2.1. System description

As illustrated in Figure 1, the study subject of this paper is a novel 5-DOF double-driven parallel mechanism, which consists of a fixed base, a moving platform, a PUU limb, and two PR(RPRR)S limbs. Herein, P, R, S, and U represent the prismatic, the revolute, the spherical, and the universal joint, respectively. P specifically refers to an actuated prismatic joint. Compared with traditional parallel mechanisms, the PR(RPRR)S limb has two active joints in one limb (which is also called the double-driven limb), reducing the number of limbs of parallel mechanisms; that is, only three limbs are needed to achieve 5-DOF of the moving platform. This double-driven strategy can effectively increase motion flexibility and reduce interference between limbs, showing good application foreground. Additionally, to increase stiffness and simplify control, actuated revolute joints in the double-driven limbs are designed as a sub-closed loop, significantly increasing the complexity and difficulty of kinematic error modeling.

Figure 1. 3D model: (a) the whole parallel mechanism, (b) PUU limb, and (c) PR(RPRR)S limb.

For the convenience of analysis, the schematic diagram of the 5-DOF double-driven parallel mechanism is shown in Figure2. The base frame $\{O_{1}\}$ is established at the intersection point $O_{1}$ of the extension lines of the three guide rails. The positive direction of $y_{1}$ axis points to the PUU limb rail’s direction, that is, $A_{1}$ . The $z_{1}$ axis is perpendicular to the plane where the three guides are located and points to the side of the moving platform, and the $x_{1}$ axis is determined by the right-hand rule. The origin $O_{2}$ of frame $\{O_{2}\}$ of the moving platform is located at the geometric center of the isosceles right triangle $C_{1}C_{2}C_{3}$ . The $y_{2}$ axis points from $O_{2}$ to $C_{1}$ . The $x_{1}$ axis is parallel to $\boldsymbol{C}_{3}\boldsymbol{C}_{2}$ and points to $C_{2}$ , and the $z_{2}$ axis can be determined by the right-hand rule. According to the above frame definition, the 5-DOF of the moving platform can be expressed as three translation motions (along $x_{1}, y_{1}$ , and $z_{1}$ axis) and two rotation motions (around $x_{1}$ and $y_{1}$ axis) [Reference Wang, Lin, Cao, Wu and Sun7]. The main geometric parameters and corresponding notations of this 5-DOF parallel mechanism are shown in Table 1.

Table 1. Geometric parameters of the 5-DOF parallel mechanism.

Figure 2. Schematic diagram of the 5-DOF parallel mechanism.

2.2. Kinematic error modeling of the parallel mechanism

The closed-loop vector method is utilized for the 5-DOF double-driven parallel mechanism under study for kinematic error modeling. Error transfer equations of each limb should be derived by using the geometric relations. Especially, an equivalent transformation method will be proposed for the limb with a sub-closed loop. Then kinematic error model of the whole parallel mechanism can be established by combining the error transfer equations and the differential form of parasitic motion.

2.2.1 Error transfer equation of PUU limb

The closed-loop vector of the PUU limb is shown in Figure3. $\boldsymbol{u}_{1}$ is the unit direction vector of the fixed-length rod $B_{1}C_{1}$ . $\boldsymbol{p}$ is the position vector of the origin $O_{2}$ of moving platform frame $\{O_{2}\}$ . $\boldsymbol{w}_{1}$ is the unit direction vector of $\boldsymbol{O}_{1}\boldsymbol{A}_{1}$ , and $\boldsymbol{c}_{1}$ is the position vector of $C_{1}$ in frame $\{O_{2}\}$ . The closed-loop vector equation of PUU limb can be expressed as

Figure 3. Schematic diagram of PUU limb.

(1) \begin{equation} \mathit{g}_{1}\boldsymbol{w}_{1}+\boldsymbol{b}_{1}+l_{1}\boldsymbol{u}_{1}=\boldsymbol{p}+\boldsymbol{R}\boldsymbol{c}_{1} \end{equation}

where $\boldsymbol{w}_{1}=\left[\begin{array}{c@{\quad}c@{\quad}c} \cos \varphi _{1} & \sin \varphi _{1} & 0 \end{array}\right]^{T}, \varphi _{1}=\frac{\pi }{2}, \boldsymbol{b}_{1}=\left[\begin{array}{c@{\quad}c@{\quad}c} 0 & 0 & h \end{array}\right]^{T}, \boldsymbol{c}_{1}=\left[\begin{array}{c@{\quad}c@{\quad}c} 0 & \frac{2}{3}d & 0 \end{array}\right]^{T}$ . $\boldsymbol{R}$ is the orientation matrix of frame $\{O_{2}\}$ with respect to frame $\{O_{1}\}$ .

According to X-Y-Z Euler angle notations, $\boldsymbol{R}$ can be expressed as

(2) \begin{equation} \boldsymbol{R}=\boldsymbol{Rot}\left(x, \alpha \right)\boldsymbol{Rot}\left(y,\beta \right)\boldsymbol{Rot}\left(z,\gamma \right)=\left[\begin{array}{c@{\quad}c@{\quad}c} \textrm{c}\beta \textrm{c}\gamma & -\textrm{c}\beta \textrm{s}\gamma & \textrm{s}\beta \\ \textrm{c}\alpha \textrm{s}\gamma +\textrm{s}\alpha \textrm{s}\beta \textrm{c}\gamma & \textrm{c}\alpha \textrm{c}\gamma -\textrm{s}\alpha \textrm{s}\beta \textrm{s}\gamma & -\textrm{c}\beta \textrm{s}\alpha \\ \textrm{s}\alpha \textrm{s}\gamma -\textrm{c}\alpha \textrm{s}\beta \textrm{c}\gamma & \textrm{s}\alpha \textrm{c}\gamma +\textrm{c}\alpha \textrm{s}\beta \textrm{s}\gamma & \textrm{c}\alpha \textrm{c}\beta \end{array}\right] \end{equation}

where $\textrm{c}\alpha =\cos \alpha, \textrm{s}\alpha =\sin \alpha$ . Then, differentiate Eq. (1), ignore the high-order infinitesimal terms, and yield

(3) \begin{equation} d\mathit{g}_{1}\boldsymbol{w}_{1}+\mathit{g}_{1}d\boldsymbol{w}_{1}+d\boldsymbol{b}_{1}+dl_{1}\boldsymbol{u}_{1}+l_{1}d\boldsymbol{u}_{1}=d\boldsymbol{p}+d\boldsymbol{R}\boldsymbol{c}_{1}+\boldsymbol{R}d\boldsymbol{c}_{1} \end{equation}

where $d\boldsymbol{w}_{1}=d_{e}\boldsymbol{w}_{1}d\varphi _{1}=[\begin{array}{c@{\quad}c@{\quad}c} {-}\sin \varphi _{1} & \cos \varphi _{1} & 0 \end{array}]^{T}d\varphi _{1}$ . To eliminate the unknown differential term d u ₁, taking dot products on both sides of Eq. (3) with $\boldsymbol{u}_{1}$ , the error transfer equation of the PUU limb can be obtained as

(4) \begin{equation} {\boldsymbol{u}_{1}}^{\!T}d\boldsymbol{p}+\left(\boldsymbol{R}\boldsymbol{c}_{1}\times \boldsymbol{u}_{1}\right)^{T}\boldsymbol{J}_{{\unicode[Arial]{x03A9}} }d\boldsymbol{\Omega }=d\mathit{l}_{1}-{\boldsymbol{u}_{1}}^{\!T}\boldsymbol{R}d\boldsymbol{c}_{1}+{\boldsymbol{u}_{1}}^{\!T}d\mathit{g}_{1}\boldsymbol{w}_{1}+\mathit{g}_{1}{\boldsymbol{u}_{1}}^{\!T}d_{e}\boldsymbol{w}_{1}d\varphi _{1}+{\boldsymbol{u}_{1}}^{\!T}d\boldsymbol{b}_{1} \end{equation}

where $\boldsymbol{J}_{{\unicode[Arial]{x03A9}} }\boldsymbol{=}\left[\begin{array}{c@{\quad}c@{\quad}c} 1 & 0 & \sin \beta \\ 0 & \cos \alpha & -\sin \alpha \cos \beta \\ 0 & \sin \alpha & \cos \alpha \cos \beta \end{array}\right], d\boldsymbol{\Omega }\boldsymbol{=}[\begin{array}{c@{\quad}c@{\quad}c} d\alpha & d\beta & d\gamma \end{array}]^{T}$ is the X-Y-Z Euler angle error vector. $d\boldsymbol{p}=[\begin{array}{c@{\quad}c@{\quad}c} dx & dy & dz \end{array}]^{T}$ is the position error vector of frame $\{O_{2}\}$ with respect to frame $\{O_{1}\}$ .

