# Hand Mk5 Coupling Laws

## Coupling variables

Below, a sketch of the finger kinematics is reported:  Hand Mk5 leverism

Unless specified otherwise, angles reported in the next sections follow the convention in figure.

• $$q_0$$ is the variable angle of $$A-C$$.
• $$q_1$$ is the variable angle of the first phalanx.
• $$q_2$$ is the variable angle of the second phalanx.

## Coupling Laws

Considering the following quantities:

$b = \left| L_{0} - P_{0} \right|$
$d = \left| P_{1} - P_{0} \right|$
$l = \left| L_{1} - P_{1} \right|$
$k = \left| L_{1} - L_{0} \right|$
$s = \left| B - P_{0} \right|$
$t = |A - B|$
$f = |A - C|$
$r = |D - C|$
$h(q_{1}) = \left| P_{1}(q_{1}) - L_{0} \right|$
$a(q_{0}) = \left| A(q_{0}) - P_{0} \right|$
$P_{1x} = P_{0x} + d\cos\left( q_{1} + q_{1off} \right)$
$P_{1y} = P_{0y} + d\sin\left( q_{1} + q_{1off} \right)$
$A_{x} = C_{x} + f\cos\left( q_{0} + q_{0off} \right)$
$A_{y} = C_{y} + f\sin\left( q_{0} + q_{0off} \right)$

We have $$q_{2}$$ that depends only on the variable $$q_{1}$$ through the implicit dependence on $$P_{1}$$ and h:

$q_{2} = \tan^{- 1}\left( \frac{P_{1y}(q_{1}) - L_{0y}}{P_{1x}(q_{1}) - L_{0x}} \right) + \cos^{- 1}{\left( \frac{l^{2} - k^{2} + h^{2}(q_{1})}{2lh(q_{1})} \right) - q_{2bias} - \pi}.$

The Jacobian that relates the variations of $$q_{2}$$ to the variations of $$q_{1}$$ is:

$\frac{\partial q_{2}}{\partial q_{1}} = \frac{1}{2 - \frac{d^{2} - b^{2}}{d^{2} - (L_{0x}P_{1x}+L_{0y}P_{1y})}} + \frac{\left( L_{0x}P_{1y} - L_{0y}P_{1x} \right)\left( l^{2} - k^{2} - h^{2}(q_1) \right)}{2lh^{3}(q_1)\sqrt{1 - \left( \frac{l^{2} - k^{2} + h^{2}(q_1)}{2lh(q_1)} \right)^{2}}}.$

The relationship between $$q_{0}$$ and $$q_{1}$$ is:

$q_{1} = \tan^{- 1}\left( \frac{A_{y}(q_{0}) - P_{0y}}{A_{x}(q_{0}) - P_{0x}} \right) + \cos^{- 1}{\left( \frac{s^{2} - t^{2} + a^{2}(q_{0})}{2sa(q_{0})} \right) - q_{1bias} - \pi}.$

## Mk5.1 coupling variables

The following table describes the coupling variables of the finger mechanisms in the hand Mk5.1.

