Abstract - This paper compares three inverse kinematic formulation methods for industrial robot manipulators. All formulation methods are based on screw theory. Screw theory is an effective way to establish a global description of rigid body and avoids singularities due to the use of the local coordinates. In these three formulation methods, two of them use quaternion algebra and the other one uses matrix algebra. Compared with the matrix algebra, the quaternion algebra is more computationally efficient and it needs less storage area. Paden-Kahan sub problems are used to derive inverse kinematic solutions. 6-DOF industrial robot manipulator’s forward and inverse kinematic equations are derived using these formulation methods and simulation results are given. Index Terms - Industrial Robot-Arms; Paden-Kahan Subproblems; Quaternion; Screw Theory; Singularity Free Inverse Kinematic;
I. INTRODUCTION
The kinematic problem of robot arms has a wide research area in robotics. Several methods are used in robot kinematics and screw theory is the one of the most important method among them. The elements of screw theory can be traced to the work of Chasles and Poinsot in the early 1800s. Using the theorems of Chasles and Poinsot as a starting point, Robert S. Ball developed a complete theory of screws which he published in 1900 [1]. In screw theory every transformation of a rigid body or a coordinate system with respect to a reference coordinate system can be expressed by a screw displacement, which is a translation by along a � axis with a rotation by a � angle about the same axis [2]. This description of transformation is the basis of the screw theory. There are two main advantages of using screw theory for describing rigid body kinematics. The first is that it allows a global description of rigid body motion that does not suffer from singularities due to the use of local coordinates. The second advantage is that the screw theory provides a geometric description of rigid motion which greatly simplifies the analysis of mechanisms [3].
There are several applications of screw theory in kinematic. Yang and Freudenstein were the first to apply line transformation operator mechanism by using the dual-quaternion as the transformation operator [4]. Yang also investigated the kinematic of special five bar linkages using
dual 3 3� matrices [5]. Pennock and Yang extended this method to robot kinematics [6]. In these methods, dual 3 3� matrices are used as a transformation operator to represent position and orientation of robot manipulators. This transformation operator has 18 parameters while just 6 parameters are needed to represent screw motion. M. Murray solved 3-DOF and 6-DOF robot manipulator kinematics by using screw theory with 4 4� homogeneous transformation matrix operators [7]. Then J.Xie et al applied this method to the 6-DOF space manipulator [8]. In this method non-singular inverse kinematic solutions are obtained by using 4 4� homogeneous transformation matrix operators. The homogenous transformation matrix operators use 16 parameters while just 6 parameters are needed for description of screw motion. J. Funda analyzed transformation operators of screw motion and he found that dual operators are the best way to describe screw motion and also the dual-quaternion is the most compact and efficient dual operator to express screw displacement [9-10]. Finally E. Sariyildiz and H. Temeltas investigated the kinematic of 6-DOF serial robot manipulators by using quaternion and dual-quaternion operators [11-12].
In this paper, we present a comparison study for three inverse kinematic formulation methods which are all based on screw theory. In these methods, two of them use quaternion algebra and the other one uses matrix algebra to express the screw motion. The first formulation method uses quaternions as a screw motion operator which combines a unit quaternion plus pure quaternion. The second formulation method uses dual-quaternion that is the most compact and efficient dual operator to express screw displacement as a screw motion operator. These two formulation methods use eight parameters to express screw motion. The last formulation method uses homogeneous transformation matrix as a screw motion operator. This formulation method uses sixteen parameters to express screw motion. Comparison results of these formulation methods are given in section 5. This paper is also included the mathematical preliminary of quaternion algebra in section 2, screw theory by using matrix and quaternion algebras in section 3, forward and inverse kinematic analysis of an industrial robot manipulator in section 4. Simulations are made by using exponential mapping and proposed methods. These methods are compared with respect to compactness of
Proceedings of the 2011 IEEEInternational Conference on Mechatronics and Automation
August 7 - 10, Beijing, China
transformation operators and computational efficiency. Conclusions and future works are drawn in the final section.
II. MATHEMATICAL PRELIMINARY
A. Quaternion
The quaternion can be represented in the form, [13]
0( , )vq q� q (1) where 0q is a scalar and 1 2 3( , , )V q q q�q is a vector. If
0 0q � then, we get pure quaternion. The sum and the product of two quaternions are then,
0 0( ), ( )a b a b v vq q q q� � � �a bq q (2)
0 0 0 0( ), ( )a b a b v v a v b v v vq q q q q q� � � � � � �a b b a a bq q q q q q (3)
where “�”, ” � ”, “� ” denote quaternion product, dot product and cross product respectively. Conjugate, norm and inverse of the quaternion can be expressed in the forms,
*0( , )q q� � vq , 2 *|| ||q q q� � and 1 *
21
|| ||q q
q� � respectively.