2.2.2 Error transfer equation of PR(RPRR)S limbs

Considering that two double-driven PR(RPRR)S limbs have the same structure and are placed symmetrically, the modeling process of only one limb is described here. Notably, for the general closed-loop vector method, a single closed-loop vector equation can only be applied for one vector closed-loop. Therefore, kinematic error modeling of limbs with a sub-closed loop by using the closed-loop vector method is a challenging problem. To solve this problem, that is, establishing the error transfer equation of the PR(RPRR)S limb, an equivalent transformation method is proposed: (1) The PR(RPRR)S limb is equivalently transformed to a PRPS limb; that is, B ₂ and $C_{2}$ are directly connected by a virtual P joint, as shown in Figure4(b). Then error transfer equation of the PRPS limb can be calculated. (2) Geometric errors of the sub-closed loop are integrated into that of the limb based on geometric conditions. The detailed derivation process is expressed as follows.

Figure 4. Schematic diagram of double-driven limb: (a) PR(RPRR)S limb and (b) equivalent PRPS limb.

From Figure4(b), the closed-loop vector equation of the equivalent PRPS limb can be obtained as

(5) \begin{equation} g_{2}\boldsymbol{w}_{2}+\boldsymbol{b}_{2}^{}{+}{}L_{22}\boldsymbol{u}_{22}=\boldsymbol{p}+\boldsymbol{Rc}_{2}^{}{}{} \end{equation}

where $\boldsymbol{w}_{2}=\left[\begin{array}{c@{\quad}c@{\quad}c} \cos \varphi _{2} & \sin \varphi _{2} & 0 \end{array}\right]^{T}, \varphi _{2}=\frac{\pi }{4}, \boldsymbol{b}_{2}=\left[\begin{array}{c@{\quad}c@{\quad}c} 0 & 0 & h \end{array}\right]^{T}$ . $\boldsymbol{u}_{22}$ is the unit direction vector of $\boldsymbol{B}_{2}\boldsymbol{C}_{2}, L_{22}$ is the corresponding length. $\boldsymbol{c}_{2}=\left[\begin{array}{c@{\quad}c@{\quad}c} d & -\frac{1}{3}d & 0 \end{array}\right]^{T}$ is represented in frame $\{O_{2}\}$ . Next, differentiate Eq. (5), ignore the high-order infinitesimal terms, and yield

(6) \begin{equation} dg_{2}\boldsymbol{w}_{2}+g_{2}d\boldsymbol{w}_{2}+d\boldsymbol{b}_{2}+dL_{22}\boldsymbol{u}_{22}+L_{22}d\boldsymbol{u}_{22}=d\boldsymbol{p}+d\boldsymbol{R}\boldsymbol{c}_{2}+\boldsymbol{R}d\boldsymbol{c}_{2} \end{equation}

where $d\boldsymbol{w}_{2}=[\begin{array}{c@{\quad}c@{\quad}c} {-}\sin \varphi _{2} & \cos \varphi _{2} & 0 \end{array}]^{T}d\varphi _{2}=d_{e}\boldsymbol{w}_{2}d\varphi _{2}$ . $\boldsymbol{u}_{22}=[\begin{array}{c@{\quad}c@{\quad}c} {-}\sin \varphi _{2}\cos \theta _{22} & \cos \varphi _{2}\cos \theta _{22} & \sin \theta _{22} \end{array}]^{T}$ , $\theta _{22}$ is the rotation angle of the revolute joint at $B_{2}$ . By taking the derivative of $\boldsymbol{u}_{22}$ and ignoring higher-order infinitesimal terms, d u ₂₂ can be represented as $\mathit{d}\boldsymbol{u}_{22}=\left[\begin{array}{c@{\quad}c} -\cos \varphi _{2}\cos \theta _{22} & \sin \varphi _{2}\sin \theta _{22}\\ -\sin \varphi _{2}\cos \theta _{22} & -\cos \varphi _{2}\sin \theta _{22}\\ 0 & \cos \theta _{22} \end{array}\right]\left[\begin{array}{l} d\varphi _{2}\\ d\theta _{22} \end{array}\right]$ . Taking dot products on both sides of Eq. (6) with $\boldsymbol{w}_{2}$ , Eq. (6) is rewritten as

(7) \begin{equation} {\boldsymbol{w}_{2}}^{T}d\boldsymbol{p}+\left(\boldsymbol{R}\boldsymbol{c}_{2}\times \boldsymbol{w}_{2}\right)^{T}\boldsymbol{J}_{{\unicode[Arial]{x03A9}} }d\boldsymbol{\Omega }=dg_{2}-{\boldsymbol{w}_{2}}^{T}\boldsymbol{R}d\boldsymbol{c}_{2}+{\boldsymbol{w}_{2}}^{T}d\boldsymbol{b}_{2}-L_{22}\cos \theta _{22}d\varphi _{2} \end{equation}

Taking dot products on both sides of the Eq. (6) with $\boldsymbol{u}_{22}$ . It is obvious that, ${\boldsymbol{u}_{22}}^{T}\boldsymbol{w}_{2}=0$ , then we obtain

(8) \begin{equation} {\boldsymbol{u}_{22}}^{T}d\boldsymbol{p}+\left(\boldsymbol{R}\boldsymbol{c}_{2}\times \boldsymbol{u}_{22}\right)^{T}\boldsymbol{J}_{{\unicode[Arial]{x03A9}} }d\boldsymbol{\Omega }=g_{2}{\boldsymbol{u}_{22}}^{T}d\boldsymbol{w}_{2}+dL_{22}-{\boldsymbol{u}_{22}}^{T}\boldsymbol{R}d\boldsymbol{c}_{2}+{\boldsymbol{u}_{22}}^{T}d\boldsymbol{b}_{2} \end{equation}

Notably, the above derivation is focused on the equivalent PRPS limb. To consider the effects of the sub-closed loop on PR(RPRR)S limb as shown in Figure4(a), based on the cosine law of triangles $D_{1}E_{1}F_{1}$ and $B_{2}C_{2}D_{1}$ , the variable $L_{22}$ can be derived as

(9) \begin{equation} L_{22}=\sqrt{\left(l_{2}+l_{5}\right)^{2}+{l_{3}}^{2}+\left(l_{2}+l_{5}\right)l_{3}\frac{{l_{4}}^{2}+{l_{5}}^{2}-{g_{4}}^{2}}{l_{4}l_{5}}} \end{equation}

Then, differentiate Eq. (9), and ignore the high-order infinitesimal terms; dL ₂₂can be calculated as

(10) \begin{equation} dL_{22}=\partial \boldsymbol{L}_{22}d\boldsymbol{l}_{22} \end{equation}

where $\partial \boldsymbol{L}_{22}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \dfrac{\partial L_{22}}{\partial l_{2}} & \dfrac{\partial L_{22}}{\partial l_{3}} & \dfrac{\partial L_{22}}{\partial l_{4}} & \dfrac{\partial L_{22}}{\partial l_{5}} & \dfrac{\partial L_{22}}{\partial g_{4}} \end{array}\right], d\boldsymbol{l}_{22}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} dl_{2} & dl_{3} & dl_{4} & dl_{5} & dg_{4} \end{array}\right]^{T}$ .

