2.1 Configuration of heavy load multi-DoF forming machine

In order to realize the multi-Dof forming, a heavy load multi-DoF forming machine with 6-PSS configuration has recently been developed. The overall configuration of the multi-DoF forming machine is shown in Figure 1, and the protoype is displayed in Figure 2. The machine mainly consists of a bed and a 6-PSS PKM system. The PKM consists of a moving platform, fixed base, and six links. Each link connects the slider to the moving platform by two spherical joints. The six sliders connect the fixed base by prismatic joints and are driven by six servo motors. The largest forming force of the machine with PKM can reach 6000 kN.

In order to bear such a heavy forming load, the following consideration is addressed in the configuration of the new PKM: (1) the sliders are set to move on the horizontal plane; (2) the vertical angles of links are set to be small and reduced along the feeding; (3) the high stiffness ball seat structure with sliding contact for S joint is used.

In the multi-DoF forming process, the six servo motors drive the sliders to move along the guider direction and make the links conduct spatial motion. Hence, the moving platform and the attached tool will realize the required multi-DoF motion driven by the six links together. The blank is placed in the fixed mold, which is attached to the bed of machine. With the multi-DoF loading of tool, the blank is forced to undergo incremental plastic deformation and the metal flows multidirectionally to fill the tool cavity until the target shape of complicated component is achieved. Compared to the common single DoF overall forming process, the technical advantages of multi-DoF local forming process include that the forming force is much smaller, the geometry shape of formed component is much more complicated, and the microstructure and mechanical properties are much better.

Besides, the proposed heavy load parallel kinematic machine works with relatively low motion speed (max to 1 r/s), and the dynamic influences are not remarkable, thus the statics of the manipulator can be used to analyze the actuator effort and deflection in this work.

Figure 3. Schematic diagram of inverse kinematics of 6-PSS PKM and layout of spherical joints. (a) Schematic diagram of inverse kinematics of 6-PSS PKM (b) Layout of spherical joints.

2.2 Inverse kinematics of PKM

The kinematics of PKM can be divided into forward kinematics [Reference Bi and Lang33, Reference Gao and Wu34] and inverse kinematics [Reference Liping, Huayang and Liwen35, Reference Zhao, Qiu, Wang and Zhang36]. In this paper, the inverse kinematics of the PKM is discussed with the geometrical constraint of constant link length. As shown in Figure 3a, the global coordinate system $\mathrm{S}_{\mathrm{A}}(\mathrm{O}_{\mathrm{A}}-\mathrm{x}_{\mathrm{A}}\mathrm{y}_{\mathrm{A}}\mathrm{z}_{\mathrm{A}})$ is located at the center of the fixed base with the $\mathrm{z}_{\mathrm{A}}-$ axis normal to the fixed base. The distance between the origin of the moving coordinate system $\mathrm{S}_{\mathrm{B}}(\mathrm{O}_{\mathrm{B}}-\mathrm{x}_{\mathrm{B}}\mathrm{y}_{\mathrm{B}}\mathrm{z}_{\mathrm{B}})$ and the center of the moving platform is $d$ . The origin of $\mathrm{S}_{\mathrm{B}}$ and the origin of $\mathrm{S}_{\mathrm{A}}$ are located on the same vertical line. Point $A_{i}$ ( $i$ =1–6) denotes the central point of the spherical joint on the fixed base in $\mathrm{S}_{\mathrm{A}}$ . Point $B_{i}$ ( $i$ = 1–6) denotes the central point of the spherical joint on the moving platform in $\mathrm{S}_{\mathrm{A}}$ . The layout of spherical joints $A_{i}$ and $B_{i}$ is presented in Figure 3b. $\theta _{Ai}$ is the angle between the position vector of point $A_{i}$ and the x-axis in $\mathrm{S}_{\mathrm{A}}$ . $\theta _{Bi}$ is the angle between the position vector of point $B_{i}$ and the x-axis in $\mathrm{S}_{\mathrm{A}}$ .

