阅读说明：本技术 不确定环境中下的小天体软着陆微分博弈控制方法 (Small celestial body soft landing differential game control method in uncertain environment ) 是由 龙嘉腾 朱圣英 梁子璇 徐瑞 修文博 于 2020-07-30 设计创作，主要内容包括：本发明公开的不确定环境中下的小天体软着陆微分博弈控制方法,属于深空探测技术领域。本发明实现方法为：在目标小天体精确引力场模型未知的条件下,确定不规则弱引力场中的着陆器运动形式,采用状态误差传播方法设计虚拟移动目标着陆点,将不规则引力场对着陆器运动的不确定性影响转化为虚拟移动目标着陆点的随机运动；基于零和随机微分博弈设计着陆器的最优着陆控制策略,并根据所述控制最优着陆控制策略控制着陆器着陆,提高着陆器在小天体表面预定位置的着陆精度。(The invention discloses a small celestial body soft landing differential game control method in an uncertain environment, and belongs to the technical field of deep space exploration. The implementation method of the invention comprises the following steps: determining the movement form of the lander in the irregular weak gravitational field under the condition that the accurate gravitational field model of the target small celestial body is unknown, designing a virtual moving target landing point by adopting a state error propagation method, and converting the uncertain influence of the irregular gravitational field on the movement of the lander into random movement of the virtual moving target landing point; and designing an optimal landing control strategy of the lander based on the zero and random differential game, and controlling the lander to land according to the optimal landing control strategy, so that the landing precision of the lander at a preset position on the surface of the small celestial body is improved.)

1. The small celestial body soft landing differential game control method in the uncertain environment is characterized by comprising the following steps of: comprises the following steps of (a) carrying out,

step 1, determining a lander motion form in an irregular weak gravitational field;

step 2, designing a virtual moving target landing point by adopting a state error propagation method, and converting the uncertain influence of the irregular gravitational field on the movement of the lander into random movement of the virtual moving target landing point;

step 3, designing optimal landing control strategy u of lander based on zero and random differential game^*And controlling the lander to land according to the optimal landing control strategy, so that the landing precision of the lander at the preset position of the surface of the small celestial body is improved.

2. The differential gaming control method for soft landing of celestial bodies in uncertain environments of claim 1, wherein: the step 1 is realized by the method that,

to be provided withFor state variable, the detector is fixedly connected with a coordinate system at the center of the small celestial bodyThe kinetic equation below is

Wherein ω is [0,0, ω ═ o]^TIs a planetary self-rotation angular velocity vector,is the local gravitational acceleration vector, T ═ T_x,T_y,T_z]^TThe vector is controlled for the thrust of the probe,is the magnitude of the thrust control vector and satisfies the condition that u is T/m, I_spSpecific impulse of aircraft thruster, g_EIs the acceleration of gravity at sea level of the earth, a_dFor unmodeled disturbances and obeying Gaussian white noise

Where q (t) is the spectral density and (t- τ) is the dirac function at t ═ τ; the inner spherical harmonic function model of the small celestial body gravitation field is

Wherein GM and r are the gravitational constant and nominal radius of the small celestial body, P_nmIs an associative Legendre polynomial, C_nmAnd S_nmIs the spherical harmonic coefficient, theta,

In the small celestial body landing control problem, the following boundary constraints need to be satisfied

Wherein r is₀And v₀Respectively, the initial time t ═ t₀Position and velocity vector of the detector, m₀For exploringThe quality of the detector at the initial time; r is_fAnd v_fRespectively at landing time (t ═ t)_f) Target position and velocity vector of the probe; during landing, the thrust amplitude of the detector needs to meet the following constraint

T_min≤||T||≤T_max(6)

Wherein, T_minAnd T_maxRespectively the minimum value and the maximum value of the amplitude of the thruster; equation of dynamics (1) determines the form of lander motion in an irregular weak gravitational field and notes equation of dynamics (1) as

3. The differential gaming control method for soft landing of celestial bodies in uncertain environments of claim 2, further comprising: the step 2 is realized by the method that,

the Fokker-Planck-Kolmogorov (FPK) equation in equation (8) describes the propagation process of the state error distribution function;

wherein the probability density function p (x, t) of the state error is approximated by a gaussian mixture model (9);

wherein, mu_i(t) and P_i(t) are Gaussian probability density functions N (x | mu) respectively_i(t),P_i(t)) means and covariance, obtained by linear covariance or unscented transformation; the constraint condition to be satisfied by the weight coefficient is determined by equation (10)

Definition error

The weight coefficient of the gaussian mixture model equation (9) is obtained by solving the optimization problem given by equation (12)

Obtaining a probability density function p (x, t) of the lander state error by solving the formula (12); then at the end time the probability density function for the lander state error is p (x, t)_f) (ii) a Let random variables Δ (t) -p (x, t)_f) Defining the state variable of the virtual target landing site asThe virtual moving target landing site motion is described as

