CN112051854B - Rapid planning method for optimal trajectory of lunar soft landing considering complex constraints - Google Patents

Rapid planning method for optimal trajectory of lunar soft landing considering complex constraints Download PDF

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CN112051854B
CN112051854B CN202011005654.5A CN202011005654A CN112051854B CN 112051854 B CN112051854 B CN 112051854B CN 202011005654 A CN202011005654 A CN 202011005654A CN 112051854 B CN112051854 B CN 112051854B
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trajectory
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CN112051854A (en
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吴伟仁
崔平远
尚海滨
张哲�
赵梓辰
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Beijing Institute of Technology BIT
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00—Control of position, course or altitude of land, water, air, or space vehicles, e.g. automatic pilot
    • G05D1/02—Control of position or course in two dimensions
    • G05D1/021—Control of position or course in two dimensions specially adapted to land vehicles
    • G05D1/0212—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory
    • G05D1/0223—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory involving speed control of the vehicle
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00—Control of position, course or altitude of land, water, air, or space vehicles, e.g. automatic pilot
    • G05D1/02—Control of position or course in two dimensions
    • G05D1/021—Control of position or course in two dimensions specially adapted to land vehicles
    • G05D1/0212—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory
    • G05D1/0221—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory involving a learning process

Abstract

The invention discloses a rapid planning method for an optimal trajectory of lunar soft landing considering complex constraints, and belongs to the technical field of aerospace. The implementation method of the invention comprises the following steps: discretizing the lunar soft landing trajectory planning problem, and converting the trajectory planning problem into a nonlinear planning problem of finite dimensional variables and constraints; carrying out equivalent transformation on the constraint in the nonlinear programming problem by using saturated function transformation, and converting the nonlinear inequality constraint into equality constraint under an expanded state space; and carrying out convex processing on the constraint of the nonlinear programming problem after the equivalent transformation, adding the confidence interval of the expanded state variable as a penalty term into the performance index, improving the convergence of sequence iteration, and finally solving a convex subproblem by applying a convex optimization solver sequence until a converged moon soft landing optimal trajectory is obtained, so that the fast planning of the moon soft landing optimal trajectory is realized, and the planning efficiency, the robustness and the constraint processing capability of the moon soft landing optimal trajectory are improved.

Description

Rapid planning method for optimal trajectory of lunar soft landing considering complex constraints
Technical Field
The invention relates to a deep space celestial body soft landing attitude and orbit optimal trajectory planning method, which is particularly suitable for a planning method of a lunar soft lander optimal trajectory considering complex constraints and belongs to the technical field of aerospace.
Background
The soft landing is a key technology for realizing lunar surface exploration and has important application value in a lunar exploration task. Because the moon has no atmosphere, the soft landing of the moon needs to be realized by braking and decelerating by using a reverse thrust engine, so that the lander can land on the surface of the moon safely. On one hand, the soft landing process of the moon is limited by complex constraints such as energy, dynamics, attitude and orbit maneuvering coupling and the like, and how to plan to obtain the optimal landing track under the complex constraints is a difficult point to solve; on the other hand, in order to improve the safety and reliability of the soft landing of the moon, how to realize the fast planning of the soft landing trajectory so as to deal with the emergency in the landing process is also a key problem to be solved. The planning of the moon soft landing trajectory is one of the hot problems concerned by current technologists.
In the developed moon soft Landing Trajectory planning method, in the prior art [1] (Cho D H, Kim D, Leeghim H, optical Lunar Landing Trajectory Design for Hybrid Engine [ J ]. physical schemes in Engineering,2015), an indirect Trajectory planning method is proposed for a moon Landing task, which is based on the Pompe's value principle to derive the Optimal control law and co-state function relationship, and the co-state initial value is obtained by solving the problem of two-point edge values, so as to determine the Optimal control law and Landing Trajectory. The method has the advantages that the calculation precision is high, and the obtained optimal track strictly meets the necessary condition of first-order optimality. The method has the defects that the initial value of the co-state is difficult to guess, so that the convergence of a numerical value solving algorithm is difficult, and the reliability of the solving process cannot be ensured. Meanwhile, the method cannot handle complex task constraints such as attitude and orbit maneuvering coupling in the soft landing process.