In this paper, the motion of the moving platform can be represented by 6 parameters. Parameters $x, y, z$ represent the positional of the tool in three directions, and parameters $\alpha, \beta, \gamma$ represent the rotation angle of the tool in three directions. In multi-DoF forming process, they are expressed as follows:

(1) \begin{align} \left\{\begin{array}{l} \alpha =\varphi \cos \left(\omega t\right)\\[3pt] \begin{array}{l} \beta =\varphi \sin \left(\omega t\right)\\[3pt] \gamma =0\\[3pt] x=0\\[3pt] y=0\\[3pt] z=kt=h \end{array} \end{array}\right. \end{align}

where $\varphi$ is the inclination angle of tool, $\omega$ is the rotational speed of tool, $k$ is feed rate, and $h$ is feed distance.

The rotation matrix of $\mathrm{S}_{\mathrm{B}}$ relative to $\mathrm{S}_{\mathrm{A}}$ can be obtained:

(2) \begin{align} _{B}^{A}\hspace{0pt}\boldsymbol{R}=\left[\begin{array}{c@{\quad}c@{\quad}c} \cos \beta \cos \gamma & \sin \alpha\sin \beta \cos \gamma -\cos \alpha \sin \gamma & \sin \alpha \sin \gamma + \cos \alpha\sin \beta \cos \gamma \\[3pt] \cos \beta \sin \gamma & \sin \alpha\sin \beta \sin \gamma +\cos \alpha \cos \gamma & \cos \alpha\sin \beta \sin \gamma -\sin \alpha \cos \gamma \\[3pt] -\sin \beta & \sin \alpha \cos \beta & \cos \alpha \cos \beta \end{array}\right] \end{align}

Besides, $\boldsymbol{p}_{OB}$ denotes the position vector of point O_B in $\mathrm{S}_{\mathrm{A}}$ , which is calculated as follows:

(3) \begin{align} \boldsymbol{p}_{OB}=\left[x,y,z+Z_{P}\right]^{T} \end{align}

where $Z_{P}$ is the height of the end of PKM relative to the fixed base.

$\boldsymbol{b}_{i}^{\prime}$ is set as the position vector of point $B_{i}$ in $\mathrm{S}_{\mathrm{B}}$ , which is calculated as follows:

(4) \begin{align} \boldsymbol{b}_{i}^{\prime}=\left[r_{B}\cos \theta _{Bi},r_{B}\sin \theta _{Bi},d\right]^{T} \end{align}

where $r_{B}$ is radius of moving platform, $\theta _{Bi}$ is the polar angle for point, and $B_{i}, d$ is the distance between end of PKM and center of moving platform.

Hence, the position vector of point $B_{i}$ in $\mathrm{S}_{\mathrm{A}}$ can be calculated as follows:

(5) \begin{align} \boldsymbol{b}_{i}=_{B}^{A}\hspace{0pt}\boldsymbol{R}\boldsymbol{b}_{i}^{\prime}+\boldsymbol{p}_{OB} \end{align}

According to the position relationship between the moving platform, fixed base, and 6 links,

the position vector of point $A_{i}$ can be calculated as follows:

(6) \begin{align} \boldsymbol{a}_{i}\left(a_{ix}\right)=\left[a_{ix},a_{ix}\tan \theta _{Ai},0\right]^{\mathrm{T}} \end{align}

where $a_{ix}$ is the projection of the position vector $\boldsymbol{a}_{i}$ onto the $\mathrm{x}_{\mathrm{A}}-$ axis and $\theta _{Ai}$ is the polar angle for point $A_{i}$ .

Based on the geometric constraints of the fixed length of the link, the following equation can be obtained:

(7) \begin{align} \left| \boldsymbol{a}_{i}\left(a_{ix}\right)-\boldsymbol{b}_{i}\right| =\left| \boldsymbol{l}_{i}\right| =l \end{align}

where $l$ is the length of the link.

