Table 17 GF expressions of kinematic pairs used in this paper

Prismatic pair

P

1

\({\text{G}}_{\text{F7}}^{\text{I}} \left( {{\text{T}}_{\text{a}} , 0 , 0 , 0 , 0 , 0} \right)\)

Revolute pair

R

1

\({\text{G}}_{\text{F21}}^{\text{II}} \left( {{\text{R}}_{{{\upalpha }}} , 0 , 0 , 0 , 0 , 0} \right)\)

Cylinder pair

C

2

\({\text{G}}_{\text{F20}}^{\text{II}} \left( {{\text{T}}_{\text{a}} , {\text{R}}_{{{\upalpha }}} , 0 , 0 , 0 , 0} \right)\)

Spherical pair

S

3

\({\text{G}}_{\text{F12}}^{\text{II}} \left( {{\text{R}}_{{{\upalpha }}} , {\text{R}}_{{{\upbeta }}} , {\text{R}}_{{{\upgamma }}} , 0 , 0 , 0} \right)\)

Universal pair

U

2

\({\text{G}}_{\text{F18}}^{\text{II}} \left( {{\text{R}}_{{{\upalpha }}} , {\text{R}}_{{{\upbeta }}} , 0 , 0 , 0 , 0} \right)\)

T-R universal pair

U^

2

\({\text{G}}_{\text{F20}}^{\text{II}} \left( {{\text{T}}_{\text{a}} , {\text{R}}_{{{\upalpha }}} , 0 , 0 , 0 , 0} \right)\)

R-T universal pair

^U

2

\({\text{G}}_{\text{F19}}^{\text{II}} \left( {{\text{R}}_{{{\upalpha }}} , {\text{T}}_{\text{a}} , 0 , 0 , 0 , 0} \right)\)

Parallelogram pair

P^R

1

\({\text{G}}_{\text{F7}}^{\text{I}} \left( {{\text{T}}_{\text{a}} , 0 , 0 , 0 , 0 , 0} \right)\)