Development and Research of Technological Equipment that Implements Dynamic Control of Process of Additive Fabrication of Parts of Complex Spatial Shapes Based on Mechanisms with a Hybrid Layout

The article is devoted to the study of the accuracy of fabrication of the surface layer of engineering products by additive methods. The analysis of the advantages and disadvantages of technology for layer-by-layer synthesis of products took place. It was revealed that with additive fabrication, the accuracy characteristics of the surface layer are significantly different from the characteristics of the surface layer of products obtained by traditional methods. The analysis of domestic and foreign works on the topic of research allowed to reveal that in order to increase the accuracy characteristics of products obtained by additive methods, it was necessary to provide dynamic control of the spatial orientation of the final link of the fabrication system of the additive installation in the process of fabrication. To control the spatial orientation of the working body of additive equipment, the use of mechanisms with a hybrid layout was proposed. The problem of parametric synthesis of a mechanism with a hybrid layout has been solved, which allows to create a space of design parameters that provide, at the early stages of design, the required formative capabilities of the mechanism for additive fabrication with a hybrid layout.


Introduction
Formulation of parts by additive methods is characterized by high values of shape error. This is due to the fact that the formation of the surface layer of a complex shape part occurs line by line (in layers), and the orientation of the final link of the fabrication system (extruder) of the additive installation is unchanged and independent of the curvature of the formed surface [1][2][3]. To reduce the error in the shape of the part, it is necessary, when formulation surface points, to ensure its orientation at which the normal at the point of the surface being formed will coincide with the axis of the final link of the formulation system ( Figure 1) [17][18][19][20][21][22][23].

Designing of Equipment for Dynamic Process Control of Additive Formulation of a Part
To solve the problem of dynamic control of the process of additive fabrication, in paper [4] it was proposed to use a mechanism with parallel kinematics -a hexapod having six degrees of freedom, on which the part is formed owing to its moving part. When the additive installation extruder approaches the part surface, the hexapod changes its orientation, ensuring that the extruder axis coincides with the normal to the part surface at the point which is to be formed. However, the use of mechanisms with parallel kinematics does not provide sufficient angles of rotation of the part relative to the vertical axis. To eliminate this drawback, it is proposed to create an installation using a mechanism with a hybrid layout ( Figure 2).

Figure 2.
Additive installation with a hybrid layout: 1 -molded part, 2 -extruder, 3 -stepper motors, 4 -rotating table, 5 -guides, 6 -hinges, 7 -lead screws The designed equipment for additive fabrication is based on a parallel structure mechanism with constant-length rods. It is proposed to install a rotating table on a moving platform, which, if there are 5 degrees of freedom, will allow the angle of rotation of the part relative to the vertical axis in the range . This will significantly improve access to the points on the surface of the formed part.
This installation has 5 degrees of freedom -movement along the axis X, Y and Z; the inclination of the platform relative to the axis X; table rotation relative to the axis Z.
The main task of the design of technological equipment is the problem of parametric synthesis, the solution of which will allow to create a space of design parameters providing the required formative capabilities of the installation at the early stages of design, such as: -the ability to set the normal to the surface of the part in the formed point for all points of the surface of the part along the axis of the extruder (Figure 1).
The solution of this problem is possible by compiling, on the basis of a generalized model, of the mechanisms with parallel kinematics [3], a system of six equations in accordance with the number of rods, as follows:  vectors determining the position of the movable joints in the coordinate system of the installation base; q1,..,q4 -controlled coordinates of the installation, determining the position of the hinges along the axis Z; -a matrix that determines the position of the movable platform at which the extruder will contact the point on the surface of the part with coordinates (u, v), and its axis will coincide with the normal to the surface of the part; -vectors that specify the position of the hinges of the moving platform in its own coordinate system.
Due to the redundancy of this system of equations for the designed installation, it can be reduced to the following one: will be determined by means of generation of the geometric closure equation (Figure 3) of the installation coordinate systems: where 1 e -versor determining the positive direction of the axis X3 of the extruder coordinate system ; Using the system of equations (2), it becomes possible to determine the points on the space of design parameters at which this system will have a solution, which, therefore, will represent the range of their permissible values.

Results of Solving the Problem of Parametric Synthesis of a Device for Dynamic Control of the Process of Additive Product Fabrication
As an example, we will consider the finding of the range of permissible values for the lengths of the installation rods with constant dimensions of the base and the movable installation platform (Figure 4). Thus, the vectors that determine the position of the movable hinges in the coordinate system of the installation base (see Fig. 4), will be set as follows: The vectors that specify the position of the hinges of the moving platform in its own coordinate system will be defined as follows: We will set the following device parameters for calculation: Device height: H = 600 mm; Extruder length: he = 100 mm; Hinges location options: HR = 500 mm, h1R = 85 mm, h2R = 85 mm, BR = 650 mm, b1R = 25 mm, b2R = 65 mm, h1S = 85 mm, h2S = 20 mm, b1S = 65 mm; hc = 20 mm. Table 1 shows the values of minimum Lmin and maximum Lmax rod lengths in the fabrication of parts such as a hemisphere described by the following equation: where z ,   surface curvilinear coordinates; R  hemisphere radius (R=50..100 mm).  0  319  387  319  363  319  339  5  313  384  315  361  316  338  15  300  380  306  357  310  335  25  284  380  297  356  303  333  40  260  388  282  356  292  331  50  235  422  271  357  284  331  75  --235  397  264  337  100  ----235  373 In Table 1, the values Lmin are determined based on the solution of the equation system (2) for various hemisphere z parameter values, which are included into the equation (9), and the value Lmax is determined by the limitations of the size of the working area imposed on the coordinates q1,…,q4. Figure 5 shows the range of permissible lengths of the rods of additive equipment for various values of the radius of the formed part in the form of a hemisphere. Thus, to ensure the hemisphere formulation conditions outlined above with R=50 mm, the range of permissible rod lengths is in the range L=319…380 mm; R=75 mm -L=319…356 mm; R=100 mm -L=319…331 mm.
In a similar way, the range of permissible values for other parameters of the designed installation can be obtained.

Conclusions
Thus, based on the method described above, parametric synthesis of equipment for dynamic control of the process of additive formulation of products can be performed, as well as a range of values of the parameters of the formed surfaces at which the extruder will be in contact with all points of the formed surface (internal points of the part and points on its surface), as well as the normal to the surface of the part will be established at the formed points of the surface along the axis of the extruder.