Kinematics/dynamics analysis of novel 3UPUR ${+}$ SP-type hybrid hand with three flexible fingers

Abstract

A novel 3UPUR $+$ SP-type hybrid hand with three flexible fingers is designed. Its prototype is built up, and its characteristics, DoF and constrained wrench are analyzed. Its kinematic formulae for solving DoF, the displacement, velocity, acceleration of the moving links are derived and analyzed. Its dynamic formulae for solving the dynamic active forces and the dynamic constrained wrench are derived. Finally, an analytic example is given to demonstrate the solution of the kinematics and dynamics, and the analytical solutions are verified by simulation model. This paper is aimed at laying a solid theoretical and technical foundation of development of the novel hybrid hand with three flexible fingers and similar hybrid hands with multi-flexible fingers.

Appendix

1. :

Derivation of ${{{\varvec{D}}}}'_{ri1}$

$$\begin{aligned} {{\varvec{D}}}'_{ri1}= & {} [{\varvec{\delta }}_{ri} \times ({\varvec{\delta }}_{ri} \times {\varvec{R}}_{i1} ){]}'\nonumber \\= & {} {{\varvec{\delta }} }'_{ri} \times ({\varvec{\delta }}_{ri} \times {\varvec{R}}_{i1} )+{\varvec{\delta }}_{ri} \times ({{\varvec{\delta }} }'_{ri} \times {\varvec{R}}_{i1} )+{\varvec{\delta }}_{ri} \nonumber \\&\times ({\varvec{\delta }} _{ri} \times {{\varvec{R}}}'_{i1} ) \nonumber \\= & {} {{\varvec{\delta }} }'_{ri} \times ({\varvec{\delta }}_{ri} \times {\varvec{R}}_{i1} )+{\varvec{\delta }}_{ri} \times ({{\varvec{\delta }} }'_{ri} \times {\varvec{R}}_{i1} ) \nonumber \\= & {} ({{\varvec{\theta }} }'_{ri} \times {\varvec{\delta }}_{ri} )\times ({\varvec{\delta }}_{ri} \times {\varvec{R}}_{i1} )+{\varvec{\delta }}_{ri} \nonumber \\&\times [({{\varvec{\theta }} }'_{ri} \times {\varvec{\delta }}_{ri} )\times {\varvec{R}}_{i1} ] \nonumber \\= & {} [({{\varvec{\theta }} }'_{ri} \times {\varvec{\delta }}_{ri} )\cdot {\varvec{R}}_{i1} ]{\varvec{\delta }}_{ri} -[({{\varvec{\theta }} }'_{ri} \times {\varvec{\delta }}_{ri} )\cdot {\varvec{\delta }}_{ri} ]{\varvec{R}}_{i1} \nonumber \\&+\,({\varvec{\delta }}_{ri} \cdot {\varvec{R}}_{i1} )({{\varvec{\theta }} }'_{ri} \times {\varvec{\delta }}_{ri} )-[{\varvec{\delta }}_{ri} \cdot ({{\varvec{\theta }} }'_{ri} \times {\varvec{\delta }}_{ri} )]{\varvec{R}}_{i1} \nonumber \\= & {} {\varvec{\delta }}_{ri} \left[ {\varvec{R}}_{i1}^T ({{\varvec{\theta }} }'_{ri} \times {\varvec{\delta }} _{ri} )\right] +({{\varvec{\theta }} }'_{ri} \times {\varvec{\delta }}_{ri} )({\varvec{\delta }}_{ri} \cdot {\varvec{R}}_{i1} ) \nonumber \\= & {} -{\varvec{\delta }}_{ri} {\varvec{R}}_{i1}^T \hat{{{\varvec{\delta }} }}_{ri} {{\varvec{\theta }} }'_{ri} -\,\left( {\varvec{\delta }}_{ri}^T {\varvec{R}}_{i1} \right) \hat{{{\varvec{\delta }} }}_{ri} {{\varvec{\theta }} }'_{ri} . \end{aligned}$$

(a1)

2. :