By substituting Eqs. (10) to (8), (8) can be rewritten as

(11) \begin{equation} {\boldsymbol{u}_{22}}^{T}d\boldsymbol{p}+\left(\boldsymbol{R}\boldsymbol{c}_{2}\times \boldsymbol{u}_{22}\right)^{T}\boldsymbol{J}_{{\unicode[Arial]{x03A9}} }d\boldsymbol{\Omega }=g_{2}{\boldsymbol{u}_{22}}^{T}\mathit{d}_{\mathit{e}}\boldsymbol{w}_{2}d\varphi _{2}+\partial \boldsymbol{L}_{22}d\boldsymbol{l}_{22}-{\boldsymbol{u}_{22}}^{T}\boldsymbol{R}d\boldsymbol{c}_{2}+{\boldsymbol{u}_{22}}^{T}d\boldsymbol{b}_{2} \end{equation}

Similarly, error transfer equations of the other PR(RPRR)S limb can be formulated as

(12) \begin{equation} {\boldsymbol{w}_{3}}^{T}d\boldsymbol{p}+\left(\boldsymbol{R}\boldsymbol{c}_{3}\times \boldsymbol{w}_{3}\right)^{T}\boldsymbol{J}_{{\unicode[Arial]{x03A9}} }d\boldsymbol{\Omega }=dg_{3}-{\boldsymbol{w}_{3}}^{T}\boldsymbol{R}d\boldsymbol{c}_{3}+{\boldsymbol{w}_{3}}^{T}d\boldsymbol{b}_{3}+L_{33}\cos \theta _{33}d\varphi _{3} \end{equation}

(13) \begin{equation} {\boldsymbol{u}_{33}}^{T}d\boldsymbol{p}+\left(\boldsymbol{R}\boldsymbol{c}_{3}\times \boldsymbol{u}_{33}\right)^{T}\boldsymbol{J}_{{\unicode[Arial]{x03A9}} }d\boldsymbol{\Omega }=g_{3}{\boldsymbol{u}_{33}}^{T}d_{e}\boldsymbol{w}_{3}d\varphi _{3}+\partial \boldsymbol{L}_{33}d\boldsymbol{l}_{33}-{\boldsymbol{u}_{33}}^{T}\boldsymbol{R}d\boldsymbol{c}_{3}+{\boldsymbol{u}_{33}}^{T}d\boldsymbol{b}_{3} \end{equation}

where $\varphi _{3}=\dfrac{3}{4}\pi, \partial \boldsymbol{L}_{33}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \dfrac{\partial L_{33}}{\partial l_{6}} & \dfrac{\partial L_{33}}{\partial l_{7}} & \dfrac{\partial L_{33}}{\partial l_{8}} & \dfrac{\partial L_{33}}{\partial l_{9}} & \dfrac{\partial L_{33}}{\partial g_{5}} \end{array}\right], d\boldsymbol{l}_{33}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} dl_{6} & dl_{7} & dl_{8} & dl_{9} & dg_{5} \end{array}\right]^{T}$ .

2.2.3 Error modeling of whole parallel mechanism considering a parasitic motion

To derive kinematic error modeling of the whole parallel mechanism, error transfer equations of the three limbs, that is, Eqs. (4)and (7) and Eqs. (11) to (13) should be combined, resulting in a matrix equation as

(14) \begin{equation} \boldsymbol{J}_{1}d\boldsymbol{X}=\boldsymbol{J}_{2}d\boldsymbol{W}_{1} \end{equation}

where $d\boldsymbol{X}=[\begin{array}{c@{\quad}c} d\boldsymbol{p} & d\boldsymbol{\Omega } \end{array}]^{T}, \ \boldsymbol{J}_{1}=\left[\begin{array}{c@{\quad}c} {\boldsymbol{u}_{1}}^{\!T} & (\boldsymbol{R}\boldsymbol{c}_{1}\times \boldsymbol{u}_{1})^{T}\boldsymbol{J}_{{\unicode[Arial]{x03A9}} }\\ {\boldsymbol{w}_{2}}^{T} & (\boldsymbol{R}\boldsymbol{c}_{2}\times \boldsymbol{w}_{2})^{T}\boldsymbol{J}_{{\unicode[Arial]{x03A9}} }\\ {\boldsymbol{u}_{22}}^{T} & (\boldsymbol{R}\boldsymbol{c}_{2}\times \boldsymbol{u}_{22})^{T}\boldsymbol{J}_{{\unicode[Arial]{x03A9}} }\\ {\boldsymbol{w}_{3}}^{T} & (\boldsymbol{R}\boldsymbol{c}_{3}\times \boldsymbol{w}_{3})^{T}\boldsymbol{J}_{{\unicode[Arial]{x03A9}} }\\ {\boldsymbol{u}_{33}}^{T} & (\boldsymbol{R}\boldsymbol{c}_{3}\times \boldsymbol{u}_{33})^{T}\boldsymbol{J}_{{\unicode[Arial]{x03A9}} } \end{array}\right]_{5\times 6}, \ \boldsymbol{J}_{2}=\left[\begin{array}{c} \begin{array}{c@{\quad}c} \boldsymbol{N}_{1} & \boldsymbol{0}_{1\times 26} \end{array}\\ \begin{array}{c@{\quad}c@{\quad}c} \boldsymbol{0}_{1\times 9} & \boldsymbol{N}_{2} & \boldsymbol{0}_{1\times 18} \end{array}\\ \begin{array}{c@{\quad}c@{\quad}c} \boldsymbol{0}_{1\times 10} & \boldsymbol{N}_{3} & \boldsymbol{0}_{1\times 13} \end{array}\\ \begin{array}{c@{\quad}c@{\quad}c} \boldsymbol{0}_{1\times 22} & \boldsymbol{N}_{4} & \boldsymbol{0}_{1\times 5} \end{array}\\ \begin{array}{c@{\quad}c} \boldsymbol{0}_{1\times 23} & \boldsymbol{N}_{5} \end{array} \end{array}\right]_{5\times 35}$ ,

$\boldsymbol{N}_{1}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} {\boldsymbol{u}_{1}}^{\!T}\boldsymbol{w}_{1} & -{\boldsymbol{u}_{1}}^{\!T}\boldsymbol{R} & {\boldsymbol{u}_{1}}^{\!T} & 1 & \mathit{g}_{1}{\boldsymbol{u}_{1}}^{\!T}d_{e}\boldsymbol{w}_{1} \end{array}\right]_{1\times 9}, \ \boldsymbol{N}_{2}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1 & -{\boldsymbol{w}_{2}}^{T}\boldsymbol{R} & {\boldsymbol{w}_{2}}^{T} & -L_{22}\cos \theta _{22} \end{array}\right]_{1\times 8}$ ,

$\boldsymbol{N}_{3}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} -{\boldsymbol{u}_{22}}^{T}\boldsymbol{R} & {\boldsymbol{u}_{22}}^{T} & \mathit{g}_{2}{\boldsymbol{u}_{22}}^{T}d_{e}\boldsymbol{w}_{2} & \partial \boldsymbol{L}_{22} \end{array}\right]_{1\times 12}, \ \boldsymbol{N}_{4}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1 & -{\boldsymbol{w}_{3}}^{T}\boldsymbol{R} & {\boldsymbol{w}_{3}}^{T} & L_{33}\cos \theta _{33} \end{array}\right]_{1\times 8}$ ,

$\boldsymbol{N}_{5}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} -{\boldsymbol{u}_{33}}^{T}\boldsymbol{R} & {\boldsymbol{u}_{33}}^{T} & \mathit{g}_{3}{\boldsymbol{u}_{33}}^{T}d_{e}\boldsymbol{w}_{3} & \partial \boldsymbol{L}_{33} \end{array}\right]_{1\times 12}$ ,

$d\boldsymbol{W}_{1}={[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} d\mathit{g}_{1} & d\boldsymbol{c}_{1} & d\boldsymbol{b}_{1} & dl_{1} & d\varphi _{1} & d\mathit{g}_{2} & d\boldsymbol{c}_{2} & d\boldsymbol{b}_{2} & d\varphi _{2} & d\boldsymbol{l}_{22} & d\mathit{g}_{3} & d\boldsymbol{c}_{3} & d\boldsymbol{b}_{3} & d\varphi _{3} & d\boldsymbol{l}_{33} \end{array}]^{T}}_{35\times 1}$ .

