an interval ${\overline{\mathbf{\lambda }}}_{{\alpha }_d}$ defined in equation (2).

$${\overline{\mathbf{\lambda }}}_{{\alpha }_{\lambda }}=\{\lambda \in \mathbf{\Lambda },{\mu }_{\lambda }(\lambda )\ge {\alpha }_{\lambda }\}$$

(2)

The interval ${\overline{\mathbf{\lambda }}}_{{\alpha }_{\lambda }}=[{\lambda }_{{\alpha }_{\lambda }l},{\lambda }_{{\alpha }_{\lambda }r}]$ of the fuzzy variable $\tilde{\mathbf{\lambda }}$ is shown in Figure 1. ${\lambda }_{{\alpha }_{\lambda }l}$ and ${\lambda }_{{\alpha }_{\lambda }r}$ are the interval bounds ${\overline{\mathbf{\lambda }}}_{{\alpha }_{\lambda }}$ with:

$$\begin{array}{c}{\lambda }_{{\alpha }_{\lambda }l}=min[\{\lambda \in \mathbf{\Lambda },{\mu }_{\lambda }(\lambda )\ge {\alpha }_{\lambda }\}]\hfill \\ {\lambda }_{{\alpha }_{\lambda }r}=max[\{\lambda \in \mathbf{\Lambda },{\mu }_{\lambda }(\lambda )\ge {\alpha }_{\lambda }\}]\hfill \end{array}$$

Fuzzy-interval analysis

This analysis considers a performance index of the manipulator’s kinematic model (see equation (3)). Therefore, the performance index y depends on the input parameters $\mathbf{\lambda }$ .

$$y=\mathbf{\Gamma }\left(\mathbf{\lambda }\right)$$

(3)

where $\mathbf{\Gamma }$ denotes a performance function based on the kinematic equations, that is, this method permits analyzing any kinematic characteristic. For example, the workspace size or the global conditioning index. ²⁴ $\mathbf{\lambda }=\left[\begin{array}{ccccc}{\lambda }_1\hfill & \dots \hfill & {\lambda }_i\hfill & \dots \hfill & {\lambda }_m\hfill \end{array}\right]$ is the set of the m input parameters. These input parameters can be the link lengths.

By considering the input parameters of equation (3) as fuzzy variables $\tilde{\mathbf{\lambda }}=\left[\begin{array}{ccccc}{\tilde{\mathbf{\lambda }}}_1\hfill & \dots \hfill & {\tilde{\mathbf{\lambda }}}_i\hfill & \dots \hfill & {\tilde{\mathbf{\lambda }}}_m\hfill \end{array}\right]$ , the output will also be a fuzzy variable, consequently:

$$\tilde{\mathbf{y}}=\tilde{\mathbf{\Gamma }}(\tilde{\mathbf{\lambda }})$$

(4)

where $\tilde{\mathbf{\Gamma }}$ is a fuzzy function of the kinematic model that expresses a relationship between inputs $\tilde{\mathbf{\lambda }}$ and fuzzy output $\tilde{\mathbf{y}}$ . It is worth mentioning that this numerical model considers multiple m inputs with $i=1,2\dots ,m$ , and a single output $\tilde{\mathbf{y}}$ .

The interval representation defines the fuzzy output $\tilde{\mathbf{y}}$ as presented in equation (5):

$$\tilde{\mathbf{y}}=\{(y,\mu (y))|y\in \mathbb{R}\}\mathrm{where}0\le \mu (\mathrm{y})\le 1$$

(5)

Moreover, the fuzzy output $\tilde{\mathbf{y}}$ (see Figure 1(c)) is modeled using $\alpha$ -levels with ${\overline{\mathbf{y}}}_{{\alpha }_{\lambda }}=\{y\in \mathbb{R},\mu (y)\ge {\alpha }_{\lambda }\}$ . Therefore, ${\overline{\mathbf{y}}}_{{\alpha }_{\lambda }}$ is an interval derived from the membership value $\mu (y)={\alpha }_{\lambda }$ :

$${\overline{\mathbf{y}}}_{{\alpha }_{\lambda }}=(y_{{\alpha }_{\lambda }l},y_{{\alpha }_{\lambda }r})\mathrm{for}\mu (y)={\alpha }_{\lambda }$$

(6)

where $y_{{\alpha }_{\lambda }l}=min(y\in \mathbb{R},\mu (y)\ge {\alpha }_{\lambda })$ and $y_{{\alpha }_{\lambda }r}=max(y\in \mathbb{R},\mu (y)\ge {\alpha }_{\lambda })$ . The assessment of $y_{{\alpha }_{\lambda }r}$ requires to solve of two optimizations to find $y_{{\alpha }_{\lambda }l}$ and $y_{{\alpha }_{\lambda }r}$ for the ${\alpha }_{\lambda }=\mu (y)$ .

It worth mentioning that the intervals ${\overline{\mathbf{y}}}_{{\alpha }_{\lambda }}$ of the fuzzy ouput $\tilde{\mathbf{y}}$ are obtaineded by maximizing and minimizing the kinematic model $\mathbf{\Gamma }$ within the search space ${\overline{\mathbf{\lambda }}}_{{\alpha }_{\lambda }}$ that corresponds to the definition of interval inputs. Therefore, the optimizations to obtain the interval outputs are presented in equation (7).

$$y_{{\alpha }_{\lambda }l}=\underset{\mathbf{\lambda }\in {\overline{\mathbf{\lambda }}}_{{\alpha }_{\lambda }}}{min}\mathbf{\Gamma }(\mathbf{\lambda })y_{{\alpha }_{\lambda }r}=\underset{\mathbf{\lambda }\in {\overline{\mathbf{\lambda }}}_{{\alpha }_{\lambda }}}{max}\mathbf{\Gamma }(\mathbf{\lambda })$$

