V. KOSTJUKOV, M. MEDVEDEV, V. PSHIKHOPOV OPTIMIZATION OF MOBILE ROBOT MOVEMENT ON A PLANE WITH FINITE NUMBER OF REPELLER SOURCES

Kostjukov V., Medvedev M., Pshikhopov V. Optimization of mobile robot movement on a plane with finite number of repeller sources. Abstract. The paper considers the problem of planning a mobile robot movement in a conflict environment, which is characterized by the presence of areas that impede the robot to complete the tasks. The main results of path planning in the conflict environment are considered. Special attention is paid to the approaches based on the risk functions and probabilistic methods. The conflict areas, which are formed by point sources that create in the general case asymmetric fields of a continuous type, are observed. A probabilistic description of such fields is proposed, examples of which are the probability of detection or defeat of a mobile robot. As a field description, the concept of characteristic probability function of the source is introduced; which allows us to optimize the movement of the robot in the conflict environment. The connection between the characteristic probability function of the source and the risk function, which can be used to formulate and solve simplified optimization problems, is demonstrated. The algorithm for mobile robot path planning that ensures the given probability of passing the conflict environment is being developed. An upper bound for the probability of the given environment passing under fixed boundary conditions is obtained. A procedure for optimizing the robot path in the conflict environment is proposed, which is characterized by higher computational efficiency achieved by avoiding the search for an exact optimal solution to a suboptimal one. A procedure is proposed for optimizing the robot path in the conflict environment, which is characterized by higher computational efficiency achieved by avoiding the search for an exact optimal solution to a suboptimal one. The proposed algorithms are implemented in the form of a software simulator for a group of ground-based robots and are studied by numerical simulation methods.

1. Introduction. The task of a mobile robot path planning is one of the most relevant problems of modern robotic science [1][2][3][4][5]. When planning paths in the conflict environment it is necessary not only to take into account the local obstacles, but also the presence of areas that impede the mobile robot from completing its tasks. The sources of such areas can be detecting equipment, sources of radioactive or chemical pollution, etc. This raises the problem of avoiding the effect of such sources by the mobile robot [6]. The popular approach to solving this problem in uncertain conditions is the use of AI technologies [7][8][9] and methods that utilize the probability theory or risk functions [10 -18].
Thus, in the papers [7][8][9] the different aspects of the neural network use related to the planning of movement tasks in the conflict environments are considered. P. Agrawal and H. Agrawal [7] solve the problem of the evasive objects group persecution. Hunters have to recognize the friendly robot groups and perform the path planning to persecute the evasive objects.
Developed neural network solves the persecution problem taking into account the delays and limitations of communication channels. The network predicts the evasive objects' motion. The proposed approach advantages in an obstructed environment are demonstrated by simulations. The paper [8] proposes a biologically inspired neural algorithm based on the dynamic threat assessment. The algorithm is a combination of a fuzzy set, neural network, artificial potential field, and Bellman optimization procedure. The effectiveness of the presented algorithm is demonstrated by experimental results. In the article [9] the problem of autonomous collision-avoidance of mobile robots in dynamic environments is considered. The system includes a threat-avoidance unit based on the neural estimation of threat degree.
The possibilities of using generalized risk functions to solve optimization tasks in case of uncertainties are thoroughly observed in the general review [10].
The work [11] sets the optimizations problems of minimizing the integral of that function along the target trajectory, basing on the definition of the risk function, which is set in each point within the space where the sources are located. This trajectory can be imposed with various restrictions, among which the minimum trajectory length l and the presence of an upper limit of this length are particularly important.
The authors use the risk function for finding an object at the point M from the sources action K in the form of In the case of a single source, this problem can be solved analytically by the methods of variations calculus. For the case of many sources, the authors proposed the nonlinear programming algorithm based on the optimization of finite transitions between the nodes of a special graph that takes into consideration the weight coefficients of edges in the form of distances between the adjacent vertices and the respective transmission costs in the form of the risk function integrals on these edges. The papers [12,13] show that in case of an object, moving with changing speed in the neighborhood with one source, the optimal trajectory and speed-changing mode are such that the current value of the summarized "signal" from the source must be constant in time. In the papers [14 -16] this property is generalized in the cases of several sources, moving and heterogeneous observers.