Based on Eqs. (5)-(7), the following equation can be obtained:

(8) \begin{align} a_{ix}=\frac{b_{ix}+\sqrt{l^{2}\tan {\theta _{Ai}}^{2}+l^{2}-\tan {\theta _{Ai}}^{2}{b_{ix}}^{2}-\tan {\theta _{Ai}}^{2}{b_{iz}}^{2}+2\tan \theta _{Ai}b_{ix}b_{iy}-{b_{iy}}^{2}-{b_{iz}}^{2}}+\tan \theta _{Ai}b_{iy}}{\tan {\theta _{Ai}}^{2}+1} \end{align}

where

\begin{align*} \begin{array}{l} b_{ix}=d\left(\sin\! \alpha \sin\! \gamma +\cos\! \alpha \cos\! \gamma \sin\! \beta \right)-r_{B}\sin\! \theta _{Bi}\left(\cos\! \alpha \sin\! \gamma -\cos\! \gamma \sin\! \alpha \sin\! \beta \right)+r_{B}\cos\! \beta \cos\! \gamma \cos\! \theta _{Bi}\\[3pt] b_{iy}=r_{B}\sin\! \theta _{Bi}\left(\cos\! \alpha \cos\! \gamma +\sin\! \alpha \sin\! \beta \sin\! \gamma \right)-d\left(\cos\! \gamma \sin\! \alpha -\cos\! \alpha \sin\! \beta \sin\! \gamma \right)+r_{B}\cos\! \beta \sin\! \gamma \cos\! \theta _{Bi}\\[3pt] b_{iz}=h+Z_{P}-r_{B}\sin\! \beta \cos\! \theta _{Bi}+\cos\! \alpha \cos\! \beta d+r_{B}\cos\! \beta \sin\! \alpha \sin\! \theta _{Bi} \end{array} \end{align*}

Based on Eq. (8), the displacement of the slider can be expressed by Eq. (9).

(9) \begin{align} s_{i}=\sqrt{{a_{ix}}^{2}+\left(a_{ix}\tan \theta _{Ai}\right)^{2}}-r_{A} \end{align}

where $r_{A}$ is the distance between the initial position of the slider and the origin of $\mathrm{S}_{\mathrm{A}}$ .

According to the velocity projection theorem, the projections of the velocity of any two points on the same rigid body onto the line between these two points are equal [Reference Antonov and Glazunov37]. So the following equation can be obtained:

(10) \begin{align} \boldsymbol{v}_{Ai}\cdot \overline{\boldsymbol{l}}_{i}=\boldsymbol{v}_{Bi}\cdot \overline{\boldsymbol{l}}_{i} \end{align}

where $\boldsymbol{v}_{Ai}$ is the velocity of the spherical joint on the fixed base, $\boldsymbol{v}_{Bi}$ is the velocity of the spherical joint on the moving platform, and $\boldsymbol{l}_{i}$ is the unit vector along the link, $\overline{\boldsymbol{l}}_{i}=\boldsymbol{l}_{i}/l$ .

The velocity of the spherical joint on the moving platform is equal to the velocity of the moving platform at this point, so $\boldsymbol{v}_{Bi}$ can be expressed by:

(11) \begin{align} \boldsymbol{v}_{Bi}=\frac{\mathrm{d}\left[x,y,z\right]^{T}}{\mathrm{d}t}+\frac{\mathrm{d}\left[\alpha, \beta, \gamma \right]^{T}}{\mathrm{d}t}\times \boldsymbol{r}_{i} \end{align}

where $\boldsymbol{r}_{i}$ denotes the vector $\overline{\boldsymbol{O}_{B}\boldsymbol{B}_{i}}$ in $\mathrm{S}_{\mathrm{A}}$ .

Substitute Eq. (11) into Eq. (10), the following equation can be obtained:

(12) \begin{align} v_{i}\left[\cos \theta _{Ai},\sin \theta _{Ai},0\right]^{T}\cdot \overline{\boldsymbol{l}}_{i}=\overline{\boldsymbol{l}}_{i}\cdot \frac{\mathrm{d}\left[x,y,z\right]^{T}}{\mathrm{d}t}+\left(\boldsymbol{r}_{i}\times \overline{\boldsymbol{l}}_{i}\right)\cdot \frac{\mathrm{d}\left[\alpha, \beta, \gamma \right]^{T}}{\mathrm{d}t} \end{align}

where $v_{i}$ is the magnitude of velocity of the slider.

