In this article, it is proposed to consider and discuss the implementation of the method of parallel convergence in space in a computer mathematics system. In this method, the pursuer's speed vector is directed arbitrarily. The pursuer's trajectory gradually approaches the movement in the plane formed by the line connecting the initial positions of the pursuer and the target, and the velocity vector. In this task, the target moves evenly and rectilinearly. The pursuer moves evenly. The points of the pursuer's trajectory are calculated sequentially. They are the intersection points of the plane containing the line of sight, the sphere and the cone. As the pursuer approaches the plane of target's movement, the algorithm for calculating the trajectory points changes. Now the point of the pursuer's trajectory is the result of the intersection of the sphere, the plane of movement of the target and the plane containing the line of sight.
to a point on the circle of Apollonius.
In Figure 1, the point is the pursuer's position, and the point is the target's position. The Apollonius circle is a set of points { }, for which it is characteristic that the ratio of distances to two fixed points (points P and K in Figure 1). Inrelation to the pursuit problem, it will look like this: | | | | = | |, | | where is the pursuer's speed, is the target's speed.
2021 Copyright for this paper by its authors. The fixed direction of target's movement allocates a single point K and a single direction of the pursuer's velocity on the Apollonian circle.
Then, for the problem of pursuit on a plane where the pursuer and the target move rectilinearly and uniformly, there is an iterative scheme presented in Figure 2: +1 = + | | ∙ | | ∙ ∆ , where ∆T is the time interval of a discrete pursuit problem. ∙ ∆ target +1 is predetermined: t: ∙ ∆ , +1 = + ∙ ∆ .
( − )
2 = 2 { = + ∙ +1 − ∙ | +1 − | .
The radius and the center of the Apollonius circle are calculated as follows: =
2 2 − 2 ∙ | − |, = +
∙ ( − ).
2
2 − 2
satisfies the solution of the system of equations (
The purpose of this article is to describe a model of the pursuit problem in space, when the velocity The task is to calculate the points of the pursuer's trajectory at a certain trajectory of the target. vectors of the pursuer and the target, in Figure 3 and , respectively, do not lie in the same plane. ∝ Φi Σ i
We will assume that at any time the movement of the target can be represented as rectilinear and uniform.
The plane formed by the line of sight at the time of the start of the pursuit in Figure 3, this is a origin at point , the axis of the abscissa directed along a straight line( ). straight line ( ) and the velocity vector , will be considered as the coordinate plane with the
It is necessary in the iterative process to ensure that the coordinates of the point of the pursuer are located in the coordinate plane
in Figure 3 this is the plane . In this case, the point of the pursuer's position belonged to the plane corresponding to the moment of time. The plane contains the next step of the target (point ), is perpendicular to the plane and parallel to the line of sight ( ).
In addition, the trajectory of the pursuer must satisfy the curvature restrictions, that is, the radius of curvature of the trajectory cannot be less than a certain threshold value.
The geometric model for constructing the trajectory of the pursuer can be conditionally divided into three parts. In the first part, the trajectory in space is calculated. In the second part, the trajectory is calculated on the plane. The third describes the smooth transition of the trajectory from space to plane. 3.1.
In the kinematic model of parallel convergence in space, considered in this article, the pursuer's trajectory is calculated for two cases. In the first case, the trajectory segment is located in space. In the second case, the task turns into a pursuit on the plane. The pursuer in Figure 3 moves along the plane . A smooth transition from space to a plane is also calculated.
The pursuit model is discrete, so the time interval is introduced, during which the participants of the iterative process make a step. The pursuer, being at the point (Fig. 3), has the opportunity to take a step within the sphere of radius ∙ Δ
with the center at the point . is the speed module of the uniform pursuer's movement . This possibility is limited by a regular cone with a solution angle ∝ and a vertex at the point .
The angle of the cone solution is equal to ∝= ∙ ∆ , ω - is the maximum frequency of angular into an iterative scheme shown in Figure 2. rotation of the pursuer equal to = ⁄ , where is the minimum radius of curvature of the (Figure 3). In the future, when moving to the plane (Figure 3), belonging to the plane is transformed
The axis of the cone is directed along the current velocity vector of the pursuer leaving the point . Thus, there is a geometric problem of calculating a point belonging to three surfaces: a sphere, a cone and a plane.
The model of the pursuit problem of this article allows you to replace the correct cone with a plane. The intersection line of a regular cone and a sphere belongs to the plane . The parameters of the plane will be as follows = ⁄
current pursuer's position , is the velocity modulus of the uniform pursuer's movement , = + ∙ ∙ ( ), where is the radius of the sphere equal to the pursuer's step ∙ Δ (Figure 3).