(7)

The implementation of the proposed fuzzy-interval analysis is showed in the flowchart in Figure 2. For computational proposes, the input parameters $\tilde{\mathbf{\lambda }}$ are discretized to evaluate a finite number of $\alpha$ -levels according to equation (2). The fuzzy-interval outputs $\tilde{\mathbf{y}}$ demand the computation of the bounds ( $y_{{\alpha }_{\lambda }l}$ and $y_{{\alpha }_{\lambda }r}$ ) of every single interval ${\overline{\mathbf{y}}}_{{\alpha }_{\lambda }}$ at the ${\alpha }_{\lambda }$ -level according to equation (7). This procedure is carried out for the considered ${\alpha }_{\lambda }$ -level with $0\le {\alpha }_{\lambda }\le 1$ and ${\alpha }_{\lambda }$ is incremented by ${\delta }_{\alpha }$ to compute every interval $\overline{y}$ of equation (6).

Figure 2.

Flowchart to implement the fuzzy-interval approach.

The interval approach is used to evaluate the manipulator’s kinematics, such as the forward kinematic model, or an algorithm to assess the workspace size subjected to interval inputs, such as the dimensional tolerances of links.

Interval sensitivity analysis

The interval methods of the present contribution were inspired by the research works suggested by Moens and Barbosa et al.^25,26 The sensitivity aims at quantifying the contribution of every fuzzy parameter on the output of the manipulator. For example, the sensitivity analysis could determine the relative contribution of the fuzzy-interval link dimensional tolerance on the interval end-effector positioning error. The sensitivity can help to determine the degree of influence of every parameter on the kinematic performance. In the present analysis, the input fuzzy parameters, $\tilde{\mathbf{\lambda }}$ , are considered at the $\alpha$ -level ${\mu }_{\lambda }$  = 0 to analyze the sensitivity for the greater degree of uncertainty dispersion. In accordance with that, the input parameters, $\overline{\mathbf{\lambda }}$ , and the output of the model, $\overline{\mathbf{y}}$ , are considered as intervals.

Initially, the interval output radius $\mathrm{\Delta }y$ is determined in equation (8).

$$\mathrm{\Delta }y=\frac{y_l-y_r}{2}$$

(8)

The output $\overline{\mathbf{y}}$ considered as an interval subjected to the inputs $\overline{\mathbf{\lambda }}$ as intervals is defined by $\delta$ as presented in equation (9). The rate of the widths of an interval output $\overline{\mathbf{y}}$ and an interval inputs $\overline{\mathbf{\lambda }}=\left[\begin{array}{c}{\overline{\mathbf{\lambda }}}_1,\dots ,{\overline{\mathbf{\lambda }}}_i,\dots ,{\overline{\mathbf{\lambda }}}_m\hfill \end{array}\right]$ determines the sensitivity.

$$\delta =\frac{\partial (\mathrm{\Delta }y)}{\partial (\mathrm{\Delta }\mathbf{\lambda })}$$

(9)

The correlation of the interval widths of $\overline{\mathbf{y}}$ and $\overline{\mathbf{\lambda }}$ quantifies the interval sensitivity of equation (10). Since several inputs ${\lambda }_i$ are examined simultaneously, the interval output takes into account multiple intervals.

$${\rho }_i=\frac{\mathrm{\Delta }{\lambda }_i}{\mathrm{\Delta }{y}_i}{\delta }_i$$

(10)

where ${\delta }_i$ represents the interval sensitivity of the $i-\mathrm{th}$ input parameter ${\lambda }_i$ according to equation (9), $\mathrm{\Delta }{y}_i=y_{\mathrm{ri}}-y_{\mathrm{li}}$ and $\mathrm{\Delta }{\lambda }_i={\lambda }_{\mathrm{ri}}-{\lambda }_{\mathrm{li}}$ are the nominal interval widths for every interval input derived from the solution of equation (11).

$$y_{\mathrm{li}}=\underset{{\lambda }_i\in {\overline{\mathbf{\lambda }}}_i}{min}\mathbf{\Gamma }(\mathbf{\lambda })y_{\mathrm{ri}}=\underset{{\lambda }_i\in {\overline{\mathbf{\lambda }}}_i}{max}\mathbf{\Gamma }(\mathbf{\lambda })$$

(11)

The normalization of the interval sensitivity is shown in equation (12); this normalized sensitivity quantifies the total effect of every interval input on the output.

$${\eta }_i=\frac{{\rho }_i}{\sum _{i=1}^m{\rho }_i}$$

(12)

The normalized sensitivity ${\eta }_i$ of equation (12) sets a comparison relationship of how influential a single interval input is on the output.

Moreover, the absolute interval sensitivity is derived from the interval radius of equation (8). Thus, ${\delta }_i$ should be obtained for ${\delta }_r$ and ${\delta }_l$ that correspond to the upper and lower bounds.

$${\delta }_i=\frac{\partial \mathrm{\Delta }y}{\partial \mathrm{\Delta }\mathbf{\lambda }}=\frac{1}{2}(\frac{\partial y_l}{\partial \mathrm{\Delta }\mathbf{\lambda }}-\frac{\partial y_r}{\partial \mathrm{\Delta }\mathbf{\lambda }})=\frac{1}{2}({\delta }_{\mathrm{li}}-{\delta }_{\mathrm{ri}})$$

(13)

$\delta$ quantifies the difference of the upper and lower bounds against the inputs defined as intervals.

The upper $y_r$ and lower variation $y_l$ of the sensitivities are quantified by ${\delta }_{\mathrm{ri}}$ and ${\delta }_{\mathrm{ri}}$ . $\mathrm{\Delta }\mathbf{\lambda }$ computes this variation that expresses the correlation between the input intervals. Therefore, the manipulator kinematics $\mathbf{\Gamma }$ are applied to compute the upper and lower sensitivities for evaluation of ${\mathbf{\lambda }}^{y_l}$ and ${\mathbf{\lambda }}^{y_r}$ . These upper and lower sensitivity limits can be estimated with equation (14).

$${\delta }_{l_i}=-\left|{\left(\frac{\partial y}{\partial \mathbf{\lambda }}\right)}_{{\mathbf{\lambda }}^{y_l}}\right|{\delta }_{r_i}=\left|{\left(\frac{\partial y}{\partial \mathbf{\lambda }}\right)}_{{\mathbf{\lambda }}^{y_r}}\right|$$

(14)

indicate that

The sensitivity limits ( ${\delta }_{l_i}$ and ${\delta }_{r_i}$ ) correspond to the gradients obtained for the last optimization iteration at ${\mathbf{\lambda }}^{y_l}$ and ${\mathbf{\lambda }}^{y_r}$ . The relative interval sensitivity of equation (10) can be derived from equation (13).

The flowchart of Figure 3 illustrates the implementation of the interval approach. The normalized interval sensitivity ${\eta }_i$ is calculated solving the optimizations of equation (7). The sensitivity indices of equations (10), (12), and (14) are determined. The solution of the global optimizations of equation (7) by means of metaheuristic methods could demand a high computational cost for the analysis of manipulators with a complex kinematic model with several input interval parameters.

Figure 3.

Flowchart to implement the interval sensitivity approach.

Case studies

This section considers the application of the fuzzy-interval approach to assessing the effect of dimensional tolerances modeled as fuzzy-intervals. First, the effect of joint error models on the positioning error of a planar parallel manipulator is analyzed. Then, the interval sensitivity is illustrated as a complementary method to illustrate the dimensional tolerance on a Delta manipulator. Finally, the fuzzy-interval approach is applied to the optimal design of a Cartesian Parallel Manipulator.

Positioning error analysis of 5R parallel manipulator

Figure 4 presents the 5R parallel manipulator. The frame O is fixed at the segment $A_1{A}_2$ midpoint. The manipulator is symmetrical since $A_1{B}_1=A_2{B}_2$ , $O{A}_1=O{A}_2$ , and $B_{1}P=B_{2}P$ . The end-effector is at $P(x,y)$ . It owns two kinematic chains $i=1,2$ , which are identical; every kinematic chain is acted by an active joint ( ${\theta }_i$ for $i=1,2$ ), a passive joint, and it has two links. Moreover, the manipulator’s geometry can be specified by three single parameters $r_1=A_i{B}_i$ , $r_3=O{A}_i$ , and $r_2=B_{i}P$ . The solution for the forward kinematic used in this contribution corresponds to the up-configuration according to Liu et al. ²⁷

Figure 4.

5R parallel planar manipulator.

The error analysis consists in solving the forward kinematics by taking into account the active joint errors $\delta {\theta }_1$ and $\delta {\theta }_2$ as fuzzy variables. For this particular case study, the link lengths are defined as $r_3=0.8$  m, $r_2=1.0$  m, and $r_1=1.2$  m. Furthermore, the positioning active joints errors are considered as the input parameters $\mathbf{\lambda }={\left[\begin{array}{cc}\delta {\theta }_1\hfill & \delta {\theta }_2\hfill \end{array}\right]}^T$ and they are set as fuzzy variables with triangular membership function $\delta {\tilde{\mathbf{\theta }}}_i=(-0.05,0,0.05{)}^o$ , for $i=1,2$ . In addition, the interval error is faced up to the positioning error derived from the Monte Carlo simulation (MCS). ²⁰ Therefore, the uncertainties in the active joints are modeled as random variables in the MCS:

$$\delta {\theta }_i(\mathrm{\Omega })={\delta }_{\theta i}\xi (\mathrm{\Omega })$$

(15)

where, ${\delta }_{\theta i}$ (for $i=1,2$ ) is the joint error dispersion with ${\delta }_{\theta i}={\frac{0.05}{3}}^o$ . $\xi (\mathrm{\Omega })$ is the unit normal random variable and $\mathrm{\Omega }$ is a random process. A normal distribution governs the unit normal random variable; this distribution was selected to evaluate the uncertain parameters. Moreover, it worth mentioning that the dispersion of the active joints error are stated as random variables $\delta {\theta }_i(\mathrm{\Omega })$ of equation (15) and the triangular fuzzy variables $\delta {\tilde{\mathbf{\theta }}}_i$ .

The positioning error is computed using the fuzzy-interval approach of Section 2.2 and the flowchart of Figure 2. Hence, two different definitions of the positioning error were considered as the kinematic function ( $\mathbf{\Gamma }(\mathbf{\lambda })$ ) of equation (4), thus:

$$p(\lambda )=\{\begin{array}{cc}x(\lambda )\mathrm{and}y(\lambda )\hfill & (16)\hfill \\ \multicolumn{1}{c}{\Vert {\left[\begin{array}{cc}x(\lambda )\hfill & y(\lambda )\hfill \end{array}\right]}^T\Vert }& (17)\hfill \end{array}$$

The positioning of the x and y axes is considered separately in equation (16), that is, the solution to the optimization problem of equation (7) should be carried out twice to obtain $\overline{\mathbf{x}}$ and $\overline{\mathbf{y}}$ at each ${\alpha }_{\lambda }$ -level (see Figure 2). Moreover, equation (17) computes the norm of the planar error. The Genetic Algorithm ²⁸ was used to solve the interval optimizations of equation (7).