Eq. (12) can be expressed by:

(13) \begin{align} \boldsymbol{J}_{s}\left[v_{1},\cdots, v_{6}\right]^{T}=\boldsymbol{J}_{p}\left[\boldsymbol{v}_{p},\boldsymbol{\omega }_{p}\right]^{T} \end{align}

where $\boldsymbol{J}_{s}=\text{diag}([[\cos \theta _{A1},\sin \theta _{A1},0]^{T}\cdot \overline{\boldsymbol{l}}_{1},\cdots, [\cos \theta _{A6},\sin \theta _{A6},0]^{T}\cdot \overline{\boldsymbol{l}}_{6}]), \boldsymbol{J}_{p} = \left[\begin{array}{c@{\quad}c} {\overline{\boldsymbol{l}}_{1}}^{T} & (\boldsymbol{r}_{1}\times \overline{\boldsymbol{l}}_{1})^{T}\\[3pt] \vdots & \vdots \\[3pt] {\overline{\boldsymbol{l}}_{6}}^{T} & (\boldsymbol{r}_{6}\times \overline{\boldsymbol{l}}_{6})^{T} \end{array}\right],$

$\overline{\boldsymbol{l}}_{i} = \left[\begin{array}{c} (b_{ix}-a_{ix})/l\\[3pt] (b_{iy}-a_{ix}\tan \theta _{Ai})/l\\[3pt] b_{iz}/l \end{array}\right],\boldsymbol{r}_{i}\times \overline{\boldsymbol{l}}_{i}=\left[\begin{array}{c} (b_{iy}b_{iz}+(a_{ix}\tan \theta _{Ai}-b_{iy})(b_{iz}-Z_{P}))/l\\[3pt] ({-}b_{ix}b_{iz}-(a_{ix}-b_{ix})(b_{iz}-Z_{P}))/l\\[3pt] (b_{iy}(a_{ix}-b_{ix})-b_{ix}(a_{ix}\tan \theta _{Ai}-b_{iy}))/l \end{array}\right]$ .

Multiply both sides of Eq. (13) by the inverse matrix of $\boldsymbol{J}_{s}$ , the speed of slider can be obtained:

(14) \begin{align} \left[v_{1},\cdots, v_{6}\right]^{T}=\boldsymbol{J}_{v}\left[\frac{\mathrm{d}\left[x,y,z\right]}{\mathrm{d}t},\frac{\mathrm{d}\left[\alpha, \beta, \gamma \right]}{\mathrm{d}t}\right]^{T} \end{align}

where $\boldsymbol{J}_{v} = \left[\begin{array}{l@{\quad}l} {\overline{\boldsymbol{l}}_{1}}^{T}/([\cos \theta _{A1},\sin \theta _{A1},0]^{T}\cdot \overline{\boldsymbol{l}}_{1}) & (\boldsymbol{r}_{1}\times \overline{\boldsymbol{l}}_{1})^{T}/([\cos \theta _{A1},\sin \theta _{A1},0]^{T}\cdot \overline{\boldsymbol{l}}_{1})\\[3pt] \vdots & \vdots \\[3pt] {\overline{\boldsymbol{l}}_{6}}^{T}/([\cos \theta _{A6},\sin \theta _{A6},0]^{T}\cdot \overline{\boldsymbol{l}}_{6}) & (\boldsymbol{r}_{6}\times \overline{\boldsymbol{l}}_{6})^{T}/([\cos \theta _{A6},\sin \theta _{A6},0]^{T}\cdot \overline{\boldsymbol{l}}_{6}) \end{array} \right]$ .

Table I. The configuration parameters of multi-DoF forming machine.

The configuration parameters of multi-DoF forming machine are shown in Table I. The feed rate $k=3$ mm/s, the process time $t=0\sim 5\mathrm{s}$ . The configuration change of PKM and multi-DoF forming motion in different process time is shown in Figure 4. The displacement of each slider is obtained based on Eq. (9), as shown in Figure 5a and b. It is found that the displacement curve of each slider is a shape of sin with different phase angles. The wave center increases with the increase of time, and the wave amplitude also increases with the increase of time. The wave center is the average value of the displacement curve of the slider in each movement period of the PKM. The velocity of each slider is also obtained by Eq. (14), as shown in Figure 5c and d. It is found that the velocity curve of each slider is also a shape of sin with different phase angles. The wave center remains unchanged with time increasing, and the wave amplitude increases with time increasing.

