Analysis of payload capability performance of a planar two degree-of-freedom parallel manipulator

Bing WANG，Zhi-hui WEI，Xu-meng NIE，Shu-kui HAN，Wei-na HAN（School of Mechanical and Electrical Engineering，North China Institute of Aerospace Engineering，Langfang 065000，China）

Analysis of payload capability performance of a planar two degree-of-freedom parallel manipulator

Bing WANG*，Zhi-hui WEI，Xu-meng NIE，Shu-kui HAN，Wei-na HAN
（School of Mechanical and Electrical Engineering，North China Institute of Aerospace Engineering，Langfang 065000，China）

This paper addresses payload capability performance analysis of a two degree-of-freedom planar parallel manipulator with planar translational motion and small inertia.Unlike the traditional analysis and optimum design method，the geometric parameters and the range of input variables of the manipulator are considered together as the kinematics parameters to obtain the new design space which is different with the traditional one by utilizing the theory of geometric model of the solution space method.The global payload capability indices are calculated in the new design space，the atlases for payload capability of the manipulator are plotted，and the relationship between the kinematics parameters and the indices is discussed.The performance atlases are very convenient for a designer to evaluate the payload performance and select the optimum kinematics parameters of the manipulator，and the new method also provides a useful analysis and design way for the planar parallel manipulators with prismatic actuators.

Parallel manipulator，Design space，Performance atlas，Payload capability

Hydromechatronics Engineering

http：//jdy.qks.cqut.edu.cn

E-mail：jdygcyw@126.com

1 Introduction

Two-degree-of-freedom parallel manipulators are an important class of manipulators that can follow arbitrary curves.Because of their usefulness in applications，these manipulators have attracted the attention of researchers who have investigated the analysis and optimum design method［1-10］.Feng Gao and Xinjun Liu［1］utilized a geometric model of the solution space to obtain analytical relationships between the link lengths of two-degree-of-freedom parallel planar manipulators and performance criteria based on the global conditioning and velocity indices.Zhang Li-jie etal.［2］discussed the relationship between link lengths and global performance indices，such as condition number of Jacobian matrix，payload and stiffness using the physical model of the solution space method.Unlike the traditional optimum design method used in parallel mechanisms，Xin-Jun Liu etal.［3］proposed a design approach utilizing a performance chart to design the PRRRP 2-DOF parallel manipulator.

Although extensive research has been direct toward the analysis and design of 2-DOF planar parallel manipulators，only the relationship between the link lengths of 2-DOF planar parallel manipulators and performance criteria has been discussed.In fact，the performances of the manipulators are closely related not only with the link lengths of the manipulators，but also with the range of the input variables.

The existing planar 2-DOF parallel manipulator is the well-known five-bar mechanism with prismatic actuators or revolute actuators.The output of the manipulator is the translational motion of a point on the end effecter.In some case，it is required to control the motion of a rigid body instead of a point on a plane with two degree of freedom.Xin-Jun Liu etal.［4］presented a novel planar 2-DOF parallel manipulator that departs from the existing designs in which a parallelogram mechanism is adopted in each of the two legs.The motion of the platform is the planar translational motion of a rigid body.In order to reduce the horizontal dimension of this kind of manipulator，Li［5］proposed an intersectional planar 2-DOF parallel manipulator.The parallelogram of each leg is cross hinged on both ends of the moving platform.

The manipulators mentioned above both hinged the parallelogram of each leg at both ends of the moving platform.The geometric parameter of the moving platform will be larger，and the corresponding mass and inertia will be larger also，that is not conducive for the manipulator to realize high speed operation task.

In this paper，a novel planar 2-DOF parallel manipulator is proposes.It departs from the existing design in that the parallelogram mechanism adopted in each of the two legs is cross hinged together at the moving platform.The geometric parameter of the moving platform is smaller，and the corresponding mass and inertia are smaller also.The kinematics problems and velocity equation of the new planar 2-DOF manipulator are given.Two types of singularities are obtained.Unlike the traditional analysis and optimum design method，not only the geometric parameters but also the range of the input variables of the manipulator are consider as the kinematics parameters，and the design space is obtained by utilizing the theory of geometric model of the solution space.The global payload capability indices are calculated in the design space，the payload capability atlases of the manipulator are plotted，and the relationship between the payload capability and the kinematics parameters of the manipulator is discussed.The atlases can be of great help in the performance evaluation and optimum design of the manipulator，and the new method also provides a comprehensive and effective way for synthesis of planar parallel manipulators with prismatic actuators.

2 Description of the manipulator

Fig.1 is the manipulator being consideredin this paper，where the base is labeled 3 and the part B1B2is the moving platform.The moving platform is connected to the base by two identical legs.Each leg consists of a planar four-bar parallelogram.The parallelogram A1A3B2B1is for the first leg and the parallelogram A2A4B1B2is for the second leg.In each planar fourbar parall-elogram，the joints are all revolute pairs. The part 1 and 2 are prismatic actuators.Motions of the moving platform are achieved by the combin-ation of movement of the part 1 and 2 that canbe transmitted to the platform by the system of the two parallelograms.Because the part B1B2is always parallel to the part 1 and 2，the moving platform has two pure translational degrees of freedom with respect to the base. Because the two parallelogram of each leg hinge together in point B1and B2of the moving platform，the geometric parameter of the moving platform is small-er than the manipulator mentioned above and the mass and inertia are smaller also.The manip-ulator will has wide application in the fields of metal cutting，part handling and 3D printing，etc.

Fig.1 The noveI pIanar 2-DOF paraIIeI manipuIator

3 Kinematics analysis

As illustrated in Figure 1，a reference frame：xoy is fixed to the base and a moving reference frame：x′o′y′is attached to the moving platform，where o′=p（x，y）is the point to be positioned by the manipulator. Displacement of the actuator x1and x2is the input of the manipulator and R3is the range of the prismatic actuators.The link length of the parallelogram is R1and the geometric parameter of the moving platform is R2.Then the inverse kinematic problem of the manipulator can be solved by writing following constraint equation

From Eqs.（1），the direct kinematic of the manipulator can be described as

Hence，the solutions of the direct kinematics of the manipulator can reach two.In the base coordinate system shown in Fig.1，we should select the“-”mode.

Equation（1）can be differentiated with respect to time to obtain the velocity equations.This leads to an equation of the form

where.x is the vector of input velocity defined as

and.p is the velocity of the output variables of the manipulator defined as

matrices A and B are the 2×2 Jacobian matrices of the manipulator and can be expressed as

In the parallel manipulator，singularities occur whenever A，B or both，become singular，as shown in Fig.2 and 3.

Fig.2 The fist singuIarity of the manipuIator

Fig.3 The second singuIarity of the manipuIator

Because singularity leads to a loss of the controllability and degradation of the natural stiffness of manipulators，the manipulator should be away from the singularity when it working，and the geometric parameters and rang of input variables should satisfy

Equation（8）establishes the relationship between geometric parameters and rangs of input variables.It makes it possible considering not only the geometric parameters but also the range of input variables of the manipulator as the kinematics parameters.

4 Design space

Previously，a geometric model of the solution space was derived for 2-DOF planar parallel manipulators and used to study performance characteristics［1-2］. The extensive research has discussed the relationship between the link lengths of 2-DOF planar parallel manipulators and performance criteria.In fact，the performances of the manipulators are closely related not only with the link lengths of the manipulators，but also with the range of the input variables.

Therefore，considering the geometric parameters and the rang of input variables together as the kinematics parameters will make it possible to study the relationship between the performance indices and kinematics parameters of planar parallel manipulator comprehensively and systematically.

Because the kinematics parameters may have a wide range of possible values，it is convenient to avoid explicit use of the physical sizes of the manipulators during analysis and design.We shall define normalized kinematics parameters of the manipulators and construct a design space as follows.

Let

Where riis the non-dimensional length of kinematics parameter Ri.From Eq.（9）and（10），one can obtain

Theoretically，from Eq.（11），the three non-dimensional parameters r1，r2and r3have any value between 0 and 3.For the parallel manipulator studied here，the analysis on the singularity shows r2is less than r3and r3is less than r1+r2.Therefore，the three parameters should be

Based on Eq.（12），one can establish a design space as show in Fig.4，in which the diamond ABCD is actually the design space.