In the prior art [2] (Remesh N, Raman R V, Lalithambika V R, Fuel optimal Lunar Soft mapping Design Using differential Solution Schemes [ J ]. International Review of Aerospace Engineering,2016), a direct Trajectory planning method Using nonlinear programming is proposed for the Lunar Soft Landing mission. The method has the advantage that the task constraint in the track planning process can be well processed. The method has the disadvantages of large non-linear planning calculation amount and low track planning efficiency.
Disclosure of Invention
In order to solve the problems of low efficiency, poor robustness, weak constraint processing capability and the like frequently occurring when the optimal trajectory planning is carried out on the moon by the conventional method, the technical problems to be solved by the invention are as follows: the method realizes the rapid planning of the optimal trajectory of the soft landing of the moon under the condition of considering complex constraints, and improves the planning efficiency, robustness and constraint processing capability of the optimal trajectory of the soft landing of the moon.
The purpose of the invention is realized by the following technical scheme:
the invention discloses a method for rapidly planning an optimal trajectory of a lunar soft landing by considering complex constraints, which comprises the following steps: firstly, discretizing a lunar soft landing trajectory planning problem, and converting the trajectory planning problem into a nonlinear planning problem of finite dimensional variables and constraints; then, carrying out equivalent transformation on the constraint in the nonlinear programming problem by utilizing a saturated function transformation technology, converting the nonlinear inequality constraint in the nonlinear programming problem into an equality constraint under an expanded state space, and making a foundation for subsequent operation of convex and sequence solution; thirdly, carrying out convex processing on the constraint of the nonlinear programming problem after the equivalent transformation, adding the confidence interval of the expanded state variable as a penalty term into the performance index, further improving the convergence of the sequence iteration, and finally solving a convex subproblem by applying a convex optimization solver sequence until a converged moon soft landing optimal trajectory is obtained, namely, realizing the fast planning of the moon soft landing optimal trajectory under the condition of considering the complex constraint, and improving the planning efficiency, robustness and constraint processing capability of the moon soft landing optimal trajectory.
The invention discloses a method for rapidly planning an optimal trajectory of a lunar soft landing by considering complex constraints, which comprises the following steps:
step one, discretizing the lunar soft landing trajectory planning problem, and converting the trajectory planning problem into a nonlinear planning problem of finite dimensional variables and constraints.
Defining the starting time and the ending time of the soft landing of the moon as t0And tfFor a time interval [ t0,tf]And performing N equal division (N is a positive integer). Defining the time, the lander state and the control quantity at the discrete nodes as ti，xiAnd uiWherein i is 0,1,2, … N.
Directly carrying out discretization processing on discrete nodes to obtain a discretization form of constraint functions s (x, u, t) not containing differential terms and performance indexes J (x, u) in the trajectory planning problem, wherein the constraint functions s (x, u, t) are less than or equal to 0
si(xi,ui,ti)≤0,i＝0,1,2,…N
J＝J(x0,x1,…,xN,u0,u1,…,uN)
Discretizing the lander dynamic model containing the differential terms by using a point matching method to obtain the following equation form constraint
Figure BDA0002695786000000021
Wherein: h represents the time difference between two adjacent discrete points, and f is a right function of the lander attitude and orbit dynamics equation.
Definition z ═ x0,x1,…,xN]And η ═ u0,u1,…,uN]Further, all equality constraints w (z, eta) ═ 0 in the trajectory planning problem are decomposed into two inequality constraint forms as follows
Figure BDA0002695786000000022
Expressing the performance index and all constraints of the trajectory planning problem as functions of state quantity and control quantity on discrete points to obtain a nonlinear planning problem of finite dimensional variables and constraints, which is described as follows:
minimize J(z,η)
subject to g(z,η)≤0
wherein: where g is the set of all inequality constraints in the trajectory planning problem.
Through discretization processing, the track planning problem is converted into the state quantity z on the optimal discrete node*And a control quantity η*The performance index set by the lunar soft landing task is minimized.
In the first step, discretization is carried out by adopting a fitting method, preferably, the fitting method carries out discretization on the fitting by applying a trapezoidal formula, and then the planning speed is improved on the premise of ensuring the precision.
