CN111846288A - Small celestial body soft landing differential game control method in uncertain environment - Google Patents

Small celestial body soft landing differential game control method in uncertain environment Download PDF

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CN111846288A
CN111846288A CN202010750835.4A CN202010750835A CN111846288A CN 111846288 A CN111846288 A CN 111846288A CN 202010750835 A CN202010750835 A CN 202010750835A CN 111846288 A CN111846288 A CN 111846288A
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CN111846288B (en
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龙嘉腾
朱圣英
梁子璇
徐瑞
修文博
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Beijing Institute of Technology BIT
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B64—AIRCRAFT; AVIATION; COSMONAUTICS
    • B64G—COSMONAUTICS; VEHICLES OR EQUIPMENT THEREFOR
    • B64G1/00—Cosmonautic vehicles
    • B64G1/22—Parts of, or equipment specially adapted for fitting in or to, cosmonautic vehicles
    • B64G1/24—Guiding or controlling apparatus, e.g. for attitude control
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B64—AIRCRAFT; AVIATION; COSMONAUTICS
    • B64G—COSMONAUTICS; VEHICLES OR EQUIPMENT THEREFOR
    • B64G1/00—Cosmonautic vehicles
    • B64G1/22—Parts of, or equipment specially adapted for fitting in or to, cosmonautic vehicles
    • B64G1/62—Systems for re-entry into the earth's atmosphere; Retarding or landing devices
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems

Abstract

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.

Description

Small celestial body soft landing differential game control method in uncertain environment
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 with
Figure BDA0002609996440000013
For state variable, the detector is fixedly connected with a coordinate system at the center of the small celestial body
Figure BDA0002609996440000012
The kinetic equation below is
Figure BDA0002609996440000021
Figure BDA0002609996440000022
Small and satisfies the condition that u is T/m, IspSpecific impulse of aircraft thruster, gEIs the acceleration of gravity at sea level of the earth, adFor unmodeled disturbances and obeying Gaussian white noise
Figure BDA0002609996440000023
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
Figure BDA0002609996440000024
Wherein GM and r are the gravitational constant and nominal radius of the small celestial body, PnmIs an associative Legendre polynomial, CnmAnd SnmIs the spherical harmonic coefficient, theta,
Figure BDA0002609996440000025
R is the longitude, the latitude and the radius of the position of the detector, and satisfies the relation shown in the formula (4).
Figure BDA0002609996440000026
In the small celestial body landing control problem, the following boundary constraints need to be satisfied
Figure BDA0002609996440000027
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 isfAnd vfRespectively 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
Tmin≤||T||≤Tmax(6)
Wherein, TminAnd TmaxRespectively 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
Figure BDA0002609996440000031
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.
Figure BDA0002609996440000032
Wherein the probability density function p (x, t) of the state error is approximated by a gaussian mixture model equation (9).
Figure BDA0002609996440000033
Wherein, mui(t) and Pi(t) are Gaussian probability density functions N (x | mu) respectivelyi(t),Pi(t)) by means of linear covariance or unscented transformation. The constraint condition that the weight coefficient needs to satisfy is determined by equation (10).
Figure BDA0002609996440000034
Definition error
Figure BDA0002609996440000035
The weight coefficients of the gaussian mixture model equation (9) are obtained by solving the optimization problem given by equation (12).
Figure BDA0002609996440000036
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
Figure BDA0002609996440000037
The virtual moving target landing site motion is described as
Figure BDA0002609996440000038
Wherein the content of the first and second substances,
Figure BDA0002609996440000039
I 6×3＝[03×3,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 ad0; the virtual moving target landing site is an escaper, and the motion thereof is determined by equation (13). Order to
Figure BDA0002609996440000041
And satisfy
Figure BDA0002609996440000042
Wherein u isvAnd 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(tf)-xt(tf)|| (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
Figure BDA0002609996440000043
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.
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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
Figure BDA00026099964400000510
For state variable, the detector is fixedly connected with a coordinate system at the center of the small celestial body
Figure BDA0002609996440000052
The kinetic equation below is
Figure BDA0002609996440000053
Figure BDA0002609996440000054
Small and satisfies the condition that u is T/m, IspSpecific impulse of aircraft thruster, gEIs the acceleration of gravity at sea level of the earth, adFor unmodeled disturbances and obeying Gaussian white noise
Figure BDA0002609996440000055
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
Figure BDA0002609996440000056
Wherein GM is 446210m3/s2And r 16km is the gravity constant and nominal radius of the small celestial body Eros 433, PnmIs an associative Legendre polynomial, CnmAnd SnmIs the spherical harmonic coefficient, theta,
Figure BDA0002609996440000057
R is the longitude, the latitude and the radius of the position of the detector, and satisfies the relation shown in the formula (4).
Figure BDA0002609996440000058
In the small celestial body landing control problem, the following boundary constraints need to be satisfied
Figure BDA0002609996440000059
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 isf＝[853,5010,45]m and vf＝[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
Tmin≤||T||≤Tmax(22)
Wherein, Tmin2N and Tmax18N 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
Figure BDA0002609996440000061
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.
Figure BDA0002609996440000062
Wherein the probability density function p (x, t) of the state error is approximated by a gaussian mixture model equation (9).
Figure BDA0002609996440000063
Wherein, mui(t) and Pi(t) are Gaussian probability density functions N (x | mu) respectivelyi(t),Pi(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).
Figure BDA0002609996440000064
Definition error
Figure BDA0002609996440000065
The weight coefficients of the gaussian mixture model equation (9) are obtained by solving the optimization problem given by equation (12).
Figure BDA0002609996440000066
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
Figure BDA0002609996440000071
The virtual moving target landing site motion is described as
Figure BDA0002609996440000072
Wherein the content of the first and second substances,
Figure BDA0002609996440000073
I 6×3＝[03×3,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 ad0; the virtual moving target landing site is an escaper, and the motion thereof is determined by equation (13). Order to
Figure BDA0002609996440000074
And satisfy
Figure BDA0002609996440000075
Wherein u isvAnd 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(tf)-xt(tf)|| (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
Figure BDA0002609996440000076
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.

