3.1. Inverse position analysis

Inverse position analysis aims to solve for the inputs of the driving joints $d_{i}\,(i=1,2,3,4)$ under the given position and orientation of the MP $(x,y,z,\phi )$ . Here, $d_{i}$ represents the displacement of each actuated joint along the $\boldsymbol{Z}$ -axis; $\phi$ is the rotation angle about the $\boldsymbol{Y}$ -axis, where clockwise rotation is defined as negative and counterclockwise as positive. As seen in Figure 2, from the origin $O$ of the global coordinate system to the geometric centers $C_{i}$ of the revolute joints, the closed-loop equation of limb i can be written as follows:

(9) \begin{align} \boldsymbol{O}\boldsymbol{C}_{i}=\boldsymbol{O}\boldsymbol{A}_{i}+\boldsymbol{A}_{i}\boldsymbol{B}_{i}+\boldsymbol{B}_{i}\boldsymbol{C}_{i} \end{align}

Projecting Eq. (9) along global coordinate axes $\boldsymbol{X},\boldsymbol{Y}$ and $\boldsymbol{Z}$ , the algebraic equation can be expressed as follows:

(10) \begin{align} \begin{cases} x=l_{1}\cos \theta _{11}\\ y-b_{2}=-a_{1}+l_{2}\cos \theta _{12}\\ z=d_{1}+l_{1}\sin \theta _{11}+l_{2}\sin \theta _{12} \end{cases} \\[-32pt] \nonumber \end{align}

(11) \begin{align} \begin{cases} x-b_{1}\cos \phi =-a_{2}+l_{2}\cos \theta _{22}\\ y=l_{1}\cos \theta _{21}\\ z+b_{1}\sin \phi =d_{2}+l_{1}\sin \theta _{21}+l_{2}\sin \theta _{22} \end{cases} \\[-32pt] \nonumber \end{align}

(12) \begin{align} \begin{cases} x=l_{1}\cos \theta _{31}\\ y+b_{2}=a_{1}-l_{2}\cos \theta _{32}\\ z=d_{3}+l_{1}\sin \theta _{31}+l_{2}\sin \theta _{32} \end{cases} \\[-32pt] \nonumber \end{align}

(13) \begin{align} \begin{cases} x=a_{2}-l_{2}\cos \theta _{42}\\ y=l_{1}\cos \theta _{41}\\ z=d_{4}+l_{1}\sin \theta _{41}+l_{2}\sin \theta _{42} \end{cases} \end{align}

where the intermediate variable $\theta _{i1}$ denotes the angle between link $A_{i}B_{i}$ and the negative direction of the $\boldsymbol{X}{-}$ axis for limbs i = 1, 3, or the angle between link $A_{i}B_{i}$ and the negative direction of the $\boldsymbol{Y}{-}$ axis for limbs i = 2, 4. Similarly, $\theta _{i2}$ is defined as the acute angle between link $B_{i}C_{i}$ and the $\boldsymbol{Y}{-}$ axis for i = 1, 3, and as the acute angle between link $B_{i}C_{i}$ and the $\boldsymbol{X}{-}$ axis for i = 2, 4.

By solving Eqs. (10)–(13), the inverse position results are as follows:

(14) \begin{align} \begin{cases} d_{1}=z-l_{1}\sin \theta _{11}-l_{2}\sin \theta _{12}\\[3pt]d_{2}=z+b_{1}\sin \phi -l_{1}\sin \theta _{21}-l_{2}\sin \theta _{22}\\[3pt]d_{3}=z-l_{1}\sin \theta _{31}-l_{2}\sin \theta _{32}\\[3pt]d_{4}=z-l_{1}\sin \theta _{41}-l_{2}\sin \theta _{42} \end{cases} \end{align}

where $\theta _{11}=a\cos \frac{x}{l_{1}},\theta _{12}=a\cos \frac{y-b_{2}+a_{1}}{l_{2}},\theta _{21}=a\cos \frac{y}{l_{1}},\theta _{22}=a\cos \frac{a_{2}+x-b_{1}\cos \phi }{l_{2}},\theta _{31}=a\cos \frac{x}{l_{1}},$

$\theta _{32}=a\cos \frac{a_{1}-y-b_{2}}{l_{2}},\theta _{41}=a\cos \frac{y}{l_{1}},\theta _{42}=a\cos \frac{a_{2}-x}{l_{2}}$ .

3.2. Forward position analysis

In contrast to the inverse position analysis, forward position analysis involves solving for the position vector $(x,y,z,\phi )$ of the MP for the given inputs $d_{i}(i=1,2,3,4)$ . The position vectors of joints $B_{i}$ and $C_{i}$ in the fixed coordinate system are expressed as follows:

(15) \begin{align} \begin{cases} \boldsymbol{B}_{1}=\left(l_{1}\cos \theta _{11},-a_{1},d_{1}+l_{1}\sin \theta _{11}\right),\boldsymbol{C}_{1}=\left(x,y-b_{2},z\right)\\[3pt]\boldsymbol{B}_{2}=\left(-a_{2},l_{1}\cos \theta _{21},d_{2}+l_{1}\sin \theta _{21}\right),\boldsymbol{C}_{2}=\left(x-b_{1}\cos \phi, y,z+b_{1}\sin \phi \right)\\[3pt]\boldsymbol{B}_{3}=\left(l_{1}\cos \theta _{31},a_{1},d_{3}+l_{1}\sin \theta _{31}\right),\boldsymbol{C}_{3}=\left(x,y+b_{2},z\right)\\[3pt]\boldsymbol{B}_{4}=\left(a_{2},l_{1}\cos \theta _{41},d_{4}+l_{1}\sin \theta _{41}\right),\boldsymbol{C}_{4}=\left(x,y,z\right) \end{cases} \end{align}

According to the geometric constraints of the mechanism:

(16) \begin{align} \left| \boldsymbol{B}_{i}\boldsymbol{C}_{i}\right| =l_{2}\left(i=1,2,3,4\right) \end{align}

