Error modeling and sensitivity analysis of a 3-P(Pa)S parallel type spindle head with parallelogram structure

Abstract

The 3-P(Pa)S mechanism, based on the 3-Prismatic-Rotational-Spherical (PRS) mechanism, but with three of its revolution joints and passive links replaced by parallelogram structures, provides improved stiffness and better constraint of undesired Degree of Freedom (DOFs). The general method to establish the error model of a 3-P(Pa)S mechanism is simplifying it to a 3-PRS mechanism, which fails to account for the influence of the errors in the parallelogram structure. In this article, the error model of a spindle head based on 3-P(Pa)S mechanism was established. By projecting the structural errors of the parallelogram structure to the structural errors of a 3-PRS mechanism which is simplified from the 3-P(Pa)S mechanism, the errors of the parallelogram can be included into the model. Then, the projecting method was verified by an experiment on the spindle head to prove its correctness. Based on the error model, sensitivity analysis of the structural errors was carried out. According to the characteristic of the mechanism, two new sensitivity indexes, standardized signed local sensitivity index and standardized signed global sensitivity index, were proposed which can better reflect the influence of the structural errors on the end effector error and can also indicate whether some structural errors have similar or opposite influence on the end effector. Then, the most sensitive errors were derived and the sensitivity distribution within the workspace of the spindle head was analyzed. Additionally, two possible applications based on the sensitivity analysis were proposed to reduce the error of the spindle head.

Keywords

Manufacturing robotics error modeling parallel mechanism sensitivity analysis parallelogram structure

Introduction

Compared to the conventional serial type mechanisms, parallel mechanisms are expected to have higher stiffness and perform with greater dexterity^1

–4 and have been widely applied in the design of machine tools.^1,5,6 In the last decades, a series of less-freedom parallel mechanisms were introduced as spindle head for machining,^7

–12 which can be installed on a serial type mechanism to realize 5-DOF movement, among which a series of the most widely used spindle heads are based on the 3-PRS mechanism.^13,14 Additionally, some parallel mechanisms have been designed based on the parallelogram structure, which can provide higher stiffness and constrain undesired degrees of freedom.^15,16 Therefore, a new less-freedom parallel mechanism is proposed based on the 3-PRS mechanism, in which three revolution joints and passive links were substituted by three parallelogram structures. Because of the existence of the parallelogram, the entire mechanism is named as 3-P(Pa)S mechanism, where “Pa” denotes the parallelogram structure.

Since there are more joints and links in the parallel mechanism that causing errors, the accuracy issue of the parallel mechanism is always one of the most important issues. Error modeling is the first step to investigate the error of a mechanism. The general error modeling method for this kind of parallel mechanisms with parallelogram structure is to replace the parallelogram by a link and a revolution joint since they are considered equivalent in the ideal condition.¹⁷ For the 3-P(Pa)S mechanism, when the parallelogram structure is replaced, its error model can be simplified to the error model of a 3-PRS mechanism, which is well established.^18

–21 However, it is an incomplete error model since the influence of the errors in the parallelogram structure is not included, which may reduce the accuracy of the model.

In order to improve the error model, the errors of the parallelogram structure must be well considered. Huang et al.²² established the error model of a 3-DOF parallel spindle head with parallelogram structure. By connecting two middle points on two opposite parallel links of the parallelogram and assuming the other two links of the parallelogram are uniformly parallel to each other, the errors of the four links of the parallelogram structure were transformed into the error between the two middle points. Chen et al.¹⁵ established the error model of a 4-DOF selective compliance assembly robot arm with parallelogram structure using the same method. However, this method cannot to be used for the 3-P(Pa)S mechanism since some parts of the parallelogram structure in 3-P(Pa)S mechanism are rigid bodies but not links. Therefore, a new integrated error model including the errors of the parallelogram structure is needed for the 3-P(Pa)S mechanism.

In this article, the error model of a parallel type spindle head based on 3-P(Pa)S mechanism was established. To include the errors of the parallelogram structure into the error model, firstly, the error model of a 3-PRS mechanism simplified from 3-P(Pa)S mechanism was established. Then, two individual chains, the PRS chain and the P(Pa)S chain, of the 3-PRS and 3-P(Pa)S mechanism were analyzed, and the united kinematic equations of the PRS and P(Pa)S chains were established. An error projecting method was proposed to project the structural errors of the parallelogram to the structural errors of the 3-PRS mechanism. During the projection, the characteristic of the parallelogram was considered and the projection equation is simplified. Therefore, the errors of the parallelogram were included into the error model. Then, a verification experiment was carried out on the 3-P(Pa)S spindle head to verify the correctness of the error projection method.

Based on the error model, sensitivity analysis is necessary for the parallel mechanism, which can estimate the influence of structural errors on the error of the end effector, so the most sensitive structural errors can be derived and be paid more attention in the design, manufacture, and assembly process. Many researchers have done meaningful researches in sensitivity analysis.²³ For instance, Fan et al.²⁴ analyzed the sensitivity of the 3-PRS mechanism and derived the most sensitive errors. Additionally, some other researchers also proposed specific sensitivity indexes for a certain kind of mechanism to guide the design process.^25,26 Li et al.²⁷ proposed a new sensitivity index, named as general global sensitivity fluctuation index, to reflect the error fluctuation within the workspace.

In this article, on the basis of the proposed error model, the sensitivity of the structural errors of the 3-P(Pa)S parallel type spindle head, especially the errors of the parallelogram structure, was analyzed. Firstly, the local sensitivity index (LSI) and the global sensitivity index (GSI) of the structural errors were calculated. Then, according to the characteristic of the 3-P(Pa)S mechanism, two new sensitivity indexes, standardized signed local sensitivity index (SSLSI) and standardized signed global sensitivity index (SSGSI), were proposed, which can better reflect the influence of the structural errors on the end effector and can also indicate whether some structural errors have similar or opposite influence on the end effector. Based on the new index, the most sensitive errors were derived. And the sensitivity distribution within the workspace of the spindle head was analyzed. Finally, two possible applications of SSLSI and SSGSI were proposed to reduce error of the spindle head.

The structure of the article is demonstrated as follows: the structure of the 3-P(Pa)S spindle head is introduced in “Structure description” section. Then the error modeling of the 3-P(Pa)S mechanism is presented in “Error modeling of the 3-P(Pa)S parallel type spindle head” section. In “Verification of the error projection” section, the proposed error projecting method was verified by an experiment on the spindle head. In “Sensitivity analysis” section, sensitivity analysis is carried out based on the error model. The final section is the conclusion.

Structure description

The structure of the 3-P(Pa)S parallel spindle head is shown in Figure 1, and Figure 2 is the schematic of the spindle head. It is composed of two platforms (steady platform and moving platform) and three chains. The chains are uniformly distributed along the circumference of the platforms, and the angle between each pair of chains is 120°. There is a parallelogram structure in each chain. One side of the parallelogram is connected to a prismatic joint and its opposite side is fixed to a bar at the end of which is a spherical joint. The guide way of the prismatic joint is located on the steady platform and the spherical joint is located on the moving platform, and the axes along the guide ways are denoted by Z_m1-, Z_m2-, and Z_m3-axis. Three actuators were installed on the prismatic joint to control the pose of the mechanism. At the equilibrium pose, the axial line of the spindle is along the Z-axis. Compared to 3-PRS mechanism, 3-P(Pa)S mechanism substitutes the passive link and rotational joint with the parallelogram structure, so the stiffness can be improved since the chains of the mechanism become stronger. Additionally, because of the existence of the parallelogram structure, the undesired DOFs of the chain can be restrained.

Figure 1.

Structure of the 3-P(Pa)S parallel type spindle head.

Figure 2.

Schematic of the 3-P(Pa)S parallel type spindle head.

