Fast generation method of reachable domain of small celestial body attachment

Abstract

The small celestial body attachment reachable domain fast generation method disclosed by the application belongs to the technical field of deep space exploration. The implementation method of the application is as follows: a small celestial body attachment trajectory optimization model is established by taking fuel consumption as an optimization performance index, a mapping function of the small celestial body surface centroid distance and the longitude and latitude is introduced to simplify the terminal position constraint of the attachment trajectory optimization problem, and then the optimal fuel consumption attachment point is obtained by optimization solution. The reachable domain boundary points in different directions are solved with the optimal fuel consumption attachment point as the center, the reachable domain boundary point search strategy is constructed, the reachable domain boundary point solution problem in a given direction is converted into a series of fuel consumption optimal fixed point attachment trajectory optimization problems and boundary interpolation, the optimization solution process of the reachable domain boundary point is simplified, and the optimization solution efficiency is improved. After part of the attachment reachable domain boundary points are searched, the B-spline curve is adopted to fit the reachable domain boundary by taking the part of the attachment reachable domain boundary points as the original data points, the small celestial body attachment reachable domain is generated, and the small celestial body attachment trajectory planning efficiency is improved.

Description

Technical Field

[0001] This invention relates to a method for generating reachable domains, and more particularly to a method for rapidly generating reachable domains attached to small celestial bodies, belonging to the field of deep space exploration technology. Background Technology

[0002] Small body attachment exploration is a crucial prerequisite for implementing sample return, small body defense, and future small body resource development and utilization. The selection of landing sites for attachment exploration missions must comprehensively consider both scientific value and engineering safety, requiring landing sites to be the optimal locations reachable by the probe and offering the highest scientific returns. The small body attachment reachability domain refers to the set of terminal locations on the surface of small celestial bodies that the probe can reach under given initial conditions and constraints. Solving for the attachment reachability domain can provide a basis for the selection of the aforementioned landing sites and is of great significance in attachment exploration mission planning.

[0003] Due to the irregular shape and non-uniform gravitational field of small celestial bodies, the boundaries of their reachable domains often exhibit irregular shapes, resulting in complex terminal position constraints in the trajectory optimization problem. Directly applying existing methods for solving this problem leads to long computation times and low efficiency. Therefore, it is necessary to research methods for rapidly generating reachable domains specifically tailored to the characteristics of small celestial bodies. Summary of the Invention

[0004] To address the issues of complex terminal position constraints and low solution efficiency in the trajectory optimization problem during the solution of the reachable domain for small celestial bodies, the main objective of this invention is to provide a rapid method for generating the reachable domain for small celestial bodies. By constructing a reachable domain boundary point search strategy, the complex non-fixed-point attachment trajectory optimization problem is transformed into fixed-point attachment fuel consumption optimization and boundary interpolation. Furthermore, B-spline curves are used to fit the reachable domain boundary, rapidly generating the reachable domain for small celestial bodies. This invention transforms the problem of generating the reachable domain for small celestial bodies into boundary point search and reachable domain boundary fitting, which simplifies the optimization solution process, improves the efficiency of the reachable domain solution, and thus improves the efficiency of small celestial body attachment trajectory planning.

[0005] The objective of this invention is achieved through the following technical solution.

[0006] This invention discloses a method for rapidly generating reachable domains for small celestial bodies. It establishes an optimization model for small celestial body attachment trajectories using fuel consumption as the performance optimization index. By introducing a mapping function between the centroid distance and latitude / longitude of the small celestial body surface, the terminal position constraints of the attachment trajectory optimization problem are simplified, leading to the optimal fuel consumption attachment point. Using the optimal fuel consumption attachment point as the center, the reachable domain boundary points in different directions are solved. By constructing a reachable domain boundary point search strategy, the problem of solving reachable domain boundary points in a given direction is transformed into a series of fuel consumption-optimal fixed-point attachment trajectory optimization problems and boundary interpolation, thereby simplifying the reachable domain boundary point optimization process and improving optimization efficiency. After obtaining some reachable domain boundary points, these are used as original data points, and B-spline curves are used to fit the reachable domain boundaries, rapidly generating the reachable domain for small celestial bodies, thus improving the efficiency of small celestial body attachment trajectory planning.

