When using the Newton-Euler method, the friction of each moving pair is not considered, then the Newton-Euler equation of each member is established. Then the dynamic model of the PM is obtained by eliminating the internal forces between the members. Finally, the relationship between the driving force and the external forces of the moving platform is obtained, which is illustrated as follows.

### 3.1 Dynamic Equation of Moving Platform

As shown in Figure 4, the gravity of the moving platform is *mg*, the constraint force of the sub-moving platform is *F*_{a1}, and the constraint force of the PSS branch chain is *F*_{a2}. The external force and moment of the moving platform are *F*_{w} and *M*_{w} respectively.

The dynamic equation of the moving platform is:

$${\varvec{F}}_{w} + {\varvec{F}}_{a1} + {\varvec{F}}_{a2} + m{\varvec{g}} = m{\varvec{a}}_{p} {,}$$

(22)

$${\varvec{M}}_{w} - \frac{1}{2}{\varvec{c}}_{p} \times {\varvec{F}}_{a1} + \frac{1}{2}{\varvec{c}}_{p} \times {\varvec{F}}_{a2} = {}^{o}{\varvec{I}}_{p} {{\varvec{\varepsilon}}}_{p} + {{\varvec{\omega}}}_{p} \times \left( {{}^{o}{\varvec{I}}_{p} {{\varvec{\omega}}}_{p} } \right)\;{,}$$

(23)

where \({}^{o}{\varvec{I}}_{p} = {}^{o}{\varvec{R}}_{p} {\varvec{I}}_{p} {}^{o}{\varvec{R}}_{p}^{{\text{T}}} {;}\) \({}^{o}{\varvec{R}}_{p}\) is the transformation matrix of the moving coordinate system to the base coordinate system;\({}^{o}{\varvec{I}}_{p}\) is the inertia tensor of the moving platform in the base frame;\({\varvec{I}}_{p}\) is the inertia tensor of the moving platform in the local frame;\({\varvec{c}}_{p}\) is the position vector from the center of mass of the moving platform to the center of the spherical joint on the moving platform.

### 3.2 Dynamic Equation of the Sub-Moving Platform

As shown in Figure 5, the dynamic equation of the sub-moving platform can be written as follows:

$$- {\varvec{F}}_{a1} + m_{l} {\varvec{g}} + {\varvec{F}}_{c11} + {\varvec{F}}_{c12} + {\varvec{F}}_{c2} = m_{l} {\varvec{a}}_{o} .$$

(24)

where *F*_{c11}, *F*_{c12}, *F*_{c2} are constraint forces of the active link *B*_{i}*C*_{i} (*i*=1,2)in the sub-moving platform; *m*_{l} is mass of the sub-moving platform; −*F*_{a1} is the reaction force of the moving platform.

### 3.3 Dynamic Equation of Connecting Rod

The R-R-link is subject to the constraint reaction force of the sub-moving platform −*F*_{ci} (*i*=1, 2), its own gravity *m*_{c}*g*, and the constraint force *F*_{bi} (*i* = 1, 2), and its force analysis is shown in Figure 6.

Therefore, the dynamic equations of the two parallel links in the parallelogram are as follows:

$$- {\varvec{F}}_{c1i} + m_{c} {\varvec{g}} + {\varvec{F}}_{b1i} = m_{c} {\varvec{a}}_{li} {,}\;\left( {i = 1{,}\;{2}} \right),$$

(25)

$$\frac{{l_{c} }}{2}{\varvec{c}}_{i} \times \left( { - {\varvec{F}}_{ci} } \right) + \frac{{l_{c} }}{2}\left( { - {\varvec{c}}_{i} } \right) \times {\varvec{F}}_{bi} = {}^{o}{\varvec{I}}_{li} {{\varvec{\varepsilon}}}_{li} + {{\varvec{\omega}}}_{li} \times \left( {{}^{o}{\varvec{I}}_{li} {{\varvec{\omega}}}_{li} } \right)\;.$$

(26)

Dynamic equation of single link (*B*_{2}*C*_{2}) is as follows:

$$- {\varvec{F}}_{c2} + m_{c} {\varvec{g}} + {\varvec{F}}_{b2} = m_{c} {\varvec{a}}_{l2} {,}$$

(27)

$$\frac{{l_{c} }}{2}{\varvec{c}}_{2} \times \left( { - {\varvec{F}}_{c2} } \right) + \frac{{l_{c} }}{2}\left( { - {\varvec{c}}_{2} } \right) \times {\varvec{F}}_{b2} = {}^{o}{\varvec{I}}_{l2} {{\varvec{\varepsilon}}}_{l2} + {{\varvec{\omega}}}_{l2} \times \left( {{}^{o}{\varvec{I}}_{l2} {{\varvec{\omega}}}_{l2} } \right)\;{,}$$

(28)

where \({}^{o}{\varvec{I}}_{li} = {}^{o}{\varvec{R}}_{li} {\varvec{I}}_{li} {}^{o}{\varvec{R}}_{li}^{{\text{T}}} {,}\;\left( {i = 1{,}\;2} \right)\),\({}^{o}{\varvec{I}}_{li}\) is the inertia tensor of the connecting rod in the base frame ;\({\varvec{I}}_{li}\) is the inertia tensor of the connecting rod in the local frame;\({}^{o}{\varvec{R}}_{li}\) is the transformation matrix from the local frames of the connecting rod to the base frame.

Further, -*S*-*S*-link (*B*_{3}*C*_{3}) is subject to the constraint reaction force of the moving platform−*F*_{a2}, the constraint force of the drive member *F*_{b3}, the self-gravity *m*_{c}′g, while *m*_{c}′ is the mass of *-S-S-* connecting rod, and its stress is shown in Figure 7.

Then, the dynamic equations of the -S-S- link are described as

$$- {\varvec{F}}_{{{\text{a}}2}} + m^{\prime}_{c} {\varvec{g}} + {\varvec{F}}_{b3} = m^{\prime}_{c} {\varvec{a}}_{l3} {,}$$

(29)

$$\frac{{l^{\prime}_{c} }}{2}{\varvec{c}}_{3} \times \left( { - {\varvec{F}}_{a2} } \right) + \frac{{l^{\prime}_{c} }}{2}\left( { - {\varvec{c}}_{3} } \right) \times {\varvec{F}}_{b3} = {}^{o}{\varvec{I}}_{l3} {{\varvec{\varepsilon}}}_{l3} + {{\varvec{\omega}}}_{l3} \times \left( {{}^{o}{\varvec{I}}_{l3} {{\varvec{\omega}}}_{l3} } \right)\;{,}$$

(30)

where \({}^{o}{\varvec{I}}_{l3} = {}^{o}{\varvec{R}}_{l3} {\varvec{I}}_{l3} {}^{o}{\varvec{R}}_{l3}^{{\text{T}}} .\)

### 3.4 Dynamic Equation of Driving Sliders

The three driving sliders are subject to the constraint reaction forces of each link −*F*_{bi} (*i* = 1, 2, 3), its own gravity, and the driving force of the driving motor *m*_{i}*g* (*i* = 1,2, 3), and the force diagram of the slider is shown in Figure 8.

The dynamic equation of slider 1 is

$${\varvec{F}}_{1} - {\varvec{F}}_{b11} - {\varvec{F}}_{b12} + m_{1} {\varvec{g}} = m_{1} {\varvec{a}}_{1} .$$

(31)

The dynamic equations of slider 2 and slide 3 are as follows:

$${\varvec{F}}_{i} - {\varvec{F}}_{bi} + m_{i} {\varvec{g}} = m_{i} {\varvec{a}}_{i} .\;\left( {i = 2{,}\;3} \right)$$

(32)

### 3.5 The Integrated Dynamic Model of the PM

The establishment of the integrated dynamic model is to eliminate the internal forces of members and to obtain the dynamic relationship between the input force, torque and output force.

Taking the dot product of the both sides of Eq. (31) with \({\varvec{e}}_{1}^{{\text{T}}}\)

$$\tau_{1} = {\varvec{e}}_{1}^{{\text{T}}} \cdot {\varvec{F}}_{1} = {\varvec{e}}_{1}^{{\text{T}}} \cdot \left( {{\varvec{F}}_{b11} + {\varvec{F}}_{b12} } \right) +m_{1} {\varvec{a}}_{1} {,}$$

(33)

where \(\tau_{i} \left( {i = 1,2,3} \right)\) is the driving force of the slider, \({\varvec{e}}_{i} \left( {i = 1,2,3} \right)\) is the unit vector for driving force.