For the convenience of identification and compensation, Eq. (14) should be expressed in standard form; that is, $d\boldsymbol{X}$ should be derived by using the inverse of $\boldsymbol{J}_{1}$ . Notably, the rank of $\boldsymbol{J}_{1}$ is five, which is less than the dimension of $d\boldsymbol{X}$ , that is, six. Therefore, to obtain a unique solution, an additional constraint equation is needed. Here, the parasitic motion of the 5-DOF parallel mechanism will be analyzed and utilized. From Figure 5, the PUU limb has a geometric constraint; that is, $\boldsymbol{O}_{H}\boldsymbol{B}_{1}, \boldsymbol{B}_{1}\boldsymbol{C}_{1}$ , and $\boldsymbol{O}_{2}\boldsymbol{C}_{1}$ are always coplanar, in other words $(\boldsymbol{O}_{2}\boldsymbol{C}_{1}\times \boldsymbol{B}_{1}\boldsymbol{C}_{1})\cdot \boldsymbol{O}_{H}\boldsymbol{B}_{1}=0$ , which can be formulated as

Figure 5. Geometric constraints on PUU limb of the mechanism.

(15) \begin{equation} -\frac{2}{3}g_{1}\left[\mathit{x}\sin \alpha \cos \gamma +\left(z-h\right)\cos \beta \sin \gamma +\mathit{x}\mathit{\cos }\mathit{\alpha }\sin \beta \sin \gamma \right]=0 \end{equation}

From Eq. (15), there is a mapping relationship between γ and other parameters. In other words, this 5-DOF parallel mechanism has a parasitic motion, which can be described as a small constrained motion in the remaining direction of motion in addition to the motion in the specified direction [Reference Sun and Huo33]. By using Eq. (15), γ can be obtained as

(16) \begin{equation} \gamma =\arctan \left(\frac{-x\sin \alpha }{\left(z-h\right)\cos \beta +\mathit{x}\sin \beta \cos \alpha }\right) \end{equation}

Therefore, the independent parameters of frame $\{O_{2}\}$ can be set as $\boldsymbol{X}'=[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} x & y & z & \alpha & \beta \end{array}]^{T}$ , and then the differential relation between $\gamma$ and parameters $\boldsymbol{X}'$ can be formulated as

(17) \begin{equation} d\gamma =\Delta _{\gamma }d\boldsymbol{X}' \end{equation}

where $\Delta _{\gamma }=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \dfrac{\partial \gamma }{\partial x} & \dfrac{\partial \gamma }{\partial y} & \dfrac{\partial \gamma }{\partial z} & \dfrac{\partial \gamma }{\partial \alpha } & \dfrac{\partial \gamma }{\partial \beta } \end{array}\right], \ d\boldsymbol{X}'=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} dx & dy & dz & d\alpha & d\beta \end{array}\right]^{T}$ .

The relationship between $d\boldsymbol{X}$ and $d\boldsymbol{X}'$ can be expressed as

(18) \begin{equation} d\boldsymbol{X}=\boldsymbol{J}_{p}d\boldsymbol{X}' \end{equation}

where $\boldsymbol{J}_{p}=\left[\begin{array}{l} \boldsymbol{E}_{5\times 5}\\ \Delta _{\gamma } \end{array}\right]_{6\times 5}, \ \boldsymbol{E}_{5\times 5}$ is the identity matrix. By combining Eqs. (14) and (18), the following equation can be derived:

(19) \begin{equation} \boldsymbol{J}_{1}\boldsymbol{J}_{p}d\boldsymbol{X}'=\boldsymbol{J}_{2}d\boldsymbol{W}_{1} \end{equation}

Thus, the error model of this parallel mechanism is formulated as

(20) \begin{equation} d\boldsymbol{X}=\boldsymbol{J}d\boldsymbol{W}_{1} \end{equation}

where $\boldsymbol{J}\boldsymbol{=}\boldsymbol{J}_{p}(\boldsymbol{J}_{1}\boldsymbol{J}_{p})^{-1}\boldsymbol{J}_{2}, \ \boldsymbol{J}_{1}\boldsymbol{J}_{p}$ is always invertible within the prescribed workspace.

2.3. Base and tool frame error modeling

As shown in Figure6, an external measurement device, that is, laser tracker, is applied in this paper to measure the moving platform pose. A measurement tool needs to be added to the moving platform for this measurement strategy. The machining and assembly errors of the measurement tool will also affect the calibration effect. In addition, the measured pose needs to be converted to the base frame $\{O_{1}\}$ from the measurement frame $\{O_{m}\}$ using a transformation matrix, that is, ${}^{O1}{\boldsymbol{T}}{_{Om}^{}}$ , which is generally unknown and may introduce errors. Therefore, it is necessary to further extend the kinematic error model, that is, Eq. (20), by considering the tool frame and base frame errors.

Figure 6. Measurement of pose error of the moving platform.

First, for tool frame error modeling, a tool frame is built as shown in Figure 6, of which the orientation is the same as frame $\{O_{2}\}$ . Then position vector of $O_{3}$ in frame $\{O_{1}\}$ can be expressed as

(21) \begin{equation}{\boldsymbol{p}}_{O3}={\boldsymbol{p}}+{\boldsymbol{Rt}}\end{equation}

where t is the position vector of tool frame $\{O_{3}\}$ with respect to frame $\{O_{2}\}$ and $\boldsymbol{R}$ is the rotation matrix of frame $\{O_{2}\}$ with respect to frame $\{O_{1}\}$ . Take differentiation on both sides of Eq. (21), resulting in

(22) \begin{equation} d\boldsymbol{p}_{O3}=d\boldsymbol{p}+d\boldsymbol{Rt}+\boldsymbol{R}d\boldsymbol{t} \end{equation}

where $d\boldsymbol{t}$ is the position error vector of frame $\{O_{3}\}$ described in frame $\{O_{2}\}$ .

According to Eq. (20) and the matrix differential theory, $d\boldsymbol{Rt}=-[(\boldsymbol{Rt})\times ]\boldsymbol{J}_{{\unicode[Arial]{x03A9}} }d\boldsymbol{\Omega }$ . Then, by substituting $d\boldsymbol{Rt}$ to Eq. (22), we can obtain

(23) \begin{equation} d\boldsymbol{p}_{O3}=\boldsymbol{J}_{pO3}d\boldsymbol{W}_{2} \end{equation}

where $\boldsymbol{J}_{pO3}=\left[\begin{array}{c@{\quad}c} \boldsymbol{J}_{a}-[(\boldsymbol{Rt})\times ]\boldsymbol{J}_{{\unicode[Arial]{x03A9}} }\boldsymbol{J}_{b} & \boldsymbol{R} \end{array}\right], \ \boldsymbol{J}_{a}$ represents the first three rows of $\boldsymbol{J}$ , while $\boldsymbol{J}_{b}$ is the last three rows. And, $[(\boldsymbol{Rt})\times ]$ is the skew-symmetric matrix of vector $\boldsymbol{Rt}$ . $d\boldsymbol{W}_{2}=[\begin{array}{c@{\quad}c} d\boldsymbol{W}_{1}^{T} & d\boldsymbol{t}^{T} \end{array}]^{T}$ .

As shown in Figure6, the measured pose of the tool frame is described in the measurement frame $\{O_{m}\}$ . For the sake of clarity, the transformation matrix between frame $\{O_{1}\}$ and frame $\{O_{m}\}$ is denoted as ${}^{O1}{\boldsymbol{T}}{_{Om}^{}}$ , which can be expressed specifically as

(24) \begin{equation} {}^{O1}{\boldsymbol{T}}{_{Om}^{}}=\left[\begin{array}{c@{\quad}c} {}^{O1}{\boldsymbol{R}}{_{Om}^{}} & {}^{O1}{\boldsymbol{p}}{_{Om}^{}}\\[5pt] \boldsymbol{0}_{1\times 3} & 1 \end{array}\right] \end{equation}

where ${}^{O1}{\boldsymbol{R}}{_{Om}^{}}$ and ${}^{O1}{\boldsymbol{p}}{_{Om}^{}}$ denote, respectively, the rotation matrix and position vector of frame $\{O_{m}\}$ with respect to frame $\{O_{1}\}$ .

