INTRODUCTION
In the manufacturing process of the aircraft Sshaped inlet, advanced automatic
fiber placement technology is used. The fiber is placed automatically on the
mandrel by the endeffector according to the preplanned trajectory (Liu,
2011; Zhu et al., 2011). The robotic manipulator
has at least 6 Degrees of Freedom (DOF) to place the fiber on mandrel but the
6DOF robotic manipulator can not track mandrel position in arbitrary orientation.
Therefore, some scholars improved the robotic manipulator’s tracking goal
capability by increasing DOF, in other words to improve the robotic manipulator’s
dexterousness, the DOF by additional is called redundant DOF; this robotic manipulator
with additional DOF is called redundant robotic manipulator (Han
et al., 2009; Yang et al., 2011).
Redundant robotic manipulator has more flexibility and better obstacle and singularity
avoidance capability than robotic manipulator has less than 6DOF (Lu,
2007; Zhang et al., 2004). So, studying
7DOF redundant robotic manipulator is more meaningful. The inverse kinematics
is the basis of planning and motion control for robotic manipulator which is
the same as redundant robotic manipulator. The first need is to solve the inverse
kinematics problem if trajectory planning and motion control are to be achieved.
Redundant robotic manipulator’s selfmotion manifold contains all the inverse
kinematics (Moll and Kavraki, 2004; Muller,
2004; Zhao et al., 2007, 2009;
Xinfeng and Dongbiao, 2012; Zhang
et al., 2011; Kazem et al., 2010),
shows the mapping relationship between the endeffector and joints and reflects
the selfmotion ability essentially.
Inverse position should be obtained in order to get selfmotion manifold, the inverse position is difficult to solve for redundant robotic manipulator whose DOF is more than 6. Inverse kinematics based on selfmotion manifold are all the inverse kinematics, therefore, analyzed the redundant automatic fiber placement robotic manipulator’s selfmotion manifolds which has some reference to analyzing inverse kinematics of other type redundant robotic manipulators.
The selfmotion manifold is used to obtain the all inverse kinematics solutions
of the redundant robotic manipulators. The position space manifold and orientation
space manifold are defined at first, then the selfmotion manifold could be
obtain by pairing the position space manifold and orientation space manifold,
at last verified by example.
7DOF AUTOMATED FIBER PLACEMENT REDUNDANT ROBOTIC MANIPULATOR’S TOPOLOGY
The automatic fiber placement robotic manipulator’s structure is shown
in Fig. 1a. From its structure, there is a 6DOF fiber placement
robotic manipulator and a rotational mandrel. According to the principle of
equivalence movement: mandrel coordinate system is fixed with the base coordinate
system, a virtual revolute joint linked the base and the mandrel together, Rotational
mandrel is equivalent to the robotic manipulator’s rotation around the
base. So the fiber placement robotic manipulator with 6DOF and the mandrel
with 1DOF becomes a 7DOF redundant robotic manipulator. The shoulder has a
revolute joint, the elbow has three prismatic 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. 1b.

Fig. 1(ab): 
Automated fibre placement robotic manipulator’s structure
and topology. 1: Tailstock, 2: Mandrel, 3: Chuck, 4: Link, 3, 5: Link, 2,
6: Link, 1, 7: Placement head, 8: Link, 6, 9: Link, 5, 10: Link, 4, 11:
Mandrel part and 12: Manipulator part 
Table 1: 
Parameters of the automated fiber placement robotic manipulator 

