Kinematic reliability and sensitivity analysis of the modified Delta parallel mechanism

Abstract

In order to improve the kinematic reliability, it is crucial to find out the influence of each error source on the kinematic reliability of the mechanism. Reliability sensitivity analysis is used to find the changing rate in the probability of reliability in relation to the changes in distribution parameters. Based on the structural response surface function method, the functional relation between the kinematic reliability of a modified Delta parallel mechanism and the original input-error vectors is described using the quadratic function with cross terms. Moreover, the partial derivatives of the functional relation with respect to the means and variances of the original input errors are derived, which can efficiently evaluate kinematic reliability sensitivity of the mechanism. The advantages of this method are as follows: First, the response surface function, which can be easily set up by the position-error model of the mechanism, is convenient for calculating the variance, partial derivative, and reliability sensitivity. Second, in this case (unlike in the traditional error-mapping model), although the input-error values are unknown, pseudorandom variables used as random input-error sources can be generated by MATLAB software. Furthermore, the kinematic reliability of the mechanism can be assessed using the Monte Carlo method.

Introduction

The Delta parallel mechanism was presented by Clavel ¹ in 1988, as a high-speed parallel mechanism with three degrees of freedom (3-DOF). Currently, it is one of the most successful parallel mechanisms for commercial applications. However, it has the drawback of possessing several spherical joints, which are difficult to assemble and whose accuracy is difficult to guarantee. Tsai ^{2 –4} invented a modified 3-DOF Delta parallel mechanism. Its most important properties are that all the kinematic joints are rotational joints and that it provides large workspace, high stiffness, low inertia, and large payload capacity.

Kinematic reliability is defined as the probability that the end effector position and orientation fall within a specified range of the desired location. Because of geometric errors in the components of the mechanism, as well as the position and pose errors of the joints, the end effector position of the mechanism will deviate from the ideal kinematic trajectory, thereby reducing the mechanism’s kinematic accuracy. ^5,6 Therefore, it is particularly important to determine whether the mechanism can accurately complete the specified motion by analyzing the kinematic reliability of this mechanism. The degree of influence of the errors on this kinematic reliability can be analyzed by means of kinematic reliability sensitivity analysis. ^{7 –16} At present, studies on reliability sensitivity analysis mainly focus on the structural system of the mechanism. Reliability sensitivity analysis research on the kinematic system of the mechanism is limited, and breakthroughs have been scarce. ^17,18

This study aims to establish a kinematic-error model and analyze the kinematic reliability and reliability sensitivity of the modified Delta parallel mechanism, thus providing theoretical support for the design of such high-precision mechanisms.

Descriptions of the modified Delta parallel mechanism

The modified Delta parallel mechanism is shown in Figure 1.

Figure 1.

Modified Delta mechanism.

The mechanism comprises a moving platform connected to a fixed platform by three parallel kinematic chains. Each chain contains a rotational joint activated by actuators on the fixed platform. The motion is transmitted to the moving platform through parallelograms formed by follower links and rotational joints, as shown in Figure 2.

Figure 2.

Descriptions of the kinematic chain.

The end effector of the mechanism is installed at the center of the moving platform; its coordinates are expressed as [x, y, z]. The rotational joints at points B_i and D_i (i =1, 2, 3) are placed in accordance with the equilateral triangle on the fixed and moving platforms.

The global coordinate system O-xyz is located at the center (point O) of the fixed platform. The xz plane coincides with the plane of the fixed platform, and the x-axis points to the center of the joint at point B ₁ and the y-axis points vertically upward, as shown in Figure 3.

Figure 3.

Coordinate system of the kinematic chain.

The reference coordinate system, A_i -x_iy_iz_i (i = 1, 2, 3), is obtained when O-xyz revolves by an angle β_i (i = 1, 2, 3) around the y-axis, where β_i is 0°, 120°, or 240°, respectively.

The body-fitted coordinate system B_i -x_iy_iz_i , C_i -x_iy_iz_i , and D_i -x_iy_iz_i are located at points B_i , C_i , and D_i , respectively. Their z_i -axes coincide with the axes of the joints B_i , C_i , and D_i , and their y_i -axes point vertically upward.

The body-fitted coordinate system of the moving platform is given by P-x_iy_iz_i , whose x_iz_i plane coincides with that of the moving platform and whose x_i -axis points to the center of joint D ₁ and the y_i -axis points vertically upward. The reference coordinate system, F_i -x_iy_iz_i (i = 1, 2, 3), is obtained when P-x_iy_iz_i revolves β_i along the y_i -axis.

