The friction cone $Ki$ associated with contact event *i* is defined as $Ki\u2261{[x,\u2009y,\u2009z]T\u2208\mathbb{R}3:\u20030\u2264x\u2009\u2009\u2227\u2009\u2009\mu ix\u2212y2+z2\u22650}$. Similarly, the polar cone $Ki\xb0$ associated with the friction cone $Ki$ is defined as $Ki\xb0\u2261{[a,\u2009b,\u2009c]T\u2208\mathbb{R}3:ax+by+cz\u22640\u2009\u2009\u2200[x,\u2009,y,\u2009z]T\u2208Ki}$. Based on Eq. (5), $\gamma i(l+1)\u2261[\gamma i,n(l+1),\u2009\gamma i,u(l+1),\u2009\gamma i,w(l+1)]T\u2208Ki$. Define next $di\u2261[1\Delta t\Phi i(l)+vi,n(l+1),\u2009vi,u(l+1),\u2009vi,w(l+1)]T\u2208\mathbb{R}3$. Then, using Eq. (5)

$diT\xb7\gamma i(l+1)=\gamma i(l+1)(1\Delta t\Phi i(l)+vi,n(l+1))+\gamma i,u(l+1)\u2009vi,u(l+1)+\gamma i,w(l+1)\u2009vi,w(l+1)=\mu i\gamma i,n(l+1)vi,T(l+1)+vi,u(l+1)\gamma i,u(l+1)+vi,w(l+1)\gamma i,w(l+1)=\gamma i,F(l+1)\u2009vi,T(l+1)+vi,u(l+1)\gamma i,u(l+1)+vi,w(l+1)\gamma i,w(l+1)=\alpha i\u2009\gamma i,F(l+1)\u2212\alpha i\u2009\gamma i,F(l+1)=0\u2009$