Combining Eqs. (15)–(16) leads to

(17) \begin{align} \begin{cases} \left(x-l_{1}\cos \theta _{11}\right)^{2}+\left(y-b_{2}+a_{1}\right)^{2}+\left(z-d_{1}-l_{1}\sin \theta _{11}\right)^{2}={l}_{2}^{2}\\[3pt]\left(x-b_{1}\cos \phi +a_{2}\right)^{2}+\left(y-l_{1}\cos \theta _{21}\right)^{2}+\left(z+b_{1}\sin \phi -d_{2}-l_{1}\sin \theta _{21}\right)^{2}={l}_{2}^{2}\\[3pt]\left(x-l_{1}\cos \theta _{31}\right)^{2}+\left(y+b_{2}-a_{1}\right)^{2}+\left(z-d_{3}-l_{1}\sin \theta _{31}\right)^{2}={l}_{2}^{2}\\[3pt]\left(x-a_{2}\right)^{2}+\left(y-l_{1}\cos \theta _{41}\right)^{2}+\left(z-d_{4}-l_{1}\sin \theta _{41}\right)^{2}={l}_{2}^{2} \end{cases} \end{align}

From Figure 2, we can obtain $\theta _{11}=\theta _{31},\theta _{21}=\theta _{41},x=l_{1}\cos \theta _{11},y=l_{1}\cos \theta _{21}$ ; consequently, all the intermediate variables in Eq. (17) can be eliminated. Then the forward kinematic solutions can be derived as:

(18) \begin{align} \begin{cases} x=\left(2a_{2}E_{1}-\sqrt{4{a}_{2}^{2}{E}_{2}^{2}{l}_{1}^{2}-{E}_{1}^{2}{E}_{2}^{2}+{E}_{2}^{4}{l}_{1}^{2}}\right)/\left(4{a}_{2}^{2}+{E}_{2}^{2}\right)\\[3pt]y=\left({d}_{1}^{2}-{d}_{3}^{2}+2M_{4}\left(d_{3}-d_{1}\right)\right)/\left(4\left(b_{2}-a_{1}\right)\right),z=M_{4}+\sqrt{{l}_{1}^{2}-x^{2}}\\[3pt]\phi =2a\tan \left(\left(K_{2}-\sqrt{{K}_{2}^{2}-{K}_{3}^{2}+{K}_{1}^{2}}\right)/\left(K_{1}+K_{3}\right)\right) \end{cases} \end{align}

The intermediate variables in the Eq. (18) are as follows: $M_{1}=4((d_{3}-d_{1})^{2}+4(b_{2}-a_{1})^{2}),$

\begin{align*} M_{2}&=4\left(d_{3}-d_{1}\right)\left({d}_{1}^{2}-{d}_{3}^{2}\right)-16\left(b_{2}-a_{1}\right)^{2}\left(d_{3}+d_{1}\right),M_{4}=\left(-M_{2}+\sqrt{{M}_{2}^{2}-4M_{1}M_{3}}\right)/\left(2M_{1}\right),\\[3pt]K_{2}&=2b_{1}\left(z-d_{2}-\sqrt{{l}_{1}^{2}-y^{2}}\right)\!, M_{3}=\left(\left({d}_{1}^{2}-{d}_{3}^{2}\right)+4\left(b_{2}-a_{1}\right)^{2}\right)^{2}-16{l}_{2}^{2}\left(b_{2}-a_{1}\right)^{2}+16{d}_{3}^{2}\left(b_{2}-a_{1}\right)^{2},\\[3pt]K_{3}&={l}_{2}^{2}-\left(x+a_{2}\right)^{2}-{b}_{1}^{2}-\left(z-d_{2}-\sqrt{{l}_{1}^{2}-y^{2}}\right)^{2}\!, E_{2}=2\left(d_{4}+\sqrt{{l}_{1}^{2}-y^{2}}\right)-2M_{4},K_{1}=-2\left(x+a_{2}\right)b_{1},\\[3pt]E_{1}&={a}_{2}^{2}+{M}_{4}^{2}+{l}_{1}^{2}-2M_{4}\left(d_{4}+\sqrt{{l}_{1}^{2}-y^{2}}\right)+\left(d_{4}+\sqrt{{l}_{1}^{2}-y^{2}}\right)^{2}-{l}_{2}^{2}.\end{align*}

3.3. Calculation example of forward and inverse position solutions

The structural dimensions are given as $a_{1}=a_{2}=306,l_{1}=308,l_{2}=342,b_{1}=110,b_{2}=90$ (unit: mm), and the initial pose of the MP is set to $(x_{0},y_{0},z_{0},\varphi )=(29.765,23.481,758.559,0)$ . With the driving function in Eq. (19) applied at the preset pose of the MP, the theoretical solutions derived from Eq. (14) and the corresponding Adams simulation results are shown in Figure 4. Meanwhile, by inputting the corresponding data of the driving joints into Adams, the simulation curves of the pose of the MP in Adams and the theoretical solutions obtained via Eq. (18) are presented in Figure 5. Figures 4 and 5 show that the theoretical curves of the driving joints and the MP are almost the same as the corresponding simulation curves in Adams, which verifies the correctness of the kinematic analytical solutions of the improved 3T1R PM.

Figure 4.

Comparison of the driving joint displacements from theoretical and simulation results (a) d₂, d₃, and d₄ and (b) d₁.

Figure 5.

Comparison between theoretical and simulated MP pose (a) x, y, and $\phi$ and (b) z.

Figure 5. Long description

Panel A: A line graph shows the displacement in millimeters of a manipulator pose in the x and y directions over time in seconds. The x-axis represents time in seconds, ranging from 0 to 8 seconds. The y-axis represents displacement in millimeters, ranging from -10 to 80 millimeters. The graph includes multiple lines representing theoretical and simulated values for x, y, and phi directions. The theoretical values for x and y are marked with blue stars and red crosses, respectively, while the simulated values are shown with blue and red lines. The phi values are marked with green stars and lines. The graph shows oscillatory behavior for x and y directions, while the phi values remain constant. Panel B: Another line graph shows the displacement in millimeters of a manipulator pose in the z direction over time in seconds. The x-axis represents time in seconds, ranging from 0 to 8 seconds. The y-axis represents displacement in millimeters, ranging from 758 to 776 millimeters. The graph includes two lines representing theoretical and simulated values for the z direction. The theoretical values are marked with orange crosses, and the simulated values are shown with an orange line. The graph shows a linear increase in displacement over time.

