Method of control over spacecraft descent in atmosphere of planets

FIELD: aircraft engineering.

SUBSTANCE: invention relates to spacecraft control in atmosphere of planet by adjusting its aerodynamics. Proposed method consists in selection of conditions for changing the angle of roll to zero at changing the spacecraft from isothermal descent section (IDS) to skip path. With spacecraft in IDS, angle of roll (γ) is, first, increased to decrease aerodynamic performances and to maintain constant temperature at critical area of spacecraft surface. As flight velocity decreases angle (γ) is decreased from its maximum. In IDS, increase in aerodynamics does not cause further temperature increase over its first peak. Therefore selection of the moment of changing to γ=0 allows efficient deceleration of spacecraft at the next step of flight. The best option is the descent of spacecraft of IDS when γ reaches its maximum. Here, angle of attack is set to correspond to maximum aerodynamic performances. This increases the duration of final flight stage and deceleration efficiency. Increase in angle of attach after descent from IDS and completion of climb results in increased in drag, hence, decrease in velocity at initiation of soft landing system.

EFFECT: minimised final velocity and maximum temperature at surface critical area, lower power consumption.

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The invention relates to space exploration, in particular to control the descent of a SPACECRAFT (SC) in the atmosphere of the planet, using controlled aerodynamic braking and minimizing the final speed of the spacecraft under the condition of minimum and maximum temperature in the critical region of its surface.

A known method of controlling the descent of a spacecraft in the atmosphere of the planet, using controlled aerodynamic braking and providing the lower end of the speed of the spacecraft, described in the book - Ivanov N. M., Martynov A. I. "the Motion of spacecraft in the atmospheres of the planets. M.: Nauka, 1985, pp. 168-173 - [1]. This method is to control the aerodynamic braking by changing the roll angle γ KA. The motion of the spacecraft is carried out with a constant value of angle of attack corresponding to maximum aerodynamic balancing. The method involves one-time switching of the roll angle γ with values equal to π glad to zero, which corresponds to the switching of the effective aerodynamic qualities from the minimum value (-K) to the maximum (+K).

The disadvantages of this method are as follows. Firstly, in case of using this program upravleniyaschitayut temperature in the critical region of the surface of the SPACECRAFT reach excessively high values. This is because the spacecraft with glide, to fly a mission on trajectories with multiple ricochets and, consequently, with multiple local maxima temperatures. Moreover, the absolute maximum temperatures, as a rule, coincides with the second or third local maxima. Used management program roll angle does not effectively prevent the rise of temperature after reaching his first local maximum. Secondly, the method is similar does not control the angle of attack of the spacecraft. This reduces the potential damping rate and lower maximum temperatures increase the angle of attack leads to an increase in the intensity of aerodynamic braking KA.

A known method of controlling the descent of a spacecraft in the atmosphere of the planet, using controlled aerodynamic braking and providing the lower end of the speed of the spacecraft, described in the book.M. Ivanov, A. I. Martynov, "motion Control of spacecraft in the atmosphere of Mars. Moscow, Nauka, Main editorial Board for physical and mathematical literature, 1977, pp. 159-169 - [2]. This method is the two-parameter control of the spacecraft roll angles of attack. Input kosmicheskogo the apparatus to the atmosphere is performed with the angle γ=π rad and angle of attack α, the corresponding maximum value of the balancing aerodynamic quality. On the initial flight phase switches of the roll angle γ to zero. After reaching the angle of the velocity vector to the local horizon zero value of the roll angle γ is determined from the condition for the provision of space vehicle flight on izolyatsia the plot (the plot with constant height). Then you can switch the roll angle γ to zero, providing the movement of the spacecraft on ricochetnaya trajectory with increasing altitude. On this site there is an increase in the angle of attack α from the value corresponding to the maximum aerodynamic lift coefficient KA to a value corresponding to the maximum aerodynamic drag coefficient KA.

The disadvantages of this method are as follows. First, its implementation does not control the angle of attack α of the spacecraft on the area of intensive growth temperature in the critical region of the surface. While increasing the angle of attack α in this area reduces as the velocity of the SPACECRAFT, and depend on the temperature of the heating surface. Secondly, the introduction of sovestkogo of flight does not reduce the first local maximum of the temperature, because this area starts after reaching the T_max. Third, the descent of the SPACECRAFT with sovestkogo plot is carried out with the maximum value of the aerodynamic lift coefficient, that is, for sufficiently small values of the aerodynamic drag coefficient and the values of the aerodynamic quality that is different from the absolute maximum. In this case, there are provisions as to increase the intensity of aerodynamic drag and increase the duration of the flight. Both these factors may contribute to a lower final velocity of the SPACECRAFT.

The closest to the technical nature of the claimed method of controlling the descent of a spacecraft in the atmosphere of the planet, using controlled aerodynamic braking and providing the lower end of the speed of the spacecraft under the condition of minimum and maximum temperature in the critical region of its surface, is a method of controlling the descent of a spacecraft in the atmosphere of the planet which minimizes the maximum temperature in the critical region of the surface of the SPACECRAFT. Specified the known method described in the patent RU №2493059, publ. 20.09.2013 - [3], which is selected by the prototype. This method of controlling the descent of a spacecraft in the atmosphere of the planet lies in its space, Breakfast is the only orientation and management of aerodynamic braking, stabilization at the entrance into the atmosphere at the corners of the roll, yaw and angle of attack for maximum glide ratio, determining current coordinates and velocities of the flight of the spacecraft and the actuation means for the landing, in the process of descent continuously measure the temperature T of the external surface of the spacecraft in its critical region, for each measured temperature value T calculate the acceleration and speed of its change by calculating in time, respectively, the first$$\stackrel{}{T}$$and the second$$\stackrel{}{T}$$derivatives; upon reaching the second derivative negative values$$\stackrel{}{T}<0$$preserving the first derivative positive values of$$\stackrel{}{T}>0$$increase the angle of attack and continue the descent to the condition of equality to zero of the first derivative of$$\stackrel{}{T}=0$$, then set the values of the Glov roll and attack, ensuring compliance with equality to zero first and second derivatives of$$\stackrel{}{T}=\stackrel{}{T}=0$$at which carry out the descent of a spacecraft on the isotemperature section, upon reaching the first derivative of the negative values of$$\stackrel{}{T}<0$$establish a zero Bank angle and angle of attack for maximum value model quality, and complete the plot of inhibition of spacecraft.

The disadvantage of this method is that its implementation is not fully implemented all control in the final section of the descent of the SPACECRAFT to reduce the final speed. So, is fairly late gathering KA with isotemperature plot (if$$\stackrel{}{T}<0$$), which leads to the reduction of the length of the final segment of the flight SPACECRAFT and low intensity quenching speed. Furthermore, the method does not control the angle of attack KA immediately before the introduction of the soft landing system, which is also what leads to the decrease in the intensity of aerodynamic deceleration of the spacecraft.

The invention consists in the rational management of the roll angles of attack which minimizes the final speed of the flight SPACECRAFT while minimizing the maximum temperature in the critical region of its surface. This is achieved by introducing a new episode of the control compared to the prototype. First, the choice of appropriate conditions switching of the roll angle to zero, which ensures the vanishing KA with isotemperature site and transfer it to rikollisuus trajectory with increasing altitude. The motion of the SPACECRAFT in isotemperature site is provided by a corresponding change in angle of Bank: first there is an increase of the roll angle γ, leading to the reduction of the aerodynamic quality and maintaining a constant temperature in the critical region of the surface of the SPACECRAFT (the absence of such a control mode leads to the initial temperature decrease with further significant growth), and then decreasing the speed of the roll angle γ KA, reaching its maximum value and starts to decrease. At this stage of the flight the increase in aerodynamic quality does not lead to further increase in temperature beyond its first maximum. Therefore, when setting the zero value of the roll angle, you can achieve the most the e effective damping rate on the subsequent flight. The most rational is the gathering KA with isotemperature plot at the moment of achievement of the roll angle γ maximum value: earlier switching of the roll angle γ to a value of zero may cause further temperature rise, and a later - to reduce the duration of the subsequent section of the descent, and consequently reduce the efficiency of the damping rate of the SPACECRAFT. Secondly, at the time of descent of a SPACECRAFT with isotemperature plot set the angle of attack KA, corresponding to the maximum value of the aerodynamic quality KA (prototype - the maximum value of the aerodynamic lift coefficient KA). This is due to the need to ensure a longer final flight and more intense inhibition of CA in the atmosphere. Thirdly, the increased angle of attack KA after his departure from the isotemperature plot and complete set flight altitude increases aerodynamic drag coefficient and to a greater reduction of speed by the time of the introduction of the soft landing system.

