Visual analysis base on computer space parallel mechanism workspace

Guang-ping CHEN, Yuan-fei HAN, Tie-zhong XIAO

(1Sichuan Equipment Manufacturing Engineering Technology Applications Laboratory of Robotics, Deyang 618000, China) (2Shool of Mechanical Engineering, Southwest Jiaotong University, Chengdu 611756,China)

Abstract: Aiming at the 3TIR spatial parallel mechanism, this paper proposes a widely-generalized discrete Monte Carlo method. Through the discretization of rotational degrees of freedom, the dimensionality reduction effect is achieved. By using the union and intersection of subspaces, The reachable workspace of the entire space parallel mechanism and the visualization effect of the flexible workspace are analyzed by taking an example of a 3T1R space parallel mechanism of type 4-PRPaRR as an example to study the effect of the specific parameters on the size of the workspace and the performance. For the first time, the work performance of space parallel mechanism is analyzed from the perspective of flexible work space and reachable work space, which provides an important theoretical foundation for the application and research of multi-freedom space parallel mechanism.

Key words： Space parallel mechanism, Workspace, Visualization, Monte carlo method

1 Introduction

The 3T1R parallel manipulators is the combination of three moving and one rotating spatial parallel mechanisms. It is widely used in high-speed sorting applications because of its excellent working characteristics and mechanical properties. The working space of the space parallel mechanism refers to the total space volume swept by the end effector of the parallel mechanism when the robot arm performs all possible actions[1-2]. The workspace is an important indicator of the working capacity of the space parallel mechanism, therefore the study of spatial visualization has very important significance for the research and application of spatial parallel mechanism.

The analytical methods and numerical methods are the main methods to solve the spatial parallel mechanism. Literature[3-6],the Monte Carlo method was used to analyze and optimize the 3-DOF Delt of parallel mechanism with different configurations. In the literature[7-10], the method of coordinate search was used to numerically analyze that the working space of a certain configuration of the 3T1R parallel manipulators. The method of limit search was used in the literature[11] to analyze the working space of the 3T1R parallel mechanism; The literature[12] uses the method of boundary value search for a type of 2-RPC/2-SPC’s 3T1R parallel mechanism performs workspace space analysis. In the literature[13], the method of the combination of the analytical method and numerical method were used to analyze the workspace. In Ge and Wu[14] the method of Quaternion Algorithm was used to visualize 4-dimensional space.

The visual research on the work space of the spatial parallel mechanism mainly focuses on parallel mechanisms with no more than 3 degrees of freedom at present, And the 3T1R parallel manipulators is more difficult to realize the visualization of workspace because it has four degrees of freedom, which is higher than the dimension of three-dimensional space. There are many topological configurations of the 3T1R parallel manipulators[15-16], The spatial parallel mechanism with type 4-PRPaRR was taken as an object of study in this paper which propose a 3T1R parallel manipulators universally applicable to 4 degrees of freedom. The numerical method of visual analysis of work space, namely the discrete fixed-step angle Monte Carlo method, is not only simple and easy to implement, but also widely applicable to the visual analysis of the working space of any 3T1R parallel manipulators. It provides a theoretical basis for the promotion and application of the 3T1R parallel manipulators.

2 The structural analysis of 4-PRPaRR

The parallel mechanism of 4-PRPaRR is a typical 3T1R parallel manipulators. As shown in Fig.1, the global coordinate system is established with the plane of the guide rail as the reference plane. The global coordinate system isOX0Y0Z0, and the unit direction vector is[ijk].

The plane where the end effector is located is the reference plane, and the position of the centroid is the origin of the coordinate system. The movement coordinate system is the local coordinate systemPXYZ, and the unit direction vector is[uvw].The parallel mechanism of 4-PRPaRR is composed of four symmetrically arranged PRPaRR robots and fixed guide rails. The size parameters of each robot arm are consistent. The input is four translational displacementsl1,l2,l3,l4along the guide rail direction. The output is the movement of the moving platform. The entire mechanism only contains the moving pair and the rotating pair. From Fig.1 to Fig.2, it is easy to know that the 4-PRPaRR has a translational motion in theX,YandZdirections and a total of four degrees of freedom about theZaxis, and the moving platform is always parallel to the plane where the guide rail is located. Taking the P-point of the centroid as the research object, the output variable is the position vectorXp=[XPYPZP]Tand its rotation azimuth angleθz. Due to the constraints of the geometry of the robot and the mechanical constraints on the joints, it can be seen that:

3 The inverse kinematics and workspace constraint equations of 4-PRPaRR

The purpose of inverse kinematics solution is to take the output parameter of the known function, that is, the position vectorXp=[XPYPZP]Tand its azimuth θz of the end effector as input variables, and obtain the original input parameters l1, l2, l3, and l4. Kinematic equations.