Derivation of ${{\varvec{D}}}_i^T {{{{\varvec{\theta }}}}}''_{ri}$

$$\begin{aligned} {\varvec{D}}_i^T {{\varvec{\theta }} }''_{ri}= & {} {\varvec{D}}_i^T \left[ \frac{{\varvec{D}}_{ri} }{d_{ri} }({{\varvec{o}}}'-\hat{{{\varvec{e}}}}_i {{\varvec{\theta }} }')\right] '\nonumber \\= & {} {\varvec{D}}_i^T \left( \frac{{\varvec{D}}_{ri} }{d_{ri} }\right) '({{\varvec{o}}}'-\hat{{{\varvec{e}}}}_i {{\varvec{\theta }} }')\nonumber \\&+\,\frac{{\varvec{D}}_i^T {\varvec{D}}_{ri} }{d_{ri} }({{\varvec{o}}}'-\hat{{{\varvec{e}}}}_i {{\varvec{\theta }} }'{)}'\nonumber \\= & {} {\varvec{D}}_i^T \frac{{{\varvec{D}}}'_{ri} d_i -{d}'_i {\varvec{D}}_{ri} }{d_{ri}^2 }({{\varvec{o}}}'-\hat{{{\varvec{e}}}}_i {{\varvec{\theta }} }')\nonumber \\&+\,{\varvec{D}}_i^T {\varvec{D}}_{ri} [{{\varvec{o}}}''-{\hat{\varvec{e}}}_i {{\varvec{\theta }} }'' -({{\varvec{\theta }} }'\times {\varvec{e}}_i )\times {{\varvec{\theta }} }']/d_{ri}\nonumber \\= & {} \left\{ {\varvec{D}}_i^T {{\varvec{D}}}'_{ri} ({{\varvec{o}}}' -{\hat{\varvec{e}}}_i {{\varvec{\theta }} }')-{d}'_i {\varvec{D}}_i^T {{\varvec{\theta }} }'_{ri}\right. \nonumber \\&\left. +\,{\varvec{D}}_i^T {\varvec{D}}_{ri} [{{\varvec{o}}}''-{\hat{\varvec{e}}}_i {{\varvec{\theta }} }''-({{\varvec{\theta }} }'\times {\varvec{e}}_i )\times {{\varvec{\theta }} }']\right\} \big /d_{ri}\nonumber \\= & {} \left\{ {\varvec{D}}_i^T {{\varvec{D}}}'_{ri} ({{\varvec{o}}}'-{\hat{\varvec{e}}}_i {{\varvec{\theta }} }')\right. \nonumber \\&-\,{\varvec{\omega }}_{ri}^T {\varvec{D}}_i [({\varvec{R}}_{i1} \times {\varvec{R}}_{i2} )^{T}({{\varvec{o}}}'+{{\varvec{\theta }} }'\times {\varvec{e}}_i )]\nonumber \\&+\,{\varvec{D}}_i^T {\varvec{D}}_{ri} \left( {{\begin{array}{cc} {\varvec{E}}&{}\quad {-{\hat{{{\varvec{e}}}}}_i } \\ \end{array} }}\right) {{\varvec{A}}}\nonumber \\&\left. +\,[({\varvec{D}}_i^T {\varvec{D}}_{ri} )\times {{\varvec{\theta }} }']\cdot ({{\varvec{\theta }} }'\times {\varvec{e}}_i )\right\} \big /d_{ri}. \end{aligned}$$

(a2)

3. :