(19) \begin{align} \begin{cases} x=x_{0}+24\sin \left(2.5t\right),y=y_{0}+24-24\cos \left(2.5t\right)\\[3pt]z=z_{0}+\dfrac{12.5t}{2\pi },\varphi =0 \end{cases} \end{align}

4.1. Velocity analysis

Velocity analysis is the basis of the singularity analysis. By differentiating both sides of Eq. (14), we can derive the mapping between the input velocity $\,\dot{\!d}_{i}(i=1,2,3,4)$ and the output velocity $(\dot{x},\dot{y},\dot{z},\dot{\phi })$ . To date, numerous studies have investigated the Jacobian matrix of the mechanism in detail, such as Ref. [Reference Dong, Sun, Song, Gao and Lian30–Reference Yuan and Zhang32]. Therefore, the derivation of the Jacobian matrix is omitted here. The derived Jacobian matrix can be expressed as follows:

(20) \begin{align} \boldsymbol{J}_{d}\,\,\dot{\!\boldsymbol{d}}=\boldsymbol{J}_{X}\,\dot{\!\boldsymbol{X}} \end{align}

The $\boldsymbol{J}_{d}$ and $\boldsymbol{J}_{X}$ are the inverse Jacobian matrix and the forward Jacobian matrix, respectively, and are given as follows:

(21) \begin{align} \boldsymbol{J}_{d}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} J_{d11} & 0 & 0 & 0\\[3pt]0 & J_{d22} & 0 & 0\\[3pt]0 & 0 & J_{d33} & 0\\[3pt]0 & 0 & 0 & J_{d44} \end{array}\right] \\[-28pt] \nonumber \end{align}

(22) \begin{align} \boldsymbol{J}_{X}=\left[\begin{array}{l@{\quad}l@{\quad}l@{\quad}l} J_{X11} & J_{X12} & J_{X13} & J_{X14}\\[3pt]J_{X21} & J_{X22} & J_{X23} & J_{X24}\\[3pt]J_{X31} & J_{X32} & J_{X33} & J_{X34}\\[3pt]J_{X41} & J_{X42} & J_{X43} & J_{X44} \end{array}\right] \end{align}

where $ J_{d11}=\sin \theta _{11}\sin \theta _{12},J_{d22}=\sin \theta _{21}\sin \theta _{22},J_{d33}=\sin \theta _{31}\sin \theta _{32},J_{d44}=\sin \theta _{41}\sin \theta _{42},$ $ J_{X11}=\cos \theta _{11}\sin \theta _{12},J_{X12}=\cos \theta _{12}\sin \theta _{11},J_{X13}=\sin \theta _{11}\sin \theta _{12},J_{X14}=0,J_{X21}=\sin \theta _{21}\cos \theta _{22},$ $J_{X22}=\cos \theta _{21}\sin \theta _{22},J_{X23}=\sin \theta _{21}\sin \theta _{22},J_{X31}=\cos \theta _{31}\sin \theta _{32},J_{X32}=-\cos \theta _{32}\sin \theta _{31},$ $ J_{X34}=0,J_{X24}=\left(b_{1}\cos \phi \sin \theta _{21}\sin \theta _{22}+b_{1}\sin \phi \sin \theta _{21}\cos \theta _{22}\right),J_{X33}=\sin \theta _{31}\sin \theta _{32},$ $ J_{X41}=-\cos \theta _{42}\sin \theta _{41},J_{X42}=\cos \theta _{41}\sin \theta _{42},J_{X43}=\sin \theta _{41}\sin \theta _{42},J_{X44}=0$

When $| \boldsymbol{J}_{d}| \neq 0$ , the Jacobian matrix for the improved PM 2PPaRR-2PPaRU equals

(23) \begin{align} \boldsymbol{J}={\boldsymbol{J}}_{d}^{-1}\boldsymbol{J}_{X} \end{align}

4.2. Singularity analysis

Singularity is an inherent property of the PMs. It significantly affects the PM’s performance, especially near its singular configuration, potentially causing damage to equipment. In this section, we will examine inverse kinematic singularity, forward kinematic singularity, and combined kinematic singularity [Reference Gosselin and Angeles33].

4.2.1. Inverse kinematics singularity

Inverse kinematic singularity may arise when the MP is at the workspace boundary. In such a configuration, the MP loses one or more DOFs. When this occurs, $| \boldsymbol{J}_{d}| =0$ and $| \boldsymbol{J}_{x}| \neq 0$ . Since $\boldsymbol{J}_{d}$ is a diagonal matrix, it is easy to obtain $| \boldsymbol{J}_{d}| =0$ when $\sin \theta _{i1}\sin \theta _{i2}=0\,(i=1,2,3,4)$ . To avoid interference between the links, the following constraints should be satisfied:

(24) \begin{align} \begin{cases} 30^{\circ }\leq \theta _{i1}\leq 150^{\circ }\\[3pt]0\leq \theta _{i2}\lt 90^{\circ } \end{cases} \end{align}

From Eq. (24), the four situations can be found as follows:

(25) \begin{align} \theta _{i2}=0\left(i=1,2,3,4\right) \end{align}

In these situations, links $B_{i}C_{i}(i=1,2,3,4)$ and the MP are coplanar and parallel with the XOY plan when $\theta _{i2}=0$ ; meanwhile, the MP lies on the boundary of its workspace. The four inverse kinematic singular configurations are presented in Figure 6.

Figure 6.

The inverse kinematic singularity configurations (a) $\theta _{12}=0$ , (b) $\theta _{22}=0$ , (c) $\theta _{32}=0$ , and (d) $\theta _{42}=0$ .

4.2.2. Forward kinematics singularity

A forward kinematic singularity, also known as a parallel singularity, occurs within the workspace. In such configurations, the MP undergoes infinitesimal motion in certain directions while the driving joints remain fixed, which implies that the MP obtains extra DOFs. The forward kinematic singularity occurs when $| \boldsymbol{J}_{d}| \neq 0$ and $| \boldsymbol{J}_{X}| =0$ . According to Eq. (22), $\det (\boldsymbol{J}_{X})=-b_{1}\sin \theta _{11}\sin \theta _{21}\sin \theta _{41}\sin (\phi +\theta _{22})\sin (\theta _{12}+\theta _{32})\sin (\theta _{42}+\theta _{31})$ can be obtained, the physical constraint is same as Eq. (24).