When 2|| || 1q � , we get unit quaternion which can be used as a rotation operator [14].
Definition 1: Let aq and bq be two pure quaternions and the product of these two pure quaternions is
( ), ( )a b v v v vq q� � � � �a b a bq q q q (4)
Then, let’s define two new functions by using the product of two pure quaternions given by
� a b v vV q q� � �a bq q
� a b v vS q q� � � �a bq q
B. Dual - Quaternion
The dual-quaternion can be represented in the form ˆˆ ˆ( , )S vq q� q or ˆ oq q q�� � (5)
where 0ˆs s sq q q�� � is a dual scalar, ˆ v v v�� � oq q q is a dual
vector, q and oq are both quaternions and � is the dual factor [15]. The sum and the product of two dual-quaternions are then
ˆ ˆ ( ) ( )o oa b a b a bq q q q q q�� � � � � (6)
ˆ ˆ ( ) ( )o oa b a b a b a bq q q q q q q q�� � � � � � � (7)
where “� ”, “� ” denote the quaternion and dual-quaternion products respectively. Conjugate, norm and inverse of the dual-quaternion can be expressed in the forms
When 2ˆ|| || 1q � , we get a unit dual-quaternion which can be used as a rigid motion operator [16].
Definition 2: Let ˆaq and ˆbq be two dual-quaternions and the product of two dual-quaternions be
ˆ ˆ ˆ ( , ) ( , )o oab a b ab ab abs v abs vq q q q q q q� �� � � � � � o
ab abq q (8) Then, let’s define four new functions by using the product of two dual-quaternions and definition 1 given by,
� ˆ ˆa b absS R q q q� �
� ˆ ˆ oa b absS D q q q� �
� ˆ ˆa b vV R q q� � abq
� ˆ ˆa b vV D q q� � oabq
III. SCREW THEORY
The general screw motion which can be seen in figure 2 can be defined by using matrix algebra as follows [15],
Figure 1: General screw motion
( , ) ( ( , ))2
1
kT
�
� �� �� ��� �� �� �
3x3R d d I R d p
0 (9)
A screw motion can be expressed by using quaternion algebra as follows
Rotation: cos ,sin2 2
q � � � �� � � � �� � � �
d (10)
Translation: *t q p q p q� � � � �� (11)
where q� is the amount of pure translation and p is the position vector of some point on the line in the pure quaternion form. A screw motion can also be expressed by using dual-quaternion as follows [17],
53
cos2ˆ
sin ( )2
k
qk
�
� �
� ��� �� �� �
� �� ��� ��� � �� �� �� �� �� �
d m
(12)
This representation is very compact and also it uses algebraically separates the angle and pitch k information. If we write ˆ k �� � and d̂ �� �d m then, the equation (12) becomes more compact as the following
ˆ ˆ ˆˆ cos sin2 2
q d � � � �� �� � � �� � � �
� � � � (13)
IV. MANIPULATOR KINEMATICS
The kinematic solution of the industrial robot manipulator which is shown in figure 2 is obtained as follows,
d
d
d
d
d d
1
2
3
5
64
ly0
lz0
z2
z1l
l
l z3
{Base}
{Tool}
Figure 2: 6-DOF serial arm robot manipulator in its reference configuration
A. Forward Kinematics
Quaternion based method
1. Determining the joint axis vector: First we attach the axis vectors which describe the motion of the joints.
2. Obtaining transformation operator: Transformation operators of all joints are obtained by using quaternions as follows
Rotation: cos ,sin2 2i i
iq � � � �� � � � �� � � �
id
(14)
Translation: *oi i i i iq p q p q� � � � (15)
where ip is an arbitrary point on the ith axis.
3. Formulization of rigid motion: Using (3) and (11) rigid transformation of serial robot manipulator can be expressed as
1 1 2n nq q q q� � � �� (16)
* *
1 1 1 1 1 1 1 1 1o o
n n n n n n n nq q p q q p q q� � �� � � � � � � (17)
where 1nq and 1o
nq indicate rotation and translation respectively. The orientation and the position of the end effector can be expressed as follows
*1 1 1o n o nl q l q� � � � (18)
*1 1 1 1
oep n ep n np q p q q� � � � �
(19)
where ,ep op l indicate the position and the orientation of the end effector before the transformation and 1 1,ep op l� � indicate the position and the orientation of the end effector after the transformation.
