Collision warning method for relative motion of three-body spacecraft based on polytope reachable set
By calculating the reachable set of spacecraft relative motion using the polymorphic cell reachable set theory, the computational efficiency and convergence problems of spacecraft collision warning in three-body orbits are solved, realizing efficient collision risk warning and avoidance.
Patent Information
- Application Number
- CN202610906982.3
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-06-23
- Publication Date
- 2026-11-03
AI Technical Summary
Existing technologies for early warning of collisions between spacecraft in a three-body orbit involve large computational loads, are difficult to handle non-convex sets under complex dynamic models, and have difficulty converging during the solution process.
By employing the theory of reachable sets of multi-cells, the system establishes the dynamic equations of relative motion of the spacecraft, calculates the state transition tensor, maps the initial error set and the control set, and directly determines the collision risk of the spacecraft, thus avoiding the two-point boundary value problem.
It improves the computational efficiency of the relative motion reachable set of spacecraft in a three-body orbit, can accurately predict collision risks, and is applicable to the formation motion of spacecraft in the Earth-Moon orbit.
Smart Images
Figure CN123033400A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to a collision warning method for relative motion between spacecraft based on reachable sets, and more particularly to a collision warning method for relative motion between spacecraft in a three-body orbit based on multi-cell calculation of the reachable set of spacecraft. It is applicable to calculating the reachable set of relative motion of spacecraft in a three-body orbit and is used for spacecraft collision risk assessment, belonging to the aerospace field. Background Technology
[0002] The exploration and utilization of lunar space has entered a new stage of rapid development. Traditional single-spacecraft models are insufficient to meet the demands of complex missions. Close-range relative motion missions involving multiple spacecraft in a three-body orbit, such as approaching, circling, and formation flying, will inevitably become an important component of future lunar exploration. Due to the complex dynamic environment of a three-body orbit, early warning of collision risks during close-range spacecraft flight is essential.
[0003] The spacecraft reachable set theory has shown great potential and application value in the fields of spacecraft maneuverability assessment, collision threat prediction, error tolerance analysis, and orbit design optimization. For spacecraft collision threat prediction, a new technology [1] (see Reachable domain for satellite relative motion along elliptic orbit with uncertainty and process noise [J] Journal of Aerospace Engineering, 2020, 234(7): 1287-1300.) proposed an algorithm for solving the relative reachable domain envelope surface for any elliptic reference orbit in near-Earth space, and performed collision detection based on the relative reachable domain. This method can solve the accurate spacecraft reachable boundary, but its solution process involves two-point boundary value problems, and the computational load is large. When the spacecraft dynamics model is more complex, there is a problem of convergence.
[0004] To address the problem of high computational cost in solving reachable sets, a method for predicting satellite collisions using zonotope was proposed in [2] (see Collision prediction and avoidance for satellite ultra-close relative motion with zonotope-based reachable sets [J] Proceedings of the Institution of Mechanical Engineers, Xu Z, Chen X, Huang Y, et al., 2019, 233(11): 3920-3937.). This method solves reachable sets based on set propagation of dynamic evolution, avoiding the problem of solving two-point boundary values and improving computational efficiency. However, zonotope cannot represent non-convex sets, and in complex dynamic environments, reachable sets are prone to evolve into non-convex sets.
[0005] Therefore, a method for calculating and propagating the orbital reachability set was proposed by the prior art [3] (see Rapid and automatic reachability estimation of electric propulsion spacecraft [J] The Journal of the Astronautical Sciences, Patel PR, Scheeres D J. 2023, 70(6): 45.). Since the polynomial can represent a non-convex set, the reachability set of the spacecraft under the complex dynamic model can be solved by mapping the state transition tensor.
[0006] However, solving the reachability set of a spacecraft using multi-cell methods depends on solving the state transition tensor of the dynamic equations, and specific derivations are required for different dynamic models. Summary of the Invention
[0007] The purpose of this invention is to provide a collision warning method for the relative motion of a three-body spacecraft based on the reachability set of a multi-cell model. This method calculates the reachability set of the relative motion of a spacecraft using multi-cell theory within a three-body dynamics model for collision warning analysis. In the Earth-Moon-Moon-centered coordinate system, the method establishes the dynamic equations of the tracking spacecraft based on the relative motion relationship of the spacecraft. Given the initial state of the tracking spacecraft, the nominal orbit is obtained by integrating the equations. Based on the nominal orbit state, the state transition tensor of the dynamic system at the corresponding integration step size is calculated. Given the initial state error of the tracking spacecraft and the maximum value of the spacecraft's small thrust control, the initial state error set and control set are obtained, and a collision warning region set is defined for the target spacecraft. The initial error set is mapped using the state transition tensor, and considering the effect of the control set, the reachability set of the tracking spacecraft at each time point is obtained. The spatial relationship between this set and the collision warning region set is determined, and based on the relationship between the sets, a collision warning is issued.
[0008] The objective of this invention is achieved through the following technical solution.
[0009] The collision warning method for relative motion of three-body spacecraft based on polymorphic cell reachability sets disclosed in this invention includes the following steps:
[0010] Step 1: Establish the dynamic equations of the tracking spacecraft by the relative motion relationship of the spacecraft. Given the initial state of the tracking spacecraft, obtain the nominal orbit by integrating the equations.
[0011] The dynamic equations for tracking spacecraft are expressed as follows:
[0012]
[0013] In the formula, the augmented vector ; To track spacecraft state variables, Among them, tracking spacecraft position vector velocity vector ; The differential equations for relative motion without external acceleration; system control vector. , These are the control components for tracking the spacecraft in various directions under the relative motion coordinate system; , .
[0014] Expanding the components of the dynamic equations as follows:
[0015]
[0016] In the formula, The gravity parameters are in the Earth-Moon-lunar coordinate system; the target spacecraft's position vector. In the Earth-Moon-centered coordinate system, the positional relationship between the target spacecraft and the tracking spacecraft satisfies ,in To track the spacecraft's position vector relative to the moon; To track the distance between spacecraft and Earth; To track the distance between the spacecraft and the moon; The distance between the target spacecraft and Earth; The distance between the target spacecraft and the moon satisfies:
[0017]
[0018] Step 2: Based on the nominal orbital state obtained in Step 1, calculate the state transition tensor of the dynamic system at the corresponding integration step size.
[0019] The state transition tensor of any order is represented as:
[0020]
[0021] In the formula: Let the order be the tensor. Represents the state components of the system; Indicates the first The system differential equations Find the partial derivative of the variable; It is the set of all non-empty combinations of variables. ;
[0022] The differential equation of the first-order state transition tensor is:
[0023]
[0024] In the formula, yes The first-order state transition tensor of the system at each time step is the identity matrix at the initial time step; It is the Jacobian matrix of the relative motion augmented system; Indicates the first There are several state variables; the specific expression is:
[0025]
[0026] in,
[0027]
[0028]
[0029] In the formula, Indicates the first Differential equations for each state quantity Find the first-order partial derivative. . The expressions for each component in the matrix are:
[0030]
[0031] The second-order state transition tensor differential equation of a system with relative motion can be expressed as:
[0032]
[0033] In the formula, This is the index of the corresponding position in the matrix; Indicates the first The differential equation for the nth state variable is related to the nth Find the first-order partial derivatives of each state variable; Indicates the first The differential equation for the nth state variable is related to the nth and the Find the second-order partial derivatives of each state variable; express The first-order state transition tensor at time t, whose subscripts correspond to the positions in the matrix; express The second-order state transition tensor at time step 1. Note that initially, the second-order state transition tensor has nine 9s. A zero matrix of 9;
[0034] From the formula In a relative motion system , , , , , The first-order partial derivatives are all constants, and their corresponding... All matrices are zero matrices; the rest are non-zero matrices. The matrix is represented as:
[0035]
[0036] In the formula, Indicates the first Differential equations for each state quantity and Find the second-order partial derivative. , The specific expressions for each component in the matrix are:
[0037]
[0038] Step 3: Given the initial state error of the tracking spacecraft and the maximum value of the spacecraft's small thrust control, obtain the initial state error set and control set, and divide the collision warning area set for the target spacecraft.