Kinematic-error model of the mechanism

According to Figure 3, a closed loop equation can be written for each chain

$\begin{array}{l}\mathit{O}\mathit{P}=\mathit{O}{\mathit{B}}_i+{\mathit{B}}_i{\mathit{C}}_i+{\mathit{C}}_i{\mathit{C}}_{ij}+{\mathit{C}}_{ij}{\mathit{D}}_{ij}+{\mathit{D}}_{ij}{\mathit{D}}_i-\mathit{P}{\mathit{D}}_i\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(i=1-3;j=1-2)\end{array}$

1

Rewriting equation (1) in the O-xyz coordinate frame leads to

$\begin{array}{c}\mathit{P}={}^o\mathit{R}{\hspace{0.17em}}_{ai}\left({\mathit{r}}_{1i}+{}^{ai}\mathit{R}{\hspace{0.17em}}_{bi}\left(a_i{\mathit{u}}_i\text{+}{\left(-1\right)}^j{}^{bi}\mathit{R}{\hspace{0.17em}}_{ci}\raisebox{1ex}{$e{}_{1ij}\hspace{0.17em}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\mathit{E}}_2\right)\right)\\ +c{\mathit{w}}_i-{}^o\mathit{R}{\hspace{0.17em}}_{fi}\left({\mathit{r}}_{2i}+{\left(-1\right)}^j{}^{fi}\mathit{R}{\hspace{0.17em}}_{di}\raisebox{1ex}{$e{}_{2ij}\hspace{0.17em}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\mathit{E}}_2\right)\\ (i=1-3;j=1-2)\end{array}$

2

where E ₂ = (0, 0, 1)^T and P is the position vector of the end effector.

Under small perturbations, the position of the end effector can be expressed as

$\begin{array}{c}\mathit{P}+\Delta \mathit{X}={}^o\mathit{R}{\hspace{0.17em}}_{ai}\left({\mathit{r}}_{1i}+\Delta {\mathit{r}}_{1i}\right)+\left(c{\mathit{w}}_i+\Delta c_{ij}{\mathit{w}}_i+c\Delta {\mathit{w}}_i\right)\\ -\mathrm{T}{}^p\mathit{R}{\hspace{0.17em}}_{fi}\left({\mathit{r}}_{2i}+\Delta {\mathit{r}}_{2i}+{\left(-1\right)}^j(\raisebox{1ex}{$e_{1ij}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\Delta e_{2ij})\mathbf{\Xi }{E}_2\right)\\ +{}^o\mathit{R}{\hspace{0.17em}}_{ai}\Psi \left(\left(a_i{\mathit{u}}_i+\Delta a_i{\mathit{u}}_i\right)+{\left(-1\right)}^j(\raisebox{1ex}{$e_{2ij}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\Delta e_{1ij})\mathbf{\Xi }{\mathit{E}}_2\right)\\ (i=1-3;j=1-2)\end{array}$

3

where E ₁ is a 3 × 3 unit matrix, $T=\left({\mathit{E}}_1+\mathit{\theta }\times \right)$ , $\Xi =\left({\mathit{E}}_1+{\mathit{\theta }}_{Ci}\times \right)$ , $\Psi =\left({\mathit{E}}_1+{\mathit{\theta }}_{Bi}\times \right)$ .

To obtain the position-error vectors Δ X of the end effector, the following is done. Substituting P from equation (2) into equation (3) and simplifying, yields

$\begin{array}{c}\Delta \mathit{X}={}^o\mathit{R}{\hspace{0.17em}}_{ai}\Lambda +\Delta c_{ij}{\mathit{w}}_i+c\Delta {\mathit{w}}_i-\mathrm{T}{}^p\mathit{R}{\hspace{0.17em}}_{fi}\left({\mathit{r}}_2+{\left(-1\right)}^j\raisebox{1ex}{$e_{1ij}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\mathit{E}}_2\right)\\ -{}^p\mathit{R}{\hspace{0.17em}}_{fi}\left(\Delta {\mathit{r}}_{2i}+\Phi \left({\left(-1\right)}^j\raisebox{1ex}{$e_{1ij}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\mathit{E}}_2\right)+\left({\left(-1\right)}^j\Delta e_{2ij}{\mathit{E}}_2\right)\right)\\ (i=1-3;j=1-2)\end{array}$

