2. Factors of uncertainty in processing trajectory measurements

In applied mathematics, the development of methods for processing trajectory measurements occupies a special place because of their complexity. General methodological problems in this area have not been solved in many respects due to the possibility of obtaining subjective conclusions or the use of excessive formalization, which makes it difficult to extract practically useful results.

An important role in solving problems associated with the development of methods for processing trajectory measurements is played by the concept of randomness. The concept of randomness is a certain type of uncertainty, characteristic of frequently observed events.

The methods used to solve problems related to the processing of trajectory measurements can be divided into sections corresponding to those deductive theories whose apparatus is used to solve problems from other fields of knowledge. If the studied problem, the solution of which provides the development of new methods for processing trajectory measurements, can be represented as a certain set of objects related by relations, then it is always possible either to find a suitable formalism to solve the problem or to create a new mathematical structure suitable for solving the problem.

It should be noted that the question of whether it is possible to single out objects forming a population in such a problem area as the development of new methods for processing trajectory measurements that allows them to be interpreted as sets connected by relations is a matter of not applied mathematics, but a part of the science of developing and testing of complex space systems.

The limited knowledge of the processes and phenomena studied does not allow the creation of absolutely accurate models of elements, including space systems and technologies. In addition, it is impossible to carry out an infinite number of experiments on the system model because of real limitations. For these reasons, the accuracy of the estimates obtained will be determined by the reliability of the information on the structure and parameters of the models being created, and also by the errors caused by the imperfection of the methods used for setting and processing the experiments carried out on the model.

If the accuracy of calculating the indicators will be sufficient for practical purposes, then the application of simulation methods can be considered justified. In practice, a series of control checks is carried out for this purpose, the main purpose of which is to establish a measure of 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 under analysis. Otherwise, the system model needs to be improved.

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 kinemat-

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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

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

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

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

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 applica-

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

tions, the object is idealized as a material point located at its center of mass.

vehicles. The main method for determining the motion of space objects is modeling.

the control actions in order to evade the collisions of space vehicles.

ics and can be supplemented by equations of control.

problem being solved.

distribution laws.

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 eliminating the danger of such a collision allows.
