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

ROBOMECH Journal

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

	\(\dot{\mathbf {X}} \ne \mathbf {0}\)	\(\dot{\mathbf {X}} = \mathbf {0}\)
	Slippage	1st Assumption: stationary	2nd Assumption: incipient slip
\(\beta _1\)	0	1	0
\(\beta _{21}\)	1	0	1
\({{\varvec{\upbeta }}}_{22}\)	\(\mathbf {A}\)	\(\mathbf {0}_{3 \times 1}\)	\(\mathbf {C}\)
Condition to be checked	–	\(B < 1\)	If 1st Assumption is not correct

\(\mathbf {X} = \left[ x,y,\theta \right] ^{\text {T}}\),
\(\mathbf {A} = \mu {\left( {{{\dot{x}}^2} + {{\dot{y}}^2} + {\lambda ^2}{{\dot{\theta } }^2}} \right) ^{ - \frac{1}{2}}}{\left[ {\dot{x}},{\dot{y}},{{\lambda ^2}\dot{\theta } } \right] ^{\text {T}}}\),
\(B = \left( {f_x^2 + f_y^2 + m_z^2/{\lambda ^2}} \right) /{\left( {\mu N} \right) ^2}\),
\({\mathbf{C}} = - \mu {\left( {\bar{f}_x^2 + \bar{f}_y^2 + \bar{m}_z^2/{\lambda ^2}} \right) ^{ - \frac{1}{2}}}{\left[ {{{\bar{f}}_x}},{{{\bar{f}}_y}},{{{\bar{m}}_z}} \right] ^{\text {T}}}\),
where \(\bar{f}_x\), \(\bar{f}_y\), and \(\bar{m}_z\) are calculated from the 1st Assumption