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Таблица 3
Результаты вычислений
Уровни факторов Число опытов Ео h.=Eh
1 2 3 4 5
С1 27 147,7 808,0
С2 27 74,6 206,1 1539,5
С3 21 119,1 525,4
В1 27 120,2 535,1
В2 26 112,7 470,1 1559,5
В3 27 122,3 554,0
А1 27 102,5 389,1
А2 27 112,3 467,1 1473,5
A3 27 129,1 617,3
Остальные факторы оказывают существенно меньшее влияние. Однако, все они существенны, поскольку для них Влияние парных взаимодействий и тройного взаимодействия малы и недостоверны, поскольку для них К<^| (34)
Таким образом, факторы А, В и С независимы, а их парные и тройные взаи-мо-действия не оказывают существенного влияния на результаты экспериментов.
Список литературы:
1. Венцель Теория вероятностей, наука, Москва, 1966. 576 с
2. Василенко. П.А, Погорелый Л.В, Основы научных исследований: (Механизация сельского хозяйства) Киев, Высшая школа, 1985. 265 с.
the analysis of non-euclidean metric for motion
planning
Valbahs Edvards
PhD Student, Rezekne Academy of Technologies, Rezekne, Latvia
Grabutfs Peter
Dr. sc. ing, professor, Rezekne Academy of Technologies, Rezekne, Latvia
ABSTRACT
In order to achieve the wide range of the robotic application it is necessary to provide iterative motions among points of the goals. For inflance, in the induflry mobile robots can replace any components between a Morehouse and an assembly department. Ammunition replacement is widely used in military services. Working place is possible in ports, airports, wafle site and etc. Mobile agents can be used for monitoring if it is necessary to observe control points in the secret place. The paper deals with path planning programme for mobile robots. The aim of the research paper is to analyse motion-planning algorithms that contain the design of modelling programme. The programme is needed as environment modelling to obtain the simulation data. The simulation data give the possibility to conduct the wide analyses for selected algorithm. Analysis means the simulation data interpretation and comparison with other data obtained using the motion-planning. The results of the careful analysis were considered for optimal path planning algorithms. The experimental evidence was proposed to demonflrate the effectiveness of the algorithm for fleady covered space. The results described in this work can be extended in a number of directions, and applied to other algorithms.
Keywords: Euclidean , metric, path planning, robotic, Simulated Annealing.
Introduction
The article is connected to the travelling salesman problem (TSP), but with some exceptions and conditions. In the case when the TSP is envisaged the following approximate path planning algorithms are used [1, 7, 9]:
• The closed neighbour algorithm;
• Simulated Annealing (SA);
• Genetic Algorithm (GA);
• Ant colony optimization.
The closed neighbour approach is the simple^ and flraightforward TSP one [13]. The way to this approach to
always visit the closefl city. The polynomial complexity of the approach is O(n2). The algorithm is relatively simple:
• choose a random city;
• find out the nearefl city unvisited and visit it;
• are there any unvisited cities left? If yes, repeat flep 2;
• return to the firfl city.
SA is successfully used and adapted to get an approximate solutions for the TSP [13]. SA is basically a randomized local search algorithm similar to Tabu Search but do not allow path exchange that deteriorates the solution. The polynomial complexity of the approach is O(n2) accordingly.
Input: ProblemSize, iterationsmax, tempmax Output:S& est
1. Scurreni*CreateInitialSolution(ProblemSize)
2. Sbest* Scurrent
3. for i=l to iterationsmcx do
4. &"^CreateNeighborSolution(Scurreni)
5. tempcurr*CalculateTemperature(i, tempmax)
6. if Cost(Si)<Cost(Scurrent) then
7. Scurrent* Si
8. if Cost(Si) < Cost(Sbest) then
9. Sbest* Si
10. end
11. else if Exp((Cost(Scurreni)-Cost(Si))/tempcurr)>Rand() then
12. Scurrent* Si
13. end
14. end
15. return Sbest
Figure 1 Pseudocode for SA
The SA method [15, 17] is widely used in applied science (see Fig. 1). The well-known traveling salesman problem has effectively solved by means of this method. For inflance, the arrangement of many circuit elements on a silicon subflrate is considerably improved to reduce interference among the elements [14, 21].
GA conducts in a way similar to the nature [7]. A basic GA flarts working with a randomly generated population of potential solution. The candidates are then mated to produce offspring and only some of them go through a mutating process. Each candidate has an optimal value demonflrating us how go it is. Choosing the mofl optimal candidates for mating and mutation the overall optimality of the candidate solutions increases. Using GA to the TSP involves implementing a crossover procedure, a measure of optimality and mutation as well. Optimality of the solution is a length of the solution.
