# 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 \(D-C\) |