Wherein the content of the first and second substances,

4. The differential gaming control method for soft landing of celestial bodies in uncertain environments of claim 3, wherein: the step 3 is realized by the method that,

the precise landing problem of the surface of the small celestial body is converted into the escape zero-pursuit and differential game problem by introducing a virtual moving target landing point, whereinLand vehicles are trackers whose motion is determined by the equation (1) of well-known dynamics, i.e. unmodeled disturbance a_d0; the virtual moving target landing point is an escaper, and the motion of the virtual moving target landing point is determined by the formula (13); order toAnd satisfy

Wherein u is_vThe virtual control quantity corresponding to the virtual moving target landing point; the performance index of the chase escape zero and differential game problem is defined as the time at the end of the game (t ═ t)_f) The relative distance between the two parties is shown in formula (15)

J＝||x(t_f)-x_t(t_f)|| (15)

The optimal strategy of the lander is to minimize the relative distance between the two parties, while the optimal strategy of the virtual moving target landing point (escaper) is to maximize the relative distance between the two parties, i.e. the lander is a virtual moving target landing point

Thus, the catch-up and differential game problem is given by equation (16); obtaining a control strategy u of the lander in the irregular gravitational field through a saddle point planning method solving formula (16)^*；

Under the condition that the accurate gravitational field model of the target small celestial body is unknown, the landing accuracy of the lander at the preset position of the surface of the small celestial body is improved by designing a corresponding landing control strategy.

Technical Field

The invention relates to a landing control method, in particular to a small celestial body soft landing control method, and belongs to the technical field of deep space exploration.

Background

The precise soft landing of the surface of the small celestial body is a necessary premise for implementing tasks such as in-situ detection, sampling return and the like of the surface of the small celestial body. Because the small celestial body has small mass, the formed weak gravitation can not agglomerate the small celestial body into a sphere-like celestial body. Therefore, the small celestial bodies in the solar system have different shapes, so that the distribution of the gravitational field around the small celestial bodies is irregular, and accurate modeling is difficult.

Under the conditions that the gravitational field environment is extremely complex and unknown and accurate modeling is difficult, the control difficulty for realizing soft landing on the surface of the small celestial body is extremely high, and the requirement on the control accuracy is extremely high. In a weak gravitational field, the escape speed of the detector is very low, and a slight control error may cause the detector to collide when landing, so that bounce is generated and even the detector is caused to escape.

Considering the requirement of high-precision landing control of the small celestial body, the accurate landing of the lander on the surface of the small celestial body is realized by designing a corresponding lander control method under the condition that an accurate model of an irregular gravitational field of the small celestial body is unknown.

Disclosure of Invention

The invention discloses a small celestial body soft landing differential game control method in an uncertain environment, which aims to solve the technical problems that: under the condition that a target small celestial body accurate gravitational field model is unknown, a virtual moving target landing point is designed through a state error propagation method, the uncertain influence of an irregular gravitational field on the movement of a lander is converted into random movement of the virtual moving target landing point, then an optimal landing control strategy of the lander is designed through a zero and random differential game method, and the landing accuracy of the lander at a preset position on the surface of a small celestial body is improved.

The purpose of the invention is realized by the following technical scheme.

The invention discloses a small celestial body soft landing differential game control method in an uncertain environment, which is used for determining the movement form of a lander in an irregular weak gravitational field, designing a virtual moving target landing point by adopting a state error propagation method and converting the uncertain influence of the irregular gravitational field on the lander movement into random movement of the virtual moving target landing point. And designing an optimal landing control strategy of the lander based on the zero and random differential game, and controlling the lander to land according to the optimal landing control strategy, so that the landing precision of the lander at a preset position on the surface of the small celestial body is improved.

The invention discloses a small celestial body soft landing differential game control method in an uncertain environment, which comprises the following steps:

step 1, determining the movement form of the lander in the irregular weak gravitational field.

To be provided withFor state variable, the detector is fixedly connected with a coordinate system at the center of the small celestial bodyThe kinetic equation below is

Small and satisfies the condition that u is T/m, I_spSpecific impulse of aircraft thruster, g_EIs the acceleration of gravity at sea level of the earth, a_dFor unmodeled disturbances and obeying Gaussian white noise

Where q (t) is the spectral density and (t- τ) is the dirac function at t ═ τ. The inner spherical harmonic function model of the small celestial body gravitation field is

Wherein GM and r are the gravitational constant and nominal radius of the small celestial body, P_nmIs an associative Legendre polynomial, C_nmAnd S_nmIs the spherical harmonic coefficient, theta,

R is the longitude, the latitude and the radius of the position of the detector, and satisfies the relation shown in the formula (4).

