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.

Joint angle feedback

The joint encoder is mounted in position \(P_0\), hence we can measure the angle \(q_1\) that in turn is used to close the control loop.

Motor encoder feedback

A quadrature encoder is mounted on the motor shaft, which is currently used only to handle the tendon wrapping.

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:

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

Let us move on to explore the relation between the motor pulling the tendon and \(q_0\). Both the tendon ends are clamped to bushings that describe circular arcs; one bushing is centered in \(D\) and its trajectory is an arc centered in \(C\) whereas the other is centered in \(F\) and its trajectory is an arc centered in \(E\).

_{Finger motor pulley (forearm)}

Be \(q_{mot}\) the variable angle of the motor pulley and assuming the steel wire to be inextensible, the arc lengths described by the two trajectories shall be equal, i.e.:

\[r_{mot} \cdot q_{mot} = r \cdot q_0\]

Hence we get

\[q_0 = \frac{r_{mot}}{r} \cdot q_{mot} \]

Mk5.2 coupling variables

The following table describes the coupling variables of the finger mechanisms in the hand Mk5.2.

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	29.09	34.18	34.18	34.18	29.09	\(x\) coordinate of the second end of the leverism
\(L_{1y}\)	mm	0.24	-0.2	-0.2	-0.2	0.24	\(y\) coordinate of the second end of the leverism
\(P_{0x}\)	mm	0	0	0	0	0	\(x\) coordinate of the axis of the joint between ground and the first phalanx (origin)
\(P_{0y}\)	mm	0	0	0	0	0	\(y\) coordinate of the axis of the joint between ground and the first phalanx (origin)
\(P_{1x}(0)\)	mm	35	40	40	40	35	\(x\) coordinate of the axis of the joint between the first and the second phalanx
\(P_{1y}(0)\)	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	-97.54	Angle of \(A-C\) when the finger is in the theoretical zero position (read note 1)
\(q_{2bias}\)	deg	-162.27	-156.18	-156.18	-156.18	-162.27	Angle between \(L_1-P_1\) and the horizontal of the second phalanx (read note 2)
\(q_{1off}\)	deg	2.45	2.15	2.15	2.15	2.45	Angle of \(P_1-P_0\) when the finger is in the theoretical zero position
\(q_{1bias}\)	deg	-90	-90	-90	-90	-90	Angle of \(B-P_0\) when the finger is in the theoretical zero position

\(q_{0max}\)	deg	-6.42	-6.42	-6.42	-6.42	-6.42	Max value of \(q_0\), i.e. when the finger is fully closed
\(q_{1max}\)	deg	98	98	98	98	98	Max value of \(q_1\), same as above
\(q_{2max}\)	deg	199.77	196.35	196.35	196.35	199.77	Max value of \(q_2\), same as above

\(q_{1rest}\)	deg	3.63	5.75	5.75	5.75	5.71*	Rest value for \(q_1\) (* read note 3)
\(q_{2rest}\)	deg	3.63	5.75	5.75	5.75	8.49*	Rest value for \(q_2\) (* read note 3)

\(k\)	mm	34.30	39.4	39.4	39.4	34.30	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	6.07	6.07	6.07	6.04	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 \(D-C\)
\(r_{mot}\)	mm	5	5	5	5	5	Distance between the axis of the bushing and the rotational axis of the finger motor

Note 1: Version Mk5.2 has shifted the hard stop of the rest position to the last moving phalanx. As a consequence, \(q_1\) is non-zero when the hand is at rest. Parameters were measured considering the following, physically compenetrating position to stay consistent with the reference system adopted until now. The new rest positions for all the fingers have been stored inside \(q_{1rest}\), \(q_{2rest}\) (see table).

Note 2: As a consequence of Note 1, \(q_{2bias}\) is now better generalised as the angle between \(L_1-P_1\) and the horizontal of the second phalanx. This new definition still works with older versions. See image to have a more intuitive notion.

Note 3: Although the pinky is mechanically identical to the thumb, its rest value depends on the ring which is mechanically coupled to.

_{Theoretical, unreachable zero for Mk5.2}

_{Actual rest position for Mk5.2}

_{Generalised definition for q2bias}

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
\(P_{1x}(0)\)	mm	35	40	40	40	35	\(x\) coordinate of the axis of the joint between the first and the second phalanx
\(P_{1y}(0)\)	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 \(D-C\)
\(r_{mot}\)	mm	5	5	5	5	5	Distance between the axis of the bushing and the rotational axis of the finger motor

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
\(P_{1x}(0)\)	mm	20	30	30	30	25	\(x\) coordinate of the axis of the joint between the first and the second phalanx
\(P_{1y}(0)\)	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 \(D-C\)