In [17], the authors introduce a functional that estimates the risk of detecting the j th moving object of a group that operates jointly (for example, to break through the enemy defenses) by one of the N sensors that can summarize the signals incoming at the same time: where  ij (t) and v j (t) are the distance to the i th sensor and the velocity of the j th moving object of a group; T j is the approximate time of this object to reach its target point. Here the "signal" on each sensor at a given moment of time reflects the respective summand, which depends from  ij (t) and v j (t). The exponent k characterizes the physical field in which the detection is carried out, and the exponent m is the dependence of the emitted signal's intensity level from the object's velocity. Thus, for example, for a primary hydroacoustic field in a shallow sea k = 1; k = 2 corresponds to a thermal field, a primary electromagnetic field or a primary hydroacoustic field in the deep sea; for the magnetic field detectors k = 3, and in case of the active detection mode for electromagnetic or hydroacoustic static fields k = 4.
Next, the various modifications of the introduced functionality are introduced to build the behavior models for the moving object groups, which differ by the information organization of their elements, i.e. by their degree of awareness about each other's actions and sensors. The resulting functionals already take into account the mentioned organization. There are the given results of a simulation of an enemy defense breakthrough by the groups with the different informational organization. The common aspect of all models constructed this way is the operation with the concept of detection risk instead of the concept of detection probability.
For example, introduced by the authors for the "non-cooperative" model risk function: where  j (x,y) is the distance from the current location of the object to the i th sensor does not correctly reflect any probability. First, the necessity to exclude the first argument of the function min(, 1), the absence of smoothness of this dependence on of the object's position indicates the artificial nature of approximation of the true probability function of the considered event -the detection of a moving object by sensors at a given point. Second, the introduced function does not allow us to evaluate the probabilities of detection by sensors or the probability of damage from fire sources or other adverse factors of the enemy when the assumption of the instantaneous action of these sources of counteraction has already been incorrect. Thus, the model constructed by the authors does not permit to accumulate correctly the probability of damage/detection during the final mission time in the general case.
In [18], the authors enhance the problem set in [17] by introducing the factor of concentration of individual moving objects in a small area, which increases the overall risk of their detection by the sensor system. The pi value, introduced here by the authors, formally reflects the probability of detecting a moving object by a group of sensors under conditions of finding other objects is also introduced by the cut-off principle of the subintegral expression of the respective risk function using the function min(, 1). The authors themselves notice that the issue of the correct representation of this probability has not been fully considered by them.
In addition, please, note that the problem of estimation of the probability of damaging the moving object in case of its detection is not raised in [17,18] as well.
The common thing of works [10][11][12][13][14][15][16][17][18] is the use of the risk function and absence of methods to calculate the probabilities of spotting, damage, etc. For example, an extreme trajectory, which can be obtained by an effective numerical method for optimizing risks on a network indirect graph [10], needs to determine this probability. Indeed, knowing the optimal trajectory dos not yet guarantee the completion of the mission by a single mobile object or a group of mobile objects if there is no way to calculate the probability of completing a mission. Having a methodology for calculating the probability of completing a mission, which is connected with passing the trajectory, will allow us in some cases to choose acceptable trajectories.
The papers [17,18] use the concept of the probability of object's damage in case of its detecting, but this probability is not considered as a function of trajectory and/or moving speed, because it is set as an external constant.
The papers [19][20][21][22] also present different methods of optimizing the risk function in the conflict environment. The method of planning of robot's behavior in an environment with threats is developed in the article [19]. It proposes the bioinspired system that realizes the global and local robot's РОБОТОТЕХНИКА, АВТОМАТИЗАЦИЯ И СИСТЕМЫ УПРАВЛЕНИЯ _____________________________________________ behavior. On the local level, the risk function is used, which is optimized with the proposed AI system. Also, in the mentioned paper, the collision probabilities are not introduced, and the risk function depends on distances and experimentally adjusted parameters. In the article [20] the problem of visiting the target points by one or several robots in the hostile environment is solved. Heuristic algorithms for obtaining optimal paths from the point of view of the probability of the task are proposed in the article. The article uses probabilities set by default, which calculation method is not proposed.