In the small celestial body landing control problem, the following boundary constraints need to be satisfied

Wherein r is₀And v₀Respectively, the initial time t ═ t₀Position and velocity vector of the detector, m₀Is the quality of the detector at the initial moment; r is_fAnd v_fRespectively at landing time (t ═ t)_f) Target position and velocity vector of the probe. During landing, the thrust amplitude of the detector needs to meet the following constraint

T_min≤||T||≤T_max(6)

Wherein, T_minAnd T_maxRespectively the minimum value and the maximum value of the amplitude of the thruster. Equation of dynamics (1) determines the form of lander motion in an irregular weak gravitational field and notes equation of dynamics (1) as

And 2, designing a virtual moving target landing point by adopting a state error propagation method, and converting the uncertain influence of the irregular gravitational field on the movement of the lander into random movement of the virtual moving target landing point.

The Fokker-Planck-Kolmogorov (FPK) equation in equation (8) describes the propagation process of the state error distribution function.

Wherein the probability density function p (x, t) of the state error is approximated by a gaussian mixture model equation (9).

Wherein, mu_i(t) and P_i(t) are Gaussian probability density functions N (x | mu) respectively_i(t),P_i(t)) by means of linear covariance or unscented transformation. The constraint condition that the weight coefficient needs to satisfy is determined by equation (10).

Definition error

The weight coefficients of the gaussian mixture model equation (9) are obtained by solving the optimization problem given by equation (12).

The probability density function p (x, t) of the lander state error is obtained by solving equation (12). Then at the end time the probability density function for the lander state error is p (x, t)_f). Let random variables Δ (t) -p (x, t)_f) Defining the state variable of the virtual target landing site as

The virtual moving target landing site motion is described as

Wherein the content of the first and second substances, I _6×3＝[0_3×3,I₃]. The motion of the virtual moving target landing site is determined by equation (13). And (3) designing a virtual moving target landing point by adopting a state error propagation method given by an FPK equation in the formula (8), and converting the uncertain influence of the irregular gravitational field on the movement of the lander into random movement of the virtual moving target landing point shown in the formula (13).

Step 3, designing optimal landing control strategy u of lander based on zero and random differential game^*And controlling the lander to land according to the optimal landing control strategy, so that the landing precision of the lander at the preset position of the surface of the small celestial body is improved.

The precise landing problem of the small celestial body surface is converted into the zero-escape pursuit and differential game problem by introducing a virtual moving target landing point, wherein the lander is a chaser, the motion of which is determined by a completely known kinetic equation (1), namely unmodeled disturbance a_d0; the virtual moving target landing site is an escaper, and the motion thereof is determined by equation (13). Order to

And satisfy

Wherein u is_vAnd the virtual control quantity corresponding to the virtual moving target landing point. The performance index of the chase escape zero and differential game problem is defined as the time at the end of the game (t ═ t)_f) The relative distance between the two parties is shown in equation (15).

J＝||x(t_f)-x_t(t_f)|| (15)

The optimal strategy for landers (chasers) is to minimize the relative distance between the two parties, while the optimal strategy for virtual moving target landing sites (escapes) is to maximize the relative distance between the two parties, i.e., to maximize the distance between the two parties

Thus, the problem of the escape from zero and differential game is given by equation (16). Obtaining a control strategy u of the lander in the irregular gravitational field through a saddle point planning method solving formula (16)^*。

Under the condition that the accurate gravitational field model of the target small celestial body is unknown, the landing accuracy of the lander at the preset position of the surface of the small celestial body is improved by designing a corresponding landing control strategy.

Has the advantages that:

1. the invention discloses a small celestial body soft landing differential game control method in an uncertain environment, which designs a virtual moving target landing point through a state error propagation method, converts the uncertain influence of an irregular gravitational field on the movement of a lander into the random movement of the virtual moving target landing point, realizes equivalent conversion of the modeling error of the irregular gravitational field of a small celestial body, and further facilitates the design of an optimal landing control strategy of the lander through a zero sum random differential game method.

2. The invention discloses a small celestial body soft landing differential game control method in an uncertain environment, which designs an optimal landing control strategy of a lander through a zero and random differential game method, and dynamically tracks a virtual moving target landing point by the lander, namely, the lander is controlled to land according to the optimal landing control strategy, and the landing precision of the lander at a preset position on the surface of a small celestial body is improved under the condition of gravitational field modeling error.

Drawings

FIG. 1 is a flow chart of a small celestial body soft landing differential game control method in an uncertain environment;

FIG. 2 is a plot of lander conditions over time.

Detailed Description

For a better understanding of the objects and advantages of the invention, reference is made to the following description, taken in conjunction with the accompanying drawings, which illustrate, by way of example, the principles of the invention.