Claims (4)

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 with
Figure FDA0002609996430000011
For state variable, the detector is fixedly connected with a coordinate system at the center of the small celestial body
Figure FDA0002609996430000012
The kinetic equation below is
Figure FDA0002609996430000013
Wherein ω is [0,0, ω ═ o]TIs a planetary self-rotation angular velocity vector,
Figure FDA0002609996430000014
is the local gravitational acceleration vector, T ═ Tx,Ty,Tz]TThe vector is controlled for the thrust of the probe,
Figure FDA0002609996430000015
is the magnitude of the thrust control vector and satisfies the condition that u is T/m, IspSpecific impulse of aircraft thruster, gEIs the acceleration of gravity at sea level of the earth, adFor unmodeled disturbances and obeying Gaussian white noise
Figure FDA0002609996430000016
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
Figure FDA0002609996430000017
Wherein GM and r are the gravitational constant and nominal radius of the small celestial body, PnmIs an associative Legendre polynomial, CnmAnd SnmIs the spherical harmonic coefficient, theta,
Figure FDA0002609996430000018
R is the longitude, latitude and radius of the position of the detector, and satisfies the relation shown in formula (4)
Figure FDA0002609996430000019
In the small celestial body landing control problem, the following boundary constraints need to be satisfied
Figure FDA0002609996430000021
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 isfAnd vfRespectively 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
Tmin≤||T||≤Tmax(6)
Wherein, TminAnd TmaxRespectively 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
Figure FDA0002609996430000022
。
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;
Figure FDA0002609996430000023
wherein the probability density function p (x, t) of the state error is approximated by a gaussian mixture model (9);
Figure FDA0002609996430000024
wherein, mui(t) and Pi(t) are Gaussian probability density functions N (x | mu) respectivelyi(t),Pi(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)
Figure FDA0002609996430000025
Definition error
Figure FDA0002609996430000026
The weight coefficient of the gaussian mixture model equation (9) is obtained by solving the optimization problem given by equation (12)
Figure FDA0002609996430000031
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 as
Figure FDA0002609996430000032
The virtual moving target landing site motion is described as
Figure FDA0002609996430000033
Wherein the content of the first and second substances,
Figure FDA0002609996430000034
I 6×3＝[03×3,I3](ii) a 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).
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 ad0; 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 to
Figure FDA0002609996430000035
And satisfy
Figure FDA0002609996430000036
Wherein u isvThe 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(tf)-xt(tf)|| (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
Figure FDA0002609996430000037
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.
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