4

where $\Phi =\left({\mathit{E}}_1+{\mathit{\theta }}_{Di}\times \right)$ and $\begin{array}{c}\Lambda =\Delta {\mathit{r}}_{1i}+\Psi \left(a_i{\mathit{u}}_i+{\left(-1\right)}^j\raisebox{1ex}{$e_{1ij}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\mathit{E}}_2\right)+\Delta a_i{\mathit{u}}_i\\ +\Xi \left({\left(-1\right)}^j\raisebox{1ex}{$e_{1ij}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\mathit{E}}_2\right)+{\left(-1\right)}^j\Delta e_{1ij}{\mathit{E}}_2\end{array}$ .

To eliminate the unit pose-error vector of the follower link, we multiply both sides of equation (4) by w _i ^T and simplify, which yields

$\begin{array}{c}{\mathit{w}}_i^{\mathrm{T}}\Delta \mathit{X}={\mathit{w}}_i^{\mathrm{T}}{}^o\mathit{R}{\hspace{0.17em}}_{ai}\Delta {\mathit{k}}_i+a_i{\left({}^o\mathit{R}{\hspace{0.17em}}_{ai}{\mathit{u}}_i\times {\mathit{w}}_i\right)}^{\mathrm{T}}{}^o\mathit{R}{\hspace{0.17em}}_{ai}{\mathit{\theta }}_{Bi}+\Delta c_{ij}\\ +{\left(-1\right)}^j\left(\raisebox{1ex}{$e_{1ij}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\left({}^o\mathit{R}{\hspace{0.17em}}_{ai}{\mathit{E}}_2\times {\mathit{w}}_i\right)}^{\mathrm{T}}{}^o\mathit{R}{\hspace{0.17em}}_{ai}{\mathit{\theta }}_i+\raisebox{1ex}{$\Delta e_{1ij}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\mathit{w}}_{{\hspace{0.17em}}_i}^{\mathrm{T}}{}^o\mathit{R}{\hspace{0.17em}}_{ai}{\mathit{E}}_2\right)\\ +{\mathit{w}}_i^{\mathrm{T}}{}^o\mathit{R}{\hspace{0.17em}}_{ai}{\mathit{u}}_i\Delta a_i-{\left({}^p\mathit{R}{\hspace{0.17em}}_{fi}\left({\mathit{r}}_{2i}+{\left(-1\right)}^j\raisebox{1ex}{$e_{2ij}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\mathit{E}}_2\right)\times {\mathit{w}}_i\right)}^{\mathrm{T}}\mathit{\theta }\\ (i=1-3;j=1-2)\end{array}$

5

where Δ k _i = Δ r _1i − Δ r _2i and θ _i = θ _Bi + θ _Ci − θ _Di.

Subtracting equation (5) at j = 2 from equation (5) at j = 1 leads to

$\begin{array}{c}{e}_{1ij}{\left({}^o\mathit{R}{\hspace{0.17em}}_{ai}{\mathit{E}}_2\times {\mathit{w}}_i\right)}^{\mathrm{T}}\mathit{\theta }=\Delta c_{id}+\Delta e_{1ij}{\mathit{w}}_{{\hspace{0.17em}}_i}^{\mathrm{T}}{}^o\mathit{R}{\hspace{0.17em}}_{ai}{\mathit{E}}_2\\ +e_{2ij}{\left({}^p\mathit{R}{\hspace{0.17em}}_{fi}{\mathit{E}}_2\times {\mathit{w}}_i\right)}^{\mathrm{T}}{}^o\mathit{R}{\hspace{0.17em}}_{ai}{\mathit{\theta }}_i\\ (i=1-3)\end{array}$

6

where Δc_id = Δc_{i 2} − Δc_{i 1.}

Simplifying equation (6), the pose-error mapping model of the mechanism can be constructed as

$\mathit{\theta }={\mathit{J}}_{\theta }{\epsilon }_{\theta }$

7

where J _θ = A ⁻¹ B , A = e _1ij [ ^o R _{a 1} E ₂ × w ₁ ^o R _{a 2} E ₂ × w ₂ ^o R _{a 3} E ₂ × w ₃]^T, B = diag[ B _i ]^T, B _i = [1 w _i ^To R _ai E ₂ e _2ij ( ^o R _ai × w _i )^To R _ai ], ε _θ = [ ε _{θ 1} ^T ε _{θ 2} ^T ε _{θ 3} ^T ]^T, ε _θi ^T = [Δc_id Δe_i θ _i ^T]^T, and Δe_i = 2(Δe _1ij − Δe _2ij ).