To construct the kinematic error model, kinematic errors of base frame $\{O_{1}\}$ are introduced as $[d\boldsymbol{p}_{0},d\boldsymbol{\Omega }_{0}]^{T}=[dx_{0}\ dy_{0}\ dz_{0}\ d\alpha _{0}\ d\beta _{0}\ d\gamma _{0}]^{T}$ , where $d\boldsymbol{p}_{0}$ and $d\boldsymbol{\Omega }_{0}$ are, respectively, the position error vector and orientation error vector. For the convenience of analysis [Reference Ma, Bazzoli, Sammons, Landers and Bristow34], we define frame $\{O_{1}'\}$ to denote the actual base frame and define frame $\{O_{3}'\}$ to denote the actual tool frame, which are different from the nominal frames, that is, frame $\{O_{1}\}$ and $\{O_{3}\}$ . The defined frames are shown in Figure7.

Figure 7. Schematic diagram of defined frames $\{O_{1}'\}$ and $\{O_{3}'\}$ .

Then, considering kinematic errors of the base frame, in different frames, we can obtain

(25) \begin{equation} {}^{O1^{\prime}}{\boldsymbol{p}}{_{O3^{\prime}}^{}}={}^{O1^{\prime}}{\boldsymbol{R}}{_{Om}^{}}{}^{Om}{\boldsymbol{p}}{_{O3^{\prime}}^{}}+{}^{O1^{\prime}}{\boldsymbol{p}}{_{Om}^{}} \end{equation}

(26) \begin{equation} {}^{O1}{\boldsymbol{p}}{_{O3^{\prime}}^{}}=\left[d\boldsymbol{\Omega }_{0}\times \right]{}^{O1^{\prime}}{\boldsymbol{p}}{_{O3^{\prime}}^{}}+{}^{O1^{\prime}}{\boldsymbol{p}}{_{O3^{\prime}}^{}}+d\boldsymbol{p}_{0} \end{equation}

where ${}^{O1^{\prime}}{\boldsymbol{p}}{_{O3^{\prime}}^{}}$ denotes the position vector of point $O_{3}'$ described in the actual base frame $\{O_{1}'\}$ , while ${}^{O1}{\boldsymbol{p}}{_{O3^{\prime}}^{}}$ is the position vector of point $O_{3}'$ with respect to the nominal base frame $\{O_{1}\}$ . ${}^{O1^{\prime}}{\boldsymbol{R}}{_{Om}^{}}$ is the orientation matrix of frame $\{O_{m}\}$ with respect to frame $\{O_{1}'\}$ .

According to Eqs. (25) and (26), $d{\boldsymbol{p}}_{O3}$ and $d{\boldsymbol{p}}_{mO3}$ can be expressed as

(27) \begin{equation} d\boldsymbol{p}_{O3}={}^{O1}{\boldsymbol{p}}{_{O3^{\prime}}^{}}-{}^{O1}{\boldsymbol{p}}{_{O3}^{}}=d\boldsymbol{p}_{0}-\left[{}^{O1^{\prime}}{\boldsymbol{p}}{_{O3^{\prime}}^{}}\times \right]d\boldsymbol{\Omega }_{0}+{}^{O1^{\prime}}{\boldsymbol{p}}{_{O3^{\prime}}^{}}-{}^{O1}{\boldsymbol{p}}{_{O3}^{}} \end{equation}

(28) \begin{equation} d\boldsymbol{p}_{mO3}={}^{O1^{\prime}}{\boldsymbol{p}}{_{O3^{\prime}}^{}}-{}^{O1}{\boldsymbol{p}}{_{O3}^{}}=d\boldsymbol{p}_{O3}-d\boldsymbol{p}_{0}+\left[{}^{O1^{\prime}}{\boldsymbol{p}}{_{O3^{\prime}}^{}}\times \right]d\boldsymbol{\Omega }_{0} \end{equation}

where $d\boldsymbol{p}_{mO3}, d\boldsymbol{p}_{O3}$ are position error vectors of the tool frame, expressed in frames $\{O_{1}'\}$ and $\{O_{1}\}$ , respectively.

The orientation error vector of the tool frame is

(29) \begin{equation} d\boldsymbol{\Omega }_{mO3}=d\boldsymbol{\Omega }-d\boldsymbol{\Omega }_{0}\boldsymbol{=}\boldsymbol{J}_{b}d\boldsymbol{W}_{2}-d\boldsymbol{\Omega }_{0} \end{equation}

By combining Eqs. (23), (28), and (29), the kinematic error model of the parallel mechanism considering base and tool frame errors can be obtained as

(30) \begin{equation} d\boldsymbol{X}_{mO3}=\boldsymbol{J}_{mO3}d\boldsymbol{W} \end{equation}

where $d\boldsymbol{X}_{mO3}=\left[\begin{array}{l} d\boldsymbol{p}_{mO3}\\ d\boldsymbol{\Omega }_{mO3} \end{array}\right], \ \boldsymbol{J}_{mO3}=\left[\begin{array}{c@{\quad}c@{\quad}c} \boldsymbol{J}_{pO3} & -\boldsymbol{E}_{3\times 3} & \left[{}^{O1^{\prime}}{\boldsymbol{p}}{_{O3^{\prime}}^{}}\times \right]\\ \boldsymbol{J}_{b} & \boldsymbol{0}_{3\times 6} & -\boldsymbol{E}_{3\times 3} \end{array}\right], \ d\boldsymbol{W}=[\begin{array}{c@{\quad}c@{\quad}c} d\boldsymbol{W}_{2} & d\boldsymbol{p}_{0} & d\boldsymbol{\Omega }_{0} \end{array}]^{T}$ .

2.4. Simulation verification of the kinematic error model

To verify the above kinematic error model, a virtual prototype with geometric errors is established in SolidWorks software, and the moving platform pose error by simulation would be compared with that calculated by the MATLAB program [Reference Sun, Zhai, Song and Zhang35]. It is notable that the numerical error inherent in SolidWorks computational solvers is set as 10⁻⁸ mm.

Without loss of generality, 20 typical configurations are selected to do this verification. The positions are selected as shown in Figure8. For each position, three typical orientations of the moving platform would be simulated, namely, Pose 1 ( $\alpha =0^{\circ }, \beta =0^{\circ }$ ), Pose 2 ( $\alpha =5^{\circ }, \beta =-5^{\circ }$ ), and Pose 3 ( $\alpha =-5^{\circ }, \beta =5^{\circ }$ ). The process is flowcharted in Figure9. It is worth pointing out that the error values of vector $d\boldsymbol{W}$ are all in the range of $[\begin{array}{c@{\quad}c} {-}0.2 & 0.2 \end{array}](\textrm{mm})$ or $[\begin{array}{c@{\quad}c} {-}0.02 & 0.02 \end{array}](^{\circ})$ .

Figure 8. Selected sample positions in the prescribed workspace.

Figure 9. Flow chart of kinematic error model validation.

The results are shown in Figs.10, 11, and 12. It is worth pointing out that the D-value represents the deviation between the simulated value and the calculated value. The position error and orientation error are expressed by the module of the position error vector and the module of the attitude error vector, that is, $| d\boldsymbol{p}| =\sqrt{dx^{2}+dy^{2}+dz^{2}}, | d\boldsymbol{\Omega }| =\sqrt{d\alpha ^{2}+d\beta ^{2}+d\gamma ^{2}}$ .

Figure 10. Verification results of Pose 1 $\alpha =0^{\circ }, \beta =0^{\circ }$ : (a) position error and (b) orientation error.

Figure 11. Verification results of Pose 2 $\alpha =5^{\circ }, \beta =-5^{\circ }$ : (a) position error and (b) orientation error.

Figure 12. Verification results of Pose 3 $\alpha =-5^{\circ }, \beta =5^{\circ }$ : (a) position error and (b) orientation error.

The calculated pose errors match with the simulation pose errors well; that is, the maximum deviations are around 0.002 mm and 0.002°, respectively. The maximum position error deviations of Pose 1, Pose 2, and Pose 3 are 0.19%, 0.25%, and 0.26%, respectively. The maximum orientation error deviations of Pose 1, Pose 2, and Pose 3 are 0.24%, 0.52%, and 0.55%, respectively. Considering that the nonlinear terms are neglected in kinematic error modeling, the results prove that the kinematic error model has enough accuracy.

3.1. Parameter identification method

To identify the kinematic errors of the mechanism, n sets of measurement configurations are selected in the workspace. From Eq. (30), 6 × n scalar equations can be obtained as

(31) \begin{equation} d\boldsymbol{Q}_{n}=\boldsymbol{H}_{n}d\boldsymbol{W} \end{equation}

where $d\boldsymbol{Q}_{n}=\left[\begin{array}{l} d{\boldsymbol{X}_{mO3}}^{(1)}\\ d{\boldsymbol{X}_{mO3}}^{(2)}\\ \ldots \\ d{\boldsymbol{X}_{mO3}}^{(n)} \end{array}\right], \ \boldsymbol{H}_{n}=\left[\begin{array}{l} {\boldsymbol{J}_{mO3}}^{(1)}\\ {\boldsymbol{J}_{mO3}}^{(2)}\\ \ldots \\ {\boldsymbol{J}_{mO3}}^{(n)} \end{array}\right], \ d{\boldsymbol{X}_{mO3}}^{(i)}$ represents the error vector of $i-th$ measurement configuration and ${\boldsymbol{J}_{mO3}}^{(i)}$ is the corresponding identification Jacobian matrix $(i=1,2,3\ldots n)$ .

When $6\times n\gt k$ , where k is the dimension of $d\boldsymbol{W}$ , by applying the least-square technique, the kinematic error vector $d\boldsymbol{W}$ can be calculated as

(32) \begin{equation} d\boldsymbol{W}=\boldsymbol{H}^+_n d\boldsymbol{Q}_n = (\boldsymbol{H}_n^T \boldsymbol{H}_n)^{-1} \boldsymbol{H}_n^T d\boldsymbol{Q}_n \end{equation}

where $\boldsymbol{H}_{\mathit{n}}^{+}=(\boldsymbol{H}_{n}^{T}\boldsymbol{H}_{\mathit{n}})^{-1}\boldsymbol{H}_{n}^{T}$ is the pseudo-inverse of $\boldsymbol{H}_{\mathit{n}}$ . For the 5-DOF double-driven parallel mechanism under study, the dimension of $d\boldsymbol{W}$ , that is, the number of kinematic errors to be identified, is 44. Therefore, at least n = 8 groups of measurement configurations are required. In addition, due to the multi-closed loop structure of the parallel mechanism, the kinematic error model is more complicated. Some kinematic errors may not affect the accuracy of the mechanism, and some ones may have a multilinear correlation with other errors. In other words, some kinematic parameter errors are redundant. Therefore, to derive the minimal model that is more suitable for practical industrial application, the parameter identifiability analysis should be completed first.

3.2. Identifiability analysis

This section focused on parameter identifiability analysis to ensure the stability and accuracy of parameter identification. And the QR decomposition technique is applied. According to QR decomposition, the identification Jacobian $\boldsymbol{H}_{n}$ can be rewritten as

(33) \begin{equation} \left(\boldsymbol{H}_{n}\right)_{m\times k}=\boldsymbol{Q}_{m\times m}\left[\begin{array}{c} \boldsymbol{R}_{k\times k}\\ \boldsymbol{0}_{\left(m-k\right)\times k} \end{array}\right] \end{equation}

where $\boldsymbol{Q}_{m\times m}$ is am × m orthogonal matrix, $\boldsymbol{R}_{k\times k}$ is an upper triangular matrix, and $\boldsymbol{0}_{(m-k)\times k}$ is a null matrix. m = 6 × n .n is the total number of measurement configurations. k is the number of parameter errors to be identified. According to the properties of the orthogonal matrix and upper triangular matrix, the parameters corresponding to the zero diagonal elements of matrix $\boldsymbol{R}_{k\times k}$ can be considered as unidentifiable. Among them, the parameters corresponding to the all-zero columns of the matrix $\boldsymbol{R}_{k\times k}$ are independent of the pose error. The parameters corresponding to the columns with zero diagonal elements and nonzero other elements are linearly related to the previous identifiable parameters. Moreover, for several linearly related parameters to be identified, the top priority is determined as identifiable parameters. Therefore, changing the order of the parameters to be identified can result in different final identifiable parameter vector [Reference Wang, Sun, Zhang, Wu, Zhao, Zhang and Meng36].

According to the QR decomposition method, a sufficient number of measurement configurations, that is, $6n\gt k$ , are randomly determined. By analyzing the identification Jacobian matrix, the rank of the identification matrix is $r=35$ , and the total number of parameters to be identified is $k=44$ , so there are nine redundant parameters. Among them, there are no all-zero columns in $\boldsymbol{R}_{k\times k}$ , and the number of columns whose diagonal is zero while other elements are nonzero are the 6th, 15th, 20th, 21st, 28th, 33rd, 34th, 37th, and 40th columns, respectively, indicating that the parameters corresponding to these columns are linearly correlated with the remaining parameters.

After reasonably adjusting the sequence of parameters to be identified, the 6th, 15th, 18th, 19th, 28th, 31st, 32nd, 37th, and 40th parameters in $d\boldsymbol{W}$ are treated as redundant parameters, namely, $db_{1y}, \ db_{2y}, \ dl_{2}, \ dl_{3}, \ db_{3y}, \ dl_{6}, \ dl_{7}, \ dt_{y}$ , and $dy_{0}$ . Therefore, the final parameter error vector to be identified is

(34) \begin{equation} \begin{array}{c} d\boldsymbol{W}'=\left[dg_{1}\ dc_{1x}\ dc_{1y}\ dc_{1z}\ db_{1x}\ db_{1z}\ dl_{1}\ d\varphi_{1}\ dg_{2}\ dc_{2x}\ dc_{2y}\ dc_{2z}\ db_{2x}\ db_{2z}\ d\varphi_{2}\ dl_{4}\ dl_{5}\ dg_{4}\ dg_{3}\right.\\ \left.dc_{3x}\ dc_{3y}\ dc_{3z}\ db_{3x}\ db_{3z}\ d\varphi _{3}\ dl_{8}\ dl_{9}\ dg_{5}\ dt_{x}\ dt_{z}\ dx_{0}\ dz_{0}\ d\alpha _{0}\ d\beta _{0}\ d\gamma _{0}\right]^{T} \end{array} \end{equation}

Then, by removing corresponding columns, the Jacobian matrix $\boldsymbol{H}_{n}$ are represented as $\boldsymbol{H}_{n}'$ . Eq. (31) and Eq. (32) can be rewritten as

(35) \begin{equation} d\boldsymbol{Q}_{n}=\boldsymbol{H}_{n}'d\boldsymbol{W}' \end{equation}

(36) \begin{equation} d\boldsymbol{W}'=\left(\left(\boldsymbol{H}_{n}'\right)^{T}\boldsymbol{H}_{n}'\right)^{-1}\left(\boldsymbol{H}_{n}'\right)^{T}d\boldsymbol{Q}_{n} \end{equation}

5.1. Numerical simulation based on virtual prototype

With a concept similar to simulation verification of the kinematic error model, numerical simulations are implemented to verify the effectiveness of the proposed kinematic calibration method, especially parameter identification in a virtual environment. Here, the SolidWorks software is utilized to establish the virtual prototype with geometric errors. Figure16 shows a flow chart of the procedure of simulation. For clarity, the detailed procedure of the kinematic calibration simulation is as follows.

Figure 16. Flow chart of the kinematic calibration simulation.

1. 1. Randomly select geometric errors $d\boldsymbol{W}$ , where the errors are all within the range of $[\begin{array}{c@{\quad}c} {-}0.2 & 0.2 \end{array}](\textrm{mm})$ or $[\begin{array}{c@{\quad}c} {-}0.02 & 0.02 \end{array}](^{\circ})$ . Then, the virtual prototype with geometric errors is established in SolidWorks software.

2. 2. The 30 optimal measurement configurations selected in Section 4 are used for simulation, which are also treated as target configurations, that is, $\boldsymbol{Q}_{n}$ . Then, drive the parallel mechanism to measure the pose vector of frame $\{O_{3}\}$ in each measurement configuration, which is treated as the measured configurations, that is, $\boldsymbol{Q}_{n}^{m}$ .