Error modeling of the 3-P(Pa)S parallel type spindle head

The 3-P(Pa)S mechanism can be simplified to a 3-PRS mechanism in the ideal condition. Because of the symmetry, we only investigate chain 1 in its own plane, as shown in Figure 3. In the figure, it can be derived that $N_{12} N_{13}$ and $N_{11} N_{14}$ have the same length and are parallel to each other. And because the orientation of $M_{14} M_{11}$ is fixed by the prismatic joint, the orientation of $N_{11} N_{14}$ is also fixed. Thus, the position of the point N₁₄ is unchanged when the parallelogram is rotating. And the rotation of the parallelogram $M_{11} M_{12} N_{13} M_{13} M_{14}$ is equal to the rotation of the line N₁₄N₁₃. Therefore, the 3-P(Pa)S mechanism is simplified to the 3-PRS mechanism. However, the structural error of the parallelogram structure is not included after the simplification. Thus, in this article, we would like to establish the relationship between the P(Pa)S chain and the PRS chain. And the structural errors of the parallelogram structure can be projected to the structural errors of the 3-PRS mechanism. So, the first step is establishing the error model of the 3-PRS mechanism simplified from the 3-P(Pa)S mechanism.

Figure 3.

Simplification of chain 1 of the 3-P(Pa)S mechanism.

Error model of the 3-PRS mechanism simplified from the 3-P(Pa)S mechanism

The structure of the 3-PRS mechanism simplified from the 3-P(Pa)S mechanism is shown in Figure 4(a), in which the original 3-P(Pa)S mechanism is shown by dot line. Except for the parallelogram structure, the two mechanisms are the same, so the nomenclatures in the 3-PRS mechanism can also be applied to the 3-P(Pa)S mechanism. The coordinate frames were established as shown in Figure 4(b). $O_{B i} (i = 1 \sim 3)$ is located at the connection point of the ith chain and the steady platform. $O_{C i} (i = 1 \sim 3)$ is located at the pedal from the center of the ith spherical joint to the rotational axis of the ith rotational joint. The fixed coordinate frame ${O : O - X Y Z}$ is established at the center of the steady platform. In which the Z-axis is vertical to the steady platform and points to the moving platform. The Y-axis is located in the plane of the steady platform and points to the opposite direction of O_B3. The X-axis is vertical to the Y-axis and Z-axis to form a right-handed orthogonal frame. The moving coordinate frame ${O' : O' - X' Y' Z'}$ is located at the center of the moving platform. The directions of the axes are identical to the fixed frame when the mechanism is at the equilibrium pose. There are three coordinate frames in each chain. Taking the first chain as an example, the coordinate frame ${O_{B 1} : O_{B 1} - X_{B 1} Y_{B 1} Z_{B 1}}$ is located at the point O_B1 and fixed to the guide way of the prismatic joint. The Z_B1-axis points upward along the guide way. In the ideal condition, the X_B1-axis locates along the direction of the line OO_B1. Then, the Y_B1-axis is determined by the right-handed rule. The coordinate frame ${O_{C 1} : O_{C 1} - X_{C 1} Y_{C 1} Z_{C 1}}$ is on the slider of the prismatic joint. The Y_C1-axis locates along the rotational axis. The X_C1-axis and Z_C1-axis are parallel to the X_B1-axis and the Z_B1-axis, respectively, in the ideal condition. The coordinate frame ${O_{D 1} : O_{D 1} - X_{D 1} Y_{D 1} Z_{D 1}}$ locates at the point O_D1 and is fixed to the passive link. The points O_D1 and O_C1 are coincident points. The Z_D1-axis is fixed along the passive link and the Y_D1-axis is coincident to the Y_C1 axis, and the X_D1 axis can be derived by the right-handed rule. For the convenience of error modeling, there are two groups of auxiliary coordinate frames. ${O_{i} : O_{i} - X_{i} Y_{i} Z_{i}} (i = 1 \sim 3)$ is established at the point O, the Z_i axis is parallel to the Z-axis, and the X_i-axis is parallel to the line OO_Bi. The frame ${{O'}_{i} : {O'}_{i} - {X'}_{i} {Y'}_{i} {Z'}_{i}} (i = 1 \sim 3)$ is established in the same way on the moving platform.

Figure 4.

Error modeling process. (a) Structure of the 3-PRS mechanism simplified from the 3-P(Pa)S mechanism, (b) establishment of the coordinate frames, and (c) establishment of the vector loop.

The vector loops were established as shown in Figure 4(c). Taking one chain as an example, P is the vector from O to O′, expressed in the fixed coordinate frame {O}, representing the position of the moving platform. a_i is the vector from O′ to the center of the ith spherical joint, which is expressed in the coordinate frame ${{O'}_{i}}$ . b_i is the vector from O to O_Bi, which is expressed in the coordinate frame {O_i}. $q_{i} e_{3}$ is the vector from O_Bi to O_Ci, expressed in the coordinate frame ${O_{B i}}$ . $e_{3} = [0, 0, 1]$ is the unit vector along the Z_Bi-axis. $l_{i} ω_{i}$ is the vector from O_Ci to the center of the ith spherical joint, which is expressed in the fixed coordinate frame {O}, representing the orientation and length of the passive link, in which ω_i is the unit vector along the passive link and l_i is the length of the passive link.

Rotational matrices were defined to represent the orientation of coordinate frames. R is the rotational matrix from {O′} to {O}, which represents the orientation of the moving platform. For the convenience of the error compensation, R is described in the form of the wuw angle, which means that {O} first rotates about its Z-axis, then rotates about its new X-axis, and finally rotates about its new Z-axis. The angles of these three coordinate axis rotations are denoted by φ, θ, and ψ. Therefore, the rotational matrix R can be expressed as follows, in which s and c represent sin and cos, respectively.

R = [\begin{matrix} c φ c ψ - s φ c θ s ψ & - c φ s ψ - s φ c θ c ψ & s φ s θ \\ s φ c ψ + c φ c θ s ψ & - s φ s ψ + c φ c θ c ψ & - c φ s θ \\ s θ s ψ & s θ c ψ & c θ \end{matrix}]

$R_{B i}$ is the rotational matrix from frame ${O_{B i}}$ to {O_i}, which represents the orientation of the ith prismatic joint. $R_{C i}$ is the rotational matrix from frame ${O_{C i}}$ to ${O_{B i}}$ . $R_{B i} \cdot R_{C i}$ represents the orientation of the ith rotational joint. $R_{D i}$ is the rotational matrix from frame ${O_{D i}}$ to ${O_{C i}}$ . $R_{B i} \cdot R_{C i} \cdot R_{D i}$ represents the orientation of the ith passive link. R_i is the rotational matrix from {O_i} to {O} and also the rotational matrix from ${O_{i}^{′}}$ to {O′}. It can be expressed as

R_{i} = [\begin{matrix} cos α_{i} & - sin α_{i} & 0 \\ sin α_{i} & cos α_{i} & 0 \\ 0 & 0 & 1 \end{matrix}]

in which, $α_{i} = - \frac{2 π}{3} (i - 1) + \frac{5 π}{6} (i = 1 \sim 3)$ .

The vector loop equation can be expressed as

P + R R_{i} a_{i} = R_{i} b_{i} + q_{i} R_{i} R_{B i} e_{3} + l_{i} ω_{i}

where $ω_{i} = R_{i} R_{B i} R_{C i} R_{D i} e_{3}$ .

Equation (3) is the kinematic model of the 3-PRS mechanism simplified from the 3-P(Pa)S mechanism. It can be expressed in a more compact format

F (p, q, r) = 0

Equation (4) describes the relationship between the pose of the end effector p , the actuator movement q , and the structural parameters r .