Adding equation (5) at j = 2 and equation (5) at j = 1 leads to

$\begin{array}{c}{\mathit{w}}_i^{\mathrm{T}}\Delta \mathit{X}=\Delta c_{Pi}+{\mathit{w}}_i^{\mathrm{T}}{}^o\mathit{R}{\hspace{0.17em}}_{ai}{\mathit{u}}_i\Delta a_i+{\mathit{w}}_i^{\mathrm{T}}{}^o\mathit{R}{\hspace{0.17em}}_{ai}\Delta {\mathit{k}}_i\\ +a_i{\left({}^o\mathit{R}{\hspace{0.17em}}_{ai}{\mathit{u}}_i\times {\mathit{w}}_i\right)}^{\mathrm{T}}{}^o\mathit{R}{\hspace{0.17em}}_{ai}{\mathit{\theta }}_{Bi}-{\left({}^p\mathit{R}{\hspace{0.17em}}_{fi}{\mathit{r}}_{2i}\times {\mathit{w}}_i\right)}^{\mathrm{T}}\mathit{\theta }\\ (i=1-3)\end{array}$

8

where Δc_Pi = (Δc_{i 2} + Δc_{i 1})/2.

Substituting θ in equation (7) into equation (8), the position-error mapping model of the mechanism can be constructed as

$\Delta \mathit{X}={\mathit{J}}_P\mathit{\epsilon }=\left[\begin{array}{cc}{\mathit{J}}_{PP}& {\mathit{J}}_{PJ}\end{array}\right]\left[\begin{array}{c}{\mathit{\epsilon }}_{PP}\\ {\mathit{\epsilon }}_{\theta }\end{array}\right]$

9

where J _PP = C ⁻¹ D , C = [ w ₁ w ₂ w ₃]^T, D = diag[ D _i ]^T, D _i = [1 w _i ^To R _ai u _i w _i ^To R _ai a( ^o R _ai u _i × w _i )^To R _ai ], J _PJ = C ⁻¹ EJ _θ , E = −[ ^o R _{a 1} r ₂ × w ₁ ^o R _{a 2} r ₂ × w ₂ ^o R _{a 3} r ₂ × w ₃]^T, ε _PP = [ ε _{PP 1} ^T  ε _{PP 2} ^T ε _{PP 3} ^T ]^T, ε _PPi = [Δc_iP Δa_i Δ k _i ^T θ _Bi ^T]^T, and Δ k _i = Δ r _1i − Δ r _2i .

Equations (7) and (9) show that 21 geometric errors affect the position and pose precision of the end effector, 9 geometric errors only affect the position precision of the end effector.

Kinematic reliability analysis of the mechanism using the Monte Carlo method

The Monte Carlo method is a numerical calculation method used to find an approximate solution to technical engineering problems by generating random statistical sampling data using a computer.

The process of evaluating kinematic reliability using the Monte Carlo method is as follows:

The mean values of the geometric errors of all components and joints were obtained according to the upper- and lower-limit deviations of all geometric errors because the geometric errors of individual parts obey the normal distribution, so the error means μ_i and variance σ_i can be obtained.

The random-error parameters Δy_ij (i = 1, 2…, m, where m is the number of error sources; j = 1, 2…, n, where n is the number of generated samples) can be generated using the normrnd(μ_i , σ_i ) function in MATLAB [version 6.5]. The kinematic errors of this mechanism can be obtained by substituting Δy_ij into equations (7) and (9). By repeating the above operation n times, n samples ΔX_i (i = 1, 2…, n) of the output kinematic error can be obtained. Keep count of numbers (s = s + 1) if the output kinematic error is less than or equal to the permissible error (ΔX_i ≤ Δ) before achieving the number of tests. According to the law of large numbers, the kinematic reliability of the mechanism is R = s/n. A flowchart of the reliability calculation is shown in Figure 4.

Figure 4.

Flowchart of kinematic reliability analysis.

Reliability sensitivity analysis

The response surface function is based on the idea that when the output response is unknown, the relation between the output response and the input random parameter vector can be described by a quadratic function with a cross term.