Name UoM Values Description
THUMB INDEX MIDDLE RING PINKY
$$L_{0x}$$ mm -5 -5 -5 -5 -5 $$x$$ coordinate of the first end of the leverism
$$L_{0y}$$ mm 4 4 4 4 4 $$y$$ coordinate of the the first end of the leverism
$$L_{1x}$$ mm 29 34 34 34 29 $$x$$ coordinate of the second end of the leverism
$$L_{1y}$$ mm 0.8 0.8 0.8 0.8 0.5 $$y$$ coordinate of the second end of the leverism
$$P_{1x}$$ mm 35 40 40 40 35 $$x$$ coordinate of the axis of the joint between the first and the second phalanx
$$P_{1y}$$ mm 1.5 1.5 1.5 1.5 1.5 $$y$$ coordinate of the axis of the joint between the first and the second phalanx
$$q_{0off}$$ deg -97.54 -97.54 -97.54 -97.54 -97.52 Angle of $$A-C$$ when the finger is fully open
$$q_{2bias}$$ deg -173.35 -173.35 -173.35 -173.35 -170.53 Angle of $$L_1-P_1$$ when the finger is fully open
$$q_{1off}$$ deg 2.45 2.15 2.15 2.15 2.46 Angle of $$P_1-P_0$$ when the finger is fully open
$$q_{1bias}$$ deg -90 -90 -90 -90 -90 Angle of $$B-P_0$$ when the finger is fully open
$$q_{0max}$$ deg -12 -12 -12 -12 -11.97 Max value of $$q_0$$, i.e. when the finger is fully closed
$$q_{1max}$$ deg 90 90 90 90 90 Max value of $$q_1$$, same as above
$$q_{2max}$$ deg 191.44 193.06 193.06 193.06 188.7 Max value of $$q_2$$, same as above
$$k$$ mm 34.15 39.13 39.13 39.13 34.18 Connecting rod length, $$\|L_1-L_0\|$$
$$d$$ mm 35.03 40.03 40.03 40.03 35.03 Distance between the two joints, $$P_1$$ and $$P_0$$
$$l$$ mm 6.04 6.04 6.04 6.04 6.08 Distance between $$L_1$$ and $$P_1$$
$$b$$ mm 6.4 6.4 6.4 6.4 6.4 Distance between $$L_0$$ and $$P_0$$
$$s$$ mm 5.5 5.5 5.5 5.5 5.5 Distance between joint $$B$$ of the connecting rod $$A-B$$ and joint $$P_0$$
$$t$$ mm 14.5 14.5 14.5 14.5 14.5 Length of the connecting rod $$A-B$$
$$f$$ mm 5.5 5.5 5.5 5.5 5.5 Length of the connecting rod $$A-C$$
$$r$$ mm 8.5 8.5 8.5 8.5 8.5 Length of the connecting rod $$D-C$$

## Mk5.0 coupling variables

The following table describes the coupling variables of the finger mechanisms in the hand Mk5.0.

Name UoM Values Description
THUMB INDEX MIDDLE RING PINKY
$$L_{0x}$$ mm -5.55 -5 -5 -5 -5 $$x$$ coordinate of the first end of the leverism
$$L_{0y}$$ mm 2.85 4 4 4 4 $$y$$ coordinate of the the first end of the leverism
$$L_{1x}$$ mm 11.5 24 24 24 19 $$x$$ coordinate of the second end of the leverism
$$L_{1y}$$ mm 1.5 0.8 0.8 0.8 0.5 $$y$$ coordinate of the second end of the leverism
$$P_{1x}$$ mm 20 30 30 30 25 $$x$$ coordinate of the axis of the joint between the first and the second phalanx
$$P_{1y}$$ mm 1.5 1.5 1.5 1.5 1.5 $$y$$ coordinate of the axis of the joint between the first and the second phalanx
$$q_{0off}$$ deg -110.71 -97.54 -97.54 -97.54 -97.54 Angle of $$A-C$$ when the finger is fully open
$$q_{2bias}$$ deg -180 -173.35 -173.35 -173.35 -170.54 Angle of $$L_1-P_1$$ when the finger is fully open
$$q_{1off}$$ deg 4.29 2.86 2.86 2.86 3.43 Angle of $$P_1-P_0$$ when the finger is fully open
$$q_{1bias}$$ deg -90 -90 -90 -90 -90 Angle of $$B-P_0$$ when the finger is fully open
$$q_{0max}$$ deg -44.68 -11.97 -11.97 -11.97 -11.97 Max value of $$q_0$$, i.e. when the finger is fully closed
$$q_{1max}$$ deg 86.35 90 90 90 90 Max value of $$q_1$$, same as above
$$q_{2max}$$ deg 135.65 189.2 189.2 189.2 183.31 Max value of $$q_2$$, same as above
$$k$$ mm 17.1 29.18 29.18 29.18 24.25 Connecting rod length, $$\|L_1-L_0\|$$
$$d$$ mm 20.06 30.04 30.04 30.04 25.04 Distance between the two joints, $$P_1$$ and $$P_0$$
$$l$$ mm 8.5 6.04 6.04 6.04 6.08 Distance between $$L_1$$ and $$P_1$$
$$b$$ mm 6.24 6.4 6.4 6.4 6.4 Distance between $$L_0$$ and $$P_0$$
$$s$$ mm 6.52 5.5 5.5 5.5 5.5 Distance between joint $$B$$ of the connecting rod $$A-B$$ and joint $$P_0$$
$$t$$ mm 13 14.5 14.5 14.5 14.5 Length of the connecting rod $$A-B$$
$$f$$ mm 6 5.5 5.5 5.5 5.5 Length of the connecting rod $$A-C$$
$$r$$ mm 7.5 8.5 8.5 8.5 8.5 Length of the connecting rod $$D-C$$