CN110562493B - Mars power descending trajectory planning method based on vector trajectory - Google Patents

Mars power descending trajectory planning method based on vector trajectory Download PDF

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CN110562493B
CN110562493B CN201910845476.8A CN201910845476A CN110562493B CN 110562493 B CN110562493 B CN 110562493B CN 201910845476 A CN201910845476 A CN 201910845476A CN 110562493 B CN110562493 B CN 110562493B
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崔平远
龙嘉腾
朱圣英
徐瑞
梁子璇
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Beijing Institute of Technology BIT
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Abstract

The invention discloses a mars power descent trajectory planning method based on a vector trajectory, and belongs to the technical field of deep space exploration. The implementation method of the invention comprises the following steps: on the basis of determining the optimal performance index and constraint conditions of Mars power reduction fuel consumption, vectorizing a kinetic equation, camera field angle constraint and geometric convex track constraint, and discretizing position, speed and acceleration variables, performance indexes and constraint conditions; converting inequality constraint and control constraint into a second-order cone constraint form; therefore, the track planning problem is converted into a second-order cone planning problem, the second-order cone planning problem is solved, landing at a target landing point is realized, the camera view angle constraint and the geometric convex track constraint are met, and meanwhile, the track planning method has instantaneity and convergence. The method can effectively guarantee the visibility of the navigation camera to the obstacles in the landing area, and applies the constraint of the geometric convex track to improve the avoidance capability of the obstacles around the landing point.

Description

Mars power descending trajectory planning method based on vector trajectory
Technical Field
The invention relates to a power descent trajectory planning method, in particular to a mars power descent trajectory planning method based on a vector trajectory, and belongs to the technical field of deep space exploration.
Background
The accurate and safe landing of the surface of the Mars is a technical premise of carrying out scientific detection tasks such as Mars surface in-situ detection, sampling return and the like. In order to obtain a rich scientific return, the Mars lander inevitably needs to land in a complex terrain area in the future, which provides challenges for the lander's ability to avoid complex obstacles and landing safety.
The obstacle avoidance technology of the landing trajectory planning method is an important technology for ensuring the landing safety of the lander in a complex terrain area. The track curvature is an important factor influencing the obstacle avoidance effect, and the landing convex track has obvious advantages on obstacle avoidance. Meanwhile, in order to ensure the real-time detection capability of the lander on the landing obstacle, the target landing area needs to be located within the field of view of the navigation camera. In addition, in order to ensure the on-line application of the trajectory planning algorithm, the mars power descent trajectory planning algorithm needs to have stronger instantaneity and convergence on the basis of meeting the camera view angle constraint and the geometric convex trajectory constraint.
Aiming at the future Mars landing detection requirement, an online track planning method with strong real-time performance and convergence is needed to be designed on the basis of meeting the camera field angle constraint and the geometric convex track constraint, so that the landing safety of the future Mars landing detection task in a complex terrain area is guaranteed.
Disclosure of Invention
The invention discloses a mars power descent track planning method based on a vector track, which aims to solve the technical problems that: landing at a target landing point is realized, the camera field angle constraint and the geometric convex track constraint are met, and meanwhile, the track planning method needs to have real-time performance and convergence.
The purpose of the invention is realized by the following technical scheme.
The invention discloses a mars power descent track planning method based on vector tracks, which is characterized in that on the basis of determining optimal performance indexes and constraint conditions of mars power descent burnup, a dynamic equation, camera field angle constraint and geometric convex track constraint are subjected to vectorization, and position, speed and acceleration variables, performance indexes and constraint conditions are discretized; and converting the inequality constraint and the control constraint into a second-order cone constraint form. Therefore, the track planning problem is converted into a second-order cone planning problem, the second-order cone planning problem is solved, landing at a target landing point is realized, the camera view angle constraint and the geometric convex track constraint are met, and meanwhile, the track planning method has instantaneity and convergence.
The invention discloses a mars power descent track planning method based on a vector track, which comprises the following steps of:
step 1, determining a dynamic equation, a fuel consumption optimal performance index and constraint conditions of a Mars power descent section.
And (3) determining a dynamic equation of the Mars power descending section as shown in the formula (1).
Figure GDA0002765600980000021
Wherein x and z are the horizontal position and height of the lander respectively; v. ofxAnd vzThe speed of the lander in the horizontal direction and the height direction respectively; a isxAnd azAcceleration of the lander in the horizontal direction and the height direction respectively; a iscxAnd aczThe control acceleration of the lander along the horizontal direction and the height direction are respectively, and the relation shown in the formula (2) is satisfied.
Figure GDA0002765600980000022
Wherein g is the Mars surface gravity acceleration. In the process of planning the Mars dynamic descent trajectory, the boundary constraint condition shown in the formula (3) needs to be satisfied.
Figure GDA0002765600980000023
Wherein x is0And z0Respectively, the initial time t ═ t0The position of the lander in the horizontal and height directions; v. ofx0And vz0Respectively, the initial time t ═ t0The speed of the lander in the horizontal and altitude directions; x is the number offAnd zfRespectively, landing time t ═ tfThe target position of the lander in the horizontal and height directions; v. ofxfAnd vzfRespectively, landing time t ═ tfThe target speed of the lander in the horizontal and altitude directions. Mars power descending section requirement vxf＝0、v zf0; placing the origin of the landing coordinate system at the landing site, then there is xf＝0、zf＝0。
The optimal performance index of the spark power reduction burnup is
Figure GDA0002765600980000024
In the Mars power descending section, in order to ensure the visibility of the navigation camera to the landing point and the obstacle avoidance capability in the landing process, the camera view angle constraint and the geometric convex track constraint need to be met.
Constraint one: camera field angle constraints.
Since the navigation camera has a field angle, the visibility of the camera to the landing area needs to be ensured, namely, the geometric relation shown in the formula (5) is satisfied.
Figure GDA0002765600980000025
Wherein alpha isFoVIs a line-of-sight angle and is defined as an included angle between a connecting line of the camera and the landing point and the optical axis of the camera; gamma is the angle of view of the camera and is defined as the included angle between the two sight edges of the camera, and gamma/2 is the included angle between the sight edges of the camera and the optical axis of the camera.
Constraint two: and (4) constraint of a geometrical convex track.
The geometric convex trajectory constraint is expressed as shown in equation (6).
Figure GDA0002765600980000026
Constraint condition three: and controlling the constraint.
The control constraint is expressed as shown in formula (7) or formula (8)
Tmin≤||T||≤Tmax (7)
ac,min≤||ac||≤ac,max (8)
Wherein T ═ macIn order to provide the thrust of the lander,
Figure GDA0002765600980000031
is the thruster amplitude, m is the lander mass, TminAnd TmaxThe minimum and maximum values of the thrust amplitude are respectively. a isc,minAnd ac,maxAre respectively controlThe minimum value and the maximum value of the acceleration amplitude, and the relation shown in the formula (9) is satisfied.
Figure GDA0002765600980000032
And 2, vectorizing the dynamic equation of the Mars power descent section, the optimal performance index and the constraint condition in the step 1, and determining a vector track, namely defining the vector track.
Defining a position vector r ═ x, z]TVelocity vector v ═ vx,vz]TAcceleration vector a ═ ax,az]TControlling the acceleration vector ac＝[acx,acz]TThe gravity acceleration vector g is [0, g ]]T. Vectorization processing is performed on the kinetic equation (1) to obtain a vectorization form shown in formula (10).
Figure GDA0002765600980000033
Vectorizing the camera angle constraint to make the optical axis direction of the navigation camera coincide with the thrust acceleration direction of the lander, and expressing the geometric relationship between the angle of view in the formula (5) and the camera angle of view as the vector included angle relationship shown in the formula (11) to obtain the vector form of the camera angle constraint.
Figure GDA0002765600980000034
Vectorizing the geometric convex track constraint to define a pseudo velocity vector
Figure GDA0002765600980000035
Equivalent transformation of formula (6) into a vector
Figure GDA0002765600980000036
And a ═ ax,az]TIn (2)Product of large quantities
Figure GDA0002765600980000037
I.e. vectors
Figure GDA0002765600980000038
And a ═ ax,az]TThe angle therebetween satisfies the condition shown in the formula (13) or (14).
Figure GDA0002765600980000039
Figure GDA00027656009800000310
Equation (13) is a vector form for geometric convex trajectory constraint. The vector track is determined through the equations (10), (11) and (13), namely, the vector track is defined.
And 3, discretizing the vectorized position, speed and acceleration vectors, the performance indexes and the constraint conditions.
The state variables in the landing process are uniformly dispersed into n points, and the corresponding position, speed, acceleration and control acceleration of the lander are shown as the formula (15)
Figure GDA0002765600980000041
According to the kinetic equation (1), a fourth-order Runge Kutta integral is adopted, and the discrete variables satisfy the following relation
Figure GDA0002765600980000042
Wherein the content of the first and second substances,
Figure GDA0002765600980000043
Ui＝ai
Figure GDA0002765600980000044
using the fourth-order Runge Kutta integral, the differential equation (16) is transformed into the following relationship
Figure GDA0002765600980000045
Wherein the content of the first and second substances,
Figure GDA0002765600980000046
and 4, converting the track optimization problem into a second-order cone planning problem through the discretized position, speed, acceleration vector, performance index and constraint condition.
Step 4.1: a second order cone constraint form of the camera field angle constraint is determined.
Defining a position vector riAt an angle to the gravitational acceleration g of
Figure GDA0002765600980000047
Defining an acceleration vector ac,iAt an angle to the gravitational acceleration g of
Figure GDA0002765600980000048
Angle of sight alphaFoV,iAnd alphar,i、αa,iThere are the following relationships
αFoV,i＝αr,i+αa,i (21)
By substituting formula (21) for formula (5)
Figure GDA0002765600980000051
Introducing an intermediate variable gamma1Then the inequality constraint in equation (22) is equivalently transformed into the inequality group shown in equation (23)
Figure GDA0002765600980000052
By substituting formula (19) and formula (20) for formula (23)
Figure GDA0002765600980000053
Since the Mars surface gravity acceleration g is a constant vector and g | | | | is g, the first two inequalities in equation (24) are constrained to be second-order cone constraints.
Step 4.2: a second order cone constraint form of the geometric convex trajectory constraint is determined.
Defining vectors
Figure GDA0002765600980000054
At an angle to the gravitational acceleration g of
Figure GDA0002765600980000055
Then angle alphacov,iAnd alphav,i、αa,iThere are the following relationships
αFoV,i＝αv,i+αa,i (26)
By substituting formula (14) for formula (26)
Figure GDA0002765600980000056
Introducing an intermediate variable gamma2Then the inequality constraint in equation (27) is equivalently transformed into the inequality group shown in equation (28)
Figure GDA0002765600980000057
By substituting formula (20) and formula (25) for formula (27)
Figure GDA0002765600980000058
The first two inequality constraints in equation (29) are second order cone constraints.
Step 4.3: a second order cone constraint form of the control constraint is determined.
Let σ become [ σ ]1,σ2,…,σn]TThe control constraint equation (8) is converted into the form shown in equation (30).
Figure GDA0002765600980000061
The first inequality constraint in equation (30) is a second order cone constraint.
Step 4.4: performance index conversion of formula (4)
Figure GDA0002765600980000062
In summary, the Mars dynamic descent segment burn-up optimization trajectory planning problem described by the formula (1), the formula (4), the formula (5) and the formula (14) is converted into the following formula [ sigma, gamma ]1,Γ2]For optimizing variables, equation (32) is used as a performance index, equation (33) is used as an inequality constraint, and equation (17) is used as a second-order cone programming form of a dynamic constraint. Therefore, the real-time performance and the convergence of the trajectory planning method are ensured.
Figure GDA0002765600980000063
So that
Figure GDA0002765600980000064
And 5: and (4) solving the second-order cone planning problem determined in the step (4), realizing landing at a target landing point, meeting camera field angle constraint, geometric convex track constraint and control constraint, and simultaneously ensuring real-time performance and convergence of the track planning method.
Has the advantages that:
1. according to the mars power descent track planning method based on the vector track, disclosed by the invention, the visibility of a navigation camera to obstacles in a landing area can be effectively ensured by applying the camera view angle constraint, and the avoidance capability to the obstacles around the landing point is improved by applying the geometric convex track constraint.
2. The mars power descent track planning method based on the vector tracks, disclosed by the invention, vectorizes a dynamic equation, an optimal performance index and a constraint condition of a mars power descent section, determines the vector tracks, namely realizes the definition of the vector tracks, and converts the vectorized constraint condition into a second-order cone constraint form, so that the power descent track planning problem is converted into a second-order cone planning problem, and the instantaneity and the convergence of a track planning algorithm can be ensured.
Drawings
FIG. 1 is a flowchart of a Mars power descent trajectory planning method based on vector trajectories;
fig. 2 shows a Mars power descent trajectory planned by the method.
Fig. 3 is a view angle variation curve of the planned track by the method.
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 mars power descent trajectory planning method based on the vector trajectory disclosed in this embodiment specifically includes the following steps:
step 1, determining a dynamic equation, a fuel consumption optimal performance index and constraint conditions of a Mars power descent section.
The dynamic equation of the Mars power descending section is shown as the formula (34).
Figure GDA0002765600980000071
Where x and z are the horizontal position and height of the landing gear, respectively, vxAnd vzSpeed of the lander in the horizontal and altitude directions, axAnd azAcceleration of the lander in the horizontal and height directions, acxAnd aczThe control acceleration of the lander in the horizontal direction and the height direction are respectively, and the relationship shown in the formula (35) is satisfied.
Figure GDA0002765600980000072
Wherein g is the Mars surface gravity acceleration. In the process of planning the Mars dynamic descent trajectory, the boundary constraint condition shown in the formula (36) needs to be met.
Figure GDA0002765600980000073
Wherein x is0And z0Respectively, the initial time t ═ t0The position of the lander in the horizontal and height directions; v. ofx0And vz0Respectively, the initial time t ═ t0The speed of the lander in the horizontal and altitude directions; x is the number offAnd zfRespectively, landing time t ═ tfThe target position of the lander in the horizontal and height directions; v. ofxfAnd vzfRespectively, landing time t ═ tfThe target speed of the lander in the horizontal and altitude directions. Mars power descending section requirement vxf＝0、v zf0; placing the origin of the landing coordinate system at the landing site, then there is xf＝0、zf＝0。
The optimal performance index of the Mars power reduction burnup is
Figure GDA0002765600980000074
In the Mars power descending section, in order to ensure the visibility of the navigation camera to the landing point and the obstacle avoidance capability in the landing process, the camera view angle constraint and the geometric convex track constraint need to be met.
Constraint one: camera field angle constraints.
Since the navigation camera has a field angle, the visibility of the camera to the landing area needs to be ensured, namely, the geometric relation shown in the formula (38) is satisfied.
Figure GDA0002765600980000081
Wherein alpha isFoVIs a line-of-sight angle and is defined as an included angle between a connecting line of the camera and the landing point sight and the optical axis of the camera; the camera view angle γ is 60 °, defined as the angle between the two viewing edges of the camera, and γ/2 is the angle between the viewing edge of the camera and the optical axis of the camera.
Constraint two: and (4) constraint of a geometrical convex track.
The geometric convex trajectory constraint is expressed as shown in equation (39).
Figure GDA0002765600980000082
Constraint condition three: and controlling the constraint.
The control constraint is expressed as shown in equation (40) or equation (41).
Tmin≤||T||≤Tmax (40)
ac,min≤||ac||≤ac,max (41)
Wherein T ═ macIn order to provide the thrust of the lander,
Figure GDA0002765600980000083
for the thrust amplitude, the mass m of the lander is 1905kg, and the minimum value and the maximum value of the thrust amplitude are respectively Tmin3200N and Tmax＝24000N。ac,minAnd ac,maxThe minimum value and the maximum value of the control acceleration amplitude are respectively, and the relation shown in the formula (42) is satisfied.
Figure GDA0002765600980000084
And 2, vectorizing the dynamic equation of the Mars power descent section, the optimal performance index and the constraint condition in the step 1, and determining a vector track, namely defining the vector track.
Defining a position vector r ═ x, z]TVelocity vector v ═ vx,vz]TAcceleration vector a ═ ax,az]TControlling the acceleration vector ac＝[acx,acz]TThe gravity acceleration vector g is [0, g ]]T. The kinetic equation (34) is expressed as a vectorized form shown in formula (43).
Figure GDA0002765600980000085
Vectorizing the camera angle constraint to make the optical axis direction of the navigation camera coincide with the thrust acceleration direction of the lander, and expressing the line-of-sight angle and the geometric relation in the formula (38) as a vector included angle relation shown in the formula (44) to obtain a vector form of the camera angle constraint.
Figure GDA0002765600980000086
Vectorizing the geometric convex track constraint to define a pseudo velocity vector
Figure GDA0002765600980000091
Equivalent transformation of formula (39) into a vector
Figure GDA0002765600980000092
And a ═ ax,az]TInner product of (2)
Figure GDA0002765600980000093
I.e. vectors
Figure GDA0002765600980000094
And a ═ ax,az]TThe angle therebetween satisfies the condition shown in the formula (46) or (47).
Figure GDA0002765600980000095
Figure GDA0002765600980000096
Equation (47) is a vector form for geometric convex trajectory constraints.
The vector locus is determined through equations (43), (44) and (46), namely, the vector locus is defined.
And 3, discretizing the vectorized position, speed and acceleration vectors, the performance indexes and the constraint conditions.
And uniformly dispersing the state variables in the landing process into n points, wherein the corresponding lander position, speed, acceleration and control acceleration are shown as a formula (48).
Figure GDA0002765600980000097
According to the kinetic equation (34), a fourth-order Runge Kutta integral is adopted, and the discrete variables satisfy the following relation
Figure GDA0002765600980000098
Wherein the content of the first and second substances,
Figure GDA0002765600980000099
Ui＝ai，
Figure GDA00027656009800000910
using the fourth-order Runge Kutta integral, the differential equation (49) is transformed into the following relationship
Figure GDA00027656009800000911
Wherein the content of the first and second substances,
Figure GDA00027656009800000912
and 4, converting the track optimization problem into a second-order cone planning problem through the discretized position, speed and acceleration vectors, the performance indexes and the constraint conditions.
Step 4.1: a second order cone constraint form of the camera field angle constraint is determined.
Defining a position vector riAt an angle to the gravitational acceleration g of
Figure GDA0002765600980000101
Defining an acceleration vector ac,iAt an angle to the gravitational acceleration g of
Figure GDA0002765600980000102
Angle of sight alphaFoV,iAnd alphar,i、αa,iThere are the following relationships
αFoV,i＝αr,i+αa,i (54)
By substituting formula (54) for formula (38)
Figure GDA0002765600980000103
Introducing an intermediate variable gamma1Then, the inequality constraint in equation (55) is equivalently transformed into the inequality group shown in equation (56)
Figure GDA0002765600980000104
By substituting formula (52) and formula (53) for formula (56)
Figure GDA0002765600980000105
Since the Mars surface gravity acceleration g is a constant vector and g | | | | is g, the first two inequalities in equation (57) are constrained to be second-order cone constraints.
Step 4.2: a second order cone constraint form of the geometric convex trajectory constraint is determined.
Defining vectors
Figure GDA0002765600980000106
At an angle to the gravitational acceleration g of
Figure GDA0002765600980000107
Then angle alphacov,iAnd alphav,i、αa,iThere are the following relationships
αFoV,i＝αv,i+αa,i (59)
By substituting formula (47) for formula (59) having
Figure GDA0002765600980000108
Introducing an intermediate variable gamma2Then the inequality constraint in equation (60) is equivalently transformed into the inequality group shown in equation (61)
Figure GDA0002765600980000111
By substituting formula (52) and formula (53) for formula (61)
Figure GDA0002765600980000112
The first two inequality constraints in equation (62) are second order cone constraints.
Step 4.3: a second order cone constraint form of the control constraint is determined.
Let σ become [ σ ]1,σ2,…,σn]TThe control constraint equation (41) is converted into the form shown in equation (63).
Figure GDA0002765600980000113
The first inequality constraint in equation (63) is a second order cone constraint.
Step 4.4: conversion of Performance index formula (37) to
Figure GDA0002765600980000114
In summary, the Mars dynamic descent segment burn-up optimization trajectory planning problem described by equation (34), equation (37), equation (38) and equation (47) is transformed into [ σ, Γ [ ]1,Γ2]For optimizing variables, equation (65) is used as a performance index, equation (66) is used as an inequality constraint, and equation (50) is used as a second-order cone programming form of a dynamic constraint. Therefore, the real-time performance and the convergence of the trajectory planning algorithm are ensured.
Figure GDA0002765600980000115
So that
Figure GDA0002765600980000116
And 5: and (4) solving the second-order cone planning problem determined in the step (4), realizing landing at a target landing point, meeting camera field angle constraint and geometric convex track constraint, and simultaneously ensuring real-time performance and convergence of the track planning method.
The simulation initial conditions are
[x0,z0,vx0,vz0]＝[3000m，1700m,-35m/s,-85m/s] (67)
Maximum value T of thrust amplitudemaxAnd a minimum value TminAre respectively Tmax＝24000N，Tmin＝3200N。
Fig. 2 shows a Mars power descent trajectory planned by the method. It can be seen from the figure that the method can realize the adjustment of the Mars power descending track, and the geometric concave track at the initial time is adjusted to the geometric convex track before landing.
Fig. 3 is a view angle variation curve of the planned track by the method. It can be seen that the field angle is always maintained within ± 30 degrees.
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. A mars power descent track planning method based on vector tracks is characterized by comprising the following steps: comprises the following steps of (a) carrying out,
step 1, determining a dynamic equation, an optimal performance index and a constraint condition of a Mars power descent section;
step 2, vectorizing the dynamic equation of the Mars power descent section, the optimal performance index and the constraint condition in the step 1, and determining a vector track, namely defining the vector track;
step 3, discretizing the vectorized position, speed and acceleration vectors, the optimal performance indexes and the constraint conditions;
step 4, converting the trajectory optimization problem into a second-order cone planning problem through the discretized position, speed, acceleration vector, optimal performance index and constraint condition;
and 5: solving the second-order cone planning problem determined in the step 4, realizing landing at a target landing point, meeting camera field angle constraint, geometric convex track constraint and control constraint, and simultaneously ensuring real-time performance and convergence of the track planning method;
wherein, the step 1 is realized by the following steps,
determining a dynamic equation of a Mars power descending section as shown in the formula
Figure FDA0002765600970000011
Wherein x and z are the horizontal position and height of the lander respectively; v. ofxAnd vzThe speed of the lander in the horizontal direction and the height direction respectively; a isxAnd azAcceleration of the lander in the horizontal direction and the height direction respectively; a iscxAnd aczRespectively controlling the acceleration of the lander along the horizontal direction and the height direction, and satisfying the relation shown in the formula
Figure FDA0002765600970000012
Wherein g is the Mars surface gravity acceleration; in the process of planning the Mars dynamic descent track, the boundary constraint condition shown by the formula needs to be met
Figure FDA0002765600970000013
Wherein x is0And z0Respectively, the initial time t ═ t0The position of the lander in the horizontal and height directions; v. ofx0And vz0Respectively, the initial time t ═ t0The speed of the lander in the horizontal and altitude directions; x is the number offAnd zfRespectively, landing time t ═ tfThe target position of the lander in the horizontal and height directions; v. ofxfAnd vzfRespectively, landing time t ═ tfTarget speed of the lander in horizontal and altitude directions; mars power descending section requirement vxf＝0、vzf0; placing the origin of the landing coordinate system at the landing site, then there is xf＝0、zf＝0；
The optimal performance index of the Mars power reduction is
Figure FDA0002765600970000014
In the Mars power descending section, in order to ensure the visibility of a navigation camera to a landing point and the obstacle avoidance capability in the landing process, the field angle constraint and the geometric convex track constraint of the camera need to be met;
constraint one: camera field angle constraints;
because the navigation camera has a field angle, the visibility of the camera to a landing area needs to be ensured, namely, the geometric relation shown in the formula is met;
Figure FDA0002765600970000021
wherein alpha isFoVIs a line-of-sight angle and is defined as an included angle between a connecting line of the camera and the landing point and the optical axis of the camera; gamma is the angle of view of the camera and is defined as the included angle between two sight edges of the camera, and gamma/2 is the included angle between the sight edges of the camera and the optical axis of the camera;
constraint two: constraint of geometric convex track;
the geometric convex trajectory constraint is expressed as shown in the formula;
Figure FDA0002765600970000022
constraint condition three: controlling constraints;
the control constraint is expressed as a formula or in the form of a formula
Tmin≤||T||≤Tmax (7)
ac,min≤||ac||≤ac,max (8)
Wherein T ═ macIn order to provide the thrust of the lander,
Figure FDA0002765600970000023
is the thruster amplitude, m is the lander mass, TminAnd TmaxRespectively the minimum value and the maximum value of the thrust amplitude; a isc,minAnd ac,maxThe control acceleration amplitude is respectively the minimum value and the maximum value, and the relation shown in the formula is satisfied;
Figure FDA0002765600970000024
2. the method for planning a Mars dynamic descent trajectory based on a vector trajectory according to claim 1, wherein: the step 2 is realized by the method that,
defining a position vector r ═ x, z]TVelocity vector v ═ vx,vz]TAcceleration vector a ═ ax,az]TControlling the acceleration vector ac＝[acx,acz]TThe gravity acceleration vector g is [0, g ]]T(ii) a Vectorizing the dynamic equation to obtain a vectorization form shown in a formula (10);
Figure FDA0002765600970000025
vectorizing the camera angle constraint to make the optical axis direction of the navigation camera coincide with the thrust acceleration direction of the lander, wherein the geometric relationship between the angle of view and the camera angle of view is expressed as the vector included angle relationship shown in the formula (11), and the vector form of the camera angle constraint is obtained;
Figure FDA0002765600970000031
vectorizing the geometric convex track constraint to define a pseudo velocity vector
Figure FDA0002765600970000032
Conversion of equation equivalence into vector
Figure FDA0002765600970000033
And a ═ ax,az]TInner product of (2)
Figure FDA0002765600970000034
I.e. vectors
Figure FDA0002765600970000035
And a ═ ax,az]TThe included angle between the two satisfies the condition shown in the formula (13) or (14);
Figure FDA0002765600970000036
Figure FDA0002765600970000037
the formula (13) is a vector form for geometric convex track constraint; the vector track is determined through the equations (10), (11) and (13), namely, the vector track is defined.
