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Table 1 Parameters \(\beta _1\), \(\beta _{21}\), and \({{\varvec{\upbeta }}}_{22}\) in different states of planar contact

From: Modeling and control of planar slippage in object manipulation using robotic soft fingers

 \(\dot{\mathbf {X}} \ne \mathbf {0}\)\(\dot{\mathbf {X}} = \mathbf {0}\)
Slippage1st Assumption: stationary2nd Assumption: incipient slip
\(\beta _1\)010
\(\beta _{21}\)101
\({{\varvec{\upbeta }}}_{22}\)\(\mathbf {A}\)\(\mathbf {0}_{3 \times 1}\)\(\mathbf {C}\)
Condition to be checked\(B < 1\)If 1st Assumption is not correct
  1. \(\mathbf {X} = \left[ x,y,\theta \right] ^{\text {T}}\),
  2. \(\mathbf {A} = \mu {\left( {{{\dot{x}}^2} + {{\dot{y}}^2} + {\lambda ^2}{{\dot{\theta } }^2}} \right) ^{ - \frac{1}{2}}}{\left[ {\dot{x}},{\dot{y}},{{\lambda ^2}\dot{\theta } } \right] ^{\text {T}}}\),
  3. \(B = \left( {f_x^2 + f_y^2 + m_z^2/{\lambda ^2}} \right) /{\left( {\mu N} \right) ^2}\),
  4. \({\mathbf{C}} = - \mu {\left( {\bar{f}_x^2 + \bar{f}_y^2 + \bar{m}_z^2/{\lambda ^2}} \right) ^{ - \frac{1}{2}}}{\left[ {{{\bar{f}}_x}},{{{\bar{f}}_y}},{{{\bar{m}}_z}} \right] ^{\text {T}}}\),
  5. where \(\bar{f}_x\), \(\bar{f}_y\), and \(\bar{m}_z\) are calculated from the 1st Assumption