HCR - Calculating the Controlled Forces

${\vec{W}}_c=\overline{\stackrel{ˉ}{G}}\cdot \left[\begin{array}{l}{F}_{c1}\\ F_{c1}\end{array}\right]$Where:

• $W_c$ is in the columns space of $\overline{\stackrel{ˉ}{G}}$.NOT_SURE_ABOUT_THIS What does it mean?
  ⇒ G has max rank so it has $1$ or $\infty$ solutions.
  ⇒ ($\infty$ solutions $\iff$ $N(G)\ne \{0\}$)

Then:$\overline{\stackrel{ˉ}{F_c}}={\overline{\stackrel{ˉ}{G}}}^{\#}\cdot {\vec{W}}_c+N_G\cdot \xi$Where:

• ${\overline{\stackrel{ˉ}{G}}}^{\#}$ : pseudo-inverse
• $N_G$ : free parameter vector ($N_G=N(G)$)
• ${\overline{\stackrel{ˉ}{G}}}^{\#}\cdot {\vec{W}}_c$ : solution of the linear system
• ==Among all the $\xi$ we chose one such that $\overline{\stackrel{ˉ}{F_c}}=[F_{c1},F_{c2}]$ belongs to the friction cones==.

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• ~ Ex.:
  • Given an object and some contact points:
    
    We want to calculate: ${\vec{W}}_c=\overline{\stackrel{ˉ}{G}}\cdot {\left[{\vec{F}}_{c1},{\vec{F}}_{c2},{\vec{F}}_{c2}\right]}^T$So:
    1. Compute $\overline{\stackrel{ˉ}{G}}$ :
       
    2. Calculate the nullspace of $G$ : $N(G)$
       
    3. Find $\xi$ such that: