2.1. Analysis of the risk of collisions of space appliances

The area where most artificial earth satellites operate is very extensive; its volume is about 1012–1013 km3 .

The density of artificial Earth satellites and space debris can be estimated by numbers in the order of 1015, and this number is constantly increasing. For example, in 2007, China tested an anti-satellite missile, sending it to one of its old satellites, adding about 3500 extra fragments in the area between 160 and 2000 km above the surface. This is a very large amount of objects and they must somehow be taken into account. The general picture of the contamination of near-Earth space is clearly shown in Figure 2 [3]. The orbital information on more than 20,000 space objects (SO) of more than 10 cm in size has been fixed and regularly updated.

Figure 2. Earth space.

Simulation of the motion of such satellites reduces to obtaining, as a rule, systems of ordinary differential equations of motion of an object and their integration by one or another method. As a result, the dependence of the motion parameters on time is obtained for the given initial conditions. These equations are a form of representation of the laws of dynamics and kinematics and can be supplemented by equations of control.

proximity between the real and simulated processes. If, according to the results of the comparison, it turns out that for all input conditions, the differences in the simulation of real processes do not exceed some given critical values, then the model is considered adequate for the system

In the process of modeling, it is possible to use various assumptions and coarsening, as a result of which the mathematical formulation of the investigated problem can only be its approximate reflection. When carrying out formalizations, there always arises the question of the adequacy of the resulting mathematical description of problems associated with the development of methods for processing trajectory measurements. If the solution of the content problem is not complicated and requires a numerical solution, an adequacy check can be made by experimental calculations using the initial data for which the desired results are obtained experimentally. The calculated results should differ from the reference data by no more than the task of assessing the risks of collision of space vehicles or carrying out preventive measures

The area where most artificial earth satellites operate is very extensive; its volume is about

The density of artificial Earth satellites and space debris can be estimated by numbers in the order of 1015, and this number is constantly increasing. For example, in 2007, China tested an anti-satellite missile, sending it to one of its old satellites, adding about 3500 extra fragments in the area between 160 and 2000 km above the surface. This is a very large amount of objects and they must somehow be taken into account. The general picture of the contamination of near-Earth space is clearly shown in Figure 2 [3]. The orbital information on more than 20,000 space

objects (SO) of more than 10 cm in size has been fixed and regularly updated.

under analysis. Otherwise, the system model needs to be improved.

eliminating the danger of such a collision allows.

1012–1013 km3

Figure 2. Earth space.

.

200 Probabilistic Modeling in System Engineering

2.1. Analysis of the risk of collisions of space appliances

By managing the catalogs of space objects (and such catalogs exist both in Russia and in the US), one can assess the mutual position and carry out the forecast of their movement. In particular, it is possible to assess the dangerous convergence and even collision of space vehicles. The main method for determining the motion of space objects is modeling.

Simulation of the motion of controlled space objects can be performed with different goals determined by the specific content of the tasks being solved. Among the tasks that involve the use of motion models for controlled objects, let us dwell on the identification of parameters and SO states based on the results of their measurements. Such a task is the basis for justifying the control actions in order to evade the collisions of space vehicles.

Any of the models used in solving the problem presented earlier is a mathematical idealization of real motion. Therefore, in modeling, of course, a special question arises about the adequacy of the mathematical description of the real movement of an object. The adequacy of the model is directly dependent on the degree of confidence in the a priori data, the completeness of their accounting for modeling, and the accuracy of the model's reproduction on a computer.

Requirements for accuracy in modeling can be considered to be dictated by the content of the problem being solved.

The a priori data on motion include the laws of kinematics and dynamics, the parameters (for example, the traction force, the nature of its variation in time and the time of the engine, the aerodynamic coefficients), the characteristics of the surrounding space (for example, the model of the atmosphere, the gravitational potential), the characteristics of the interaction of the object with the surrounding medium. Sometimes it becomes possible to use the data about the software path (program parameters) of the movement of the object. The a priori information may include the law governing the apparatus (for example, the method of parallel approach) and the features of its implementation by the control system (lag, other management errors). These data can include both deterministic and random parameters with known distribution laws.

The most reliable are usually the laws of dynamics and kinematics of the motion of objects. The validity of kinematic constraints such as "the linear velocity vector of the object is the first time derivative of the vector of its position" is beyond doubt. At the same time, the use of the dynamics relations should be carried out taking into account that they are sufficiently close to the real movement only under the conditions of their formulation. In many practical applications, the object is idealized as a material point located at its center of mass.

If by the conditions of the solution of the problem this assumption is relatively rough, then the degree of confidence in describing the trajectory of the movement of the object as a material point can turn out to be low. If necessary, the object can be considered as a system of rigid bodies and clarify the description. If it is required to take into account more subtle effects that have a noticeable effect on the components of the object's motion that are of interest, for example, on orientation in space, then the movements of the fuel components in the supply lines and the like can be modeled.