The Monte Carlo Simulation is carried out to compute the positioning error following two steps: $i)$ the uncertain parameters (see equation (15)) are sampled to obtain a set of $n_s$ random inputs; $\mathrm{ii})$ the positioning $p(\mathbf{\lambda })$ of equation (14) is computed for each random input, these results are presented in Figure 6(c)).

Initially, Figure 5 shows the y-axis end-effector coordinate $\mathbf{p}={\left[\begin{array}{cc}x\hfill & y\hfill \end{array}\right]}^T={\left[\begin{array}{cc}-0.5\hfill & 1.70\hfill \end{array}\right]}^T$ derived from the fuzzy-interval approach (see Figure 5(a)) and the Monte Carlo simulation (MCS) (see Figure 5(b)). The fuzzy-interval position of the $\tilde{\mathbf{y}}$ was computed for six different $\alpha$ -levels, thus $\mu (y)$  = 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0 as presented in Figure 5(a). Moreover, $\tilde{\mathbf{y}}$ was also approximated as a triangular fuzzy variable by solving the interval problem only at the $\alpha$ -level $\mu (y)$  = 0.0; thus, the obtained triangular fuzzy output for this approximation is $\tilde{\mathbf{y}}=(y_l,y_d,y_r)$ (see Figure 5(a), red dashed line); it observed that this approximation estimates successfully the fuzzy positioning error $\tilde{\mathbf{y}}$ . Finally, the fuzzy-interval method is face up to the Monte Carlo simulation; the Monte Carlo Simulation can approximate the probability density function (PDF) by considering 5000 samples, that is, by computing 5000 times the forward kinematic model as presented in the histogram of Figure 5(b).

Figure 5.

y-Axis end-effector position evaluated with the fuzzy-interval approach and MCS: (a) workspace and (b) error evaluation.

The end-effector position at $\mathbf{p}={\left[\begin{array}{cc}x\hfill & y\hfill \end{array}\right]}^T={\left[\begin{array}{cc}-0.5\hfill & 1.70\hfill \end{array}\right]}^T$ obtained with the fuzzy-interval approach and the Monte Carlo simulation (MCS) of Figure 6; the results of Figure 6(a) and (b) were generated using the fuzzy-interval approach of Figure 2 and the definition of the positioning errors of equations (16) and (17), respectively. Figure 6(a) and (b) show the end-effector error $\tilde{\mathbf{p}}$ by considering the definitions of equation (4) (the limits of the interval positioning errors for the x and y coordinates are represented as a rectangular in the plane) and equation (17) (the interval positioning error, computed as the norm of x and y coordinates, is represented as a circle centered in the nominal position of the end-effector at the plane), respectively. In addition, Figure 6(c) shows the end-effector derived from the Monte Carlo simulation. Finally, Figure 6(d) presents the last three approaches comparison, the fuzzy-interval approached were computed at the $\alpha$ -level $\mu$  = 0. One can observe that the fuzzy-inteval approach is more conservative than the Monte Carlo simulation leading to an over estimation of the Cartesian error. Nevertheless, the fuzzy-interval approach permits evaluating the amplitude of the error for different $\alpha$ -levels, that is, it permits assessing how the positioning error varies for several dispersion levels of the active joint error $\delta \tilde{\mathbf{\theta }}$ .

Figure 6.

End-effector error by using the fuzzy-interval approach: (a) fuzzy-interval with equation (16), (b) fuzzy-interval with equation (17), (c) MCS, and (d) approaches comparison.

The end-effector error will be evaluated at six different positions within the usable workspace (see Figure 7(a)). Figure 7(b) shows the results at four different end-effector positions (( $a)$ , ( $b)$ , ( $c)$ , and ( $e)$ ), the fuzzy-interval approach was evaluated at the $\alpha -$ level $\mu =$ 0. As additional results, Table 1 shows the outputs by using the fuzzy-interval approach with the definition of equation (17) and the Monte Carlo simulation; the fuzzy-interval approach demands the forward kinematics computation $p(\mathbf{\lambda })$ fewer times that the Monte Carlo simulation.

Figure 7.

Error evaluation for the 5R parallel manipulator: (a) workspace and (b) error evaluation.

Table 1.

Positioning error analysis: fuzzy-interval and Monte Carlo simulation.

$\mathbf{p}$ (m)	$\overline{p}$ (m)	$n_o$	${\mu }_x$ (m)	${\sigma }_x$ (m)	${\mu }_y$ (m)	${\sigma }_y$ (m)	$n_s$
${\mathbf{p}}_a={\left[\begin{array}{cc}0.0\hfill & 1.7\hfill \end{array}\right]}^T$	[0,0.00063]	1040	0.0000	0.00014	1.7	0.00012	5000
${\mathbf{p}}_b={\left[\begin{array}{cc}0.5\hfill & 1.7\hfill \end{array}\right]}^T$	[0,0.00055]	1040	0.5	0.00012	1.7	0.0001	5000
${\mathbf{p}}_c={\left[\begin{array}{cc}-0.5\hfill & 1.7\hfill \end{array}\right]}^T$	[0,0.00055]	1040	−0.5	0.00012	1.7	0.0001	5000
${\mathbf{p}}_e={\left[\begin{array}{cc}0.5\hfill & 1.45\hfill \end{array}\right]}^T$	[0,0.00053]	1040	0.5	0.00013	1.45	0.0001	5000

Sensitivity analysis of the delta parallel manipulator with fuzzy parameters

The evaluation of dimensional tolerance of links on end-effector error is carried out through the interval sensitivity approach. The proposed interval sensitivity is compared to derivative-based local methods that have already been used to assess the length links variations of parallel manipulators. ¹⁰