[0007] The present invention discloses a method for rapidly generating reachable domains for small celestial bodies, comprising the following steps:

[0008] Step 1: Using fuel consumption as the performance optimization index, and combining dynamic constraints, initial state constraints, terminal state constraints, and path constraints, establish a small celestial body attachment trajectory optimization problem; solve this small celestial body attachment trajectory optimization problem to obtain the minimum fuel consumption required for attachment under a given initial state and the corresponding optimal fuel consumption attachment point L0.

[0009] A fixed coordinate system Oxyz is defined with the center of mass of the small celestial body as the origin. The z-axis coincides with the axis of maximum inertia, the x-axis coincides with the axis of minimum inertia, and the y-axis satisfies the right-hand rule with respect to the x and z axes. The attachment dynamics equations of the small celestial body are as follows:

[0010]

[0011] Where r is the position vector, v is the velocity vector, ω is the spin angular velocity vector of the small celestial body, g(r) is the gravitational acceleration acting on the detector, and a c It controls acceleration. m is the detector mass, I... sp Let g be the thrust specific impulse, g0 be the standard gravitational acceleration at sea level, and T be the thrust vector, satisfying a c =T / m.

[0012] Using the known latitude and longitude of a point on the surface of a small celestial body and its corresponding centroid distance, and combining this with interpolation, the centroid distance of the point on the surface of the small celestial body at a given latitude and longitude is obtained, thus yielding the three-axis coordinates corresponding to the given latitude and longitude; the longitude θ and latitude θ are then used to calculate the three-axis coordinates. The mapping relationship between the distance r between the centroid of a point on the surface of a small celestial body and the distance r between the centroids is denoted as: The terminal position constraint is then expressed as

[0013]

[0014] Among them, t f r(t) is the terminal time at which the detector attaches. f ) for t f The distance between the detector position and the centroid at any given time is R(θ(t)). f ), For t f The distance from the center of mass of the sub-satellite point on the surface of the small celestial body corresponding to the time-of-motion detector. Using burnup as the performance optimization index, and combining dynamic constraints, initial state constraints, terminal state constraints, and path constraints, the optimization model for the small celestial body attachment trajectory is obtained as follows:

[0015] min J=-m(t f )

[0016]

[0017] Among them, T max Let t0 be the maximum thrust of the detector, t0 be the initial time, r0 and v0 be the initial position and velocity vectors respectively, and m0 be the initial total mass of the detector.

[0018] Solving the trajectory optimization problem described in equation (3) yields the optimal terminal quality. and the corresponding position vector of the optimal fuel consumption adhesion point L0 Optimal fuel consumption C min0 satisfy

[0019]

[0020] Maximum available fuel consumption C max satisfy

[0021] C max =m0-m net (5)

[0022] Where, m net Let C be the net mass of the detector. min0 >C max When C is unreachable, there is no reachable domain; when C is unreachable, there is no reachable domain. min0 ≤C max Then proceed to step two.

[0023] Step 2: Using the optimal burnup attachment point L0 obtained in Step 1 as the reference point, search for two points on the surface of the small celestial body with the same longitude as L0, located within and outside the reachable domain respectively. Interpolate the coordinates of the two points to obtain the coordinates of the boundary point P0.

[0024] Using the optimal burnup attachment point L0 as the reference point, the orientation of a point L on the surface of the small celestial body relative to L0 is described by the direction angle α and the centroid angle β. The longitude line where L0 is located is taken as the reference longitude line, the direction angle α is the angle between LL0 and the reference longitude line, and β is the angle between OL0 and OL.

[0025] With P 0,j For the search point, P 0,j The coordinates are (0, β) 0,j ), where β 0,j =j×λ0, (j=1,2...), where λ0 is the step size of the centroid angle change; through P 0,j The coordinates are used to obtain their corresponding latitude and longitude, and the latitude and longitude are substituted into... P was calculated 0,j The corresponding centroid distance, further through P 0,j The latitude, longitude, and centroid distance of P are obtained 0,j The corresponding position r on the surface of the small celestial body t , will r t As the terminal position constraint for attachment trajectory optimization, a new optimization problem is derived.

[0026] min J=-m(t f )

[0027]

[0028] Equation (6) yields P 0,j The optimal adhesion fuel consumption C at the target landing point min,j By continuously increasing j and iterating, a series of optimal adhesion fuel consumption C are obtained. min,j Until C min,j and C min,j-1 The loop stops when equation (7) is satisfied.