It can be obtained $b_{1},\sin \theta _{11},\sin \theta _{21}$ and $\sin \theta _{41}$ are not equal to 0. According to the forward kinematic singularity condition $| \boldsymbol{J}_{d}| \neq 0\boldsymbol{,}\ | \boldsymbol{J}_{X}| =0$ and Eq. (24), it can be easily derived that $\sin (\theta _{12}+\theta _{32})\neq 0$ . Thus, $| \boldsymbol{J}_{X}| =0$ is equal to:

(26) \begin{align} \sin \left(\theta _{42}+\theta _{31}\right)=0\text{ or }\sin \left(\phi +\theta _{22}\right)=0 \end{align}

When $\sin (\theta _{42}+\theta _{31})=0$ , combined with the physical constraint $x_{C3}=x_{C4}$ (where $x_{C3}$ and $x_{C4}$ are the projections of the geometric centers of revolute joints C ₃ and C ₄ onto the X -axis), the following equation can be obtained:

(27) \begin{align} \begin{cases} \theta _{31}+\theta _{42}=\pi \\[3pt]l_{1}\cos \theta _{31}=a_{2}-l_{2}\cos \theta _{42} \end{cases} \end{align}

It can be derived from Eq. (27) that $\theta _{42}=a\cos (a_{2}/(l_{2}-l_{1}))$ , which means $l_{2}-l_{1}\gt a_{2}$ . This will lead to an excessively long link $B_{i}C_{i}\,(i=1,2,3,4)$ , reducing the PM’s stiffness and rendering it impractical for control. When $\sin (\theta _{22}+\phi )=0$ , it can be derived that $\phi =-\theta _{22}$ . In this singular configuration, the axis of link B ₂ C ₂ is coplanar with the MP. Figure 7 presents one of the singular configurations associated with forward kinematics.

Figure 7.

The forward kinematic singularity configuration $\theta _{22}=-\phi$ .

Figure 7. Long description

A diagram of a parallel manipulator. The diagram shows a structure with multiple vertical columns connected by horizontal and diagonal links. The vertical columns are labeled with black tops and have orange connectors at the bottom. The horizontal and diagonal links are connected to a central moving platform. The central moving platform is connected to the vertical columns via orange connectors. The diagram includes labels for the Y-axis, points B2 and C2, and angles theta22 and phi. The Y-axis is shown as a red arrow pointing horizontally. B2 and C2 are the geometric centers of the corresponding joints. theta22 and phi are the angles corresponding to link B2C2 and the moving platform (MP), respectively. The diagram illustrates the motion and rotation capabilities of the parallel manipulator.

4.2.3. Combined kinematic singularity

In the combined kinematic singular configurations of the mechanism, the MP can move in one or several directions when the actuated joints are locked. Conversely, when the MP is constrained, the actuated joints can undergo finite motion. The corresponding conditions are $| \boldsymbol{J}_{d}| =0$ and $| \boldsymbol{J}_{X}| =0$ . Based on the analysis in Sections 4.2.1 and 4.2.2, three types of combined singular configurations can be identified, as shown in Figure 8.

Figure 8.

Combined kinematic singular configurations (a) $\theta _{22}=-\phi$ and $\theta _{12}=0$ , (b) $\theta _{22}=-\phi$ and $\theta _{32}=0$ , and (c) $\theta _{22}=-\phi$ and $\theta _{42}=0$ .

5.1. Local motion/force transmission analysis

Liu et al. [Reference Wang, Wu and Liu22, Reference Chen, Xie and Liu23] proposed the local transmission index (LTI) based on the transmission angle theory [Reference Balli and Chand34]. The transmission index reflects the energy transfer efficiency of the PM. The closer the transmission index is to unity, the higher the transmission efficiency. The LTI can be defined as follows:

(28) \begin{align} \begin{cases} \mathrm{ITI}_{i}=\dfrac{\left| \boldsymbol{\$}_{i1}\circ \boldsymbol{\$}_{Ti}\right| }{\left| \boldsymbol{\$}_{i1}\circ \boldsymbol{\$}_{Ti}\right| _{\max }}\\[8pt]\mathrm{OTI}_{i}=\dfrac{\left| \boldsymbol{\$}_{oi}\circ \boldsymbol{\$}_{Ti}\right| }{\left| \boldsymbol{\$}_{oi}\circ \boldsymbol{\$}_{Ti}\right| _{\max }}\\[8pt]\mathrm{LTI}=\min \left\{\mathrm{ITI}_{i},\mathrm{OTI}_{i}\right\} \end{cases} \end{align}

where $\boldsymbol{\$}_{i1},\boldsymbol{\$}_{Ti}$ and $\boldsymbol{\$}_{Oi}$ denote the input twist screw, transmission wrench screw and the output twist screw of the limb i, respectively. $\mathrm{ITI}_{i},\mathrm{OTI}_{i}$ and LTI represent the input transmission index, output transmission index, and the minimum value of all the input and output transmission index, that is, the LTI of the PM. LTI ranges from zero to unity, and a higher LTI indicates better transmissibility.

