CN115202380B - Extraterrestrial celestial body landing track planning method based on piecewise polynomial - Google Patents
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Abstract
The invention discloses an extraterrestrial celestial body landing track planning method based on a piecewise polynomial, and belongs to the technical field of deep space exploration. The implementation method of the invention comprises the following steps: on the basis of traditional polynomial programming, two normalized time segmentation points are introduced in the track programming process, the landing track of the detector is divided into three stages, a segmentation polynomial equation is constructed according to initial and final positions and speed constraints, a function relation between thrust and terminal time is established by a thrust amplitude constraint condition, the value of the terminal time meeting the thrust amplitude constraint is solved under the condition of a given time segmentation point, then the burnup is used as an optimization index, the time segmentation points are optimized and solved, an acceleration polynomial equation with optimal burnup is obtained, the real-time track programming meeting the thrust amplitude constraint and realizing the landing track with optimal burnup is obtained in online programming, namely the real-time track programming meeting the thrust limiting and the landing of the extraterrestrial celestial body with optimal burnup is realized, and further the accurate fixed-point soft landing of the detector is ensured.
Description
Technical Field
The invention relates to an extraterrestrial celestial body landing track planning method, in particular to an extraterrestrial celestial body landing track planning method based on a piecewise polynomial, and belongs to the technical field of deep space exploration.
Background
Mars detection and asteroid detection are research hot spots in the field of deep space exploration, and research and analysis of the Mars detection and the asteroid detection are beneficial to human exploration of life origins, evolution processes of solar systems and other important scientific problems, and promote the development of space technology. In order to obtain celestial data and mineral samples of mars and asteroids, a landing probe is required. The power down segment is a critical phase of the landing of the probe and relates to whether the probe can safely and accurately reach the desired landing site. The trajectory of the power down leg needs to meet constraints such as position and speed at the landing site, as well as thrust constraints during landing. Because the Mars pneumatic environment and the asteroid gravitation environment are complex, the initial condition of the power descent section is uncertain, and the track of the offline optimization design is difficult to meet the high-precision landing requirement. Therefore, there is a need to develop extraterrestrial celestial body landing online trajectory planning techniques.
Related researches exist for the problem of landing track planning of a power descent section, wherein the polynomial track planning method has an analytic form, can realize quick solution and has high calculation efficiency. However, the conventional polynomial trajectory planning method fails to consider thrust constraint and fuel consumption, which may cause unnecessary fuel consumption in practical application, and increases the cost of the detection task. In order to meet the thrust constraint and reduce the fuel consumption in unknown environments as much as possible, the traditional polynomial method needs to be improved, and a novel piecewise polynomial trajectory planning method is developed.
Disclosure of Invention
Aiming at the problem of real-time track planning of the extraterrestrial celestial body landing of the Mars and asteroid detectors, starting from meeting the requirements of the thrust amplitude constraint of the detectors, reducing the fuel consumption and ensuring the accurate fixed-point soft landing, the method provides an extraterrestrial celestial body landing track planning method based on a piecewise polynomial.
The aim of the invention is achieved by the following technical scheme.
According to the extraterrestrial celestial body landing track planning method based on the piecewise polynomials, two normalized time piecewise points are introduced in the track planning process, the detector landing track is divided into three stages, a piecewise polynomial equation is constructed according to initial and final positions and speed constraints, a function relation between thrust and terminal time is built according to thrust amplitude constraint conditions, the value of the terminal time meeting the thrust amplitude constraint is solved under the condition of the given time piecewise points, then the burnup is used as an optimization index, the time piecewise points are optimized and solved, an acceleration polynomial equation with optimal burnup is obtained, the landing track meeting the thrust amplitude constraint and achieving optimal burnup is obtained through online planning, and real-time track planning of extraterrestrial celestial body landing meeting the thrust limiting and optimal burnup is achieved.
The invention discloses an extraterrestrial celestial body landing track planning method based on a piecewise polynomial, which comprises the following steps:
step one, dividing a detector landing track into three stages according to time nodes and according to landing terminal time t f With two normalized time segment points lambda 1 And lambda is 2 And (3) designing switching time of three stages, constructing a three-stage polynomial acceleration equation for landing track planning, and integrating to obtain a polynomial equation of speed and position. The normalized two time segmentation points are introduced in the track planning process, so that the adjustability of the polynomial track is enhanced.