[0039] The initial state error range is selected using Mahalanobis distance. The error ellipsoid is represented as a set using a polycell. Any state variable inside the ellipsoid can be regarded as the initial deviation state of the spacecraft. For input control, a sphere centered at the origin and with the maximum permissible acceleration as its radius is used in acceleration space to represent the set of control inputs that the spacecraft can apply at any given time. Based on the relative motion requirements of the spacecraft, a minimum warning distance is selected. A no-fly zone (BOX) is defined by a cuboid centered on the target spacecraft, with length equal to the minimum warning distance in each coordinate direction of the location space. This region is represented by a set. .
[0040] Step 4: Map the initial error set through the state transition tensor, while considering the effect of the control set, to obtain the reachable set of the tracked spacecraft at each time point, and determine the set relationship between it and the collision warning area set in Step 3. Based on the relationship between the sets, determine whether to issue a collision warning.
[0041] Deviation of state variables at any given time:
[0042]
[0043] The formula With a Taylor expansion to second order, the deviations of the various state variables can be re-expressed as:
[0044]
[0045] In the formula, Represents the order of the tensor, when When the state transition tensor degenerates into the state transition matrix; Indicates the first The deviation of each state variable at the initial time; Indicates the first The state transition tensor of order 1 has a dimension of 0. ,in Let be the dimension of the system state variables. For the first The dimension of the augmented state.
[0046] The formula Represented using a polymorphic cell set, for any time... The deviation set at the next moment can be represented as:
[0047]
[0048] In the formula, Tensor mapping operations on sets; ,in, for The state deviation set of the spacecraft at any given moment for Spacecraft control set This represents the Cartesian product operation of two sets.
[0049] Translate the obtained deviation set center to The nominal state of relative motion of the spacecraft at a given moment ,get The reachable set of time:
[0050]
[0051] In the formula, This represents a translation operation to the center of the set.
[0052] Through Japanese style By iterating continuously, the reachable set of the spacecraft at any given time can be calculated; then, the reachable set of the spacecraft at each time point can be compared with the no-fly zone mentioned in step three. Determine the positional relationship:
[0053]
[0054] In the formula: A function to determine the relationship between sets;
[0055] Since the reachable set boundary is the farthest distance that a spacecraft can reach by continuously applying control, if the judgment result returns 0, there is no risk of collision between the two spacecraft at the current moment, and no collision warning is triggered. The spacecraft continues to propagate the reachable set along the nominal orbit. If the judgment result returns 1, there is a risk of collision between the spacecraft at the current moment, and the current position of the nominal orbit of the spacecraft is recorded as the warning position.
[0056] Beneficial effects:
[0057] 1. The collision warning method for relative motion of three-body orbital spacecraft based on polycell reachable sets disclosed in this invention uses polycell theory to characterize the reachable sets of relative motion of three-body orbital spacecraft, and directly performs forward propagation calculation through sets. Compared with the traditional method, it avoids the problem of convergence difficulty of two-point boundary value problems in solving optimal control problems under complex dynamic models, and improves the calculation efficiency of reachable sets.
[0058] 2. The collision warning method for relative motion of three-body orbital spacecraft based on polycell reachable sets disclosed in this invention derives the high-order state transition tensor of the dynamic model of relative motion of three-body orbital spacecraft, and applies polycell theory to solve the reachable set of relative motion of spacecraft in three-body orbit through tensor mapping. It can be used to calculate the reachable set of relative motion of spacecraft formation in Earth-Moon space orbit.
