2.2. Differential equations of the spatial motion of space systems

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

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

As criteria of comparison and estimation of models of movement of SO, it is expedient to apply

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

t∈½ � O;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

> ð T

½ � <sup>Δ</sup>λMð Þ<sup>t</sup> <sup>2</sup>

dt:

O

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

Confidence indicators are complex and combine a confidence interval and a confidence prob-

The confidence interval is the range of values of some characteristic β of the function

The confidence probability P is the probability that the calculated value of the characteristic

j j ΔλMð Þt

determined, as a rule, experimentally and are therefore known with some errors.

α1½ �¼ λMð Þt max

α2½ �¼ λMð Þt

lines and the like can be modeled.

202 Probabilistic Modeling in System Engineering

those or other modeling functional errors.

ΔλMð Þt , for example, a module ΔλMð Þt :

square error of modeling:

system, and so on.

ΔλMð Þt : B ¼ βmin; βmax

� �.

β ¼ β½ � ΔλMð Þt in the confidence interval B.

ability.

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 characterizes the orientation of the object in space.

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

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

$$\dot{V} = \sum F\_{\mathbf{x}} / m; \dot{V}\_{w\_{\mathbf{z}\_k}} = \sum F\_{\mathbf{y}\_k} / m; \dot{V}\_{w\_{\mathbf{y}\_k}} = -\sum F\_{\mathbf{z}\_k} / m.$$

where PFx, <sup>P</sup>Fyk , PFzk are the projections on the axis of the specified coordinate system of the resultant force acting on the center of mass of the object; ωyk , ωzk are the projections of the 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 orientation angles of the trajectory coordinate system with respect to the normal one:

$$
\omega\_{\chi\_k} = \dot{\Psi}\sin\theta\_i\\\omega\_{y\_k} = \dot{\Psi}\cos\theta\_i\\\omega\_{z\_k} = \theta\_i
$$

Here θ and Ψ are the angles of the path and the slope of the trajectory. As a result of the substitution, we obtain a system of equations of the form

$$\dot{V} = \sum F \mathbf{x}\_{\mathbb{R}} / m\_{\prime} \dot{V} = \sum F \mathbf{y}\_{\mathbb{R}} / m\_{\prime} \, V \dot{\psi} \cos \theta = \sum F \mathbf{z}\_{\mathbb{R}} / m\_{\prime}$$

In the right-hand side of the equations, we include the components of the traction force, gravity, and control forces.

When obtaining an expression for the aerodynamic force, we use the velocity coordinate system 0xayaza, and then from it, we proceed to the trajectory system. Projections of this force on the coordinate axis of the last system are represented in the form

If we consider the problem in posing the risk of collision of an uncontrolled SO with a controlled spacecraft (SC), then it is necessary to consider the process of mutual proximity in

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

In this three-dimensional space, the spacecraft structure is considered as a sphere of a given radius. The region of possible position of the SO is represented in the form of an ellipsoid

CSCþSO ¼ CSC þ CSO

In the picture plane, the SV and the SO region are represented as a circle and an ellipse,

The problem reduces to the search for the probability of hit of a random vector whose density is given by the ellipsoid of scattering errors into the sphere of a given cone in Figure 3, where

control of the spacecraft, necessary maneuvers should be provided for the purpose of evasion. To assess the characteristics of the collision risk, an archive of dangerous convergence (ADC) between all objects in the catalog is maintained in the Information and Analytical Center for Near-Earth Space Monitoring, taking into account the data of the catalog of the American

The ADC gathers all potentially dangerous convergences between all cataloged SOs. "Potentially dangerous" means either convergence of two SOs to a distance less than a given distance

Figure 3. Geometric interpretation of the relative location of spacecraft and space objects in outer space at the time of

, the collision risk is high enough, therefore, when planning the

. For the value of Pc in

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

205

whose parameters and orientation are determined by the total error matrix

Tdc is the time of dangerous convergence, and V and Dv are the speed. For example, the approach of a SO to an ISS is considered safe if P<sup>c</sup> < 10�<sup>5</sup>

three-dimensional space and in the picture plane.

respectively.

the range between 10�<sup>4</sup> and 10�<sup>5</sup>

dangerous convergence.

USSSN; they are available on the Internet [6, 7].

$$\begin{aligned} \mathbf{R} \mathbf{A} \mathbf{x}\_{R} &= -\mathbf{x}\_{R} = -\mathbf{x}\_{d}; \\\\ \mathbf{R} \mathbf{A} \mathbf{y}\_{R} &= \mathbf{y}\_{R} = \mathbf{y}\_{a} \cos \gamma\_{a} \mathbf{-}\_{a} \sin \gamma\_{a}; \\\\ \mathbf{R} \mathbf{A} \mathbf{z}\_{R} &= \mathbf{z}\_{R} = \mathbf{y}\_{a} \cos \gamma\_{a} + \mathbf{z}\_{a} \sin \gamma\_{a} \end{aligned}$$

where γ<sup>а</sup> is an angle of rotation of the velocity system relative to the trajectory system.