When considering the structural errors, equation (3) can be expressed as

P + δ P + R R (θ) R_{i} (a_{i} + δ a_{i}) = R_{i} (b_{i} + δ b_{i}) + (q_{i} + δ q_{i}) R_{i} R_{B i} R (θ_{B i}) e_{3} + (l_{i} + δ l_{i}) R_{i} R_{B i} R (θ_{B i}) R_{C i} R (θ_{C i}) R_{D i} R (θ_{D i}) e_{3}

By subtracting equation (3) from equation (5) and considering $R_{B i} = R_{C i} = I$ in the ideal condition, we obtain

δ P + (R θ) \times (R R_{i} a_{i}) = - R R_{i} δ a_{i} + R_{i} δ b_{i} + q_{i} (R_{i} θ_{B i}) \times (R_{i} e_{3}) + δ q_{i} R_{i} e_{3} + l_{i} (R_{i} θ_{B i}) \times ω_{i} + l_{i} (R_{i} θ_{C i}) \times ω_{i} + l_{i} (R_{i} R_{D i} θ_{D i}) \times ω_{i} + δ l_{i} ω_{i}

Multiplying both sides of equation (6) by $ω_{i}^{T}$ , we derive

ω_{i}^{T} δ P + {(R R_{i} a_{i} \times ω_{i})}^{T} \cdot (R θ) = - ω_{i}^{T} \cdot R R_{i} δ a_{i} + ω_{i}^{T} \cdot R_{i} δ b_{i} + q_{i} {(R_{i} e_{3} \times ω_{i})}^{T} \cdot (R_{i} θ_{B i}) + δ q_{i} ω_{i}^{T} \cdot R_{i} e_{3} + δ l_{i}

Letting $v_{i} = R_{i} e_{2}$ be the unit vector along the rotational axis, and multiplying both sides of equation (6) by $v_{i}^{T}$ , we derive

v_{i}^{T} δ P + {(R R_{i} a_{i} \times v_{i})}^{T} \cdot (R θ) = - v_{i}^{T} \cdot R R_{i} δ a_{i} + v_{i}^{T} \cdot R_{i} δ b_{i} + q_{i} {(R_{i} e_{3} \times v_{i})}^{T} \cdot (R_{i} θ_{B i}) + l_{i} {(ω_{i} \times v_{i})}^{T} \cdot (R_{i} θ_{B i}) + l_{i} {(ω_{i} \times v_{i})}^{T} \cdot (R_{i} θ_{C i}) + l_{i} {(ω_{i} \times v_{i})}^{T} \cdot (R_{i} R_{D i} θ_{D i})

Equations (7) and (8) can be integrated into a matrix equation

δ p = J δ r

where J is the Jacobi matrix, which can be expressed as

J = A^{- 1} B

in which

A = [\begin{matrix} ω_{1}^{T} & {[R R_{1} a_{1} \times ω_{1}]}^{T} R \\ ω_{2}^{T} & {[R R_{2} a_{2} \times ω_{2}]}^{T} R \\ ω_{3}^{T} & {[R R_{3} a_{3} \times ω_{3}]}^{T} R \\ v_{1}^{T} & {[R R_{1} a_{1} \times v_{1}]}^{T} R \\ v_{2}^{T} & {[R R_{2} a_{2} \times v_{2}]}^{T} R \\ v_{3}^{T} & {[R R_{3} a_{3} \times v_{3}]}^{T} R \end{matrix}]

B = diag (\begin{matrix} σ_{1} & σ_{2} & σ_{3} & τ_{1} & τ_{2} & τ_{3} \end{matrix})

and

σ_{i} = [\begin{matrix} - ω_{i}^{T} \cdot R R_{i} & ω_{i}^{T} \cdot R_{i} & q_{i} {(R_{i} e_{3} \times ω_{i})}^{T} \cdot R_{i} \end{matrix}

\begin{matrix} 0_{1 \times 3} & 0_{1 \times 3} & ω_{i}^{T} \cdot R_{i} e_{3} & 1 \end{matrix}]

τ_{i} = [\begin{matrix} - v_{i}^{T} \cdot R R_{i} & v_{i}^{T} \cdot R_{i} & q_{i} {(R_{i} e_{3} \times v_{i})}^{T} \cdot R_{i} + l_{i} {(ω_{i} \times v_{i})}^{T} \cdot R_{i} \end{matrix}

\begin{matrix} l_{i} {(ω_{i} \times v_{i})}^{T} \cdot R_{i} & l_{i} {(ω_{i} \times v_{i})}^{T} \cdot R_{i} R_{D i} & 0 & 0 \end{matrix}]

Thus, the error model of the 3-PRS mechanism is derived as expressed by equation (9), where $δ p = {[\begin{matrix} δ P^{T} & δ θ^{T} \end{matrix}]}^{T} = {[\begin{matrix} δ p_{x} & δ p_{y} & δ p_{z} & δ φ & δ θ & δ ψ \end{matrix}]}^{T}$ denotes the pose error of the end effector. $δ r = {[\begin{matrix} δ r_{1}^{T} & δ r_{2}^{T} & δ r_{3}^{T} \end{matrix}]}^{T}$ denotes the structural errors of the 3-PRS mechanism, in which $δ r_{i} = {[\begin{matrix} δ a_{i}^{T} & δ b_{i}^{T} & θ_{B i}^{T} & θ_{C i}^{T} & θ_{D i}^{T} & δ q_{i}^{T} & δ l_{i}^{T} \end{matrix}]}^{T}$ denotes the structural errors of the ith chain. However, some structural errors are redundant errors, for example, θ_Di and θ_Ci both represent the orientation error of the passive link, and we removed θ_Di . δb_i along the Z_Bi-axis is described as $δ b_{i z}$ . $δ b_{i z}$ and $δ q_{i}$ both represent the actuator movement error of the ith prismatic joint, so $δ b_{i z}$ is eliminated. $θ_{C i y}$ and $θ_{B i z}$ are the self-rotation errors of the rotational joint and prismatic joint, respectively, so they are eliminated. There are 11 independent structural errors for each chain and the total number of the errors is 33, listed in Table 1.

Table 1.

Structural errors of the 3-PRS mechanism simplified from 3-P(Pa)S mechanism.

No.	Structural errors	Description of the structural errors
1–3	$[\begin{matrix} δ a_{i x} & δ a_{i y} & δ a_{i z} \end{matrix}]$	Position error of the ith spherical joint to the point O′
4–5	$[\begin{matrix} δ b_{i x} & δ b_{i y} \end{matrix}]$	Position error of the point O_Bi to the point O
6–7	$[\begin{matrix} θ_{B i x} & θ_{B i y} \end{matrix}]$	Orientation error of coordinate frame ${O_{B i}}$ to {O_i}
8–9	$[\begin{matrix} θ_{C i x} & θ_{C i z} \end{matrix}]$	Orientation error of coordinate frame ${O_{C i}}$ to ${O_{B i}}$
10	$δ q_{i}$	Actuator movement error of the ith prismatic joint
11	$δ l_{i}$	Length error of the ith passive link

Error projecting method

In this section, the structural errors of the parallelogram structure were projected to the 3-PRS mechanism. The schematic of the ith parallelogram structure is shown in Figure 5. $M_{i 2} N_{i 3} M_{i 3}$ is considered as a rigid body and the entire parallelogram structure can move vertically along the guide way of the prismatic joint. One coordinate frame is established. In the ideal condition, there is an intersection point of the rotational axis of rotational joint M_i1 and the plane that the parallelogram locates in. And the original point of the coordinate frame is established at the intersection point on the plane. Its y_i-axis is parallel to the guide way of the prismatic joint. The x_i-axis is parallel to the X_i-axis. The eight structural parameters of the parallelogram structure are denoted by $d_{i 1} \sim d_{i 8}$ . d_i1 and d_i2 describe the length of the two parallel links $M_{i 4} M_{i 3}$ and $M_{i 1} M_{i 2}$ . The line $M_{i 2} M_{i 3}$ is perpendicular to the line $N_{i 2} N_{i 3}$ , so d_i3, d_i4, and d_i5 can describe the relative position of N_i3 to M_i2 and M_i3. M_i4 and M_i1 are both located on the slider of the prismatic joint, so d_i6 and d_i8 describe the relative position of M_i4 to M_i1. d_i7 is the offset value of point M_i1 to the original point of the coordinate frame along the x_i-axis, which describes the offset of the entire parallelogram structure. In the ideal condition, d_i7 is zero.

