Study of a New 5-DOF Parallel Mechanism for Multi-directional 3D Printing

This paper ﬁrst designs a new 5-DOF parallel mechanism with 5PUS-UPU for multi-directional 3D printing, and then analyses its DOF by traditional Grubler-Kutzbach and motion spiral theory. It theoretically shows that the mechanism meets the requirement of 5 dimensions of freedoms including three-dimensional movement and two-dimensional rotation. Basing on this, the real mechanism is built, but unfortunately it is found unstable in some positions. Grassmann line geometry method is applied to analyze its unstable problem caused by singular posture, and then an improving method is put forward to solve it. With the improved mechanism, closed loop vector method is employed to establish the inverse position equation of the parallel mechanism, and kinematics analysis is carried out to get the mapping relationships between position, speed and acceleration of moving and ﬁxed platform, Monte Carlo method is used to analyze the workspace of the mechanism, to explore the inﬂuencing factors of workspace, and then to get the better workspace. Finally an experiment is designed to verify the mechanism working performance to satisfy the spatial motion requirements of multi-directional 3D printing.


Introduction
Parallel mechanism (PM) can be defined as a closedloop mechanism in which the moving platform and the fixed platform are connected by at least two independent kinematic chains. The mechanism has two or more degrees of freedom, and is driven in parallel.
Common parallel mechanisms are Delta and Stewart.
Unlike traditional series structure, parallel structure is widely used in automobile, aviation, medical treatment, education and other fields due to its small cumu-pacity of the parallel mechanism, and becomes a hot spot in robot research and application in the world [4][5][6][7][8]. At present, many scholars have applied parallel mechanisms with different postures to 3D printers and achieved many results. Among them, Delta mechanism has been successfully applied in cases of 3D printing technology based on three-dimensional translation and high accuracy [9]. Guo Xiaobo [10] and others analyzed the workspace of 3-HSS parallel 3D printer and Bi Changfei [11] did kinematics analysis and Simulation of 3-HSS parallel 3D printer. These 3D printers based on parallel mechanism have higher precision and better motion performance than those based on series mechanism, but for these 3D printers based on a 3-freedom parallel mechanism, their printing heads can only maintain one direction. However, during the printing process, it was found that they cannot print parts with complex surfaces or slopes with high quality. Therefore, with the development of 3D printers, a concept of multi-directional printing has been proposed. This printing method can accumulate materials in different directions on the formed surface, which improves the quality of the surface of complex printed parts, but requires the mechanism with 5 degrees of freedom. Keating S et al. [12] designed a five-degree-offreedom 3D printer to achieve multi-directional printing. However, this robot adopts a serial mechanism, and the inertia and error of the entire mechanism are large. At present, there are relatively few studies on multi-directional 3D printing based on parallel mechanisms. Song et al. [13] applied the Stewart parallel mechanism to a 3D printer, but the six-degree-offreedom Stewart mechanism achieves a relatively small working space and has one redundant degree of freedom to rotate around the print head. Fang Yuefa et al. [14] designed a four-degree-of-freedom parallel multidirectional 3D printer and performed kinematic analy-sis. Wei Ye et al. [15] designed a reconfigurable parallel mechanism for multidirectional additive manufacturing, which has four motion modes to choose from, but can only achieve up to four degrees of freedom in one mode. Pan Ying et al. [16] designed a five-degree-offreedom parallel 3D printer, but with a compound dynamic platform, the complex structure is not suitable for promotion.
With the development of parallel mechanism, the analysis of singularity and optimization of workspace of parallel mechanism have become a hot research topic. J. Wang [17] used a 4 × 4 Jacobian matrix to analyze the singularity of a spatial 4-DOF parallel mechanism; J. Gallardo-Alvarado et al. [18] analyzed the singular configuration of a 4-DOF parallel robot by using the screw theory; Sheng Guo et al. [19] analyzed the singularity of a 4-RRCR parallel mechanism using srew theory; Thomas, M.J. [20] used a numerical based algorithm to determine the singular points within the workspace of the 3-PRUS mechanism by determining points where the determinant of the Jacobian becomes zero; Wolf [21] analyzed the singularity of two kinds of 3-DOF parallel robots by using the Grassmann line geometry method; Chunxu Tian et al. [22] analyzed singularities of a partially decoupled generalized parallel mechanism for 3T1R motion; Zongjie Tao et al. [23] deveoped system design methodology (LIDeM) for reducing interference among links and increasing workspace of mechanism; Liping Wang et al.
In this paper, a 5-DOF parallel mechanism based on multi-directional 3D printer is designed. The mechanism adopts parallel structure completely and the printer head is installed on the moving platform of the parallel mechanism. During the printing process, as the posture of the moving platform changes, the print head can realize multi-directional printing. As the driving part of the parallel mechanism, two methods of constant-length link drive and telescopic link drive are currently adopted. Comparing the two methods, the fixed-link length driving mode can greatly reduce the structural size of the link, which is not easy to produce interference in the process of movement [25]. Therefore, this driving mode is used to build parallel mechanism in this paper. Finally, we study the 5PUS-UPU five-degree-of-freedom parallel mechanism from aspects of scheme design, freedom degree verification, test equipment construction, singularity analysis, scheme improvement, mechanism motion analysis, workspace analysis, and experimental verification.
2 Design and verification of the scheme