And step two, carrying out equivalent transformation on the constraint in the nonlinear programming problem, and converting the nonlinear inequality constraint in the nonlinear programming problem into an equality constraint under an expanded state space, so that the problem in the step three is conveniently emphasized.
The value range of inequality constraint g in the nonlinear programming problem is [ - ∞,0 [ ]]Selecting monotonous increasing and smooth, and the value range is [ - ∞,0 [ ]]Saturation function in range
Figure BDA0002695786000000031
Varying the inequality constraints in the nonlinear programming problem into
Figure BDA0002695786000000032
Wherein: xi ∈ [ - ∞, + ∞ ] is an unconstrained extension variable.
By the above transformation, inequality constraints in the nonlinear programming problem are transformed into equality constraints. It should be noted that whatever the value of the original inequality constraint g (z, η), the extended variable ξ can always be found to hold the new equality constraint, so the transformation is equivalent.
Through an equivalent transformation, a new nonlinear programming problem can be described as
Figure BDA0002695786000000033
Through discretization processing, the track planning problem is converted into the state quantity z on the optimal discrete node*Control quantity eta*And extended state quantity xi*The performance index required by the lunar soft landing trajectory is minimized.
And step three, carrying out convex processing on the constraint of the nonlinear programming problem transformed in the step two, and adding the expanded confidence interval of the state variable as a penalty term into the performance index, thereby improving the convergence of the sequence convex optimization algorithm in the step four.
And (4) based on the current reference track, utilizing Taylor first-order expansion to expand the constraint in the nonlinear programming problem obtained in the step two, and obtaining the approximate convex form of the constraint
Figure BDA0002695786000000034
Wherein:
Figure BDA0002695786000000035
and
Figure BDA0002695786000000036
the state quantity, the control quantity and the expansion variable at each discrete point of the current reference track are obtained.
When the first step iteration solution of the track planning problem is carried out, the reference track is guessed and given by the initial value, and when the subsequent iteration solution is carried out, the reference track is selected as the track obtained by the previous step iteration.
If the performance index of the nonlinear programming problem is a non-convex function, the non-convex function also needs to be subjected to Taylor first-order expansion and expressed as a convex form
Figure BDA0002695786000000037
In addition, in order to improve the convergence of iterative solution, an extension variable is introduced into a performance index for punishment, and a new performance index is obtained as
Figure BDA0002695786000000038
Through the convex processing, the current convex sub-problem can be obtained as
Figure BDA0002695786000000041
And step four, solving the convex subproblem by using a convex optimization solver sequence until a converged moon soft landing optimal trajectory is obtained, namely realizing the rapid planning of the moon soft landing optimal trajectory under the condition of considering complex constraints, and improving the planning efficiency, robustness and constraint processing capacity of the moon soft landing optimal trajectory.
Solving the convex subproblem obtained in the fourth step by using a convex optimization solver to obtain an optimal solution (z) of the convex subproblemk,ηk,ξk)。
Defining a convergence tolerance epsilon0The convergence condition is
Figure BDA0002695786000000042
If ε > ε0Then the optimal solution (z) of the convex sub-problem is utilizedk,ηk,ξk) Steps three and four are repeated instead of the reference track.
If ε is not more than ε0Then the iterative computation terminates and the optimal solution (z) for the current convex subproblem is obtainedk,ηk,ξk) Namely the optimal solution of the lunar soft landing trajectory planning problem is output
(z*,η*,ξ*)＝(zk,ηk,ξk)
And (3) until the converged optimal trajectory of the soft landing of the moon is obtained, namely, the rapid planning of the optimal trajectory of the soft landing of the moon is realized under the condition of considering complex constraints, and the planning efficiency, robustness and constraint processing capacity of the optimal trajectory of the soft landing of the moon are improved.
Has the advantages that:
1. the invention discloses a rapid planning method for an optimal trajectory of lunar soft landing considering complex constraints, which comprises the steps of firstly carrying out discretization processing on a lunar soft landing trajectory planning problem, converting the trajectory planning problem into a nonlinear planning problem of finite dimensional variables and constraints, then carrying out equivalent transformation on the constraints in the nonlinear planning problem by using a saturation function, converting nonlinear inequality constraints in the nonlinear planning problem into equality constraints under an expanded state space, wherein the saturation function can effectively process strong nonlinear constraints of attitude-orbit maneuvering coupling, dynamics and the like, and further prepares for subsequent convex operation and sequential convex optimization solution. Further, carrying out convex processing on the constraint of the transformed nonlinear programming problem, and adding the confidence interval of the expanded state variable as a penalty term into the performance index, thereby improving the convergence of the sequence convex optimization algorithm; and finally, solving the convex subproblem by using a convex optimization solver sequence until a converged moon soft landing optimal trajectory is obtained, namely, realizing the rapid planning of the moon soft landing optimal trajectory under the condition of considering complex constraints, and improving the planning efficiency, robustness and constraint processing capacity of the moon soft landing optimal trajectory.