The confidence level of the modeling quality is the interval B, in which the error characteristic

The Approach of Probabilistic Risk Analysis and Rationale of Preventive Measures for Space Systems…

In modeling, the influence of the atmosphere, the rotation of the earth, reactive forces, etc., can be taken into account to some extent. To simplify the model, a number of factors that determine the movement of the object, but are insignificant, are combined and replaced in the noise component model ("useful noise"). Useful noise is a random amount included in the model. The resulting solution of the equations is the deterministic basis of the simulated motion. It can be supplemented by random specially imitated components, which are introduced additively,

Let us consider the system of equations of translational and rotational motions of SO. The first subgroup of equations characterizes the displacement of the center of mass, and the second

In doing so, we will use the normal terrestrial coordinate system. For simplicity, we assume that the acceleration of gravity is constant in magnitude and direction, Coriolis acceleration is

The simplest way is that the velocity of the translational motion of the center of mass of the rocket relative to the Earth is described in the projections on the axis of the trajectory coordi-

angular velocity of rotation of the trajectory coordinate system Oxk yk zk relatively fixed system Oxgygzg on the axis of the trajectory coordinate system. To determine them, we use the relations that reveal the relationship of these components of the angular velocity with the

<sup>ω</sup><sup>Χ</sup><sup>k</sup> <sup>¼</sup> <sup>Ψ</sup>\_ sin <sup>θ</sup>iωyk <sup>¼</sup> <sup>Ψ</sup>\_ cos <sup>θ</sup>iωzk <sup>¼</sup> <sup>θ</sup>

Here θ and Ψ are the angles of the path and the slope of the trajectory. As a result of the

<sup>V</sup>\_ <sup>¼</sup> <sup>X</sup>FxR=m,V\_ <sup>¼</sup> <sup>X</sup>FyR=m, Vψ\_ cos <sup>θ</sup> <sup>¼</sup> <sup>X</sup>FzR=<sup>m</sup>

In the right-hand side of the equations, we include the components of the traction force,

orientation angles of the trajectory coordinate system with respect to the normal one:

<sup>=</sup>m; <sup>V</sup>\_ wyk ¼ �XFzk=m,

, ωzk are the projections of the

http://dx.doi.org/10.5772/intechopen.74212

203

PFzk are the projections on the axis of the specified coordinate system of

absent; the curvature of the Earth is neglected, the wind is not taken into account.

falls with a confidence probability.

multiplicatively or additively-multiplicatively.

characterizes the orientation of the object in space.

where PFx,

<sup>P</sup>Fyk ,

gravity, and control forces.

2.2. Differential equations of the spatial motion of space systems

nate system Oxk yk zk, since υxk ¼ υ, υyk ¼ υzk ¼ 0 in this case. Then,

the resultant force acting on the center of mass of the object; ωyk

substitution, we obtain a system of equations of the form

<sup>V</sup>\_ <sup>¼</sup> <sup>X</sup>Fx=m;V\_ wzk <sup>¼</sup> <sup>X</sup>Fyk

The degree of confidence in the mathematical model is determined not only by the nature of the idealization of the object (material point, aggregate of rigidly bound bodies, etc.), but also careful consideration when modeling individual factors affecting the movement of the apparatus. The adequacy of the model also depends on the accuracy of knowledge of the parameters appearing in the formulas (ballistic coefficient, aerodynamic coefficients). These values are determined, as a rule, experimentally and are therefore known with some errors.

As criteria of comparison and estimation of models of movement of SO, it is expedient to apply those or other modeling functional errors.

Denote λð Þt is the real change in the motion parameter on the time interval, ½ � O; Т , λMð Þt are the process simulating this motion. Then, the absolute deviation of the model motion from the real is a function of time ΔλMðÞ¼ t λMð Þ� t λð Þt , describing the modeling errors in time. As the values that generalize such errors, extreme, averaged and confidence indicators can be used.

By extreme exponents, we mean the largest (least) values of some characteristics of a function ΔλMð Þt , for example, a module ΔλMð Þt :

$$\alpha\_1[\lambda\_M(t)] = \max\_{t \in [O, T]} |\Delta \lambda\_M(t)|$$

When the average is implemented, this or that characteristic of the function ΔλMð Þt is averaged over a time interval ½ � O; T . An example of averaged index of modeling quality is the average square error of modeling:

$$\alpha\_2[\lambda\_M(t)] = \int\_0^T [\Delta \lambda\_M(t)]^2 dt.$$

The introduction of confidence indicators in quality is associated with the random nature of the modeling errors caused, for example, by the presence of random components in the movement of the object. The appearance of such components is due to fluctuations in the properties of the environment surrounding the SO, the parameters of various elements of the flight control system, and so on.

Confidence indicators are complex and combine a confidence interval and a confidence probability.

The confidence interval is the range of values of some characteristic β of the function <sup>Δ</sup>λMð Þ<sup>t</sup> : <sup>B</sup> <sup>¼</sup> <sup>β</sup>min; <sup>β</sup>max � �.

The confidence probability P is the probability that the calculated value of the characteristic β ¼ β½ � ΔλMð Þt in the confidence interval B.

The confidence level of the modeling quality is the interval B, in which the error characteristic falls with a confidence probability.

In modeling, the influence of the atmosphere, the rotation of the earth, reactive forces, etc., can be taken into account to some extent. To simplify the model, a number of factors that determine the movement of the object, but are insignificant, are combined and replaced in the noise component model ("useful noise"). Useful noise is a random amount included in the model.

The resulting solution of the equations is the deterministic basis of the simulated motion. It can be supplemented by random specially imitated components, which are introduced additively, multiplicatively or additively-multiplicatively.