Figure 5.

Schematic of the parallelogram structure.

The structural parameters of one PRS chain related to the parallelogram structure are shown by $M_{i 1} N_{i 5} N_{i 4} N_{i 3}$ in Figure 5. It is considered that the parallelogram structure and the PRS chain have the same trajectory, thus the structural errors of the parallelogram can be projected to the structural errors of the PRS chain.

Considering a certain circumstance in which the rotation angle of the parallelogram is φ_pi, the ideal vector loops between parallelogram structure and PRS chain can be established, expressed in equations (11) and (12)

- d_{i 6} e_{2} - d_{i 8} e_{1} + d_{i 1} d_{i 1} = d_{i 2} d_{i 2} + d_{i 3} d_{i 3} + d_{i 5} d_{i 3}

- {b ′}_{i x} e_{1} + q_{i} e_{2} + l_{i} l_{i} = - d_{i 6} e_{2} - d_{i 8} e_{1} + d_{i 1} d_{i 1} - d_{i 5} d_{i 3} + d_{i 4} R_{s} d_{i 3}

Then, the structural errors can be regarded as the differential of structural parameters, so the vector loops including error were established, expressed in equations (13) and (14). In these two equations, there is an approximation that the orientation of the link d_i2 is parallel to the link l_i. The verification for the feasibility of this approximation is illustrated in the Appendix 1.

- (d_{i 6} + δ d_{i 6}) e_{2} - (d_{i 8} + δ d_{i 8}) e_{1} + (d_{i 1} + δ d_{i 1}) (d_{i 1} + δ d_{i 1}) = (d_{i 2} + δ d_{i 2}) d_{i 2} + (d_{i 3} + δ d_{i 3}) (d_{i 3} + δ d_{i 3}) + (d_{i 5} + δ d_{i 5}) (d_{i 3} + δ d_{i 3})

- ({b ′}_{i x} + δ {b ′}_{i x}) e_{1} + q_{i} e_{2} + (l_{i} + δ l_{i}) l_{i} = δ d_{i 7} e_{1} - (d_{i 6} + δ d_{i 6}) e_{2} - (d_{i 8} + δ d_{i 8}) e_{1} + (d_{i 1} + δ d_{i 1}) (d_{i 1} + δ d_{i 1}) - (d_{i 5} + δ d_{i 5}) (d_{i 3} + δ d_{i 3}) + (d_{i 4} + δ d_{i 4}) R_{s} (d_{i 3} + δ d_{i 3})

Vector loops expressed by equations (13) and (14) are illustrated in Figure 6, in which e₁ and e₂ are the unit vectors along the x_i-axis and y_i-axis, respectively. $d_{i j} (i = 1 \sim 3, j = 1 \sim 8)$ and $l_{i} (i = 1 \sim 3)$ denote the unit vector along d_ij and l_i, respectively. The differential of the unit vector (e.g. $δ d_{i j}$ ) denote the orientation error of the vector, and the differential of the length (e.g. $δ d_{i j}$ ) denote the length error of the vector. $R_{s} = [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}]$ is the rotational matrix from d_i3 to d_i4 .

Figure 6.

Vector loops of the error projecting.

By subtracting equations (11) and (12) from equations (13) and (14), and ignoring high order infinitesimal errors, we obtain

d_{i 1} δ d_{i 1} - (d_{i 3} + d_{i 5}) δ d_{i 3} = - d_{i 1} δ d_{i 1} + d_{i 2} δ d_{i 2} + d_{i 3} δ d_{i 3} + d_{i 3} δ d_{i 5} + e_{2} δ d_{i 6} + e_{1} δ d_{i 8}

l_{i} δ l_{i} - e_{1} δ {b ′}_{i x} - d_{i 1} δ d_{i 1} + (d_{i 5} - d_{i 4} \cdot R_{s}) δ d_{i 3} = d_{i 1} δ d_{i 1} + R_{s} d_{i 3} δ d_{i 4} - d_{i 3} δ d_{i 5} - e_{2} δ d_{i 6} + e_{1} δ d_{i 7} - e_{1} δ d_{i 8}

Define an operator $W (φ) = {[\begin{matrix} - sin φ & cos φ \end{matrix}]}^{T}$ . Considering the characteristic of the parallelogram, we obtain

{\begin{matrix} l_{i} = d_{i 1} = d_{i 2} = W (φ_{p i}) \\ d_{i 3} = W (α_{i}) \\ δ d_{i 1} = | δ d_{i 1} | W (φ_{p i} + \frac{π}{2}) \\ δ d_{i 3} = | δ d_{i 3} | W (α_{i} + \frac{π}{2}) \end{matrix}

where α_i is the angle between d _i3 and the y_i -axis, which is a constant value.

Thus, equations (15) and (16) can be written as

δ γ = J_{p} \cdot δ γ_{p}

in which $δ γ = {[\begin{matrix} δ γ_{1}^{T} & δ γ_{2}^{T} & δ γ_{3}^{T} \end{matrix}]}^{T}$ . We also have $δ γ_{i} = {[\begin{matrix} δ l_{i} & - δ {b ′}_{i x} & | δ d_{i 1} | & | δ d_{i 3} | \end{matrix}]}^{T}$ , in which $- δ {b ′}_{i x} = δ b_{i x}$ . δl_i and δb_ix are structural errors in the ith PRS chain. $| δ d_{i 1} |$ and $| δ d_{i 3} |$ describe the small rotation of the rotational joints in the parallelogram structure.

$δ γ_{p} = {[\begin{matrix} δ γ_{p 1}^{T} & δ γ_{p 2}^{T} & δ γ_{p 3}^{T} \end{matrix}]}^{T}$ , in which $δ γ_{p i} = {[\begin{matrix} δ d_{i 1} & δ d_{i 1} & \dots & δ d_{i 8} \end{matrix}]}^{T}$ represents the structural errors in the ith parallelogram structure. And we further have

J_{p} = d i a g (\begin{matrix} J_{p 1} & J_{p 2} & J_{p 3} \end{matrix})

in which

J_{p i} = A_{p i}^{- 1} B_{p i}

where

A_{p i} = [\begin{matrix} 0_{2 \times 1} & 0_{2 \times 1} & d_{i 1} W (φ_{p i} + \frac{π}{2}) & - (d_{i 3} + d_{i 5}) W (α_{i} + \frac{π}{2}) \\ W (φ_{p i}) & e_{1} & - d_{i 1} W (φ_{p i} + \frac{π}{2}) & (d_{i 5} - d_{i 4} R_{s}) W (α_{i} + \frac{π}{2}) \end{matrix}]

B_{p i} = [\begin{matrix} - W (φ_{p i}) & W (φ_{p i}) & W (α_{i}) & 0_{2 \times 1} & W (α_{i}) & e_{2} & 0_{2 \times 1} & e_{1} \\ W (φ_{p i}) & 0_{2 \times 1} & 0_{2 \times 1} & R_{s} W (α) & - W (α_{i}) & - e_{2} & e_{1} & - e_{1} \end{matrix}]

Equation (18) is the error projecting function of parallelogram structures. In each chain, eight structural errors of the parallelogram structure were projected to two errors in PRS chain. Since the rest part of the 3-P(Pa)S and 3-PRS mechanisms are the same, the total number of structural errors of a P(Pa)S chain is 17, as shown in Table 2. And the total number of structural errors of the 3-P(Pa)S mechanism is 51. Additionally, the ideal values of the structural parameters in one chain are illustrated in Table 3, in which there are only displacement structural parameters since all the angular structural parameters are zero based on the establishment of the coordinate frames.

Table 2.

Structural errors in one chain of the 3-P(Pa)S mechanism.