The study [21] considers the problem of finding a reliable path in an uncertain conflict environment, which is solved by minimizing the expected risk. To solve the problem, a topological map is compiled.
The authors of [22] propose the method of path planning in the stochastic environment. This method maximizes the probability of arriving at a given point within the given time interval. The authors describe the probability of passing the area using Levi's distribution. The problem is solved for an environment represented by the graph.
Thus, the problem of calculation of passing probability for random trajectories in conditions of random sources, also including their heterogeneity, is relevant and not well studied.
It should be mentioned that the problem of moving in the area with the sources that create threats for completing the assigned mission has much in common with movement in the environment with the obstacles. To solve this problem a significant number of methods are used which are based on potential fields [23][24][25][26], dynamic (including unstable) forces [4,5,27], graph-searching algorithms [28][29][30][31], and assessment of geometrical complexity of the environment [32].
Potential fields allow us to find the optimal path with a rigorous justification of the stability of the impact of the introduced virtual forces. However, the method restrictions are that it can guarantee obstacle avoidance, but the particular look of the planned trajectories cannot be exactly forecasted.
Dynamic (including unstable) fields make it possible to efficiently solve the problem of local minima; however, predicting the robot's trajectory in advance also seems problematic.
In many cases the discretized graph-searching algorithms allow us to find paths in the conflict and uncertain environments more effectively. Thus, [28] presents the algorithms of calculating the shortest path basing on the piecewise-linear discrete model of the configuration space, which permits to use it for random topology and dimensionality. The proposed method builds the geodesic path in the metrical space more precisely than the Dijkstra's algorithm.
The article [29] considers the multicomponent procedure that includes ant colony optimization and A*. The 3-D space is considered. There is a given set of target points for which the ant colony optimization calculated the sequence of passage on large cells. The path plan is clarified using A*, which takes into account the local obstacles.
The paper [30] presents the algorithm of planning in the 3-D environment that is presented in the form of the probability graph. To build the graph, a sufficient number of points-nodes are generated, the coordinates of which are random variables with a uniform distribution law inside an admissible region of space. On this graph, the permissible movements are highlighted without taking into account the obstacles. Then, using the algorithm А* and potential fields, the obstacles are taken into account and the path is formed.
The discrete searching procedures allow us to find the suboptimal paths with insignificant calculation expenses, but the obtained trajectory may need further processing, for example, path smoothing. From this point of view, the continuous methods can at once take into account the characteristic peculiarities of passing the conflict areas by the robots.
The present article is organized in the following way. Section 2 formulates the problem to be solved and gives a probabilistic description of the source with a local field of action. The description of the characteristic probability function is introduced and the basic assumptions are made. In section 3, based on the assumptions made, the characteristic probability function in continuous form for one source is proposed. In section 4, based on the finite-difference introduction, the function is introduced, which allows us to approximately calculate the probability of robot's nondetecting when it is moving by a random trajectory in the area of the source. The obtained result is generalized in the case of several repeller sources. Also, the expression for describing the probability of a robot's nondetecting is given in continuous form. Section 5 presents the examples of optimization problems connected with robot's movement within the area of one or several repeller sources. Chapter 6 contains the examples of numerical simulation.

Problem statement.
Let us assume that each repeller source S  has a center O in small neighborhood of which its radiation is maximal. For definiteness, we assume that the source creates probing radiation and we estimate the probability of the object passing undetected in the field of this source. However, all subsequent conclusions can be transferred to the sources of other types, in particular, creating damaging factors.
We will single out the sources whose scope is the entire subspace of the Euclidean plane that is valid for laying target trajectories of a continuous type, and sources with a scope limited by some figure inside the specified subspace, for example, by a circular sector.
Basing on the information about technical characteristics of this source it is possible to construct a dependence of the probability q s (Tr), q s 1 of a mobile robot passing undetected/undamaged from the particular trajectory Tr defined as a vector function of time. We will consider only the trajectories that can be presented as a set of a finite number of parameters.
Let us divide the initial object's movement trajectory Tr by separate sections r i , i=1,2,…,N; each section corresponding to an elementary trajectory (Tr i ).