Figure 4. Illustration for the motion of multi-DoF forming. (a) t = 0.25 s (b) t = 0.5 s (c) t = 0.75s (d) t = 1 s.

Figure 5. Displacement and velocity of each slider with time. (a) Displacement of each slider with $\varphi$ = 1 (b) Velocity of each slider with $\varphi$ = 2 (d) Velocity curve of each slider with $\varphi$ = 1. (c) Displacement of each slider with $\varphi$ = 2 (d) Velocity curve of each slider with $\varphi$ = 2.

2.3 The geometric workspace

When the PKM is in some special postures during the motion, the multi-DoF forming machine will be out of control or suffer changes in DoF. The configuration of the machine at this time is called singular configuration. Singular configurations represent extreme cases of the postures of PKM, in which the machine does not work properly. In the singular pose, the PKM has the stiffest pose against the loadings along the axis; however the normal force is offset in operation, not applied to the center of the platform which will cause added torque, so the singular pose cannot be used for real work. Therefore, the singular configurations should be excluded when calculating the workspace of PKM. Based on the above analysis of slider velocity, it can be concluded that when the link is perpendicular to the guide rail, the determinant of $\boldsymbol{J}_{s}$ becomes zero, indicating that the PKM is in a singular configuration. At this time, the moving platform does not move, but the slider still has instantaneous motion. To ensure the kinematic stability, the determinant of $\boldsymbol{J}_{s}$ , namely $\det (\boldsymbol{J}_{s})$ should not equal zero. Besides, the factors that affect the workspace of PKM include the displacement of slider and the limitation of the angle of spherical joint. Due to the limitation of the length of the guideway of the PKM, the displacements of the sliders in their respective direction of movement can not exceed the permissible ranges. The maximum and minimum values of slider displacement are represented by $s_{\max }$ and $s_{\min }$ , respectively. The sliders and links as well as links and moving platform are connected by spherical joint, and the angle of the spherical joint is limited. The maximum value of the angle of the spherical joint is represented by $\theta _{\max }$ . Hence, the displacement of the $i$ th slider and the angle of spherical joint should meet the following conditions.

(15) \begin{align} \left\{\begin{array}{l} s_{\min }\leq s_{i}\leq s_{\max }\\[3pt] \theta _{ai}=\arccos \left(\overline{\boldsymbol{l}}_{i}\cdot \left[0,0,1\right]^{T}\right)\leq \theta _{\max }\\[3pt] \theta _{bi}=\arccos \left(\overline{\boldsymbol{l}}_{i}\cdot \boldsymbol{R}\left(\alpha, \beta, \gamma \right)\left[0,0,1\right]^{T}\right)\leq \theta _{\max }\\[3pt] \det \left(J_{s}\right)\neq 0 \end{array}\right. \end{align}

In this paper, the feed distance and inclination angle of the tool are chosen as the key processing parameters to express the workspace of PKM. Based on the constraints in Eq. (15), the range of each parameter can be calculated, namely the workspace of PKM. For the multi-DoF forming machine, the limit position of the slider is $s_{\min }=-50$ and $s_{\max }=150$ , and the maximum range of motion of the spherical joint is $\theta _{\max }=\pi /4$ . According to the motion condition of Eq. (1), the feed distance $h=[h_{\min }^{s},h_{\max }^{s}]$ can be solved with the inclination angle setting as $\varphi =[\varphi _{\min }^{g},\varphi _{\max }^{g}]$ , as shown in the Figure 6.

Figure 6. The geometric workspace of parallel kinematic machine.