The Delta manipulator has three identical kinematic chains between the fixed base and the moving platform. The frames $\{B\}$ and $\{P\}$ are located at the fixed base and the moving platform, respectively (see Figures 8 and 9). Each kinematic chain has the following configuration $\underset{¯}{R}\mathrm{UU}$ where $\underset{¯}{R}$ denotes the active rotational joint at the point $B_i$ of the fixed base, this active joint is defined by the angle ${\theta }_i$ for $i=1,2,3$ . The universal passive joints U are fixed at $A_i$ and $P_i$ for $i=1,2,3$ (see Figure 8). The lower links of length $l_i$ were considered as the 4 bar parallelogram mechanism to provide a translational motion with constant orientation in the moving platform. The forward kinematic solution used in this contribution adopts the solution presented in Williams et al. ²⁹

Figure 8.

Delta manipulator and its geometric parameters.

Figure 9.

Delta manipulator: fixed base and moving platform geometric definition. (a) Fixed base and (b) Moving platform.

The fixed base is defined based on the equilateral triangle in which the Cartesian frame $\{B\}$ is fixed in its centroid (see Figure 9(a)). Similarly, the moving platform is also defined as an equilateral triangle (see Figure 9(b)); the frame $\{P\}$ is fixed at the centroid of the moving platform. The moving platform has translation motion with fixed orientation with respect to the fixed base. Thus, the frames $\{P\}$ and $\{B\}$ have the same orientation, and the orientation matrix of $\{P\}$ with respect $\{B\}$ ${}_P{}^B\mathbf{R}={\mathbf{I}}_3$ is constant. The vector of the active joint variables is defined as ${\left[\begin{array}{ccc}{\theta }_1\hfill & {\theta }_2\hfill & {\theta }_3\hfill \end{array}\right]}^T$ and the moving platform Cartesian position is stated as $\mathbf{p}={\left[\begin{array}{ccc}{p}_x\hfill & p_y\hfill & p_z\hfill \end{array}\right]}^T$ . According to the kinematic modeling defined by, ³⁰ the geometry of every kinematic chain of the manipulator can be set by six parameters; therefore, $s_{\mathrm{Pi}}$ , $w_{\mathrm{Pi}}$ , $u_{\mathrm{Pi}}$ define the geometry of the moving base, $w_{\mathrm{Bi}}$ defines the fixed basis and $l_i$ and $L_i$ define the length of the upper and lower links, respectively for $i=1,2,3$ . Figure 9(a) presents the numerical values of the geometric parameters.

Application of the interval sensitivity analysis

The interval sensitivity quantifies how the interval geometric parameters affect the positioning error of the kinematic chains. The sensitivity analysis algorithm of Figure 3 is applied for every kinematic chain j = 1,2,3 considering the interval inputs $\overline{\mathbf{\lambda }}$ at the $\alpha$ -level ${\mu }_{\lambda }$  = 0.0, separately. Thus, six geometric parameters for every chain are considered, thus m = 6.

The geometric parameters of the manipulator are defined as triangular fuzzy variables $\tilde{\mathbf{\lambda }}=({\mathbf{\lambda }}_l,{\mathbf{\lambda }}_d,{\mathbf{\lambda }}_r)$ . Every fuzzy parameter is defined as follows: ${\tilde{\mathbf{\lambda }}}_i=({\lambda }_i-\mathrm{\Delta }d,{\lambda }_i,{\lambda }_i+\mathrm{\Delta }d)$ ; that is, the value of every geometric parameters ${\lambda }_i$ (see Table 2) varies $\pm \mathrm{\Delta }d$ . For this application, $\mathrm{\Delta }d$  = 0.001 m.

Table 2.

Geometric parameters of the Delta manipulator.

Parameter	$S_{\mathrm{Bi}}$	$S_{\mathrm{Pi}}$	$L_i$	$l_i$	$h_i$	$w_{\mathrm{Bi}}$	$u_{\mathrm{Bi}}$	$w_{\mathrm{Pi}}$	$u_{\mathrm{Pi}}$
Value (m)	0.567	0.076	0.524	1.244	0.131	0.164	0.327	0.022	0.044

The kinematic model computes the positioning error for every kinematic chain as a function of the geometric parameters and the end-effector position, $\mathbf{\lambda }$ and $\mathbf{p}$ of equation (A1), respectively. As presented in equation (18), the end-effector error $e_j$ of the $j-\mathrm{th}$ kinematic chain is stated by the difference of the position norm considering the nominal $\mathbf{\lambda }$ parameters and the fuzzy parameters $\overline{\mathbf{\lambda }}$ evaluated at the $\alpha -$ level, ${\mu }_{\lambda }$  = 0; in addition, the positioning error $e_j$ is evaluated at a specific end-effector position ${\mathbf{p}}_k$ .

$$e_j=||\mathbf{p}(\mathbf{\lambda },{\mathbf{p}}_k)-\mathbf{p}(\overline{\mathbf{\lambda }},{\mathbf{p}}_k)||$$

(18)

Therefore, according to equation (3), the model for the present application is $y=e_j(\mathbf{\lambda },{\mathbf{p}}_k)$ .

The optimization problems of equations (7) and (11) are solved by using the Genetic Algorithm (GA) ²⁸ of the Matlab optimization toolbox. The population was fixed as 20 individuals for every input interval parameter; the upper and lower limits for the search space are ${\mathbf{\lambda }}_l$ and ${\mathbf{\lambda }}_r$ , respectively; the other settings were considered as default to run the GA.

