INTRODUCTION
Binocular stereo (robot eyes) can be carried on all kinds of mobile robots,
underwater robots and flying robots (Zhixiang et al.,
2011) because of the following advantages: small size, light weight and
compact volume, etc. And it can be used in various environments that are dangerous
or unsuitable for human such as traffic, military, counter terrorism, reconnaissance,
disaster relief and so on. But bionic eye is often affected by unstructured
and bumpy environments, for example, uneven ground for mobile robots (Mehrjerdi
et al., 2010), surge for underwater robots, wind speed for flying
robots, etc. Those not only cause instability or distortion of the receiving
images but also increase the difficulties of observing images by human eyes
and processing images by computers (Maohai et al.,
2011; Quan and Ming, 2011). So far, many bionic
eyes have been designed. But in many cases, these designed bionic eyes only
achieve onedimensional or twodimensional rotation and lack of torsion (Itoh
et al., 2006; Maini et al., 2008;
Batista et al., 2000; Yu
and Wang, 2004). One or two axes of bionic eyes could provide images that
can be digitalized and vision processed to compensate for distortions. However,
the torsion axes are crucial for realizing accurate human eye captured images
with a proper sampling rate. Besides, the software compensation process for
a missing torsional rotation is computationally complex and simultaneously disables
data acquisition. Only with the third torsion axis can the received images be
realized in 3D properly. But little attention has been paid to emulate the actual
mechanics of the eye (Bang et al., 2006).
Spherical Parallel Mechanism (SPM) has 3 rotational degrees of freedom (3DOF)
which can offer excellent dynamic performance, light weight, high speed and
high precision (Gosselin and StPierre, 1997). Those
just coincide with the characteristics of bionic eye requirements. So it is
appropriate that SPM is applied in bionic eye design. The method of global dimensional
synthesis was often applied for previous design of spherical parallel mechanism
which optimized the global performance indexes and got orthogonal spherical
parallel mechanism (Gosselin and StPierre, 1997; Liu,
1999; Jin and Rong, 2007). However, bionic eye has
special space and structure requirements which is different from those fields.
If bionic eye is designed with orthogonal spherical parallel mechanism, it would
result in being a larger prototype, no meeting the working space requirements
and interfering easily between components.
System description: As shown in Fig. 1, the SPM is
composed of a moving platform (endeffector) and a fixed base that are connected
by three equally spaced limbs, each consisting of revolute joints only. The
axes of all joints intersect at a common point O which is named the SPM center.

Fig. 1: 
The structures of SPM 
The motion of any point in the mechanism rotates about the point O thereby
a spherical parallel manipulator provides three degrees of freedom of pure rotations.
According to the method of DH (Zhangqi et al., 2011)
linkage coordinate system, a fixed coordinate system Ox_{0} y_{0}
z_{0}, moving coordinate system Ox_{0}' y_{0}' z_{0}'
and linkage coordinate system Ox_{ij} y_{ij} z_{ij}
(i, j =1, 2, 3,..., i, j, denotes the jth revolute joint of ith limb) are set
up. The plane paralleling to the fixed platform and passing the SPM center is
called middle plane (Fig. 2). Since, the manipulator should
be symmetric, the structural parameters of a 3DOF SPM can be reduced to four
geometrical angles α_{1}, α_{2}, β_{1}
and β_{2}, which is shown in Fig. 1. α_{1},
α_{2} are, respectively defined as the angle between axis z_{1i}
and z_{2i}, axis z_{3i} and z_{3i}, β_{1},
β_{2}, respectively means semicone angle of the fixed base, semicone
angle of the moving platform. u_{i}, w_{i}, v_{i} (i
= 1, 2, 3) denotes unit vector of coordinateaxis z_{1i}, z_{2i},
z_{3i} (i = 1, 2, 3) for fixed coordinate system, respectively.
Moving platform (the eye) attitude can be expressed by direction cosine matrix
R_{XYZ} of moving coordinate system Ox_{0}' y_{0}'
z_{0}' to fixed coordinate system Ox_{0} y_{0} z_{0}
(Gosselin and StPierre, 1997):
where, φ_{x}, φ_{y}, φ_{z} denotes yaw,
pitch, roll angles of bionic eye, respectively. S^{•} and C^{•}
represents sin and cos, respectively.