The specific implementation method of the first step is as follows:
first, according to the terminal time t f And normalized two time segment points lambda 1 And lambda is 2 And constructing a piecewise acceleration equation. According to the optimal idea of burnup, designing the acceleration equation into a Bang-Bang form expression
Wherein a represents the acceleration vector of the detector under the celestial center fixedly connected coordinate system, and t f Representing the terminal time, the planning process is divided into three time periods, 0 to lambda respectively 1 t f Corresponding first segment lambda 1 t f To lambda 2 t f Corresponding second segment and lambda 2 t f To t f A corresponding third segment. a, a 1 And a 3 The polynomial undetermined coefficients of the first and third segments are represented, respectively.
Integrating acceleration equations of three time periods respectively to obtain a velocity polynomial equation expressed as
Wherein v represents the velocity vector of the detector under the celestial center fixedly connected coordinate system, v 1 、v 2 And v 3 Multiple representing three time periods respectivelyThe polynomial is the coefficient to be determined.
The velocity polynomial equations for the three time periods are integrated separately to obtain the position polynomial equation expression as follows
Wherein r represents the position vector of the detector under the celestial center fixedly connected coordinate system, and r 1 、r 2 And r 3 And respectively represent the polynomial undetermined coefficients of three time periods.
The normalized two time segmentation points are introduced in the track planning process, a segmentation polynomial equation is constructed, and the adjustability of the polynomial track is enhanced.
Substituting the initial and terminal speed and position constraints into a constructed polynomial equation to calculate polynomial coefficients of each time period to obtain a time segmentation point lambda 1 And lambda is 2 Terminal time t f Parameter dependent polynomial equations.
The specific implementation method of the second step is as follows:
the piecewise polynomial needs to satisfy the primary and final velocity and position constraints, as well as the equality constraints of the various time-piecewise points.
At time t=0, the three-axis speed constraint and the three-axis position constraint, which are required to satisfy the initial point, are expressed as follows
Wherein v is 0 Indicating the initial speed of the detector, r 0 Indicating the initial position of the detector.
At t=λ 1 t f The time, speed and position equations need to satisfy the following equation relationship
At t=λ 2 t f The time, speed and position equations need to satisfy the following equation relationship
At t=t f To meet the requirements of accurate guidance and soft landing, a speed constraint with a terminal triaxial speed of 0 and a triaxial position constraint of the terminal landing point are established according to the target landing point of the detector, and are expressed as follows
Wherein r is f Indicating the detector terminal position, the terminal landing speed is zero.
Substituting the equation relation between the initial and final speed and position constraint conditions and each time segment point into a polynomial equation to solve the time segment point lambda 1 And lambda is 2 Terminal time t f The result of the polynomial undetermined coefficients of (2) is as follows
Constructing a time segment point lambda using the polynomial coefficients obtained by the solution 1 And lambda is 2 Terminal time t f Polynomial equation of (c).
Step three, calculating to obtain a thrust expression according to the dynamics equation of the detector and the polynomial equation obtained in the step two, and establishing a relation of the terminal time t by the thrust amplitude constraint f Inequality constraint of (a) and solving to obtain given time segmentation point lambda 1 And lambda is 2 Terminal time t meeting thrust limiting constraint under initial value condition f Namely, the polynomial coefficient is adjusted to meet the constraint condition of the thrust amplitude of the detector, so that the planned detector track reduces the occurrence of the thrust saturation condition as much as possible.
The specific implementation method of the third step is as follows:
during track planning, the terminal time t meeting the thrust limiting constraint condition is needed f And solving.
Firstly, under the asteroid fixed connection coordinate system, a detector dynamics equation is constructed, and a relation of the position, the speed, the acceleration and the thrust of the detector is obtained.