Dual-Quaternion based method
1. Determining joint axis vector and moment vector: First we attach the axis vectors which describe the motion of the joints. Then the moment vectors of these axes are obtained.
2. Obtaining transformation operator: Transformation operators of all joints are obtained by using dual-quaternions as follows
ˆ ˆ ˆ( , )i is ivq q q� or ˆ oi i iq q q�� � (20)
wher cos ,sin2 2i i
iq � � � �� � � � �� � � �
id � �*12
oi i i i i iq p q p q q� � � � �
3. Formulization of rigid motion: Using (7) rigid transformation of serial robot manipulator can be expressed as
1 1 2ˆ ˆ ˆ ˆn nq q q q� � � �� (21)
The orientation and the position of the end effector can be expressed as follows,
*1 1 1o n o nl q l q� � � �
� � � �� �1 1 1 1 1 1 1 1
o o� �� � � � �ep V n V n V n V n V n V np q q q q q q
(22)
B. Inverse Kinematics
Paden-Kahan subproblems which can be found in [7] and [18] are used to obtain the inverse kinematic solution.
Quaternion based method
The orientation and the position of the end effector are defined using inq and o
inq respectively. Then the solution is as the following,
Step 1: Firstly, we put two points at the intersection of the axes. The first one is wp which is at the intersection of the
54
wrist axes and the second one is bp which is at the intersection of the first two axes. The last three joints do not affect the position of the point wp and the first two joints do not affect the position of the point bp . Then we can easily write the following equations,
* *
13 13 13 16 16 16o o
w wq p q q q p q q� � � � � � � (23)
*12 12 12
ob bq p q q p� � � � (24)
The position of the point bp is free from the angles of first two joints. If we subtract these two equations from each other then, we get
* *13 13 13 12 12 12
* *16 13 13 12 12 12
o ow b
o ow b
q p q q q p q q
q p q q q p q q
� � � � � � �
� � � � � � � � (25)
If we take the end effector position 0 1 2 3( , , , )o o o o oinq q q q q� at the
intersection of the wrist axes then, we have ow inp q� . Hence
we can write * *
3 3 16 16 16o o
w w inq p q pb q p q q pb q pb� � � � � � � � � � (26)
Using the property that distance between points is preserved by rigid motions; take the magnitude of both sides of equation (26), we get subproblem 3. The parameters of sub problem 3 are � o
w inp = p q and bq = p . 3 can be found by using subproblem 3.
Step 2: If we translate wp by using known 3 we obtain a new point p . We get subproblem 2 by using the point p as the initial position and the point q as the final position of the subproblem 2 motion. The points p and q can be formulized as follows
* *12 3 3 12 13( ) o o
w inq q p q q q pb q pb� � � � � � (27)
* * *12 3 3 3 3 3 3 12( ) o
w inq q p q p q p q q q� � � � � � � (28)
Hence, * *3 3 3 3 3 3wp q p q p q p q� � � � � � � and o
inq q� .
1 and 2 can be found by using subproblem 2.
Step3: To find wrist angles we put a point cp which is on the 6d axis and it does not intersect with 4d and 5d axes. Thus the point cp is not affected from the last joint angle. Fourth and fifth joints angles determine the position of the point cp . For known 1 2, and 3 angles, we can write the following equation
The equation (29) gives us the subproblem 2. To obtain the parameters of subproblem 2 we should find the points
p and q . The point p is equal to cp . We should just find q . The point q can be found given by
* * *
0 1 1 2m c m m m m m t tq q p q q p q q p q q q� � � � � � � � � � � (30)
where *1 2 3( )m inq q q q q� � � � , *
1 3 3 3 3tq q p q p� � � � and * *
2 3 2 3 2 3 3 2 3*
2 3 1 2 3
( ) ( )
( ) ( )tq q p q q q p q q
q q p q q
� � � � � � � � �
� � � � (31)
Step4: The first five joint angles are obtained. Only the last joint angle is unknown. Of course, we can find the last joint angle by using given subproblems however; it can be also easily solved by using algebraic equations for the known joint angles. In this step we will find the last joint angle by using algebraic equations. The last joint angle can be found from the orientation part of the kinematic equation given by,
� �
122
6 arctan 2 , 1S k S k � �
� �� �� �
(32)
where *1 2 3 4 5( ) ink q q q q q q� � � � � � .