According to Figure 5, when the inclination angle of tool increases, the wave amplitude of the slider displacement for each cycle increases. As the feed distance increases, the wave center of the slider displacement for each cycle increases. Therefore, to ensure that the slider does not exceed the travel limit, when the wave amplitude of the slider displacement for each cycle is larger, the wave center will necessarily be smaller, which means that the maximum feed distance of the PKM becomes smaller. Conversely, when the wave amplitude of the slider displacement for each cycle decreases, the maximum feed distance of the PKM increases. So it can be observed from Figure 6 that the maximum feed distance of the PKM gradually decreases as the inclination angle of tool increases. The feed distance can reach its maximum value of 32.4 mm. When the inclination angle of tool is 2°, the feed distance can reach its minimum value of 15.8 mm.

3.1 Stiffness modeling of PKM

The static stiffness of PKM reflects the posture error caused by elastic deformation of components between the fixed base and moving platform under external load. The main methods for static stiffness analysis of PKM are finite element analysis, structure matrix analysis, and so on [Reference Liu, Huang, Chetwynd and Kecskemethy38, Reference Tian, Ma, Xia, Ma and Li19]. The stiffness model considering axial deformation of links is established in this paper.

The forming load applied to the PKM is set as $(\boldsymbol{f},\boldsymbol{m})$ , where $\boldsymbol{f}$ is the forming force and $\boldsymbol{m}$ is the forming torque. The synthesized deformation of the moving platform is set as $(\Delta \boldsymbol{X},\Delta \boldsymbol{\theta })$ , where $\Delta \boldsymbol{X}$ is linear deformation along direction of x, y and z axis and $\Delta \boldsymbol{\theta }$ is angular deformation along direction of x, y and z axis. The relation between synthesized deformation and the forming load meets the following relationship:

(16) \begin{align} \left[\begin{array}{l} \boldsymbol{f}\\[3pt] \boldsymbol{m} \end{array}\right]=\boldsymbol{K}\left[\begin{array}{l} \Delta \boldsymbol{X}\\[3pt] \Delta \boldsymbol{\theta } \end{array}\right] \end{align}

where $\boldsymbol{K}$ is stiffness matrix.

As the multi-DoF forming machine is under heavy load, the links will produce axial deformation. The rigidity of the moving platform and the fixed base is much higher than that of link, which can be regarded as rigid bodies. The deformation of the moving platform hence can be regarded as that it is caused by the link deformation. $f_{i}, \Delta l_{i}$ and $k_{i}$ are set as the axial force, deformation and stiffness of the link. The axial force of link can be presented by:

(17) \begin{align} f_{i}=k_{i}\Delta l_{i}=\frac{EA}{L}\Delta l_{i}\left(i=1\cdots 6\right) \end{align}

where $A$ and $E$ represent the cross-section area of each link and elastic modulus of link material, respectively.

According to the principle of virtual work, the following equation can be obtained:

(18) \begin{align} \left[\boldsymbol{f}^{T},\boldsymbol{m}^{T}\right]\left[\begin{array}{l} \Delta \boldsymbol{X}\\[3pt] \Delta \boldsymbol{\theta } \end{array}\right]=\sum _{i=1}^{6}f_{i}\Delta l_{i}={\boldsymbol{F}_{l}}^{T}\Delta \boldsymbol{l} \end{align}

where $\boldsymbol{F}_{l}=[f_{1},f_{2},f_{3},f_{4},f_{5},f_{6}]^{T}, \Delta \boldsymbol{l}=[\Delta l_{1},\Delta l_{2},\Delta l_{3},\Delta l_{4},\Delta l_{5},\Delta l_{6}]^{T}$ and $\boldsymbol{F}_{l}=\text{diag}([k_{1}\cdots k_{6}])\Delta \boldsymbol{l}$ .

Figure 7. Force diagram of the moving platform.

As shown in Figure 7, the force balance condition of the moving platform can be obtained:

(19) \begin{align} \left[\begin{array}{l} \sum _{i=1}^{6}f_{i}\overline{\boldsymbol{l}}_{i}\\[3pt] \sum _{i=1}^{6}f_{i}\left(\boldsymbol{r}_{i}\times \overline{\boldsymbol{l}}_{i}\right) \end{array}\right]=\left[\begin{array}{l} \boldsymbol{f}\\[3pt] \boldsymbol{m} \end{array}\right]=\boldsymbol{J}_{f}\boldsymbol{F}_{l} \end{align}

where $\boldsymbol{J}_{\boldsymbol{f}} = \left[\begin{array}{l@{\quad}l@{\quad}l} \overline{\boldsymbol{l}}_{1} & \cdots & \overline{\boldsymbol{l}}_{6}\\[3pt] \boldsymbol{r}_{1}\times \overline{\boldsymbol{l}}_{1} & \cdots & \boldsymbol{r}_{6}\times \overline{\boldsymbol{l}}_{6} \end{array} \right]$ .