The kinetic equation of the probe is as follows
Wherein w is the planetary rotation angular velocity, 2w×v represents the coriolis acceleration, w× (w×r) represents the centripetal acceleration, the control acceleration and disturbance acceleration of the detector are ignored, g is the gravitational acceleration vector, T is the engine thrust, F is the aerodynamic force (for asteroid, the term is 0), I sp Denote engine specific impulse, g e Is the gravitational acceleration constant.
And (3) subtracting the gravitational acceleration and various accelerations of the celestial body from the polynomial equation of the acceleration of the detector according to the dynamic equation of the detector and the polynomial equation obtained in the second step to obtain a thrust acceleration expression.
From the time segment point lambda 1 And lambda is 2 Respectively calculating the thrust T and the terminal time T of the first section initial time and the third section initial time f Is a relation of (3). Solving and obtaining the terminal time t meeting the limiting constraint of each sectional thrust by Newton iteration method f . Comparing t calculated from the initial moments of the two segments f Selecting the largest t f As the terminal time of the piecewise polynomial trajectory. The terminal time t obtained at this time f Satisfy at a given time segment point lambda 1 And lambda is 2 And the thrust amplitude constraint under the initial value condition enables the planned detector track to reduce the occurrence of the thrust saturation condition as much as possible.
Step four, taking fuel consumption as an optimization index of the polynomial track, and dividing the time into points lambda 1 And lambda is 2 Expanding parameter optimization to obtain polynomial coefficients with optimal parameters, and solvingAnd solving to obtain an acceleration curve with optimal burnup, integrating to obtain a polynomial landing track, and realizing real-time track planning of extraterrestrial celestial body landing meeting the requirements of thrust limiting and optimal burnup, thereby ensuring accurate fixed-point soft landing of the detector.
The specific implementation method of the fourth step is as follows:
selecting the fuel consumption delta m as an optimization index of a polynomial track, and calculating through a polynomial equation to obtain the fuel consumption delta m relative to a time segmentation point lambda 1 And lambda is 2 For a target parameter lambda 1 And lambda is 2 And (5) carrying out optimization solution.
Thus, an optimized model of the time segment point parameter design problem is established
min J＝Δm
s.t.0≤λ 1 ≤λ 2 ≤1 (10)
Setting a time segment point lambda 1 And lambda is 2 Setting target optimization precision to determine iteration termination conditions, and performing iteration optimization to obtain a parameter value lambda meeting optimal fuel consumption 1 And lambda is 2 Substituting the acceleration curve into a polynomial equation to obtain an acceleration curve, integrating to obtain a polynomial landing track, and realizing real-time track planning of extraterrestrial celestial body landing meeting the requirements of thrust limiting and optimal burnup, thereby ensuring accurate fixed-point soft landing of the detector.
Preferably, in the fourth step, the inner point method is adopted for time segmentation point lambda 1 And lambda is 2 And (5) optimizing unfolding parameters.
The beneficial effects are that:
1. according to the extraterrestrial celestial body landing track planning method based on the piecewise polynomial, on the basis of traditional polynomial track planning, time piecewise points are introduced, a functional relation between the thrust of the detector and the terminal time is established according to the piecewise polynomial model and the dynamic equation of the detector, the amplitude limiting constraint of the thrust can be met by adjusting the size of the terminal time, the occurrence of the thrust saturation condition of the detector in the track planning process is reduced as much as possible, and the landing track planning efficiency and precision are improved.
2. The invention discloses an underground outside based on a piecewise polynomialCelestial body landing track planning method, fuel consumption is used as optimization index of polynomial track, and time segmentation point lambda is used as time segmentation point lambda 1 And lambda is 2 And (3) expanding the parameters to optimize, obtaining polynomial coefficients with optimal parameters, solving to obtain an acceleration curve with optimal fuel consumption, and integrating to obtain a polynomial landing track, so that the fuel consumption of the polynomial track can be effectively reduced.
3. The method for planning the landing track of the extraterrestrial celestial body based on the piecewise polynomial disclosed by the invention can realize real-time track planning of the landing of the extraterrestrial celestial body which meets the requirements of thrust limiting and optimal burnup on the basis of realizing the beneficial effects 1 and 2, thereby ensuring the accurate fixed-point soft landing of the detector.
