Hand Mk5 Coupling Laws
Coupling variables
We report below a sketch of the finger's kinematics:
Hand Mk5 leverism
The following table describes the finger mechanism's coupling variables of the hand Mk5.
| Measure | UoM | Fingers | Description | Notes | ||||
|---|---|---|---|---|---|---|---|---|
| THUMB | INDEX | MIDDLE | RING | PINKY | ||||
| L0x | mm | -5.55 | -5 | -5 | -5 | -5 | X coordinate of first end of leverism | Those measurements are taken with respect to a coordinate system located in P0 (joint between the palm and first phalanx) |
| L0y | mm | 2.85 | 4 | 4 | 4 | 4 | Y coordinate of first end of leverism | Those measurements are taken with respect to a coordinate system located in P0 (joint between the palm and first phalanx) |
| L1x | mm | 11.5 | 24 | 24 | 24 | 19 | X coordinate of second end of leverism | Those measurements are taken with respect to a coordinate system located in P0 (joint between the palm and first phalanx) |
| L1y | mm | 1.5 | 0.8 | 0.8 | 0.8 | 0.5 | Y coordinate of second end of leverism | Those measurements are taken with respect to a coordinate system located in P0 (joint between the palm and first phalanx) |
| P1x | mm | 20 | 30 | 30 | 30 | 25 | X coordinate of the axis of the joint between first and second phalanx | Those measurements are taken with respect to a coordinate system located in P0 (joint between the palm and first phalanx) |
| P1y | mm | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | Y coordinate of the axis of the joint between first and second phalanx | Those measurements are taken with respect to a coordinate system located in P0 (joint between the palm and first phalanx) |
| q0off | deg | -20.71 | -7.54 | -7.54 | -7.54 | -7.54 | Angle of the lever ACD respect to the vertical in position 0. | |
| q2bias | deg | 180 | 173.35 | 173.35 | 173.35 | 170.54 | Angle of L1-P1 respect to the horizontal in position 0. | |
| q1off | deg | 4.29 | 2.86 | 2.86 | 2.86 | 3.43 | Angle of P1-P0 respect to the horizontal in position 0. | |
| q1bias | deg | 0 | 0 | 0 | 0 | 0 | Angle of B-P0 respect to the horizontal in position 0. Not drawn in the picture because null. | |
| q0 | deg | 45.32 | 78.03 | 78.03 | 78.03 | 78.03 | Absolute angle of the lever ACD respect the palm, which has its fulcrum at C and transmits motion to the connecting rod AB by reversing the motion of the cable connected at D. | |
| q1 | deg | 82.06 | 90 | 90 | 90 | 90 | Absolute angle of the first phalanx with respect to the palm. | |
| q2 | deg | 68.31 | 189.2 | 189.2 | 189.2 | 183.31 | Absolute angle of the second phalanx with respect to the palm. | |
| beta | deg | 135.65 | 82.33 | 82.33 | 82.33 | 80.54 | Angle between the line P1-L0 and the horizontal and represents an intermediate step for the calculation of q2. | |
| k | mm | 17.1 | 29.18 | 29.18 | 29.18 | 24.25 | Connecting rod length L1-L0 (constant). | |
| d | mm | 20.06 | 30.04 | 30.04 | 30.04 | 25.04 | The distance between the two joints P1-P0 (constant). | |
| l | mm | 8.5 | 6.04 | 6.04 | 6.04 | 6.08 | The distance between the attachment point L1 of the connecting rod L1-L0 on the second knuckle and the second joint P1 (constant). | |
| b | mm | 6.24 | 6.4 | 6.4 | 6.4 | 6.4 | The distance between the attachment point L0 of the connecting rod L1-L0 on the palm and the first joint P0 (constant). | |
| h | mm | - | - | - | - | - | The distance between the second joint P1 and the connection L0 of the connecting rod on the palm (variable, depends on q1). | Not needed for now |
| s | mm | 6.52 | 5.5 | 5.5 | 5.5 | 5.5 | the distance between joint B of the connecting rod A-B and joint P0 (constant). | |
| t | mm | 13 | 14.5 | 14.5 | 14.5 | 14.5 | The length of the connecting rod A-B. | |
| f | mm | 6 | 5.5 | 5.5 | 5.5 | 5.5 | The length of the connecting rod A-C. | |
| r | mm | 7.5 | 8.5 | 8.5 | 8.5 | 8.5 | The length of the connecting rod D-C. | |
| a | mm | - | - | - | - | - | The distance between the connecting rod joint A and the joint P0 (variable, depends on q0). | Not needed for now |
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:
\[\frac{\partial q_{2}}{\partial q_{1}} = \frac{1}{2 - \frac{d^{2} - b^{2}}{d^{2} - L_{0} \bullet P_{1}}} + \frac{\left( L_{0x}P_{1y} - L_{0y}P_{1x} \right)\left( l^{2} - k^{2} - h^{2} \right)}{2lh^{3}\sqrt{1 - \left( \frac{l^{2} - k^{2} + h^{2}}{2lh} \right)^{2}}}.\]
The relationship between \(q_{0}\) and \(q_{1}\) is:
\[q_{1} = \tan^{- 1}\left( \frac{A_{y}(q_{0}) - P_{0y}}{A_{x}(q_{0}) - P_{0x}} \right) + \cos^{- 1}{\left( \frac{s^{2} - t^{2} + a^{2}(q_{0})}{2sa(q_{0})} \right) - q_{1bias}}.\]