Fixing $\boldsymbol{\$}_{i1}$ (the input twist screw), the newly added constraint wrench screw represents the transmission wrench screw of limb i. According to the reciprocity between the twist screw system and the wrench screw system [Reference Dai and Jones35], we can obtain the transmission wrench screws in limbs 1, 2, 3, and 4 as follows (see Appendix A):

(29) \begin{align} \begin{cases} \boldsymbol{\$}_{T1}=\left(l_{11},m_{1},n_{1};\,p_{1},q_{1},r_{1}\right)\\[3pt]\boldsymbol{\$}_{T2}=\left(l_{21},m_{2},n_{2};\,p_{2},q_{2},r_{2}\right)\\[3pt]\boldsymbol{\$}_{T3}=\left(l_{31},m_{3},n_{3};\,p_{3},q_{3},r_{3}\right)\\[3pt]\boldsymbol{\$}_{T4}=\left(l_{41},m_{4},n_{4};\,p_{4},q_{4},r_{4}\right) \end{cases} \end{align}

With the input joint in limb i actuated and the input joints in the other three limbs fixed, the unit twist screw of the MP is equivalent to the output twist screw of that limb. According to the reciprocity between the twist screw $\boldsymbol{\$}_{Oi}(i=1\sim 4)$ and the wrench screw $\boldsymbol{\$}_{Ci},\boldsymbol{\$}_{Tk}(i=1,2;\,k=1\sim 4)$ :

(30) \begin{align} \boldsymbol{\$}_{Oi}\circ \boldsymbol{\$}_{Ci}=0,\boldsymbol{\$}_{Oi}\circ \boldsymbol{\$}_{Tk}=0\left(i\neq k\right) \end{align}

The PM’s output twist screws can be determined. As indicated by the analysis in Ref. [Reference Zhang, Dong, Dai, Chen and Zhao12], it is evident that the cross-section profile of the mechanism’s reachable workspace remains consistent at different heights when the orientation of the MP is fixed, without accounting for the limitations of actuated displacements. Therefore, this section focuses on analyzing the distribution of LTI within the cross-section of the workspace at a specific operating height (Z = 1,000 mm). Given that the 3T1R PM is primarily used in light-load scenarios, the configuration of the improved 3T1R PM is picked into GTW when $\mathrm{LTI}\geq 0.5$ . The structural parameters remain consistent with those in Section 3.3. The LTI distribution across the reachable workspace under various orientations is symmetrical about the X -axis, as depicted in Figure 9, aligning with the symmetry of the mechanism architecture about the X -axis. Compared to the LTI distribution of the original design in Ref. [Reference Zhang, Dong, Dai, Chen and Zhao12], the region with poor transmission performance is deviated from the home configuration.

Figure 9.

LTI distribution (a) $\phi =-45^{\circ }$ , (b) $\phi =0$ , (c) $\phi =45^{\circ }$ , (d) $\phi =-90^{\circ }$ , and (e) $\phi =90^{\circ }$ .

Figure 9. Long description

Panel A: A heat map representing LTI distribution with a gradient from blue to red. The x-axis ranges from -35 to 110 millimeters, and the y-axis ranges from -125 to 125 millimeters. The color scale on the right indicates values from 0.05 to 0.35, with red representing higher values and blue representing lower values. The distribution shows a concentrated region of higher values around the center-right. Panel B: A heat map representing LTI distribution with a gradient from blue to red. The x-axis ranges from -35 to 145 millimeters, and the y-axis ranges from -125 to 125 millimeters. The color scale on the right indicates values from 0 to 0.6, with red representing higher values and blue representing lower values. The distribution shows a concentrated region of higher values around the center, with a labeled region indicating LTI values greater than or equal to 0.5. Panel C: A heat map representing LTI distribution with a gradient from blue to red. The x-axis ranges from -35 to 110 millimeters, and the y-axis ranges from -125 to 125 millimeters. The color scale on the right indicates values from 0.1 to 0.6, with red representing higher values and blue representing lower values. The distribution shows a concentrated region of higher values around the center. Panel D: A heat map representing LTI distribution with a gradient from blue to red. The x-axis ranges from -35 to 35 millimeters, and the y-axis ranges from -125 to 125 millimeters. The color scale on the right indicates values from 0.1 to 0.4, with red representing higher values and blue representing lower values. The distribution shows a concentrated region of higher values around the center. Panel E: A heat map representing LTI distribution with a gradient from blue to red. The x-axis ranges from -35 to 35 millimeters, and the y-axis ranges from -125 to 125 millimeters. The color scale on the right indicates values from 0.1 to 0.4, with red representing higher values and blue representing lower values. The distribution shows a concentrated region of higher values around the center.

5.2. Overall good transmission workspace analysis

To provide an overall measure of the good transmission region within the entire reachable workspace, the GTWR [Reference Xie, Liu, Wang and Wang36] is defined as follows:

(31) \begin{align} \Gamma =\frac{\int \gamma d\mathrm{s}}{\int ds} \end{align}

Here, $\gamma$ refers to the point in the GTW, $\int \gamma ds$ refers to the set of $\gamma$ in the overall reachable workspace and $\int ds$ refers to the total discrete points in the reachable workspace. The overall reachable workspace comprises the cross-sections obtained under various orientations at the operating height Z = 1,000 mm. Clearly, $\Gamma$ ranges from 0 to 1, with higher values indicating better global transmission performance. In Figure 9, the 3T1R PM exhibits GTW only when the orientation of the MP equals 0 and 45°, resulting in GTWR values of 0.223 and 0.239, respectively.

6.1. Stiffness matrix of the parallelogram

Since the links are slender and flexible, they can be modeled as cantilever beams in the stiffness analysis. Based on the Euler–Bernoulli equation [Reference Filipovic37], the stiffness matrix of a link can be derived as:

(32) \begin{align} \boldsymbol{K}_{bar}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} k_{11} & 0 & 0 & 0 & 0 & 0\\[3pt]0 & k_{22} & 0 & 0 & 0 & k_{26}\\[3pt]0 & 0 & k_{33} & 0 & k_{35} & 0\\[3pt]0 & 0 & 0 & k_{44} & 0 & 0\\[3pt]0 & 0 & k_{53} & 0 & k_{55} & 0\\[3pt]0 & k_{62} & 0 & 0 & 0 & k_{66} \end{array}\right] \end{align}

where $k_{11}=GI_{x}/l_{i},\ k_{22}=4EI_{y}/l_{i},\ k_{33}=4EI_{z}/l_{i},\ k_{44}=EA/l_{i},\ k_{55}=12EI_{z}/{l}_{i}^{3},\ k_{66}=12EI_{y}/{l}_{i}^{3}, k_{26}=k_{62}=6EI_{y}/{l}_{i}^{2}, k_{35}=k_{53}=-6EI_{z}/{l}_{i}^{2}$ . The $l_{i}$ is the link length, $A,\ E$ and $G$ are the cross-section area, the Young’s and shear modules of the link, $I_{y},I_{z}$ and $I_{x}$ are the quadratic and polar moments of inertia of the cross section, respectively.