Drawings
FIG. 1 is a flow chart of a method for land celestial body landing track planning based on a piecewise polynomial.
FIG. 2 is a schematic illustration of a segmented acceleration profile for an extraterrestrial celestial body landing.
Fig. 3 shows thrust curves corresponding to the piecewise polynomial trajectories at different terminal times.
Fig. 4 is a graph of the lander triaxial acceleration corresponding to the optimal time segment point.
Fig. 5 is a graph of lander thrust corresponding to an optimal time segment point.
Detailed Description
For a better description of the objects and advantages of the present invention, the following description of the invention will be taken in conjunction with an example and the accompanying drawings.
Taking the lander mass m=1000 kg, the asteroid gravitational acceleration g= [ -0.0014, -0.0052,0.0011] T (unit: m/s) 2 ) Asteroid self-rotation angular velocity w= 3.3139e -4 rad/s, gravitational acceleration constant g e ＝9.807m/s 2 Specific impulse I of engine sp =300, detector initial position r 0 ＝[2320,5257,-1707] T (unit: m), initial velocity v 0 ＝[0,0,0] T The method comprises the steps of carrying out a first treatment on the surface of the Detector terminal position r f ＝[2127,4819,-1565] T (unit: m), terminal speed v f ＝[0,0,0] T . The maximum thrust was 20N.
As shown in fig. 1, the method for planning the landing track of the extraterrestrial celestial body based on the piecewise polynomial disclosed in this embodiment specifically comprises the following implementation steps:
step one, dividing a detector landing track into three stages according to time nodes and according to landing terminal time t f With two normalized time segment points lambda 1 And lambda is 2 And (3) designing switching time of three stages, constructing a three-stage polynomial acceleration equation for landing track planning, and integrating to obtain a polynomial equation of speed and position. The normalized two time segmentation points are introduced in the track planning process, so that the adjustability of the polynomial track is enhanced.
First, according to the terminal time t f And normalized two time segment points lambda 1 And lambda is 2 And constructing a piecewise acceleration equation. According to the optimal idea of burnup, designing the acceleration equation into a Bang-Bang form expression
Wherein a represents the acceleration vector of the detector under the celestial center fixedly connected coordinate system, and t f Representing the terminal time, the planning process is divided into three time periods, 0 to lambda respectively 1 t f Corresponding first segment lambda 1 t f To lambda 2 t f Corresponding second segment and lambda 2 t f To t f A corresponding third segment. a, a 1 And a 3 The polynomial undetermined coefficients of the first and third segments are represented, respectively. Taking a certain axis as an example, the segmented acceleration curve is shown in fig. 2.
Integrating acceleration equations of three time periods respectively to obtain a velocity polynomial equation expressed as
Wherein v represents the velocity vector of the detector under the celestial center fixedly connected coordinate system, v 1 、v 2 And v 3 And respectively represent the polynomial undetermined coefficients of three time periods.
And integrating the velocity polynomial equations of the three time periods respectively to obtain a position polynomial equation, wherein the expression is as follows:
wherein r represents the position vector of the detector under the celestial center fixedly connected coordinate system, and r 1 、r 2 And r 3 And respectively represent the polynomial undetermined coefficients of three time periods.
The normalized two time segmentation points are introduced in the track planning process, a segmentation polynomial equation is constructed, and the adjustability of the polynomial track is enhanced.
Substituting the initial and terminal speed and position constraints into a constructed polynomial equation to calculate polynomial coefficients of each time period to obtain a time segmentation point lambda 1 And lambda is 2 Terminal time t f Parameter dependent polynomial equations.
The piecewise polynomial needs to satisfy the primary and final velocity and position constraints, as well as the equality constraints of the various time-piecewise points.
