HCR - Grasp Matrix

Grasp Matrix

• For each contact point we have a wrench $\overrightarrow{W_i}$ of dimensions $[6\times 1]$
• We define the Grasp Matrix $\overline{\overline{G}}$ as:$\overrightarrow{W_i}=\overline{\overline{G}}\cdot {\left[\overrightarrow{F_1},\overrightarrow{F_2},\dots ,\overrightarrow{F_n},\right]}_{3n\times 1}^T$Where:
  • $W_c$ has dimensions: $[6\times 1]$
  • ${\left[\overrightarrow{F_1},\overrightarrow{F_2},\dots ,\overrightarrow{F_n},\right]}^T$ has dimensions: $[3n\times 1]$
  • So $\overline{\overline{G}}$ has dimensions: $[6\times 3n]$

Instead of using the Moment of each contact point, we use the arm $\overrightarrow{b}$ to calculate it using the grasp matrix: Where:

• $\overline{\overline{I}}$ : Identity matrix $[3\times 3]$
• $\overline{\overline{S(b)}}$ : Skew matrix $[3\times 3]$
• $\overrightarrow{b}$ : arm (distance of the contact point $C_i$ from the RF $O$ : $\overline{O{C}_i}$) $[3\times 1]$

NOTE: The Grasp Matrix depend on the reference frame, so we can write:${}^0\overline{\overline{G}}\text{or}{\overline{\overline{G}}}_0$Meaning the Grasp matrix is associated to the $0$-th reference frame