Derivation of ${{{\theta }}}''_i$

$$\begin{aligned} {{{\theta }} }''_i= & {} \left[ \frac{{\varvec{D}}_i^T \big ({{\varvec{\theta }} }'-{{\varvec{\theta }} }'_{ri} \big )}{d_i }\right] ' \nonumber \\= & {} \big \{\big [{\varvec{D}}_i^T \big ({{\varvec{\theta }} }'-{{\varvec{\theta }} }'_{ri} \big ){\big ]}'d_i -{d}'_i {\varvec{D}}_i^T \big ({{\varvec{\theta }} }'-{{\varvec{\theta }} }'_{ri} \big )\big \}/d_i ^{2} \nonumber \\= & {} \big [\big ({\varvec{D}}_i^T\big )'\big ({{\varvec{\theta }} }'-{\varvec{J}}_{{{\theta }} ri} {\varvec{V}}\big )+{\varvec{D}}_i^T \big ({{\varvec{\theta }} }''-{{\varvec{\theta }} }''_{ri} \big ) \nonumber \\&-\,\big ({{\varvec{D}}}'_i \cdot {\varvec{R}}_i +{\varvec{D}}_i \cdot {{\varvec{R}}}'_i \big ){{{\theta }} }'_i \big ]/d_i \nonumber \\= & {} \big \{\big ({\varvec{J}}_{{ Di}} {\varvec{V}}\big )^{T}\big ({{\varvec{\theta }} }'-{\varvec{J}}_{{{\theta }} ri} {\varvec{V}}\big )+{\varvec{D}}_i^T {{\varvec{\theta }} }''-{\varvec{D}}_i^T {{\varvec{\theta }} }''_{ri} \nonumber \\&-\,\big [\big ({\varvec{J}}_{{ Di}} {\varvec{V}}\big )^{T}{\varvec{R}}_i +\big (-\hat{{{\varvec{R}}}}_i {{\varvec{\theta }} }'\big )^{T}{\varvec{D}}_i \big ]{{{\theta }} }'_i \big \}/d_i \nonumber \\= & {} \big \{{\varvec{D}}_i^T \left( {{\begin{array}{cc} {\varvec{0}}&{}\quad {\varvec{E}} \\ \end{array} }} \right) {{\varvec{A}}}-{\varvec{D}}_i^T {\varvec{J}}_{{{\theta }} ri} {{\varvec{A}}}+{\varvec{V}}^{T}{\varvec{J}}_{{ D i}}^T \big ({{\varvec{\theta }} }'-{\varvec{J}}_{{{\theta }} ri} {\varvec{V}}\big ) \nonumber \\&+\,{\varvec{V}}^{T}\frac{{\varvec{D}}_i^T {\varvec{R}}_{i1} {\varvec{J}}_{{{ Ri}}2}^T +{\varvec{J}}_{{{ Ri}}2}^T {\varvec{D}}_i {\varvec{D}}_{ri1}^T +{\varvec{D}}_i^T {\varvec{R}}_{i2} {\varvec{J}}_{{{ Dr}}i1}^T }{d_{ri} }\nonumber \\&\left( {{\begin{array}{cc} {\varvec{E}}&{} {-{\hat{\varvec{e}}}_i } \\ \end{array} }} \right) {\varvec{V}} \nonumber \\&+\,\frac{{\varvec{V}}^{T}\big ({\varvec{J}}_{{{\theta }} ri}^T \big )_{3\times 6} {\varvec{D}}_i \big ({\varvec{R}}_{i1} \times {\varvec{R}}_{i2} \big )^{T}}{d_{ri} }\left( {{\begin{array}{cc} {{\varvec{E}}_{3\times 3} }&{} {-{\hat{\varvec{e}}}_i } \\ \end{array} }} \right) {\varvec{V}} \nonumber \\&-\,\frac{1}{d_{ri} }{\varvec{V}}^{T}\left( {{\begin{array}{cc} {{{\varvec{0}}}_{{3}\times {3}} }&{} {{{\varvec{0}}}_{{3}\times {3}} } \\ {{{\varvec{0}}}_{{3}\times {3}} }&{} {{\hat{\varvec{e}}}_i s\big ({\varvec{D}}_{ri}^T {\varvec{D}}_i \big )} \\ \end{array} }} \right) {\varvec{V}} \nonumber \\&-\,\big ({\varvec{V}}^{T}{\varvec{J}}_{{{ Di}}}^T {\varvec{R}}_i +\big (-\hat{{{\varvec{R}}}}_i \left( {{\begin{array}{cc} {{{\varvec{0}}}_{3\times 3} }&{}\quad {{\varvec{E}}_{3\times 3} } \\ \end{array} }} \right) {\varvec{V}}\big )^{T}{\varvec{D}}_i \big ){{\varvec{\theta }} }'_i \big \}/d_i \nonumber \\ {{{\theta }} }''_i= & {} \big \{\big [{\varvec{D}}_i^T \left( {{\begin{array}{cc} {{{\varvec{0}}}_{3\times 3} }&{}\quad {{\varvec{E}}_{3\times 3} } \\ \end{array} }} \right) -{\varvec{D}}_i^T \big ({\varvec{J}}_{{{\theta }} ri} \big )_{3\times 6} \big ]{{\varvec{A}}} \nonumber \\&+\,{\varvec{V}}^{T}{\varvec{J}}_{{{ Di}}}^T \big [\left( {{\begin{array}{cc} {{{\varvec{0}}}_{3\times 3} }&{}\quad {{\varvec{E}}_{3\times 3} } \nonumber \\ \end{array} }} \right) -{\varvec{J}}_{{{\theta }} ri} \big ]{\varvec{V}} \nonumber \\&+\,{\varvec{V}}^{T}\frac{1}{d_{ri} }\big [\big ({\varvec{D}}_i^T {\varvec{R}}_{i1} \big )\big ({\varvec{J}}_{{{ Ri}}2}^T \big )_{3\times 6} +\big ({\varvec{J}}_{{{ Ri}}2}^T \big )_{3\times 6} {\varvec{D}}_i {\varvec{D}}_{ri1}^T \nonumber \\&+\,\big ({\varvec{D}}_i^T {\varvec{R}}_{i2} \big )\big ({\varvec{J}}_{{{ Dri}}1}^T \big )_{3\times 6} \big ]\left( {{\begin{array}{cc} {{\varvec{E}}_{3\times 3} }&{} {-{\hat{\varvec{e}}}_i } \\ \end{array} }} \right) {\varvec{V}} \nonumber \\&+\,{\varvec{V}}^{T}\frac{\big ({\varvec{J}}_{{{\theta }} ri}^T \big )_{3\times 6} {\varvec{D}}_i \big ({\varvec{R}}_{i1} \times {\varvec{R}}_{i2} \big )^{T}}{d_{ri} }\left( {{\begin{array}{cc} {{\varvec{E}}_{3\times 3} }&{} {-{\hat{\varvec{e}}}_i } \\ \end{array} }} \right) {\varvec{V}} \nonumber \\&-\,\frac{1}{d_{ri} }{\varvec{V}}^{T}\left( {{\begin{array}{cc} {{{\varvec{0}}}_{{3}\times {3}} }&{}\quad {{{\varvec{0}}}_{{3}\times {3}} } \\ {{{\varvec{0}}}_{{3}\times {3}} }&{}\quad {{\hat{\varvec{e}}}_i s\big ({\varvec{D}}_{ri}^T {\varvec{D}}_i \big )} \\ \end{array} }} \right) {\varvec{V}} \nonumber \\&-\,{\varvec{V}}^{T}\big [{\varvec{J}}_{{{ Di}}}^T {\varvec{R}}_i +\left( {{\begin{array}{cc} {\varvec{0}}&{}\quad {\varvec{E}} \\ \end{array} }} \right) ^{T}\hat{{{\varvec{R}}}}_i {\varvec{D}}_i \big ]{\varvec{J}}_{{{\theta }} i} {\varvec{V}}\big \}\,/d_i . \end{aligned}$$

(a3)

4. :

Derivation of ${{\varvec{W}}}_{{i}}^{\prime \prime }$

$$\begin{aligned}&{\varvec{R}}_i \times \big [{\varvec{R}}_i \times \big ({\varvec{e}}_{Wi} -{\varvec{e}}_i \big )\big ]\nonumber \\&\quad =\big [{\varvec{R}}_i \cdot \big ({\varvec{e}}_{Wi} -{\varvec{e}}_i \big )\big ]-\big ({\varvec{R}}_i \cdot {\varvec{R}}_i \big )\big ({\varvec{e}}_{Wi} -{\varvec{e}}_i \big )\nonumber \\&\quad ={\varvec{e}}_i -{\varvec{e}}_{Wi} ,\nonumber \\&{{\varvec{e}}}'_{Wi} ={{\varvec{W}}}'_i -{{\varvec{o}}}'\nonumber \\&\quad ={{\varvec{\theta }}}'\times {{\varvec{e}}}_{Wi} -{{{\theta }} }'_i {\varvec{R}}_i \times \big ( {\varvec{e}}_{Wi}-{\varvec{e}}_i \big ), \nonumber \\&{\varvec{W}}_i ^{\prime \prime }=\big [{{\varvec{o}}}'+{{\varvec{\theta }} }'\times {\varvec{e}}_{Wi} -{{{\theta }} }'_i {\varvec{R}}_i \times \big ({\varvec{e}}_{Wi} -{\varvec{e}}_i \big )\big ]'\nonumber \\&\quad ={{\varvec{o}}}''+{{\varvec{\theta }} }''\times {\varvec{e}}_{Wi} +{{\varvec{\theta }} }'\times {{\varvec{e}}}'_{Wi} -{{\varvec{\theta }} }''_i {\varvec{R}}_i \times \big ({\varvec{e}}_{Wi} -{\varvec{e}}_i \big )\nonumber \\&\quad \quad -\,{{{\theta }} }'_i {{\varvec{R}}}'_i \times \big ({\varvec{e}}_{Wi} -{\varvec{e}}_i \big )-{{{\theta }} }'_i {\varvec{R}}_i \times {{{\varvec{l}}}}'_{Wi}\nonumber \\&\quad ={{\varvec{o}}}''+{{\varvec{\theta }} }''\times {\varvec{e}}_{Wi} -{{\varvec{\theta }} }''_i {\varvec{R}}_i \times \big ({\varvec{e}}_{Wi} -{\varvec{e}}_i \big )\nonumber \\&\quad \quad +\,{{\varvec{\theta }} }'\times \big [{{\varvec{\theta }} }'\times {\varvec{e}}_{Wi} -\omega _i {\varvec{R}}_i \times \big ({\varvec{e}}_{Wi} -{\varvec{e}}_i \big )\big ]\nonumber \\&\quad \quad -\,\omega _i \big ({{\varvec{\theta }} }'\times {\varvec{R}}_i \big ) \times \big ({\varvec{e}}_{Wi} -{\varvec{e}}_i \big )\nonumber \\&\quad \quad -\,\omega _i {\varvec{R}}_i \times \big [\big ({{\varvec{\theta }} }' -{\varvec{R}}_i {{{\theta }} }'_i \big )\times \big ({\varvec{e}}_{Wi} -{\varvec{e}}_i \big )\big ]\nonumber \\&\quad ={{\varvec{o}}}''+{{\varvec{\theta }} }''\times {\varvec{e}}_{Wi} +{{\varvec{\theta }} }'\times \big ({{\varvec{\theta }} }'\times {\varvec{e}}_{Wi} \big )\nonumber \\&\quad \quad -\,{\theta }'_i {\varvec{\omega }} \times \big [{\varvec{R}}_i \times \big ({\varvec{e}}_{Wi} -{\varvec{e}}_i \big )\big ]-{\theta }''_i {\varvec{R}}_i \times \big ({\varvec{e}}_{Wi} -{\varvec{e}}_i \big )\nonumber \\&\quad \quad -\,{{{\theta }} }'_i \big ({{\varvec{\theta }} }'\times {\varvec{R}}_i \big ) \times \big ({\varvec{e}}_{Wi} -{\varvec{e}}_i \big )-{{{\theta }} }'_i {\varvec{R}}_i \times \big [{{\varvec{\theta }} }'\nonumber \\&\quad \quad \times \, \big ({\varvec{e}}_{Wi} -{\varvec{e}}_i \big )\big ]\nonumber \\&\quad \quad +\,{{{\theta }} }'_i {\varvec{R}}_i \times \big [{\varvec{R}}_i {{{\theta }} }'_i \times \big ({\varvec{e}}_{Wi} -{\varvec{e}}_i \big )\big ]\nonumber \\&\quad ={{\varvec{o}}}''-\hat{{{{\varvec{e}}}}}_{Wi} {{\varvec{\theta }} }''-{{{\theta }} }''_i {\varvec{R}}_i \times \big ({\varvec{e}}_{Wi} -{\varvec{e}}_i \big )+{{\varvec{\theta }} }'\nonumber \\&\quad \quad \times \, \big ({{\varvec{\theta }} }'\times {\varvec{e}}_{Wi} \big )\nonumber \\&\quad \quad -\,{{{\theta }} }'_i {{{\theta }} }'_i \big ({{\varvec{e}}}_{Wi} -{{\varvec{e}}}_i \big )-2{{{\theta }} }'_i {{\varvec{\theta }} }'\times \big [{\varvec{R}}_i \times \big ({\varvec{e}}_{Wi} -{\varvec{e}}_i \big )\big ].\nonumber \\ \end{aligned}$$