Fig. 2: 
The moving platform and β_{2} 
The inverse kinematics solution: The inverse kinematics is that three
input angles of motor (θ_{1}, θ_{2}, θ_{3})
are solved through the known three attitude angles of bionic eye (φ_{x},
φ_{y}, φ_{z}) (Zhangqi et al.,
2011). Closed loop equation can be obtained from Alternate angle Law (Cong
et al., 2011):
Direction vector of u_{i}, w_{i}, v_{i} in fixed coordinate
system Ox_{0} y_{0} z_{0} can be obtained by attitude
transformation. The inverse kinematics solution can be figured out from formula
2 (Gosselin and StPierre, 1997).
Jacobian matrix: The Eq. 3 is established by differentiation of formula 2 and utilizing the mixed product of vectors:
As also showed in matrix form:
where,
represents angular velocity vector of motor, ω = (ω_{x}, ω_{y},
ω_{z})^{T} represents angular velocity vector of moving
platform, J_{1} = diag [b_{1}, b_{2}, b_{3}
], b_{i} = (u_{i}xw_{1})^{T}•v_{i}
represents the first kind of Jacobian matrix, J_{2} = [w_{1}xv_{1},
w_{2}xv_{2}, w_{3}xv_{3}]^{T} represents
the second kind of Jacobian matrix. J denotes Jacobian matrix which is determined
by four structural parameters α_{1}, α_{2}, β_{1},
β_{2} and three attitude angles φ_{x}, φ_{y},
φ_{z}.
The dexterity performance index: Salisbury and Craig
(1981) defined the dexterity of a robotic manipulator as the kinematic accuracy.
Mathematically, they defined the dexterity as the condition number of the Jacobian
matrix of the robot. The dexterity of the 3DOF SPM, noted k, can then be written
as follow (Gosselin and StPierre, 1997):
where, J denotes any norm of its matrix.
It is bounded as follows:
And hence, the reciprocal of the condition number (noted and referred to here as the dexterity of the manipulator) is used instead.
where, 0≤ζ≤1, ζ vary with the configuration of the robot and the structural parameters of the robot. A value of ζ equal or close to 1 corresponds to a configuration with a very good kinematic accuracy while a value of ζ equal to 0 is obtained when the manipulator is in a singular configuration. The bigger the value of ζ is, the better the dexterity and control accuracy of the robot are.
Dynamic performance index: Dynamic performance index can be expressed
as below (Liu, 1999):
where, ξ denotes dynamic performance index of the robot. The bigger the value of ξ is, the better the dynamic performance of the robot is.
The target working space and preliminary optimization of structural parameters
The target working space: The eyeball is assumed to be modeled as a homogeneous
sphere of radius R, having 3 rotational degrees of freedom about its center
and the largest range of human eyes can be pointed in a vision cone of ±38°
with ±15° in torsion (Cannata and Maggiali, 2006).
There exist errors where in the process of SPM movement, key parts processing
and assembling, revolute joints clearance, servo control, position drift for
angular pyramid vertex of moving platform relative to fixed base, component
thermoelastic distortion (Zeng et al., 2008;
AbRahman et al., 2011; Suhail
et al., 2011) which would make the actual working space is slightly
bigger than the target working space. Therefore, the target working space for
the prototype is designed a cone of ±38° with ±15° in
torsion (in a cone of 38°, any position can achieve at least 15° in
torsion). The maximum semicone angle approximately reached 40° that was
shown in the final experimental results.
Preliminary optimization of structural parameters: According to the special structure requirements of bionic eye, relative parameters are preliminarily optimized before optimizing the structural parameters.
α_{1} and α_{2} (structure angles of driving bar and passive bar): theoretically, the range of α_{1} and α_{2} are both (0, 180°). Excessive small value of α_{1} and α_{2} would cause the working space too small and does not meet the design requirements but also bring difficulty to assemble the bionic eye. Excessive large value of α_{1} and α_{2} would result in a larger prototype of bionic eye, interfering easily between components, besides, it may make the passive bar locate above middle plane in the movement course which leads to block the camera view and interfere with robot’s face. Thus, the range of α_{1} and α_{2} are set as: (50°, 80°).
