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

$$\dot{\mathbf {X}} \ne \mathbf {0}$$$$\dot{\mathbf {X}} = \mathbf {0}$$
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