2. The invention discloses a rapid planning method for an optimal trajectory of a lunar soft landing considering complex constraints, which processes complex strong nonlinear constraints through equivalent transformation and relaxation capacity of a saturation function, and the problem of a lunar soft landing trajectory convex sub-problem always has a feasible solution. Meanwhile, the efficient solving capability of the convex optimization solver is utilized to realize the sequence solution of the track planning problem. Due to the fact that the saturation function has the gradual saturation characteristic, the convergence of the sequence convex optimization algorithm is remarkably improved, namely the sensitivity of initial value guessing of the sequence convex optimization algorithm is reduced, complex types of constraints such as attitude-orbit coupling can be handled, and meanwhile the method has high-efficiency solving speed.
Drawings
FIG. 1 is a schematic flow chart of a method for rapidly planning an optimal trajectory of a lunar soft landing in consideration of complex constraints according to the present invention;
FIG. 2 is a simulation diagram of a three-dimensional lunar soft landing trajectory and an initial reference trajectory
FIG. 3 is a simulation diagram of constraint of two-dimensional trajectory and space obstacle for lunar soft landing
Detailed Description
For a better understanding of the objects and advantages of the present invention, reference should be made to the following detailed description taken in conjunction with the accompanying drawings and examples.
Example 1:
the method for rapidly planning the optimal trajectory of the lunar soft landing in consideration of the complex constraint disclosed by the embodiment comprises the following specific steps:
step one, discretizing the lunar soft landing trajectory planning problem, and converting the trajectory planning problem into a nonlinear planning problem of finite dimensional variables and constraints.
The surface of a given moon of the mission is a landing reference surface, the Z-axis direction is parallel to the earth center distance vector, the north and east directions of the moon surface are respectively consistent with the directions of the X-axis and the Y-axis, namely the XY plane is a moon landing track reference surface. Setting the initial intersatellite point of the lander as the center of an XY plane, setting the initial height as 15km, namely setting the initial position of the lander as [0, 0, 15km ], and setting the initial speed in the corresponding direction as [0, 1.2, 1] km/s respectively. The task requires the lander to realize the lunar surface soft landing, namely the speed of reaching the lunar surface is zero, the landing target position given by the task is [0, 40, 25] km, and the target speed is [0, 0, 0] km/s. Discretizing the lander track from an initial state to a target state, selecting N-50 discrete points, and obtaining the discrete variables to be optimized of the lander.
Rn＝R(tn),Vn＝V(tn)
Qn＝Q(tn),Ωn＝Ω(tn),n＝1,2,...,N
Wherein: rn，Vn，Qn，ΩnRespectively representing the discretized position, speed, attitude angle and angular velocity vector of the lander, tnRepresenting the time values corresponding to the discrete points. Given an initial time t0Terminal time t of 0sf130s, then there are
Figure BDA0002695786000000051
Translation kinetic equation considering lunar soft landing orbit
Figure BDA0002695786000000052
Figure BDA0002695786000000053
Wherein: r and V respectively represent the position and the velocity vector of the lander, and mu is 4902.8026km3/s2Representing the moon center gravity constant, U represents the center of mass translational acceleration vector imposed by the lander, which needs to be coupled with the lander attitude dynamics, i.e. satisfying the following equality constraints
U/||U||＝Qu
Wherein: quRepresenting the fixed pointing direction of the landing gear body coordinate system, defining the attitude angle of the landing gear as Q, the corresponding attitude angular velocity as omega, the torque applied by the landing gear as T, the rotational inertia of the landing gear as I, and the attitude dynamics of the landing gear is
Figure BDA0002695786000000061
Figure BDA0002695786000000062
Discretizing the kinetic equation by using a trapezoidal discrete formula to obtain
Figure BDA0002695786000000063
Figure BDA0002695786000000064
Figure BDA0002695786000000065
Figure BDA0002695786000000066
Wherein: Δ t ═ tn+1-tnRepresenting the step of time between two adjacent discrete points.
Considering the actual path obstacle avoiding requirement, defining a cylindrical no-fly area, and restraining the lander track to be strictly outside the area, namely the lander state needs to meet the requirement
Figure BDA0002695786000000067
Wherein: rx，RyRepresenting the components of the lander position vector in the X-axis and Y-axis directions, R, respectivelyox，RoyRespectively represent the center coordinates of the bottom surface of the no-fly zone, RcRepresenting the forbidden zone radius. Discretization of the path constraints is similar to the attitude and orbit coupling constraints described above, with discrete points being directly substitutedInto a constrained, discrete form
U(tn)/||U(tn)||＝Qu(tn)
Figure BDA0002695786000000068
In summary, the above equations are integrated into a compact form, and then the original problem is discretized into the following nonlinear programming problem
minimize J＝||R(tN)-Rf||
subject to V(tN)＝Vf,x(t0)＝x0
x(tn+1)-x(tn)＝(f(tn+1)+f(tn))Δt/2,n＝2,...,N
g(tn)≤0,h(tn)＝0,n＝1,2,...,N
Wherein x ═ RT,VT,QT,ΩT]TAnd representing state variables, g and h respectively represent inequality and equality path constraints, and J represents a performance index for minimizing the target terminal error.
Step two, selecting a single boundary saturation function to prepare for the following steps
Before processing the path constraints in the model in the step one, a proper saturation function needs to be selected. For ease of discussion, the saturation function needs to satisfy the zero-point single boundary property, i.e., given the extension variable ξ, the saturation function
Figure BDA0002695786000000069
Then there is
Figure BDA00026957860000000610
In addition, the function needs to have saturation, bijection and monotonicity, i.e. for an arbitrary variable ξ, there is one and only one saturation function value
Figure BDA00026957860000000611
Corresponding thereto, and the gradient of the saturation function is constantly satisfied
Figure BDA00026957860000000612
When ξ → + ∞ is reached, the saturation function value satisfies
Figure BDA0002695786000000071
Numerous studies have shown that the saturation function in exponential form usually has good effect in dealing with specific problems, and in this implementation step, the saturation function in negative exponential is selected
Figure BDA0002695786000000072
And step three, applying the saturation function in the step two, performing equivalent transformation on the nonlinear programming problem in the step one, and converting nonlinear inequality constraints in the nonlinear programming problem into equality constraints under an expanded state space, so that the problem in the step five is conveniently emphasized.
For the nonlinear programming problem described above, its generally compact form is first obtained. Constrain any equation to h i0 is decomposed into two inequality constraints, hiLess than or equal to 0 and-hiLess than or equal to 0. The resulting inequality constraints are then merged with the original inequality constraints, i.e. the original problem is equivalent to a non-linear programming problem containing only inequality path constraints, in its compact form as follows.
minimize J(x)
subject to G(x)≤0
For any component G in the inequality constrainti(x) Less than or equal to 0, taking the saturation function
Figure BDA0002695786000000073
So that it satisfies the following equation
Figure BDA0002695786000000074
Selecting a proper reference track and a proper reference expansion variable, and improving the convergence speed of the sequence convex optimization
At any given discrete point in time tnIn the above, the reference control quantity is defined as constant zero, i.e. the spacecraft is in an uncontrolled landing stage, with
Figure BDA0002695786000000075
Integrating the lander dynamic differential equation to obtain the uncontrolled orbit of the lander
Figure BDA0002695786000000076
Figure BDA0002695786000000077
Note that: in the implementation step, from the angle of robustness test, the state quantity is randomly selected near the uncontrolled track and is used as a reference state track
Figure BDA0002695786000000078
Calculating orbit
Figure BDA0002695786000000079
Corresponding constraint function value Gi(xn) If the track does not satisfy the constraint, an extended variable reference value is taken near the zero point, namely
if Gi(xn)＞0,ξi→0
Otherwise, if the orbit satisfies the constraint, its corresponding saturation function value is calculated, i.e. the value of the saturation function is calculated
Figure BDA00026957860000000710
To sum up, let
Figure BDA00026957860000000711
Then
Figure BDA00026957860000000712
I.e. the taken reference trajectory.
And step five, carrying out convex processing on the constraint of the nonlinear programming problem after conversion in the step three based on the reference track in the step four, and adding the expanded confidence interval of the state variable as a penalty term into the performance index so as to improve the convergence of the sequence convex optimization algorithm in the step six.
At a given reference trajectory
Figure BDA00026957860000000713
And nearby, calculating an approximate form of the discretized moon soft landing trajectory planning model. To constraint
Figure BDA0002695786000000081
Performing first-order Taylor approximation and neglecting high-order terms thereof to obtain linear expansion equality constraint
Figure BDA0002695786000000082
In order to ensure the approximation degree of the constraint and enhance the convergence of the algorithm, a penalty term is introduced
Figure BDA0002695786000000083
Because the original performance index is a convex function, additional convex processing is not needed to be carried out on the original performance index, and the performance index of the approximate problem is changed into the performance index of the approximate problem
Figure BDA0002695786000000084
And step six, solving the convex subproblem by using a convex optimization solver sequence, and quickly obtaining a global optimal solution of the subproblem, namely an approximate solution for realizing the optimal trajectory of the moon soft landing under the condition of considering complex constraint.
Step seven, judging whether the subproblem sequence is converged
Defining a convergence condition of
Figure BDA0002695786000000085
Wherein: epsilon>0 is a given tolerance, if the above formula is satisfied, step eight is executed, if not, the optimal solution { ξ ] obtained in step six is taken*,x*,F*Is a reference track
Figure BDA0002695786000000086
And then jumping to the step five and continuing to solve.
And step eight, outputting the convergence approximate solution obtained in the step seven, namely realizing the optimal trajectory of the lunar soft landing under the condition of considering complex constraint.
Under the condition of an initial reference trajectory given randomly, the lunar soft landing trajectory is solved based on the implementation steps, and as shown in fig. 2, the lander successfully completes a landing task and meets all dynamics and end point constraints. As shown in fig. 3, the lander successfully avoids the cylindrical no-fly zone, and the total calculation time is about 0.7s, so as to meet the requirement of autonomous planning.
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 (2)

1. A method for rapidly planning the optimal trajectory of the lunar soft landing in consideration of complex constraints is characterized by comprising the following steps: comprises the following steps of (a) carrying out,
the method comprises the following steps that firstly, discretization processing is carried out on a lunar soft landing trajectory planning problem, and the trajectory planning problem is converted into a nonlinear planning problem of finite-dimension variables and constraints;
performing equivalent transformation on the constraint in the nonlinear programming problem, and converting the nonlinear inequality constraint in the nonlinear programming problem into equality constraint under an expanded state space, so as to facilitate the projection of the problem in the third step;
step three, carrying out convex processing on the constraint of the nonlinear programming problem transformed in the step two, and adding the confidence interval of the expanded state variable as a penalty term into a performance index, thereby improving the convergence of the sequence convex optimization algorithm in the step four;
solving a convex subproblem by using a convex optimization solver sequence until a converged moon soft landing optimal trajectory is obtained, namely realizing the rapid planning of the moon soft landing optimal trajectory under the condition of considering complex constraints, and improving the planning efficiency, robustness and constraint processing capacity of the moon soft landing optimal trajectory;
the first implementation method comprises the following steps of,
defining the starting time and the ending time of the soft landing of the moon as t0And tfFor a time interval [ t0,tf]Carrying out N equal division; defining the time, the lander state and the control quantity at the discrete nodes as ti，xiAnd uiWherein i is 0,1,2, … N;
directly carrying out discretization processing on discrete nodes to obtain a discretization form of constraint functions s (x, u, t) not containing differential terms and performance indexes J (x, u) in the trajectory planning problem, wherein the constraint functions s (x, u, t) are less than or equal to 0
si(xi,ui,ti)≤0,i＝0,1,2,…N
J＝J(x0,x1,…,xN,u0,u1,…,uN)
Discretizing the lander dynamic model containing the differential terms by using a point matching method to obtain the following equation form constraint
Figure FDA0003083261730000011