(a4)

5. :

Derivation of $l_i^{\prime \prime }$

Some items are represented as follows:

$$\begin{aligned}&d_{ri} ={\varvec{r}}_i \cdot \big ({\varvec{R}}_{i1} \times {\varvec{R}}_{i2} \big ),\quad {\varvec{D}}_i ={\varvec{R}}_{i3} \times {\varvec{R}}_{i4} ,\quad \nonumber \\&d_i ={\varvec{D}}_i \cdot {\varvec{R}}_i ,\quad d_{li} ={{{\varvec{l}}}}_i \cdot \big ({\varvec{R}}_{i1} \times {\varvec{R}}_{i2} \big ), \nonumber \\&{\varvec{D}}_{ri1} ={\varvec{\delta }}_{ri} \times \big ({\varvec{\delta }}_{ri} \times {\varvec{R}}_{i1} \big )=\hat{{{\varvec{\delta }} }}_{ri}^2 {\varvec{R}}_{i1} , \nonumber \\&{\varvec{D}}_{ri1}^T ={\varvec{R}}_{i1}^T \hat{{{\varvec{\delta }} }}_{ri}^2 ,\quad {\varvec{D}}_{ri} =-\big ({\varvec{R}}_{i1} {\varvec{R}}_{i2}^T +{\varvec{R}}_{i2} {\varvec{D}}_{ri1}^T \big ), \nonumber \\&{\varvec{D}}_{li1} ={\varvec{\delta }}_{li} \times \big ({\varvec{\delta }}_{li} \times {\varvec{R}}_{i1} \big )=\hat{{{\varvec{\delta }} }}_{li}^2 {\varvec{R}}_{i1} , \nonumber \\&{\varvec{D}}_{li1}^T ={\varvec{R}}_{i1}^T \hat{{{\varvec{\delta }} }}_{li}^2 ,\quad {\varvec{D}}_{li} =-\big ({\varvec{R}}_{i1} {\varvec{R}}_{i2}^T +{\varvec{R}}_{i2} {\varvec{D}}_{li1}^T \big ), \end{aligned}$$

(a5)