We introduce the following probabilistic space ( Tr , F, P), where  Tr is the set of elementary events corresponding to the given trajectory Tr, F is the admissible set of the subsets  Tr ; P is the countably additive function on the set FF: P(F)1, P( Tr )=1, P()=0.
The set  Tr contains of: where q s (Tr i ) is the probability of the object's non-detecting on the elementary section A i A i+1 of the trajectory Tr irrelatively to the already passed section AA i of this trajectory; q s (AA i ) is the probability of the object's non-detecting when passing the section AA i . If for each section Tr i we determine its passing probability q si (Tr i ) then the value of the probability function of passing the mobile robot undetected along the entire trajectory q s (Tr) is calculated using the expression [33]: It needs to be mentioned that for the sources of continuous type we should use the division mentioned above to the entire trajectory, since, according to the definition of such sources, the entire trajectory is within their scope (see Fig. 1). In case of discrete sources it is enough to divide by sections only those parts of the trajectory that are within the respective figure which is the combination of scopes of these sources (see Fig. 2). In case of rather small sections r i the probability q s (Tr) can be calculated using the limit Thus, the problem can be reduced to determining the values of the function q s on rather small sections of the trajectory Tr within each of which the change in the distance from the object to each source can be neglected, as well as the change of the object's velocity. Assume that q si is the monotonically increasing function of the distance d i between the geometrical center M i of the area Tr i and the source's center O. This assumption is based on the fact that the probability of the robot spotting is monotonically increasing with decreasing distance to the center O. Assume that the source is directed. For simplicity, we assume that there is one characteristic direction of the maximum of its influence on the object and it is described by the unit vector n s . If  i is the angle between n s and direction drawn from the source's center O to the point M i , then the function q si is monotonically increasing with the increase of the module  i . In case of the discrete source we assume that the vector n s simultaneously sets the direction of the axis of symmetry of the circular cone -the range of action of such source according to the accepted assumption.
Let us describe the characteristic change of the source's influence depending on the distance using the parameter d m , and the respective change by angle using the parameter  m .
Next, assume that the source has response inertia and detection error. The latter can be caused by the influence of interference and errors in the identification system associated with the source. Let  be the effective frequency of the source, which is the greater, the lower the indicated inertia is. Thus, in case of fixed , the function q si depends on the time T i of passing the trajectory Tr i . Let us assume that q si is the monotonically decreasing function T i . The bigger is,  the lower is the probability of the objects' passing given the fixed time T i of being within the source's field.
Notice, that in general case q si depends on the orientation of the vector r i in relation to the direction OM i . However, in case of rather small r the influence of orientation can be neglected.
For the discrete type sources it might be appropriate to introduce the lower boundaries for values d and T: d min и T min . For example, the restrictions on minimum detection distance can be connected with the peculiarities of the beam pattern. The minimum duration of the object's presence in the field of the source can be determined by the necessity of signal accumulation or inertia of identification or guidance systems.
Thus, we will characterize the sources of both types by the set of , moreover S (the scope) for the continuous type source coincides with the plane subspace indicated above, within which the movements of the object are permissible, and for the discrete type source it coincides with the indicated above circled cone with the center O, opening angle , directing vector of the symmetry axis n s , minimum and maximum radiuses d min and d max (d min can be zero).
Definition 1. The value q 0s (d i ,  i , T i ) we will call the characteristic probability function of the source , if for rather small sections Tr i it determines the probability of their passing, i.e q si (Tr i )=q 0s (d i ,  i , T i ).

Synthesis of the characteristic probability function.
According to the Definition 1, the function q 0s (d i ,  i , T i ) satisfies the following Statement 1. Statement 1. The probability of robot detection by the considered source remains unchanged in two cases: 1. The mobile robot is at rest for a time T at some point M determined by the distance d and the angle ; 2. The mobile robot is moving for a time T along the circular arc determined by the radius d and the angle .
The indicated cases are presented on the example of continuous type source in Figure 3.
Let us introduce two events. Event 1 means that the robot is not detected while being at rest. Event 2 means that the robot is not detected while moving along the respective circular arc.  Fig. 3. Conditions for the constancy of the probability of detecting a robot appearing in Statement 1 Radial arrows in Figure 3 show that the source's scope is exceeding the circle with the radius d m that has the meaning of characteristic influence of the source over the distance to the object.