Dual-Quaternion based method
The orientation and the position of the end effector are defined using ˆ ( , )o
in in inq q q� . Then the solution is as the following,
Step 1: Firstly, we put two points at the intersection of the axes. The first one is wp which is at the intersection of the wrist axes and the second one is bp which is at the intersection of the first two axes. The last three joints do not affect the position of the point wp and the first two joints do not affect the position of the point bp . Then we can easily write the following equations,
* *13 6 13 13 6 13
* *13 5 13 13 5 13
* *13 6 13 13 6 13
* *12 2 12 12 2 12
* *12 1 12 12 1 12
12
ˆ ˆˆ ˆ ˆ ˆ( )
ˆ ˆˆ ˆ ˆ ˆ( )
ˆ ˆˆ ˆ ˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ( )
ˆ ˆˆ ˆ ˆ ˆ( )
ˆ
V R q l q V D q l q
V R q l q V D q l q
V R q l q V R q l q
V R q l q V D q l q
V R q l q V D q l q
V R q
� �� � � � � �� �� �� �� � � � � � �� �� �
� � � � �� �� �
� � � � � �
� � � � � �
* *2 12 12 2 12ˆ ˆˆ ˆ ˆl q V R q l q
� �� �� �� � � �� �� �
� � � � �� �� �
oin bq p
(33)
If we take the magnitude of both sides of equation (33) then, we get the subproblem 3. The parameters of the
subproblem 3 can be defined as � �6 6ˆ ˆV R l V D l� � �a
� � 5 5 6 6ˆ ˆ ˆ ˆV R l V D l V R l V R l� �� �� �
� � � 2̂V R l �b =
55
� � � 2 1 1 2 2ˆ ˆ ˆ ˆ ˆV D l V R l V D l V R l V R l� �� � �� �
� �, l is
the axis of the joint 3 that is 3d and � �oin b� q p . 3 can be
found by using the subproblem 3.
Step 2: If we use known 3 in the equation (33), we obtain
� � � �
* *12 6 12 12 6 12
* *12 5 12 12 5 12
* *12 6 12 12 6 12
ˆ ˆˆ ˆ ˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ(
ˆ ˆˆ ˆ ˆ ˆ)
V R q l q V D q l q
V R q l q V D q l q
V R q l q V R q l q
� �� � � � � �
� �� � � � � �
� �� � � � � � oinq
(34)
where *6 3 6 3ˆ ˆˆ ˆl q l q� � � � and *
5 3 5 3ˆ ˆˆ ˆl q l q� � � � . The equation (34)
gives the subproblem 2. The parameters of subproblem 2 are
� � � � 6 6 5 5 6ˆ ˆ ˆ ˆ ˆV R l V D l V R l V D l V R l� �� � � � �� � � � �� �
� �a
6̂V R l�� , 1l is axis of the joint 1 that is 1d , 2l is the axis of
the joint 2 that is 2d and � oinb q . 1 and 2 can be found by
using the subproblem 2.
Step3: To find the wrist angles, let’s consider a point 6 �� �i 6p p d (initial point) that is on the axis 6d , and it is
not coincident with the 4d and 5d axes. Two imaginer axes are used to find ep (end point) that is the position of the point
ip after the rotation by 4 and 5 angles. The point ip is the intersection point of the two imaginer axes. Let’s define the two imaginer axes which are on the 6d axis and intersect on the point 6 0 1 6 0 1 2 6[ , , ]x y zd ly ly d lz lz lz d� � �� � � � � �ip given
by [0,1,0]�7d and [0,0,1]�8d . Then we can easily write,
parameters of subproblem 2 are � �8 8ˆ ˆV R l V D l� �� � �a
� � 7 7 8 8ˆ ˆ ˆ ˆV R l V D l V R l V R l� �� � � �� �� �
� �1l is the imaginer
axis of 7d , 2l is the imaginer axis 8d and
�� �oin 6b q d . 4 and 5 can be found by using the
subproblem2. Step4: To find the last joint angle, we need a point which
is not on the last joint axis. We call it 5 �� �d 5p p d . Two imaginer axes are used to find �dp that is the position of the point dp after the rotation by 6 . The point dp is the
intersection point of the two imaginer axes. Let’s define the two imaginer axes 9d and 10d which are on the 5d Then we can easily write,
� � � �
* *6 10 6 6 10 6
* * *6 9 6 6 9 6 6 10 6
*6 10 6
ˆ ˆˆ ˆ ˆ ˆ
ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ
ˆˆ ˆ
V R q l q V D q l q
V R q l q V D q l q V R q l q
V R q l q �
� �� � � � � �
� �� � �� � � � � � � �� �� �
�� � � �oin 5q d
(36)
The equation (36) gives us the subproblem 1. The
parameters of subproblem 1 are � �10 10ˆ ˆV R l V D l� �� �a
� � 9 9 10 10ˆ ˆ ˆ ˆV R l V D l V R l V R l� �� � � �� � �� �
� �, l is axis of
the joint 6 that is 6d , and �� �oin 5b q d .