The essence of the present invention, the method of controlling the descent of a spacecraft in the atmosphere of the planet lies in the spatial orientation of the spacecraft and the management of its aerodynamic deceleration, stabilization of the spacecraft is ri it enters the atmosphere of the planet on the roll angles γ, yaw and angle of attack α of the spacecraft, providing maximum glide ratio, the determination of the current coordinates and velocities of the flight of the spacecraft, continuous measurement of the temperature T of the external surface of the spacecraft in its critical region, we compute the velocity and acceleration of its change by computing time, respectively, the first$$\stackrel{}{T}$$and the second$$\stackrel{}{T}$$derivatives; upon reaching the second derivative negative values$$\stackrel{}{T}<0$$preserving the first derivative positive values of$$\stackrel{}{T}>0$$increase the angle of attack α of the spacecraft and continue the descent to the condition of equality to zero of the first derivative of$$\stackrel{}{T}=0$$after which the set values of roll angles γ and attack α of the spacecraft, providing the conditions of equality to zero of the first and second proizvodi is x $$\stackrel{}{T}=0$$and$$\stackrel{}{T}=0$$providing the descent of a spacecraft on the isotemperature plot, while in the process of motion of a spacecraft on the isotemperature plot continuously comparing values of roll angles γ in the current t_iprevious t_i-1times and compute their difference Δγ by the formula

Δγ=γ_i-γ_i-1,

where γ_i- the value of the roll angle of the spacecraft at time t_ii=1, 2, 3 ...,;

γ_i-1- the value of the roll angle of the spacecraft at time t_i-1i=1, 2, 3 ...,

if the condition

Δγ<0,

set the value of the roll angle γ KA, equal to about 0 rad and the angle of attack α KA, corresponding to the maximum value of its aerodynamic quality, ensuring the vanishing of the spacecraft with isotemperature plot and movement ricochetnaya trajectory; carry out continuous measurement of the angle of inclination of the velocity vector θ spacecraft to the local horizon, when the condition

h_i<h_max,

where h_i- the current value of the altitude of the spacecraft at time t_iatmosphere planet;

h_max- maximum flight altitude of the spacecraft during its movement on ricochetnaya trajectory

set the value of the angle of attack α of a spacecraft in accordance with the mathematical expression

,

where V is the current value of the speed of the spacecraft;

V₀- the speed of the spacecraft when it reaches its maximum altitude at ricochetnaya trajectory after leaving the isotemperature area;

θ is the angle of the velocity vector of the spacecraft to the local horizon;

C_x0- is the aerodynamic drag coefficient at zero angle of attack of the spacecraft;

S - the area of the fuselage mid-section of the spacecraft;

M is the constant obtained after the introduction of assumptions and transform the original system of differential equations;

l, n, A - constant coefficients of the approximation of the dependency of the aerodynamic coefficients on the angle of attack to the analytical mind,

to achieve the angle of attack α of the spacecraft of α*, corresponding to the maximum value of the aerodynamic coefficient of drag, fly a mission with this value of angle of attack α* before commissioning the soft landing system.

The technical result of the bretania - the method of controlling the descent of a SPACECRAFT in the atmosphere of the planet is to minimize the final velocity of the spacecraft and the minimization of the maximum temperature in the critical region of its surface, which leads to a weight reduction of thermal protection of SPACECRAFT and to decrease the required energy costs for the implementation of space missions to study the planets of the Solar system and, consequently, to increase the share of the payload in the overall weight balance of the spacecraft. The application of the proposed method, depending on the design and ballistic characteristics of SC, boundary conditions and parameters of the planets destination allows to reduce the total weight of the fuel and heat-resistant coating lander ~ 15% compared to using the prototype method.

This technical result is achieved by testing the rational management of the roll angles and attack KA, namely due to the fact that in the method of controlling the descent of a spacecraft in the atmosphere of the planet, the selected prototype consists of the spatial orientation of the spacecraft and its management aerodynamic deceleration, stabilization when entering the atmosphere of the planet on the roll angles γ, yaw and angle of attack α of the spacecraft, providing maximum glide ratio, defined the research Institute of current coordinates and velocities of the space vehicle flight continuous measurement of the temperature T of the external surface of the spacecraft in its critical region, we compute the velocity and acceleration of its change by computing time, respectively, the first$$\stackrel{}{T}$$and the second$$\stackrel{}{T}$$derivatives; upon reaching the second derivative negative values$$\stackrel{}{T}<0$$preserving the first derivative positive values of$$\stackrel{}{T}>0$$increase the angle of attack α of the spacecraft and continue the descent to the condition of equality to zero of the first derivative of$$\stackrel{}{T}=0$$after which the set values of roll angles γ and attack α of the spacecraft, providing that the conditions for equality to zero first and second derivatives of$$\stackrel{}{T}=0$$and$$\stackrel{}{T}=\mathrm{}$$providing the descent of a spacecraft on the isotemperature area, additionally, the motion of the spacecraft on the isotemperature plot continuously comparing values of roll angles γ in the current t_iprevious t_i-1times and compute their difference Δγ by the formula