No.	Structural errors	Description of the structural errors
1–8	$δ d_{i 1} \sim δ d_{i 8}$	Structural errors of the ith parallelogram structure
9–11	$[\begin{matrix} δ a_{i x} & δ a_{i y} & δ a_{i z} \end{matrix}]$	Position error of the ith spherical joint to the point O′
12	$δ b_{i y}$	Position error of the point O_Bi to the point O along the Y_B1-axis
13	$δ q_{i}$	Actuator movement error of the ith prismatic joint
14–15	$[\begin{matrix} θ_{B i x} & θ_{B i y} \end{matrix}]$	Orientation error of the guide way of the ith prismatic joint
16–17	$[\begin{matrix} θ_{C i x} & θ_{C i z} \end{matrix}]$	Orientation error of the ith parallelogram structure

Table 3.

Ideal structural parameters in one chain of the 3-P(Pa)S mechanism.

Structural parameters	d _i1	d _i2	d _i3	d _i4	d _i5	d _i6	d _i7
Value (mm)	487	487	70	190	70	70	0
Structural parameters	d _i8	a_ix	a_iy	a_iz	b_iy	q_i
Value (mm)	121.244	200	0	75	0	129.544

Thus, the whole error model of the 3-P(Pa)S mechanism is expressed by combining equations (4), (9), and (18) into an equation set. Compared to the general error model, this model includes the structure error of the parallelogram, so it can fully describe the errors of the 3-P(Pa)S mechanism. The error model has a lot of applications. For example, if the actuator movement q and the 51 structural errors $δ r_{p}$ are given, by solving the equation set, the pose of the end effector p and its error δ_p can be derived, which is called the forward-problem. When a series of actuator movements q and the poses of the end effector p are given, a series of equation sets can be derived, by which 51 structural errors $δ r_{p}$ can be derived, which is called the inverse-problem.

Verification of the error projection

Since the errors of the parallelogram structure were projected to the errors of the 3-PRS mechanism in the new error model, the error projection method needs to be verified. Thus, an experiment was carried out, in which two important structure errors of the parallelogram structure, $δ d_{i 1}$ and $δ d_{i 2}$ , were investigated. The experiment setup is shown in Figure 7, and the flow chart of the experiment is shown in Figure 8, which is explained below.

Figure 7.

Experiment setup.

Figure 8.

Flowchart of the experiment.

In the experiment, in order to generate structural errors $δ d_{i 1}$ and $δ d_{i 2}$ , two heating plates were placed on both sides of the link to heat it uniformly and swiftly to ensure that other parts of the spindle head incur much less temperature change. When the temperature stabilizes, it is supposed that the thermal expansion of the link is equal to the corresponding structural error. Although there are original structural errors, according to small deformation theory, they will not influence the increased error of the end effector caused by thermal expansion.

Then, three temperature sensors were placed along the width of the link that corresponding to $δ d_{i 1}$ and $δ d_{i 2}$ to measure the temperature distribution of the link. The thermal expansion of the link can be derived using Finite Element Analysis (FEA): by adjusting the boundary conditions in the FEA model to ensure that the temperature distribution of the link derived by FEA is equal to the measured temperature value, the relatively accurate thermal expansion of the link is derived, which is considered as $δ d_{i 1}$ and $δ d_{i 2}$ . Then, $δ d_{i 1}$ and $δ d_{i 2}$ were inputted into the error model to derive the position error of the end effector.

Next, the position error of the end effector was measured. A ball-ended measuring bar was installed on the end of the motor spindle to make sure the position error of the end effector can be measured along the three axes. Two Charge Coupled Device (CCD) non-contact laser displacement sensors (type: HK-020, Keyence Corp., Japan) were installed along the X-axis and Y-axis to measure the position error of these two axes. A dial indicator (type: 2109S, Mitutoyo Corp., Japan) was placed along the Z-axis to measure the position error of the Z-axis. The needle of the dial indicator and the laser of the CCD sensor are pointed to the center of the ball, so the slope of the spherical surface at the measurement point is zero, making sure that the spherical surface of the ball would not generate error in the measurement.

In the end, the measured error was compared to the error calculated by the error model. Owing to the symmetry of the three chains, only the first chain of the spindle head was analyzed in the verification experiment. The experiment was carried out at four different poses illustrated in Figure 9, in which the angle on the radius represents the tilt angle of the spindle head, and the angle on the circumference represents the orientation that the spindle head tilt to. At each pose, two structural errors, δd₁₁ and δd₁₂, were investigated, making the total number of experimental groups eight, as shown in Table 4. In the table, the pose of the end effector for each group was illustrated. Additionally, the investigated structural error ( $δ d_{i 1}$ or $δ d_{i 2}$ ), as well as its temperature rise and thermal expansion (value of structural error) derived by the FEA, was also illustrated.

Figure 9.

Four poses of the mechanism in the verification.

Table 4.

Experiment groups.

Group	Pose of the end effector $[φ, θ]$	Investigated structural error	Temperature of the sensors before heating $[t_{1}, t_{2}, t_{3}]$ (°C)	Temperature of the sensors after heating $[t_{1}^{'}, t_{2}^{'}, t_{3}^{'}]$ (°C)	Thermal expansion (structural error) derived by the FEA (mm)
1	[0°, 0°]	δd ₁₁	[21.1, 21.1, 21.1]	[26.3, 26.6, 25.7]	0.030
2	[0°, 0°]	δd ₁₂	[19.1, 19.0, 19.1]	[22.4, 23.7, 22.9]	0.023
3	[150°, 23.9°]	δd ₁₁	[20.5, 20.7, 20.5]	[26.1, 26.8, 25.2]	0.035
4	[150°, 23.9°]	δd ₁₂	[20.6, 20.8, 20.6]	[24.0, 25.3, 23.5]	0.021
5	[30°, 23.9°]	δd ₁₁	[16.5, 16.5, 16.5]	[24.5, 26.3, 24.9]	0.049
6	[30°, 23.9°]	δd ₁₂	[17.7, 17.7, 17.6]	[25.6, 26.1, 25.9]	0.048
7	[330°, 23.9°]	δd ₁₁	[17.2, 17.2, 17.3]	[25.6, 25.9, 25.5]	0.045
8	[330°, 23.9°]	δd ₁₂	[16.9, 16.9 17.1]	[21.1, 21.9, 21.9]	0.027

The results of the verification were shown in Figure 10. The red columns indicate the measured absolute position error of the end effector, and the blue columns indicate the absolute position error of the end effector derived from the error model. Although the calculation of FEA would induce some error, the difference between the error by error model and measured error is small and the average residual error is less than 10 µm. So it can be concluded that the error projection method is reasonable and effective.

Figure 10.

Results of the verification.

Sensitivity analysis

Sensitivity analysis is one of the most important applications based on the error model, which can derive the influences of the structural errors on the pose error of the end effector, contributing to the evaluation of the goodness of the mechanism as well as assisting the designing process. In this section, based on the error model proposed above, sensitivity analysis of the structural errors of the 3-P(Pa)S parallel type spindle head was carried out. Firstly, two basic indexes, LSI and GSI, were introduced.

LSI and GSI

The forward-problem of the error model can be expressed as

[\begin{matrix} p & δ p \end{matrix}] = G (δ r_{p}, q)

in which $δ r_{p}$ represents 51 structural errors of the 3-P(Pa)S mechanism, and they were designated from $δ r_{p 1}$ to $δ r_{p 51}$ .

And we can get $p = [\begin{matrix} P^{T} & θ^{T} \end{matrix}] = {[\begin{matrix} p_{x} & p_{y} & p_{z} & φ & θ & ψ \end{matrix}]}^{T}$ ; $δ p = [\begin{matrix} δ P^{T} & δ θ^{T} \end{matrix}] = {[\begin{matrix} δ p_{x} & δ p_{y} & δ p_{z} & δ φ & δ θ & δ ψ \end{matrix}]}^{T}$ .