V. SIMULATION RESULTS
Here, a comparative study of the presented and exponential mapping methods is worked out. These solutions are implemented to the 6-DOF industrial serial robot manipulator which is shown in figure 2. Exponential mapping method uses homogeneous transformation matrices as a screw motion operator. Homogeneous transformation matrices require sixteen memory locations for definition of rigid body motion while the quaternions require eight memory locations. The storage requirement affects the computational time because the cost of fetching an operand from memory exceeds the cost of performing a basic arithmetic operation. The performance comparison of homogeneous transformation matrix and quaternion operators can be seen in table I.
TABLE I PERFORMANCE COMPARISON OF RIGID MOTION OPERATORS
In order to obtain the rigid body transformation operator for a n-link serial robot manipulator � � �64 1n � multiplies and � �48 1n � additions must be used if screw theory with exponential mapping method is used. � � �57 1n � multiplies and � �42 1n � additions must be used if screw theory with quaternion method is used.
� � �48 1n � multiplies and � �40 1n � additions must be used if screw theory with dual-quaternion method is used.
Figure 3 shows that as the degrees of freedom increase the method which uses dual-quaternion as a rigid body transformation operator becomes more advantageous.
56
0 5 10 15 200
500
1000
1500
2000
2500
Degrees of Freedom
Num
bers
of T
otal
Cal
cula
tions
Hom. Trans. MatrixQuaternionDual-Quaternion
Figure 3: Performance comparison of the rigid body transformation
chaining operations
Simulation results of these methods are given in figures 4 and 5. The computation time is evaluated using Matlab’s tic-toc commands.
Figure 4: Simulation times of the forward kinematic solutions (second)
00.51
Single�Solutin Successive�Solution
Exponential�map.
Quaternion
Figure 5: Simulation times of the inverse kinematic solutions (second)
As it can be seen from the figures 3, 4 and 5, the methods which use quaternion algebra as a screw motion operator, are more computationally efficient than exponential mapping method. And also the dual-quaternion based method gives the best result in both forward and inverse kinematics. Since the formulation of the dual-quaternion based method is more compact than quaternion based method. Running environment is as table II.
TABLE II RUNNING ENVIRONMENT
Cpu Cpu Memory Operating System
Software
Core2Duo 2.2 GHz 2 GB Windows XP Matlab 7
V. CONCLUSION
In this paper, three screw theory based formulation methods of the kinematic problem of serial robot manipulators have been compared in terms of computational efficiency. The main advantage of screw theory in robot kinematics lies in its geometrical representation of link and joint axes of a manipulator, giving better understanding of its configuration
in the workspace and avoiding singularities due to the use of the local coordinates. In these formulation methods, the first one uses quaternions as a screw motion operator which combines a unit quaternion plus pure quaternion, the second one uses dual-quaternions as a screw motion operator and the last one uses 4x4 homogeneous matrices as a screw motion operator. Screw theories with quaternion and dual-quaternion methods are more computationally efficient than exponential mapping method. And also screw theory with dual-quaternion is the best method. On these accounts, the wider use of the screw theory based methods with dual-quaternion in robot kinematic studies has to be considered by robotics community.
REFERENCES [1] R. S. Ball, The Theory of Screws. Cambridge, U.K.: Cambridge
Univ.Press, 1900. [2] J.D. Adams and D.E. Whitney, Application of Screw Theory to
Constraint Analysis of Mechanical Assemblies Joined by Features, J.Mech.Des.--March 2001--Volume 123,Issue 1,26 (7 pages) DOI:10.1115/1.1334858
[4] A.T. Yang and F. Freudenstein, Application of dual-number quaternions to the to the analysis of spatial mechanism, ASME Trans. J. Applied Mechanics, 300-306 (1964)
[5] A.T. Yang, Displacement analysis of spatial five-link mechanism using (3 x 3) matrices with dual number elements, ASME Trans. J. Engr. For Industry Feb.1969,152-157
[6] G.R. Pennock and A.T Yang, Application of dual-number matrices to the inverse kinematic problem of robot manipulators, ASME Trans. J. Mech. Trans. Auto. In Design,107,201-208(1985)
[7] M. Murray, Z. Li, and S.S. Sastry, A mathematical introduction to robotic manipulation, Boca Raton FL:CRC Press,1994.
