HCR - Lecture 10 (Original)

Green Robot

Robot covered in green screen to be used in the metaverse for moving real objects, the green screen allows an easy replacement of the robot image, it can be invisible to the eye of the beholder (wearing a VR headset).

Ideas

Metaverse with physical interactions (enabled by robotics) then you can get many new ideas of paradigms of interactions

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Calculate the Forces

$$\overrightarrow{W_c}=\overline{\overline{G}}\cdot \left[\begin{array}{l}{F}_{c1}\\ F_{c1}\end{array}\right]$$

→ Where:

• $W_c$ is in the columns space of $\overline{\overline{G}}$ → G has max rank so it has $1$ or $\infty$ solutions → ($\infty$ solutions $\iff$ $N(G)\ne \{0\}$)

Then:

$$\overline{\overline{F_e}}={\overline{\overline{G}}}^{\#}\cdot \overrightarrow{W_c}+N_G\cdot \xi$$

→ Where:

• ${\overline{\overline{G}}}^{\#}$ : pseudo-inverse
• $N_G$ : free parameter vector ($N_G=N(G)$)
• ${\overline{\overline{G}}}^{\#}\cdot \overrightarrow{W_c}$ : solution of the linear system

Among all the $\xi$ we chose one such that $\overline{\overline{F_e}}=[F_{c1},F_{c2}]$ belongs to the friction cones.

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~ Ex.:

Given an object and some contact points:

$$\overrightarrow{W_c}=\overline{\overline{G}}\cdot {\left[\overrightarrow{F_{c1}},\overrightarrow{F_{c2}},\overrightarrow{F_{c2}},\right]}^T$$

1. Compute $\overline{\overline{G}}$ :

2. Calculate the null space of $G$ : $N_G$

3. Find $\xi$ such that: