CN105929844B - Barrier-avoiding method under a kind of more Obstacles Constraints environment of objects outside Earth soft landing - Google Patents
Barrier-avoiding method under a kind of more Obstacles Constraints environment of objects outside Earth soft landing Download PDFInfo
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Abstract
Barrier-avoiding method under a kind of more Obstacles Constraints environment of objects outside Earth soft landing, belongs to Navigation, Guidance and Control technical field.The present invention, for the optimal Guidance problem under more Obstacles Constraints environment, will propose a kind of optimal soft landing method of guidance of the improvement based on convex optimization on the basis of the former.Firstly, establishing the optimal Second-order cone programming model of fuel that Obstacles Constraints are not added, and standardized；Secondly, carrying out analysis modeling for martian surface protrusion obstacle, and linear transformation is carried out, convert convex constraint for non-convex constraint, be dissolved among Second-order cone programming problem, and establishes the optimal Second-order cone programming model of complete consideration Obstacles Constraints；Carry out the correctness of verification algorithm finally by the simulation analysis of three kinds of different type Obstacles Constraints.Apply it in soft lunar landing research, simulation result shows that new algorithm realizes effectively evading for three-dimensional space obstacle, be rationally utilized around and above obstacle can flight space, while it is optimal to meet fuel.
Description
Technical field
The invention belongs to Navigation, Guidance and Control technical field, it is related to the optimal system of avoidance under a kind of more Obstacles Constraints environment
Guiding method.
Background technique
Currently, for objects outside Earth soft landing process optimum guidance problems there are many approach applications wherein, but more
Avoidance problem under Obstacles Constraints environment does not have preferable solution to propose yet.
Given below is the general solution for the soft landing process optimum guidance problems of objects outside Earth, process
Are as follows:
In the Approach phase of entire landing mission, lander is close from menology, can ignore the shadow of moon autobiography at this time
It rings, and entire kinetic model is established in a menology fixed coordinate system using landing point as origin.Menology fixed coordinates
Shown in the definition of system such as Fig. 1 shows, the vertical menology of ox axis, ox, oy, tri- axis of oz composition right-handed coordinate system.
Since menology does not have air, and complicated control problem is not considered, only consider guidance problems, can obtain the dynamic of Approach phase
Mechanical equation:
Wherein,
R --- it is position vector of the lander under oxyz system, r=[rx ry rz]T；
V --- it is velocity vector of the lander under oxyz system, v=[vx vy vz]T；
A --- it is acceleration of the lander under oxyz system, a=[ax ay az]T；
gm--- it is Mars acceleration of gravity；
Tc--- for the net thrust vector (being directed toward lander central axis) of lander engine, Tc=| | Tc| |, it is that it is big
It is small；
M --- it is the quality at lander per moment；
α --- it is lander engine burn-up rate, is greater than 0,(geFor terrestrial gravitation acceleration, φ hair
The angle of motivation established angle, i.e. jet direction and lander central axis, IspFor engine/motor specific impulse).
Consider the rail that boundary condition, control force size constraint, the constraint of lander attitude angle and ground level constrain
Mark fuel optimal problem are as follows:
Objective function:
Meet formula (1) constraint equation and following constraint:
0<T1≤Tc≤T2 (3)
rx≥0 (4)
Wherein T1、T2For control force bound；θaltAs shown in Figure 1, the inclination angle of referred to as lander,It is given for it
The upper limit,The effect of the constraint is to guarantee that lander is landed with certain landing angle, without touching before landing
The small protrusion of upper ground surface.From the foregoing it is appreciated that, the soft landing optimal Guidance process of existing objects outside Earth is directed to more Obstacles Constraints
Avoidance problem under environment provides the scheme of efficiently solving.
Summary of the invention
The object of the present invention is to provide barrier-avoiding methods under a kind of more Obstacles Constraints environment of objects outside Earth soft landing, with solution
The certainly avoidance problem during the soft landing optimal Guidance of objects outside Earth under more Obstacles Constraints environment.
The technical solution adopted by the present invention to solve the above technical problem is:
Raised Disorder Model selection
Firstly, a model space geometric is selected for the raised obstacle on menology, to be depicted with the form of mathematics
Come, obstacle is made to become a state constraint in entire track optimizing problem.Selection for model, it is necessary to meet on the whole
The gabarit of constraint, and there is the ability that can portray most raised obstacles.Herein, using the method for convex optimization come into
Row obstacle avoidance also needs to consider selected 3-D geometric model, after mathematical description and certain mathematic(al) manipulation, energy
Enough and energy be converted to convex constraint more conveniently.Menology protrusion obstacle is described as pyramid type constraint by comprehensive considerations above,
As shown in Figure 2.
(a), (b) shown in Fig. 2, (c) respectively indicate three kinds of unacceptable the barrer types, and (a) is big apex angle obstacle, Qi Banding
Angle is larger, and general 30 ° or more；It (b) is small apex angle obstacle, general 30 ° or less；(c) be then (b) a kind of special case, when circular cone
Height acquirement is very big, and much higher than the height for the spatial point that lander may pass through, and its semiapex angle obtains very little, at this moment, circular cone
The constraint of type can approximately regard that cylindrical type constrains as again, for describing the raised obstacle that those are approximately cylinder.Pass through tune
The height and semiapex angle size of full circle cone carry out the raised obstacle of the overwhelming majority on approximate description menology, it can be seen that hinder protrusion
Hinder be described as pyramid type constraint there is versatility.
Raised Disorder Model mathematical description
After lander scan process, can obtain these raised obstacles position and rough elevation information,
Middle elevation information do not need accurately to know, generally takes a more conservative estimated value；The semiapex angle of circular cone can also be according to need
The safety margin wanted takes an estimated value.In short, the circular cone volume ratio practical obstacle volume taken is bigger, safety margin is higher,
More can obstacle avoidance, but can also reduce lander can flight space size.
If cone height be H_h, semiapex angle α,In the fixed landing coordinate system of menology, conical tip is sat
It is designated as H=(H_h H_y H_z)T, any time, coordinate of the lander in menology fixed coordinate system is P=(x y z)T, then
The cosine value of lander and the angle by the high unit vector of circular cone are as follows:
Wherein n is the unit vector on cone height direction, i.e. n=(1 0 0)T；β is the folder of P-H vector and n vector
Angle.
Lander is set to avoid these raised obstacles, the flight path of lander cannot pass through these circular cones, such as Fig. 3 and 4
Shown, lander cannot be below horizontal plane flight first, followed by its line with obstacle vertex must circle element of a cone it
Outside, then it needs to meetI.e.
(6) formula deform
Since the final landing point of lander is lower than the height of obstacle certainly,ToFormula
(7) just become
Notice that the described constraint of formula (8) is a nonconvex property constraint, the addition of this constraint will be so that entire track
Optimization and avoidance problem can not be solved with convex optimization method, therefore, it is necessary to be carried out convexification conversion.
The conversion of Obstacles Constraints convexification
Convexification conversion is carried out to nonconvex property Obstacles Constraints obtained above, is asked so that entire problem turns to a convex optimization
Topic, it is more exact that turning to a Second-order cone programming problem.
It is to take single order to the norm constraint contained that by formula (8), this non-convex constraint, which turns to the general thought of convexity constraint,
Taylor expansion item makes entirely constraint become a linear restriction, since linear restriction is one kind of convex constraint, also just will
Obstacles Constraints are converted to convex constraint.
For convenience of solution, first vector is write as component form:
P-H=(x-H_h y-H_y z-H_z)
If f (P)=norm (P-H),
F (P) can be obtained to component first derivative each in P and to the first derivative f ' (P) of P,
Continue derivation, obtaining its second dervative is
F " (P) is a Hessian matrix, wherein
The first order Taylor of available f (P)
In above formula, 0 < ξ < 1, Pξ=P0+ξ(P-P0).Its single order linear term is taken to obtain to (14)
F (P)=f (P0)+f′(P0)(P-P0) (15)
At this moment, formula (8) constraint becomes a linear restriction
In the case where ignoring second-order remained, linear restriction described in (16) formula, the P in formula are obtained0Can take does not have
The lander position at the optimization each timing node in track obtained when avoidance constraint is added, takes P in this way0It can reduce as much as possible
Remainder, the linear restriction after making conversion are more constrained close to original.