[0029] (C min,j -C max (C) min,j-1 -C max )≤0 (7)

[0030] β 0,j ,β 0,j-1 Interpolation is performed to obtain the centroid angle β0 corresponding to the boundary point P0, and the coordinates of P0 are further obtained as (0, β0); the interpolation formula is:

[0031]

[0032] Step 3: Based on the coordinates of P0 obtained in Step 2, only change the direction angle to obtain α, which is the same as the centroid angle of P0. i Point P in the direction of =i×Δα (i = 1, 2, ..., N) i,0 Where Δα is the step size of the direction angle change; with P i,0 Using α as the baseline, search i The centroid angle β corresponding to the boundary point of the reachable domain attached in the direction i The boundary point P is obtained. iFinally, a series of boundary points P1, P2, ..., P in different directions are obtained. N .

[0033] P i,0 The coordinates are (α) i ,β0), with P i,0 Solving equation (6) for the terminal position of the corresponding small celestial body surface point yields P. i,0 The corresponding optimal adhesion fuel consumption is C min,0 When C min,0 =C max ,β i =β0, P i,0 That is, the boundary point; when C min,0 ≠C max When only the centroid angle is changed, the result is the same as P. i,0 Point P with the same direction angle i,j This is used as a search point to perform a search for the reachable domain boundary points.

[0034] When C min,0 >C max At that time, P i,j The corresponding coordinates are (α) i ,β i,j ), where β i,j =β0-j×λ, (j=1,2,…), where λ is the step size of the centroid angle change; with P i,j Solving equation (6) for the terminal position corresponding to the surface point of the small celestial body yields C. min,j The process involves continuously increasing j and iterating until C is reached. min,j <C max Stop the loop when the time is right; use equation (9) for β i,j-1 and β i,j Interpolation can be used to obtain the boundary point P. i Corresponding centroid angle β i Further, P was obtained. i The coordinates are (α) i ,β i ).

[0035]

[0036] When C min,0 <C max At that time, P i,j The corresponding coordinates are (α) i ,β i,j ), where β i,j =β0+j×λ, (j=1,2,...), where λ is the step size of the centroid angle change; with P i,j Solving equation (6) for the terminal position corresponding to the surface point of the small celestial body yields C. min,j The solution is obtained by continuously increasing j and iterating until C... min,j>C max Stop the loop when the time is right; use equation (9) for β i,j-1 and β i,j Interpolation is performed to obtain the boundary point P. i Corresponding centroid angle β i Further, P was obtained. i The coordinates are (α) i ,β i ).

[0037] Search point P i,j The centroid angle β i,j The selection method is as follows

[0038]

[0039] In obtaining α i Boundary point P in the direction i Then, the same search strategy is used to solve for α. i+1 Boundary point P in the direction i+1 Ultimately, the boundary points P1, P2, ..., P can be obtained through the search. N The coordinates. Due to different α i Direction P i The searches for points are independent of each other; therefore, parallel computing can be used to improve search efficiency when searching for boundary points.

[0040] Step 4: Based on the searches in Steps 2 and 3, obtain the boundary points P0, P1, P2, ..., P of the reachable domain. N Then, using this as the original data, a cubic uniform B-spline curve is used to fit the boundary of the attachment reachable domain. The fitted B-spline curve is the boundary of the attachment reachable domain of the small celestial body.

[0041] The definition of a cubic uniform B-spline curve is:

[0042]

[0043] Among them, Q i (i = 0, 1, ..., n) are control points, and the basis function N i,k (u) is a set of piecewise polynomials of degree k defined on the node vector U, where U = {u0, u1, ..., u} m Let} be a non-decreasing sequence of m+1 uniformly distributed numbers over the domain u∈[0,1], satisfying

[0044] u0=0≤u1≤u2…≤u m =1 (12)

[0045] For a cubic uniform B-spline curve, k=3. According to the Cox-Deboor recursive formula, the basis function N... i,k(u) is defined as

[0046]

[0047] Where i = 0, 1, 2, ..., n, k ≥ 1. When k = 0, the following condition is satisfied:

[0048]

[0049] As can be seen from the above construction method of B-spline curves, the number of nodes m must satisfy...

[0050] m = n + k + 1 (15)

[0051] The fitting method for cubic uniform B-spline curves uses interpolation fitting. First, control points are calculated from the feature points, and then the curve is generated from these control points according to the definition of a cubic uniform B-spline curve. For a cubic uniform B-spline curve, the feature points {V... i} and control point {Q i The relationship between} is represented as

[0052] s i (0)=(Q i-1 +4Q i +Q i+1 ) / 6=V i ,i=1,2,...,n (16)

[0053] To ensure the curve is closed, the following boundary conditions are added.