3. 3. By comparing $\boldsymbol{Q}_{n}$ with $\boldsymbol{Q}_{n}^{m}$ , the measured pose error vector of frame $\{O_{3}\}$ is calculated as $d\boldsymbol{Q}_{n}=\boldsymbol{Q}_{n}^{m}-\boldsymbol{Q}_{n}$ , which is also treated as the pose error vector of frame $\{O_{3}\}$ before calibration.

4. 4. By substituting $d\boldsymbol{Q}_{n}$ into Eq. (36), identified kinematic error vector $d\boldsymbol{W}'$ can be obtained.

5. 5. To compensate the kinematic errors, the target configurations are adjusted as $\boldsymbol{Q}_{n}^{a}\boldsymbol{=}\boldsymbol{Q}_{n}-d\boldsymbol{Q}_{n}'$ , where $d\boldsymbol{Q}_{n}'$ is calculated according to Eq. (35), that is, $d\boldsymbol{Q}'_{n}=\boldsymbol{H}_{n}'d\boldsymbol{W}'$ , and $\boldsymbol{Q}_{n}^{a}$ is the adjusted configurations. Then, drive the parallel mechanism to measure the pose vector of frame $\{O_{3}\}$ in each adjusted target configuration, which is treated as the measured configurations after calibration, that is, $\boldsymbol{Q}_{n}^{ma}$ .

6. 6. By comparing $\boldsymbol{Q}_{n}$ and $\boldsymbol{Q}_{n}^{ma}$ , the measured pose error vector of frame $\{O_{3}\}$ after calibration is calculated as $d\boldsymbol{Q}_{n}^{a}$ .

Pose errors of frame $\{O_{3}\}$ before and after calibration are shown in Figure17. From Figure17, the parallel mechanism’s accuracy has been dramatically enhanced after kinematic calibration. The average position error has been reduced from 0.515 mm to 0.004 mm, and the average orientation error has been reduced from 0.249° to 0.001°, which validates the effectiveness of the proposed kinematic calibration method.

Figure 17. Kinematic calibration simulation results: (a) position error and (b) orientation error.

To further analyze the robustness of the proposed kinematic calibration method to measurement errors, noise errors following a normal distribution with $N(0,20^{2})(\mu m)$ or $N(0,0.02^{2})(^{\circ })$ are added to measured configurations, that is, $\boldsymbol{Q}_{n}^{m}$ , in the simulation above, that is, Step (2). Then the corresponding pose errors after calibration are shown in Figure18. After introducing measurement noise, the average position error after kinematic calibration is 0.015 mm, and the average orientation error is 0.019°. Measurement noise can affect the results. However, the average pose errors after calibration are all within limits of acceptability, which also confirms the effectiveness of redundant parameter reduction and measurement configurations optimization in Section 4.

Figure 18. Kinematic calibration simulation results with measurement noise: (a) position error and (b) orientation error.

5.2. Experiment verification

To further verify the effectiveness of the proposed kinematic calibration method in practical application, a calibration experiment is conducted in this section. The prototype is built by our team, and the measurement device is an API OT2 laser tracker with the observed error of 15 μm + 0.5 μm/m. Notably, the full pose of the tool (position and orientation) should be measured in the experiment, while the laser tracker with one sphere reflector generally only offers the position, that is, the center of a sphere reflector. Therefore, a specific measurement tool has been designed, as shown in Figure19 (a).

Figure 19. Measurement tool for kinematic calibration: (a) 3D model and (b) tool frame.

The tool can be bolted to the moving platform of the parallel mechanism, and it contains three reflector holders, the centers of which make an equilateral triangle. For simplicity, a tool frame is defined as shown in Figure19 (b), and the original point of frame $\{O_{3}\}$ is located on the center point of the equilateral triangle.

Considering that the manufacturing errors of measurement tools are relatively small, the nominal transformation matrix from frame $\{O_{2}\}$ to frame $\{O_{3}\}$ , that is, ${}^{O2}{\boldsymbol{T}}{_{O3}^{}}$ , is treated as known. Figure20 shows the experimental site for kinematic calibration. When an external measurement device, that is, laser tracker, is applied, all the measured poses of frame $\{O_{3}\}$ are with respect to the measurement frame $\{O_{m}\}$ . The measured pose of frame $\{O_{3}\}$ with respect to frame $\{O_{m}\}$ is described as

Figure 20. Experimental site and equipment for kinematic calibration experiment.

(38) \begin{equation} {}^{Om}{\boldsymbol{T}}{_{O3}^{}}=\left[\begin{array}{c@{\quad}c} {}^{Om}{\boldsymbol{R}}{_{O3}^{}} & {}^{Om}{\boldsymbol{p}}{_{O3}^{}}\\ \boldsymbol{0}_{1\times 3} & 1 \end{array}\right]=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \boldsymbol{r}_{x} & \boldsymbol{r}_{y} & \boldsymbol{r}_{z} & {}^{Om}{\boldsymbol{p}}{_{O3}^{}}\\ & \boldsymbol{0}_{1\times 3} & & 1 \end{array}\right] \end{equation}

where ${}^{Om}{\boldsymbol{p}}{_{O3}^{}}=({}^{Om}{\boldsymbol{M}}{_{1}^{}}+{}^{Om}{\boldsymbol{M}}{_{2}^{}}+{}^{Om}{\boldsymbol{M}}{_{3}^{}})/3$ denotes the position vector of frame $\{O_{3}\}, {}^{Om}{\boldsymbol{M}}{_{i}^{}}$ is the measured position vector of the center of reflector, and ${}^{Om}{\boldsymbol{R}}{_{O3}^{}}$ represents the orientation matrix of frame $\left\{O_{3}\right\}, \boldsymbol{r}_{y}=\left({}^{Om}{\boldsymbol{M}}{_{1}^{}}-\frac{1}{2}\left({}^{Om}{\boldsymbol{M}}{_{2}^{}}+{}^{Om}{\boldsymbol{M}}{_{3}^{}}\right)\right)/\left\| {}^{Om}{\boldsymbol{M}}{_{1}^{}}-\frac{1}{2}\left({}^{Om}{\boldsymbol{M}}{_{2}^{}}+{}^{Om}{\boldsymbol{M}}{_{3}^{}}\right)\right\|, \boldsymbol{r}_{z}=\boldsymbol{r}_{x}\times \boldsymbol{r}_{y},\ \boldsymbol{r}_{x}=$ $({}^{Om}{\boldsymbol{M}}{_{2}^{}}-{}^{Om}{\boldsymbol{M}}{_{3}^{}})/\| {}^{Om}{\boldsymbol{M}}{_{2}^{}}-{}^{Om}{\boldsymbol{M}}{_{3}^{}}\|$ . All position and orientation vectors are described in frame  $\{O_{m}\}$ .

Because the kinematic error model of this parallel mechanism is established in the base frame $\{O_{1}\}$ , the measured pose needs to be converted to the base frame $\{O_{1}\}$ from the measurement frame $\{O_{m}\}$ . In other words, the transformation matrix from frame $\{O_{1}\}$ to frame $\{O_{m}\}$ , that is, ${}^{O1}{\boldsymbol{T}}{_{Om}^{}}$ , must be derived. In this paper, the SVD method [Reference Li, Jiang, Luo, Yang, Guo and Hu41] is utilized to calculate the transformation matrix ${}^{O1}{\boldsymbol{T}}{_{Om}^{}}$ . Construct the following objective function:

(39) \begin{equation} F=\sum _{i=1}^{n}\left\| \left({}^{O1}{\boldsymbol{R}}{_{Om}^{}} {}^{Om}{\boldsymbol{p}}{_{O3}^{}}^{(i)}+{}^{O1}{\boldsymbol{p}}{_{Om}^{}}\right)-{}^{O1}{\boldsymbol{p}}{_{O3}^{}}^{(i)}\right\| _{2}^{2} \end{equation}

where ${}^{Om}{\boldsymbol{p}}{_{O3}^{}}^{(i)}$ is the i-th measured position vector in the measurement frame $\{O_{m}\}, {}^{O1}{\boldsymbol{p}}{_{O3}^{}}^{(i)}$ is the i-th nominal position vector in the base frame $\{O_{1}\}$ , and n is the number of measurement configurations. Centralize all vectors ${}^{Om}{\boldsymbol{p}}{_{O3}^{}}^{(i)}$ and ${}^{O1}{\boldsymbol{p}}{_{O3}^{}}^{(i)}$ , resulting in