Based on Eq. (16), Eq. (18), and Eq. (19), the stiffness matrix can be obtained:

(20) \begin{align} \boldsymbol{K}=\boldsymbol{J}_{f}\text{diag}\left(\left[k_{1}\cdots k_{6}\right]\right){\boldsymbol{J}_{f}}^{T} \end{align}

Where the specific calculation method for $\boldsymbol{K}$ is given in the Appendix.

Based on Eq. (16) and Eq. (20), the deformation of PKM can be calculated by Eq. (21):

(21) \begin{align} \left[\begin{array}{l} \Delta \boldsymbol{X}\\[3pt] \Delta \boldsymbol{\theta } \end{array}\right]=\boldsymbol{K}^{-1}\left[\begin{array}{l} \boldsymbol{f}\\[3pt] \boldsymbol{m} \end{array}\right] \end{align}

When the multi-DoF forming machine is used for forming, the tool suffers the forming load of $\boldsymbol{f}=[0,0,F_{z}]^{T},\boldsymbol{m}=[F_{z}R\cos (2\pi t),F_{z}R\sin (2\pi t),0]^{T}$ , where $F_{z}$ is the magnitude of forming load and $R$ is the arm of forming load.

Set $\boldsymbol{f}=[0,0,5000]^{T}\mathrm{kN}$ and $\boldsymbol{m}=[750\cos (2\pi t),750\sin (2\pi t),0]^{T}\mathrm{kN}\cdot \mathrm{m}$ , the deformation of tool is presented in Figure 8. It is found that the linear deformation of tool is between 0.15 and –0.15 mm, and the deformation of tool in z direction is the largest. The angular deformation of tool is between 1 and –1 mrad, and the angular deformation of tool in z direction is the smallest. It indicates that the linear deformation along z axis and the angular deformation along x axis and y axis are the main factors of tool deformation.

Figure 8. The deformation of tool with time. (a) Linear deformation (b) Angular deformation.

3.2 Deformation error of the PKM

Figure 9. Illustration of the deformation error of the parallel kinematic machine.

Since the multi-DoF forming machine works under heavy load conditions, the elastic links will produce large elastic deformation. Due to the relatively small deformation of the tool, it can be treated as rigid body. Therefore, the tool motion can deviate from the predetermined trajectory under the coordinated deformation of 6 links. As shown in Figure 9, the tool composed of spherical joint $B_{1}\cdots B_{6}$ represents the theoretical position without deformation error, and the tool composed of spherical joints $B_{1}^{\prime}\cdots B_{6}^{\prime}$ represents the real position with deformation error. $\mathrm{S}_{\mathrm{B}}$ is the coordinate system of the theoretical position of the tool and $\mathrm{S}_{\mathrm{B}}^{\prime}$ is fixed on the real position of tool. Therefore, deformation error of the PKM is the difference between the theoretical trajectory and the real trajectory of the reference point $C_{i}$ placed on the tool. According to the stiffness model of the PKM, the deformation error of the PKM can be expressed by:

(22) \begin{align} \Delta e=\left| \Delta \boldsymbol{X}+\left(\boldsymbol{R}\left(\alpha +\Delta \theta _{x},\beta +\Delta \theta _{y},\gamma +\Delta \theta _{z}\right)-\boldsymbol{R}\left(\alpha, \beta, \gamma \right)\right)\boldsymbol{c}_{i}\right| \end{align}

where $\mathbf{c}_{\mathbf{i}}=[r_{t}\cos \phi, r_{t}\sin \phi, 1](r=0\sim r_{t},\phi =0\sim 2\pi )$ is the position vector of point $C_{i}$ in the tool, $r_{t}$ is the radius of the tool and $\phi$ is the polar angle of point in the tool.