By constructing the response surface function, the mapping relation between the random variables of the input errors and the position error of the mechanism can be established; namely, the functional relation between the position error of the mechanism, ΔX, and the input-error vector, y = (y ₁, y ₂,…, y_n ), can be described by a quadratic function containing the cross term, as shown below

$\begin{array}{c}\Delta X=C_0+{\displaystyle \sum _{i=1}^n{C}_i{y}_i}+{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=i}^n{C}_{ij}{y}_i{y}_j}}\\ (i=1\dots n;j=i\dots n)\end{array}$

10

where C ₀, C_i , C_ij (i = 1…n; j = i…n) are undetermined coefficients.

When the permissible error obeys the normal distribution, the limit state function of kinematic reliability is

$\begin{array}{c}G\left(X\right)=\Delta -\Delta X=\Delta -\left(C_0+{\displaystyle \sum _{i=1}^n{C}_i{y}_i}+{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=i}^n{C}_{ij}{y}_i{y}_j}}\right)\\ (i=1\dots n;j=i\dots n)\end{array}$

11

According to the Monte Carlo method, the mean and variance of the output position error of the mechanism can be obtained as

$\{\begin{array}{c}\mu =\frac{1}{s}{\displaystyle \sum _{i=1}^s\Delta X_i}\\ {\sigma }^2=\frac{1}{s-1}{\displaystyle \sum _{i=1}^s{(\Delta X_i-\mu )}^2}\end{array}$

12

The reliability index is obtained as

$\beta =\frac{\mu }{\sqrt{D}}$

13

Because G(X) follows a normal distribution, the kinematic reliability of the mechanism can also be expressed as

$R=P(\Delta X<\Delta )=\Phi (\beta )$

14

where Φ(•) is the standard normal distribution function.

Differentiating equation (14) with respect to the mean matrix, μ _i , and the variance matrix, D _i , of the random-parameter error, the kinematic reliability sensitivity is obtained

$\frac{\partial R}{\partial {\mu }_i^{\mathrm{T}}}=\frac{\partial R}{\partial \beta }\left(\frac{\partial \beta }{\partial \mu }\frac{\partial \mu }{\partial {\mu }_i^{\mathrm{T}}}+\frac{\partial \beta }{\partial D}\frac{\partial D}{\partial {\mu }_i^{\mathrm{T}}}\right)(i=1\dots n)$

15

$\frac{\partial R}{\partial {\mathit{D}}_i^{\mathrm{T}}}=\frac{\partial R}{\partial \beta }\left(\frac{\partial \beta }{\partial \mu }\frac{\partial \mu }{\partial {\mathit{D}}_i^{\mathrm{T}}}+\frac{\partial \beta }{\partial D}\frac{\partial D}{\partial {\mathit{D}}_i^{\mathrm{T}}}\right)(i=1\dots n)$

16

where $\frac{\partial R}{\partial \beta }=\Phi \left(\beta \right)$ , $\frac{\partial \beta }{\partial \mu }=\frac{1}{\sqrt{D}}$ , $\frac{\partial \beta }{\partial D}=-\frac{\mu }{2}{D}^{-\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ , $\frac{\partial \mu }{\partial {\mu }_i^{\mathrm{T}}}={\left[\frac{\partial \mu }{\partial {\mu }_1},\frac{\partial \mu }{\partial {\mu }_2},\text{K},\frac{\partial \mu }{\partial {\mu }_n}\right]}^{\mathrm{T}}$ , $\frac{\partial \mu }{\partial {\mathit{D}}_i^{\mathrm{T}}}={\left[\frac{\partial \mu }{\partial D_1},\frac{\partial \mu }{\partial D_2},\text{K},\frac{\partial \mu }{\partial D_n}\right]}^{\mathrm{T}}$ , $\frac{\partial D}{\partial {\mu }_i^{\mathrm{T}}}={\left[\frac{\partial D}{\partial {\mu }_1},\frac{\partial D}{\partial {\mu }_2},\text{K},\frac{\partial D}{\partial {\mu }_n}\right]}^{\mathrm{T}}$ , and $\frac{\partial D}{\partial {\mathit{D}}_i^T}={\left[\frac{\partial D}{\partial D_1},\frac{\partial D}{\partial D_2},\text{K},\frac{\partial D}{\partial D_n}\right]}^{\mathrm{T}}$ .