[0054]

[0055] According to equation (16), the formula for calculating the control points inversely is as follows:

[0056]

[0057] The search results are used to define the boundary points P0, P1, P2, ..., P of the attached reachable domain. N As control point {V i The control point {Q} is obtained by solving equations (17) and (18). i The control points {Q} obtained from the solution will be... i Substituting into the definition (11) yields the B-spline curve, which is the boundary of the small celestial body attachment reachable domain obtained by fitting.

[0058] The process also includes step five: using the boundary of the small celestial body attachment reachability domain obtained in step four as the basis for selecting landing points, comprehensively considering scientific value and engineering safety, and selecting the location with the highest scientific return as the landing point. This rapid generation of the small celestial body attachment reachability domain provides a basis for online planning of attachment and exploration missions, improves the efficiency of small celestial body attachment trajectory planning, and enhances the safety of small celestial body attachment.

[0059] Beneficial effects:

[0060] 1. The fast generation method for the attachment reachability domain of small celestial bodies disclosed in this invention simplifies the terminal position constraint of the optimal attachment trajectory optimization problem by introducing a mapping function between the centroid distance and latitude and longitude of the surface of the small celestial body, thereby reducing the complexity of the optimization problem and improving the efficiency of solving the optimal attachment point.

[0061] 2. The fast generation method for small celestial body attachment reachable domains disclosed in this invention takes the optimal burnout attachment point as the center and solves the reachable domain boundary points in different directions. By designing a reachable domain boundary point search strategy, the problem of solving the reachable domain boundary points in a given direction is transformed into a series of burnout optimal fixed-point attachment trajectory optimization problems and boundary interpolation, thereby simplifying the optimization solution process and improving the optimization solution efficiency.

[0062] 3. The method for rapidly generating the reachable domain of small celestial bodies disclosed in this invention, after searching for some boundary points of the reachable domain, uses them as original data points and uses B-spline curves to fit the boundary of the reachable domain, thereby rapidly generating the reachable domain of small celestial bodies and improving the efficiency of small celestial body attachment trajectory planning. Attached Figure Description

[0063] Figure 1 Flowchart of a method for rapidly generating reachable domains for small celestial bodies;

[0064] Figure 2 Flowchart for searching the reachable domain boundary point P0 for small celestial bodies;

[0065] Figure 3 For small celestial bodies, attach the reachable domain boundary point P. i Search flowchart;

[0066] Figure 4 This is a schematic diagram of the reachable domain boundary attached to the surface of the small celestial body 433Eros, where... Figure 4 (a) represents the boundary of the attached reachable domain in the fixed coordinate system of a small celestial body. Figure 4 (b) represents the boundary of the attached reachable domain in polar coordinates; Detailed Implementation

[0067] To better illustrate the purpose and advantages of the present invention, the following description, in conjunction with an embodiment and corresponding drawings, further explains the invention.

[0068] To verify the feasibility of the method, the attachment mission to asteroid 433Eros was used as an example to solve for the reachability region of the attachment. It is assumed that the asteroid rotates uniformly around its axis of maximum inertia, with a rotational angular velocity ω = 3.31 × 10⁻⁶. -4 rad / s. The initial mass of the detector is m0 = 800 kg, and the net mass is m net=796kg. Engine specific impulse I sp =300s, maximum thrust T max =25N. The detector's initial state r0 = [0, 0, 15000] T m, v0 = [0,0,0] T m / s.

[0069] like Figure 1 As shown in the figure, the method for rapidly generating reachable domains for small celestial bodies disclosed in this embodiment has the following specific implementation steps:

[0070] Step 1: Using fuel consumption as the performance optimization index, and combining dynamic constraints, initial state constraints, terminal state constraints, and path constraints, establish a small celestial body attachment trajectory optimization problem; solve this small celestial body attachment trajectory optimization problem to obtain the minimum fuel consumption required for attachment under a given initial state and the corresponding optimal fuel consumption attachment point L0.

[0071] A fixed coordinate system Oxyz is defined with the center of mass of the small celestial body as the origin. The z-axis coincides with the axis of maximum inertia, the x-axis coincides with the axis of minimum inertia, and the y-axis satisfies the right-hand rule with respect to the x and z axes. The attachment dynamics equations of the small celestial body are as follows:

[0072]

[0073] Where r is the position vector, v is the velocity vector, ω is the spin angular velocity vector of the small celestial body, g(r) is the gravitational acceleration acting on the detector, and a c It controls acceleration. m is the detector mass, I... sp Let g be the thrust specific impulse, g0 be the standard gravitational acceleration at sea level, and T be the thrust vector, satisfying a c =T / m.