Establishing DH coordinate system and its structural parameters as shown
in Table 1.
REDUNDANT ROBOTIC MANIPULATOR SELFMOTION MANIFOLD
Position space manifold of redundant robotic manipulator: The endeffector
position can be obtained easily by the 7th joint coordinate system original
position. The definition of moving coordinate system does not affect analyzing
the robotic manipulator’s kinematics, so the endeffector coordinate system
can be assumed coinciding with the 7th joint coordinate system. The automated
fibre placement robotic manipulator’s three prismatic joints intersect
at one point and determined the endeffector’s spatial position. Then the
endeffector position is determined by the ,
the
is the vector between intersection of the 5, 6 and 7th joint axis and origin
of the base coordinate system, all such
constitutes the redundant robotic manipulator’s position work space, so
the three prismatic joints is also called position joints.
According to the equivalent motion of the automatic fiber placement robotic manipulator, waist, shoulder and elbow’s configuration space can be defined as:
The redundant robotic manipulator’s position space U_{p} can be expressed as:
Here, f_{p} is the map from C_{WSC} to U_{p}; θ_{p} = [θ_{1}, d_{2}, d_{3}, d_{4}]^{T}.
Either take
∈ U_{P} position space manifold is defined which the vector
is corresponding to:
C_{p} is directly called position space manifold.
The redundant fiber placement robotic manipulator orientation joint manifold: Three revolute joints axis on wrist intersect at point O_{4}, endeffector orientation adjustment can be carried out only by the 5, 6 and 7th joints’ rotation and position of point O_{4} does not change when endeffector orientation adjusts. Three revolute joints on wrist are also called orientation joints. Then the redundant robotic manipulator is decoupled in position and orientation.
The wrist joints configuration space is defined as:
Assuming
and
for the given θ_{pk}εC_{p}. When the robotic manipulator
is in some singularity, kinematics redundant will occur when adjusting wrist
joints orientation. Assuming:
orientation joints manifold which the vector pair (θ_{pk}, z7O_{4}, y7O_{4}) is corresponding to is defined:
Similarly, C_{o} is directly called orientation joint manifold.
The redundant fiber placement robotic manipulator selfmotion manifold:
The redundant fiber placement robotic manipulator’s endeffector position
workspace can be expressed by the position space manifold; endeffector orientation
can be expressed by two coordinate axes vector z_{7} and y_{7}
of the 7th joint. Any one point on the position workspace manifold corresponds
to a orientation space, the orientation space can be described as all the vector
pairings (z_{7}, y_{7}) which matched by axis z_{7}
and axis y_{7} of the 7th joint coordinate system on the point, endeffector
orientation space can be described as all the vector pairings of .
The joint space of the ith joint can be expressed as C_{i} and then the joint configuration space C can be expressed as seven separated product space of C_{i}:
The role of robotic joint is equivalent to the role of functions in mathematics which implements mapping and inverse mapping between the joint configuration space and the endeffector position and orientation workspace. Assuming θ = [θ_{1}, θ_{2}, L, θ_{7}]^{T}, so the redundant robotic endeffector position and orientation workspace can be described as:
Where, f can be obtained by direct kinematics of redundant robotic manipulator.
Either takes vector uεU, so the selfmotion manifold which corresponded to by vector u can be described as a pairing that composed by a configuration subspace C_{SO} and a set of mappings F_{SO}:
f is easy to obtain by θ for the redundant robotic manipulator, therefore, the main task which solves redundant robotic manipulator selfmotion manifold is to solve C_{SO} and C_{SO} is called the redundant robotic manipulator self motion manifold.
THE AUTOMATIC FIBER PLACEMENT REDUNDANT ROBOTIC MANIPULATOR SELFMOTION MANIFOLD
Three revolute joint axes of the automatic fibre placement robotic manipulator wrist intersect at one point and determined the robotic manipulator endeffector’s orientation in the work space; three prismatic joint axes intersect at one point and determined the robotic manipulator endeffector’s position in the workspace. The endeffector’s orientation can be achieved only by three wrist joints and the endeffector position can not be moved, so the robotic manipulator’s selfmotion manifold can be divided position space manifold and orientation space manifold. The position space manifold and orientation space manifold can be solved, respectively; the robotic manipulator selfmotion manifold can be obtained by pairing them.
The redundant automatic fiber placement robotic manipulator position space
manifold: Assuming position and orientation transformation matrix is
which coordinate system T fixed the endeffector relative to coordinate system
fixed the mandrel S, there have:
The position joint can be calculated according to Eq. 9:

Fig. 2: 
Automatic fiber placement redundant robotic manipulator position
space manifold 
According to the equivalent movement principle, assuming the automatic fiber placement redundant robotic manipulator endeffector tracked a spatial circular in its work space, the automatic fiber placement redundant robotic manipulator’s position space manifold is shown in Fig. 2.
But in actual fiber placing process, the robotic manipulator endeffector actual route is a spatial curve in Fig. 2, in addition to the part to be paved size limits; the position space manifold is a portion of Fig. 2.
The automatic fiber placement redundant robotic manipulator orientation joint manifold: The three revolute joints of the robotic manipulator intersect at one point, the geometric structure as shown in Fig. 3. The desired coordinate of the endeffector is (x_{ee}, y_{ee}, z_{ee}) under the coordinate system X_{0}Y_{0}Z_{0}, the distance from the point to the endeffector is l , ψ is an angle between two planes, one is a vertical plane through the first joint, the other is that contain the endeffector and its projection on the X_{0}Y_{0} plane.
The following equations were established according to the automatic fibre placement redundant robotic manipulator orientation joint geometric structure:
The inverse kinematics can be solved according to Eq. 1113
and for given (x_{ee}, y_{ee}, z_{ee}) and ψ:

Fig. 3: 
Automatic fibre placement redundant robotic manipulator orientation
joint geometric structure 
The automatic fiber placement redundant robotic manipulator selfmotion
manifold: Redundant robotic manipulator selfmotion manifold is the sub
manifold of the configuration space manifold. The redundant automatic fiber
placement robotic manipulator selfmotion manifold is paired by the position
space manifold and the orientation joint manifold. The redundant automatic fiber
placement robotic manipulator orientation joint manifold is a plane curve and
the position space manifold is a hollow cylinder at θ_{5} = 0,
therefore the redundant automatic fiber placement robotic manipulator selfmotion
manifold is a plane curve in the hollow cylinder.

Fig. 4: 
Desired trajectory of endeffector 
The redundant automatic fiber placement robotic manipulator orientation joint
manifold is a spatial curve at θ_{5} ≠ 0, the redundant automatic
fiber placement robotic manipulator selfmotion manifold is a spatial curve
in the hollow cylinder, consequently, the 7DOF redundant fiber placement robotic
manipulator can complete the fiber placement task in any orientation.
VERIFICATION THE REDUNDANT AUTOMATIC FIBER PLACEMENT ROBOTIC MANIPULATOR SELFMOTION MANIFOLD
In order to verify the correctness of the selfmotion manifold solved, the redundant automatic fiber placement robotic manipulator placing some Sshaped inlet mandrel which is taken as an example, using the inverse kinematics solved by selfmotion manifold to verify. Figure 4 is desired trajectory of the redundant automatic fiber placement robotic manipulator endeffector on the mandrel, Fig. 5 shows the simulation trajectory of the redundant automatic fiber placement robotic manipulator endeffector which using the inverse kinematics solved and desired trajectory.
According to the desired fiber placement trajectory, the simulation fiber placement
trajectory of the endeffector can be obtained by selfmotion manifold and shown
in Fig. 5. Compared to Fig. 4, simulation
fiber placement trajectory obtained is good agreement with desired fiber placement
trajectory, the redundant automatic fiber placement robotic manipulator selfmotion
manifold solved in this paper is correct. Ma (2005)
proposed the actual joint configuration and the desired joint configuration
is consistent in the selfmotion manifold; Yao et al.
(2009) and Zhao et al. (2007, 2009)
planned the robotic motion by using selfmotion manifold.

Fig. 5: 
Simulation trajectory of endeffector 
CONCLUSION
The 7DOF redundant automatic fiber placement robotic manipulator is the redundant robotic manipulator which its position and orientation are decoupled and its selfmotion manifolds can be seen as which the position space manifold and the orientation joint manifold paired. When solving the redundant robotic manipulator selfmotion manifold whose position joint and orientation joint decoupled, the redundant robotic manipulator selfmotion manifold can be decomposed the position workspace manifold and orientation workspace manifolds, then analyzed and solved, respectively. Finally, the redundant robotic manipulator selfmotion manifold can be obtained by pairing the position workspace manifold and the orientation workspace manifold.
ACKNOWLEDGMENT
This study is financially supported by National Natural Science Foundation of China (51275479), to express my gratitude.