Substituting Eq. (25) into Eq. (33), we can write:

$$\tau_{1} = {\varvec{e}}_{1}^{{\text{T}}} \cdot \sum\limits_{i = 1}^{2} {\left( {{\varvec{F}}_{c1i} - m_{c} {\varvec{g}} + m_{c} {\varvec{a}}_{li} } \right)} + m_{1} {\varvec{a}}_{1} .$$

(34)

According to Eq. (26),we can write:

$$l_{c} {\varvec{c}}_{1} \times \left( {{\varvec{F}}_{{c_{11} }} + {\varvec{F}}_{{c_{12} }} } \right) = {\varvec{C}}_{1} {,}$$

(35)

$${\varvec{C}}_{1} = l_{c} {\varvec{c}}_{1} \times \left( {m_{c} {\varvec{g}} - m_{c} {\varvec{a}}_{l1} } \right) - 2{}^{o}{\varvec{I}}_{l1} {{\varvec{\varepsilon}}}_{l1} - 2{{\varvec{\omega}}}_{l1} \times \left( {{}^{o}{\varvec{I}}_{l1} {{\varvec{\omega}}}_{l1} } \right)\;.$$

(36)

Taking the cross product of the two sides of Eq. (35) with *e*_{1} gives:

$$l_{c} {\varvec{e}}_{1} \times {\varvec{c}}_{1} \times \left( {{\varvec{F}}_{c11} + {\varvec{F}}_{c12} } \right) = {\varvec{e}}_{1} \times {\varvec{C}}_{1} .$$

(37)

Then, we can write:

$${\varvec{F}}_{c11} + {\varvec{F}}_{c12} = \frac{{{\varvec{c}}_{1} \left[ {{\varvec{e}}_{1}^{{\text{T}}} \left( {{\varvec{F}}_{c11} + {\varvec{F}}_{c12} } \right)} \right]}}{{{\varvec{e}}_{1}^{{\text{T}}} \cdot {\varvec{c}}_{1} }} - \frac{{{\varvec{e}}_{1} \times {\varvec{C}}_{1} }}{{l_{c} {\varvec{e}}_{1}^{{\text{T}}} {\varvec{c}}_{1} }}.$$

(38)

According to Eq. (34), we can get:

$${\varvec{e}}_{1}^{{\text{T}}} \left( {{\varvec{F}}_{c11} + {\varvec{F}}_{c12} } \right) = \tau_{1} + 2{\varvec{e}}_{1}^{{\text{T}}} \left( {m_{c} {\varvec{g}} - m_{c} {\varvec{a}}_{l1} } \right) - m_{1} {\varvec{a}}_{1} .$$

(39)

Substituting Eq. (39) into Eq. (38) gives:

$${\varvec{F}}_{c11} + {\varvec{F}}_{c12} = \frac{{{\varvec{c}}_{1} \left[ {\tau_{1} + 2{\varvec{e}}_{1}^{{\text{T}}} \left( {m_{c} {\varvec{g}} - m_{c} {\varvec{a}}_{l1} } \right) - m_{1} {\varvec{a}}_{1} } \right]}}{{{\varvec{e}}_{1}^{{\text{T}}} \cdot {\varvec{c}}_{1} }} - \frac{{{\varvec{e}}_{1} \times {\varvec{C}}_{1} }}{{l_{c} {\varvec{e}}_{1}^{{\text{T}}} {\varvec{c}}_{1} }}.$$

(40)

Similarly,

$${\varvec{F}}_{c2} = \frac{{{\varvec{c}}_{2} \left[ {\tau_{2} + {\varvec{e}}_{2}^{{\text{T}}} \left( {m_{c} {\varvec{g}} - m_{c} {\varvec{a}}_{l2} } \right) - m_{2} {\varvec{a}}_{2} } \right]}}{{{\varvec{e}}_{2}^{{\text{T}}} \cdot {\varvec{c}}_{2} }} - \frac{{{\varvec{e}}_{2} \times {\varvec{C}}_{2} }}{{l_{c} {\varvec{e}}_{2}^{{\text{T}}} {\varvec{c}}_{2} }}{,}$$