Design of the primary scheme
At present, 6-DOF Stewart platform [26] is used as the prototype for 5-DOF parallel mechanism. The minimum degrees of freedom of the 3D printer structure that meets multi-directional printing is 5-3 movements and 2 rotations. In the case of satisfying the degree of freedom requirement, the less degree of freedom of the mechanism, the simpler mechanical structure, the lower manufacturing and control costs. Therefore, we choose to remove one of the six active limbs in the Stewart platform, and then rearrange the spatial positions of the five active limbs. However, at this time, there are only 5 driving modules, while the number of degrees of freedom of the moving platform is 6. The number of driving modules is less than the degrees of freedom of the mechanism, which will cause the motion of the moving platform to be uncertain. The redundant degree of freedom of the moving platform is limited by adding a restraint limb between the moving platform and the fixed platform. At this time, the number of degrees of freedom of the moving platform is 5, which meets the requirement of the degree of freedom.
As for the drive part of parallel mechanism, fixedlength link drive and telescopic link drive are the two most widely used ways at present. The driving mode of the mobile pair with a fixed-length link can greatly reduce the structural dimension of the link and avoid interference during the movement [24]. Therefore, we choose the driving method of the mobile pair with a fixed-length link in this paper. The five mobile pairs are used as actuators, which are symmetrically distributed in the shape of regular pentagons in space. The moving platform also adopts the symmetrical structure of the positive pentagon. As shown in    In three-dimensional space, when a component is free before it is connected with other components by kinematic pairs. The n components including the frame have 6(n−1) degrees of freedom. When one component is connected to another component with a kinematic pair, the component is constrained to a certain extent and its degree of freedom is reduced. When the two components are connected by a kinematic pair with f i degrees of freedom, the degree of freedom of the original system is reduced by 6 − f i due to increased constraints. When the number of kinematic pairs con-tinues to increase to g, the degree of freedom of the system is reduced by At this time, the degree of freedom of the mechanism is M = 6(n − g − 1) Where M is the degree of freedom of the mechanism (Mobility); n is the total number of components of the mechanism; g is the number of kinematic pairs of the mechanism; f i is the number of degrees of freedom of the ith kinematic pair.
In this mechanism, n = 14, g = 18, That means the mechanism has 5 degrees of freedom, and the five sliders are used as the driving part, and the mechanism can achieve a certain movement.

Verification of degree of freedom by screw theory
Traditional Grubler-Kutzbach method can only be calculated as 5 degrees of freedom of the mechanism, which is not clear about the specific situation of the degree of freedom. Therefore, after a long-term development, the idea of solving the degree of freedom of parallel mechanism by screw theory emerged. The degree of freedom of 5-PUS-UPU parallel mechanism is further analyzed by using screw theory. In screw theory, the screw is generally expressed as (S; S 0 )or plücker coordinates (a b c; l m n) [27]. S is the original part of the rotation and S 0 is the dual part of the rotation [16].The reciprocal product of two screws ($ 1 = (S 1 ; S 0 1 ), $ 2 = (S 2 ; S 0 2 )) is defined as the sum of the dot products after the exchange of the original and dual part When the reciprocal product formed by $ 1 and $ 2 is 0: They are reciprocal, and the screw $ 2 is called the reciprocal screw of $ 1 . Therefore, the reciprocal screw can be obtained based on the fact that the reciprocal product of the two screws is 0.
The parallel mechanism designed in this paper uses  The screw system consisting of $ 1 , $ 2 , $ 3 , $ 4 , $ 5 , and $ 6 has been simplified to: The twist of the active limb has a full rank, so there is no reciprocal screw. Similarly, the other four active limbs have the same analysis results. It can be seen that the five active limbs have no restriction on the moving platform [29].
When analyzing the constraint of the constraint limb to the moving platform, constrained limb coordinate system o 6 − x 6 y 6 z 6 is established. Similarly, the centre point of Hook joint is set as origin o, and the two axes of Hook joint are set as x 6 -axes and y 6 -axes respectively, and the z 6 axes are perpendicular to the plane of x 6 o 6 y 6 to establish coordinate system, as shown in Figure 3.
Then , the screw system of the restrained limb is represented as: Where l 3 , m 3 , n 3 , m 4 , l 5 represent coordinate parameters and are not 0.
The screw system consisting of $ 1 , $ 2 , $ 3 , $ 4 , and $ 5 has been simplified to: Finally, the reciprocal screw of the constrained limb is: $ r = (0 0 0; 0 0 1). From the physical meaning of the screw and the reciprocal screw, when the screw represents motion, the reciprocal screw represents constraints, that means, the constraint limb gives the moving platform a constraint moment rotating around the z-axis, while the remaining degrees of free- around the x-axis and y-axis.