The position error $δ p_{p o s}$ and the orientation error $δ p_{o r i}$ are the two factors that influence the accuracy of the end effector.

$δ p_{p o s}$ is expressed as

δ p_{p o s} = | δ P | = \sqrt{δ p_{x}^{2} + δ p_{y}^{2} + δ p_{z}^{2}}

$δ p_{o r i}$ is expressed as

δ p_{o r i} = | (R^{e} - R) e_{3} |

in which R and R^e are rotational matrices that composed of θ and δ θ , reflecting the ideal orientation of the end effector and the orientation of the end effector with error, respectively.

Thus, the sensitivity of the jth structural error can be described as

ω_{j} = lim_{δ r_{p j} \to 0} \frac{δ \tilde{p}}{δ r_{p j}} (j = 1 \sim 51)

in which $δ \tilde{p}$ represents $δ p_{p o s}$ or $δ p_{o r i}$ . It is observed that ω_j is relative to the pose of the mechanism since ω_j is different in every single poses within the workspace, so it is named as LSI. However, it is expected that the sensitivity index is not affected by the pose of the end effector. By integrating the LSI over the workspace of the mechanism V, the sensitivity is no longer relative to the pose of the mechanism and is denoted by the GSI

Ω_{j} = \frac{\int_{V}^{} ω_{j} d V}{\int_{V}^{} d V}

The GSIs of the structural errors are listed in Table 5.

Table 5.

List of GSIs of structural errors.

GSI to the position error of the end effector ( $δ p_{p o s}$ )	Displacement structural errors	$δ d_{i 1}$	$δ d_{i 2}$	$δ d_{i 3}$	$δ d_{i 4}$	$δ d_{i 5}$
		2.440	0.408	0.220	1.811	1.142
		$δ d_{i 6}$	$δ d_{i 7}$	$δ d_{i 8}$	$δ a_{i x}$	$δ a_{i y}$
		2.439	0.107	0.089	0.461	0.652
		$δ a_{i z}$	$δ b_{i y}$	$δ q_{i}$
		1.945	0.639	2.028
	Angular structural errors	$θ_{B i x}$	$θ_{B i y}$	$θ_{C i x}$	$θ_{C i z}$
	Angular structural errors	288.890 mm	55.072 mm	155.225 mm	16.036 mm
GSI to the orientation error of the end effector ( $δ p_{o r i}$ )	Displacement structural errors	$δ d_{i 1}$	$δ d_{i 2}$	$δ d_{i 3}$	$δ d_{i 4}$	$δ d_{i 5}$
		4.188 × 10⁻³ mm⁻¹	7.008 × 10⁻⁴ mm⁻¹	3.779 × 10⁻⁴ mm⁻¹	3.108 × 10⁻³ mm⁻¹	1.960 × 10⁻³ mm⁻¹
		$δ d_{i 6}$	$δ d_{i 7}$	$δ d_{i 8}$	$δ a_{i x}$	$δ a_{i y}$
		4.185 × 10⁻³ mm⁻¹	1.841 × 10⁻⁴ mm⁻¹	1.532 × 10⁻⁴ mm⁻¹	6.275 × 10⁻⁵ mm⁻¹	8.7610 × 10⁻⁵ mm⁻¹
		$δ a_{i z}$	$δ b_{i y}$	$δ q_{i}$
		3.333 × 10⁻³ mm⁻¹	6.116 × 10⁻⁴ mm⁻¹	3.489 × 10⁻³ mm⁻¹
	Angular structural errors	$θ_{B i x}$	$θ_{B i y}$	$θ_{C i x}$	$θ_{C i z}$
	Angular structural errors	0.2845	0.0945	0.1485	0.0166

GSI: global sensitivity index.

SSLSI and SSGSI

LSI and GSI are the basic indexes for all kinds of mechanisms. However, based on LSI and GSI, it is preferred to propose a particular index that can better describe the characteristics of a specific mechanism such as the 3-P(Pa)S mechanism. The following factors need to be considered.

Firstly, although some errors have larger GSIs, their corresponding structural parameters are small. In manufacturing, the magnitude of the errors is usually proportional to the length of their corresponding structural parameters or positioning dimensions according to ISO-286, so the structural errors that have larger GSI or LSI may not have a large influence on the end effector in the real application. Secondly, it is noticed that GSI and LSI are all positive number, only reflecting the absolute extent of the influence. The new index is expected to have a sign (positive or negative), so we can figure out whether some structural parameters have similar or opposite influence on the end effector. Thirdly, in LSI and GSI, the error of the end effector is reflected by the position error and the orientation error. It is expected to establish a synthetic error that include both position error and orientation error, so the new index is more concise. Fourthly, the LSI and GSI of the displacement error and angular error don’t have the same unit to be compared together. It is expected that they have the same unit.

Therefore, the new index is established as follows.

Firstly, a synthetic error that reflects the error of the end effector is established. Only one chain was investigated as an example because of the symmetry. By analyzing the structure of the 3-P(Pa)S mechanism, it is mainly composed of three chains. For each chain, it can be regarded as a planer mechanism in the plane of the chain. Therefore, the structural errors of the chain can be classified into two categories: (1) structural errors within the plane of the chain and (2) structural errors perpendicular to the plane of the chain.

For the first kind of structural errors, these structural errors basically “increase” or “reduce” the length of the chain, inducing the error of the end effector by “rotating” the end effector in the plane of the chain (because the parasitic motion of this kind of mechanism is very small). Thus, we can project the error of the end effector to a unit vector e _pi that is in the plane. Since the equilibrium pose of the end effector is along the Z-axis, e _pi is defined to be perpendicular to the Z-axis, so the projected error can retain most of the information. For the second kind of structural errors, these errors basically induce the errors of the end effector that are perpendicular to the plane of the chain. Therefore, the error of the end effector was projected to a unit vector e _ni that is perpendicular to the plane. The projection process is shown in Figure 11.

Figure 11.

Projection of the position error and the orientation error of the end effector.

According to Figure 11, expressions of e _pi and e _ni are derived. e _pi is expressed as

e_{p i} = {[\begin{matrix} cos (β_{i} + π) & sin (β_{i} + π) & 0 \end{matrix}]}^{T}

in which β_i is the angle between the plane of chain i and the X-axis

β_{i} = \frac{5 π}{6} + (i - 1) \frac{2 π}{3} (i = 1 \sim 3)

e _ni is expressed as

e_{n i} = {[\begin{matrix} cos (β_{i} + \frac{π}{2}) & sin (β_{i} + \frac{π}{2}) & 0 \end{matrix}]}^{T}

There are 13 of 17 structural errors ( $δ d_{i 1}$ ∼ $δ d_{i 8}$ , $δ a_{i x}$ , $δ a_{i z}$ , $δ q_{i}$ , $θ_{B i y}$ , and $θ_{C i z}$ ) within the plane of the chain, so the end effector errors caused by these structural errors were projected to e _pi. According to Figure 11, the projection of the position error of the end effector is expressed as

δ p_{p o s}^{'} = δ P^{T} \cdot e_{p i}

The projection of orientation error of the end effector is expressed as

δ p_{o r i}^{'} = C \cdot {(R^{e} e_{3} - R e_{3})}^{T} \cdot e_{p i}

in which $R^{e} e_{3} - R e_{3}$ represents the orientation error vector and its length is equal to the value of orientation error. C is the average length of the cutting tool. By multiplying the length of the cutting tool, orientation error of the end effector is transformed to the position error of the tool tip. In this article, the average length of the cutting tool is set to 80 mm.