Numerical example

The parameters of the mechanism are specified as follows: a_i = c = 1000 mm, e _1ij /2 = 50 mm, and r _1i = r _2i = [500 0 0]^T mm. The initial output position coordinate of the end effector is [0 1000 0]^T mm. The input angular accelerations of three driving arms are 0.5°/s², −0.5°/s², and 1°/s².

Numerical example of error analysis of the mechanism

The length errors of the driving arms are given as Δa ₁ = 0.5 mm, Δa ₂ = −1 mm, and Δa ₃ = −0.5 mm, and other errors are ignored. The output position error, Δ X , of the mechanism is shown in Figure 5.

Figure 5.

Effect of driving-arm-length error on output errors of the mechanism as a function of time.

The length errors of follower links are given as Δc ₂₁ = 0.5 mm, Δc ₂₂ = 0 mm, Δc ₂₁ = 0 mm, Δc ₂₂ = −0.5 mm, Δc ₃₁ = 1 mm, and Δc ₃₂ = −1 mm, and other errors are ignored. Δ X of the mechanism is shown in Figure 6.

Figure 6.

Effect of the follower link’s length error on output errors of the mechanism as a function of time.

The configuration-position-error vector of the joint at points B_i and D_i are given by Δ r ₁₁ = [1 0.5 −0.5]^T, Δ r ₂₁ = [−1 −0.5 0.5]^T, Δ r ₁₂ = [1 0.5 −0.5]^T, Δ r ₂₂ = [−1 −0.5 0.5]^T, Δ r ₁₃ = [1 0.5 −0.5]^T, and Δ r ₂₃ = [−1 −0.5 0.5]^T, and other errors are ignored. Δ X of the mechanism is shown in Figure 7.

Figure 7.

Effect of each chain’s position on the mechanism’s position errors.

The pose errors of the coordinate systems are given by B_i -x_iy_iz_i , C_i -x_iy_iz_i , and D_i -x_iy_iz_i : θ _Bi = θ _Ci = θ _Di = [0.5*pi/180 0.5*pi/180 0]^T, and other errors are ignored. Δ X of the mechanism is shown in Figure 8.

Figure 8.

Effect of the components’ attitude error on the mechanism’s position errors.

The length errors of the shafts C_iC_ij (   j = 1, 2) are given as Δe ₁₁₁ = 0.5, Δe ₁₂₁ = −0.5, Δe ₁₁₂ = 0.5, Δe ₁₂₂ = −0.5, Δe ₁₁₃ = 0.5, and Δe ₁₂₃ = −0.5, and other errors are ignored. Δ X of the mechanism is shown in Figure 9.

Figure 9.

Effect of the connecting-shaft-length error on the mechanism’s position errors.

The following conclusions are obtained from Figures 5 to 9.

The length errors of the driving arms have little influence on the position error of the mechanism. The effect of the follower links’ length errors on the position error is larger than that of the driving arms. They affect both the position and pose accuracy of the mechanism when the lengths of two follower links differ. The length errors of shafts C_iC_ij have a great influence on the position error of the mechanism. This effect will gradually increase with the lateral movement of the end effector. These factors, including the configuration-position errors and the pose errors of the joints, seriously affect the mechanism’s position and pose errors.

Numerical example of kinematic reliability analysis of the mechanism

The kinematic reliability of the mechanism can be analyzed using the Monte Carlo method when nominal dimensions and geometric tolerances of the parts are given, such as setting the lengths of the driving arms and follower links as a_i = 1000 ± 1 mm and c_i = 1000 ± 1 mm; the vector components of the configuration-position errors of the joints at B_i and D_i as r _1ix = r _2ix = ±1 mm and r _1iy = r _2iy =r _1iz = r _2iz = ±0.5 mm; the lengths of links C_iC_ij (j = 1, 2) as e _1i = e _2i = 100 ± 0.5 mm; and the vector components of the pose errors of the joints at points B_i , C_i , and D_i as θ_Bix = θ_Biy = ±0.3°, θ_Cix = θ_Ciy = ±0.3°, and θ_Dix = θ_Diy = ±0.3°. Three joints can rotate freely around the z-axis; therefore, the z components of pose errors are 0. The number of samples is n = 400.

When the mechanism has a 7-s motion and the maximum permissible limit of the position error of the mechanism is Δ = [5 −5 8]^T, kinematic reliability is determined based on the Monte Carlo method, as shown in Table 1.

Table 1.

Reliability calculation.