$$\begin{aligned}&\big ({\varvec{R}}_{i1} \times {\varvec{R}}_{i2} \big )'={\varvec{R}}_{i1} \times \big ({{{\theta }} }'_{ri1} {\varvec{R}}_{i1} \times {\varvec{R}}_{i2} \big ) \nonumber \\&\quad ={{{\theta }} }'_{ri1} \big [\big ({\varvec{R}}_{i1} \cdot {\varvec{R}}_{i2} \big ){\varvec{R}}_{i1} -\big ({\varvec{R}}_{i1} \cdot {\varvec{R}}_{i1} \big ){\varvec{R}}_{i2} \big ] \nonumber \\&\quad =-{\theta }'_{ri1} {\varvec{R}}_{i2} , \nonumber \\&{d}'_{ri} =\big ({\varvec{R}}_{i1} \times {\varvec{R}}_{i2} \big )^{T}\big ({{\varvec{o}}}'+{{\varvec{\theta }} }'\times {\varvec{e}}_i \big ), \nonumber \\&{d}'_{li} =\big [{{{l}}}'_i {\varvec{\delta }}_{li} +{{l}}_i \big ({{\varvec{\theta }} }'_{li} \times {\varvec{\delta }}_{li} \big )\big ]\cdot \big ({\varvec{R}}_{i1} \times {\varvec{R}}_{i2} \big ), \nonumber \\&{{\varvec{\delta }} }'_{ri} ={{\varvec{\theta }} }'_{ri} \times {\varvec{\delta }}_{ri} ,\nonumber \\&{{l}}_i ^{\prime \prime } =\big ({{{\varvec{b}}}}'_{i2} \cdot {\varvec{\delta }}_{li} \big )'={{{\varvec{b}}}}''_{i2} \cdot {\varvec{\delta }}_{li} +{{{\varvec{b}}}}'_{i2} \cdot {{\varvec{\delta }} }'_{li} \nonumber \\&\quad ={{\varvec{o}}}''\cdot {\varvec{\delta }}_{li} +\big ({{\varvec{\theta }} }''\times {\varvec{e}}_{i2} \big )\cdot {\varvec{\delta }} _{li} -{{{\theta }} }''_i \big ({\varvec{R}}_i \times {{{\varvec{l}}}}_{hi} \big )\cdot {\varvec{\delta }}_{li} \nonumber \\&\qquad +\,\big [{{\varvec{\theta }} }'\times \big ({{\varvec{\theta }} }'\times {\varvec{e}}_{i2} \big )\big ]\cdot {\varvec{\delta }}_{li} -{{{\theta }} }'_i {{{\theta }} }'_i {{{\varvec{l}}}}_{hi} \cdot {\varvec{\delta }}_{li} \nonumber \\&\qquad -\,2{{{\theta }} }'_i \big [{{\varvec{\theta }} }'\times \big ({\varvec{R}}_i \times {\varvec{l}}_{hi} \big )\big ]\cdot {\varvec{\delta }}_{li} +{{{\varvec{b}}}}'_{i2} \cdot \big ({{\varvec{\theta }} }'_{li} \times {\varvec{\delta }}_{li} \big ) \nonumber \\&\quad ={\varvec{\delta }}_{li}^T {{\varvec{o}}}''-{\varvec{\delta }}_{li}^T \hat{{{\varvec{e}}}}_{i2} {{\varvec{\theta }} }''+{\varvec{\delta }}_{li}^T \hat{{{{\varvec{l}}}}}_{hi} {\varvec{R}}_i {{{\theta }} }''_i \nonumber \\&\qquad +\,\big ({\varvec{\delta }}_{li} \times {{\varvec{\theta }} }'\big )\cdot \big ({{\varvec{\theta }} }'\times {\varvec{e}}_{i2} \big )-{{\varvec{\theta }}}^{\prime T}_i {\varvec{\delta }}_{li}^T {{{\varvec{l}}}}_{hi} {{{\theta }} }'_i \nonumber \\&\qquad -\,2{{\varvec{\theta }}}^{\prime T}_i \big ({\varvec{\delta }}_{li} \times {{\varvec{\theta }} }'\big )\cdot \big ({\varvec{R}}_i \times {{{\varvec{l}}}}_{hi} \big )-{{\varvec{v}}}_{bi2}^T {{\varvec{G}}}_i {{{\varvec{b}}}}'_{i2} \nonumber \\&\qquad ={\varvec{\delta }}_{li}^T \left( {{\begin{array}{cc} {{\varvec{E}}}&{}\quad {-\hat{{{\varvec{e}}}}_{i2} } \\ \end{array} }} \right) {{\varvec{A}}}+{\varvec{\delta }}_{li}^T \hat{\varvec{l}}_{hi} {\varvec{R}}_i {{{\theta }} }''_i \nonumber \\&\qquad -\,\big (\hat{{{\varvec{\delta }} }}_{li} {{\varvec{\theta }} }'\big )^{T}\big (\hat{{{\varvec{e}}}}_{i2} {{\varvec{\theta }} }'\big )-\big ({{\varvec{J}}}_{{{\theta i}}} {{\varvec{V}}}\big )^{T}{\varvec{\delta }}_{li}^T {\varvec{l}}_{hi} {{\varvec{J}}}_{{{\theta i}}} {{\varvec{V}}} \nonumber \\&\qquad +\,2\big ({{\varvec{J}}}_{{{\theta i}}} {{\varvec{V}}}\big )^{T}\big (\hat{\varvec{l}}_{hi} {\varvec{R}}_i \big )^{T}\big (\hat{{{\varvec{\delta }}}}_{li} {{\varvec{\theta }} }'\big )-{{{\varvec{b}}}}^{\prime T}_{i2} {{\varvec{G}}}_i {{{\varvec{b}}}}'_{i2} \nonumber \\&\quad ={\varvec{\delta }}_{li}^T \left( {{\begin{array}{cc} {{\varvec{E}}}&{}\quad {-\hat{{{\varvec{e}}}}_{i2} } \\ \end{array} }} \right) {{\varvec{A}}} \nonumber \\&\qquad +\,\big ({\varvec{\delta }}_{li}^T \hat{\varvec{l}}_{hi} {\varvec{R}}_i \big )\big ({{\varvec{J}}}_{{{\theta i}}} {{\varvec{A}}}+{{\varvec{V}}}^{T}{{\varvec{h}}}_{{{\theta i}}} {{\varvec{V}}}\big )+{{\varvec{\theta }} }'^{T}\hat{{{\varvec{\delta }} }}_{li} \hat{{{\varvec{e}}}}_{i2} {{\varvec{\theta }} }' \nonumber \\&\qquad -\,{{\varvec{V}}}^{T}{{\varvec{J}}}_{{{\theta i}}}^T {\varvec{\delta }}_{li}^T {\varvec{l}}_{hi} {{\varvec{J}}}_{{{\theta i}}} {{\varvec{V}}} \nonumber \\&\qquad -\,2{{\varvec{V}}}^{T}{{\varvec{J}}}_{{{\theta i}}}^T {\varvec{R}}_i ^{T}\hat{\varvec{l}}_{hi} \left( {{\begin{array}{cc} {{{\varvec{0}}}_{3\times 3} }&{}\quad {\hat{{{\varvec{\delta }} }}_{li} } \\ \end{array} }} \right) \nonumber \\&\qquad {{\varvec{V}}}-{{\varvec{V}}}^{T}{{\varvec{J}}}_{bi2}^T {{\varvec{G}}}_i {{\varvec{J}}}_{bi2} {{\varvec{V}}}, \end{aligned}$$

