CN113722821B - Projection method for spacecraft intersection docking trajectory planning event constraint - Google Patents
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Abstract
The invention discloses a projection method for planning event constraint of a spacecraft rendezvous and docking trajectory, and belongs to the technical field of aerospace. The implementation method of the invention comprises the following steps: discretizing the event constraint solution space to obtain a series of space discrete grids; calculating constraint function values at each discrete grid boundary point, and interpolating constraint function data to obtain a linear expression form of the constraint function; constructing an event detection function, relaxing the event constraint function, and raising the grid judgment constraint to obtain a raised event constraint function form; constructing a switch function, describing an original event trigger function, and obtaining a convex expression form of the event trigger function; solving the convex expression form to obtain an optimal trajectory of spacecraft intersection and butt joint, and further improving the convergence and efficiency of spacecraft intersection and butt joint trajectory planning with event constraint on the premise of not changing the original problem solution space.
Description
Technical Field
The invention relates to a method for projecting a spacecraft trajectory planning event constraint, which is particularly suitable for projecting the spacecraft intersection docking trajectory planning event constraint, and belongs to the technical field of aerospace.
Background
The spacecraft trajectory planning is a key technology for realizing deep space exploration and geospatial tasks, and has important application value in spacecraft intersection docking tasks. Because of the docking event in the spacecraft docking task, the docking track of the spacecraft is discontinuous, and discontinuous event constraint exists in planning the docking track. On the one hand, how to overcome the discontinuity and process the event constraint, and obtaining the constraint form which is convenient for track planning is a difficulty to be solved. On the other hand, the trajectory planning problem of spacecraft intersection butt joint is essentially an optimal control problem, and for constraint in the optimal control problem, the constraint is provided with convex properties, so that the planning efficiency of the problem can be effectively improved, the problem solving difficulty is reduced, and how to realize the convex event constraint is also a key problem to be solved. The processing method for event constraint is one of the hot spot problems of current technological staff.
In the developed event constraint processing method, a convex method based on zero-one variable is proposed for discontinuous event constraint in the prior art [1] (Jeweison C M. "Guidance and control for multi-stage rendezvous and docking operations in the presence of uncertainty," Massachusetts Institute of Technology, 2017), and the zero-one variable is used for describing two cases of event triggering and non-triggering and is coupled in a dynamics equation. The method has the advantages of simple operation and no introduction of excessive optimization variables. However, in the method, introducing zero-variable can cause difficulty in track planning problem NP, and although a convex constraint form is obtained, the problems of poor convergence and low convergence efficiency of the track planning problem are further caused, so that the method is difficult to be applied to the problem of meeting and butting of spacecraft with high requirements on robustness.
Prior Art [2] (Lu, P. "Convex-Concave Decomposition of Nonlinear Equality Constraints in Optimal Control," Journal of Guidance, control, and Dynamics, 2020), proposes a concave-Convex decomposition method that handles event constraints as a zero-to-planning problem and uses non-integer optimization variables to sequence approximations of the zero-to-variable. The method can effectively solve the NP difficult problem caused by zero-variable, but can cause the sequence iteration characteristic of the track planning method, reduce the calculation efficiency, and is difficult to be suitable for the spacecraft intersection butt joint problem with higher calculation rapidity.
Disclosure of Invention
In order to solve the problems of low calculation efficiency and poor convergence which often occur when the prior method carries out trajectory planning event constraint salification, the invention discloses a spacecraft intersection butt joint trajectory planning event constraint salifying method, which aims to solve the technical problems that: on the premise of not adopting a sequence approximation method, the event constraint is equivalently converted into a convex form, and a real variable is used for carrying out lossless convex on the event constraint function, so that the convergence and the efficiency of the spacecraft intersection butt joint trajectory planning with the event constraint are improved on the premise of not changing the original problem solution space.
The invention aims at realizing the following technical scheme:
the invention discloses a projection method for planning event constraints on a spacecraft rendezvous and docking trajectory, which comprises the steps of firstly selecting a variable search range containing an original optimal design variable solution space according to related design variables of an event constraint function, and performing grid discretization on the whole variable search space. Event function values at each discretized grid boundary are calculated according to the constraint function form. And obtaining a linear constraint function form inside the discretized grid by adopting low quadratic center of gravity interpolation. Performing traversal calculation on all discretized grids in the search space, and converting constraint function values in the whole search space into event constraints in the form of grid judgment constraints and linear interpolation. Subsequently, a series of real variables are introduced, and the new variables are applied to describe grid triggering conditions in the search space by adopting a volume detection method. And introducing a real variable into the event constraint function through a relaxation-penalty mechanism, realizing the convexity of the grid judgment constraint, and obtaining a linear event constraint expression form on the premise of not introducing sequence approximation, namely obtaining the convexity expression form of the event trigger function. Solving the convex expression form to obtain an optimal trajectory of spacecraft intersection and butt joint, and further improving the convergence and efficiency of spacecraft intersection and butt joint trajectory planning with event constraint on the premise of not changing the original problem solution space.
The invention discloses a projection method for spacecraft intersection docking trajectory planning event constraint, which comprises the following steps:
step one, discretizing the event constraint solution space to obtain a series of space discrete grids.
Defining a spacecraft track planning event constraint function as e (x), wherein x is a related variable of the spacecraft track planning event constraint. In the trajectory planning problem, the constraint function e (x) exists among the problems in the following form.
if p(x)＝0,then e(x)＝0
Wherein: p (x) represents a trigger function of the event constraint.
In the trajectory planning problem, when the condition p (x) =0 is satisfied, meaning that the event is triggered, the event constraint condition e (x) =0 needs to be satisfied; when the condition p (x) =0 is not satisfied, meaning that the event is not triggered, whether the event constraint condition e (x) =0 is satisfied or not does not affect the spacecraft trajectory.
Consider x as a one-dimensional continuous design variable whose solution space is contained by the linear search space as follows.
Where Ω represents the solution space of the original event constraint function, x min Representing the minimum search boundary of the variable x, x max Representing the maximum search boundary for variable x.
Given a linear search space [ x ] min ,x max ]Is divided into a series of linear small grids, denoted omega j
Ω j :＝[x min,j ,x max,j ]
Wherein: omega shape j Represents the j-th linear small grid, x min,j Represents the lower bound, x, of the jth linear small grid max,j Representing the upper bound of the jth linear small grid.
x min,j And x max,j Satisfy the following mathematical relationship
Thus, a spatial grid and a position parameter of the specific grid are obtained.
And secondly, calculating constraint function values at each discrete grid boundary point, and interpolating constraint function data to obtain a linear expression form of the constraint function.
And according to the constraint function form, in each given grid region, calculating to obtain constraint function values at grid boundaries.
e l,j ＝e(x min,j )
e u,j ＝e(x max,j )
Wherein e l,j Representing a constraint function value, e, at the lower boundary of the jth linear small grid u,j Representing the constraint function value at the upper boundary of the jth linear small grid.
Given a linear small grid region
Ω j :＝[x min,j ,x max,j ]
Constraint function value e in the region l,j And e u,j Obtaining the interpolation form of the constraint function in the region
Thus, a linear event constraint function is obtained over the entire task area
Thirdly, constructing an event detection function, relaxing the event constraint function, and raising the grid judgment constraint to obtain a raised event constraint function form.
A relaxation variable k is introduced, which is a vector of dimension n-1. Using the relaxation variables to express event-triggered constraints, define κ j The j-th component representing the relaxation variable builds the inequality constraint as follows
|x-x min,j |+|x-x max,j |≤κ j
Meanwhile, penalty terms are additionally introduced into the performance index, so that the inequality constraint time is taken and the like
The event constraint function is subjected to the following mathematical transformation
Wherein D represents a sufficiently large constant.
The constraint function has the meaning that when the variable x is located in a regionWhen the lower bound of the relaxation factor is 0, namely kappa j 0, wherein the event constraint function is equivalent to the original function; conversely, when the variable x is not located in the region +.>When the lower bound of the relaxation factor is larger than 0, the event constraint function is met, and the problem of track planning is not entered.
And step four, constructing a switching function, describing an original event triggering function, and obtaining a convex expression form of the event triggering function.
For a given event trigger function p (x), calculate at each linear small regionAt the boundary, the value of the trigger function.
p l,j ＝p(x min,j )
p u,j ＝p(x max,j )
Wherein p is l,j Representing an event trigger function value, p, at the j-th linear small grid lower boundary u,j Representing the event trigger function value at the upper boundary of the jth linear small grid.
For the event trigger constraint p (x) =0, using the zero point theorem, one can get: when the area isWhen p (x) zero is present, there is
p l,j ·p u,j ≤0
On the contrary, there are
p l,j ·p u,j ＞0
Thus, the expression form of Guan Hanshu ε is defined as
if p l,j ·p u,j ≤0,ε＝0
if p l,j ·p u,j ＞0,ε＝1
Subsequently, the switching function is introduced into the event detection function to obtain
ε(|x-x min,j |+|x-x max,j |)≤κ j
Finally, a convex expression form of the event constraint function is obtained
Subjectto
ε(|x-x min,j |+|x-x max,j |)≤κ j
When the number of space discrete points is large enough, the convex optimization problem is equivalent to the original problem, namely, the event constraint is equivalently converted into a convex form on the premise of not adopting a sequence approximation method.