Therefore, the synthetic error can be derived as

δ p_{s y n} = δ p_{p o s}^{'} + δ p_{o r i}^{'}

There are only 4 of 17 structural errors ( $δ a_{i y}$ , $δ b_{i y}$ , $θ_{B i x}$ , and $θ_{C i x}$ ) perpendicular to the plane of the chain, so the end effector error caused by these structural errors are projected to e _ni. And the synthetic error can be similarity derived

δ p_{s y n} = δ P^{T} \cdot e_{n i} + C \cdot {(R^{e} e_{3} - R e_{3})}^{T} \cdot e_{n i}

Therefore, the new local index of the jth structural error can be derived. For displacement structural errors, since the magnitude of the errors are usually proportional to their corresponding structural parameters, their corresponding structural parameters or positioning dimensions were introduced as a multiplier

{ω ′}_{j} = r_{p j} \cdot lim_{δ r_{p d j} \to 0} \frac{δ p_{s y n}}{δ r_{p d j}}

in which $δ r_{p d j}$ is the jth displacement structural error and r_pj is the corresponding structural parameter or positioning dimension. For the parameter whose corresponding structural parameter is zero, the positioning dimension was used. For example, the value of a_iy is zero, but its positioning dimension is the distance between two adjacent spherical joints along the Y_i-axis.

As for angular structural errors, it was firstly multiplied by the length of the corresponding part. So it can be transformed into the displacement structural error. Then, the index was multiplied by the positioning dimension corresponding to the transformed displacement structural error. For instance, the angular error of the guide way was multiplied by the length of the guide way to be transformed to the position error of the end of the guide way. Then, the index was multiplied by the positioning dimension of the end of the guide way. Therefore, the index can be written as

{ω ′}_{j} = r_{p j} \cdot lim_{δ r_{p a j} \to 0} \frac{δ p_{s y n}}{δ r_{p a j} \cdot r_{p j}^{′}}

in which $r_{p j}^{'}$ is the length of the corresponding part. r_pj is the positioning dimension corresponding to the transformed displacement structural error. $δ r_{p a j}$ is the corresponding angular structural error.

Therefore, the new local index is established and named as SSLSI. The new global index can also be derived by integrating SSLSI within the whole workspace, V, named as SSGSI

{Ω'}_{j} = \frac{\int_{V}^{} ω_{j}^{'} d V}{\int_{V}^{} d V}

Thus, the SSGSIs and SSLSIs of the structural errors can be derived. The SSGSIs of all the structural errors are listed in Table 6. According to the table, it can be observed that the SSGSI of displacement structural errors and the angular structural errors have the same unit to be compared together. Additionally, there is only one index for each structural error, making the new index more concise.

Table 6.

List of SSGSIs of the structural errors.^a

SSGSI	Displacement structural errors	$δ d_{i 1}$	$δ d_{i 2}$	$δ d_{i 3}$	$δ d_{i 4}$	$δ d_{i 5}$
		1262.6 mm	−212.75 mm	16.494 mm	370.10 mm	86.167 mm
		$δ d_{i 6}$	$δ d_{i 7}$	$δ d_{i 8}$	$δ a_{i x}$	$δ a_{i y}$
		−184.01 mm	−12.389 mm	11.668 mm	9.3021 mm	−109.99 mm
		$δ a_{i z}$	$δ b_{i y}$	$δ q_{i}$
		−94.763 mm	120.34 mm	163.75 mm
	Angular structural errors	$θ_{B i x}$	$θ_{B i y}$	$θ_{C i x}$	$θ_{C i z}$
	Angular structural errors	−49.671 mm	−58.288 mm	26.958 mm	−0.815 mm

SSGSI: standardized signed global sensitivity index.

^aFour structural errors (δa_iy , δb_iy , θ_Bix , and δ_Cix ) are perpendicular to the plane of the Chain i. The rest structural errors are located in the plane of the Chain i.

Since $δ d_{i 1}$ , $δ d_{i 2}$ , $δ d_{i 4}$ , $δ d_{i 6}$ , $δ q_{i}$ , $δ b_{i y}$ $δ a_{i y}$ , and $δ a_{i z}$ have the largest SSGSIs and their combined SSGSI account for 90.9% of the total amount. Their SSLSIs of these structural errors were mainly analyzed and shown in Figure 12. The SSLSIs were represented by a polar diagram in which the coordinates along the radius represent the value of the tilt angle of the end effector, and the coordinates on the circumference represent the direction that the end effector tilt to, named as roll angle. Only the SSLSI of the structural errors in chain 1 (i = 1) were illustrated because of the symmetry. For each graph, the plane of chain 1 is marked by a dash-dot line.

Figure 12.

SSLSI of (a) δd₁₁, (b) δd₁₂, (c) δd₁₄, (d) δd₁₆, (e) δq₁, (f) $δ b_{1 y}$ , (g) $δ a_{1 y}$ , and (h) $δ a_{1 z} .$ SSLSI: standardized signed local sensitivity index.

Analysis of the sensitivity index

Some conclusions can be drawn based on the SSGSIs and SSLSIs:

Arrange all the structural errors according to their absolute SSGSIs, as shown in Figure 13. It can be concluded from the figure that, among all the 17 errors, the most influential ones are $δ d_{i 1}$ , $δ d_{i 4}$ , $δ d_{i 2}$ , $δ d_{i 6}$ , $δ q_{i}$ , $δ b_{i y}$ $δ a_{i y}$ , and $δ a_{i z}$ . And they account for about 90.9% of the total SSGSI. Additionally, it can be observed that the structural errors that have largest SSGSI are mostly located in the parallelogram structure. Among these errors, $δ d_{i 1}$ and $δ d_{i 2}$ represent the error of the two long parallel links of the parallelogram, which are the main part of the parallelogram structure. $δ d_{i 4}$ represents the error of the link that connecting the parallelogram and the moving platform. $δ d_{i 6}$ represents the error between the center of two rotational joints in the parallelogram. Thus, when manufacturing and assembling these parts, their accuracy should be paid more attention to.

Since the relative error is usually considered during the machining, we should not only consider the magnitude of the error but also pay attention to the variation of the error. According to the SSLSI shown in Figure 12, it can be observed that most SSLSIs vary rapidly at the boundary of the workspace (where the tilt angle of the spindle head is larger). However, near the center of the workspace (where the tilt angle is smaller), the variation of the SSLSIs are slower. Therefore, this spindle head has a better performance near the center of the workspace where the influence of the structure errors varies a little. Additionally, since the standard deviation can reflect the extent of the variation of the data, the standard deviations of the SSLSIs of the structural errors were calculated, as shown in Table 7. According to the table, $δ a_{i x}$ , $δ b_{i y}$ , $δ a_{i y}$ , $δ d_{i 1}$ , and $δ d_{i 2}$ have the largest standard deviations. Therefore, if we considering more about the relative errors, these four structural errors should be carefully considered.

According to Figure 12, it can be further observed that some structural errors have similar contour of SSLSI but totally opposite sign (e.g. $δ d_{i 1}$ and $δ d_{i 2}$ , $δ d_{i 1}$ and $δ d_{i 6}$ , etc.), which indicates that some structural errors may have opposite influence on the end effector. Especially for $δ d_{i 1}$ and $δ d_{i 2}$ , although they represent the error of two paralleled links in the parallelogram structure which have the same shape, they have almost opposite influences on the end effector.

Figure 13.

Arrangement of structural errors according to their absolute SSGSIs (from the maximum to the minimum). SSGSI: standardized signed global sensitivity index.

Table 7.

Standard deviations of the SSLSIs.

Structural error	$δ d_{i 1}$	$δ d_{i 2}$	$δ d_{i 3}$	$δ d_{i 4}$	$δ d_{i 5}$	$δ d_{i 6}$	$δ d_{i 7}$	$δ d_{i 8}$
Standard deviation of the SSLSI (mm)	48.612	38.826	1.773	12.242	3.055	7.085	7.101	0.413
Structural error	$δ a_{i x}$	$δ a_{i y}$	$δ a_{i z}$	$δ b_{i y}$	$δ q_{i}$	$θ_{B i x}$	$θ_{B i y}$	$θ_{C i x}$	$θ_{C i z}$
Standard deviation of the SSLSI (mm)	57.593	61.952	6.584	69.109	4.267	25.219	27.323	14.692	6.361

SSLSI: standardized signed local sensitivity index.