Ant colony optimization is the algorithm that is inspired by the nature [11]. It is based on ant colony moving behaviour. Good results can be achieved by means of the algorithm but not for complex problems.
We managed to use SA algorithm rather successfully in our previous work [20] taking into account the specific side of this work (it will be discussed in detail further). Therefore, it is necessary to discuss some principles of SA realization in detail. In order to calculate the total path it is necessary to know the shortefl route among all the cities. As we do not know the diflance, we mufl apply one of the algorithms to define the shortefl route among all the cities. It is Dijkflra's algorithm [10] that gives the possibility to get the shortefl path tree. The polynomial complexity of the Dijkflra's algorithm is O(n2).
Goals
The aim of the research paper is to analyze motion-planning algorithms that contain the design of modelling programme. The programme is needed as environment modelling to obtain the simulation data. The simulation data give the possibility to conduct the wide analyses for selected algorithm. Analysis
means the simulation data interpretation and comparison with other data obtained using the motion-planning.
The use in practice and the necessity of it is greatly connected to optimal algorithm and methodological work out for autonomous agents that move in the space and are able to plan route on their own [2, 3, 4, 5 ,6, 19]. One of such agent-samples exiting in our everyday life is autonomous vacuum cleaner. Autonomous vacuum cleaners do not usually use the motion-planning algorithm. They are based on some simple algorithms, for example cleaning in a spiral, crossing the premises avoiding the walls and their moving is casual after touching the walls. The philosophy of this design was offered by the scientifls of Massachusetts Inflitute of Technology. Agents mufl behave as insects having primitive controlling devices in accordance to the environment. As a result, though an autonomous vacuum cleaner is very effective in cleaning premises, it is required much more time as compared with work made by a human. There is a drawback, the autonomous vacuum cleans some spaces many times but other spaces only once. The use of motion-planning algorithms can raise the effectiveness of an autonomous vacuum cleaner.
Assumptions
In order to fulfill the aim of the research paper the following conditions are introduced for:
• premises where an object moves;
• robot (or object) moves around the premises;
• path the robot moves on in the premises.
The premises are presented as two-dimensional plane. The plane of premises is equally divided into the cells. The cell dimensions are equal to agent dimension that moves in the premises. The space can be represented as a graph with two kinds of edges (see Fig. 2). Horizontal and vertical edges are marked with unbroken lines they are of similar length, but others are longer and marked with dash lines. It is linked with agent movement possibilities.
Figure 2
The example of the graph and 3 x 3 space
The object moves only one cell forward and back i.e. during one motion the object can move to the one cell from empty eight ones (eight cells around one cell) paying attention to that cell is
not occupied by the obflacle but if it is occupied, the graph will not have the relevant vertex (see Fig. 3).
Figure 3
The example of agent's motion (where vij is relevant vertex)
As opposed to classical TSP we take a number of additional rules and it means that the agent can cross the one the same cell several times in succession (it mufl cross any cell one time obligatory). Thus, the agent's initial vertex does not coincide with its final vertex of total route.
In this research paper both algorithms were compared practically using and combining different placement of obflacles in the unchangeable two-dimensional space. All the results were
obtained on one and the same computer (2.66 GHz processor and 2GB RAM), operating syflems (Ubuntu 15.04 Linux were used). The following information was collected about:
• the number of covering for each cell;
• the time which was necessary for both algorithms to plan the route.
The given illuflrations (see Fig. 5) show coverage density (it is an example that was obtained in our previous work [20]).
Figure 4
Density scale (white - uncovered; black - covered the mofl)
The density scale (see Fig. 4) is the same for all coverage densities. Coverage density shows how often the robot covers each cell.
1 1) 1: 1 -I 1.1
Figure 5
Coverage density for the space without obflacles for SA
Mathematical analysis
Let's discuss Euclidian (or hypotenuse of right-angled triangle) and non-Euclidian (calculated with offered/developed
approach) diflance between two graph points that are mutually moved along vertical and longitude (see Fig. 6).
Figure 6
Example for agents moving from junction v0,3 to junction v5,0
Let's label the diflance calculated by means of non-Euclidian geometry with L, then L = K + 205*A. You can see that these diflances numerically differ and not always are equal. Usually L is a bit bigger than Euclidian diflance (or hypotenuse C). L matches with Euclidian diflance at following conditions (see Fig. 6):
• if catheti of a triangle are equal and then K = 0 (in case of equilateral triangle);
• or junctions are not mutually moved along vertical or longitude. For example if one of the catheti of triangle is equal with zero.