[0074] Using the known latitude and longitude of a point on the surface of a small celestial body and its corresponding centroid distance, and combining this with interpolation, the centroid distance of the point on the surface of the small celestial body at a given latitude and longitude is obtained, thus yielding the three-axis coordinates corresponding to the given latitude and longitude; the longitude θ and latitude θ are then used to calculate the three-axis coordinates. The mapping relationship between the distance r between the centroid of a point on the surface of a small celestial body and the distance r between the centroids is denoted as: The terminal position constraint can then be expressed as:

[0075]

[0076] Among them, t f r(t) is the terminal time at which the detector attaches. f ) for t f The distance between the detector's position and its centroid at any given time. For t fThe distance from the center of mass of the nadir point on the surface of the small celestial body corresponding to the time-of-motion detector. Using burnup as the performance optimization index, and combining dynamic constraints, initial state constraints, terminal state constraints, and path constraints, the optimization model for the small celestial body attachment trajectory is obtained as follows:

[0077] min J=-m(t f )

[0078]

[0079] Among them, T max Let t0 be the maximum thrust of the detector, t0 be the initial time, r0 and v0 be the initial position and velocity vectors respectively, and m0 be the initial total mass of the detector.

[0080] Solving the trajectory optimization problem described in equation (21) yields the optimal terminal quality. and the corresponding position vector of the optimal fuel consumption adhesion point L0 Latitude and longitude corresponding to the optimal fuel consumption adhesion point L0 The optimal fuel consumption is (6.59°, 89.73°). The maximum available fuel consumption is C max =m0-m net = 4kg, where m net C represents the net mass of the detector. min0 ≤C max Proceed to step two.

[0081] Step 2: Using the optimal burnup attachment point L0 obtained in Step 1 as the reference point, search for two points on the surface of the small celestial body with the same longitude as L0, one located within the reachable region and the other outside. Interpolate the coordinates of the two points to obtain the coordinates of the boundary point P0. The search process for the boundary point P0 of the small celestial body's reachable region is as follows: Figure 2 As shown.

[0082] Using the optimal burnup attachment point L0 as the reference point, the orientation of a point L on the surface of the small celestial body relative to L0 is described by the direction angle α and the centroid angle β. The longitude line where L0 is located is taken as the reference longitude line, the direction angle α is the angle between LL0 and the reference longitude line, and β is the angle between OL0 and OL.

[0083] A coordinate system Ox'y'z' is defined with the center of mass of the small celestial body as the origin: the z' axis coincides with OL0, the x' axis lies in the plane of the reference longitude line and is perpendicular to the z' axis, and the y' axis satisfies the right-hand rule with respect to the x' and z' axes. The transformation matrix M from the small celestial body fixed coordinate system Oxyz to the coordinate system Ox'y'z' is defined. xx' for

[0084]

[0085] With P 0,jLet P be the search point. 0,j The coordinates are (0, β) 0,j ), where β 0,j =j×λ0, (j=1,2...), the centroid angle variation step size λ0=15°. Let P... 0,j Having the same latitude and longitude and a centroid distance of If the point is P', then the coordinates of P' are (0, β). 0,j The position vector r of P' in coordinate system Ox'y'z' can be obtained from the coordinates of P' and the distance between the centroids. x'y'z' After coordinate transformation, the position vector r of P' in the fixed coordinate system Oxyz of the small celestial body is obtained. xyz Further derived from the position vector r xyz Obtain the latitude and longitude corresponding to P', which is the coordinates of point P. 0,j Latitude and longitude; using mapping relationships Can be made by P 0,j P was obtained from the calculation of latitude and longitude. 0,j Corresponding to the centroid distance, further through P 0,j The latitude, longitude, and centroid distance of P are obtained 0,j The corresponding vector r of the surface point of the small celestial body t , will r t As the terminal position constraint for attachment trajectory optimization, a new optimization problem is derived.

[0086] min J=-m(t f )

[0087]

[0088] Equation (23) yields P 0,j The optimal adhesion fuel consumption C at the target landing point min,j By continuously increasing j and iterating, a series of optimal adhesion fuel consumption C are obtained. min,j Until C min,j and C min,j-1 The loop stops when equation (24) is satisfied.