Δγ=γ_i-γ_i-1,

where γ_i- the value of the roll angle of the spacecraft at time t_ii=1, 2, 3 ...,;

γ_i-1- the value of the roll angle of the spacecraft at time t_i-1i=1, 2, 3 ...,

if the condition

Δγ<0,

set the value of the roll angle γ KA, equal to about 0 rad and the angle of attack α KA, corresponding to the maximum value of its aerodynamic quality, ensuring the vanishing of the spacecraft with isotemperature plot and movement ricochetnaya trajectory; carry out continuous measurement of the angle of inclination of the velocity vector θ spacecraft to the local horizon, when the condition is met:

h_i≤h_max,

where h_i- the current value of the altitude of the spacecraft at time t_iin the atmosphere of the planet;

h_max- maximum flight altitude of the spacecraft during its movement on ricochetnaya trajectory

set Zn is an increase in the angle of attack α of a spacecraft in accordance with the mathematical expression

,

where V is the current value of the speed of the spacecraft;

V₀- the speed of the spacecraft when it reaches its maximum altitude at ricochetnaya trajectory after leaving the isotemperature area;

θ is the angle of the velocity vector of the spacecraft to the local horizon;

C_x0- is the aerodynamic drag coefficient at zero angle of attack of the spacecraft;

S - the area of the fuselage mid-section of the spacecraft;

M is the constant obtained after the introduction of assumptions and transform the original system of differential equations;

l, n, A - constant coefficients of the approximation of the dependency of the aerodynamic coefficients on the angle of attack to the analytical mind,

to achieve the angle of attack α of the spacecraft of α*, corresponding to the maximum value of the aerodynamic coefficient of drag, fly a mission with this value of angle of attack α* before commissioning the soft landing system.

The claimed method of controlling the descent of a spacecraft in the atmosphere of the planet is illustrated by the following figures.

In Fig.1 for the prototype method shows the graphs of the dependencies of the temperature T of the heating body of the SPACECRAFT in the critical region of its outer surface, karasti flying V the roll angles γ and attack α from the time of the descent into the Martian atmosphere while minimizing the maximum temperature values.

In Fig.2 for the inventive method shows the graphs of the dependencies of the temperature T of the heating body of the SPACECRAFT in the critical region of its outer surface, the velocity V and altitude h, the roll angles γ and attack α from the time of the descent into the Martian atmosphere while minimizing the final speed while minimizing the maximum temperature values.

The comparison is shown in Fig.1 and 2 are graphs we can conclude that almost the same final speed of the AC, the temperature T of the heating body of the SPACECRAFT in the critical region of its outer surface in the present invention is less than in the method prototype.

According to [2], page 194 aerodynamic coefficients of drag and lift forces with a high degree of accuracy can be approximated by the following analytical dependencies:

C_x=C_x0+Asin²(nα+l),

C_y=C_y0+Asin(nα+l)cos(nα+l).

In particular, when using forms spacecraft type bearing housing C_x0=0,2; C_y0=-0,1; A=2,3; n=1,125; l=5,625°.

For other types of forms can be used similar dependencies for other values of coefficients [2], page 194.

We will show the possibility of carrying out the invention, i.e. in the possibility of its industrial applications. A feature of the conduct of space activities in many countries is to enhance the study of the planets of the Solar system. In the framework of the Federal space program 2016-2025, provided the work to create a spacecraft to explore Mars, Venus, Jupiter, mercury, including design landers. However, one of the major problems is the development of key management technologies that reduce the mass-energy costs in all areas of interplanetary flights. In these circumstances favourable factor is the lower cost of fuel for damping of the speed during the operation of the soft landing system and weight reduction of thermal protection of SPACECRAFT. The successful solution of this problem is largely provided when placing aboard the landers control systems aerodynamic braking, using the principles of management of roll angles and attack KA set forth in the present invention.