Results and analysis

Eighth different end-effector positions over the workspace are considered in this analysis as presented in Figure 10. Figure 10 presents the workspace and the analyzed end-effector positions, thus the plane xy for $z=-1$  m (see Figure 10(a)) and the plane xz for $y=0$  m (see Figure 10(b)).

Figure 10.

Workspace of the Delta manipulator: (a) plane xy for $z=-1$  m and (b) plane xz for $y=0$  m.

Initially, the positioning error sensitivity to the dimensional tolerances is computed by using the derivative-based local (see Appendix A) and the interval sensitivity to compare these two approaches. Figure 11 shows the sensitivity coefficients $\partial \mathbf{p}/\partial {\lambda }_i$ by using the derivative-based local sensitivity and the indices obtained with interval sensitivity ${\eta }_i$ . The derivative-based local method and the interval method quantify relatively similar sensitivities for the analyzed end-effector positions. Nevertheless, the intensity of the local coefficients and interval indices has not the same comparative proportion because the derivative-based method expresses the sensitivity at a specific position and the interval approach considers the sensitivity for the input dimensional tolerances. There are numerical differences between the local and interval method; although, the dimensional tolerances in the present analysis are relatively small ±0.001 m for every parameter.

Figure 11.

Sensitivity indices by using derivative-based local and interval sensitivity approaches: (a) at ${\mathbf{p}}_A$ , (b) at ${\mathbf{p}}_G$ , and (c) at ${\mathbf{p}}_D$ .

For the comparative proposed, the dimensional tolerance of the geometric parameters was considered small regarding the numerical values of link lengths. The local method cannot evaluate large variations of input parameters within the kinematic models, such as variations of joint positions. For these cases, the interval sensitivity has a potential advantage over the local methods since it can include a wide variation of the inputs to the sensitivity computation.

Finally, the interval sensitivity is computed for the workspace trajectory that passes at the positions: ${\mathbf{p}}_G$ , ${\mathbf{p}}_F$ , and ${\mathbf{p}}_D$ (see Figure 12). The interval sensitivity approach quantifies the sensitivity of every dimensional tolerance of the geometric parameters defined as intervals for every kinematic chain of the manipulator. The interval sensitivities vary as a function of the workspace position of the end-effector. These interval sensitivities show the end-effector positions in which the dimensional tolerances are more sensitive to the positioning error. Therefore, the interval sensitivity analysis could be used as a complementary design procedure of the parallel manipulator that determines which dimensional tolerances should be minimized to reduce the positioning error.

Figure 12.

Interval sensitivity indices for every kinematic chain: (a) chain j = 1, (b) chain j = 2, and (c) chain j = 3.

As a benefit, the interval sensitivity approach permits the assessment of the effect of the interval tolerances of geometric parameters on the positioning error of the end-effector differently from the derivative-based method that quantifies the local sensitivity.

Fuzzy-interval assessment of the workspace size

This case study assesses how the fuzzy geometric parameters affect the workspace size by using the fuzzy interval definition and solving the interval sensitivity analysis. Therefore, the workspace size sensitivity against the variation of the geometric parameters modeled as fuzzy-intervals is evaluated. This sensitivity analysis evaluates how every geometric parameter defined within a maximum limit and a minimum limit influences the workspace size.

The Cartesian parallel manipulator is showed in Figure 13. The manipulator is composed of three symmetric kinematic chains; that is, the first and second link lengths ( $l_1$ and $l_2$ ) of every kinematic chain are equal. Each kinematic chain owns three revolute passive joints ( ${\theta }_{j,i}$ , $j=x,y,z$ , and for $i=1,2,3$ ) and an active prismatic joint ( $\underset{¯}{P}$ ); thus $\underset{¯}{P}\mathrm{RRR}$ (see Figure 13(b)). The three prismatic joints act along the X, Y, and Z axes of the inertial frame O. The moving platform has constant orientation and it has three translational DOFs $(x,y,z)$ . These prismatic active joints are defined as $\mathbf{q}={\left[\begin{array}{ccc}{q}_x\hfill & q_y\hfill & q_z\hfill \end{array}\right]}^T$ . The prismatic active joints directly defines the Cartesian position P. Thus, $q_x=x$ , $q_y=y$ , and $q_z=z$ . Therefore, the Jacobian matrix $\mathbf{J}={\mathbb{I}}_{3x3}$ .

Figure 13.

Cartesian Parallel Manipulator (CPM): (a) CAD model and design variables and (b) kinematic chains and workspace.

The geometry of the manipulator can be defined by four parameters: the link lengths of every kinematic chain $l_1$ , $l_2$ , the prismatic joints extension $d_q$ , and the moving platform length $l_p$ (see Figure 13(a)). Consequently, the parameters that define the manipulator’s geometry can be set in $\mathbf{\lambda }$ :

$$\mathbf{\lambda }={\left[\begin{array}{cccc}{l}_1\hfill & l_2\hfill & l_p\hfill & d_q\hfill \end{array}\right]}^T$$

(19)

An initial definition of the geometric parameters of the Cartesian parallel manipulator is presented in Table 3. This definition was adopted to illustrate the CAD model of Figure 13(a).

Table 3.

Geometric parameters of the CPM.

Parameter	$l_1$	$l_2$	$l_p$	$d_q$
Value (m)	0.075	0.075	0.015	0.100

Flowchart to determine the workspace size

The kinematic chain’s symmetry (see Figure 13(b)) makes the workspace shape, w, to be a cube with $\mathrm{\Delta }w$ edge length.

$$w=\mathrm{\Delta }w\times \mathrm{\Delta }w\times \mathrm{\Delta }w$$

(20)

The flowchart of Figure 14 determines the edge length $\mathrm{\Delta }w$ of the workspace. The algorithm crosses the moving platform (from the Cartesian position ${\mathbf{p}}_i$ until ${\mathbf{p}}_f$ see Figure 13(b)) tracking a diagonal motion with the following coordinates: $x=\delta$ , $y=\delta$ and $z=\delta$ with $\delta \in [0,d_q]$ .

