Research Journal of Applied Sciences, Engineering and Technology 4(23): 5212-5216, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: April 25, 2012
Accepted: May 13, 2012
Published: December 01, 2012
Study of the Redundant Robotic Manipulator’s Kinematics Performance Index
1, 2
Ge Xinfeng, 1Zhao Dongbiao, 1Lu Yonghua and 1Liu Kai
College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and
Astronautics, Nanjing 210016, China
2
College of Electrical and Information Engineering, Xuchang University, Xuhang 461000, China
1
Abstract: The kinematics equations of robotic manipulator is derived according to moving frame system
method in this study; volume element is defined employing exterior differential and moving frame system and
is taken as kinematics performance index measuring redundant robotic manipulator’s manipulability. Then take
the 7-DOF automatic fiber placement robotic manipulator as an example and obtain the volume element
function of it. Compared the volume element function which derived with Yoshikawa's manipulability function,
it shows the volume element function and the manipulability function are the same, so the volume element as
a kinematics performance index is feasible.
Keywords: Exterior differential, kinematics performance index, manipulability, moving frame system, volume
element
INTRODUCTION
Robotic manipulator’s manipulability is a
comprehensive measure of robotic manipulator’s
manipulability in all directions, is one of key performance
indices of redundant robotic manipulator’s overall
dexterity (René Mayorga and Johnatan, 2005). Therefore,
study on manipulability indices have been one of the main
researches in studying the redundant robotic
manipulator’s kinematics (Samer and Moghavvemi, 2011)
Yoshikawa (Tsuneo, 1984) proposed manipulability as an
overall dexterity index of the redundant robotic
manipulators and draw the conclusion that w is the greater
and the redundant robotic manipulator’s dexterity is the
better. But the manipulability is based on Jacobian matrix
and dependent on the Euclidean measures, however, the
Euclidean measure is changeable with the coordinate
system, that is to the different coordinate systems
connected end-effector, the minimum singular value
points of the corresponding Jacobian matrix are different,
This shows that manipulability index based on the
minimum singular value of Jacobian matrix is incorrect
(Felix Reinhart and Jochen, 2009). Zhang (2004)
proposed volume element as a performance index to
measure the performance of robot kinematics, But for the
redundant robotic manipulator kinematics performance
index, the author did not give definition. The volume
element function is defined on the basis of Zhang
Liandong’s proposed and as a measurement of redundant
robotic manipulator kinematics manipulability index and
7-DOF automated fiber placement robotic manipulator,
for example, finds its volume function, compared the
obtained volume element function and manipulability
function. The volume element function and manipulability
functions are different in the form due to excessive joint
variables of redundant robotic manipulator, the same trend
of the curve drawn by the manipulability function and the
volume element function comparing their curves. It shows
that the volume element function reflected the
manipulability of the redundant robotic manipulator endeffector and can be used as performance index to measure
the kinematics of redundant robotic manipulator.
THE ROBOTIC MANIPULATOR MOVING
FRAME SYSTEM RECURRENCE FORMULA
AND THE EXTERIOR DIFFERENTIAL
Moving frame system recurrence formulas: The link
moving frame system established end of the link, which
is established the next joint. So the moving frame system
on the end-effector is just on the last link, the moving
frame system on last link is obtained by recursive (Zhang
et al., 2010).
Figure 1 shows the robotic manipulator three adjacent
joints i-1, i, i+1 and the moving frame system on the link
i-1 and the link i, li denote the length of link, ai denote the
offset, "i(i+1) denote torsion angle between joint i-1 and
joint i.
The revolute joint moving frame system recurrence
formula:
Corresponding Author: GE Xinfeng, College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and
Astronautics, Nanjing 210016, China
5212
Res. J. Appl. Sci. Eng. Technol., 4(23): 5212-5216, 2012
0
⎡1
⎢ 0 cosα
i ( i + 1)
i −1
Ti = ⎢
⎢ 0 sin α i (i +1)
⎢
0
⎣0
0
− sin α i (i +1)
cosα i (i +1)
0
li
⎤
− ai +1 sin α i (i +1) ⎥
⎥
d i + ai + 1 cosα i (i +1) ⎥
⎥
1
⎦
(5)
So the position and orientation of robotic manipulator
end-effector is:
{e1, e2, e3; r} = 0T1(21)1T2(21)2...n!1T2(2n)
The exterior differential: The coordinate in n dimension
Euclidean space Rn is (x1,x2,…,xn), the real vector based
on (dx1, dx2, …, dxn) is V and its space is G (V) (Chen,
2006). The element in Vp (p = 1, …, n) can be expressed
as:
Fig. 1: The moving frame systems on adjacent links
[
]
[
( i − 1)
⎧ r = r l cosθ + a sin α
+ li sin θi + ai +1 sin α i ( i +1) cosθi
i −1 i
i
i +1
i ( i + 1) sin θi e1
⎪ i
⎪ e2(i −1) + ai +1 cos αi ( i +1) e3(i −1)
⎪⎪
(i )
( i − 1)
( i − 1)
⎨ e1 = cosθi e1 + sin θi e2
⎪ (i −1)
( i − 1)
( i − 1)
( i − 1)
⎪ e2 = − cosα i (i +1) sin θi e1 + cos α i (i +1) cosθi e2 + sin α i ( i + 1) e3
( i − 1)
( i − 1)
( i − 1)
⎪ e (i −1) = sin α
− sin α i (i +1) cosθi e2 + cosα i (i +1) e3
i ( i + 1) + sin θi e1
⎪⎩ 3
∑
]
1≤i1<L<ip≤n
(1)
The displacement joint moving frame system
recurrence formula:
(2)
where, 2, di denote the joint variables of the revolute joint
and the displacement joint, respectively, e1(i), e2(i-1), e3(i)
denote the orientation vectors of link i, ri denote the
position vector of link i moving frame system origin. The
moving frame system on the link i determine the position
and orientation of the links completely.
Expressed as matrix form:
{r ; e
i
(i )
1
, e2(i −1) , e3(i ) } = {ri −1 ; e1(i −1) , e3(i −1) , e3(i −1) }
( i − 1)
Ti
(3)
To revolute joint:
i −1
⎡ cosθi
⎢ sin θ
i
Ti = ⎢
⎢ 0
⎢
⎣ 0
− cosα i (i +1) sin θi
cosα i (i +1) cosθi
sin α i (i +1) sin θi
− sin α i (i +1) cosθi
sin αi (i + 1)
0
cosα i (i +1)
0
li cosθi + α i +1 sin α i (i +1) sin θi ⎤
li sin θi − α i +1 sin α i (i +1) cosθi ⎥
⎥
⎥ (4)
ai +1 cosα i (i +1)
⎥
1
⎦
ai1Lip (x1, x2,L, xn )dxi1 ∧L∧ dxip
(7)
(7) is called p-order exterior differential form in the field
of Rn. In particular, the element T = 3"i (x1, x2, …, xn) dxi
in the field of V1 = V called the first order exterior
differential form.
The set of moving frame systems comprise the
robotic manipulator workspace surface, the surface
reflects the robotic manipulator kinematics characteristics.
Therefore, the exterior differential and the moving frame
systems are applied to define invariants on the surface and
make it reflect the robotic manipulator kinematics
performance.
The robotic manipulator kinematics performance
index-the volume element function: Volume element
function is essentially the robotic manipulator workspace
geometric description, reflecting the end-effector motion
density. The robotic manipulator quantitative workspace
can be obtained after integrating, which reflects the
manipulability of the robotic manipulator end-effector.
The moving frame system is established on the robotic
manipulator end-effector in order to study the robotic
manipulator kinematics manipulability. With the endeffector motion, the set of the moving frame systems is
the robotic manipulator workspace. The robotic
manipulator workspace has a structure of differential
manifold and the relative component of the moving frame
system is the differential manifold tangent space. T1, T2,
T3, T23, T31, T12 are the relative components of the
moving frame system, the largest linearly independent
group is found from the moving frame system’s six
relative components, the external product of elements in
largest linearly independent group constitutes volume
element dV:
dV=T1vT2vT3vT23vT31vT12
To displacement joint:
(6)
The displacement volume element is:
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Res. J. Appl. Sci. Eng. Technol., 4(23): 5212-5216, 2012
Fig. 2: The automated fiber placement robotic manipulator’s structure
dT = T1vT2vT3
The revolute volume element is:
dR = T23vT31vT12
Volume element reflects the comprehensive
kinematics of the robotic manipulator end-effector
translational and rotational movement, which is the
robotic manipulator manipulability that can reach a series
of position and orientation, translational and rotational
volume are corresponding to position and orientation.
THE VOLUME ELEMENT OF THE 7-DOF
AUTOMATED FIBER PLACEMENT ROBOTIC
MANIPULATOR
The structure and parameters of the automated fiber
placement robot: The 7-DOF automated fiber placement
robotic manipulator developed by Nanjing University of
Aeronautics and Astronautics as an example to solve the
volume of redundant degrees of freedom robot in this
study, the 7-DOF automated fiber placement robotic
manipulator structure as shown in Fig. 2. From its
structure, there is a 6-DOF fiber placement robotic
manipulator and a rotational mandrel. The following
equivalent transformation in getting the robotic
manipulator workspace using volume element can be
done: mandrel as stationary and the coordinate system
fixed mandrel coincides with the base coordinate system,
a virtual revolute joints linked the base of the fiber
placement robotic manipulator and the mandrel together,
The rotational motion of the mandrel is equivalent to the
robotic manipulator’s rotation around the mandrel. So the
Table 1:The automated fiber placement robotic manipulator’s D-H
parameters
ai-1
"iThe scope of the
Link i
(mm)
(°)
di(mm)
2i(°)
joint variables
1
0
0
d1
0
-150-150
2
0
90
d2
-90
-110-110
3
a2
90
d3
0
-100-100
4
0
0
c
24
-210-210
5
0
90
0
25
-150-150
6
0
90
0
26
-260-260
fiber placement robotic manipulator with 6-DOF and the
mandrel with 1-DOF becomes a 7-DOF redundant robotic
manipulator. The shoulder has a revolute joint, the elbow
has three displacements joint, wrist has three revolute
joint. The three revolute joint axes of the wrist intersect at
one point, the automatic fiber placement robotic
manipulator’s topology after equivalent motion as shown
in Fig. 2. Establishing D-H coordinate system and its
structural parameters as shown in Table 1.
The volume element of the 7-DOF automated fiber
placement robotic manipulator: Plugging the
parameters of the 7-DOF automated fiber placement
robotic manipulator in Table 1 into (4) and (5), we obtain
the matrix representation of the moving frame system on
each joint:
⎡ cosθ1
⎢ sin θ
1
0
T1 = ⎢
⎢ 0
⎢
⎣ 0
3
5214
⎡1
⎢0
T4 = ⎢
⎢0
⎢
⎣0
− sin θ1
cosθ1
0
0
0 a0 ⎤
⎡1
⎢0
0 0⎥ 1
⎥; T = ⎢
2
⎢0
1 0⎥
⎥
⎢
0 1⎦
⎣0
0⎤
⎡ 0
⎢
1 0 0⎥ 2
⎥, T = ⎢ 0
3
⎢− 1
0 1 d2 ⎥
⎢
⎥
0 0 0⎦
⎣ 0
0 0
0
a3 ⎤
⎡ cosθ5 − sin θ5
⎢
0 − 1 − d4 ⎥ 4
⎥ T = ⎢ sin θ5 cosθ5
0
1 0
0 ⎥ 5 ⎢ 0
⎢
⎥
0
0 0
1 ⎦
⎣ 0
0
⎡ cosθ6
0 0⎤
⎢ 0
0 0⎥ 5
⎥ T = ⎢
1 c ⎥ 6 ⎢ sin θ6
⎥
⎢
0 1⎦
⎣ 0
0
0 ⎤
0 − 1 − d3 ⎥
⎥
0 0
0 ⎥
⎥
0 0
1 ⎦
1
− sin θ6
0
cosθ6
0
0 0⎤
− 1 0⎥
⎥
0 0⎥
⎥
0 1⎦
Res. J. Appl. Sci. Eng. Technol., 4(23): 5212-5216, 2012
⎡ cosθ7
⎢ 0
6
T7 = ⎢
⎢ − sin θ7
⎢
⎣ 0
− sin θ7
0
− cosθ7
0
The external product of the components constitutes
automated fiber placement robotic manipulator’s volume
element dV:
0 0⎤
1 0⎥
⎥
0 0⎥
⎥
0 1⎦
Multiplying the each formula above, we can get the
matrix representation of the moving frame system on the
robotic manipulator’s end-effector:
⎡R
g= ⎢
⎣0
⎡ g11
p⎤ ⎢ g 21
= ⎢
1 ⎥⎦ ⎢ g 31
⎢
⎣ 0
g12
g13
g 22
g 32
0
g 33
g 33
0
g14 ⎤
g 24 ⎥
⎥
g 34 ⎥
⎥
1 ⎦
(8)
where,
g11 = s21 s25c26 c27 !c21 s26s27+ s21 c25s27
g12 = !s21 s25c26 s27+ c21 s26s27+ s21 c25c27
g13 = ! s21 s25s26 ! c21 c26, g14=-(c+d4)c21+d3s21+a0
g21 = ! c21 s25c26 c27- s21 s26 c27 ! c21 c25s27
g22 = c21 s25c26 c27+ s21 s26 s27 ! c21 c25c27
g23 = c21 s25s26 ! s21c26, g24 = !(c+d4) s21 !d3c21
g31 = !c25c26 c27+ s25s27, g32= c25c26 s27+ s25c27, g33=
c25s26 , g34 = !a3+d2
s2 = sin2, c2 = cos2
The generalized velocity of the 7-DOF automated
fiber placement robotic manipulator’s end-effector:
R −1dp⎤ ⎡ Ω 1
⎥= ⎢
0 ⎦ ⎣ 0
⎡ R −1 dR
g −1dg = ⎢
⎣ 0
Ω 2⎤
0 ⎥⎦
(9)
Suppose the object speed of the 7-DOF automated
fiber placement robotic manipulator’s end-effector is:
V = [T1,T2,T3,T23,T31,T12]T
− ω12
0
ω 23
ω 31 ⎤
⎥
− ω 23 ⎥ , Ω 2 =
0 ⎥⎦
Integrating (12), we obtain volume element function.
Compared with manipulability: Manipulability as a
performance index measuring the robotic manipulator
kinematics has been proven. Next manipulability and
volume element of the 7-DOF automated fiber placement
robotic manipulator were solved and compared them.
Jacobian matrix of the 7-DOF automated fiber placement
robotic manipulator is:
⎡ J11 J12 J13 J14 0 0
⎢J J
⎢ 21 22 J23 J24 0 0
⎢J J J J
0 0
J = ⎢ 31 32 33 34
⎢ J41 0 0 0 J45 J 46
⎢J
0 0 0 J55 J56
⎢ 51
⎢⎣ J61 0 0 0 J65 0
0⎤
0⎥⎥
0⎥
⎥
0⎥
0⎥
⎥
1⎥⎦
where,
J11 = !d3s26 c27+(c+d4)( s25c26 c27+c25s27), J12= !c25c26
c2 7 + s 2 5 s 2 7
J13 = s25c26 c27+ c25s27, J14= J45= s26 c27,
J21 = d3s26 s27+(c+d4)( s25c26 s27+ c25c27), J23= -s25c26
s 2 7 + c2 5 c2 7
J24= J55= -s26 s27, J31= -d3c26 +(c+d4)s25s26, J33= s25s26,
J34= J65= c26,
J41= -c25c26 c27+ s25s27, J46= -s27, J51= J22=c25c26 s27+
s25c27, J56= -c27,
J61= J32=-c25s26
s2= sin2, c2= cos2
There is no way to compare similarities and differences of
the manipulability function and volume element function
for more variables. Then, make the curve of the
manipulability function and volume element function,
Assuming:
⎡ 0
⎢
Ω 1 = ⎢ ω12
⎢⎣ − ω 31
dv = (g11dg14+g21dg24+g31dg34+)v(g12dg14+g22dg24 +g23dg24)
v (g 13 dg 14 +g 23 dg 24 +g 33 dg 34 )v (g 13 dg 12 +g 23 dg 22 +g 33 dg 32 )
v(g12dg11+g21dg22+g32dg31)v(g11dg13+g21dg21+g31dg33)
⎡ ω1 ⎤
⎢ ⎥
⎢ω 2 ⎥
⎢⎣ ω 3 ⎥⎦
Plugging (8) into (9), we obtain the six linearly
independent relative component of the moving frame
system:
⎧ω1 = g 11 dg 14 + g 21 dg 24 + g 31d g 3 4
⎪
⎪ω 2 = g 12 d g 14 + g 22 d g 2 4 + g 2 3 dg 34
⎪⎪ω 3 = g13 d g 14 + g 23 d g 24 + g 33 d g 34
⎨
⎪ω 23 = g 13 d g12 + g 23 d g 22 + g 33 dg 32
⎪ω12 = g 12 d g 11 + g 22 d g 21 + g 32 d g 31
⎪
⎪⎩ω 31 = g 11 dg 13 + g 2 1d g 21 + g 31 d g 33
Fig. 3: Manipulability function curve
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Res. J. Appl. Sci. Eng. Technol., 4(23): 5212-5216, 2012
ACKNOWLEDGMENT
This study is financially supported by National
Natural Science Foundation of China (51175261;
51005122), Air Fund (2008ZE52049), to express my
gratitude.
REFERENCES
Fig. 4: Volume element function curve
respectively and compared their trend. The trend of the
curves as shown in Fig. 3 and 4. The trend and extreme
points of the curve of the manipulability function and
volume element function are the same. So the volume
element as the performance index measuring the 7-DOF
automated fiber placement robotic manipulator kinematics
is feasible.
CONCLUSION
Kinematics performance index measuring redundant
robotic manipulator’s manipulability is proposed
employing exterior differential and moving frame system.
The 7-DOF automated fiber placement robotic
manipulator developed by Nanjing University of
Aeronautics and Astronautics as an example, the
calculation of the volume element function is given, the
volume element function as kinematics performance index
measuring redundant robotic manipulator’s manipulability
is feasible comparing the manipulability.
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