In Fig.4，the diamond ABCD is restricted by r1，r2and r3.Therefore it can be figured in another form as show in Fig.5，which referred to as the planar-closed configuration of the design space.

Fig.4 The design space

Fig.5 The pIanar-cIosed configuration of the design space

For convenience，two orthogonal coordinate x and y are utilized to express r1，r2and r3.Thus，by using

coordinate r1，r2and r3can be transformed into x and y.Eq.（13）is useful for constructing a payload capability performance atlas.

5 Global payload index and its atlas

Payload capability is one of the most important performance indices of the parallel manipulator.It determines whether the manipulator can support the external force acting on the moving platform.We use the global maximum and minimum payload indices in this paper to measure the payload capability of the planar 2-DOF parallel manipulator［2-3］，that are

where G is the force Jacobian matrix of the parallel manipulator and it is the transpose of Jacobian matrix J.The σpmaxand σvminare the maximum and minimum singular values of force matrix G.Where w is the reachable workspace of the manipulator.For the manipulator studied here，the reachable workspace is the space where direct kinematics is existence in while the prismatic actuators move from zero to R3（r3）.The ηpmaxand ηpminare the global maximum and minimum payload indexes of the parallel manipulator.The payload capability of the parallel manipulator is better if the values of the payload indexes are larger.

Fig.6 and 7 show the atlases of the ηpmaxandηpmin，respectively.From the atlases one can see that：

Fig.6 The atIas of gIobaI maximum payIoad index

Fig.7 The atIas of gIobaI minimum payIoad index

The ηpmaxand ηpminare proportional to r1.

The ηpmaxand ηpminare inverse proportional to r2.

While 1.5≤r1＜3 and 0＜r2≤0.75，the payload performance of the parallel manipulator is better.

For example，one can select r1=2.0，r2=0.2 and r3=0.8 as the kinematics parameters of the planar 2-DOF parallel manipulator with better payload capability.

6 Conclusions

In this paper，a novel planar 2-DOF parallel manipulator with planar translational motion and small inertia is presented.The direct kinematics is developed，the velocity equation is given and the singularity of the parallel manipulator is analyzed.Not only the geometric parameters but also the range of input variables of the manipulator are considered as the kinematics parameters to obtain the design space and payload capability atlases of the parallel manipulator.The relationship between the payload capability and the kinematics parameters of the manipulator is discussed.One important advantage of the atlases is that it can give designers global and visual information of what kind of kinematics parameters of the parallel manipulator can have better payload capability performance.For the atlases can be used in optimum design process to identify an optimum region，they are very useful for analysis and design.The new method mentioned in this paper also provides a comprehensive and effective way for analysis and synthesis of planar parallel manipulators with prismatic actuators.

Acknowledgements

This paper are supported by the Hebei Education Department Fundation（No.Q2012059）and Hebei Science and Technology Research and Development Program（No.13211824）.

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一種平面二自由度并聯機器人承載能力分析

王 冰*，魏志輝，聶旭萌，韓書葵，韓偉娜
北華航天工業學院機電工程學院，河北廊坊 065000

以一種可實現末端運動平臺平動且具有較小運動慣量的平面二自由度并聯機器人機構為研究對象，分析其承載能力性能。與傳統的分析和優化設計方法不同，綜合考慮機器人機構的幾何尺寸和運動變量的變化范圍，作為機器人機構的運動學參數，利用空間模型理論，建立了機器人機構的設計空間。在設計空間內計算機器人機構的承載能力性能指標并繪制了相應的承載能力性能圖譜，并探討了機構承載能力性能指標與運動學參數之間的關系。這些圖譜為設計者評價該機器人機構的承載能力性能和選擇優化的運動學參數提供了幫助。提出的新方法也為以移動副為驅動的平面并聯機器人分析與優化設計提供了一種有效的新途徑。

并聯機器人；設計空間；性能圖譜；承載能力

10.3969/j.issn.1001-3881.2015.12.018Document code：A

TP11

1 September 2014；revised 28 November 2014；accepted 1 March 2015

*Corresponding author：Bing WANG，Associate professor.

E-mail：wbrobot@163.com