Experimental verification
After the above scheme of 5-DOF parallel mechanism is verified by Grubler-Kutzbach and screw theory, the prototype is built for experiment in this study.
The frame of the prototype is built with the 20millimetre-long profiles. The screw module and profiles are connected by connecting plates. The 5-DOF parallel mechanism is composed of a moving platform, upper and lower plates, ball screws, driving limbs, restrainting limb, connecting plates, Hook joints and spherical hinges, as shown in Figure 4. In this paper, the lower plate is the fixed platform, and the geometric center of the fixed platform is set as the zero position.
However, after the construction of the prototype, when the moving platform is directly above the zero point, it is found that the movement of the moving platform is unstable and the degree of freedom is uncertain. The experimental results show that the scheme needs to be improved.

Problem analysis of parallel mechanism
Parallel mechanism has the characteristics of strong bearing capacity, large stiffness and high motion accuracy. However, in practical application, it appears stiffness degradation, motion instability, bearing capacity reduction and other phenomena when it is at some special positions, which is revealed in the experimental verification. In the study of this paper, the uncertainty of degree of freedom appears when the moving platform is directly above the zero point. The reduction of mechanism performance is not only related to the design, manufacture and assembly of the mechanism, but also closely related to the unique singularity and the dynamic stability of the singularity. The singularity of the mechanism generally refers to the situation that the inverse kinematics solution of the mechanism does not exist, the mechanism motion is unstable, and the driving force in the limb suddenly increases at some positions [30]. The analysis of singularity is the basis to verify the design and avoid the motion in singular space.

Singularity analysis
The essence of singularity of parallel mechanism is force screw failure of partial limbs acting on the mov-

Scheme improvement
Based on the singular defect caused by the same shape of fixed platform and moving platform in the preliminary scheme, the moving platform is changed from pentagon to quadrilateral. The mechanism sketch is shown in Figure 6.
In the improved scheme, the fixed platform also uses 5 mobile pairs as input and presents a positive pentagonal symmetrical distribution in space, while the moving platform uses a rectangular structure. 3 Kinematic analysis

Establishment of inverse kinematics equations
The coordinate systems of the mechanism consists of a moving coordinate system and a fixed coordinate system, as shown in Figure 6. The transformation matrix Where x, y and z represent the moving distance of the moving coordinate system along the X, Y and Z axes of the fixed coordinate system respectively; α, β and γ represent the rotating angles of the coordinate system along the X, Y and Z axes of the fixed coordinate system respectively. Any point in the moving coordinate system can be converted into a specific coordinate value in the fixed coordinate system by  Figure   8: Where X U i , Y U i , and Z U i represent the X, Y and Z coordinates of U i ; X S i , Y S i , and Z S i represent the X, Y and Z coordinates of S i .
The mapping equation for the position of five driving sliders and the posture coordinates of the moving platform is simplified as follows: Where U iz (i = 1, 2, ..., 5) represents the coordinate of the Z axis of the five sliders in the static coordinate system.

Algorithm simulation
In this mechanism, radius of outer circle of geometric center of Hook joint fixed on five sliders R = 135 mm.
The geometric center target track of the moving platform is shown Figure 9.

Research on the return home algorithm
As the initial position of the whole mechanism, the zero position corresponds to the reference point in the motion model. The accuracy of the zero position has a great influence on the motion accuracy of the mechanism. Therefore, it is very important to design reliable return home function and strictly guarantee the accuracy of return home position. Different from the actual coordinate axis of series mechanism, parallel mechanism is also called virtual axis mechanism, and there is no actual X, Y, Z axis, so the method of setting reference point on X, Y, Z axis is not feasible. In this paper, one position is chosen in each of the five driving parts of the parallel mechanism as the reference point for the return home of each axis of the parallel mechanism, as shown in Figure 13. Through the zero return button in the control program, the five sliders driven by the driving motor move to a fixed limit position in the coordinate system of the parallel mechanism. The fixed limit position is a reference point of the zero point of the parallel mechanism, and is a fixed value relative to the origin of the fixed platform mechanism. The specific method is to set the motor speed so that the five sliders move in the same direction and at the same speed. If the sliders move to the limit position first, they will stop moving. Until all the sliders move to the corresponding limit position, the mechanism will return to the zero reference point.   (4) Limit of link interference. In the actual prototype, each link has a certain size, so interference should be avoided.

Monte Carlo method
Monte Carlo method is based on the idea of random sampling to solve mathematical problems. The flowchart of solving mechanism workspace by Monte Carlo method is shown in Figure 14. The analysis process is as follows: Step 1. The random function is used to traverse the value in the initially set workspace.

Experiment
The whole prototype is made of profiles with side length of 20 mm, and the lead screw module and profiles are connected by connecting plates. The whole 5-DOF parallel mechanism is composed of moving platform, upper and lower plates, ball screws, driving limbs, restraining limb, connecting plates, Hook joints and spherical hinges. The lower plate in this paper is also called fixed platform. The 3D model and physical model are shown in Figure 21.
The size of the profile is shown in Figure 22; the diameter of the link is 8 mm, the length of the link Therefore, how to further improve the motion accuracy of the mechanism will be the focus of the next work.