ANALYTICAL IDENTIFICATION OF THE UNMANNED AERIAL VEHICLES' SURFACES FOR THE IMPLEMENTATION AT A 3D PRINTER

Based on the R-functions theory, new approaches to analytical identification of drone surfaces for realization of 3D printing technology have been developed. The R-functions theory allows one to describe geometric objects of a complex shape with a single analytical expression, that is, obtain a mathematical model of the object in a form of an equation. To derive such equations, we used both the well-known standard primitive (sphere, ellipsoid, cylinder, cone, pyramid, etc.) procedure and a new approach, blending on a frame, which enables derivation of multiparameter equations with specified properties. Multiparameter equations of surfaces of drones of various types and purposes have been derived and visualized. Adequacy of the results to the designed objects was confirmed by visualization, both in conditions of operation of the RFPreview program and by realization on a 3D printer. The use of literal parameters when specifying geometric information in an analytical form makes it possible to promptly change size and shape of the designed objects which helps reduce time required to build computational models. The proposed method can reduce labor input in operation of CAD systems by months in cases when it is necessary to view a large number of design variants in a search for an optimal solution. Having the object equation, one can easily obtain equation of any of its sections which is useful for numerical calculations, namely, when building computational meshes. This can have a great effect on reducing complexity in construction of computational models for determining aero-gas-dynamic and strength characteristics. Characterization is also often associated with the need to account for changes in the aircraft shape. This leads to the fact that establishment of aerodynamic characteristics just because of the need to build a large number of computational models to account for this factor increases work duration by months. When specifying parameters, change of the rated operating conditions is made almost instantly.


Introduction
Drones (unmanned aerial vehicles) are becoming increasingly popular all over the world every day.This is due to a series of advantages over manned aircraft, namely, absence of crew, relatively low cost of the drone at a long duration and large range of flight.Design characteristics of drones differ which diversifies areas of their application.
Three-dimensional (3D) printing is one of the new technologies gaining growing popularity at present.3D printing allows one to create three-dimensional models of any objects using special equipment: a 3D printer.Advantages of using modern 3D printers include reduction of cost of the product manufacture and the time in which the product enters the market, modeling objects of any shape and complexity, high speed and precision of manufacture and freedom of material choice.
NASA engineers are building a rocket from a special plastic capable of reaching other planets in the solar system using a 3D printer with a technology of selective laser sintering.Students at Southampton University have created the world's first aircraft called Sulsa (Southampton University Laser Sintered Aircraft) with the use of a 3D printer [5] and the British Navy got down to using it for ice reconnaissance in Antarctica [6].
Airbus Company has presented at the Berlin Air Show a drone of Thor model printed on a 3D printer [7].Its length is about 4 meters, the wingspan is the same.The device weighs about 25 kg.The company explains that the new drone was developed for aerodynamic studies associated with high risk, hence the use of 3D printing was preferred as it makes it possible to create the device in several weeks and at much lower costs.
The drone, however, is not the only Airbus innovation having connection with 3D printing.The company also carries out aerodynamic tests of wing sections printed with a 3D printer.In this case, 3D printing reduces production time by 90 % and costs by 75 %.Besides, Airbus plans installation of partitions separating passenger cabin from kitchen printed with a 3D-printer.Due to the design features and owing to 3D printing, the partition will be 50 % lighter.
The given examples indicate that creation of both models and actual drones with the use of 3D printers is a promising project.Therefore, it is necessary to develop appropriate hardware support in a form of high-tech computer systems based on the most advanced achievements of the present-day mathematics.

Literature review and problem statement
Designing drones and control systems for them cannot be imagined without mathematical simulation of drone flight.The modern approach to designing involves creation of models that best of all reflect properties and behavioral patterns of real drones.Many tasks of using drones that are posed before researchers can be solved at the design stage.Major technical potential has been created in a form of high-speed computers and advanced software to create mathematical models, e. g. such software packages as Solid Works, Ansys CFX, POLYE, etc.In all software systems, the problem of geometry data setting arises and is solved in different ways to create a mathematical and computer model of the drone being designed.For example, the wing cross section profile is specified in [9] by standard Ansys modeling tools entailing building of a new model at any slightest change of the profile appearance.NASA standard wing profiles specified point-by-point are used in [10].In addition, TsAGI standard profiles are used in [11].
Numerical CFD methods of the LES type as well as panel methods and the method of discrete vortexes are used in [12] for aerodynamic analysis and then, based on the results, changes are introduced to the aircraft shape.In essence, this is how the inverse problem is solved.
The authors of [13] took advantage of an existing drone for their studies.However, this drone was designed for a common use.Presumably, if the authors could introduce changes to the shape of the drone surface, the study would be more effective.
Study [14] is devoted to correction for wind effect on the drone flight.A special control algorithm was proposed for such correction.However, this algorithm would probably be simpler if the wind effect was taken into consideration during the drone design stage.To this end, one can change its shape to make the flight more stable.
To compile component equations of a complicated system of a drone complex, the theory of dynamic systems of randomly changing structures most fully reflecting the random stick-slip nature of impacts to which the system is exposed was used in [15].To reduce the degree of impact randomness, it is necessary to correct shape of the drone surface.This will make the drone flight steadier.
The examples show that shape of the drone surface is essential in all studies and requires constant adjustment when dealing with it.Any slightest change in appearance of the surface shape in the considered methods and approaches entails construction of a new model while insufficient attention is paid to parametric specification of the drone surfaces.Developers must be skilled in building mathematical models of surfaces which will respond promptly to changes in shape of the object under study.
Calculations and tests of a new aircraft should be given considerable attention.It should be noted that 3D printing can be of great help in this.There are different approaches to creation of 3D models depending on the data being set for printing: 1. 3D scanning of an existing object.
2. Construction of a 3D model using a rather wide range of software products that have primitive forms of various types in the arsenal of their libraries.
3. Creation of a 3D-model with the help of the mathematical apparatus of analytical geometry.
In cases when the modeled object exists physically, use of a 3D scanner is the most convenient way of obtaining 3D models.However, in the case when the modeled object has impressive dimensions or does not exist at all, this approach is inapplicable.In such cases, creation of 3D models can be realized using software products.However, this method also has a series of drawbacks, namely lack of a history of model construction, limited set of primitives in the library program modules, lack of possibility of analytical record, etc.
One of the ways for solving this problem is application of the R-functions theory, which allows one to describe geometric objects of a complex shape with a single analytical expression [16][17][18] which necessitates conducting studies in this direction.

The aim and objectives of the study
This study objective was creation of mathematical and computer models of a series of drones based on the R-functions theory for realization of the 3D printing technology.
To attain this objective, it was necessary to solve the following tasks: -explore mathematical apparatus of the R-functions theory and its application to creation of multiparameter equations of the three-dimensional drone surfaces and their realization on a 3D printer; -develop an algorithm of phased construction of the drone surface equations; -explore possibilities of using both the well-known procedure of standard primitives and blending on a frame for constructing multiparameter equations of drone surfaces and realization on a 3D printer; -realize the constructed mathematical model of a drone on a 3D printer.

1. Engineering environment of the R-functions theory
When constructing mathematical models using the R-function method, both simplest R-operations [16][17][18]

FR fk fl = ρ − −
Both well-known procedure of standard primitives [16,17] and the new approach, blending on a frame, were used.To guide the reader's eye, let us first consider the case when the frame is a line segment.
As a result, we obtain a body of rotation (Fig. 1, a) and four controlled parameters.Consider also the case when the frame is a circle arc.

(
) As a result, we obtain a cylindrical body (Fig. 1, b) with a guide similar to the Zhukovsky profile and four controlled parameters.It should be noted that in this case, the use of R-operations and full normalization to the first order of functions, ff1, [10] has made it possible to realize the procedure of blending on a frame.

2. 3D printing based on the technology of fused deposition modeling (FDM)
In order to print an object on a 3D printer, it is necessary to model it and prepare for printing.The software packages enabling creation of the three-dimensional graphics, that is, simulate objects of virtual reality and create images on the basis of these models vary greatly.In this study, we used the RFPreview program which allowed us to save the object model in a STL format.It was necessary to prepare it for layered printing (generate a G-code).The software that performs this conversion is called a slicer.The slicer allows one to see how a model is positioned on a 3D printer platform before printing, whether supports are necessary for structural elements, select all printing parameters and settings.In this study, the authors used the Cura slicer to generate G-code for 3D printers.The program has a wide selection of settings and plug-ins.With the help of Cura, one can dynamically change temperature during printing, printout objects with fine details, etc.
The technology of 3D printing is based on the technology of layer-by-layer "growing" solid objects from various materials.The FDM technology is a prototyping method using industrial grade thermoplasts.а b

Results of analytical identification of drone surfaces
A mathematical and computer model of the drone mock-up shown in Fig. 2 was constructed.For this purpose, we used both the method of blending on a frame and known equations of standard primitives, namely, equations of cylinder, triangular pyramid and ellipsoid.