The virtual spring model of the parallelogram structure is shown in Figure 10. The elastic deformations of the upper link are equivalent to the elastic deformations of the virtual spring at the end of the link, and the motion of the passive joints R1, R3 is considered. The elastic deformation $\delta \boldsymbol{t}^{up}$ of the end of the upper limb equals:

(33) \begin{align} \delta \boldsymbol{t}^{up}={\boldsymbol{J}}_{\Pi }^{\theta }\delta \boldsymbol{\theta }_{\Pi }+{\boldsymbol{J}}_{\varphi }^{up}\delta \boldsymbol{q}_{\Pi } \end{align}

where ${\boldsymbol{J}}_{\Pi }^{\theta }$ is the Jacobian matrix that corresponds to the deflection $\delta \boldsymbol{\theta }_{\Pi }$ of the upper link, and ${\boldsymbol{J}}_{\varphi }^{up}$ is the Jacobian matrix that corresponds to the motion of the passive revolute joints $\delta \boldsymbol{q}_{\Pi }$ . The ${\boldsymbol{J}}_{\Pi }^{\theta }$ and ${\boldsymbol{J}}_{\varphi }^{up}$ are expressed as:

(34) \begin{align} {\boldsymbol{J}}_{\Pi }^{\theta }=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 1 & 0 & 0 & 0 & 0 & 0\\[3pt]0 & 1 & 0 & 0 & 0 & 0\\[3pt]0 & 0 & 1 & 0 & 0 & 0\\[3pt]0 & dSq/2 & 0 & 1 & 0 & 0\\[3pt]-dSq/2 & 0 & -L-dCq/2 & 0 & 1 & 0\\[3pt]0 & L+dCq/2 & 0 & 0 & 0 & 1 \end{array}\right],\ {\boldsymbol{J}}_{\varphi }^{up}=\left[\begin{array}{c@{\quad}c} 0 & 0\\[3pt]1 & 1\\[3pt]0 & 0\\[3pt]dSq/2 & dSq/2\\[3pt]0 & 0\\[3pt]dCq/2 & L+dCq/2 \end{array}\right] \end{align}

where $Sq=\sin q,Cq=\cos q$ . According to the static equilibrium equation [Reference Pashkevich, Chablat and Wenger25]:

(35) \begin{align} \left[\begin{array}{c@{\quad}c} {\boldsymbol{J}}_{\Pi }^{\theta }{\boldsymbol{K}}_{bar}^{-1}{\boldsymbol{J}}_{\Pi }^{\theta ^{T}} & {\boldsymbol{J}}_{\varphi }^{up}\\[6pt] {\boldsymbol{J}}_{\varphi }^{up^{T}} & \boldsymbol{0}_{2\times 2} \end{array}\right]\left[\begin{array}{l} \boldsymbol{f}_{up}\\[3pt] \delta \boldsymbol{\varphi }^{up} \end{array}\right]=\left[\begin{array}{l} \delta \boldsymbol{t}^{up}\\[3pt] \boldsymbol{0}_{n\times n} \end{array}\right] \end{align}

The stiffness matrix of the upper limb is:

(36) \begin{align} {\boldsymbol{K}}_{\Pi }^{up}=\left[\begin{array}{c@{\quad}c} {\boldsymbol{J}}_{\Pi }^{\theta }{\boldsymbol{K}}_{bar}^{-1}\left({\boldsymbol{J}}_{\Pi }^{\theta }\right)^{T} & {\boldsymbol{J}}_{\varphi }^{up}\\[6pt] \left({\boldsymbol{J}}_{\varphi }^{up}\right)^{T} & \boldsymbol{0}_{2\times 2} \end{array}\right]^{-1} \end{align}

Figure 10.

The virtual spring model of the parallelogram.

The stiffness matrix ${\boldsymbol{K}}_{\Pi }^{up}$ of the upper limb is the first six-dimensional entry extracted from Eq. (36). Due to the passive joints’ motion compensation, the translational stiffness along the $z_{Pa}-$ axis vanishes, and the stiffness matrix of the parallelogram structure can be represented by the first 5-dimensional submatrix as follows:

(37) \begin{align} {\boldsymbol{K}}_{\Pi }^{up}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} k_{11}+k_{55}d^{2}{S}_{q}^{2}/4 & 0 & k_{55}d^{2}S_{2q}/8 & 0 & 0\\[3pt] 0 & k_{44}d^{2}{S}_{q}^{2}/4 & 0 & 0 & 0\\[3pt] k_{55}d^{2}S_{2q}/8 & 0 & k_{33}+k_{55}d^{2}{C}_{q}^{2}/4 & 0 & k_{35}\\[3pt] 0 & 0 & 0 & k_{44} & 0\\[3pt] 0 & 0 & k_{35} & 0 & k_{55} \end{array}\right]_{5\times 5} \end{align}

where $k_{ij}$ denotes the $(\text{\em i}\mathit{,}\text{\em j})$ entry of the matrix ${\boldsymbol{K}}_{\Pi }^{up}$ . Similarly, the stiffness matrix of the lower limb can be obtained. The overall stiffness matrix of the parallelogram structure equals:

(38) \begin{align} \boldsymbol{K}_{\Pi }={\boldsymbol{K}}_{\Pi }^{up}+{\boldsymbol{K}}_{\Pi }^{down}=2{\boldsymbol{K}}_{\Pi }^{up} \end{align}

6.2. Stiffness modeling of the PM

The 3T1R PM 2PPaRR-2PPaRU consists of two PPaRR limbs and two PPaRU limbs. Before establishing the stiffness matrix of the PM, the stiffness matrices of the two different types of limbs should be built first. The stiffness matrix of the PM is equivalent to the sum of the stiffness matrices of all four limbs.

6.2.1. Stiffness modeling of the PPaRU limb

The virtual spring model of the PPaRU limb is presented in Figure 11, where a 1-DOF virtual spring denotes the actuator stiffness, while a 5-DOF and two 6-DOF virtual springs correspond to the stiffness of the parallelogram structure, the $B_{i}C_{i}$ and the $U_{i}(i=1,3)$ links, respectively.

Figure 11.

The virtual spring model of the PPaRU limb.

According to the static equilibrium equation:

(39) \begin{align} \left[\begin{array}{c@{\quad}c} {\boldsymbol{J}}_{\theta }^{i}\left({\boldsymbol{K}}_{Li}^{i}\right)^{-1}\left({\boldsymbol{J}}_{\theta }^{i}\right)^{T} & {\boldsymbol{J}}_{q}^{i}\\[3pt] \left({\boldsymbol{J}}_{q}^{i}\right)^{T} & \boldsymbol{0}_{4\times 4} \end{array}\right]\left[\begin{array}{c} \boldsymbol{f}\\[3pt] \delta \boldsymbol{q}_{i} \end{array}\right]=\left[\begin{array}{c@{\quad}c} \boldsymbol{K}_{Li} & *\\[3pt] * & * \end{array}\right]\left[\begin{array}{c} \boldsymbol{f}\\[3pt] \delta \boldsymbol{q}_{i} \end{array}\right]=\left[\begin{array}{c} \delta \boldsymbol{t}_{Li}\\[3pt] \boldsymbol{0} \end{array}\right] \end{align}

The stiffness matrix of the PPaRU limb is the first $6\times 6$ submatrix extracted from the following matrix:

(40) \begin{align} \boldsymbol{K}_{Li}=\left[\begin{array}{c@{\quad}c} {\boldsymbol{J}}_{\theta }^{i}\left({\boldsymbol{K}}_{Li}^{i}\right)^{-1}\left({\boldsymbol{J}}_{\theta }^{i}\right)^{T} & {\boldsymbol{J}}_{q}^{i}\\[3pt] \left({\boldsymbol{J}}_{q}^{i}\right)^{T} & \boldsymbol{0}_{4\times 4} \end{array}\right]^{-1} \end{align}

where $\boldsymbol{K}_{Li}(i=1,3)$ is the stiffness matrix of the PPaRU limb, $\boldsymbol{f}$ is the force applied at the free end of the limb, $\delta \boldsymbol{q}_{i}$ is the motion of passive joints, ${\boldsymbol{K}}_{Li}^{i}=diag(\mathrm{K}_{ac},\boldsymbol{K}_{Pa},\boldsymbol{K}_{rod1},\boldsymbol{K}_{rod2})$ represents the stiffness in the joint space, ${\boldsymbol{J}}_{q}^{i}=({\boldsymbol{\$}}_{q1}^{i}\ {\boldsymbol{\$}}_{q2}^{i}\ {\boldsymbol{\$}}_{q3}^{i}\ {\boldsymbol{\$}}_{q4}^{i})$ is the Jacobian matrix of the passive joints, ${\boldsymbol{J}}_{\theta }^{i}=(\boldsymbol{\$}_{ac}\ {\boldsymbol{\$}}_{Pa1}^{i}\ {\boldsymbol{\$}}_{Pa2}^{i} \ldots {\boldsymbol{\$}}_{Rod12}^{i})$ is the Jacobian matrix of the links (see Appendix B), $\boldsymbol{r}_{C1}=(x,y-b2+l_{u},z)$ and $\boldsymbol{r}_{C3}=(x,y+b2-l_{u},z)$ are the end points of the U₁ and U₃ joints, respectively.

6.2.2. Stiffness modeling of the PPaRR limb

The virtual spring model of the PPaRR limb is presented in Figure 12, where the 1-DOF virtual spring denotes the stiffness of the actuator, and the 5-DOF and 6-DOF virtual springs represent the stiffness of the Pa joint and the link, respectively.

Figure 12.

The virtual spring model of the PPaRR limb.

Similarly, the stiffness matrix of the PPaRR limb is the first $6\times 6$ submatrix extracted from the following matrix:

(41) \begin{align} \boldsymbol{K}_{Li}=\left[\begin{array}{c@{\quad}c} {\boldsymbol{J}}_{\theta }^{i}\left({\boldsymbol{K}}_{Li}^{i}\right)^{-1}\left({\boldsymbol{J}}_{\theta }^{i}\right)^{T} & {\boldsymbol{J}}_{q}^{i}\\[3pt] \left({\boldsymbol{J}}_{q}^{i}\right)^{T} & \boldsymbol{0}_{3\times 3} \end{array}\right] \end{align}

where $\boldsymbol{K}_{Li}(i=2,4)$ is the stiffness matrix of the PPaRR limb, ${\boldsymbol{K}}_{Li}^{i}=diag(\mathrm{K}_{ac},\boldsymbol{K}_{Pa},\boldsymbol{K}_{rod})$ represents the stiffness matrix in the joint space, ${\boldsymbol{J}}_{q}^{i}=({\boldsymbol{\$}}_{q1}^{i}\ {\boldsymbol{\$}}_{q2}^{i}\ {\boldsymbol{\$}}_{q3}^{i})$ is the Jacobian matrix of the passive joints and ${\boldsymbol{J}}_{\theta }^{i}=(\boldsymbol{\$}_{ac}\ {\boldsymbol{\$}}_{Pa1}^{i}\ {\boldsymbol{\$}}_{Pa2}^{i} \ldots {\boldsymbol{\$}}_{Rod5}^{i}\ {\boldsymbol{\$}}_{Rod6}^{i})$ is the Jacobian matrix of the links (see Appendix B).

The overall stiffness matrix $\boldsymbol{K}$ of the 3T1R PM is

(42) \begin{align} \boldsymbol{K}=\boldsymbol{K}_{L1}+\boldsymbol{K}_{L2}+\boldsymbol{K}_{L3}+\boldsymbol{K}_{L4} \end{align}

6.3. Simulation case

The structural parameters are the same as those in Section 3.3. Table II lists the cross-sectional radius of the links and material properties of the 2PPaRR-2PPaRU PM, where K_ac denotes the stiffness of the actuator, r₁ denotes the link radius, and r_u denotes the radius of the U joint. To verify the correctness of the stiffness model of the PM, an external force $\boldsymbol{f}=(1\mathrm{N}.\mathrm{m};1\mathrm{N}.\mathrm{m};1\mathrm{N}.\mathrm{m};12\mathrm{N};12\mathrm{N};12\mathrm{N})$ is applied to the MP at the given pose $(34.417\text{ mm},8.211\mathrm{mm},838.575\text{ mm},0)$ . The results of the finite element model (FEM) and the analytical results are shown in Table III, and Figure 13 presents the deformation of the PM. The absolute values of the relative errors between the FEM results and the theoretical results in Table III are within 5%, which validates the correctness of the theoretical model.

Table II.

The cross-sectional area of the links and material properties

Table II. Long description

A table listing structural parameters of a mechanical system. The table has six rows and eight columns. Column headers are Kac/N/m, E/MPa, r1/mm, ru/mm, lu/mm, dPa/mm, and Poisson ratio/mu. Row 1: Kac, 5e4; E, 2e5; r1, 12; ru, 11; lu, 28; dPa, 110; Poisson ratio, 0.3.

Table III.

The result comparison of the FEA model and the analytical model of the PM

Table III. Long description

A table comparing analytical solutions, FEA solutions, and relative error for structural parameters. The table has three columns labeled delta x (mm), delta y (mm), and delta z (mm), and four rows labeled Analytical solutions, FEA solutions, and Relative error (percent). Row 1: Analytical solutions, delta x 0.0746, delta y 0.1969, delta z 0.0380. Row 2: FEA solutions, delta x 0.0770, delta y 0.1993, delta z 0.0388. Row 3: Relative error (percent), delta x 3.12, delta y 1.20, delta z 2.06.

Figure 13.

FEA model of the 2PPaRR-2PPaRU PM. (a) Linear deformation along the x-axis, (b) linear deformation along the y-axis, and (c) linear deformation along the z-axis.

6.4. The stiffness performance analysis of the 2PPaRR-2PPaRU

The stiffness performance of the PM is correlated with its configuration. Similar to the kinematic indices based on the Jacobian matrix, the typical index based on the stiffness matrix has unclear physical significance for the unit inconsistency of the rotational stiffness and translational stiffness [Reference Yan, Ong and Nee28]. The virtual stiffness index (VSI) [Reference Yan, Ong and Nee28] reflects the stiffness of a PM in terms of energy, which is used to evaluate the local stiffness of the PM. When a force $\boldsymbol{f}$ is applied to the origin of the MP, the virtual work $W_{V}$ can be written as:

(43) \begin{align} W_{V}\,=\, \boldsymbol{f}^{T}\delta \boldsymbol{t} \end{align}

where $\delta \boldsymbol{t}$ is the virtual displacement vector caused by the force $\boldsymbol{f}$ . Substituting $\delta \boldsymbol{t}=\boldsymbol{K}^{-1}\boldsymbol{f}$ into Eq. (43), the virtual work can be rewritten as

(44) \begin{align} VW=\frac{1}{\boldsymbol{f}^{T}\boldsymbol{K}^{-1}\boldsymbol{f}} \end{align}

Within the linear elastic range, the relationship between virtual displacement and the force $\boldsymbol{f}$ satisfies Hooke’s law. Under a certain external force, a larger $\boldsymbol{f}^{T}\boldsymbol{K}^{-1}\boldsymbol{f}$ corresponds to larger virtual deflections and virtual work, thus indicating poor stiffness performance, and vice versa. Because the stiffness index is closely related to the direction of the wrench applied on the MP [Reference Yan, Ong and Nee28], the stiffness performance distribution of the 2PPaRR-2PPaRU over the reachable workspace under a unit force $\boldsymbol{f}=(\sqrt{6}/6\mathrm{N}.\mathrm{m};\sqrt{6}/6\mathrm{N}.\mathrm{m};\sqrt{6}/6\mathrm{N}.\mathrm{m};\sqrt{6}/6\mathrm{N};\sqrt{6}/6\mathrm{N};\sqrt{6}/6\mathrm{N})$ is presented in Figure 14. It can be seen from Figure 14 that the stiffness performance of the PM with $\phi =45^{\circ }$ and $90^{\circ }$ is better than that of the PM with $\phi =-45^{\circ }$ and $-90^{\circ }$ , demonstrating that the PM exhibits higher stiffness performance when $\phi$ is positive.

Figure 14.

The VSI distribution over the reachable workspace under different orientations. (a) $\phi =-45^{\circ }$ , (b) $\phi =0$ , (c) $\phi =45^{\circ }$ , (d) $\phi =-90^{\circ }$ , and (e) $\phi =90^{\circ }$ .

Figure 14. Long description

Panel A: A heat map showing the VSI distribution over the reachable workspace. The x-axis ranges from -35 to 110 millimeters, and the y-axis ranges from -125 to 125 millimeters. The color scale ranges from 0 to 700, with darker colors indicating lower values and lighter colors indicating higher values. The distribution shows a gradient from left to right, with higher values concentrated on the right side. Panel B: A heat map showing the VSI distribution over the reachable workspace. The x-axis ranges from -35 to 145 millimeters, and the y-axis ranges from -125 to 125 millimeters. The color scale ranges from 0 to 2200, with darker colors indicating lower values and lighter colors indicating higher values. The distribution shows a gradient from left to right, with higher values concentrated on the left side. Panel C: A heat map showing the VSI distribution over the reachable workspace. The x-axis ranges from -35 to 110 millimeters, and the y-axis ranges from -125 to 125 millimeters. The color scale ranges from 1500 to 3200, with darker colors indicating lower values and lighter colors indicating higher values. The distribution shows a central concentration of higher values. Panel D: A heat map showing the VSI distribution over the reachable workspace. The x-axis ranges from -35 to 35 millimeters, and the y-axis ranges from -125 to 125 millimeters. The color scale ranges from 1700 to 2700, with darker colors indicating lower values and lighter colors indicating higher values. The distribution shows a central concentration of higher values. Panel E: A heat map showing the VSI distribution over the reachable workspace. The x-axis ranges from -35 to 35 millimeters, and the y-axis ranges from -125 to 125 millimeters. The color scale ranges from 2500 to 4400, with darker colors indicating lower values and lighter colors indicating higher values. The distribution shows a gradient from left to right, with higher values concentrated on the right side.