Figure 14.

Flowchart to determine the workspace size $\mathrm{\Delta }w$ .

The bounds of the workspace are ${\mathbf{p}}_i$ and ${\mathbf{p}}_f$ (see Figure 13(b)). To belong to the workspace, the end-effector position $(x,y,z)$ should fulfill the statement of equation (21).

$$c(x,y,z):\{\begin{array}{c}-1\le \frac{{(x-l_p)}^2+y^2-l_1^2-l_2^2}{2l_1{l}_2}\le 1\hfill \\ \multicolumn{1}{c}{-1\le \frac{{(x-l_p)}^2+z^2+l_1^2-l_2^2}{2l_1\sqrt{{(x-l_p)}^2+z^2}}\le 1}\\ \multicolumn{1}{c}{-1\le \frac{{((x-d_q)-l_p)}^2+{(y-d_q)}^2-l_1^2-l_2^2}{2l_1{l}_2}\le 1}\end{array}$$

(21)

This statement ensures a feasible solution to the kinematic chain’s inverse kinematics within the workspace.

Application of the interval sensitivity analysis

This analysis aims to determine the geometric parameters’ effect on the workspace size. Every geometric parameter of the vector $\mathbf{\lambda }$ of equation (19) is defined as an triangular fuzzy variable, thus ${\tilde{\mathbf{l}}}_1=(l_{1l},l_1,l_{1r})=(0.050,0.075,0.100)$  m, ${\tilde{\mathbf{l}}}_2=(l_{2l},l_2,l_{2r})=(0.050,0.075,0.100)$  m, ${\tilde{\mathbf{l}}}_p=(l_{\mathrm{pl}},l_p,l_{\mathrm{pr}})=(0.005,0.015,0.025)$ and ${\tilde{\mathbf{d}}}_q=(d_{\mathrm{ql}},d_q,d_{\mathrm{qr}})=(0.050,0.100,0.150)$  m. Thus, the vector $\mathbf{\lambda }$ of equation (19) is defined as fuzzy vector $\tilde{\mathbf{\lambda }}=\left[\begin{array}{c}{\tilde{\mathbf{l}}}_1,{\tilde{\mathbf{l}}}_2,{\tilde{\mathbf{l}}}_p,{\tilde{\mathbf{d}}}_q\hfill \end{array}\right]$ ; for the sensitivity analysis, $m=$ 4.

For the present analysis, the numerical model corresponds to the workspace size according to the definition of equation (3) and the flowchart to compute the dimension of the cuboid $\delta w$ presented in Section 3.3.1 is $y=\mathrm{\Delta }w(\mathbf{\lambda })$ . The definition of the workspace size does not depend on the pose of the end-effector. The optimization problems of equations (7) and (11) are also solved by using the Genetic Algorithm (GA) ²⁸ of the Matlab optimization toolbox with the same settings in the analysis of Section 3.2.

Results and analysis

As a previous analysis, before the sensitivity analysis, the workspace size defined by $\mathrm{\Delta }w(\mathbf{\lambda })$ is evaluated by considering the variations of the geometric parameters within maximum and minimum limits. For this analysis, the maximum extension of the prismatic joints, and the link lengths are considered in two different $\alpha$ -levels: ${\mu }_{\lambda }=$ 0.5 and ${\mu }_{\lambda }=$ 0.0; thus, two interval geometric parameters are considered ${\overline{\mathbf{l}}}_i={\overline{\mathbf{l}}}_1={\overline{\mathbf{l}}}_2$ , ${\overline{\mathbf{d}}}_q$ , and the moving platform length is defined as a constant.

It is worth to mention that this interval definition of the geometric parameters can be assumed as the design space, that is, a search domain in which the workspace size can be maximized. Consequently, two design spaces are considered that were selected to evaluate the variation of parameters around the numerical values of Table 3, ${\mu }_{\lambda }=$ 0.5: ${\overline{\mathbf{l}}}_i=[0.0625,0.0875]$  m, ${\overline{\mathbf{d}}}_q=[0.075,0.125]$  m, and $l_p=0.015$  m; ${\mu }_{\lambda }=$ 0.0: ${\overline{\mathbf{l}}}_i=[0.050,0.100]$  m, ${\overline{\mathbf{d}}}_q=[0.050,0.150]$  m, and $l_p=0.015$  m.

The results of Figure 15 show that for ${\mu }_{\lambda }=$ 0.5 and ${\mu }_{\lambda }=$ 0.0, the workspace size $\mathrm{\Delta }w$ reaches its maximum value when the link lengths ( $l_i$ ) and the maximum extension of the prismatic joints ( $d_q$ ) increase. Nevertheless, this analysis is limited to examine two variables simultaneously to obtain a workspace size plot as a function of $l_i$ and $d_q$ . Moreover, the workspace size shows a great variability over the design space defined by the interval geometric parameters (see Figure 15).

Figure 15.

Evaluation of the simplified design space of the workspace size ( $\mathrm{\Delta }w$ ): (a) ${\mu }_{\lambda }=$ 0.5 and (b) ${\mu }_{\lambda }$ = 0.0.

The four fuzzy. Figure 16 shows the interval sensitivity indices of four different cases in Table 4. Case 1 shows that the sensitivity of the four geometric parameters is equivalent (see Figure 16(a)); nevertheless, the interval workspace output $\overline{\delta \mathbf{w}}$ is the smallest of the considered cases. For case 2, by considering this search space, the bounds of $\overline{\delta \mathbf{w}}$ are increased; this implies that the obtained maximum workspace size is greater than case 1 (see Figure 16(b)).

