 Research
 Open Access
Analysis and Optimization of a Spatial Parallel Mechanism for a New 5DOF Hybrid SerialParallel Manipulator
 DongSheng Zhang^{1},
 YunDou Xu^{1, 2},
 JianTao Yao^{1, 2} and
 YongSheng Zhao^{1, 2}Email author
https://doi.org/10.1186/s1003301802514
© The Author(s) 2018
 Received: 19 March 2016
 Accepted: 27 November 2016
 Published: 26 June 2018
Abstract
Hybrid manipulators have potentially application in machining industries and attract extensive attention from many researchers on the basis of high stiffness and high dexterity. Therefore, in order to expand the application prospects of hybrid manipulator, a novel 5degreeoffreedom (DOF) hybrid serialparallel manipulator (HSPM) is proposed. Firstly, the design plan of this manipulator is introduced. Secondly, the analysis of this manipulator is carried out in detail, including kinematics analysis, statics analysis, and workspace analysis. Especially, an amplitude equivalent method of disposing the overconstrained force/couple to the nonoverconstrained force/couple is used in the statics analysis. Then, three performance indices are used to optimize the PM. Two of them have been widely used, and the third one is a new index which considers the characteristics of the actuated force. Based on the performance indices, the performance atlas is drawn and the optimal design of the PM is investigated. In order to satisfy the anticipant kinetic characteristics of the PM, the verification of the optimized physical dimension is done and the workspace based on the optimized physical dimension is carried out. This paper will lay good theoretical foundations for application of this novel HSPM and also can be applied to other hybrid manipulators.
Keywords
 Parallel mechanism
 Performance indices
 Optimal design
 Performance atlas
1 Introduction
Since the appearance of parallel mechanisms (PMs), PMs have been studied widely, mainly because of PMs owning the characteristics of compact structure, high stiffness, and high load capacity compared with serial mechanisms [1, 2]. In the early stage, Klaus Cappel developed the first flight simulator, which was based on a 6degreeoffreedom (DOF) GoughStewart platform. Hereafter, this device has been widely used in the fields of flight simulator, medical treatment, and communication. Then, as the application extension of PMs, the lowermobility (two to five DOFs) PMs appeared. The lowermobility PMs have become a research hotspot due to their desirable characteristics, such as simple structure, low kinematic coupling, low cost, and easy control [3–5]. For instance, Xie et al. [3], Huang et al. [4] and Joshi et al. [5] researched the type synthesis for different types of lowermobility PM. Clavel et al. [6], invented the 3DOF robot DELTA, which was a successful application case of the lowermobility PMs. Except that, one of the most important lowermobility PMs is the PM with one translational DOF and two rotational DOFs (2R1T): the 3RPS PM analyzed by Zhang et al. [7], is a typical 2R1T PM (R, P, and S stand for revolute, prismatic, and spherical joints, respectively). Actually, the 2R1T PMs have different properties about their rotational axes. Huang, et al. [8] analyzed the motion characteristics of rotational axis. For the two rotational DOFs of 2R1T PMs, the rotational axes of PMs can be divided into two types: continuous rotational axis (CRA) and instantaneous rotational axis (IRA). Here, CRA/IRA (continuous rotational axis) represents rotation axis. For the CRA, the moving platform (MP) can rotate around the rotation axis continually. For the IRA, the MP can only rotate around the rotational axis in a specific pose [9]. Compared with the general 2R1T PMs, the 2R1T PMs with two CRAs are easier to implement trajectory planning, parameter calibration, and motion control, which allows for a variety of application prospects. Nevertheless, there are only a few types of 2R1T PMs with two CRAs, which has limited the development and engineering application of this kind of mechanism.
In the processing industry, most of the complex space surfaces (such as the surfaces of engine blade, propeller blade, and nuclear evaporator head) need fiveaxis simultaneous machining center. Consequently, using the five or six DOFs PMs is an appropriate choice, but the multiDOF PMs are composed of multihinge and multichain. Under the influence of the complicated selfstructure, the orientation capability of multiDOF PMs is limited. And the high kinematic coupling, complicated dynamics model, hard control also should be noted. Taking these disadvantages into account, the structure of machines should be improved. Therefore, the hybrid serialparallel manipulators (HSPMs) based on the 2R1T PMs may be a trend, which present a compromise between the high stiffness of PMs and the good flexibility, large workspace of serial mechanisms [10, 11]. For example, the 5axis FSW proposed by Li [12], is installed by concatenating a 2DOF PP serial mechanism on the 2R1T PM 2SPR/RPS. The Tricept [13], Trivariant 5DOF HSPM [14] and Exechon fiveaxis machining center [15] are constructed by concatenating the 2DOF sway heads on the MPs of the 2R1T PMs: 3UPS/UP, 2UPS/UP, and 2UPR/SPR, respectively (U stands for universal joints).