(41)

$${\varvec{F}}_{a2} = \frac{{{\varvec{c}}_{3} \left[ {\tau_{3} + {\varvec{e}}_{3}^{{\text{T}}} \left( {m^{\prime}_{c} {\varvec{g}} - m^{\prime}_{c} {\varvec{a}}_{l3} } \right) - m_{3} {\varvec{a}}_{3} } \right]}}{{{\varvec{e}}_{3}^{{\text{T}}} \cdot {\varvec{c}}_{3} }} - \frac{{{\varvec{e}}_{3} \times {\varvec{C}}_{3} }}{{l_{c} {\varvec{e}}_{3}^{{\text{T}}} {\varvec{c}}_{3} }}{,}$$

(42)

where

$${\varvec{C}}_{2} = l_{c} {\varvec{c}}_{2} \times \left( {m_{c} {\varvec{g}} - m_{c} {\varvec{a}}_{l2} } \right) - {}^{o}{\varvec{I}}_{l2} {{\varvec{\varepsilon}}}_{l2} - {{\varvec{\omega}}}_{l2} \times \left( {{}^{o}{\varvec{I}}_{l2} {{\varvec{\omega}}}_{l2} } \right){,}$$

$${\varvec{C}}_{3} = l^{\prime}_{c} {\varvec{c}}_{3} \times \left( {m^{\prime}_{c}{\varvec{g}} - m^{\prime}_{c} {\varvec{a}}_{l3} } \right) - {}^{o}{\varvec{I}}_{l3} {{\varvec{\varepsilon}}}_{l3} - {{\varvec{\omega}}}_{l3} \times \left( {{}^{o}{\varvec{I}}_{l3} {{\varvec{\omega}}}_{l3} } \right)\;.$$

Substituting Eqs. (40) and (41) into Eq. (24), we can get

$${\varvec{F}}_{a1} = m_{l} {\varvec{g}} + {\varvec{F}}_{c11} + {\varvec{F}}_{c12} + {\varvec{F}}_{c2} - m_{l} {\varvec{a}}_{o} .$$

(43)

Substituting Eqs. (42) and (43) into Eq. (22) and Eq. (23), we have

$$\left[ {\begin{array}{*{20}c} {\varvec{D}} \\ {\varvec{E}} \\ \end{array} } \right]_{6 \times 1} = {\varvec{J}}_{\tau } \cdot {{\varvec{\tau}}} + \left[ {\begin{array}{*{20}c} {{\varvec{F}}_{w} } \\ {{\varvec{M}}_{w} } \\ \end{array} } \right]_{6 \times 1} {,}$$

(44)