At time t=0, the three-axis speed constraint and the three-axis position constraint of the initial point need to be satisfied, expressed as follows:
wherein v is 0 ＝[0,0,0] T ，r 0 ＝[2320,5257,-1707] T 。
At t=λ 1 t f The time, speed and position equations need to satisfy the following equation relationship
At t=λ 2 t f The time, speed and position equations need to satisfy the following equation relationship
At t=t f To meet the requirements of accurate guidance and soft landing, a speed constraint with a terminal triaxial speed of 0 and a triaxial position constraint of the terminal landing point are established according to the target landing point of the detector, and are expressed as follows
Wherein r is f ＝[2127,4819,-1565] T 。
Substituting the equation relation between the initial and final speed and position constraint conditions and each time segment point into a polynomial equation to solve the time segment point lambda 1 And lambda is 2 Terminal time t f The result is as follows:
constructing a time segment point lambda using the polynomial coefficients obtained by the solution 1 And lambda is 2 Terminal time t f Polynomial equation of (c).
Step three, calculating to obtain a thrust expression according to the dynamics equation of the detector and the polynomial equation obtained in the step two, and establishing a relation of the terminal time t by the thrust amplitude constraint f Inequality constraint of (a) and solving to obtain given time segmentation point lambda 1 And lambda is 2 Terminal time t meeting thrust limiting constraint under initial value condition f Namely, the polynomial coefficient is adjusted to meet the constraint condition of the thrust amplitude of the detector, so that the planned detector track reduces the occurrence of the thrust saturation condition as much as possible.
When track planning, the final requirement for meeting the limiting constraint condition of the thrust is neededEnd time t f And solving.
Firstly, under the asteroid fixed connection coordinate system, a detector dynamics equation is constructed, and a relation of the position, the speed, the acceleration and the thrust of the detector is obtained.
The kinetic equation for the probe is as follows:
wherein w is the planetary rotation angular velocity, 2w×v represents the coriolis acceleration, w× (w×r) represents the centripetal acceleration, the control acceleration and disturbance acceleration of the detector are ignored, g is the gravitational acceleration vector, T is the engine thrust, F is the aerodynamic force (for asteroid, the term is 0), I sp Denote engine specific impulse, g e Is the gravitational acceleration constant.
And (3) subtracting the gravitational acceleration and various accelerations of the celestial body from the polynomial equation of the acceleration of the detector according to the dynamic equation of the detector and the polynomial equation obtained in the second step to obtain a thrust acceleration expression. Setting the initial value of the time segment point as lambda 1 =0.2 and λ 2 =0.7, select different terminal times t f And respectively 400s, 450s and 500s, substituting the two into the thrust acceleration expression to obtain thrust curves corresponding to the segment polynomial tracks under different terminal time as shown in figure 3.
From the time segment point lambda 1 And lambda is 2 Respectively calculating the thrust T and the terminal time T of the first section initial time and the third section initial time f Is a relation of (3). Solving and obtaining the terminal time t meeting the limiting constraint of each sectional thrust by Newton iteration method f . Comparing t calculated from the initial moments of the two segments f Selecting the largest t f As the terminal time of the piecewise polynomial trajectory. The terminal time t obtained at this time f Satisfy at a given time segment point lambda 1 And lambda is 2 And the thrust amplitude constraint under the initial value condition enables the planned detector track to reduce the occurrence of the thrust saturation condition as much as possible.
Step four, using fuelConsumption as an optimization index of polynomial trajectories, time segment point λ 1 And lambda is 2 And (3) expanding parameter optimization to obtain polynomial coefficients with optimal parameters, solving an acceleration curve with optimal fuel consumption, integrating to obtain a polynomial landing track, and realizing real-time track planning of extraterrestrial celestial body landing meeting thrust limiting and optimal fuel consumption, thereby ensuring accurate fixed-point soft landing of the detector.
Selecting the fuel consumption delta m as an optimization index of a polynomial track, and calculating through a polynomial equation to obtain the fuel consumption delta m relative to a time segmentation point lambda 1 And lambda is 2 For a target parameter lambda 1 And lambda is 2 And (5) carrying out optimization solution.