Set the feed rate $k=1$ mm/s, the magnitude of forming load $F_{z}=$ 4000, 6000 kN, the arm of forming load $R=$ 100,200 mm, the contours of the deformation error distribution of the PKM in the workspace are shown in Figure 10.

Figure 10. Distribution of deformation error of the PKM in the workspace. (a) $Fz=4000\mathrm{kN},{}R=100\mathrm{mm}$ (b) $Fz=4000\mathrm{kN},R=200\mathrm{mm}$ . (c) $Fz=6000\mathrm{kN},R=100\mathrm{mm}$ (d) $Fz=6000\mathrm{kN},{}R=200\mathrm{mm}$ .

As shown in Figure 10, it can be seen that the maximum value of deformation error is distributed in the region formed by smaller inclination angle and lower feed distance within the workspace. Moreover, as the magnitude of forming load or arm of forming load increases, the the maximum value of deformation error gradually increases.

Deformation error $\Delta e$ of the PKM is generally determined based on actual operational requirements. When the $\Delta e$ is large, the accuracy of the multi-DoF forming machine will decrease and it will not meet the processing technology requirements of multi-DoF forming. Thus, the deformation error of the PKM and geometric constraint of PKM should satisfy the following conditions.

(23) \begin{align} \left\{\begin{array}{l} s_{\min }\leq \sqrt{{a_{ix}}^{2}+\left(a_{ix}\tan \theta _{Ai}\right)^{2}}-r_{A}\leq s_{\max }\\[3pt] \arccos \left(\left(h+Z_{P}-r_{B}\sin \beta \cos \theta _{Bi}+\cos \alpha \cos \beta d+r_{B}\cos \beta \sin \alpha \sin \theta _{Bi}\right)/l\right)\leq \theta _{\max }\\[3pt] \arccos \left(\left(\cos \alpha \sin \beta \left(b_{ix}-a_{ix}\right)-\sin \alpha \left(b_{iy}-a_{ix}\tan \theta _{Ai}\right)+\cos \alpha \cos \beta b_{iz}\right)/l\right)\leq \theta _{\max }\\[3pt] \left(\left(b_{ix}-a_{ix}\right)\cos \theta _{Ai}+\left(b_{iy}-a_{ix}\tan \theta _{Ai}\right)\sin \theta _{Ai}\right)/l\neq 0\\[3pt] \Delta e\lt =0.35mm \end{array}\right. \end{align}

Based on the constraint in Eq. (23), the feed distance $h=[h_{\min }^{s},h_{\max }^{s}]$ can be calculated for different force conditions. The workspace utilization rate $\lambda ^{s}$ considering deformation error is the ratio of the workspace considering the mechanism deformation to the geometric workspace of the PKM. In order to quantitatively analyze the effect of forming load on workspace, $\lambda ^{s}$ is introduced as follows:

(24) \begin{align} \lambda ^{s}=\int _{\varphi _{\min }^{g}}^{\varphi _{\max }^{g}}\left(h_{\max }^{s}-h_{\min }^{s}\right)\mathrm{d}\varphi /\int _{\varphi _{\min }^{g}}^{\varphi _{\max }^{g}}\left(h_{\max }^{g}-h_{\min }^{g}\right)\mathrm{d}\varphi \end{align}

Take the magnitude of forming load as 3000 to 6000 kN and the arm of forming load as 100 to 200 mm, the workspace utilization rate $\lambda ^{s}$ is obtained and shown in Figure 11. It can be observed from Figure 11 that the workspace utilization rate decreases as the magnitude of forming load or arm of forming load increases. When the magnitude of forming load or arm of forming load is small, the workspace utilization rate can be up to 100%. When the magnitude of forming load is 6000 kN and arm of forming load is 200 mm, the workspace utilization rate can decrease up to 21.7 %. The results show that with the increase of forming load, the workspace of the PKM can decrease to 1/5 of the theoretical one.

Figure 11. Workspace utilization rate $\lambda ^{s}$ with deformation error constraint.