β_{1} (semicone angle of the fixed base): The center of
the revolute joints of each limb locating on the different radius of the sphere
rotating the point O, In order that the limbs can’t be interfered with
each other. The different radius of sphere mentioned above are denoted as
respectively and moreover, .
This does not affect the dexterity. In this case, the greatest impact on overall
size of bionic eye is the radius of fixed base ().
In the case of
being constant, the greatest impact on overall size of bionic eye is the angle
of β_{1}, with the decreasing of the angle of β_{1},
R_{base} will decrease exponentially. In addition, in order to avoid
the interference with the three motors when β_{1} is too small,
the range of β_{1} is set as: (25°, 35°).
β_{2} (semicone angle of the moving platform): Many papers
(Cong et al., 2011; Liu, 1999;
Jin and Rong, 2007) have put forward the range of β_{2}
are (0, 90°). But this is not appropriate for the design of bionic eye.
On one hand, it is ensure that the imaging focus of builtin camera is just
located in the point O, so that the deflection of bionic eye is consistent with
that of the eyeball. On the other hand, it is also ensure that the driving bars
and the passive bars located below the middle plane in the movement course which
do not block the camera view and do not interfere with the robot’s face.
Based on above considerations, the value of β_{2} must be more
than 90°, i.e., the moving platform must locate blow the middle plane (Fig.
2). Besides, it is considered that the three passive bars connecting with
the bionic eye can not interfere with each other. The range of β_{2}
is set as: (125°, 135°).
STRUCTURAL PARAMETERS OPTIMIZATION DESIGN
The minimum values of dexterity and dynamic performance indexes in target working space: The method of global dimensional synthesis takes the average performance index of workspace as the optimizing object. With this method, the design parameters are determined and the average performance index is very good but it does not make arbitrary posture dexterity of target working space better. However, the method optimized the minimum performance index can make the dexterity and dynamic performance of bionic eye all over the target workspace better.
Put force Jacobian matrix G = (J^{1})^{T} (Liu,
1999) into formula 9, we can obtain:
Similarly, dynamic performance can be expressed as below:
According to formula 10, properties of matrix and norm, we can obtain:
(•_{1}, •_{2}, •_{F},
•_{∞} denote 1norm, 2norm, Fnorm, infinitynorm,
respectively). That is to say, the values of k(J) and k(G) under 2norm and
Fnorm are equal, respectively. The values of k(J) under infinitynorm and the
values of k(G)under 1norm are equal, respectively. The values of k(J) under
1norm and the values of k(G) under infinitynorm are equal, respectively.
In order to obtain the minimum values of ζ and ξ from Eq.
7 and 8, the maximum values of velocity Jacobian matrix
condition number k(J) from Eq. 5 and force Jacobian matrix
condition number k(G) from Eq. 10 should be obtained. In
the case that target working space is determined, ranges of four parameters
are optimized preliminarily and each element absolute value in velocity and
force Jacobian matrix is mostly between 0 and 1. The values of k(J) and k(G)
under 1norm, 2norm, Fnorm and infinitynorm are compared and lots of experimental
data is shown that the maximum values of k(J) and k(G) always arise under 1norm
and infinitynorm. Parts of data are only listed in Table 1.
So, we defined the minimum values of dexterity and dynamic performance indexes in target working space as:
Dexterity and dynamic performance are all considered under 1norm and infinitynorm in formula 11 and 12 which ensures that the values of dexterity and dynamic performance indexes are the minimum.
Structural parameter optimization with step method: The purpose of this design is to determine four design parameters α_{1}, α_{2}, β_{1}, β_{2} and make values ζ_{min} and ξ_{min} to be maximum by satisfying the special structure requirements and the target working space of the bionic eye.