According to the definition 1, statement 1, as well as the assumptions made in section 2 regarding the properties of the function q 0s (d, , T) on rather small trajectory Tr, we propose the following function: where , ; values d m and  m are the normalizing divisors for the distance and the angle, respectively.
If we accept the hypothesis of a uniform distribution of the probability of non-detection of the robot in azimuth, then expression (3) will take the form: Figure 4 shows the dependencies of the characteristic probability function of the type (3) at =0 from the distance d for different time spent by the robot in the field T=1, 2, 4, 10 sec, 2p=2, 2p=4, d m =2 m, =10 sec -1 .  (Fig. 1), which travels through the scope of the source.
Let us divide the part of the trajectory into rather small intervals Obviously, for sufficiently big N we can neglect the change of the distance d and approximately assume: where q si (A i A i+1 ) is the desired probability of the object passing the area For the correct use of the indicated approximation, we require that for each r i the following condition is fulfilled: where k d <<1 is the parameter that characterize the allowable maximum value of projection r i on the на radius-vector connecting the cource's center with the initial point A i of the area A i A i+1 ;  i is the angle between the direction from the source to the point A i and the directed segment r i . The fulfillment of condition (6) allows us to use of the expression (5) for an approximate calculation of the probability of the robot non-detection.
If the obtained division fulfills the condition (6) then, according to the product theorem of event probabilities following one another [33], we obtain that the probability of the robot's passing along the trajectory AD, taking into account (5), (6) and (3), determined in the following way: The expression (7) can be generalized in case of a finite number of sources of the observed type under the assumption of their independence from each other.
, k=1,2,…,K. In this case, the probability of the robot's passing undetected along the trajectory AD is determined by the expression  Now let us obtain the expression for the probability of non-detection of the robot moving along the random trajectory in the continuous form. By tending the upper limit for the lengths of division sections in (7) to zero, in the limit we transform (7) to the following expression:
For the source with the function q 0s (d, , T) that does not depend on the azimuth, basing on the expression (4), in the same way we obtain 2 ( ) exp exp ( ( )/ ) .

Optimization problems of the robot passing in the field of one or several sources. 5.1. The problem of movement in the field of the sources that does not depend on the angle .
Let's consider the case of several continuous type sources, where each source is described by the characteristic probability function of the type (4).
Let us assume that the movement is made along the piecewise-linear trajectory Tr with the points {A l }, l=1,2,…,N 0 and constant velocity v.
For several independent sources the probability of passing the considered trajectory without being detected q(Tr) is determined by the following multiplication  (11) and (12) and the fact that at a constant movement velocity it is possible to go from the differential of time to the differential of movement, for the piecewise-linear trajectory we obtain: where  (k) and are the characteristic frequency and distance of the k th where ( )  is the angle between the direction O (k) A l and the vector A l A l+1 . The respective geometrical constructions are given in the Figure 5.

Fig. 5. Determination of the characteristic geometrical values
By performing integration in (14), the summation over l can be converted to: Here we use the designation of the probability integral [34]: It is obvious that the function (13) on the set of all possible piecewise-linear trajectories has no global extremum if the number of sources of the observed type is finite. Therefore, the quality functional should include, along with (13), at least one more component. Let such component describe the the requirement of minimizing deviations from the given initial trajectory Tr (0) with the fixed endpoints A 1 =A, A N0 =D. Then we can propose the following optimization function: is the normalizing factor for equalizing the dimensions and orders of magnitude of the terms in (19); the value  is the distance between the target and the initial trajectory described by the coordinates 0 : Let us consider an approximate solution to the problem of optimal motion in the field of repeller sources according to the criterion (19).
Due to the high complexity of the function q s (Tr) in the functional (19), we will consider the other function, which behavior is close to the q s (Tr). As such a function, we choose the distance from the current point to the geometric center of the repeller sources .