Further equations that take into account the features of the motion of the SO will lead to a system of equations of motion.

Solving this system, we find all the characteristics of the motion of the rocket or SO:

$$V(t), \theta(t), \psi(t), \mathbf{x}\_{\mathcal{S}}(t), \mathbf{y}\_{\mathcal{S}}(t), r(t), \mathcal{S}(t), \mathcal{v}(t), a(t), \beta(t), \mathbf{y}\_{\mathcal{A}}(t), \sigma\_{\mathbf{x}}(t), \sigma\_{\mathbf{y}}(t), \sigma\_{\mathbf{z}}(t), m(t), z\_{\mathcal{S}}(t), \psi(t).$$

Naturally, the initial conditions for integration must be given.

With a rigorous theoretical approach to the solution of the problem of modeling, it is obviously impossible to separate the equations describing only the translational motion of the center of mass or only the rotational motion of the relative center of mass, and the equations of longitudinal and transverse motion. The relationship between translational and rotational movements is manifested through so-called cross-links.

The probability that the calculated values of the parameters of two SO movements β ¼ β½ � ΔλMð Þt in the confidence interval is estimated. In each of them, one can predict the risk of a dangerous convergence.

The same task is directly related to the definition of collision risk with space debris (CD). For probabilistic modeling of collision risks with space debris, special programs are used, for example:

Model SDPA [4] is a semi-analytic stochastic model for medium- and long-term forecasting of technogenic SG larger than 1 mm in low Earth orbits (LEOs) and geosynchronous Earth orbits (GEOs), for constructing the spatial distribution of concentration and velocity characteristics, as well as estimating collision risk [5].

The model uses summary data on SO of various sizes (including space debris without "binding" them to a specific source of pollution).

The measurement errors are estimated on the basis of averaging of the last measurements of the orbital parameters. In calculations of dangerous convergence, errors are calculated in several models, for example, in the orbital coordinate system, in the elements of the orbit, and in models of direct integration. The error matrix is used to calculate the collision probability.

If we consider the problem in posing the risk of collision of an uncontrolled SO with a controlled spacecraft (SC), then it is necessary to consider the process of mutual proximity in three-dimensional space and in the picture plane.

When obtaining an expression for the aerodynamic force, we use the velocity coordinate system 0xayaza, and then from it, we proceed to the trajectory system. Projections of this force

RAx<sup>R</sup> ¼ –x<sup>R</sup> ¼ –xa;

RAy<sup>R</sup> ¼ y<sup>R</sup> ¼ y<sup>a</sup> cos γa–z<sup>a</sup> sin γa;

RAz<sup>R</sup> ¼ z<sup>R</sup> ¼ y<sup>a</sup> cos γ<sup>a</sup> þ z<sup>a</sup> sin γ<sup>a</sup>

Further equations that take into account the features of the motion of the SO will lead to a

V tð Þ, θð Þt ,ψð Þt , xgð Þt , ygð Þt ,r tð Þ, ϑð Þt , γð Þt , αð Þt , βð Þt , γað Þt ,ϖxð Þt ,ϖyð Þt ,ϖzð Þt ,m tð Þ, zgð Þt ,ψð Þt :

With a rigorous theoretical approach to the solution of the problem of modeling, it is obviously impossible to separate the equations describing only the translational motion of the center of mass or only the rotational motion of the relative center of mass, and the equations of longitudinal and transverse motion. The relationship between translational and rotational movements

The probability that the calculated values of the parameters of two SO movements β ¼ β½ � ΔλMð Þt in the confidence interval is estimated. In each of them, one can predict the risk

The same task is directly related to the definition of collision risk with space debris (CD). For probabilistic modeling of collision risks with space debris, special programs are used, for

Model SDPA [4] is a semi-analytic stochastic model for medium- and long-term forecasting of technogenic SG larger than 1 mm in low Earth orbits (LEOs) and geosynchronous Earth orbits (GEOs), for constructing the spatial distribution of concentration and velocity characteristics,

The model uses summary data on SO of various sizes (including space debris without "bind-

The measurement errors are estimated on the basis of averaging of the last measurements of the orbital parameters. In calculations of dangerous convergence, errors are calculated in several models, for example, in the orbital coordinate system, in the elements of the orbit, and in models of direct integration. The error matrix is used to calculate the collision probability.

where γ<sup>а</sup> is an angle of rotation of the velocity system relative to the trajectory system.

Solving this system, we find all the characteristics of the motion of the rocket or SO:

on the coordinate axis of the last system are represented in the form

Naturally, the initial conditions for integration must be given.

is manifested through so-called cross-links.

of a dangerous convergence.

as well as estimating collision risk [5].

ing" them to a specific source of pollution).

example:

system of equations of motion.