To assess numerically the difference between both diflances, you can invent the following coherence: L corresponds to 100% and we have to calculate for how many percent L is bigger than C, i.e., how big percent refers to (L - C) difference (let's label the difference percent from L with P, then P = ((L-C)*100)/L). Differences and L values depend on K and A values. Then P dependence on K and A can be expressed with a function with two arguments (see Fig. 7).
Figure 7
Example for agents moving from junction v0,3 to junction v5,0
If A = confl, then P is a function with one argument that depends on K (see Fig. 8).
Figure 8 P dependence on K, if A = 1
You can see that at all A values maximum P value is equal with ~7.612%. You can express (analytically) K value from the following equitation (considering that L = K + 205*A): (((K + 205*A)-C)*100)/( K + 205*A) = 7.611. Solution of equitation is quite simple: K = 1.414*A (it mufl be marked that K is a whole number, i.e., it will always be bigger or smaller that the given value). You can also calculate value of angle a at the maximum value P that will remain conflant irrespective of value A. Considering that K is equal with 1.414*A, then a = arcsin((A + 1.414*A)/(((A + 1.414*A)2) + A2)05) = arcsin(2.414*A/ (5.827*A2+A2)05) = arcsin(2.414*A/(6.827*A2)05) = arcsin(2.414*A/2.613*A) = arcsin(2.414/2.613). After all simplifications angle a is equal with 67.65°.
Results
Taking into account the fact that the diflance among all the vertexes (cities) are unknown in the beginning, it is necessary to define the shortefl paths among those vertexes mentioned above.
Dijkflra's algorithm can be used but increasing the measures of the premises, the time is proportionally increases accordingly that is necessary for evaluating path tree. Therefore, it is needed to simplify the calculation of the shortefl path, which is possible, provided the peculiarities and nuances of the task are taken into consideration. In addition, the empty premises should be observed. If all the mentioned above remains valid, the simple algorithm can be worked out to define the shortefl paths.
Let us consider the agent's general moving paths. If there are no vertexes between the current initial and goal vertexes, the agent can move only to eight possible positions (cells) depending on goal vertex (see Fig. 3). Admitting that firfl vertex index i define the vertical position and the second vertex j defines the horizontal position we can draw a line either horizontally or vertically. And one of the vertexes will have the index with common value (see Fig. 9).
Figure 9
The example of agent moving horizontally (where i index value is common for both vertexes)
Another situation can be seen if the current initial and goal vertexes are neither on the horizontal nor vertical lines (see Fig. 10-12).
Figure 10
Three examples of agent moving (where A, B and C are sections among the vertexes): agent moves from v0,2 to v3,0 crossing
v1,2
7
6
5
4
3
2
10
15
20
25
30
А
b.)
Figure 11
Three examples of agent moving (where A, B and C are sections among the vertexes): agent moves from v0,2 to v4,0 crossing
v2,2
A c.)
Figure 12
Three examples of agent moving (where A, B and C are sections among the vertexes agent moves from v0,3 to v4,0 crossing
v1,3)
All cases of Fig. 10-12 have common characteriflics that unites them. The shortefl path from initial vertex to goal vertex is section C but for the agent this path is unavailable because of current task conditions and peculiarities. These cases can be described by the right-angled triangle where C is a side of the triangle. In addition, side B is longer than side A. One of the shortefl paths among the relevant (corresponding) vertexes:
• the agent moves along the longefl side B of the right-angled triangle until the gap between the covered path and side B is equal to side A;
• if gap between the covered path and side B is equal to side A, then the agent moves along the angle allowed (along the section D) to the goal vertex (let us mark that this action corresponds to the case when side B is equal to side A i.e. the right-angled triangle is the isosceles triangle, too (see Fig. 10) in
case initial vertex is v1,2, 4, (see Fig. 8) in case initial vertex is v2 2 and (see Fig. 12) in case initial vertex is vj 3)).
We can follow that the path is longer than optimal side C. And it can be calculated by the use of following formulae: L = B-A+205*A, where L is the length of the path from initial vertex to goal vertex. By turn, C can be calculated from C = (A2+B2)05. It is possible to calculate how match percent L is longer than C (if L is equal to 100 %), then the final result is equal to P=((L-C)*100)/L. Our goal premises are 100 x 100 cells. The value of P is reflected with contour line for the given premises depending on A and B (see Fig. 13).
The method/algorithm mentioned was applied inflead of Dijkflra's algorithm to calculate total path or covering of 100 x 100 cells big premises and it is obflacles free (see Fig. 14).