where

$$\begin{gathered} {\varvec{D}} = - \frac{{{\varvec{c}}_{1} \left[ {2{\varvec{e}}_{1}^{{\text{T}}} \left( {m_{c} {\varvec{g}} - m_{c} {\varvec{a}}_{l1} } \right) - m_{1} {\varvec{a}}_{1} } \right]}}{{{\varvec{e}}_{1}^{{\text{T}}} {\varvec{c}}_{1} }} - \frac{{{\varvec{c}}_{2} \left[ {{\varvec{e}}_{2}^{{\text{T}}} \left( {m_{c} {\varvec{g}} - m_{c} {\varvec{a}}_{l2} } \right) - m_{2} {\varvec{a}}_{2} } \right]}}{{{\varvec{e}}_{2}^{{\text{T}}} {\varvec{c}}_{2} }} \hfill \\ \qquad + \sum\limits_{i = 1}^{2} {\frac{{{\varvec{e}}_{i} \times {\varvec{C}}_{i} }}{{l_{c} {\varvec{e}}_{i}^{{\text{T}}} {\varvec{c}}_{i} }}} - \frac{{{\varvec{c}}_{3} \left[ {{\varvec{e}}_{3}^{{\text{T}}} \left( {m^{\prime}_{c} {\varvec{g}} - m^{\prime}_{c} {\varvec{a}}_{l3} } \right) - m_{3} {\varvec{a}}_{3} } \right]}}{{{\varvec{e}}_{3}^{{\text{T}}} {\varvec{c}}_{3} }} + \frac{{{\varvec{e}}_{3} \times {\varvec{C}}_{3} }}{{l_{c} ^{\prime}{\varvec{e}}_{3}^{{\text{T}}} {\varvec{c}}_{3} }} - m_{l} \left( {{\varvec{g}} - {\varvec{a}}_{o} } \right) \hfill \\ \qquad - m\left( {{\varvec{g}} - {\varvec{a}}_{p} } \right)\;{,} \hfill \\ {\varvec{E}} = \frac{1}{2}{\varvec{c}}_{p} \times \left[ {\frac{{{\varvec{c}}_{1} \left[ {2{\varvec{e}}_{1}^{{\text{T}}} \left( {m_{c} {\varvec{g}} - m_{c} {\varvec{a}}_{l1} } \right) - m_{1} {\varvec{a}}_{1} } \right]}}{{{\varvec{e}}_{1}^{{\text{T}}} {\varvec{c}}_{1} }} + \frac{{{\varvec{c}}_{2} \left[ {{\varvec{e}}_{{2}}^{{\text{T}}} \left( {m_{c} {\varvec{g}} - m_{c} {\varvec{a}}_{l2} } \right) - m_{2} {\varvec{a}}_{2} } \right]}}{{{\varvec{e}}_{2}^{{\text{T}}} {\varvec{c}}_{2} }} - } \right. \hfill \\ \qquad \left. {\sum\limits_{i = 1}^{2} {\frac{{{\varvec{e}}_{i} \times {\varvec{C}}_{i} }}{{l_{c} {\varvec{e}}_{i}^{{\text{T}}} {\varvec{c}}_{i} }} + m_{l} {(}{\varvec{g}} - {\varvec{a}}_{o} {)}} } \right] - \frac{1}{2}{\varvec{c}}_{p} \times \left[ {\frac{{{\varvec{c}}_{3} \left[ {{\varvec{e}}_{3}^{{\text{T}}} \left( {m^{\prime}_{c} {\varvec{g}} - m^{\prime}_{c} {\varvec{a}}_{l3} } \right) - m_{3} {\varvec{a}}_{3} } \right]}}{{{\varvec{e}}_{3}^{{\text{T}}} {\varvec{c}}_{3} }} - \frac{{{\varvec{e}}_{3} \times {\varvec{C}}_{3} }}{{l^{\prime}_{c} {\varvec{e}}_{3}^{{\text{T}}} {\varvec{c}}_{3} }}} \right] - \hfill \\ \qquad {}^{o}{\varvec{I}}_{p} {{\varvec{\varepsilon}}}_{p} - {{\varvec{\omega}}}_{p} \times \left( {{}^{o}{\varvec{I}}_{p} {{\varvec{\omega}}}_{p} } \right)\;{,} \hfill \\ {\varvec{J}}_{\tau } = \left[ {\begin{array}{*{20}c} {\frac{{{\varvec{c}}_{1} }}{{{\varvec{e}}_{1}^{{\text{T}}} {\varvec{c}}_{1} }}} & {\frac{{{\varvec{c}}_{2} }}{{{\varvec{e}}_{2}^{{\text{T}}} {\varvec{c}}_{2} }}} & {\frac{{{\varvec{c}}_{3} }}{{{\varvec{e}}_{3}^{{\text{T}}} {\varvec{c}}_{3} }}} \\ { - \frac{{{\varvec{c}}_{p} \times {\varvec{c}}_{1} }}{{2{\varvec{e}}_{1}^{{\text{T}}} {\varvec{c}}_{1} }}} & { - \frac{{{\varvec{c}}_{p} \times {\varvec{c}}_{2} }}{{2{\varvec{e}}_{2}^{{\text{T}}} {\varvec{c}}_{2} }}} & {\frac{{{\varvec{c}}_{p} \times {\varvec{c}}_{3} }}{{2{\varvec{e}}_{3}^{{\text{T}}} {\varvec{c}}_{3} }}} \\ \end{array} } \right]_{6 \times 3} {,} \hfill \\ \end{gathered}$$

$${{\varvec{\tau}}} = \left[ {\begin{array}{*{20}c} {\tau_{1} } & {\tau_{2} } & {\tau_{3} } \\ \end{array} } \right]_{3 \times 1}^{{\text{T}}} .$$

According to Eq. (44), we can get:

$${{\varvec{\tau}}} = {\varvec{J}}_{\tau }^{ - 1} \cdot \left[ {\begin{array}{*{20}c} {\varvec{D}} \\ {\varvec{E}} \\ \end{array} } \right] - {\varvec{J}}_{\tau }^{ - 1} \cdot \left[ {\begin{array}{*{20}c} {{\varvec{F}}_{w} } \\ {{\varvec{M}}_{w} } \\ \end{array} } \right]\,.$$

(45)

When the motion law of the moving platform and the external force and torque are known, the driving force of each driving pair can be obtained from Eq. (45).