Step five: and (3) solving the convex expression form in the step four to obtain an optimal trajectory of spacecraft intersection and docking, and further improving the convergence and efficiency of spacecraft intersection and docking trajectory planning with event constraint on the premise of not changing the original problem solution space.
The beneficial effects are that:
the invention discloses a projection method for planning event constraints on a spacecraft rendezvous and docking trajectory, which comprises the steps of firstly selecting a variable search range containing an original optimal design variable solution space according to related design variables of an event constraint function, and performing grid discretization on the whole variable search space. Event function values at each discretized grid boundary are calculated according to the constraint function form. And obtaining a linear constraint function form inside the discretized grid by adopting low quadratic center of gravity interpolation. Performing traversal calculation on all discretized grids in the search space, and converting constraint function values in the whole search space into event constraints in the form of grid judgment constraints and linear interpolation. Subsequently, a series of real variables are introduced, and the new variables are applied to describe grid triggering conditions in the search space by adopting a volume detection method. Finally, a real variable is introduced into an event constraint function through a relaxation-penalty mechanism, so that the convexity of grid judgment constraint is realized, and the convergence and calculation speed loss caused by solving the track planning problem in the traditional convexity method are avoided. The method can improve the convergence and the calculation efficiency of the spacecraft intersection-docking trajectory planning with event constraint on the premise of not changing the original problem solution space.
Drawings
FIG. 1 is a simulation diagram of spacecraft rendezvous and docking trajectory planning with event constraints.
Fig. 2 is a flowchart of a method for projecting a spacecraft rendezvous and docking trajectory planning event constraint according to the present disclosure.
Detailed Description
For a better description of the objects and advantages of the present invention, the following description will be given with reference to the accompanying drawings and examples.
Example 1: problem protrusion for planning of intersecting and butting tracks of near-earth spacecraft
As shown in fig. 2, the embodiment discloses a method for projecting a spacecraft rendezvous and docking trajectory planning event constraint, which specifically includes the following steps
Step one, constructing an event constraint basic form according to problem definition, and selecting an event constraint solution space boundary.
Considering the process of the intersection and butt joint of two spacecrafts, the position vectors of the two spacecrafts are r respectively 1 And r 2 Defining an event trigger function as
p(r 1 ,r 2 )＝∑|r 1 -r 2 |
Wherein p (r 1 ,r 2 ) Representing a spacecraft rendezvous docking event trigger function.
The trigger function means that when the positions of the two spacecrafts are consistent, the meeting and docking operation is judged to be needed; if the positions are inconsistent, the butting operation is not performed.
Correspondingly, define the event constraint function as e (m), in the specific form of
e(m 1 ,m 2 ,m 3 )＝m 1 +m 2 -m 3
Wherein m is 1 、m 2 Respectively represent the mass, m of two spacecrafts 3 The total mass after the two spacecraft meet is represented, and the mass is constant.
Thus, define m 1 、m 2 The solution space boundaries of the related variables are respectively m 1,min 、m 1,max 、m 2,min 、m 2,max Wherein m is 1,min As variable m 1 Minimum value of m 1,max As variable m 1 Maximum value of m 2,min As variable m 2 Minimum value of m 2,max As variable m 2 Is a maximum value of (a).
And step two, discretizing the event constraint solution space to obtain a series of space discrete grids.
Define the following variable solution spaces
Where Ω represents the solution space of the original event constraint function.
Given a linear search spaceN of discrete points of (2) 1 Linear search space->N of discrete points of (2) 2 The region Ω is divided into a series of linear small grids, denoted Ω j
Ω j :＝[m 1,min,j ,m 1,max,j ]×[m 2,min,j ,m 2,max,j ]
Wherein: omega shape j Represents the j-th linear small grid, m 1,min,j Representing variable m in the jth linear small grid 1 Lower bound of m 1,max,j Representing variable m in the jth linear small grid 1 Lower bound of m 2,min,j Representing variable m in the jth linear small grid 2 Lower bound of m 2,max,j Representing variable m in the jth linear small grid 2 Is defined below.
And thirdly, calculating constraint function values at each discrete grid boundary point.
Dividing a square small grid area into two triangular areas
Ω j - :＝Δ{(m 1,min,j ,m 2,max,j ),(m 1,min,j ,m 2,min,j ),(m 1,max,j ,m 2,min,j )}
Ω j + :＝Δ{(m 1,min,j ,m 2,max,j ),(m 1,max,j ,m 2,max,j ),(m 1,max,j ,m 2,min,j )}
Wherein Ω j - Is the lower triangle in the square, omega j + The operator delta represents a triangle region consisting of three feature points, which is the upper triangle within the square.