3. The method for planning a Mars dynamic descent trajectory based on a vector trajectory according to claim 2, wherein: the step 3 is realized by the method that,
the state variables in the landing process are uniformly dispersed into n points, and the corresponding position, speed, acceleration and control acceleration of the lander are shown in the formula
Figure FDA0002765600970000038
According to the kinetic equation, adopting four-order Runge Kutta integral, the above-mentioned discrete variables must satisfy the following relationship
Figure FDA0002765600970000039
Wherein the content of the first and second substances,
Figure FDA00027656009700000310
Ui＝ai
Figure FDA00027656009700000311
by using the fourth-order Runge Kutta integral, the differential equation is converted into the following relationship
Figure FDA00027656009700000312
Wherein the content of the first and second substances,
Figure FDA0002765600970000041
4. a mars power descent trajectory planning method based on vector trajectories as claimed in claim 3, wherein: step 4, the method is realized by the following steps,
step 4.1: determining a second-order cone constraint form of camera field angle constraint;
defining a position vector riAt an angle to the gravitational acceleration g of
Figure FDA0002765600970000042
Defining an acceleration vector ac,iAt an angle to the gravitational acceleration g of
Figure FDA0002765600970000043
Angle of sight alphaFoV,iAnd alphar,i、αa,iThere are the following relationships
αFoV,i＝αr,i+αa,i (21)
Substituting formula into formula I
Figure FDA0002765600970000044
Introducing an intermediate variable gamma1The inequality constraint in the formula is equivalently converted into the inequality group shown in the formula
Figure FDA0002765600970000045
Substituting the formula sum into the formula
Figure FDA0002765600970000046
Because the gravity acceleration g on the surface of the mars is a constant vector and g is equal to g, the first two inequalities in the formula are constrained to be second-order cone constraints;
step 4.2: determining a second-order cone constraint form of geometric convex track constraint;
defining vectors
Figure FDA0002765600970000047
At an angle to the gravitational acceleration g of
Figure FDA0002765600970000051
Then angle alphacov,iAnd alphav,i、αa,iThere are the following relationships
αFoV,i＝αv,i+αa,i (26)
The formula (14) is substituted by
Figure FDA0002765600970000052
Introducing an intermediate variable gamma2The inequality constraint in the formula is equivalently converted into the inequality group shown in the formula
Figure FDA0002765600970000053
Substituting the formula sum into the formula
Figure FDA0002765600970000054
The first two inequality constraints in the formula are second-order cone constraints;
step 4.3: determining a second-order cone constraint form of the control constraint;
let σ become [ σ ]1,σ2,…,σn]TConverting the control constraint expression into a form shown by a formula;
Figure FDA0002765600970000055
the first inequality constraint in the formula is a second-order cone constraint;
step 4.4: optimum performance index formula conversion
Figure FDA0002765600970000056
In summary, the Mars dynamic descent segment burnup optimization trajectory planning problem described by equation (14) is converted into [ sigma, gamma ]1,Γ2]For optimizing variables, taking an equation as an optimal performance index, an equation as inequality constraint and an equation as a second-order cone programming form of dynamic constraint; thereby ensuring the real-time performance and convergence of the trajectory planning method;
Figure FDA0002765600970000057
so that
Figure FDA0002765600970000061
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