[0089] (C min,j -C max (C) min,j-1 -C max )≤0 (24)

[0090] β 0,j ,β 0,j-1 Interpolation is performed to obtain the centroid angle β0 corresponding to the boundary point P0, and the coordinates of P0 are further obtained as (0, β0); the interpolation formula is:

[0091] Finally, the latitude and longitude of the boundary point P0 can be obtained as (6.59°, 42.15°).

[0092] Step 3: Based on the coordinates of P0 obtained in Step 2, only change the direction angle to obtain α, which is the same as the centroid angle of P0. i Point P in the direction of =i×Δα (i = 1, 2, ..., N) i,0 Where N = 8, and the direction angle change step size Δα = 40°. With P... i,0 Using α as the baseline, search i The centroid angle β corresponding to the boundary point of the reachable domain attached in the direction i The boundary point P is obtained. i Finally, a series of boundary points P1, P2, ..., P in different directions are obtained. N The boundary point P of the reachable domain for small celestial bodies. i The search process is as follows: Figure 3 As shown.

[0093] Let P i,0 The coordinates are (α) i ,β0), with P i,0 The optimization problem is described by equation (23) for solving the terminal position of the corresponding small celestial body surface point, resulting in P. i,0 The corresponding optimal adhesion fuel consumption is C min,0 When C min,0 =C max ,β i =β0, P i,0 That is, the boundary point; when C min,0 ≠C max When only the centroid angle is changed, the result is the same as P. i,0 Point P with the same direction angle i,j This is used as a search point to perform a search for the reachable domain boundary points.

[0094] When C min,0 >C max At that time, P i,j The corresponding coordinates are (α) i ,β i,j ), where β i,j =β0-j×λ, (j=1,2,...), where λ is the step size of the centroid angle change; with P i,j Solving equation (23) for the terminal position of the corresponding small celestial body surface point yields C. min,j The process involves continuously increasing j and iterating until C is reached. min,j <C max Stop the loop when the time is right; use equation (26) for β i,j-1 and β i,j Interpolation can be used to obtain the boundary point P. i Corresponding centroid angle β i .

[0095]

[0096] When C min,0 <C max At that time, P i,j The corresponding coordinates are (α) i ,β i,j ), where β i,j =β0+j×λ, (j=1,2,…), where λ is the step size of the centroid angle change; with P i,j Solving equation (23) for the terminal position of the corresponding small celestial body surface point yields C. min,j The solution is obtained by continuously increasing j and iterating until C... min,i >C max Stop the loop when the time is right; use equation (26) for β i,j-1 and β i,j Interpolation is performed to obtain the boundary point P. i Corresponding centroid angle β i .

[0097] Search point P i,j The centroid angle β i,j The selection method is as follows

[0098]

[0099] In obtaining α i Boundary point P in the direction i Then, the same search strategy is used to solve for α. i+1 Boundary point P in the direction i+1 Ultimately, the boundary points P1, P2, ..., P can be obtained through the search. N The coordinates. Due to different α i Direction P i The searches for points are independent of each other; therefore, parallel computing can be used to improve search efficiency when searching for boundary points.

[0100] Step 4: Based on the searches in Steps 2 and 3, obtain the boundary points P0, P1, P2, ..., P of the reachable domain. N Then, using this as the original data, a cubic uniform B-spline curve is used to fit the boundary of the attachment reachable domain. The fitted B-spline curve is the boundary of the attachment reachable domain of the small celestial body.

[0101] The definition of a cubic uniform B-spline curve is:

[0102]

[0103] Among them, Q i (i = 0, 1, ..., n) are control points, and the basis function N i,k (u) is a set of piecewise polynomials of degree k defined on the node vector U, where U = {u0, u1, ..., u}m Let} be a non-decreasing sequence of m+1 uniformly distributed numbers over the domain u∈[0,1], satisfying

[0104] u0=0≤u1≤u2…≤u m =1 (29)

[0105] For a cubic uniform B-spline curve, k = 3. According to the Cox-Deboor recursive formula, the basis function N... i,k (u) can be defined as

[0106]

[0107] Where i = 0, 1, 2, ..., n, k ≥ 1. When k = 0, it satisfies

[0108]

[0109] As can be seen from the above construction method of B-spline curves, the number of nodes m must satisfy...