According to the constraint function form, in each given grid region, the constraint function value at the grid boundary can be calculated, and three point divisions in the grid are definedAre respectively M 1 、M 2 、M 3 And the corresponding constraint function values at three points in the grid are expressed as e 1 、e 2 、e 3 。
And step four, interpolation is carried out on constraint function data, and a linear expression form of the constraint function is obtained.
Computing a form of interpolation of the constraint function within the region
Where the function S represents the area linear calculation function of a triangle given its three vertices.
Thus, a linear event constraint function within the entire task area can be obtained
if(m 1 ,m 2 )∈[m 1,min,j ,m 1,max,j ]×[m 2,min,j ,m 2,max,j ]
And fifthly, constructing an event detection function, and describing the position of the event constraint related variable in the solution space.
A relaxation variable k is introduced, which is a vector of dimension n-1. Using the relaxation variables to express event-triggered constraints, define κ j The j-th component representing the relaxation variable builds the inequality constraint as follows
S{(m 1 ,m 2 ),M 1 ,M 2 }+S{(m 1 ,m 2 ),M 1 ,M 3 }+S{(m 1 ,m 2 ),M 2 ,M 3 }≤κ j
Meanwhile, penalty terms are additionally introduced into the performance index, so that the inequality constraint time is taken and the like
Step six, relaxing the event constraint function, and convecting the grid judgment constraint to obtain a convex event constraint function form in the specific area.
The event constraint function is subjected to the following mathematical transformation
Wherein D represents a sufficiently large constant.
The constraint function has the meaning that when the variable x is located in a regionWhen the lower bound of the relaxation factor is 0, namely kappa j 0, wherein the event constraint function is equivalent to the original function; conversely, when the variable x is not located in the region +.>When the lower bound of the relaxation factor is larger than 0, the event constraint function is met, and the problem of track planning is not entered.
And step seven, constructing a switching function, and describing an original event trigger function.
For a given event trigger function p (r 1 ,r 2 ) Calculating a trigger function value at each small region boundary, denoted as p ll,j 、p ru,j 、p lu,j 、p rl,j . Wherein p is ll,j Representing the event trigger function value, p, at the lower left boundary of the jth linear small grid ru,j Representing an event trigger function value, p, at the upper right boundary of the jth linear small grid lu,j Representing an event trigger function value, p, at the upper left boundary of the jth linear small grid rl,j Representing the event trigger function value at the lower right boundary of the j-th linear small grid.
For event trigger constraint p (r 1 ,r 2 ) =0, using the zero point theorem, one can obtain: when the area isPresence of p (r 1 ,r 2 ) At zero point, there are
p ll,j ·p ru,j ≤0
On the contrary, there are
p ll,j ·p ru,j ＞0
Thus, the expression form of Guan Hanshu ε is defined as
if p ll,j ·p ru,j ≤0,ε＝0
if p ll,j ·p ru,j ＞0,ε＝1
And step eight, constructing a convex event constraint form according to the switching function.
Introducing the switching function into the event detection function to obtain
ε(|x-x min,j |+|x-x max,j |)≤κ j
Finally, a convex expression form of the event constraint function is obtained
Subjectto
ε(S{(m 1 ,m 2 ),M 1 ,M 2 }+S{(m 1 ,m 2 ),M 1 ,M 3 }+S{(m 1 ,m 2 ),M 2 ,M 3 })≤κ j
The constraint obtained by the salifying of the embodiment is added into the spacecraft intersection and docking trajectory planning problem, and the intersection and docking trajectory is shown in figure 1.
While the foregoing is directed to embodiments of the present invention, other and further embodiments of the invention may be devised without departing from the basic scope thereof, and the scope thereof is determined by the claims that follow.