The calculation of the intersection points of the planes and with a sphere of radius with the center at the point is more preferable than the calculation of the intersection points of the sphere, the cone of the plane from the point of view of computational difficulties.
For the plane , the reference point is the point , the normal is the vector (Figure 3): is the unit vector of the plane normal , the velocity vector for the
,
{ +1 = + ℎ ∙
[ × ]
|[ × ]| (
The found value of ℎ is substituted into the first equation of the system (
Thus, the iterative process of calculating the trajectory of the pursuer's movement in space can be considered formed. 3.2.
Calculation of the trajectory of the pursuer's movement on the plane
In the case of the transition of the pursuit to a plane, it is necessary to reduce the problem to a parallel
convergence, as shown in Figure 1. In this case, the pursuer's speed is always directed to a point on the
circle of Apollonius (in Figure 1, this is the point ). Then an iterative scheme is used, represented by
the system of equations (
If the pursuer's speed when moving to the plane is not directed at the point , as in Figure 1, then if
certain conditions are met in the direction of movement of the pursuer, you can apply the same iterative
scheme described by the system of equations (
The condition for the movement direction is as follows. It is necessary that the angle γ (Figure 4) between the speeds and +1was less than or equal to the angle ∝ = ∙ ∆ , where ω is the permissible speed of rotation of the pursuer, ∆ is the time period of the iterative process.
If the angle γ is greater than∝, but the pursuer's velocity modulus is greater than the target's velocity modulus , then the following can be proposed as an iterative scheme (Figure 4).
As a one-parameter set of parallel sight lines ( ) (Figure 4), a set of composite parallel lines { ( )} is proposed, which is formed as follows: +1 ( ) = ( ) + ( +1 − ).
point line +1( ) (Figure 5).
The next pursuer's step +1 is the point of intersection of a circle of radius ∙ ∆ centered at the
The first line of one-parameter subgroups of the set of lines { ( )} is formed from a circle of minimum radius and direct a tangent line passing through the point (Figure 6). is the initial pursuer's position,
is the minimum radius of curvature of the pursuer's trajectory, ( ̆ point of contact with the circle. is a unit vector perpendicular to the pursuer's velocity vector . A composite line consists of an arc ) and a rectilinear segment[ ], where is the initial position of the target, and is the 3.3.
into account the transition of the pursuit process to the plane . The points of the intersection of the plane (the plane), the plane and the plane (Figure 3) and the intersection points of the cone, sphere and the plane (the plane of parallel convergence) are displayed on the screen.
During the entire iterative process, the mutual location of the points of the pursuer's position and the plane of the target's movement is analyzed. Since the plane of the target's movement coincides with the coordinate plane
, it is sufficient to analyze the pursuer's application to the sign. As soon as the application sign changes, it returns to the previous calculated point of the trajectory and the calculation is performed according to another iterative scheme. angle of solution ∝= ∙ Δ , as in Figure 3, and the plane of parallel motion .
It is shown that the application of the point −1 has a positive value, and the application of the point ∗ has a negative value. The coordinates of the point ∗ (Figure 8) are obtained as a result of the intersection of the sphere ( , ∙ Δ ), a cone with an axis of rotation along the vector −1with the ∗ Π Σ vector of the pursuer and the specified circle. and the plane
of the target movement is searched for. for switching to a plane.
There is a return to the point −1 and the intersection point with the plane of parallel movement The test program, written based on the materials of the article, implements exactly such a criterion
In the proposed model for calculating the trajectory of the pursuer, the plane Π of the target movement is determined by the initial line of sight (PT) and the velocity vector of the target movement. The plane Π in this case is the bounding surface. The test program implements such a model of pursuit: the transition from pursuit in space to pursuit on a plane without going beyond the plane of restriction and with restrictions on the curvature of the pursuer's trajectory.
In the test mode, also during the transition to the plane, the use of the point ∗ as the center of the sphere intersecting the planes Π and was tried, and the construction of the pursuit trajectory from it.
In this article, a kinematic model of the pursuit problem in space was proposed. With the development of technologies, artificial intelligence systems, satellite positioning technologies for moving objects, modeling of pursuit tasks has become important.
There are many tasks and conditions in which modeling of iterative processes is required. The research results may be in demand by developers of unmanned aerial vehicles with elements of artificial intelligence.
The work was carried out with the financial support of the innovation grant of the Buryat State University in 2021 "Control of a four-link manipulator by signals received from a neurointerface". Scientific supervisor Dubanov Alexander. 7. References