Fig.1 The diagram of 4-PRPaRR parallel manipulators

Fig.2 The Three-dimensional entity mode of space parallel mechanism of 4-PRPaRR

From Fig.1, the position vector ofPpoint in the global coordinate systemOX0Y0Z0can be expressed

as:

Xp=XAi+XAiBi+XBiCi+XCiDi-XDiP,

i=1,…,4

(1)

In the local coordinate systemPXYZfixed on the moving platform:

(2)

The equation can be obtained by coordinate transformation：

XPDi=0RPUPDi

(3)

One of them：

(4)

constraint equation：

(5)

By the formula of(1)～(5), We can solve the inverse kinematics position solution of the 4-PRPaRR space parallel mechanism as:

l1=Xp+dcosθza+c±

l2=Yp+dcosθz-a+c±

l3=-Xp+dcosθz-a+c±

l4= -Yp+dcosθz-a+c±

According to the formula of(6)and(9)，In order to make the inverse kinematic position solution exist，The constraint conditions to satisfy the working space are:

(10)

(11)

(12)

(13)

4 The realization of workspace visibility based on discrete monte carlo algorithm

The workspaces include reachable workspace and dexterous workspace. The reachable workspace refers to the set of all points where the manipulators can reach, while the dexterous workspace refers to the set of all points where the manipulators can reach in arbitrary position. With the guidance of theory on probability statistics, Monte Carlo algorithm is a numerical method that solves computational problems in an “experimental” way by random number (or pseudo-random number)[17]. The 3T1R parallel manipulators can’t be visualized directly in 3D space, due to having four degrees of freedom, namely three translational degrees of freedom and one rotational degree of freedom. At each step, to obtain the reachable sub-workspace, denoted asSi, this paper used a discrete Monte Carlo algorithm solves the three positions degree of freedom parameters of the fixed step attitude angleθziby the constraint equations (10)～(13). According to the definition of the two workspaces, we can get a complete reachable workspaceWrof the 4-PRPaRR parallel manipulators by taking the intersection of the sub-workspaceSi, and also can obtain the complete flexible workspaceWfof the 4-PRPaRR parallel manipulators by taking the union ofSi. They can be expressed as:

Then the point clouds of reachable workspace and flexible workspace about the 3T1R parallel manipulators are plotted respectively in Mathematica, so as to realize the visualization of the 4-DOF workspace.

This algorithm’s realizing function and specific implementation steps had described as follows:

(1) While the assumption of drawing pace is π/N,θziis obtained by discretizing the attitude angleθz.

(2) The random real of random function is used to generate input variables (xjyjzj),j=1,…,n.

(3) Fori≤Ndo

Substitutes the fixed attitude angleθziand the random numbers(xjyjzj) into the constraint equations (10)～(13) of inverse kinematics position solution.

Forj≤N

If the restriction equation were satisfied

Assign (xjyjzj) to[XPYPZP][k]

End

End

Si=[XPYPZP]

End

The unionfunction is used to obtain the point coordinate matrixWrby taking the unionofSi, while the intersection function is used to obtain the point coordinate matrixWfby taking the intersectionofSi.

(4) Plot the point coordinate matrixWrandWfrespectively, we obtained the reachable workspaceWrand flexible workspaceWfof parallel manipulators.

5 Instance simulation

Table 1 The parameters of 4-PRPaRR parallel manipulators

Parametersa/cmc/cmd/cmr/cmli/cmValues52225≤35

Calculate the union and intersection of the discrete space ofθzto obtain the workspace of the entire parallel manipulators, as shown in Fig.6 and Fig.7:

(1) From the top view of the sub-workspaceSi, it shows whenθzchanged from 0 to π/2, the trend of the boundary of the workspace inXOYplane is that the corresponding coordinates alongXaxis from 20 gradually decrease to 18, while the corresponding coordinate values alongYaxis from 20 gradually increase to 22 (the reduction is the initial value of d), there is no change along theZaxis (observe its front view). From the default view, it shows that while the change ofθz, the basic shape of the workspace has almost unchanged.

(2) Fig.6 and Fig.7 shows that, as the selected parameters in this instance, the coincidence degree of 3D point cloud map betweenWrandWfis relatively large and close to 90%, which could directly display that the 4-PRPaRR parallel manipulator has very good motion characteristics.

(3) From Fig.3 to Fig.5, we can also intuitively find the direct influence of the initial parameters on the workspace of {a,c,d,r,li}.

① |r-a| reflects the basic symmetrical size of the workspace in the plane ofOXY.

② The choice of the initial valuedwill affect the ratio of the reachable workspaceWrand the flexible workspaceWf, That is, the value ofdis larger, the range of change in the sub-workspace will be greater, the flexible workspace will be smaller.Wr/Wfwill decrease and the 4-PRPaRR parallel manipulators will have reduced.

③ The value of |a-c| is inversely proportional toli.

④ The maximum height of the workspace in theZ-axis direction will be reflected in|r|.

Fig.3 When θz=0、π/6、π/2，the front view of subspace Si

Fig.4 When θz=0、π/6、π/2, the top view of subspace Si

Fig.5 When θz=0、π/6、π/2，the default view of sub-workspace Si

Fig.6 The 3D point cloud map of reachable workspace Si

6 Conclusion

(1) Taking the 4-PRPaRR spatial parallel manipulators for example, this paper proposed an universally applicable discrete Monte Carlo algorithm for 3TIR spatial parallel manipulators, which is simple and easy to implement, and also can process 3TIR spatial parallel manipulators for visualization.

(2) Through analyzing 4-PRPaRR parallel manipulators workspace visualization, this paper reveals the direct relationship between the workspace and its parameters, providing theoretical basis for 4-PRPaRR spatial parallel manipulators design and application.

(3) The reachable workspace and flexible workspace were obtained through the union and intersection of subspace. This paper first analyzed the working performance of spatial parallel manipulators successfully from its reachable workspace and flexible workspace, this method displayed its performance indexes more directly.

Acknowledgements

Supported by Sichuan science and technology support plan(No.2017RZ0062).