King’s College London TOMSY Project Report on The Kinematics of KCL Five Fingered Metamorphic Hand Evangelos Emmanouil Dr. Guowo Wei Dr. Ketao Zhang Prof. Jian S. Dai Centre for Robotics Research School of Natural & Mathematical Sciences King’s College London Strand WC2R 2LS March 12, 2012
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The Kinematics of KCL Five Fingered Metamorphic Hand
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King’s College London
TOMSY Project Report on
The Kinematics of KCL FiveFingered Metamorphic Hand
The KCL five fingered metamorphic hand, shown in Fig. 1, comprises ofa metamorphic palm and five fingers. Each finger is very simple, just somelinks connected by revolute joints whose axes are parallel. The interestingpart is the palm. It is a spherical five-bar linkage. It is made out of five linksin a circular configuration with every joint axis zi passing through the centreof the sphere, as shown in Fig. 2.
Of the five joints θi on the palm, only θ1 and θ5 are actuated. Theremaining joints, θ2, θ3 and θ4 are rotating freely, based on the constraintsimposed by the geometry of the spherical linkage. Each finger is actuated byonly one tendon and move on it’s own finger operation plane, as shown inFig. 3.
2 Palm Kinematics
The coordinates of the points A, B, C, D and E as well as joint angles θ2,θ3 and θ4 should be computed first. We start by assuming that point E ispE = [0, 0, 1]. Then the coordinates of points A, B and D are computedfrom the known and actuated joint angles θ5 and θ1. Then, angle θ3 can becomputed by applying the cosine law for spherical triangles on the triangle4BCD. Computing angle θ4 can be done by adding together angles 6 EDB,
2
Figure 2: Metamorphic Palm and Finger Attachment Points.
6 BDC and subtracting π. Angle θ3 can be computed in a similar way, byadding 6 ABD, 6 DBC and subtracting π. This indicates that the distance‖BD‖ has to be computed.
2.1 Angles α1 to α5
The values for the angles α1 to α5 are as follows:
α1 = 25° (1)
α2 = 40° (2)
α3 = 70° (3)
α4 = 112° (4)
α5 = 113° (5)
2.2 Points A, B, D, E
The coordinates for points A, B, D and E can be computed by performingthe rotations described in equations 6, 7, 8 and 9.
3
Figure 3: Finger Operational Planes.
pA = R(y5, α5) k (6)
pB = R(y5, α5) R(z1, θ1) k (7)
pD = R(z5,−θ5) R(y4,−α4) k (8)
pE =
00R
(9)
2.3 Joint Angles θ2, θ3, θ4
First, the distance ‖BD‖ and the angle αbd are computed.
4
bd = pb − pd (10)
‖BD‖ =√bd′ bd (11)
αBD = arccos
(1− ‖BD‖
2
2
)(12)
Then, θ3 is computed by the spherical law of cosines.
6 BCD = arccos
(cosαBD − cosα2 cosα3
sinα2 sinα3
)(13)
θ3 = 6 BCD − π (14)
Next, αBE is computed and then used to compute θ4.
be = pb − pe (15)
‖BE‖ =√be′ be (16)
αBE = arccos
(1− ‖BE‖
2
2
)(17)
By using αBE, θ4 is computed.
6 EDB = arccos
(cosαBE − cosα4 cosαBD
sinα4 sinαBD
)(18)
6 BDC = arccos
(cosα2 − cosα3 cosαBD
sinα3 sinαBD
)(19)
θ4 = ( 6 EDB + 6 BDC)− π (20)
Finally, αAD is computed and then used to compute θ3. It must be notedthat the angles of a spherical triangle don’t add up to π. Because of thisfact, angle θ3 has to be computed in the same fashion as angles θ2 and θ4.
ad = pa − pd (21)
‖AD‖ =√ad′ ad (22)
αAD = arccos
(1− ‖AD‖
2
2
)(23)
5
By using αAD, θ4 is computed.
6 ABD = arccos
(cosαAD − cosα1 cosαBD
sinα1 sinαBD
)(24)
6 DBC = arccos
(cosα3 − cosα2 cosαBD
sinα2 sinαBD
)(25)
θ2 = π − (6 ABD + 6 DBC) (26)
2.4 Point C
Point C can now be located in two different ways. One way is to move in aclockwise direction (E to D to C) and the other way is to move in a counterclockwise direction (E to A to B to C). The first way is chosen since it isthe least computationally intensive. One can use both ways and see if theequations produce the same coordinates in order to verify their correctness.
pC = R(z5,−θ5) R(y4,−α4) R(z4,−θ4) R(y3,−α3) k (27)
3 MCP Joints
After determining angles θ2, θ3 and θ4 from equations 26, 14 and 20, thepoints the fingers attach to the palm, as depicted in Fig. 4 have to be deter-mined.
Angles δ1 to δ5 determine points F1 to F5 where each MCP joint attacheson the palm and angles γ2 to γ5 are determined and fixed so that each finger,except for the thumb, is parallel and co-linear with the arm. γ1 is 0.
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Figure 4: MetaCarpoPhalangeal Joints.
3.1 Angles δ1 to δ5
The values of the angles δ1 to δ5 are given by equations 28, 29, 30, 31 and32.
δ1 = α21
2(28)
δ2 = α31
4(29)
δ3 = α41
7(30)
δ4 = α43
7(31)
δ5 = α45
7(32)
7
3.2 Angles γ1 to γ5
The values of the angles γ1 to γ5 are given by equations 34, 34, 35, 36 and37.
γ1 = 0 (33)
γ2 = −δ2 − (α4 −π
2) (34)
γ3 = δ3 − (α4 −π
2) (35)
γ4 = δ4 − (α4 −π
2) (36)
γ5 = δ5 − (α4 −π
2) (37)
3.3 Points Fi and Mi
In order to obtain the coordinates that describe the MCP joints, the homo-geneous transform matrices to points Fi need to be determined.
After obtaining the rotation matrices RFi, the coordinates of points Fi
are easily obtained by eq. 43.
fi = RFik (43)
Then, the homogeneous transform matrices for points Fi are given by eq.44.
DFi=
[RFi
RFik
0 1
](44)
Next, matrices DFiare multiplied by the homogeneous transform matrix
DFMigiven by eq. 45.
8
DFMi=
cos γi 0 − sin γi −ai0 sin γi
0 1 0 0sin γi 0 cos γi ai0 cos γi
0 0 0 1
(45)
The homogeneous transform matrices to points M1 to M5 are given byeq. 46 and eq. 47.
DM1 = DF1 DFM1 D10 (46)
DMi= DFi
DFMi(47)
where
D10 =
cos θ10 − sin θ10 0 0sin θ10 cos θ10 0 0
0 0 1 00 0 0 1
(48)
Finally, the coordinates for the points M1 to M5 can be computed fromeq. 49.
[ri00
]= DMi
0001
(49)
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References
[1] Guowu Wei, Jian S. Dai, Shuxin Wang, and Haifeng Luo (2011) Kine-matic analysis and prototype of a metamorphic anthropomorphic handwith a reconfigurable palm. International Journal Of Humanoid Robotics ,8 (3), 459–479.
[2] Lei Cui and Jian S. Dai (2011) Posture, workspace and manipulabilityof the metamorphic multifingered hand with an articulated palm. ASMEJournal of Mechanisms and Robotics , 3(2), 021001 1–7.
[3] Lei Cui and Jian S. Dai (2010) Singular-value decomposition based kine-matic analysis of the metamorphic multifingered hand. pp. DETC2010–28653–1–6.
[4] H. Laimon and Jian S. Dai (2010) Wireless bluetooth remote control ofa mlultifingered metamorphic hand. pp. DETC2010–28688–1–5.
[5] Lei Cui, Jian S. Dai, and De Lun Wang (2009) Workspace analysis ofa multifingered metamorphic hand. International Conference on Recon-figurable Mechanisms and Robots, London, ENGLAND 22-JUN-2009 -24-JUN-2009 GENOVA: KC EDIZIONI , pp. 619–625.
[6] Jian S. Dai, Delun Wang, and Lei Cui (2009) Orientation andworkspace analysis of the multifingered metamorphic hand-metahand.IEEE TRANSACTIONS ON ROBOTICS , 25 (4), 942–947.
[7] Lei Cui, D. Wang, and Jian S. Dai (2009) Dimensional synthesis of palm ofmultifingered metamorphic dexterous hand. Journal of Dalian Universityof Technology , 49 (3), 380–386.
[8] Jian S. Dai and Delun Wang (2007) Geometric analysis and synthesis ofthe metamorphic robotic hand. Journal of Mechanical Design, 129 (11),1191–1197.
72 %% Palm Joint Coordinates (Points A, B, D and E)73 % Z−axis Unit Vector74 z = [0; 0; 1];75 % Z−axis Vector including palm radious.76 k = [0; 0; R];77
127 % Theta2 needs some work128 %129 AD v = pa ccw − pd cw;130 AD = sqrt(AD v' * AD v);131 cAD = 1 − ADˆ2 / 2;132 ad = acos( cAD );133 sAD = sin( ad );134