O  Then the functional (19), taking into account the expression (20) takes the form: After calculating the partial derivatives of the function (21), using the coordinates (x l , y l ), we find the stationery point A l : In case of fulfillment of the condition (23), according to the Sylvester's criterion [35], we have the alternating signs of all the subminors of the corresponding Hessian matrix, starting with the negative one, which is easily established due to the diagonal form of this matrix. This indicates that at the point (22) there is the local maximum of the function (21). Moreover, this maximum is global, because this function has no other stationery points besides (22). That is why are the coordinates of the suboptimal trajectory of the first iteration.
If we now consider the trajectory of the first iteration as the original trajectory, and apply the transformations described above, then we obtain the expression: Points A l2 are the coordinates of the suboptimal trajectory of the second iteration.
By carrying out further calculations using the method of mathematical induction we obtain the following expression: By simplifying the last expression, we obtain: Under the condition (24) the expression (27) is greater than one, therefore, with an increase in the iteration number, the coordinates A lu becomes more distant from the geometric center of the sources .
O  We show that as the iteration number u increases, the first term of the criterion (21) monotonically increases. Let us consider the first term of the criterion (21) as the function of parameter b=s u . By substituting the expression (26) to the first term (21) we obtain: By differentiating (28) we obtain: According to (29), the function (28) reaches the minimum at the point b=0 and monotonically increases as b increases. Moreover, as the iteration number increases, the probability q s (Tr) in (19) should also increase, since the increase of means distancing the path points from the sources. In this regard, we consider the asymptotic behavior of the function q s (Tr).
Note, that the coordinates of all points A lu , i=1,2,N-1, in accordance with (26) and when u are infinitely increasing, so all the segments of the trajectory A lu , except the first and the last one, are asymptotically leaving the scope of repeller sources. This corresponds to the fact that in the sum (16) for each source it is necessary to take only the first and last terms.
We consider these terms at the iteration u, and then move on to the corresponding limits for u. For the first term we obtain:

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Taking into account the latter expressions and the properties of the function (18) erf()=1, we obtain: Similarly, for the last term of the sum (16) we calculate: Then the limit value of the probability q s,lim is determined by the expression: Figure 6 shows the geometric constructions, basing on which the expressions and the passages to the limits were obtained in (30) and (31). Fig. 6. Geometric constructions to the determination of values (30,31) Thus, the procedure (26) allows us to build up from the initial trajectory A l0 optimally by the criterion (21). That allows solving the problem of the path finding that insures the given probability of passing undetected by the sources and has minimum difference from the initially planned path. In this case, the robot trajectory belongs to the piecewiselinear linear class, and the movement velocity is assumed to be constant. 5.2. Problem of accounting for repeller sources on trafficability maps. The characteristic probability function for sources enables to account the repeller sources on trafficability maps, for example, during the path planning that uses procedures that divide space into cells [36][37][38][39]. To perform this it is necessary to endow each i th cell of the map with a certain weight c i that reflects the probability of passing this cell by an object that moves in the source's field. The invariance of the weights c i relative to the movement trajectory is a necessary condition.
Let us assume that in some area UR 2 there are K repeller sources and the obstacles. Depending on the locations of the obstacles, for each area of cell U there are given weights (1) , 1,2,.., that characterize the geometric passability, and N is the number of all cells. We assume that any robot's trajectory Tr completely belongs to the area U.
Let us consider the weight (2) , 1,2,.., that reflects the probability of the robot's detection in this cell. Then we can consider the total weight in the form of: (1) (1) (2) max (1 ) , 1, 2,.., , where (1) max с is the maximum allowable value of partial weight (1) i с ;  is the weight fraction of the first type in the total weight c i .
In order to make weights (2) i с invariant to the trajectory, let us make the following assumption. We assume that the robot passes through each cell during the same time T e . This assumption is valid at a constant velocity v e and the same cell sizes L in . Then T e is determined by the expression: Assuming the characteristic size of the cell L in sufficiently small compared to the distance from the sources to the path, we can replace the distance from the source to the path point by the distance from the source to the center of the cell, i.e.: where O i is the center of the i th cell. Under the made assumptions, the probability of passing the i th cell without detection takes the form: Then we choose the weight (2) i с of each cell so that it decreases with increasing probability (36), i.e.: Since the optimization algorithms build the trajectory that passes through the cells with minimum total weight, then the proposed method of forming weights takes into account both the geometric passability of the cells and the probability of detecting a robot in this cell. Figure 7 gives a geometric explanation of the search for the optimal path using search procedures, for example, D*.