As shown in fig. 1, the method for controlling a differential game of soft landing of a celestial body in an uncertain environment disclosed in this embodiment includes the following steps:

step 1, determining the movement form of the lander in the irregular weak gravitational field.

To be provided with

For state variable, the detector is fixedly connected with a coordinate system at the center of the small celestial body

The kinetic equation below is

Small and satisfies the condition that u is T/m, I_spSpecific impulse of aircraft thruster, g_EIs the acceleration of gravity at sea level of the earth, a_dFor unmodeled disturbances and obeying Gaussian white noise

Where q (t) is the spectral density and (t- τ) is the dirac function at t ═ τ. The small celestial body Eros 433 is used as a target celestial body, and the spherical harmonic function model in the small gravitational field is

Wherein GM is 446210m³/s²And r 16km is the gravity constant and nominal radius of the small celestial body Eros 433, P_nmIs an associative Legendre polynomial, C_nmAnd S_nmIs the spherical harmonic coefficient, theta,

R is the longitude, the latitude and the radius of the position of the detector, and satisfies the relation shown in the formula (4).

In the small celestial body landing control problem, the following boundary constraints need to be satisfied

Wherein r is₀＝[10117,6956,8256]m and v₀＝[-25,-12,-17]m/s is respectively the initial time t ═ t₀Position and velocity vector of the detector, m₀300kg is the mass of the probe at the initial moment; r is_f＝[853,5010,45]m and v_f＝[0,0,0]m/s is landing time (t ═ t)_f) Target position and velocity vector of the probe. During landing, the thrust amplitude of the detector needs to meet the following constraint

T_min≤||T||≤T_max(22)

Wherein, T_min2N and T_max18N are the minimum and maximum values of the thruster amplitude, respectively. The kinetic equation (1) determines the form of lander motion in an irregular weak gravitational field. Let kinetic equation (1) be written

And 2, designing a virtual moving target landing point by adopting a state error propagation method, and converting the uncertain influence of the irregular gravitational field on the movement of the lander into random movement of the virtual moving target landing point.

The Fokker-Planck-Kolmogorov (FPK) equation in equation (8) describes the propagation process of the state error distribution function.

Wherein the probability density function p (x, t) of the state error is approximated by a gaussian mixture model equation (9).

Wherein, mu_i(t) and P_i(t) are Gaussian probability density functions N (x | mu) respectively_i(t),P_i(t)) mean and covariance, obtained by linear covariance or unscented transformation methods. The constraint condition that the weight coefficient needs to satisfy is determined by equation (10).

Definition error

The weight coefficients of the gaussian mixture model equation (9) are obtained by solving the optimization problem given by equation (12).

The probability density function p (x, t) of the lander state error is obtained by solving equation (12). Then at the end time the probability density function for the lander state error is p (x, t)_f). Let random variables Δ (t) -p (x, t)_f) Defining the state variable of the virtual target landing site as

The virtual moving target landing site motion is described as

Wherein the content of the first and second substances, I _6×3＝[0_3×3,I₃]. The motion of the virtual moving target landing site is determined by equation (13).

Step 3, designing optimal landing control strategy u of lander based on zero and random differential game^*And controlling the lander according to the optimal landing control strategyAnd landing, namely, the landing precision of the lander at a preset position on the surface of the small celestial body is improved.

The precise landing problem of the small celestial body surface is converted into the zero-escape pursuit and differential game problem by introducing a virtual moving target landing point, wherein the lander is a chaser, the motion of which is determined by a completely known kinetic equation (1), namely unmodeled disturbance a_d0; the virtual moving target landing site is an escaper, and the motion thereof is determined by equation (13). Order toAnd satisfy

Wherein u is_vAnd the virtual control quantity corresponding to the virtual moving target landing point. The performance index of the chase escape zero and differential game problem is defined as the time at the end of the game (t ═ t)_f) The relative distance between the two parties is shown in equation (15).

J＝||x(t_f)-x_t(t_f)|| (31)

The optimal strategy for landers (chasers) is to minimize the relative distance between the two parties, while the optimal strategy for virtual moving target landing sites (escapes) is to maximize the relative distance between the two parties, i.e., to maximize the distance between the two parties

Thus, the catch-up and differential game problem is given by equation (16). Obtaining a control strategy u of the lander in the irregular gravitational field through a saddle point planning method solving formula (16)^*。

Under the condition that the accurate gravitational field model of the target small celestial body is unknown, the landing accuracy of the lander at the preset position of the surface of the small celestial body is improved by designing a corresponding landing control strategy.

The above detailed description is intended to illustrate the objects, aspects and advantages of the present invention, and it should be understood that the above detailed description is only exemplary of the present invention and is not intended to limit the scope of the present invention, and any modifications, equivalents, improvements and the like made within the spirit and principle of the present invention should be included in the scope of the present invention.