Wherein: h represents the time difference between two adjacent discrete points, and f (x, u) is a right function of the lander attitude and orbit dynamics equation;
definition z ═ x0,x1,…,xN]And η ═ u0,u1,…,uN]Further, all equality constraints w (z, eta) ═ 0 in the trajectory planning problem are decomposed into two inequality constraint forms as follows
Figure FDA0003083261730000012
Expressing the performance index and all constraints of the trajectory planning problem as functions of state quantity and control quantity on discrete points to obtain a nonlinear planning problem of finite dimensional variables and constraints, which is described as follows:
minimize J(z,η)
subject to g(z,η)≤0
wherein: wherein g (z, η) is the set of all inequality constraints in the trajectory planning problem;
through discretization processing, the track planning problem is converted into the state quantity z on the optimal discrete node*And a control quantity η*The performance index set by the lunar soft landing task is minimized;
the second step is realized by the method that,
the value range of inequality constraint g in the nonlinear programming problem is [ - ∞,0 [ ]]Selecting monotonous increasing and smooth, and the value range is [ - ∞,0 [ ]]Saturation function in range
Figure FDA0003083261730000021
Varying the inequality constraints in the nonlinear programming problem into
Figure FDA0003083261730000022
Wherein: xi ∈ [ - ∞, + ∞ ] is an unconstrained extension variable;
transforming inequality constraints in the nonlinear programming problem into equality constraints through the transformation; it should be noted that no matter what value the original inequality constraint g (z, η) takes, the extended variable ξ can always be found to make the new equality constraint hold, so the transformation is equivalent;
through an equivalent transformation, a new nonlinear programming problem can be described as
minimize J(z,η,ξ)
Figure FDA0003083261730000023
Through discretization processing, the track planning problem is converted into the state quantity z on the optimal discrete node*Control quantity eta*And extended state quantity xi*The performance index required by the lunar soft landing trajectory is minimized;
the third step is to realize the method as follows,
and (4) based on the current reference track, utilizing Taylor first-order expansion to expand the constraint in the nonlinear programming problem obtained in the step two, and obtaining the approximate convex form of the constraint
Figure FDA0003083261730000024
Wherein:
Figure FDA0003083261730000025
and
Figure FDA0003083261730000026
state quantity, control quantity and expansion variable on each discrete point of the current reference track;
when the first-step iterative solution of the track planning problem is carried out, the reference track is given by initial value guess, and when the subsequent iterative solution is carried out, the reference track is selected as the track obtained by the previous-step iteration;
if the performance index of the nonlinear programming problem is a non-convex function, the non-convex function also needs to be subjected to Taylor first-order expansion and expressed as a convex form
Figure FDA0003083261730000027
In addition, in order to improve the convergence of iterative solution, an extension variable is introduced into a performance index for punishment, and a new performance index is obtained as
Figure FDA0003083261730000028
Through the convex processing, the current convex sub-problem can be obtained as
Figure FDA0003083261730000029
Figure FDA00030832617300000210
The implementation method of the fourth step is that,
solving the convex subproblem obtained in the fourth step by using a convex optimization solver to obtain an optimal solution (z) of the convex subproblemk,ηk,ξk)；
Defining a convergence tolerance epsilon0The convergence condition is
Figure FDA0003083261730000031
If ε > ε0Then the optimal solution (z) of the convex sub-problem is utilizedk,ηk,ξk) Substitute ginsengExamining the orbit, and repeating the third step and the fourth step;
if ε is not more than ε0Then the iterative computation terminates and the optimal solution (z) for the current convex subproblem is obtainedk,ηk,ξk) Namely the optimal solution of the lunar soft landing trajectory planning problem is output
(z*,η*,ξ*)＝(zk,ηk,ξk)
And (3) until the converged optimal trajectory of the soft landing of the moon is obtained, namely, the rapid planning of the optimal trajectory of the soft landing of the moon is realized under the condition of considering complex constraints, and the planning efficiency, robustness and constraint processing capacity of the optimal trajectory of the soft landing of the moon are improved.
2. The method for rapidly planning the optimal trajectory of the lunar soft landing in consideration of the complex constraints as recited in claim 1, wherein: the distribution point method applies a trapezoidal formula to carry out discretization processing on the distribution point method, and then the planning speed is improved on the premise of ensuring the precision.
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