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

From: A \({{\varvec{\Phi}}}\) 6-m Tunnel Boring Machine Steel Arch Splicing Manipulator

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