The Research about Prescribed Workspace for Optimal Design of 6 R Robot

Based on the D-H notation, kinematics model and inverse kinematics model of 6R industrial robots are established. Using graphical method, the boundary curve equations of the 6R industrial robot workspace are obtained. Based on the prescribed workspace, the D-H parameter optimization method of 6R industrial robots is proposed. Using the genetic algorithm to determine the structural dimensions of a 6R robot, we make sure that its workspace can exactly contain the prescribed workspace. This method can be used to reduce the overall size of the robot, save materials and reduce the power consumption of the robot during its work time.


Introduction
The workspace of a robot is defined as the sets of points that can be reached by the end-effecter [1], and sometimes is known as reachable space.It is one of the main ways in the design and optimization processes of robots [2].Based on graphical approach, Xie Jun et al. [3] analyzed the workspace of a novel series Chinese medical massage arm, and learned that the upper arm and the forearm of this kind of Chinese medical massage arm must be equal.Chen Zaili [4] presented a genetic algorithm approach for the synthesis of spatial 6-DOF parallel manipulators whose workspace must include a desired workspace with orienting capabilities on a given 3D region.M.A. Laribi [5] proposed an optimal dimensional synthesis method of the DELTA parallel robot for a prescribed workspace.
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In the practical working conditions, when a robot accomplishes some special tasks, the range of motion of its end-effecter is not the whole workspace, but must be included in the whole workspace.As the range of motion usually has an irregular shape, it can be substituted by a prescribed workspace that includes this range of motion in the synthesis process of the robot [6] [7].The prescribed workspace must satisfy two basic requirements: 1) be a regular geometric model, such as a cuboid, a cylinder or a sphere and so on; 2) must contain the entire range of motion of a robot's end-effecter and has the minimum volume.
The method which is usually used to determine the structure parameters has the property of try and blindness.It is not only a waste of time, but also hard to obtain a robot with a compact structure.Different structure parameters are different information states of every link.Aiming at obtaining a compact structure, we can determine the structure parameters which make the robot with the shortest information distance [8].This paper, based on a prescribed workspace, presents an optimum design method of D-H parameters for a 6R robot.This method can obtain a 6R robot with a compact structure, the sum of whose links' lengths is smallest so that the material can be saved.

D-H Coordinate Frames and Homogeneous Transformations of 6R Robot
A commonly used convention for selecting frames of reference in robotics applications is the Denavit and Hartenberg (D-H) convention which was firstly introduced by Jacques Denavit and Richard S. Hartenberg [9].The D-H coordinate frames can be laid out as follows: 1) the Z-axis is in the direction of the joint axis; 2) the X-axis is parallel to the common normal, X n = Z n × Z n-1 ; 3) the Y-axis follows from the X-axis and Z-axis by choosing it to be a right-handed coordinate system.Four parameters known as D-H parameters can be obtained.They are θ, d, a, α.
The model of 6R robot is shown in Figure 1 which is used on a welding production line.The D-H coordinate frames for it are established as shown in Figure 2.And the D-H parameters are shown in Table 1.
Let i j T be the homogeneous transformation matrix from the ith D-H coordinate frame to the jth one, then the following equations can be got： where, The transformation matrix from the base to the end-effecter is given as follows: 0 0 1 2 3 4 5 6 1 2 3 4 5 6 0 0 0 1

Analysis and Simulation of the Workspace
Twelve messages are given by the matrix with Equation (1).n x , n y , n z , o x , o y , o z , a x , a y and a z are nine pose information that determine the posture of the robot, p x , p y and p z are three position information that determine the position of the robot.As the scope of the workspace of a 6R robot is determined by position information, so we can just calculate p x , p y and p z when we analysis the workspace.p x , p y and p z are given with Equation (2).Table 1.The D-H parameters of the 6R robot.
where c 23 = cos(θ 2 + θ 3 ), s 23 = sin(θ 2 + θ 3 ).From Equation (2), we can infer that the position of the end-effecter of a 6R robot is only defined by the first three joints (Joint 1, Joint 2 and Joint 3 in Figure 1).So we can define the end of the third link (the fourth joint point) as the reference work point of the 6R robot, and the set of points which the reference work point can arrive at can be defined as the workspace of the 6R robot [10].
Let θ 1 = 0˚, according to the graphical method, the cross-section of the workspace of the 6R robot on the plane X 0 O 0 Z 0 can be obtained [11].It is the area which is surrounded by the arc line Г 1 , Г 2 , Г 3 and Г 4 in Figure 3.The workspace of the 6R robot can be obtained by rotating the cross-section around the axis O 0 Z 0 .The three-dimensional view and the three-dimensional cross-sectional view of the workspace are respectively shown in the Figure 4 and Figure 5. From the Figure 4, we can see that the workspace is similar to a sphere.From the Figure 5, we can see that there is a cavity in the workspace.When designing 6R robots, we must make sure that the cavity doesn't intersect with the reference work point of 6R robots.

Object function of Robot Structure Optimizing Based on Prescribed Workspace
According to the given n end-effecter working points of robot, the minimum rectangular can be determined which includes these n working points [6].As shown in the Figure 6, the prescribed workspace of Δ 1 is the minimum rectangular which includes all of the working points of 6R.The size of Δ 1 is L 1 × L 2 × L 3 .Δ 1 is symmetry relative to X 0 O 0 Y 0 plane andY 0 O 0 Z 0 plane.Figure 7 shows the position of Δ 1 and the workspace.
The optimization objective function of Robot link length is as follows, And the constraint conditions are, λ 2 are constants, λ 1 , λ 2 ∈(0,1),and λ 1 + λ 2 = 1.F 1 (X) is sum of links length.F 2 (X) is the distance between the prescribed workspace and the origin O 0 of the coordinate O 0 X 0 Y 0 Z 0 .x i is the length of ith link, m is the number of the links, q j is the jth joint angle, n is the number of the joints.X is m dimensional vector, Because the positions of 6R robot end-effecters are affected by angles of the first three joints (Joint 1, Joint 2 and Joint 3 in Figure 1), the length of the first four links (Link 1, Link 2, Link 3 and Link 4 in Figure 1) can be optimized based on the prescribed workspace.The object function is： ( ) ( ) ( ) ) is the distance between the plane of ADD'A' in Figure 6 of prescribed workspace and the origin O 0 of the coordinate O 0 X 0 Y 0 Z 0 in Figure 6.The value of D(x 1 , x 2 , x 3 , x 4 ) is determined by the size of the internal cavity of work space.
The angle between the first axis of link 1 in Figure 1 and the horizontal plane is γ.According to the rules of D-H coordinate system, a 1 /cos(γ) = x 1 , a 2 = x 2 , a 3 = x 3 and d 4 = x 4 .And the object function can be deformed as follows, then the MinF(X) is equal to Equation ( 4), that is Min , , , Min , , , cos , , ,

Length of the Links Constraining
Considering the actual work requirements, the length of the links can't be small too much.The ranges of a 1 , a 2 , a 3 and d 4 are satisfied the following rules.

Joint Angles Constraining
In Figure 3, the length of links and the joint angles were certain size, the point of O 42 is in the extended line of O 1 and O 2 when the section area of workspace is the biggest.At the same time, ( ) Considering the actual situations and the interference problems of the structure, the rotation range of the joints should be limited.The ranges of θ 2 and θ 3 can be set as follows.

Internal Cavity Constraining
In Figure 3, let M 1 be the radius of Г 2 and Г 3 , let M 2 be the radius of Г 4 .Then ( ) ( ) D a a a d x x = (7) where,

Key Points Constraining
Let the y coordinate of O 41 less than or equal to the value of D(x 1 , x 2 , x 3 , x 4 ) with the internal cavity constraining.
Then the section which is on the right of the connection line of A 1 and A 2 is symmetry relative to the O 0 Y 0 axis.The space of Δ 2 can be got to revolve the section around the O 0 Z 0 axis.Δ 2 is symmetry relative to X 0 O 0 Y 0 plane and Y 0 O 0 Z 0 plane.If the point of ( ) x y z ′ ′ ′ ′ is in Δ 2 , then Δ 1 must be included in Δ 2 and in the workspace of 6R robot.And the points constraining is as follows.

Application
According to the location and distribution of a car body welding, the space sizes of prescribed workspace of one 6R robot are In order to avoid the movement interference, the links length and the joints angle robot are limited.l 1 = 100 mm, l 2 = l 3 = l 4 = 150 mm, -120˚ ≤ θ 2 ≤ +120˚, -70˚ ≤ θ 3 .Let λ 1 = 0.65, λ 2 = 0.35 in 3th equation.Based on the genetic algorithm [12], the optimal solution of the constraint conditions are shown in Table 2.And the distance between the prescribed workspace to the plane of O 0 X 0 Y 0 Z 0 is D(a 1 ,a 2 ,a 3 ,d 4 ) = 483.94mm.
According to the optimal solution, the workspace can be got by the reference [13].In market, the workspace of robot of IRB 2400/10 can meet the 10th equation.The contrast of these two kinds of robots shows in Table 3.In Table 3, the 2 kinds of workspace volume are small difference.The sum of link length of the optimized robot is shorter than that of IRB 2400/10 robot by 85 mm.And the weight of the former is 7.89% lighter than that of the latter.

Conclusion
It analyzes the workspace of 6R robot to ensure the joints that affect 6R robot's work space.And the edge curve of work space had been got by the graphic method.Matlab was used to establish the simulation model of 6R robot work space.With the prescribed workspace, the D-H parameters were optimized with GA to achieve the optimized solution meeting the constraining.The workspace and the prescribed workspace were modeled with Pro/E to prove the rationality of optimization result.

Figure 1 .
Figure 1.The three-dimensional model of the 6R robot.

Figure 2 .
Figure 2. The D-H coordinate frames of the 6R robot.

Figure 3 .
Figure 3.The section shape of the workspace of 6R robot on the Z O X 0 0

Figure 4 .
Figure 4.The three-dimensional view of the workspace.

Figure 5 .
Figure 5.The three-dimensional cross-section view of the workspace.

Figure 7 .
Figure 7.The prescribed workspace and the workspace.

3 .
of a 1 , a 2 , a 3 and d 4 is the smallest, then the shadow of Δ 1 on the plane of Y 0 O 0 Z 0 should be on the right to the connection of A 1 and A 2 in Figure To ensure the cavity do not cross Δ 1 , the follows should be met.

γ 1
is the included angle of the line of 42 O′ and O 22 and the line of O 0 Y 0 in Figure 3. 1 Δ 1 was created by the methods above mentioned showing Figure 8.And Δ 1 was cut respectively by the surfaces of A, B, C showing from Figure 9 to Figure 12.Δ 1 was just included in the workspace and Δ 1 doesn't intersect with the internal cavity.It proves the rationality of optimization result.Rounding the optimized size in Table 2, a 1 = 100 mm, a 2 = 670 mm, a 3 = 150 mm, d 4 = 690 mm.Then the actual robot can be created with the optimized size showing in Figure 13.

Figure 8 .
Figure 8.The prescribed workspace and the workspace of optimized robot.

Figure 9 .
Figure 9.The section of the optimized workspace on surface A.

Figure 10 .
Figure 10.The section of the optimized workspace on surface B.

Figure 11 .
Figure 11.The section of the optimized workspace on surface C.

Figure 12 .
Figure 12.The section of the optimized workspace on surface D.

Table 2 .
The optimal solution.

Table 3 .
The optimal solution.