Fig. 2. Schematic drone mock-up
The main body.( ) ( ) The fairing in front of the rocket in the underbelly   The results presented in Fig. 3 were obtained for the following values of literal parameters: a=5; p1=20; p2=0; a1=5.To carry out experimental realization of the model of the constructed drone, the RFPreview software package [18] based on the R-functions theory with literal parameters was used.It has allowed us to write the following program and obtain an stl file for subsequent 3D printer operation.( ) The left wing.
The tail assembly.
( ) ( ) Finally, the following is obtained: The model of the drone with wings constructed according to the first variant is presented in Fig. 5, a, and that for the second variant is presented in Fig. 5, b.The ten-parameter equation of the drone fuselage with two different multiparameter equations of the wing profile derived using standard primitives makes it possible to promptly change the drone shape when values of the literal parameters change.
A fuselage equation was derived using blending on frames in a form of straight line segments of which one had 2a length and the other had 2b length and was shifted to the nose by xc after which they were combined using R-disjunctions.Thus, the p2 parameter affects smoothing of the tail and head parts of the fuselage.
The resulting equations of object models were visualized using the RFPreview program [18].

Discussion of results obtained in the analytical identification of drone surfaces
Construction of mathematical models of objects is an important issue in realization of 3D printing.In this study, it is creation of a 3D model using the mathematical apparatus of analytical geometry, respectively.The R-functions theory solves the inverse problem of analytical geometry allowing one to describe geometric objects of a complex shape with a single analytical expression.Summing up, it should be noted that thanks to the R-functions theory, equations of surfaces of a drone of a plane type were derived in this study.They have form of general analytical expressions with literal parameters applicable both in solving strength and aerohydrodynamic problems and in designing and manufacturing products (in this case, by means of a 3D-printer).The use of literal parameters enables quick change of characteristic dimensions and shape of the object under study.The feature of positiveness of the constructed function in internal points of the object is very convenient for realization of 3D printing.Having equation of an object, one can easily obtain equation of its any cross section which is useful in numerical calculations using CFD-methods (e. g. [19]), namely, when constructing computational meshes.
We used both the well-known procedure of standard primitives (sphere, ellipsoid, cylinder, cone, pyramid, etc.) as well as the new approach, blending on a frame, which makes it possible to derive multiparameter equations with desired properties.Wings and experimental fuselage of the drone were built in this way.
As disadvantages, it is necessary that the engineering researcher shall be skilled in the mathematical apparatus of the R-functions theory and a limited set of standard primitives in library modules of the programs aimed at using the R-functions theory.The authors are planning to implement automation of the process of deriving equations of a drone of a plane type composed from standard primitives which will make the derivation process more accessible to the user.Also, further development of this procedure in relation to drones of helicopter and convertaplane types is planned.
The proposed method of specifying the shape of products using a limited number of parameters can significantly reduce complexity of work in CAD systems in cases when it is necessary to view a large number of design variants in a search for an optimal solution.This can have a great effect on reducing complexity in construction of computational models for determining aero-gas-dynamic and strength characteristics.For example, the process of building a perturbed flow region around an aircraft of a complex shape can take from several working days to weeks.At the same time, successive review of a large number of structure design variants is required during the design process in order to optimize the structure characteristics.Following deciding on the aircraft shape, characterization is also often associated with the need to take into consideration changes in the aircraft shape, e.g. if it has controls that change shape in the flight.This leads to a situation when defining aerodynamic characteristics, just the need of building a large number of computational models to account for this factor extends the work duration for months.The change in rated operating conditions is almost instantaneous in parameter specification.

Conclusions
1. Mathematical apparatus of the R-functions theory which has hitherto been used mainly in 2D problems has been studied.It has also appeared to be effective when applied in 3D problems as it was demonstrated by the visualized equations and the models fabricated on a 3D printer.
2. An algorithm of phased construction of drone equations has been developed and implemented with the help of R-functions which makes it possible to check and introduce adjustments to the model at each stage of its construction.
3. The method of standard primitives can only be used in the case of completeness of the base of standard primitives.Otherwise, it is necessary to develop methods for constructing surfaces based on other approaches.Blending on a frame used in the study can be considered as one of possible approaches.
4. The constructed mathematical model of the drone was realized on the Anet A8 3D printer working with the use of the FDM technology.The use of other, more advanced technologies of 3D printing will improve appearance of products.
and R-operations for smoothing sharp edges and corners were used[17]:

Fig. 1 .
Fig. 1.Surfaces constructed using the method of blending on a frame : body of rotation (a); cylindrical body with a guide similar to the Zhukovsky profile (b) with rounded ends,

-
complete assembly of the drone presented in Fig.3.

Fig. 5 .
Fig. 5. Drones with different wing profiles: with the use of standard primitives and R-operations for smoothing sharp edges (a); with the use of blending on a frame in a form of a circle arc (b)

Fig. 6 .
Fig. 6.Drones designed for obtaining intelligence information and its delivery to destination: p1=40 (a); p1=60 (b); p1=80 (c); p1=100 (d)Thus, value of the p1 parameter significantly affects appearance of the tail and head parts of the fuselage.Fig.7shows effect of value of the p2 parameter on appearance of the drone surface.