(40) \begin{equation} \left\{\begin{array}{l} {}^{Om}{\boldsymbol{p}}{_{O3}^{}}^{(i)\prime}={}^{Om}{\boldsymbol{p}}{_{O3}^{}}^{(i)}-\overline{{}^{Om}{\boldsymbol{p}}{_{O3}^{}}}\\ {}^{O1}{\boldsymbol{p}}{_{O3}^{}}^{(i)\prime}={}^{O1}{\boldsymbol{p}}{_{O3}^{}}^{(i)}-\overline{{}^{O1}{\boldsymbol{p}}{_{O3}^{}}} \end{array}\right. \end{equation}

where $\overline{{}^{Om}{\boldsymbol{p}}{_{O3}^{}}}=\frac{1}{n}\sum _{i=1}^{n}{}^{Om}{\boldsymbol{p}}{_{O3}^{}}^{(i)}, \overline{{}^{O1}{\boldsymbol{p}}{_{O3}^{}}}=\frac{1}{n}\sum _{i=1}^{n}{}^{O1}{\boldsymbol{p}}{_{O3}^{}}^{(i)}$ , and they satisfy the following relationship:

(41) \begin{equation} {}^{O1}{\boldsymbol{R}}{_{Om}^{}}\overline{{}^{Om}{\boldsymbol{p}}{_{O3}^{}}}+{}^{O1}{\boldsymbol{p}}{_{Om}^{}}=\overline{{}^{O1}{\boldsymbol{p}}{_{O3}^{}}} \end{equation}

Substituting Eqs. (40) and (41) into Eq. (39) yields

(42) \begin{equation} F=\sum _{i=1}^{n}\left(\left({}^{Om}{\boldsymbol{p}}{_{O3}^{}}^{(i)\prime}\right)^{T}{}^{Om}{\boldsymbol{p}}{_{O3}^{}}^{(i)\prime}-2\left({}^{O1}{\boldsymbol{p}}{_{O3}^{}}^{(i)\prime}\right)^{T}{}^{O1}{\boldsymbol{R}}{_{Om}^{}}{}^{Om}{\boldsymbol{p}}{_{O3}^{}}^{(i)\prime}+\left({}^{O1}{\boldsymbol{p}}{_{O3}^{}}^{(i)\prime}\right)^{T}{}^{O1}{\boldsymbol{p}}{_{O3}^{}}^{(i)\prime}\right) \end{equation}

To solve ${}^{O1}{\boldsymbol{R}}{_{Om}^{}}$ and ${}^{O1}{\boldsymbol{p}}{_{Om}^{}}$ , it is necessary to minimize the value of the objective function F. Considering that $({}^{Om}{\boldsymbol{p}}{_{O3}^{}}^{(i)\prime})^{T}{}^{Om}{\boldsymbol{p}}{_{O3}^{}}^{(i)\prime}$ and $({}^{O1}{\boldsymbol{p}}{_{O3}^{}}^{(i)\prime})^{T}{}^{O1}{\boldsymbol{p}}{_{O3}^{}}^{(i)\prime}$ are constant for each measurement, to minimize the value of $F$ is equivalent to maximize function $F'$ , which is expressed as

(43) \begin{equation} F'=\sum _{i=1}^{n}\left(\left({}^{O1}{\boldsymbol{p}}{_{O3}^{}}^{(i)\prime}\right)^{T}{}^{O1}{\boldsymbol{R}}{_{Om}^{}}{}^{Om}{\boldsymbol{p}}{_{O3}^{}}^{(i)\prime}\right)=\textrm{trace}\left({}^{O1}{\boldsymbol{R}}{_{Om}^{}}\boldsymbol{z}\right) \end{equation}

where $\boldsymbol{z}\boldsymbol{=}\sum _{i=1}^{n}({}^{O1}{\boldsymbol{p}}{_{O3}^{}}^{(i)\prime}({}^{Om}{\boldsymbol{p}}{_{O3}^{}}^{(i)\prime})^{T}), \textrm{trace}({}^{O1}{\boldsymbol{R}}{_{Om}^{}}\boldsymbol{z})$ represents the trace of the matrix ${}^{O1}{\boldsymbol{R}}{_{Om}^{}}\boldsymbol{z}$ .

According to SVD, $\boldsymbol{z}$ can be expressed as $\boldsymbol{z}\boldsymbol{=}\boldsymbol{u}{\unicode[Arial]{x03A3}} \boldsymbol{V}^{T}$ . To maximize $F'$ , we can obtain

(44) \begin{equation} {}^{O1}{\boldsymbol{R}}{_{Om}^{}}=\boldsymbol{V}\boldsymbol{u}^{T} \end{equation}

By substituting Eq. (44) into Eq. (41), ${}^{O1}{\boldsymbol{p}}{_{Om}^{}}$ can be calculated. Then ${}^{O1}{\boldsymbol{T}}{_{Om}^{}}$ is obtained according to Eq. (24). It is worth pointing out that the derived transformation matrix ${}^{O1}{\boldsymbol{T}}{_{Om}^{}}$ is not the accurate transformation matrix between the nominal base frame $\{O_{1}\}$ and measurement frame $\{O_{m}\}$ . Therefore, for clarity, the estimated transformation matrix is denoted by ${}^{O1^{\prime}}{\boldsymbol{T}}{_{Om}^{}}$ . According to matrix transformation, the measured poses of frame $\{O_{3}\}$ with respect to frame $\{O_{1}\}$ , that is, ${}^{O1^{\prime}}{\boldsymbol{T}}{_{O3m}^{}}$ , can be obtained as

(45) \begin{equation} {}^{O1^{\prime}}{\boldsymbol{T}}{_{O3m}^{}}={}^{O1^{\prime}}{\boldsymbol{T}}{_{Om}^{}}{}^{Om}{\boldsymbol{T}}{_{O3}^{}} \end{equation}

where ${}^{O1^{\prime}}{\boldsymbol{T}}{_{O3m}^{}}=\left[\begin{array}{c@{\quad}c} {}^{O1^{\prime}}{\boldsymbol{R}}{_{O3m}^{}} & {}^{O1^{\prime}}{\boldsymbol{p}}{_{O3m}^{}}\\ \boldsymbol{0}_{1\times 3} & 1 \end{array}\right], {}^{O1^{\prime}}{\boldsymbol{R}}{_{O3m}^{}}$ and ${}^{O1^{\prime}}{\boldsymbol{p}}{_{O3m}^{}}$ denote the measured rotation matrix and position vector of frame $\{O_{3}\}$ with respect to the estimated base frame $\{O_{1}'\}$ , respectively.

In the experiment, 30 optimal measurement configurations selected in Section 4 are used for parameter identification, that is, identification configurations. To further validate the effectiveness of the kinematic calibration method, 20 additional verification configurations are randomly selected in the prescribed workspace. The parallel mechanism is commanded to undergo the measurement configurations successively, and the measurement at each configuration is repeated three times, and the mean value is retained. The position and orientation errors before and after kinematic calibration are shown in Figure21.

Figure 21. Comparison of errors before and after calibration: (a) position error and (b) orientation error.

From Figure21, the proposed kinematic calibration can improve the absolute accuracy of the 5-DOF double-driven parallel mechanism. For both the identification configurations and verification configurations, the average position error decreases from 2.778 mm to 0.263 mm, and the average orientation error decreases from 1.115° to 0.176°. Meanwhile, we found that gravity has a significant impact on the positioning accuracy of parallel mechanisms in our related research. In future research, we will study the kinematic calibration considering the gravity factor. In addition, although there are flexible deformations and some neglected geometric errors, the accuracy after kinematic calibration is still within an acceptable range. In other words, the proposed kinematic calibration method for the 5-DOF double-driven parallel mechanism with the sub-closed loop on limbs is validated.