Except for the configuration properties of PMs, optimal design is also a hot topic [16–18]. A successful optimization can effectively improve the motion/force transmission and orientation capability [19–21], etc. Ref. [19] mentioned that the condition number of Jacobian matrix and global condition index were not used to parallel robots with mixed types of DOFs. So for the 2R1T PMs with combined DOFs, the suggested indices [19, 22] are independent of any coordinate frame and more appropriate for HSPMs. Basically, the linear actuated unit of the above mentioned PMs is the ball screw system, the fluctuations of its friction moment will have a direct impact on the stationarity of the driving system. Generally speaking, the axial load has direct effect on friction moment. So the fluctuations of actuated force will affect the friction moment, and then affect mechanical motion properties and dynamic behavior in the practical application [23, 24]. Other than motion/force transmission and orientation capability indices, the stability of the actuated force in each limb also needs to be paid attention.
In this paper, a novel 5DOF HSPM is introduced whose parallel part is an overconstrained lower mobility 2R1T PM. The remainder contents of this paper are organized as follows: In Section 2, the design plan and description of the 5DOF HSPM is performed detailedly. In Section 3, the systematical analysis of this hybrid manipulator is carried out, including kinematics, statics and workspace analysis. Section 4 reports the dimensional optimization on the basis of three indices including transmission capacity, good orientation capability and force stability. Section 5 presents conclusions.
2 Novel Spatial 5DOF HSPM
2.1 Design Plan of a New 5DOF HSPM
The core part of the Exechon fiveaxis machining center is the 2UPR/SPR PM with ten singleDOF joints, so 2UPR/SPR PM has a high stiffness. On the basis of this reason, a 3DOF 2RPU/UPR PM with nine singleDOF joints is used. Here, one point should be mentioned, the 2RPU/UPR PM in this paper is different from the 2RPU/UPR PM in Ref. [25]. Actually, the MP and the base of the PM are inverse, which is the same as the case of 3SPR and 3RPS PMs.
The 3D model of the spatial 5DOF HSPM is shown as Figure 1(a), the schematic diagram is shown as Figure 1(b).
2.2 Configuration Description of the 5DOF HSPM
As shown in Figure 1(b), features of the parallel part are as follows: the cross points of two axes within each U joint are denoted by A_{2}, a_{1} and a_{3} respectively; A_{1} and A_{3} are the projections of a_{1} and a_{3} onto the base along the direction of corresponding P joints; a_{2} is the projection of A_{2} onto the MP along the direction of P joint. The triangles a_{1}a_{2}a_{3} and A_{1}A_{2}A_{3} are isosceles and similar. The reference frame A: OXYZ is attached to the base with Xaxis pointing along vector OA_{3} and Yaxis pointing along vector A_{2}O, where O is the midpoint of A_{1}A_{3}. The moving frame a: oxyz is attached to the MP with xaxis pointing along vector oa_{3} and yaxis pointing along vector a_{2}o, where o is the midpoint of a_{1}a_{3}.
Limb_{1} and limb_{3} are identical RPU limbs (P represents an actuated prismatic joint), which are restricted in the plane A_{1}A_{3}a_{3}a_{1}. Limb_{2} is a UPR limb, which is restricted in the plane A_{2}a_{2}oO, and plane A_{2}a_{2}oO is always perpendicular to plane A_{1}A_{3}a_{3}a_{1}. For the two RPU limbs, the axes of R joints are parallel to plane defined by the axes of the U joints connected to the limbs and perpendicular to the plane A_{1}A_{3}a_{3}a_{1}. The translational direction of actuated P joints is perpendicular to the axes of R joints. The axes of two U joints connected to the MP are aligned. For the UPR limb, the axis of R joint is parallel to line a_{1}a_{3}. The axis of U joint connected to the base is parallel to the axes of R joints in limb_{1} and limb_{3}. The translational direction of actuated P joint is perpendicular to the axis of R joint and the adjacent axis of U joint.
For the singleDOF sway head, its rotational axis is always parallel to line oa_{2}. For the translational table, its translational direction is along line OA_{2}.
3 Analysis of the Hybrid Manipulator
3.1 DOF Analysis of 2RPU/UPR PM
According to the reciprocal screw theory [27], we know that every limb of 2RPU/UPR PM exists a constraint force and a constraint couple. In each limb, the constraint force passes through the U joint and is parallel to the axis of R joint; and the constraint couple is perpendicular to all of the rotational axes. Therefore, the direction of constraint couples is identical, in other words, this PM has one common constraint.
As the relationships among these joints are invariable at arbitrarily instantaneous pose of this PM, the constraint force/couple are all conform to the above analysis. Then, the parameters in Eq. (1) can be obtained as d = 6 − λ =5 (λ is the number of common constraint), \( n = 8, \) \( g = 9, \) \( \sum {f_{i} } = 12, \) \( v = 1, \) \( \zeta = 0. \) Substituting the above parameters into Eq. (1), the DOF of PM is obtained: \( M = 3. \) As for these three DOFs, the translational DOF is along Zaxis. One of the rotational DOFs is around Yaxis; the other one is around xaxis. It should be noted that the Yaxis is r_{1}axis and the xaxis is r_{2}axis, both of them are CRAs.
3.2 Inverse Kinematics Analysis of 2RPU/UPR PM
For a given pose \( (\theta_{1} ,\;\theta_{2} ,\;z), \) the limb length can be obtained by means of Eq. (3), i.e., the inverse kinematics of the PM is solved.
3.3 Statics Analysis of the 2RPU/UPR PM
3.4 Workspace Analysis of the Hybrid Manipulator
Workspaces can be divided into reachable workspace, dexterous workspace and constantorientation workspace [28, 29], etc. Among which, one of the most commonly used workspaces is the constantorientation workspace, which can be obtained as the location of point o when the MP is kept at a constant orientation. Actually, some methods, like discretization method, geometrical method and numerical method [30, 31], are often used to solve the workspaces. Thereinto, the discretization method is a simple way to determine the workspaces of PMs. But problems will occur when the workspaces possess voids. In the following, a modified discretization method is used to solve the voids problem.
In fact, the threedimensional space is composed of many uniform and adjacent cubes. In order to obtain a perfect workspace, the side length ε of cube becomes the search step size, and the position of cube becomes the search space. After that, in solving process, we can orderly judge if the central point of each cube meets the conditions, then printing and recording the right points. As this method is time wasting, some knacks should be utilized. By the way, it is worth noting that the solving process can be accelerated by some strategies such as reasonable initial step size choice, dynamic step size adjustment, etc.
From the above solving process, we know that this method applies to solve the workspaces of serial robots, PMs and hybrid mechanisms with the characteristics of single connected region and multiply connected regions. Although it is a bit of time wasting and maybe not a quite accurate method, it is a simple and effective way by choosing a reasonable step size after considering the contour shape of the workspace.
Volume distribution of 2RPU/UPR PM with different step sizes
Step size ε (mm)  Volume V (mm^{3})  Average volume V (mm^{3}) 

2.0  11 276  11 290 
2.5  11 288  
3.0  11 268  
3.5  11 318  
4.0  11 220  
4.5  11 370 
Through observing the volume trend of the workspace, we know that a smaller search step size can reduce the fluctuation of the result and the trend of relative error should tend to zero with decreasing step size.
4 Optimal Design of the 2RPU/UPR PM
Next, the 2RPU/UPR PM will be optimized by the above three performance indices. In order to guarantee the PM with a good working ability, we set the search space satisfy: the MP can rotate around r_{1}axis (−20°, 20°) and rotate around r_{2}axis (−45°, 20°), and the MP can fluctuate 150 mm along Zaxis. The following optimization will be completed in this search space.
4.1 Introduction of Three Performance Indices
A larger χ indicates a better motion/force transmission, For the purpose of high quality of motion/force transmission, the most widely accepted range for the transmission angle is (45°, 135°) or (40°, 140°) [20]. Here, we make \( \sin (TA) > \sin ({\pi /}4), \) and the following three performance indices are all based on this LTI.
With the purpose of improving the orientation capability and flexibility of the MP, the second index γ shown as in Figure 4(b) stands for the good orientation capability (GOC). GOC represents the maximum orientation capability of the MP. A larger γ indicates a good orientation capability; and this one is a simple index.
The above two indices consider the motion/force transmission and orientation capability of the MP, but these cannot explain the PM with a good stability of the actuated force. Maybe the fluctuation of actuated force is quite large and the variational axial load will make a negative effect on friction moment, which acts on the ball screw. As the fluctuations of friction moment will enlarge the difficulty in setting pretightening force, and affect the positioning accuracy of ball screw [24], it is necessary to consider the following index for the optimal design.
A smaller \( \sigma \) indicates a good force stability, which is good for improving the mechanical property of actuated system.
4.2 Parameter Design Space
4.3 Performance Atlas

Step 1. In order to narrow the optimum region in the parameter design space and guarantee the PM with a good working ability, we make \( \text{GTI} > 0.94 \), GOC > 110°, and \( \text{GFS < } 0. 8 6 5. \) As shown in Figure 7(d), the overlapping region of the three indices with the desired requirements is obtained. It will help us to select the eligible parameters.

Step 2. Select a set of dimensionless parameters from the optimum region which contains the suitable solutions. As the different coordinates in the optimum region are all eligible, a comparatively better one should be confirmed. So this part of the core work is the comparative analysis of different coordinates.

Step 3. Determine the normalization factor D by means of the practical work requirements, and then three parameters can be sure.

Step 4. Check the optimum result. If the obtained parameters conform to the design requirements, the optimizing can stop, or go back to step 2 until the optimizing is successful.
4.4 Results and Validation
With the purpose of better understanding the optimization procedure, an example is given. Here the coordinate (1.4, 0.7) in the optimum region is selected, the three dimensionless parameters are (0.7, 1.438, 0.862). Let a = 200 mm, then \( D = \frac{200}{0.7} = 285.714 \), so b = r_{2}·D = 410.857 mm, c = r_{3}·D = 246.285 mm and the initial distance of point o to the base is 720.41 mm. As the triangles a_{1}a_{2}a_{3} and A_{1}A_{2}A_{3} are similar, \( e = \frac{bc}{a} = 505.940 \) mm.
Considering potential applications, the machine should have a subset workspace in which it can keep an optimum working performance. In order to make the MP be able to move 150 mm along Zaxis, we try to change the distance of the point o to the base. According to the above analysis, we know that the initial distance is 821.465 mm. At this height, the \( \text{GTI} = 0.961, \) GOC = 113.770°, and \( \text{GFS = } 0. 8 2 5 \). For the purpose of enlarging the effective movement, let the distance of point o to the base be 850 mm; then we can get \( \text{GTI} = 0.962 \), GOC = 114.377° and \( \text{GFS = } 0. 8 5 5 \). By using the same method, when the distance of the point o to the base is 680 mm, the \( \text{GTI} = 0.950 \), GOC = 110.206° and \( \text{GFS = 0.677} \). After observing and comparing the above data, the effective movement can reach 170 mm. The above results are all satisfy the optimization conditions, so this set of optimization data is available.
Using the above parameters, the structure sketch can be obtained which is shown as Figure 1(a), and this picture is just used to intuitively show the proportional relation of 2RPU/UPR PM after optimal design. By means of the optimized results, the workspace obtained from MATLAB software is shown in Figure 7. Figure 7(a) is the 3D view, Figure 7(b)‒(c) are the corresponding projected views.
The overall dimensions can be intuitively recognized from the above workspace, which will contribute to analyze the property of this manipulator.
5 Conclusions and Future Work
 (1)
A novel 5DOF HSPM is proposed which is based on the 2RPU/UPR PM. The 2RPU/UPR PM is a new 2R1T PM with two CRAs, so it is easier to implement trajectory planning, parameter calibration, and motion control, compared with the general 2R1T PMs.
 (2)
The structure of this machine is described in detail and the kinematics, statics and workspace are all analyzed, which help us to better understanding the structural characteristics and is convenient to optimal design.
 (3)
A modified discretization method is used to reveal the workspace of the PM which also could incidentally solve the volume of the workspace.
 (4)
The optimal design of 2RPU/UPR PM is presented basing on three indices: GTI, GOC and GFS. The GTI and GOC indices are based on the classical concept of transmission angle, the GFS is a new index which considers the stability of actuated force and is good for improving the mechanical property of actuated system. All of these three indices are independent of any coordinate frame. The optimal results are validated and it is satisfied.
 (5)
In our following work, the prototype of the novel 5DOF HSPM with outstanding performances will be manufactured, and experimental studies will be performed.
Declarations
Authors’ Contributions
YSZ was in charge of the whole trial; DSZ wrote the manuscript; DSZ, YDX and JTY assisted with sampling and laboratory analyses. All authors read and approved the final manuscript.
Authors’ Information
DongSheng Zhang, born in 1988, is currently a PhD candidate at Yanshan University, China. He received his bachelor degree on mechatronics from Yanshan University, China, in 2013. His research interests include parallel manipulator and robotics.
YunDou Xu, born in 1985, is currently an associate professor at School of Mechanical Engineering, Yanshan University, China. He received his bachelor degree and PhD degree from Yanshan University, China, in 2007 and 2012 respectively. His research interests include parallel manipulator and robotics.
JianTao Yao, born in 1980, is currently a professor at School of Mechanical Engineering, Yanshan University, China. He received his bachelor degree and PhD degree from Yanshan University, China, in 2004 and 2010 respectively. His research interests include sixaxis force sensors, parallel manipulators, and mechatronics.
YongSheng Zhao, born in 1962, is currently a professor at Yanshan University, China. He received his PhD degree from Yanshan University, China, in 1999. His research interests include mechatronics engineering, robotics and parallel manipulator.
Competing Interests
The authors declare no competing financial interests.
Funding
Supported by National Natural Science Foundation of China (Grant No. 51405425), Key Basic Research Program of Hebei Province’s Applied Basic Research Plan of China (Grant No. 15961805D), and Natural Science Foundation of Hebei Province (Grant No. E2017203387).
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