Approach phase landing fuel optimal problem after Obstacles Constraints are added
DefinitionShared vc=n=[1 0 0]T, then constraint can be write as
Wherein,
The constraint is added in former problem, obtains that existing fuel is optimal, and can be realized the rail of three-dimensional space avoidance
Mark optimization problem:
Target function:
Meet:
Avoidance method of guidance considers future under the more Obstacles Constraints environment of a kind of objects outside Earth soft landing proposed by the present invention
The reliability and safety of moon exploration task, pinpoint soft landing and effective obstacle avoidance will play conclusive work
With.Have correlation technique before and the optimum value derivation algorithm based on convex programming is proposed with regard to accurate soft landing problem, the present invention
, for the optimal Guidance problem under more Obstacles Constraints environment, a kind of changing based on convex optimization will be proposed on the basis of the former
Into optimal soft landing guidance algorithm.
Main advantages of the present invention are embodied in: being given the circular cone geometry teaching model of menology protrusion obstacle, given this
The mathematical description of circular cone Disorder Model, then by non-convex to linear transfor, so that circular cone Obstacles Constraints are dissolved into second order cone rule
In the problem of drawing, accomplishes to consider simultaneously that fuel is optimal and obstacle avoidance, the obstacle avoidance of three-dimensional space may be implemented, efficiently use
Around and above obstacle can flight space, be no longer the realization obstacle avoidance in the form of plane restriction, waste obstacle about
The upper space of beam.
Innovative point of the present invention is: analysis modeling carried out for the raised obstacle on ground day surface, and completes linear transfor, it will
Non-convex constraint is converted into convex constraint, is dissolved among Second-order cone programming problem, and establishes the optimal of complete consideration Obstacles Constraints
Second-order cone programming model demonstrates the correctness of method by the simulation analysis of three kinds of different type Obstacles Constraints, has and actually answer
With value.
The present invention firstly, establishing the optimal Second-order cone programming model of fuel that Obstacles Constraints are not added, and is standardized；Its
It is secondary, analysis modeling is carried out for martian surface protrusion obstacle, and carry out linear transformation, converts convex constraint for non-convex constraint, melt
Enter among Second-order cone programming problem, and establishes the optimal Second-order cone programming model of complete consideration Obstacles Constraints；Finally by
The simulation analysis of three kinds of different type Obstacles Constraints carrys out the correctness of verification algorithm.It applies it in soft lunar landing research,
Simulation result shows that new algorithm realizes effectively evading for three-dimensional space obstacle, is rationally utilized around and above obstacle and flies
Row space, while it is optimal to meet fuel.
To objects outside Earth, typically raised obstacle has carried out mathematical modeling to the present invention；Non-linear arrive has been carried out to raised obstacle
Linear conversion, it is non-convex to convex conversion；The Approach phase landing fuel optimal problem after Obstacles Constraints are added is established, to tradition
Soft landing optimal Guidance problem propose new thinking；By the selection to three kinds of raised Obstacles Constraints, mentioned side is demonstrated
The correctness of method and feasibility in use.
Detailed description of the invention
Fig. 1 is menology fixed coordinate system, and Fig. 2 is restricted model schematic diagram, and (a), (b), (c) respectively indicate three kinds in Fig. 2
Unacceptable the barrer types, (a) are big apex angle obstacle, (b) are small apex angle obstacle, are then (c) a kind of special cases of (b).Fig. 3 constraint
Model schematic；It can flight range comparison before Fig. 4 approximate processing and processing；Fig. 5 is speed change curves figure；Fig. 6 is reasoning valve
Gate koji-making line chart；Fig. 7 is no avoidance track top view；Fig. 8 is that have avoidance track top view；Fig. 9, which is that 1 avoidance of obstacle is contour, to be dissipated
Point diagram；Figure 10 is obstacle 1 without the high scatter plot such as avoidance；Figure 11 is the high scatter plots such as 2 avoidance of obstacle；Figure 12 is obstacle 2 without avoidance
Contour scatter plot.
Specific embodiment
Specific embodiment 1: as shown in Figures 2 to 4, a kind of more obstacles of objects outside Earth soft landing described in present embodiment
The realization process of barrier-avoiding method under constraint environment are as follows:
Step 1: the raised obstacle for objects outside Earth surface is analyzed and constructs raised obstacle mathematical model；
Step 2: the raised obstacle mathematical model is carried out linear transfor, convex constraint is converted by non-convex constraint；
Step 3: the raised obstacle mathematical model after linear transfor is dissolved among Second-order cone programming problem, and establish
The optimal Second-order cone programming model of complete consideration Obstacles Constraints；
Step 4: more using the optimal Second-order cone programming model realization objects outside Earth soft landing of complete consideration Obstacles Constraints
Optimal avoidance under Obstacles Constraints environment.
Specific embodiment 2: in present embodiment, the raised obstacle described in step 1 for objects outside Earth surface
It is analyzed and constructs raised obstacle mathematical model, detailed process are as follows:
Step 1 one, raised Disorder Model selection
The raised Disorder Model of selection meets the gabarit of constraint on the whole, and hinders with that can portray most protrusions
The ability hindered；
Obstacle avoidance is carried out using the method for convex optimization, selected 3-D geometric model can be converted to convex constraint,
Moonscape protrusion obstacle is described as pyramid type constraint；
By adjusting the height and semiapex angle size of circular cone, carry out the raised obstacle of the overwhelming majority on approximate description menology, it will
It is with versatility that raised obstacle, which is described as pyramid type constraint,；
Step 1 two, raised Disorder Model mathematical description
After lander scan process, position and the elevation information of raised obstacle are obtained；
The semiapex angle of circular cone safety margin as needed takes an estimated value；If cone height be H_h, semiapex angle α,In the fixed landing coordinate system of menology, conical tip coordinate is H=(H_h H_y H_z)T, any time,
Coordinate of the land device in menology fixed coordinate system is P=(x y z)T, then lander and the folder by the high unit vector of circular cone
Cosine of an angle value are as follows:
Wherein n is the unit vector on cone height direction, i.e. n=(1 0 0)T；β is the folder of P-H vector and n vector
Angle；
Lander is set to avoid the raised obstacle, the flight path of lander cannot pass through these circular cones, and lander is first
It first cannot be below horizontal plane flight, followed by it then must need to meet with the line on obstacle vertex except circle element of a coneI.e.
(6) formula deform
Since the final landing point of lander is lower than the height of obstacle certainly,ToFormula
(7) just become
The described constraint of formula (8) is a nonconvex property constraint, and the addition of nonconvex property constraint will be so that entire track optimizing
And avoidance problem can not be solved with convex optimization method, therefore, it is necessary to be carried out convexification conversion.
Other steps are same as the specific embodiment one.
Specific embodiment 3: as shown in Figures 2 to 4, present embodiment is described by the raised obstacle in step 2
Mathematical model carries out linear transfor, converts convex constraint (conversion of Obstacles Constraints convexification), detailed process for non-convex constraint are as follows:
Convexification conversion is carried out to nonconvex property Obstacles Constraints, so that entire problem turns to a convex optimization problem, it is more acurrate
Be to turn to a Second-order cone programming problem；
It is that one is taken to the norm constraint contained by the general thought that non-convex constraint turns to the constraint of a convexity shown in formula (8)
Rank Taylor expansion item makes entirely constraint become a linear restriction, since linear restriction is one kind of convex constraint, also just will
Obstacles Constraints are converted to convex constraint；
For convenience of solution, first vector is write as component form:
P-H=(x-H_h y-H_y z-H_z)
If f (P)=norm (P-H),
F (P) can be obtained to component first derivative each in P and to the first derivative f ' (P) of P,
Continue derivation, obtaining its second dervative is
F " (P) is a Hessian matrix, wherein
The first order Taylor of available f (P)
In above formula, 0 < ξ < 1, Pξ=P0+ξ(P-P0)；Its single order linear term is taken to obtain to (14)
F (P)=f (P0)+f′(P0)(P-P0) (15)
At this moment, formula (8) constraint becomes a linear restriction
In the case where ignoring second-order remained, linear restriction described in (16) formula, the P in formula are obtained0It takes and is not added
The lander position at the optimization each timing node in track that avoidance obtains when constraining, takes P in this way0It can reduce as much as possible remaining
, the linear restriction after making conversion is more constrained close to original.
F (P) is the Taylor expansion of pure mathematics, is explained without what meaning.Here P is exactly to represent lander to consolidate in menology
Three-dimensional coordinate under position fixing system.f″(Pξ) it is exactly 2 order derivatives；ξ(P-P0) it is in P0A certain neighborhood in.
Other steps are the same as one or two specific embodiments.
Specific embodiment 4: present embodiment is in step 3, the raised obstacle mathematics by after linear transfor
Model is dissolved among Second-order cone programming problem, and establishes the optimal Second-order cone programming model of complete consideration Obstacles Constraints, tool
Body process are as follows:
DefinitionShared vc=n=[1 0 0]T, then constraint is write as
Formula (17) is exactly to derive to the deformation constrained above, and Q is intermediate variable, and it is following described for being worth；
Wherein,
The constraint is added in former problem, obtains that fuel is optimal, and realizes that the track optimizing of three-dimensional space avoidance is asked
The target function of topic:
Target function:
Meet:
Wherein:
Eu=[I3 03×1],Ex=[I6 06×1],γk=[04×4k I6 04×4(n-k)]4×4(n+1), (k=0,1 ..., n)
A∈R7×7,B∈R7×4
Other steps are identical as specific embodiment one, two or three.
The present invention and its technical effect generated are described in more detail with reference to the accompanying drawing:
Simulation parameter design
It is emulated using MATLAB yalmip Optimization Toolbox.Simulation parameter such as table 1.
The setting of 1 initial parameter of table
Table1 Initial parameters:
Take two obstacles as multiple obstacle simulation examples here.It is as follows:
Obstacle 1: apex coordinate [500 40-83], 10 ° of obstacle semiapex angle, relaxation factor 1.
Obstacle 2: apex coordinate [1500 50-350], 10 ° of obstacle semiapex angle, relaxation factor 1.
Analysis of simulation result
Fig. 5,6 give the velocity variations and thrust valve door controlling curve of lander.Final hovering coordinate is
[30.8091 -0.477208 -0.266704] m, speed are [- 0.49179 0.028592 0.081788]m/s.Meet the requirements model
It encloses, it was demonstrated that the correctness of constraint.
Fig. 7,8 give not plus Obstacles Constraints and have added Obstacles Constraints optimization track in three dimensions and its plane
Projection.It apparent can be observed from two top views, before adding avoidance, the journey overwhelming majority is same behind original optimization track
In one plane, two size obstacles are passed through, are not passed through the center of obstacle certainly；In addition avoidance optimization track exists after avoidance
Journey not in the same plane, but makees to evade among two size obstacles flight afterwards, and substantially it is seen that with around
Winged form carrys out obstacle avoidance.
Four contour scatter plots shown in Fig. 9~12, which give, to be added avoidance and does not add avoidance flight path respectively with respect to two
The change in location of obstacle.Not plus before avoidance, the point between x=961.5261 to x=673.482 is inside obstacle 2, x=
366.541 to the point between x=186.1168 inside obstacle 1, namely former optimization track has greatly in two barriers
Hinder inside；In addition all the points press the tendency direction of each node all except the circle of corresponding height after avoidance, adjacent two
The line of point does not all have intersection point with inner circle (the smaller circle of line side radius), namely does not have two o'clock line across obstacle
Situation illustrates that lander successfully avoids obstacle, and flight path is bonded compared with obstacle, and taking full advantage of can flight space.
Can to sum up obtain, by the way that conical Obstacles Constraints are added, using Second-order cone programming method, realize fuel it is optimal and
Two purposes of obstacle avoidance.Emulation gives the simulation result of three kinds of small apex angle, big apex angle, more obstacles different type obstacles, from
Simulation result above finds out that this method is more satisfactory.
In addition, the very low obstacle of some height can also use model above to add entire optimization problem as Obstacles Constraints
In, but general restriction acts on very little, what effect do not had in exhausted big a part of flight course, is only contacted to earth in lander
One section of preceding track can just play effect of contraction, and at this moment these constraints can be handled as plane restriction substantially.