$$\begin{aligned}&{{l}}_i ^{\prime \prime }={\varvec{\delta }}_{li}^T \big [\left( {{\begin{array}{cc} {{{\varvec{E}}}_{3\times 3} }&{} {-\hat{{{\varvec{e}}}}_{i2} } \\ \end{array} }} \right) +\hat{\varvec{l}}_{hi} {\varvec{R}}_i {{\varvec{J}}}_{{{\theta i}}} \big ]{{\varvec{A}}} \nonumber \\&\qquad +\,{{\varvec{V}}}^{T}\big ({\varvec{\delta }}_{li}^T \hat{\varvec{l}}_{hi} {\varvec{R}}_i \big ){{\varvec{h}}}_{{{\theta i}}} {{\varvec{V}}}+{{\varvec{V}}}^{T}\left( {{\begin{array}{cc} {{{\varvec{0}}}_{3\times 3} }&{}\quad {{{\varvec{0}}}_{3\times 3} } \\ {{{\varvec{0}}}_{3\times 3} }&{}\quad {\hat{{{\varvec{\delta }} }}_{li} \hat{{{\varvec{e}}}}_{i2} } \\ \end{array} }} \right) {{\varvec{V}}} \nonumber \\&\qquad -\,{{\varvec{V}}}^{T}{{\varvec{J}}}_{{\varvec{\theta }} i}^T {\varvec{\delta }}_{li}^T {\varvec{l}}_{hi} {{\varvec{J}}}_{{{\theta i}}} {{\varvec{V}}}-2{{\varvec{V}}}^{T}{{\varvec{J}}}_{{{\theta i}}}^T {\varvec{R}}_i^T \hat{\varvec{l}}_{hi}\nonumber \\&\qquad \left( {{\begin{array}{cc} {{{\varvec{0}}}_{3\times 3} }&{}\quad {\hat{{{\varvec{\delta }} }}_{li} } \\ \end{array} }} \right) {{\varvec{V}}} \nonumber \\&\qquad -\,{{\varvec{V}}}^{T}{{\varvec{J}}}_{bi2}^T {{\varvec{G}}}_i {{\varvec{J}}}_{bi2} {{\varvec{V}}}, \nonumber \\ {{\varvec{G}}}_i= & {} {\hat{{{\varvec{\delta }} }}_{li} {\varvec{D}}_{li} }/{d_{li} ,}\quad \big (i=1,2,3\big ). \end{aligned}$$

(a6)

6. :

Simulation mechanism of novel hybrid hand and its dynamics simulation

See Figs. 6, 7 and 8.

Fig. 6

Simulation mechanism of novel hybrid hand

Fig. 7

Dynamics simulation of novel hybrid hand

Fig. 8

A block model of novel hybrid hand before (a) and after (b) grasping object