A $${{\varvec{\Phi}}}$$ 6-m Tunnel Boring Machine Steel Arch Splicing Manipulator

Chinese Journal of Mechanical Engineering

Table 4 Motion/force transmission performance of the closed-loop structure of the grasping module

Type of grasping module	Expression of $\user2{\$ }_{{\text{T}}}$	Variable range	$\chi {\text{ = min}}\left\{ {\varsigma ,\sigma } \right\}$	${\varvec{\varGamma}}$
PRRR_P	$\user2{\$ }_{{\text{T}}} = \left( {\begin{array}{*{20}c} {x_{2} - x_{1} } & {y_{2} } & {0;} & 0 & 0 & {x_{1} y_{2} } \\ \end{array} } \right)$	$\theta \in \left[ {45^\circ ,90^\circ } \right]$	${\text{min}}\left\{ {\frac{2\sqrt 3 }{3}\left\| {\cos \left( {\gamma + \theta } \right)} \right\|,\sin \gamma } \right\}$	0.8803
PRPR_P	$\user2{\$ }_{{\text{T}}} = \left( {\begin{array}{*{20}c} { - \sin \gamma } & {\cos \gamma } & {0;} & 0 & 0 & {\frac{{y_{1} }}{\sin \gamma }} \\ \end{array} } \right)$	$\gamma \in \left[ {45^\circ ,90^\circ } \right]$	${\text{min}}\left\{ {\sin \gamma ,\frac{1}{\sqrt 2 \sin \gamma }} \right\}$	0.7563
PPP_P	$\user2{\$ }_{{\text{T}}} = \left( {\begin{array}{*{20}c} { - y_{1} } & {x_{1} } & {0;} & 0 & 0 & 0 \\ \end{array} } \right)$	$x_{1} \in \left[ {0,l_{2o} \sin \gamma } \right]$	${\text{min}}\left\{ {1,\frac{{x_{1} }}{{l_{2o} \sin \gamma }}} \right\}$	0.5000
PRRP_P	$\user2{\$ }_{{\text{T}}} = \left( {\begin{array}{*{20}c} { - x_{1} } & {y_{2} } & {0;} & 0 & 0 & {x_{1} y_{2} } \\ \end{array} } \right)$	$\gamma \in \left[ {45^\circ ,90^\circ } \right]$	${\text{min}}\left\{ {\sqrt 2 \cos \gamma ,\sin \gamma } \right\}$	0.4957

Type of grasping module	Expression of \(\user2{\$ }_{{\text{T}}}\)	Variable range	\(\chi {\text{ = min}}\left\{ {\varsigma ,\sigma } \right\}\)	\({\varvec{\varGamma}}\)
PRRR_P	\(\user2{\$ }_{{\text{T}}} = \left( {\begin{array}{*{20}c} {x_{2} - x_{1} } & {y_{2} } & {0;} & 0 & 0 & {x_{1} y_{2} } \\ \end{array} } \right)\)	\(\theta \in \left[ {45^\circ ,90^\circ } \right]\)	\({\text{min}}\left\{ {\frac{2\sqrt 3 }{3}\left\| {\cos \left( {\gamma + \theta } \right)} \right\|,\sin \gamma } \right\}\)	0.8803
PRPR_P	\(\user2{\$ }_{{\text{T}}} = \left( {\begin{array}{*{20}c} { - \sin \gamma } & {\cos \gamma } & {0;} & 0 & 0 & {\frac{{y_{1} }}{\sin \gamma }} \\ \end{array} } \right)\)	\(\gamma \in \left[ {45^\circ ,90^\circ } \right]\)	\({\text{min}}\left\{ {\sin \gamma ,\frac{1}{\sqrt 2 \sin \gamma }} \right\}\)	0.7563
PPP_P	\(\user2{\$ }_{{\text{T}}} = \left( {\begin{array}{*{20}c} { - y_{1} } & {x_{1} } & {0;} & 0 & 0 & 0 \\ \end{array} } \right)\)	\(x_{1} \in \left[ {0,l_{2o} \sin \gamma } \right]\)	\({\text{min}}\left\{ {1,\frac{{x_{1} }}{{l_{2o} \sin \gamma }}} \right\}\)	0.5000
PRRP_P	\(\user2{\$ }_{{\text{T}}} = \left( {\begin{array}{*{20}c} { - x_{1} } & {y_{2} } & {0;} & 0 & 0 & {x_{1} y_{2} } \\ \end{array} } \right)\)	\(\gamma \in \left[ {45^\circ ,90^\circ } \right]\)	\({\text{min}}\left\{ {\sqrt 2 \cos \gamma ,\sin \gamma } \right\}\)	0.4957