Time (s)	1	2	3	4	5	6	7
R (%)	100	100	99.75	99.00	98.75	94.00	72.50

According to the data in Table 1, we find that the kinematic reliability of the mechanism was better when its end effector was near the initial position. With the lateral movement of the end effector, the kinematic reliability of the mechanism gradually reduced. In particular, when a driving arm moves to a horizontal position, the kinematic reliability reaches its lowest value, which is only 72.50%; therefore, the position of the mechanism should not be listed as being in a reasonable working space.

The calculations of position reliability in Table 1 are consistent with the kinematic-error distribution in Figures 5 to 9. When the position error of the mechanism increases, the kinematic reliability of the mechanism decreases.

Numerical example of kinematic reliability sensitivity analysis of the mechanism

According to equation (10), the response surface function of the kinematic position error can be constructed as

$\begin{array}{c}\Delta \mathit{X}=C_1\delta c_{1P}+C_2\delta a_1+C_3\delta {\mathit{k}}_1{\hspace{0.17em}}^{\mathrm{T}}+C_4{\mathit{\theta }}_{B1}^{\mathrm{T}}+C_5\delta c_{2P}\\ +C_6\delta a_2+C_7\delta {\mathit{k}}_2{\hspace{0.17em}}^{\mathrm{T}}+C_8{\mathit{\theta }}_{B2}^{\mathrm{T}}+C_9\delta c_{3P}+C_{10}\delta a_3\\ +C_{11}\delta k_3{\hspace{0.17em}}^{\mathrm{T}}+C_{12}{\mathit{\theta }}_{B3}^{\mathrm{T}}+C_{13}\delta c_{1d}+C_{14}\delta e_1+C_{15}{\mathit{\theta }}_1^{\mathrm{T}}\\ +C_{16}\delta c_{2d}+C_{17}\delta e_2+C_{18}{\mathit{\theta }}_2^{\mathrm{T}}+C_{19}\delta c_{3d}+C_{20}\delta e_3+C_{21}{\mathit{\theta }}_3^{\mathrm{T}}\end{array}$

17

where C ₁ = J_PP (3,[1]), J_PP (3,[1]) represents the first column of the 3 × 24 matrix J_PP , C ₂ = J_PP (3,[2]), C ₃ = J_PP (3,[3,4,5]), J_PP (3,[3,4,5]) represents columns 3, 4, and 5 of the 3 × 24 matrix J_PP , C ₄ = J_PP (3,[6,7,8]), C ₅ = J_PP (3,[9]), C ₆ = J_PP (3,[10]), C ₇ = J_PP (3,[11,12,13]), C ₈ = J_PP (3,[14,15,16]), C ₉ = J_PP (3,[17]), C ₁₀ = J_PP (3,[18]), C ₁₁ = J_PP (3,[19,20,21]), C ₁₂ = J_PP (3,[22,23,24]), C ₁₃ = J_PJ (3,[1]), J_PJ represents a 3 × 15 matrix, C ₁₄ = J_PJ (3,[2]), C ₁₅ = J_PJ (3,[3,4,5]), C ₁₆ = J_PJ (3,[6]), C ₁₇ = J_PJ (3,[7]), C ₁₈ = J_PJ (3,[8,9,10]), C ₁₉ = J_PJ (3,[11]), C ₂₀ = J_PJ (3,[12]), and C ₂₁ = J_PJ (3,[13,14,15]).

If Δ is the maximum permissible limit of the position error of the mechanism, according to equation (11), the limit state function of the kinematic reliability can be obtained

$\begin{array}{c}G\left(\mathit{X}\right)=\Delta -\Delta \mathit{X}=\Delta -C_0-C_1\delta c_{1P}-C_2\delta a_1-C_3\delta {\mathit{k}}_1{\hspace{0.17em}}^{\mathrm{T}}\\ -C_4{\mathit{\theta }}_{B1}^{\mathrm{T}}-C_5\delta c_{2P}-C_6\delta a_2-C_7\delta {\mathit{k}}_2{\hspace{0.17em}}^{\mathrm{T}}-C_8{\mathit{\theta }}_{B2}^{\mathrm{T}}-C_9\delta c_{3P}\\ -C_{10}\delta a_3-C_{11}\delta {\mathit{k}}_3{\hspace{0.17em}}^{\mathrm{T}}-C_{12}{\mathit{\theta }}_{B3}^{\mathrm{T}}-C_{13}\delta c_{1d}-C_{14}\delta e_1-C_{15}{\mathit{\theta }}_1^{\mathrm{T}}\\ -C_{16}\delta c_{2d}-C_{17}\delta e_2-C_{18}{\mathit{\theta }}_2^{\mathrm{T}}-C_{19}\delta c_{3d}-C_{20}\delta e_3-C_{21}{\mathit{\theta }}_3^{\mathrm{T}}\end{array}$