Claims (2)
1. A projection method for spacecraft intersection docking trajectory planning event constraint is characterized by comprising the following steps: comprises the following steps of the method,
step one, discretizing an event constraint solution space to obtain a series of space discrete grids;
the first implementation method of the step is that,
defining a spacecraft track planning event constraint function as e (x), wherein x is a related variable of the spacecraft track planning event constraint; in the trajectory planning problem, the constraint function e (x) exists among the problems in the following form;
if p(x)＝0,then e(x)＝0
wherein: p (x) represents a trigger function of the event constraint;
in the trajectory planning problem, when the condition p (x) =0 is satisfied, meaning that the event is triggered, the event constraint condition e (x) =0 needs to be satisfied; when the condition p (x) =0 is not satisfied, meaning that the event is not triggered, whether the event constraint condition e (x) =0 is satisfied or not does not affect the spacecraft trajectory;
consider x as a one-dimensional continuous design variable whose solution space is contained by the linear search space as follows;
where Ω represents the solution space of the original event constraint function, x min Representing the minimum search boundary of the variable x, x max Representing the maximum search boundary for variable x;
given a linear search space [ x ] min ,x max ]Is divided into a series of linear small grids, denoted omega j
Ω j :＝[x min,j ,x max,j ]
Wherein: omega shape j Represents the j-th linear small grid, x min,j Represents the lower bound, x, of the jth linear small grid max,j Representing the upper bound of the jth linear small grid;
x min,j and x max,j Satisfy the following mathematical relationship
Thus, a spatial grid and a position parameter of a specific grid are obtained;
calculating constraint function values at each discrete grid boundary point, and interpolating constraint function data to obtain a linear expression form of the constraint function;
the implementation method of the second step is that,
according to the constraint function form, in each given grid area, calculating to obtain constraint function values at grid boundaries;
e l,j ＝e(x min,j )
e u,j ＝e(x max,j )
wherein e l,j Representing a constraint function value, e, at the lower boundary of the jth linear small grid u,j Representing constraint function values at the upper boundary of the jth linear small grid;
given a linear small grid region
Ω j :＝[x min,j ,x max,j ]
Constraint function value e in the region l,j And e u,j Obtaining the interpolation form of the constraint function in the region
Thus, a linear event constraint function is obtained over the entire task area
Step three, constructing an event detection function, relaxing the event constraint function, and raising the grid judgment constraint to obtain a raised event constraint function form;
the implementation method of the third step is that,
introducing a relaxation variable κ, the relaxation variable being a vector of dimension n-1; using the relaxation variables to express event-triggered constraints, define κ j The j-th component representing the relaxation variable builds the inequality constraint as follows
|x-x min,j |+|x-x max,j |≤κ j
Meanwhile, penalty terms are additionally introduced into the performance index, so that the inequality constraint time is taken and the like
The event constraint function is subjected to the following mathematical transformation
Wherein D represents a sufficiently large constant;
the constraint function has the meaning that when the variable x is located in a regionWhen the lower bound of the relaxation factor is 0, namely kappa j 0, wherein the event constraint function is equivalent to the original function; conversely, when the variable x is not located in the region +.>When the lower bound of the relaxation factor is larger than 0, the event constraint function is met, and the problem of track planning is not entered;
step four, constructing a switch function, describing an original event trigger function, and obtaining a convex expression form of the event trigger function;
the realization method of the fourth step is that,
for a given event trigger function p (x), calculate at each linear small regionAt the boundary, the value of the trigger function;
p l,j ＝p(x min,j )
p u,j ＝p(x max,j )
wherein p is l,j Representing an event trigger function value, p, at the j-th linear small grid lower boundary u,j Representing an event trigger function value at the upper boundary of the jth linear small grid;
for the event trigger constraint p (x) =0, using the zero point theorem, one can get: when the area isWhen p (x) zero is present, there is
p l,j ·p u,j ≤0
On the contrary, there are
p l,j ·p u,j ＞0
Thus, the expression form of Guan Hanshu ε is defined as
if p l,j ·p u,j ≤0,ε＝0
if p l,j ·p u,j ＞0,ε＝1
Subsequently, the switching function is introduced into the event detection function to obtain
ε(|x-x min,j |+|x-x max,j |)≤κ j
Finally, a convex expression form of the event constraint function is obtained
Minimize
Subject to
ε(|x-x min,j |+|x-x max,j |)≤κ j
When the number of space discrete points is large enough, the convex optimization problem is equivalent to the original problem, namely, the event constraint is equivalently converted into a convex form on the premise of not adopting a sequence approximation method.
2. A method of protrusion of spacecraft rendezvous and docking trajectory planning event constraints as claimed in claim 1, wherein: the method also comprises the following steps: and (3) solving the convex expression form in the step four to obtain an optimal trajectory of spacecraft intersection and docking, and further improving the convergence and efficiency of spacecraft intersection and docking trajectory planning with event constraint on the premise of not changing the original problem solution space.
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| Pingyuan Cui,Tong Qin,Shengying Zhu,Yang Liu,Rui Xu,Zhengshi Yu.Trajectory curvature guidance for Mars landings in hazardous terrains.Automatica.2018,第93卷161-171. * |
| 基于凸优化的自主交会迭代制导方法;池贤彬;岳晓奎;李鹏;;中国空间科学技术(01);26-34 * |
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