Thus, an optimized model of the time segment point parameter design problem is established
min J＝Δm
s.t.0≤λ 1 ≤λ 2 ≤1 (10)
Setting a time segment point lambda 1 And lambda is 2 Setting target optimization precision to determine iteration termination conditions, and performing iteration optimization to obtain a parameter value lambda meeting optimal fuel consumption 1 And lambda is 2 Substituting the acceleration curve into a polynomial equation to obtain an acceleration curve, integrating to obtain a polynomial landing track, and realizing real-time track planning of extraterrestrial celestial body landing meeting the requirements of thrust limiting and optimal burnup, thereby ensuring accurate fixed-point soft landing of the detector.
Setting the target optimization precision to be 1 multiplied by 10 according to the optimization model -6 Iterative optimization by adopting an interior point method to obtain a parameter value lambda meeting optimal burnup 1 =0.43 and λ 2 =0.86, and the terminal time t satisfying the thrust amplitude constraint is obtained by solving the newton iteration method f =576 s. Substituting the three-axis acceleration curve of the lander corresponding to the optimal time segment point into a polynomial equation, wherein the obtained three-axis acceleration curve of the lander corresponding to the optimal time segment point is shown in fig. 4, and the thrust curve of the lander corresponding to the optimal time segment point is shown in fig. 5. It can be seen that the thrust force satisfies the maximum amplitude constraint.
While the foregoing is directed to embodiments of the present invention, other and further embodiments of the invention may be devised without departing from the basic scope thereof, and the scope thereof is determined by the claims that follow.
Claims (1)
1. The extraterrestrial celestial body landing track planning method based on the piecewise polynomial is characterized by comprising the following steps of: the method comprises the following steps:
step one, dividing a detector landing track into three stages according to time nodes and according to landing terminal time t f With two normalized time segment points lambda 1 And lambda is 2 The switching time of three stages is designed, a three-stage polynomial acceleration equation for landing track planning is constructed, and a polynomial equation of speed and position is obtained through integration; the normalized two time segmentation points are introduced in the track planning process, so that the adjustability of the polynomial track is enhanced;
the specific implementation method of the first step is that,
first, according to the terminal time t f And normalized two time segment points lambda 1 And lambda is 2 Constructing a sectional acceleration equation; according to the optimal idea of burnup, designing the acceleration equation into a Bang-Bang form expression
Wherein a represents the acceleration vector of the detector under the celestial center fixedly connected coordinate system, and t f Representing the terminal time, the planning process is divided into three time periods, 0 to lambda respectively 1 t f Corresponding first segment lambda 1 t f To lambda 2 t f Corresponding second segment and lambda 2 t f To t f A corresponding third section; a, a 1 And a 3 Polynomial undetermined coefficients respectively representing the first segment and the third segment;
integrating acceleration equations of three time periods respectively to obtain a velocity polynomial equation expressed as
Wherein v represents the velocity vector of the detector under the celestial center fixedly connected coordinate system, v 1 、v 2 And v 3 Polynomial undetermined coefficients respectively representing three time periods;
the velocity polynomial equations for the three time periods are integrated separately to obtain the position polynomial equation expression as follows
Wherein r represents the position vector of the detector under the celestial center fixedly connected coordinate system, and r 1 、r 2 And r 3 Polynomial undetermined coefficients respectively representing three time periods;
introducing two normalized time segmentation points in the track planning process, constructing a segmentation polynomial equation, and enhancing the adjustability of the polynomial track;
substituting the initial and terminal speed and position constraints into a constructed polynomial equation to calculate polynomial coefficients of each time period to obtain a time segmentation point lambda 1 And lambda is 2 Terminal time t f A parameter dependent polynomial equation;
the specific implementation method of the second step is that,
the segmentation polynomial needs to meet the constraint conditions of the primary and the tail speeds and the positions, and the equality constraint relation of each time segmentation point;
at time t=0, the three-axis speed constraint and the three-axis position constraint of the initial point need to be satisfied, expressed as follows:
wherein v is 0 Representing a probeInitial velocity of measuring instrument, r 0 Representing the initial position of the detector;
at t=λ 1 t f The time, speed and position equations need to satisfy the following equation relationship