(2) m d (2) Fig. 7. Geometric explanations of the search for the optimal path in the environment with obstacles and repeller sources Figure 7 shows the cells passed by the robot. For these cell the internal numeration i=n(i AD ) is introduced. It also shows four sources and their characteristics ( ) ( ) , k k i i d  that determine the influence on the robot in this cell. Red squares indicate the obstacles impassable for the object.
Thus, the correct estimation of the "average" probability of passing each partition cell by a moving object under the conditions of counteraction sources when solving the problems of path planning by the methods of the D* family of algorithms makes it possible to take into account the influence of these sources in the respective patency maps. Therewith, it is required to bring correctly to the same weight the two partial weights of each cell using a reasonable choice of parameters (1) max , с  in the expression (33). 6. Example of identifying the parameters of the characteristic probability function. Let us consider an example of determining the parameters of the characteristic probability function (CPF) for the Abrams tank, which has the following specifications: the M256 directing gun; the rate of fire is  = 8 rounds per minute, effective firing range: 2500-3000 m. The average effective firing range is D 0 =2750 m.
Let the "effective damage" mean damage with a probability from the range of 0.9p0.95. Then the respective probability range for non-defeat is q 2 =0.05qq 1 =0.1. The average probability of non-defeat is: By solving the obtained system with respect to x,y and then returning to the variables d m and 2p, we obtain: When we make calculations (4) using the equations for the distance d m and exponent 2p according to (45) and taking into account Т=Т 0 =1min., = 8 rounds per minute, we obtain d m =2531 m, 2p=1.449. The dependence of CPF is shown in Fig. 8 with the dash-dotted line.
If we reduce the probability value q 2 , for example, to the level of q 2 =0,03, then the corresponding calculation by formulas (4) and (45) will result in: d m =2651 m, 2p=3,282. The respective CPF dependence is shown in Figure 8 with the dashed line. The obtained results show the possibility to approximate the effective adjustment of the parameters of characteristic probabilistic functions to the already known or obtained data on the specifications that determine the probabilistic properties of this weapon. A similar technique for identifying the CPF parameters is also valid for other types of sources.  (26) has been investigated. The goal is to construct a trajectory that provides a given probability q g and which deviates least from the original path. The problem of obtaining the highest possible probability of passing under fixed boundary conditions has also been studied.
In the simulation the criterion for robot stopping is the fulfillment of one of three conditions: 1) the passing probability q g is reached; 2) the difference in the passing probabilities between the adjacent iterations is less than a given value  kr ; 3) the maximum deviation of the average distance between the original and current trajectories  Trmax is reached.
The total number of points of the piecewise-linear trajectory for all numerical experiments is N=5. Figure 9 and Figure   The initial trajectory is marked in blue; the trajectories of the following iterations are marked in green. The probability of passing the initial iteration is q 0 =0.2554. The probability of passing the trajectory in the last iteration is q=0.8567< q g ; The maximum possible probability is q lim =0.8588. Since q lim < q g , the given probability of passing cannot be reached. Now let us investigate the influence of the characteristic frequencies of the sources  (k) . To do this let's carry out the experiment "B", the conditions of which differ from experiment "A" by the values of the vector =[5; 5; 5; 5]1/sec. The probability of passing each elementary section of the trajectory should be higher than in case of the sources in the experiment "A". The simulation results are shown in Figure 11 and Figure 12.
In case of the experiment "B" the limit value of probability is increased in comparison with the experiment "A" q lim =0.9267. The given probability of passing the path undetected was reached in 12 iterations q=0.902>q g . The probability of passing the initial trajectory is q 0 = 0.505. Now let us consider the case when the initial trajectory crosses the effective circle of at least one source. The conditions for the experiment "C" are the following.

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The results are shown in Figure 13 and Figure 14. The probability of passing q=0.9>q g is reached in 14 iterations. The probability of passing the initial path is q 0 =1.610 -4 . The maximum possible probability is q lim = 0.987. Рис. 13. Movement trajectories for the simulation of the experiment "C"