As for the technical means to ensure control glide KA, i.e. the management of its roll angles and attack, they are known - see, for example, [1], page 37, [2], page 57, 270, and Navigation flight of the orbital complex "SALYUT-6" - "SOYUZ" - "PROGRESS"" responsible editors B. N. The PE the ditch, I. K. Bazhinov, Moscow, Nauka, 1985, Chapter 1 - [3].

Notes. 1. The applicant was placed in the Annex to the application materials rationale used them (in the description and the claims) of the mathematical expression to calculate values of angle of attack SPACECRAFT into its final section of the descent in the atmosphere of the planet that do not unnecessarily overload the description of the invention. However, if the expert deems appropriate, the applicant will not argue for its inclusion in the description.

2. According to p. 2.3.1 of the Guidelines for examination of applications for inventions from 25.07.2011 G.: the use in the claims of the sign "about" in describing the values of the numeric parameters are valid.

3. The applicant in the application materials have used two identical term "switch" the value of the roll angle γ KA (is used to describe analogues) and "set" the value of the roll angle γ KA (in the claims), as, in his opinion, is more preferable. While believing that the unity of terminology in this case is not broken.

App. Refers to the application for the invention "Method of controlling the descent of a spacecraft in the atmospheres of the planets (using controlled aerodynamic braking and minimizing the final speed of the spacecraft under the condition of minimum and maximum temperature in the critical region of the th surface (note the Applicant).

Conclusion the mathematical dependences for calculation of angle of attack on the final section of the descent of a SPACECRAFT (SC) in the atmosphere of the planet

The motion of the SPACECRAFT in the atmosphere according to the works [1, 2] is described by the system of differential equations in the velocity coordinate system with the influence of gravitational, aerodynamic, centrifugal and Coriolis forces in the assumption of the centrality of the gravitational field

Here V is the velocity of the SPACECRAFT, θ is the angle of the velocity vector to the local horizon, ε is the heading angle, r is the radius - vector connecting the center of the planet and the position of the KA, λ and φ is the longitude and latitude of subsatellite points KA, respectively, m is the SPACECRAFT mass, t is time, ρ is the air density, C_xand C_y- aerodynamic coefficients of drag and lift forces, respectively, R is the planetary radius, h is the altitude, g is the acceleration of gravity, µ is the product of the constant of gravity on the mass of the planet, S - the area of the fuselage mid-section.

The values of the control parameters α and γ can vary

0≤α≤α_max, -π≤γ≤π.

Transform the original equation (1) given the introduction of the assumptions previously used in a number of domestic and foreign operations, in particular in the works [1, 2]

h<<R, ρ=ρ₀exp(-βh), F_to+F_C<<F_g<<F_and,

where ρ₀- p is otnesti atmosphere on the surface of Mars, β is the logarithmic rate of change of the density of the atmosphere with height, F_toF_CF_gF_and- Coriolis, centrifugal, gravitational and aerodynamic forces, respectively.

We will consider only the final section of the descending SPACECRAFT, beginning with the time the device maximum height after the flight on ricochetnaya trajectory and ending on the date of entry into force of the soft landing system.

Specific features of the dynamics of the motion of the SPACECRAFT at this stage of the descent, due to a significant reduction in the speed of flight, are increasing gravity and reducing the aerodynamic forces acting on the SPACECRAFT. On stage after the Assembly of the apparatus with isotemperature area and maximum height of rebound gravitational forces take the values one order of magnitude larger than this model.

Using these assumptions, considering the motion of the SPACECRAFT in the plane of entry into the atmosphere and taking into account that the target area is descent with zero roll angle, convert the system of equations to a form

where M is a piecewise constant function, according to the works [1, 2].

The solution of the problem of finding the optimal SPACECRAFT control with the minimum final speed was carried out using the principle of Macs is Imola Pontryagin. Let us write the Hamiltonian H and the conjugate variables Ψ_i

,

,

,.

Comparing the equations for the functions H, Ψ₁, Ψ₃convert formulas for the adjoint variables as follows:

,.

From the transversality conditions at the end point the SPACECRAFT trajectory, it follows that

Given that the Hamiltonian does not depend explicitly on time of flight is legitimate to write the equation

H≡0.

It allows to represent the dependencies for the calculation of the adjoint variables of the form

,.

Integrating these equations, taking into account formulas (3), we obtain

,

, Ψ₃(t)=a₃=const.

The dependency analysis for the calculation of conjugate variables taking into account the equality to zero of the Hamiltonian has shown that Ψ₁(t) is negative monotonically increasing function, reaching the end point of the trajectory values of -1; Ψ₂(t) is a positive monotone decreasing function, reaching the end point of the trajectory values of zero; Ψ₃(t) is a continuous function is th, having a negative value.