Parameters optimization program is developed in MATLAB. The best structural parameters are searched with step method. Based on the known values of β_{1} and β_{2}, we change the values of α_{1} and α_{2} from small to large number with 1° for step. Finally, the maximum values of ζ_{min} and ξ_{min} are obtained.
Figure 3 is the relationship between ζ_{min}
and α_{1}, α_{2}, β_{1}, β_{2}
with different values of β_{1} and β_{2}, Fig.
4 is the relationship between ξ_{min} and α_{1},
α_{2}, β_{1}, β_{2} with different values
of β_{1} and β_{2}.
Table 1: 
Values of k(J) and k(G) under four norm with different values
of α_{1}, α_{2}, β_{1}, β_{2} 


Fig. 3(ai): 
The relationship between ζ_{min} and α_{1},
α_{2}, β_{1}, β_{2} 

Fig. 4(ai): 
The relationship between ξ_{min} and α_{1},
α_{2}, β_{1}, β_{2} 

Fig. 5(ac): 
Experimental platform based on tracked robot 

Fig. 6(ab): 
Experimental results under harsh environment 
From Fig. 3 and 4, it is known that when
25°≤β_{1}≤35° and 125°≤β_{2}≤135°,
the maximum values of ζ_{min} and ξ_{min} are mostly
located in α_{1} = 75°, 80° and α_{2} = 75°
nearby. Meanwhile, ζ_{min} and ξ_{min} decrease along
with the increase of β_{1} and β_{2}. According to
the analysis, it was found that when β_{1} = 25° β_{2}
125°, β_{1} = 25° β_{2} 130° and β_{1}
= 30° β_{2} 125°, the maximum values of ζ_{min}
are all better. When β_{1} = 30° β_{2} 125°,
the maximum values of ζ_{min} are best. Finally, considering dexterity
and dynamic performance indexes, we select α_{1} = 75°, α_{2}
= 75°, β_{1} = 30°, β_{2} = 125°.
ROBOT EXPERIMENT
Some experiments based on a tracked robot equipping with bionic eye are implemented in a bumpy environment. The experimental platform is developed, as shown in Fig. 5. A robot control and image processing module is shown in Fig. 5a. The tracked robot carrying binocular stereo is presented in Fig. 5b. An enlargement of bionic eye with SPM is demonstrated in the Fig. 5c.
One can obviously find two interesting results in the experiment. First, binocular stereo have good dexterity and dynamics performance, operating flexibility, no mutual interference in the target working space for any posture. In the actual operation process, we found that the actual workspace of the bionic eye is slightly bigger than the target working space and semicone angle can approximately reach 40°. Second, binocular stereo have very good adaptive ability and it can actively compensate the visual error caused by robot attitude variation. Stabilization image can be obtained even under bumping environment.
Parts of the experimental results are shown in Fig. 6. Figure
6 a and b are the experimental results when the tracked
robot attitude occurs roll (a, compensating for the attitude; b, without compensation
for the attitude.
CONCLUSION
In this study, a 3DOF spherical parallel manipulator is presented and the prototype of the 3RRR spherical parallel binocular Stereo has been built in the lab (Fig. 5b). Specifically, the worst dexterity and dynamic performance index for bionic eye is defined by the maximum condition number of Jacobian matrix. The chart method is used to search for optimal design parameters and reasonable structural parameters are chosen in the case of satisfying the special structure requirements of bionic eye. The techniques used in this study can also be applied to he optimization of other parallel manipulators. Finally, the experimental results show that bionic eye with the optimized parameters is good coinciding with the design requirements. The size of bionic eye is slightly bigger than that of the actual eyeball in consideration of camera size (31x31 mm), those radius are 45 and 30 mm, respectively. It was found that the violent vibration of robot body and visual inertial navigation drifts would affect the accuracy of the bionic eye. Those would be taken into account for designing bionic eye in the future.
ACKNOWLEDGMENTS
This project is supported by National Hitech Research and Development Program of China (863 Program, Grant No. 2009AA04Z211), National Natural Science Foundation of China (Grant No. 50975168, Grant No. 60975068) and Ph.D. Programs Foundation of Ministry of Education of China (Grant No. 20093108110007).