18

When the end effector of the mechanism reaches point P (22.8851, −907.9187, 110.9301)^T in the workspace, the mean and variance of the output kinematic-position error can be obtained as μ = [0.5847, −4.1119, 4.2196] and D = [11.0853, 1.5325, 10.0855], respectively, according to the Monte Carlo method. At this moment, the kinematic reliability sensitivities of the means and variances are shown in Tables 2 and 3 according to equations (15) and (16), respectively.

Table 2.

Kinematic reliability sensitivity to the mean of each error component (×10⁻³).

∂R/∂μ T		δciP	δai	θ Bi	δcid	δei	θ i
Rx	i = 1	−0.18	0.16	8.90	−0.18	−0.02	−8.45
	i = 2	0.10	−0.10	2.05	0.99	−0.04	34.87
	i = 3	0.09	−0.08	2.64	−0.74	0.06	−39.49
Ry	i = 1	−0.50	0.23	49.59	0.98	0.11	−85.40
	i = 2	−0.39	0.14	−11.49	−0.45	0.02	42.11
	i = 3	−0.55	0.29	−38.10	−0.51	0.04	43.18
Rz	i = 1	0.02	0.002	−13.70	0.89	0.09	0
	i = 2	0.19	−0.01	−151.01	−0.61	0.02	0
	i = 3	−0.14	0.01	110.55	−0.27	0.02	0

Table 3.

Kinematic reliability sensitivity to the variance of each error component (×10⁻³).

∂R/∂D T		δciP	δai	θ Bi	δcid	δei	θ i
Rx	i = 1	0.01	−0.01	−0.23	0.01	0.01	0.22
	i = 2	−0.01	0.01	−0.05	−0.03	0.01	−0.92
	i = 3	−0.01	0.01	−0.07	0.02	−0.01	1.04
Ry	i = 1	−0.67	0.39	66.54	1.31	0.15	−114.57
	i = 2	−0.52	0.30	−15.43	−0.60	0.02	56.49
	i = 3	−0.73	0.43	−51.11	−0.69	0.05	57.93
Rz	i = 1	−0.01	0.01	2.87	−0.19	−0.02	0
	i = 2	−0.04	0.02	31.61	0.13	−0.01	0
	i = 3	0.03	−0.02	−23.14	0.06	−0.01	0

When the kinematic reliability sensitivity is 0, the reliability of the mechanism does not change with random errors. The magnitudes of the sensitivity’s absolute value indicate the rate at which the reliability changes with random errors.

The results of kinematic reliability sensitivity analysis can be seen in Tables 2 and 3.

Reliability sensitivity data in Tables 2 and 3 are in good agreement with the position errors of the mechanism in Figures 5 to 9.

We can see from the kinematic reliability sensitivity analysis in Tables 2 and 3 that the mean of the pose-error vectors, θ _Bi , and the composite-pose-error vectors, θ _i , have a larger influence on the kinematic reliability of the mechanism. In the z direction, the mean of θ _Bi has the largest influence on the kinematic reliability, but the mean of θ _i has no effect.

The variances of the pose-error vectors θ _Bi and θ _i have a larger influence on the kinematic reliability of the mechanism. In the y direction, the variances of θ _i have the largest influence on the kinematic reliability. In the z direction, the variance of θ _i has no effect on the kinematic reliability.

The influence of the pose errors of the joints on the kinematic reliability of the mechanism is much larger than that of the dimension errors of the components.

Conclusion

Using MATLAB software to generate random input errors that obey the normal distribution, the kinematic reliability of the modified Delta parallel mechanism was analyzed according to the Monte Carlo method under the condition of multiple errors. At the same time, a significant amount of experimental data for statistical analysis was easily obtained using this method.

The reliability sensitivity of the modified Delta parallel mechanism was analyzed using the method of structuring the response surface function. Thus, the degree of influence of each error source on the kinematic reliability of the mechanism was determined.

The assembly position and pose errors exhibited greater effect than dimension errors on the kinematic reliability of the modified Delta parallel mechanism.