204 Probabilistic Modeling in System Engineering

In this three-dimensional space, the spacecraft structure is considered as a sphere of a given radius. The region of possible position of the SO is represented in the form of an ellipsoid whose parameters and orientation are determined by the total error matrix

$$\mathsf{C}\_{\mathsf{SC}+\mathrm{SO}} = \mathsf{C}\_{\mathsf{SC}} + \mathsf{C}\_{\mathrm{SO}}.$$

In the picture plane, the SV and the SO region are represented as a circle and an ellipse, respectively.

The problem reduces to the search for the probability of hit of a random vector whose density is given by the ellipsoid of scattering errors into the sphere of a given cone in Figure 3, where Tdc is the time of dangerous convergence, and V and Dv are the speed.

For example, the approach of a SO to an ISS is considered safe if P<sup>c</sup> < 10�<sup>5</sup> . For the value of Pc in the range between 10�<sup>4</sup> and 10�<sup>5</sup> , the collision risk is high enough, therefore, when planning the control of the spacecraft, necessary maneuvers should be provided for the purpose of evasion.

To assess the characteristics of the collision risk, an archive of dangerous convergence (ADC) between all objects in the catalog is maintained in the Information and Analytical Center for Near-Earth Space Monitoring, taking into account the data of the catalog of the American USSSN; they are available on the Internet [6, 7].

The ADC gathers all potentially dangerous convergences between all cataloged SOs. "Potentially dangerous" means either convergence of two SOs to a distance less than a given distance

Figure 3. Geometric interpretation of the relative location of spacecraft and space objects in outer space at the time of dangerous convergence.

Δ, or a closer distance to a greater probability with a collision probability p<sup>c</sup> greater than the threshold pmin. Such convergences are about ≈15,000 per day. The archive is more than 20 years old. For each convergence, the following characteristics are stored in the ADC: the convergence time, trajectory and non-trajectory parameters of the objects convergence, the residuals at the moment of convergence and their probabilistic characteristics, the probability of collision. Briefly, the algorithm for the supporting ADC is described in the book [8]. There, a method for evaluating various risk characteristics using ADC is also described. For more detail, see Ref. [9].

events in large part, methods of probabilistic risk analysis and justification of preventive measures of damage reducing become most common for cosmic systems and technologies.

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

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

207

The movement of each element of the system of space objects can be divided into two components. First, the orbital object moves on a trajectory that can be represented in the elliptical form in the general case in the current time, which oriented in space in a certain way (osculating orbit). Second, the trajectory of the orbital object changes over time (generally, form and orientation are changing). Meanwhile, trajectories of motion of orbital objects change much slower than the position of orbital objects on these trajectories. Therefore, it is proposed to model changes of trajectories and identify the parts of the trajectories for the current moment in time, which are located from each other at a dangerous distance, from the point of view of possibility of collisions of orbital objects (nodes of mechanical conflicts). In other words, to simulate the nodes of mechanical conflicts, speed changing of which corresponds to speed changing of trajectories. For orbital objects, trajectories of which form a node conflict, it is necessary to determine the time intervals of their movement through the node conflicts without a significant investment of time (on the dangerous part of trajectory). Hence, the method of modeling system of orbital objects is based on the method of modeling the nodes of mechanical conflicts and the method of determining the time intervals of movement of the orbital

Tasks of the analysis of conflicts of orbital objects can be divided into two classes. In the first class, there are the tasks, where it is possible to analyze only the risk of collisions and not to predict specific orbital conflicts. Their solution is based on the consideration of the altitude-

In the second class, there are the tasks that demand prediction of orbital collisions. This prediction boils down to the prediction of convergence of pairs of orbital objects at a dangerous distance, from the point of view of their collision at possible deviations of objects from their calculated trajectories (these can be called dangerous or conflict convergences). In many cases, it is sufficient to predict only dangerous convergence of the orbital objects, and not to simulate the effects of conflicts, which change the trajectories of the colliding objects and form new orbital objects. Such tasks are solved when it is necessary to predict dangerous collisions for spacecraft, which can make the maneuver to avoid collisions. The task of prediction of dangerous convergence can be used as a base for models of near-Earth space contamination by orbital objects. The direct deterministic method is the most common. It is based on the formation of an archive of dangerous convergences of all possible pairs of orbital bodies at a specified time interval, which is included in the considered set of orbital objects (for each dangerous convergence, the passing time interval, the geometric characteristics of convergence

The traditional method to predict dangerous convergence is based on modeling the movement of objects and analyzing the current distance between them. There is a difficulty in this method. The relative speed of orbital objects can be more than 10 km/s. Meanwhile, the convergence at a dangerous distance of several kilometers lasts less than 1 s. Therefore, the prediction of dangerous convergences requires modeling with a correspondingly small time

step. At larger sizes of sets of orbital objects, it leads to significant time consumption.

latitudinal density distribution of the orbital objects at a specific point in time.

object through a node of conflicts.

and the probability of collision are determined).