At t=λ 2 t f The time, speed and position equations need to satisfy the following equation relationship
At t=t f To meet the requirements of accurate guidance and soft landing, a speed constraint with a terminal triaxial speed of 0 and a triaxial position constraint of the terminal landing point are established according to the target landing point of the detector, and are expressed as follows
Wherein r is f The terminal position of the detector is represented, and the terminal landing speed is zero;
substituting the equation relation between the initial and final speed and position constraint conditions and each time segment point into a polynomial equation to solve the time segment point lambda 1 And lambda is 2 Terminal time t f The result of the polynomial undetermined coefficients of (2) is as follows
Constructing a time segment point lambda using the polynomial coefficients obtained by the solution 1 And lambda is 2 Terminal time t f Polynomial equation of (2);
step three, obtaining according to the dynamics equation of the detector and the step twoThe calculated polynomial equation obtains a thrust expression, and the thrust amplitude constraint is used for establishing a relation to the terminal time t f Inequality constraint of (a) and solving to obtain given time segmentation point lambda 1 And lambda is 2 Terminal time t meeting thrust limiting constraint under initial value condition f Namely, the polynomial coefficient is adjusted to meet the constraint condition of the thrust amplitude of the detector, so that the planned detector track reduces the occurrence of the thrust saturation condition as much as possible;
the specific implementation method of the third step is as follows:
during track planning, the terminal time t meeting the thrust limiting constraint condition is needed f Solving;
firstly, under an asteroid fixedly connected coordinate system, a detector dynamics equation is constructed to obtain a relation of the position, the speed, the acceleration and the thrust of the detector;
the kinetic equation of the probe is as follows
Wherein w is the planetary rotation angular velocity, 2w×v is the Ke's acceleration, w× (w×r) is the centripetal acceleration, the control acceleration and disturbance acceleration of the detector are ignored, g is the gravitational acceleration vector, T is the engine thrust, F is the aerodynamic force, I sp Denote engine specific impulse, g e Is the gravitational acceleration constant;
subtracting the gravitational acceleration and various accelerations of the celestial body from the polynomial equation of the acceleration of the detector according to the dynamic equation of the detector and the polynomial equation obtained in the second step to obtain a thrust acceleration expression;
from the time segment point lambda 1 And lambda is 2 Respectively calculating the thrust T and the terminal time T of the first section initial time and the third section initial time f Is a relation of (2); solving and obtaining the terminal time t meeting the limiting constraint of each sectional thrust by Newton iteration method f The method comprises the steps of carrying out a first treatment on the surface of the Comparing t calculated from the initial moments of the two segments f Selecting the largest t f As a piecewise polynomial trajectoryTerminal time; the terminal time t obtained at this time f Satisfy at a given time segment point lambda 1 And lambda is 2 The thrust amplitude constraint under the initial value condition enables the planned detector track to reduce the occurrence of the thrust saturation condition as much as possible;
step four, taking fuel consumption as an optimization index of the polynomial track, and dividing the time into points lambda 1 And lambda is 2 Expanding parameters to obtain polynomial coefficients with optimal parameters, solving an acceleration curve with optimal fuel consumption, integrating to obtain a polynomial landing track, and realizing real-time track planning of extraterrestrial celestial body landing meeting thrust limiting and optimal fuel consumption, thereby ensuring accurate fixed-point soft landing of a detector;
the specific implementation method of the fourth step is that,
selecting the fuel consumption delta m as an optimization index of a polynomial track, and calculating through a polynomial equation to obtain the fuel consumption delta m relative to a time segmentation point lambda 1 And lambda is 2 For a target parameter lambda 1 And lambda is 2 Carrying out optimization solution;
thus, an optimized model of the time segment point parameter design problem is established
min J＝Δm
s.t.0≤λ 1 ≤λ 2 ≤1(10)
Setting a time segment point lambda 1 And lambda is 2 Setting target optimization precision to determine iteration termination conditions, and performing iteration optimization to obtain a parameter value lambda meeting optimal fuel consumption 1 And lambda is 2 Substituting the acceleration curve into a polynomial equation to obtain an acceleration curve, integrating to obtain a polynomial landing track, and realizing real-time track planning of extraterrestrial celestial body landing meeting the requirements of thrust limiting and optimal burnup, thereby ensuring accurate fixed-point soft landing of the detector.
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