Figure 13
P value depending on B and A, if A > 1 and B > A
Density of covering
Figure 14
The density of covering for 100 x 100 cells big
Density of covering changes from 1 up to 40 (there are the cells which were covered only once and there are the cells covered maximum 40 times). Totally, the agent performed 192666 fleps in order to cross each cell of the premises.
Comparison with other solutions
This section presents several search algorithms. Mofl of these are jufl classical graph search algorithms [8, 16].
Breadth firfl - selects flates using the firfl-come, firfl-serve principle. This causes the search frontier to grow uniformly and is therefore referred to as breadth-firfl search. The asymptotic running time of breadth-firfl search is O(|V|+|E|), in which |V| and |E| are the numbers of vertices and edges, respectively, in the flate transition graph [16].
Depth firfl - the resulting variant is called depth-firfl search because the search dives quickly into the graph. The running time of depth firfl search is also O(|V|+|E|) [16].
Dijkflra's algorithm - syflematic search algorithm that finds optimal plans because it is also useful for finding feasible plans. The running time is O(|V| lg |V|+|E|), in which |V| and |E| are the numbers of edges and vertices, respectively, in the graph representation of the discrete planning problem [16].
A-flar algorithm - works in exactly the same way as Dijkflra's algorithm. The only difference is the function used to sort a priority queue. It has some advantages, but in some problems it is difficult or impossible to find a heuriflic that is both efficient to evaluate and provides good search guidance [12, 16, 18].
Befl firfl - the priority queue is sorted according to an eflimate of the optimal cofl-to-go. The solutions obtained in this way are not necessarily optimal, therefore, it does not matter whether the eflimate exceeds the true optimal cofl-to-go, which was important to maintain optimality for A* search. The worfl-case performance of befl-firfl search is worse than that of A* search and dynamic programming [16].
Iterative deepening - approach is usually preferable if the search tree has a large branching factor. This could occur if there are many actions per flate and only a few flates are revisited. The iterative deepening method has better worfl-case performance than breadth- firfl search for many problems [16,18].
Our approach - implementation is very simple according to the task (which is envisaged in the chapter "Results" in detail). The running time is O(C), in which C is a conflant. The value of running time is bounded by a value that does not depend on the size of the input.
Conclusions
It can be concluded that the algorithm offered is rather simple and it replaced Dijkflra's algorithm effectively according to the task. The algorithm allows decreasing the time of calculation, which is necessary to define the shortefl route among graph vertexes.
The shortefl path can be defined in a simple way (even in such cases mentioned in Fig. 7-9), provided that it is necessary to know the gap between the indexes of initial and goal vertexes.
For inflance, if initial vertex is v..,. and goal vertex is v._... the firfl gap is A1=|i1-i2| and the second gap is A2=|j1-j2|. As to the next flep, it is needed to calculate the biggefl gap between both the gaps. The shortefl path is equal to the biggefl gap. For inflance, Fig. 7 reflects the shortefl path which occupies 4 cells, but in other cases (see Fig. 8-9) it is 5 cells big.
It mufl be marked that total path can be a bit longer it is connected to the specific task which was envisaged in the chapters "Assumptions" and "Results" in detail. The worfl case can be evaluated theoretically for the premises of 100 x 100 cells. If we take into consideration that the total route will consifl of path pieces, which are longer than 7.61 % in comparison with C value (see Fig. 10), the total path will be longer than optimal 7.61 % (actually, it is the worfl maximal variant. We mufl pay attention to the fact that SA provides only approximate solution).
The algorithm can be successfully used e.g. in autonomous public transport reflricted by means of rules, technical requirements in autonomous robots and military equipment. In addition, the algorithm can be used in various computer games where a path planning is done in dynamic environment.
It is possible to conclude that the algorithm offered can be used in the different application areas not only for path planning of a robot.
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антикоррозионным лакокрасочный материал на основе акриловых сополимеров
Абросимова Лилия Федоровна
аспирант, Комсомольский-на-Амуре Государственный Технический Университет, г. Комсомольск-на-Амуре
Шакирова Ольга Григорьевна
к.х.н., профессор, Комсомольский-на-Амуре Государственный Технический Университет, г. Комсомольск-на-Амуре
АННОТАЦИЯ
Разработан антикоррозионный лакокрасочный материал на основе акриловых сополимеров. Исследованы технологические, потребительские и эксплуатационные свойства полученного лакокрасочного материала (ЛКМ) и покрытия на его основе (ЛКП).
ABSTRACT
Anticorrosive paint based on acrylic copolymers have been developed. Technology , consumer and exploitative properties of the paint and coatings based on it have been researched.
