HCR - Lecture 8

HCR - Grasping vs Manipulation

• GRASPING: Holding an object (it moves with me but is relatively still)
• MANIPULATION: Hold an object and control its trajectory (specific to the object)

We can see:

• Manipulation → active motion
• Grasping → passive motion

→ To manipulate objects humans have specifically developed the thumb

• Really important when talking about manipulation.

There are 2 kinds of manipulation:

• IN-HAND MANIPULATION
• POWER GRASP

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Brain prospective

From our brain prospective when we grasp an object (power grasp) it does not cast anything (it doesn’t think about the hand receptors).

• For power grasping we do have control over the object in our hand, but it is not accurate, tho it is fast to process (for our brain).
• For in-hand manipulation, like writing with a pen we use 3 contact point (more generally a small number of contact points) and we need to control each of them very well. (While for power grasping we have many contact point but no need to control them)

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HCR - IN-HAND MANIPULATION

Move the object with respect to the palm. Also called “dexteritious manipulation” ~ Ex.: Writing with a pen

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HCR - POWER GRASP

I don’t care about the specific motion of the object. The object moves with respect to the arm (not the palm) ~ Ex.: Holding air-pods

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HCR - Friction Cone

==A way to visualize friction==

The friction cone depends on the $F_D$ (Force Down-ward) and $\mu$ the friction coefficient.

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• Friction Cone:
  
  
  • Where:$\begin{array}{l}\Theta =\\ \eta :\text{friction coefficent}\end{array}$(The friction coefficent ($\eta$) is specific for each pair of object interaction)
  • A force that want to break the friction is applied as shown:
    
    
    • $F_1$ is too small to break friction (the object doesn’t move).
    • $F_2$ is big enough (the object begins to move).
    • Graphically $F_1$ is inside the friction cone.
      While $F_2$ is outside the friction cone
  • Knowing where the friction cone is, and how big it is, is extremely crucial, because if i want to apply force to the object without it slipping out (break friction) i need to apply all the forces such that they stay inside the friction cones.

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• ~Ex.: Position of the friction cones
  • Suppose we have this object:
    
  • Let’s see how the position of the friction cones is useful.
    I apply to the previous object 3 forces: $F_1,F_2,F_3$, resulting in this 3 friction cones:
    
  • Take now an opposite force $F$ that tries to break friction.
    It is clear that where i apply the previous 3 forces changes the intensity of the Force $F$ needed to break friction:
    

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HCR - Representation of Forces on the Reference Frame

Every force on an object can be represented in the object RF (Reference Frame) usually defined as its barycenter by another Force plus a Torque.

Also every interaction the object has with the world can be summarized by an external Force and an external Moment (or Torque)

• To compute the Torque ${\tau }^{\prime }$

in the RF (reference frame) $0$ →

$${\left(\begin{array}{l}{M}_{x_0}\\ M_{y_0}\\ M_{z_0}\end{array}\right)}_0={\overrightarrow{b}}_0\wedge {\overrightarrow{F}}_0$$

• The vector product can be computed using the Skew Matrix:

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HCR - Definition of 'Wrench'

Formula:$W:={\left[\begin{array}{l}F\\ M\end{array}\right]}_{6\times 1}$Where:

• $F$ : Force vector $[3\times 1]$
• $M$ : Moment vector $[3\times 1]$

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HCR - Definition of 'External Wrench'

The external wrench $W_e$ is so defined:$$ W_{\kern-3px e} := \left[ \begin{array}{l} F_e \ M_e \end{array} \right]_{6 \times 1}

- $F_e$ : Sum of all external Forces - $M_e$ : Sum of all external Moments So if we have an external wrench we can define the system as **In Equilibrium** (or grasping equilibrium) if we apply to it a *Controlled Wrench* $W_{\kern-3px c}$ , such that:

W_{\kern-3px c} = - W_{\kern-3px e}

- $W_{\kern-3px c}$ can be imposed by a human hand (IRL), a robotic hand (Industry) or a virtual hand (VR). - $W_{\kern-3px c}$ the controlled wrench (3 forces and 3 moments) represents the overall effect in term of forces and moments on the barycenter of the object, given by all the **contact forces** applied by the human/robotic/virtual hand at the contacts established with the object.Link to original

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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

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HCR - Change of the Normal during Grasping

Suppose we posses an object and we want to find the best possible grasp between this two configurations:

We look at the normal of the surface at the contact points:

• Notice how if the contact points in the first case change a little the normal changes a lot.
• While in the second case if we change the contact points a little (or we perturbate them) the normal does not change at all:
  

So we can say:

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