[8] J.Xie, W. Qiang, B. Liang and C. Li, Inverse Kinematics Problem for 6- DOF Space Manipulator Based On The Theory of Screw, International Conference on Robotics and Biomimetics ,Dec.2007
[9] J. Funda and R.P. Paul, A computational analysis of screw transformations in robotics, IEEE Trans. Robot. Automat., vol.6, pp. 348–356,June 1990
[10] J. Funda, R.H. Taylor and R.P. Paul, “On homogeneous transforms, quaternions, and computational efficiency”, IEEE Transactions Robotics and Automation, vol.6, pp. 382–388, June 1990.
[11] E. Sariyildiz and H. Temeltas, “Solution of Inverse Kinematic Problem for Serial Robot Using Dual Quaterninons and Plücker Coordinates”, Advanced intelligent mechatronics, 2009
[12] E. Sariyildiz and H. Temeltas, “Solution of Inverse Kinematic Problem for Serial Robot Using Quaterninons”, International Conference on Mechatronics and Automation, 2009
[13] W.R. Hamilton, Elements of Quaternions, Vol. I,Vol II New York,Chelsea,1869.
[14] Qing Tan; Balchen, J.G. “General quaternion transformation representation for robotic application”, International Conference on Volume , Issue , 17-20 Oct 1993 Page(s):319 - 324 vol.3
[15] H. Bruyninckx and J. D. Schutter , “Introduction to Intelligent Robotics” 7th edition 1October 2001
[16] Y-L. Gu and J.Y.S. Luh “Dual Number Transformation and Its Application to Robotics” , IEEE Journal of Robotics and Automation, vol. RA-3, no. 6, December 1987
[17] Konstantinos Daniilidis ,”Hand-Eye Calibration Using Dual Quaternions” , The International Journal of Robotics Research Vol. 18, No. 3, March 1999, pp. 286-298
[18] B. Paden, Kinematics and Control Robot Manipulators, PhD thesis Department of Electrical Engineering and Computer Science, University of California,Berkeley,1986
![Page 1: [IEEE 2011 IEEE International Conference on Mechatronics and Automation (ICMA) - Beijing, China (2011.08.7-2011.08.10)] 2011 IEEE International Conference on Mechatronics and Automation](https://reader036.documents.pub/reader036/viewer/2022073012/575082461a28abf34f985641/html5/thumbnails/1.jpg)
![Page 2: [IEEE 2011 IEEE International Conference on Mechatronics and Automation (ICMA) - Beijing, China (2011.08.7-2011.08.10)] 2011 IEEE International Conference on Mechatronics and Automation](https://reader036.documents.pub/reader036/viewer/2022073012/575082461a28abf34f985641/html5/thumbnails/2.jpg)
![Page 3: [IEEE 2011 IEEE International Conference on Mechatronics and Automation (ICMA) - Beijing, China (2011.08.7-2011.08.10)] 2011 IEEE International Conference on Mechatronics and Automation](https://reader036.documents.pub/reader036/viewer/2022073012/575082461a28abf34f985641/html5/thumbnails/3.jpg)
![Page 4: [IEEE 2011 IEEE International Conference on Mechatronics and Automation (ICMA) - Beijing, China (2011.08.7-2011.08.10)] 2011 IEEE International Conference on Mechatronics and Automation](https://reader036.documents.pub/reader036/viewer/2022073012/575082461a28abf34f985641/html5/thumbnails/4.jpg)
![Page 5: [IEEE 2011 IEEE International Conference on Mechatronics and Automation (ICMA) - Beijing, China (2011.08.7-2011.08.10)] 2011 IEEE International Conference on Mechatronics and Automation](https://reader036.documents.pub/reader036/viewer/2022073012/575082461a28abf34f985641/html5/thumbnails/5.jpg)
![Page 6: [IEEE 2011 IEEE International Conference on Mechatronics and Automation (ICMA) - Beijing, China (2011.08.7-2011.08.10)] 2011 IEEE International Conference on Mechatronics and Automation](https://reader036.documents.pub/reader036/viewer/2022073012/575082461a28abf34f985641/html5/thumbnails/6.jpg)