Claims (1)
1. barrier-avoiding method under a kind of more Obstacles Constraints environment of objects outside Earth soft landing, which is characterized in that the realization of the method
Journey are as follows:
Step 1: the raised obstacle for objects outside Earth surface is analyzed and constructs raised obstacle mathematical model；
Step 2: the raised obstacle mathematical model is carried out linear transfor, convex constraint is converted by non-convex constraint；
Step 3: the raised obstacle mathematical model after linear transfor is dissolved among Second-order cone programming problem, and establish complete
The considerations of Obstacles Constraints optimal Second-order cone programming model；
Step 4: utilizing the more obstacles of optimal Second-order cone programming model realization objects outside Earth soft landing of complete consideration Obstacles Constraints
Optimal avoidance under constraint environment；
Raised obstacle mathematical model is analyzed and constructed to raised obstacle described in step 1 for objects outside Earth surface, tool
Body process are as follows:
Step 1 one, raised Disorder Model selection
The raised Disorder Model of selection meets the gabarit of constraint on the whole, and has and can portray most raised obstacles
Ability；
Obstacle avoidance is carried out using the method for convex optimization, selected 3-D geometric model can be converted to convex constraint, by the moon
Ball surface protrusion obstacle is described as pyramid type constraint；
By adjusting the height and semiapex angle size of circular cone, carry out the raised obstacle of the overwhelming majority on approximate description menology, it will be raised
It is with versatility that obstacle, which is described as pyramid type constraint,；
Step 1 two, raised Disorder Model mathematical description
After lander scan process, position and the elevation information of raised obstacle are obtained；
The semiapex angle of circular cone safety margin as needed takes an estimated value；If cone height be H_h, semiapex angle α,In the fixed landing coordinate system of menology, conical tip coordinate is H=(H_h H_y H_z)T, any time,
Coordinate of the land device in menology fixed coordinate system is P=(x y z)T, then lander and the folder by the high unit vector of circular cone
Cosine of an angle value are as follows:
Wherein n is the unit vector on cone height direction, i.e. n=(1 0 0)T；β is the angle of P-H vector and n vector；
Lander is set to avoid the raised obstacle, the flight path of lander cannot pass through these circular cones, and lander is first not
It can fly lower than horizontal plane, followed by it then must need to meet with the line on obstacle vertex except circle element of a coneI.e.
(6) formula deform
Since the final landing point of lander is lower than the height of obstacle certainly,ToFormula (7) is just
Become
The described constraint of formula (8) is a nonconvex property constraint, and the addition of nonconvex property constraint will be so that entire track optimizing and keep away
Barrier problem can not be solved with convex optimization method, therefore, it is necessary to be carried out convexification conversion；
It is described that the raised obstacle mathematical model is subjected to linear transfor in step 2, convex constraint is converted by non-convex constraint,
Detailed process are as follows:
Convexification conversion is carried out to nonconvex property Obstacles Constraints, so that entire problem turns to a convex optimization problem, it is more exact that
Turn to a Second-order cone programming problem；
It is to take single order safe to the norm constraint contained by the general thought that non-convex constraint shown in formula (8) turns to the constraint of a convexity
Expansion item is strangled, so that entirely constraint is become a linear restriction, since linear restriction is one kind of convex constraint, also just by obstacle
Constraints conversion is at convex constraint；
For convenience of solution, first vector is write as component form:
P-H=(x-H_h y-H_y z-H_z)
If f (P)=norm (P-H),
F (P) can be obtained to component first derivative each in P and to the first derivative f ' (P) of P,
Continue derivation, obtaining its second dervative is
F " (P) is a Hessian matrix, wherein
The first order Taylor of available f (P)
In above formula, 0 < ξ < 1, Pξ=P0+ξ(P-P0)；Its single order linear term is taken to obtain to (14)
F (P)=f (P0)+f′(P0)(P-P0) (15)
At this moment, formula (8) constraint becomes a linear restriction
In the case where ignoring second-order remained, linear restriction described in (16) formula, the P in formula are obtained0It takes and avoidance is not added about
The lander position at the optimization each timing node in track obtained when beam, takes P in this way0Remainder can be reduced as much as possible, make to turn
Linear restriction after changing more is constrained close to original；
In step 3, the raised obstacle mathematical model by after linear transfor is dissolved among Second-order cone programming problem, and
Establish the optimal Second-order cone programming model of complete consideration Obstacles Constraints, detailed process are as follows:
DefinitionShared vc=n=[1 0 0]T, then constraint is write as
Wherein,
The constraint is added in former problem, obtains that fuel is optimal, and realize the track optimizing problem of three-dimensional space avoidance
Target function:
Target function:
Meet:
||SΨkp+S(Φky0+Λkg4)||≤-cTΨkp-cT(Φky0+Λkg4)
Wherein:
C=[- tan (θalt) 0 0 0 0 0 0]T
eσ=[01×31]T,ez=[01×6 1]T,eh=[1 01×6]T,Eu=
[I3 03×1],Ex=[I6 06×1],Υk=[04×4k I6 04×4(n-k)]4×4(n+1), (k=0,1 ..., n) A ∈ R7×7,B∈R7×4。
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| CN113467241B (en) * | 2021-07-06 | 2022-09-16 | 北京理工大学 | Method for optimizing burn-up of convex curvature landing track |
| CN113917943B (en) * | 2021-10-14 | 2022-07-12 | 哈尔滨工业大学 | Moon soft landing optimal guidance method and system based on safe landing passage and storage medium |
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