Figure 16.

Sensitivity indices of the workspace size ( $\mathrm{\Delta }w$ ) against the design variables: (a) case 1, (b) case 2, (c) case 3, and (d) case 4.

Table 4.

Sensitivity outputs for the corresponding $\alpha$ -levels of $\tilde{\mathbf{\lambda }}$ .

Case	${\mu }_{l_1}$	${\mu }_{l_2}$	${\mu }_{l_p}$	${\mu }_{d_q}$	$\overline{\mathrm{\Delta }\mathbf{w}}$ (m)
1	0.5	0.5	0.5	0.5	[0.0200, 0.1090]
2	0.5	0.5	0.5	0.0	[0.0000, 0.1130]
3	0.0	0.0	0.0	0.0	[0.0000, 0.1360]
4	1.0	0.8	0.0	0.0	[0.0000, 0.1360]

The definition of the search space of case 3 increases the maximum workspace; nevertheless, the sensitivity of the parameters is not equivalent (see Figure 16(c)). These specific conditions imply that the interval ${\overline{\mathbf{l}}}_1$ parameter has a larger influence on the workspace size than the interval the platform length ${\overline{\mathbf{l}}}_p$ .

The case 4 (see Figure 16(d)) shows a satisfactory definition of the search space. The workspace size attaints the maximum value among the cases analyzed, and the sensitivity of the interval parameters are equalized. This condition indicates that the interval parameters have the same relative influence on the workspace size.

Finally, the workspace size optimization is carried out based on the design space considered in the four cases of Table 4. The optimization problem definition is given by:

$$\underset{\mathbf{\lambda }\in \overline{\mathbf{\lambda }}}{max}\mathrm{\Delta }w(\mathbf{\lambda })$$

(22)

The optimal design variables derived from the optimization solution ${\mathbf{\lambda }}_{\max }$ are presented in Table 5. As predicted by the interval sensitivity, the maximum workspace $\mathrm{\Delta }{w}_{\max }$ is obtained for cases 3 and 4.

Table 5.

Results of the Cartesian parallel manipulator (CPM) optimization.

Case	${\mathbf{\lambda }}_{\max }$ (m)	$\mathrm{\Delta }{w}_{\max }$ (m)
1	${\left[\begin{array}{cccc}0.083\hfill & 0.081\hfill & 0.010\hfill & 0.129\hfill \end{array}\right]}^T$	0.1090
2	${\left[\begin{array}{cccc}0.085\hfill & 0.085\hfill & 0.010\hfill & 0.134\hfill \end{array}\right]}^T$	0.1130
3	${\left[\begin{array}{cccc}0.099\hfill & 0.099\hfill & 0.006\hfill & 0.149\hfill \end{array}\right]}^T$	0.1360
4	${\left[\begin{array}{cccc}0.099\hfill & 0.099\hfill & 0.005\hfill & 0.146\hfill \end{array}\right]}^T$	0.1360

The four fuzzy input parameters of equation (3) are considered as intervals weighted by their respective $\alpha -$ levels; thus, the $\alpha -$ levels of Table 5 are evaluated. Therefore, this analysis aims at assessing the workspace size variation for different interval definitions of the design variables by solving the sensitivity analysis. Figure 17 shows the evolution of the best solution along the generations for the solution of the optimization problem of equation (22) that is solved by using the genetic algorithm (GA). For the case 4, the optimization reaches the best solution with less generations than the other cases because the correspondent sensitive indices indicate an equivalent influence on the objective function $\mathrm{\Delta }w$ (see Figure 16(d)).

Figure 17.

Optimization results.

It observed that the main benefit of the proposed approach is to improve the optimal design by evaluating the influence of the search space sensitivity on the optimal solution. For this case study, the sensitivity analysis permitted to improve the optimal solution by evaluating the sensitivity of several interval inputs that correspond to the search space of the optimization. The best optimal design was obtained by defining an equivalent sensitivity of the design variables over the search space.

Conclusion

This contribution presented an approach based on the fuzzy-interval analysis to assess how dimensional tolerances modeled as fuzzy variables affect the performance of parallel manipulators. In addition, the sensitivity analysis based on interval analysis permitted computing sensitivity indices by comparing the interval inputs and outputs. Initially, the positioning error of a 5R parallel manipulator was analyzed. The second case study illustrated the geometric parameters sensitivity on the end-effector positioning of a Delta manipulator. The third case study considered the Cartesian parallel manipulator to evaluate the design space aiming at maximizing the workspace size.

The proposed fuzzy-interval analysis showed to be a complementary procedure in the optimal design applied on parallel manipulators. The fuzzy-interval approach permitted to model the dimensional tolerance as intervals weighted by the membership function; thus, different scenarios can be obtained to assess the impact of these dimensional tolerances on the analyzed kinematic output variation. Moreover, the interval approach permitted to compute the input parameters sensitivity modeled as intervals, that is, the sensitivity is computed not only for a single input but for a range of input models as intervals. Moreover, the interval sensitivity demonstrated to be useful to determine the influence or sensitivity of every design variable on a given design criterion to be optimized. For example, in some cases, the design criterion was not sensitive or this sensitivity was low to a design variable. Therefore, this design variable can be removed from the design space to enhance the optimization procedure.

Future work will analyze of performance indices of manipulators, such as dexterity or manipulability and, the results will be verified through experiments. Moreover, the fuzzy-interval analysis of dynamic performance to include these methodologies in the optimization and control of manipulators.