From the condition of maximum of the Hamiltonian determine the law of variation of the aerodynamic drag coefficient C_xat the final stage of the flight SPACECRAFT

C_x=-signΨ₁.

Given that the associated variable Ψ₁is negative, the aerodynamic coefficient C_xwill take the maximum value at the end of the flight path.

To determine the dynamics of changes in the value of C_xdivide the first equation of system (2) on the second and get the following equation

.

As a result of its decision will write down the formula for determining the aerodynamic coefficient C_xdepending on the changing values of V and θ:

where V₀- the speed of the spacecraft when it reaches its maximum altitude at ricochetnaya trajectory after leaving the isotemperature plot.

According to the works [1, 2], the aerodynamic coefficients of drag and lift forces with a high degree of accuracy can be approximated by the following analytical dependencies:

C_x=C_x0+Asin²(nα+l),

C_y=C_y0+Asin(nα+l)cos(nα+l).

For landers-type bearing housing C_x0=0,2; C_y0=-0,1; A=2,3; n=1,125; l=5,625°.

Considering elisavietta formula (4) is converted to the following:

,

The analysis of this equation showed that the angle of attack α in the area of flight KA increases monotonically with increasing intensity and reaches the end of the segment size equal to ≈70÷85°, which corresponds to the maximum value of the aerodynamic coefficient of drag.

Sources of information

1. Ivanov N. M., Martynov A. I. "the Motion of spacecraft in the atmospheres of the planets. M.: Nauka, 1985, pp. 168-173.

2. N. M. Ivanov, A. I. Martynov, "motion Control of spacecraft in the atmosphere of Mars. Moscow, Nauka, Main editorial Board for physical and mathematical literature, 1977, pp. 159-169.

The method of controlling the descent of a spacecraft in the atmosphere of the planet, lies in its spatial orientation and management of aerodynamic deceleration, stabilization of the SPACECRAFT (SC) when entering the atmosphere of the planet on the roll angles γ, yaw and angle of attack α for maximum glide ratio in the determination of the current coordinates and velocities of the flight SPACECRAFT, continuous measurement of the temperature T of the external surface of the SPACECRAFT in its critical region, we compute the velocity and acceleration of the temperature changes by computing time, respectively, the firstand the seconda derivative is, however, achievement of the second derivative negative valueswith retention of the first derivative positive valuesincrease the angle of attack α KA and continue the descent to the condition of equality to zero of the first derivativeafter which the set values of roll angles γ and attack α KA, ensuring compliance with equality to zero first and second derivativesandproviding the descent KA on isotemperature section, characterized in that during the motion of the SPACECRAFT on the isotemperature plot continuously comparing values of roll angles γ in the current t_iprevious t_i-1times and compute their difference Δγ according to the formula
Δγ=γ_i-γ_i-1,
where γ_i- the value of the roll angle CA at time t_i, i=1, 2, 3 ...,
γ_i-1- the value of the roll angle CA at time t_i-1, i=1, 2, 3 ..., and if the condition is true
Δγ<0
set the value of the roll angle KA γ is equal to about 0 rad and angle of attack KA α corresponding to the maximum value of its aerodynamic qualities, providing a gathering KA with isotemperature plot and movement ricochetnaya trajectory, carry out continuous measurement of the angle θ of inclination of the vector soon the ti KA to the local horizon, and if the condition is true
h_i< h_max,
where h_i- the current value of the altitude of the SPACECRAFT at time t_iin the atmosphere of the planet,
h_max- maximum altitude SPACECRAFT during its movement on ricochetnaya path,
set the value of the angle of attack α in accordance with the mathematical expression
,
where V is the current speed value KA,
V₀- the speed of the SPACECRAFT when it reaches its maximum altitude at ricochetnaya trajectory after leaving the isotemperature plot
θ is the angle of the velocity vector of the SPACECRAFT to the local horizon,
C_x0- is the aerodynamic drag coefficient at zero angle of attack KA,
S - the area of the fuselage mid-section KA,
M is the constant obtained after the introduction of assumptions and transform the original system of differential equations;
l, n, A - constant coefficients of the approximation of the dependency of the aerodynamic coefficients on the angle of attack to the analytical mind,
and to achieve the angle of attack KA α α*, corresponding to the maximum value of the aerodynamic coefficient of drag, fly a mission with this value of angle of attack α* before commissioning the soft landing system.