The above conclusions are expected to have further applications. The first possible application is the tolerance design of the part of the spindle head. Take δd₁₁ and δd₁₂ as an example. Generally, the ideal value of structural parameter is within its upper deviation and lower deviation. Additionally, since d₁₁ and d₁₂ have the same length and shape, the tolerance grade for d₁₁ and d₁₂ is the same, for example

d_{11} = 487_{- 0.0135}^{+ 0.0135} mm (fundamental deviation: JS, international tolerance grade: IT5)

d_{12} = 487_{- 0.0135}^{+ 0.0135} mm (fundamental deviation: JS, international tolerance grade: IT5)

Since the contour of SSLSIs for $δ d_{i 1}$ and $δ d_{i 2}$ are similar and their influence on the end effector are opposite, it can be derived that the approximate maximum absolute position error of the end effector would happen when d₁₁ is equal to its upper bound and d₁₂ is equal to its lower bound, or vice versa. And the minimum absolute position error of the end effector is zero (when d₁₁ and d₁₂ are equal to the ideal value). Thus, the range of the absolute position error of the end effector at each pose within the workspace can be derived, as shown in Figure 14(a)

Figure 14.

Range of the absolute position error of the end effector at each pose within the workspace. (a) Before modifying the range of the tolerance and (b) after modifying the range of the tolerance.

However, since the ratio of the influence of $δ d_{i 1}$ and $δ d_{i 2}$ on the end effector is about 6:1 according to their SSLSIs and SSGSIs, we can modify the range of their tolerance by slightly improve the tolerance grade of d₁₁ and greatly reduce the tolerance grade of d₁₂

d_{11} = 487_{- 0.01}^{+ 0.01} mm (fundamental deviation: JS, international tolerance grade: IT4)

d_{12} = 487_{+ 0.0315}^{+ 0.0315} mm (fundamental deviation: JS, international tolerance grade: IT7)

We can also derive that the approximate maximum absolute position error of the end effector would happen when d₁₁ is equal to its upper bound and d₁₂ is equal to its lower bound, or vice versa. And the minimum absolute error of the end effector is zero. So the range of the absolute position error of the end effector at each pose within the workspace can be derived as shown in Figure 14(b).

It can be derived that after the modification, the cost for manufacturing these two links can be reduced. Because the tolerance grade of d₁₁ increases one level (IT5 to IT4, accuracy improved by 26%), but the tolerance grade of d₁₂ can be reduced by two levels (IT5 to IT7, accuracy reduced by 133%). However, the range of the absolute position error of the end effector do not increase but become smaller according to Figure 14(a) and (b), which means that the accuracy becomes better.

The second possible application is to alleviate the error caused by the temperature change of the mechanism (e.g. the temperature change caused by the friction of the joints or the change of the environment). Since the links corresponding to $δ d_{i 1}$ and $δ d_{i 2}$ have the same shape and are also parallel and very close to each other, the temperature change of these two links is supposed to be close. The original material of the two links are the same (HT300 gray iron, thermal expansion coefficient: 9.2 × 10⁻⁶°C⁻¹). Take chain 1 as an example, assuming that the average temperature rise of d₁₁ and d₁₂ is 5°C, the absolute position error of the end effector within the workspace can be derived, as shown in Figure 15(a). However, it has been derived that δd₁₁ and δd₁₂ have almost opposite SSLSIs, and the absolute SSGSI of δd₁₁ is about six times as large as that of δd₁₂. So if the link that correspondent to δd₁₂ can generate more thermal expansion, it would compensate the error caused by δd₁₁. Supposing that the material of the link that correspond to $δ d_{i 2}$ was changed to 7075 Aluminum alloy whose thermal expansion coefficient is 23.6 × ¹⁰⁻⁶°C⁻¹ (2.6 times as large as that of HT300 grey iron), although δd₁₂ becomes larger, the absolute position error of the end effector is expected to be reduced. The simulated absolute position error after changing the material is shown in Figure 15(b).

Figure 15.

Absolute position error of the end effector before and after changing the material. (a) Before changing the material of the link and (b) after changing the material of the link.

According to the figure, it can be derived that the absolute position error of the end effector were greatly reduced within the workspace. The average position error in the whole workspace can be reduced from 0.0449 to 0.0309 mm. Especially for the near-center area (tilt angle within [0°, 20°]) where the spindle head mostly work at, the average error can be reduced from 0.0431 to 0.0285 mm, with the accuracy being improved by 33.9%.

Conclusions

The error model of a novel 3-DOF parallel spindle head based on 3-P(Pa)S mechanism was established. By projecting the structural errors of the parallelogram structure to the errors of the 3-PRS mechanism simplified from the 3-P(Pa)S mechanism, the errors of the parallelogram structure were considered into the error model. Then, the projecting method was verified by an experiment on the spindle head to prove its correctness.

Sensitivity analysis of the spindle head was carried out based on the proposed error model. According to the characteristic of the 3-P(Pa)S mechanism, two new sensitivity indexes, SSLSI and SSGSI, were proposed which can better reflect the influence of the structural errors to the end effector error.

By analyzing the SSLSIs and SSGSIs, structural errors that have the largest influence on the end effector were derived, and it is observed that the structural errors that have the larger influence are mainly located in the parallelogram structure. Additionally, the magnitude of the variation of SSLSIs within the workspace were also analyzed. Moreover, it was found that some structural errors have similar contour of SSLSI but totally opposite sign, indicating that they may have opposite influence on the end effector. Based on that, two possible applications to reduce the error of the end effector were proposed and simulated. The error projection method and the new sensitivity indexes proposed in this article are expected to be used in the error modeling and sensitivity analysis of other parallel mechanisms.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (no. 51757260), the National Science and Technology Major Project of China (no. 2015ZX04001002), and the Tsinghua University Initiative Scientific Research Program (no. 2014z22068).

Appendix 1

In the error projection, there is an approximation in which the orientation of one of the link (d_i1 or d_i2) of the parallelogram is parallel to l_i. Thus, we can use only one parameter, φ_pi, to describe the orientation of the parallelogram structure and the orientation of the link l_i, and the error model becomes simpler to solve.

It needs to verify whether this approximation would influence the accuracy of the model. There are two models (two cases) for this approximation: (1) d_i1 is parallel to the link l_i, d_i2 is not parallel to l_i; (2) d_i2 is parallel to the link l_i, d_i1 is not parallel to l_i. These two models are two extreme cases, and the real shape of the parallelogram is between these two cases, as shown in Figure 1A. So, if there is almost no difference between the outputs of these two models, it can be concluded that the approximation is feasible.

Therefore, we derived the error of the end effector at different poses of the spindle head derived by these two models, as shown in Table 1A. In the table, without loss of generality, three structural errors ( $δ d_{i 1}, δ d_{i 2}, δ d_{i 3}$ ) were assigned different values (because these three structural errors can approximately represent other structural errors to cause the difference of the output of the models: $δ d_{i 3}$ can represent $δ d_{i 5}$ and $δ d_{i 8}$ , $δ d_{i 1}$ and $δ d_{i 2}$ can represent $δ d_{i 6}$ . $δ d_{i 4}$ and $δ d_{i 7}$ would not cause the difference of the output of the models). It can be observed that when the value of structural errors are around 10–20 µm, the difference between the output of two models is only about 0.0005 µm. Therefore, it can be concluded that there is almost no difference between the output of these two models, and the approximation is feasible.

Thus, in this article, model 2 was selected, so d_i2 is parallel to the link l_i . And the orientation of d_i2 can be described by φ_pi in the local coordinate frame of the parallelogram structure, {x_i , y_i }. Therefore, there is no orientation error, $δ d_{i 2}$ , in equations (13) and (14). Additionally, since the errors of the parallelogram were projected to the length error of l_i and b_ix, there is no $δ l_{i}$ and $δ q_{i}$ either.

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