[0110] m = n + k + 1 (32)

[0111] The fitting method for cubic uniform B-spline curves uses interpolation fitting. First, control points are calculated from the feature points, and then the curve is generated from these control points according to the definition of a cubic uniform B-spline curve. For a cubic uniform B-spline curve, the feature points {V... i} and control point {Q i The relationship between} can be represented as

[0112] s i (0)=(Q i-1 +4Q i +Q i+1 ) / 6=V i ,i=1,2,...,n (33)

[0113] To ensure the curve is closed, the following boundary conditions are added.

[0114]

[0115] According to equation (16), the formula for calculating the control points inversely is as follows:

[0116]

[0117] The boundary points P0, P1, P2, ..., P of the attached reachable domain obtained from the search are... N As control point {V i The control point {Q} is obtained by solving equations (34) and (35). i The control points {Q} obtained from the solution will be... iSubstituting into the definition (28), we obtain the B-spline curve, which is the fitted boundary of the small celestial body's reachable domain. The boundary of the reachable domain in the fixed coordinate system of the small celestial body is as follows: Figure 4 As shown in (a), the boundary of the attached reachable domain in polar coordinates is as follows: Figure 4 As shown in (b).

[0118] Compared with existing methods, the boundary of the small celestial body attachment reachable domain calculated by the present invention is basically the same as that calculated by existing methods; however, the present invention has a shorter calculation time, improves the solution efficiency of the small celestial body attachment reachable domain, and realizes the rapid calculation of the small celestial body attachment reachable domain.

[0119] The above detailed description further illustrates the purpose, technical solution, and beneficial effects of the invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for rapidly generating reachable domains for small celestial bodies, characterized in that: Includes the following steps, Step 1: Using fuel consumption as the performance optimization index, and combining dynamic constraints, initial state constraints, terminal state constraints, and path constraints, establish a small celestial body attachment trajectory optimization problem; solve this small celestial body attachment trajectory optimization problem to obtain the minimum fuel consumption required for attachment under a given initial state and the corresponding optimal fuel consumption attachment point. ; The implementation method for step one is as follows: Define a fixed coordinate system for the small celestial body with its center of mass as the origin. , The axis coincides with the axis of maximum inertia of the small celestial body. The axis coincides with the axis of minimum inertia. shaft and axis, The axes satisfy the right-hand rule; the dynamic equations for the attachment of small celestial bodies are as follows: in, It is a position vector. It is a velocity vector. It is the spin angular velocity vector of a small celestial body; It is the gravitational acceleration experienced by the probe. It controls acceleration; For detector quality, For the specific impulse of the thruster, The standard gravitational acceleration at sea level is... It is the thrust vector, satisfying ; Using the known latitude and longitude of a point on the surface of a small celestial body and its corresponding centroid distance, and combining this with interpolation, the centroid distance of the point on the surface of the small celestial body at a given latitude and longitude is obtained, thus yielding the three-axis coordinates corresponding to the given latitude and longitude; longitude... ,latitude Distance from the center of mass of a point on the surface of a small celestial body The mapping relationship between them is denoted as The terminal position constraint is then expressed as: in, The endpoint of detector attachment. for The distance between the detector's position and its centroid at any given time. for The distance from the center of mass of the sub-satellite point on the surface of the small celestial body corresponding to the time-of-motion detector; using burnup as the performance optimization index, and combining dynamic constraints, initial state constraints, terminal state constraints, and path constraints, the optimization problem model for the small celestial body attachment trajectory is obtained as follows: in, For the maximum thrust of the probe, At the initial moment, and These are the initial position and velocity vectors, respectively. This represents the initial total mass of the detector; Solution The trajectory optimization problem yields the optimal terminal quality. and the corresponding optimal fuel consumption adhesion point position vector Optimal fuel consumption satisfy Maximum available fuel consumption satisfy in, The net mass of the detector; when At that time, there was no reachable domain; when Then proceed to step two; Step 2: Using the optimal fuel consumption adhesion point obtained in Step 1 Using this as a reference point, the surface of the small celestial body and its relationship with the target celestial body are obtained through a search. For two points with the same longitude, one located within the reachable region and the other outside, interpolate their coordinates to obtain the boundary point. coordinate; Step 3, obtained in Step 2 Based on the coordinate system, by only changing the direction angle, we obtain the result... Same centroid angle Points in direction ,in The step size for the change in direction angle; Using the baseline, search The centroid angle corresponding to the boundary point of the reachable domain attached in the direction Obtain the boundary points Finally, a series of boundary points in different directions are obtained. ; Step 4: Obtain the reachable domain boundary points through searches in Steps 2 and 3. Then, using this as the original data, a cubic uniform B-spline curve is used to fit the boundary of the attachment reachable domain. The fitted B-spline curve is the boundary of the attachment reachable domain of the small celestial body.

2. The method for rapidly generating reachable domains for small celestial bodies as described in claim 1, characterized in that: It also includes step five: using the boundary of the small celestial body attachment reachable domain obtained in step four as the basis for selecting the landing point, and comprehensively considering scientific value and engineering safety to select the landing point.

3. The method for rapidly generating reachable domains for small celestial bodies as described in claim 1, characterized in that: The second step is implemented as follows: With optimal fuel consumption adhesion point Using the reference point, the direction angle is... and central angle Describe a point on the surface of a small celestial body Compared to The location; by The longitude line in question is the baseline longitude line, and the direction angle is... for The angle with the baseline longitude, for and The angle between them; by For the search point, The coordinates are ,in , The step size for the change in the centroid angle; through The coordinates are used to obtain their corresponding latitude and longitude, and the latitude and longitude are substituted into... Calculated The corresponding centroid distance, further through The latitude, longitude and centroid distance are obtained Corresponding to the position of the surface point of the small celestial body ,Will As the terminal position constraint for attachment trajectory optimization, a new optimization problem is derived. Through Get to Optimal adhesion fuel consumption for the target landing point ; continuously increasing Through iteration, a series of optimal adhesion fuel consumption were obtained. ;until and Satisfaction Stop the loop Will Interpolation is performed to obtain boundary points. Corresponding centroid angle Further obtained The coordinates are The interpolation formula is: 。 4. The method for rapidly generating reachable domains for small celestial bodies as described in claim 3, characterized in that: The method for implementing step three is as follows: The coordinates are ,by The solution formula for the terminal position of the corresponding small celestial body surface point is as follows ,get The corresponding optimal adhesion fuel consumption is ;when , , That is, the boundary point; when When only the centroid angle is changed, the result is the same as Points with the same direction angle Use it as a search point to perform a search for the boundary points of the reachable domain; when hour, Corresponding coordinates are ,in , The step size of the change in the centroid angle; The solution formula for the terminal position of the corresponding small celestial body surface point is as follows get Continuously increasing Perform the solution iteratively until... Stop the loop when the time is right; use the formula right and Interpolation can be used to obtain boundary points. Corresponding centroid angle Further obtained The coordinates are ; when hour, Corresponding coordinates are ,in , The step size of the change in the centroid angle; The solution formula for the terminal position of the corresponding small celestial body surface point is as follows get Continuously increasing Perform a loop to solve the problem, when Stop the loop when the time is right; use the formula right and Interpolation is performed to obtain boundary points. Corresponding centroid angle Further obtained The coordinates are ; Search point central angle The selection method is as follows In obtaining Boundary points in the direction Then, the same search strategy is used to solve the problem. Boundary points in the direction Ultimately, the boundary points can be obtained through searching. Coordinates; due to different Direction The searches for points are independent of each other; therefore, parallel computing can be used to improve search efficiency when searching for boundary points.

5. The method for rapidly generating reachable domains for small celestial bodies as described in claim 1, characterized in that: Step four is implemented as follows: The definition of a cubic uniform B-spline curve is: in, It is a control point. basis functions It is defined in the node vector The upper part of the segment A polynomial of degree 1, wherein... For the domain On A non-decreasing sequence of uniformly distributed numbers satisfies For a cubic uniform B-spline curve According to the Cox-Deboor recurrence relation, the basis functions Defined as in, , ;when When, satisfy As can be seen from the above construction method of B-spline curves, the number of nodes Must meet The fitting method for cubic uniform B-spline curves uses interpolation fitting. First, control points are calculated from feature points, and then the curve is generated from these control points according to the definition of a cubic uniform B-spline curve. For a cubic uniform B-spline curve, the feature points... and control points The relationship is represented as To ensure the curve is closed, the following boundary conditions are added. According to the formula The formula for calculating the control points inversely is: The searched boundary points of the attached reachable domain As a control point , by formula Japanese style The control points are obtained by solving. The control points obtained from the solution Substitute into the definition This yields the B-spline curve, which is the boundary of the small celestial body's reachable domain obtained through fitting.

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