[0059] 3. The collision warning method for relative motion of three-body orbital spacecraft based on multi-cell reachable sets disclosed in this invention can directly determine whether a spacecraft has issued a collision risk warning by means of the relationship between sets, and record the collision warning triggering event and location. It does not require analysis of the state between spacecraft and can be used for spacecraft collision avoidance. Attached Figure Description
[0060] Figure 1 This is a flowchart of the collision warning method for relative motion of a three-body orbital spacecraft based on a multi-cell reachable set disclosed in this invention.
[0061] Figure 2 This is a schematic diagram of spacecraft reachability set propagation in a multi-cell computing example of the present invention.
[0062] Figure 3 This is a schematic diagram illustrating the tracking of the nominal orbit of a spacecraft and the reachable set of the spacecraft at various times in an example of the present invention.
[0063] Figure 4 This is a schematic diagram showing the relationship between the spacecraft's reachability set and the collision warning area at the moment of collision warning in an example of the present invention. Detailed Implementation
[0064] To better illustrate the purpose and advantages of the present invention, the invention will be further described below in conjunction with the accompanying drawings and examples.
[0065] Example 1:
[0066] To verify the feasibility of the method, a spacecraft formation relative motion mission in a 2:1 DRO orbit is performed, given the initial states of the target spacecraft and the tracking spacecraft, the control input magnitude, and the initial state error of the tracking spacecraft. The specific parameters are shown in the table below:
[0067] Table 1 Task Parameters
[0068] For ease of calculation, the spacecraft status and control input values in the table are dimensionless values, and the distance units in the dimensionless coordinate system are... Time unit Gravity parameters .
[0069] like Figure 1 As shown, the specific implementation steps of the collision early warning method for relative motion of three-body spacecraft based on multi-cell reachable sets are as follows:
[0070] Step 1: The spacecraft state is described using a circular restricted three-body problem, and the dynamic equations of the tracking spacecraft are established through the relative motion relationships of the spacecraft. Given the initial state of the tracking spacecraft, the nominal orbit is obtained by integrating the equations.
[0071] The dynamic equations for tracking spacecraft are expressed as follows:
[0072]
[0073] In the formula, the augmented vector ; To track spacecraft state variables, Among them, tracking spacecraft position vector velocity vector ; The differential equations for relative motion without external acceleration; system control vector. , These are the control components for tracking the spacecraft in various directions under the relative motion coordinate system; , .
[0074] Set the integration step size to 0.02 in the dimensionless coordinate system, the total integration time to 3, and the system control input to... Starting from the initial state, integrate to obtain the nominal orbital state at each integration step size. .
[0075] Step 2: Calculate the state transition tensor of the dynamic system at the corresponding step size based on the nominal orbital state.
[0076] The differential equations of the state transition tensors of the system at each order can be expressed as:
[0077]
[0078] When solving for spacecraft reachability sets using polycells, relatively accurate results can be obtained using second-order state transition tensor mappings. Higher-order tensor mappings can optimize the boundaries of reachability sets, but the computational cost increases significantly, reducing the efficiency advantage of set computation. Therefore, this invention will focus on the computation of first- and second-order state transition tensors.
[0079] The differential equation of the first-order state transition tensor is:
[0080]
[0081] The second-order state transition tensor differential equation of a system with relative motion can be expressed as:
[0082]
[0083] Based on the state of the nominal orbit at each time point obtained in step one, combined with the formula... Japanese style By integrating, we can obtain the first-order state transition tensor of the system at each integration step. and second-order state transition tensor Here, the superscript of the state transition tensor indicates the tensor order. Note that the first-order state transition tensor has an initial identity matrix at each integral time step, while the second-order state transition tensor has nine 9s at each integral time step. A zero matrix of 9.
[0084] Step 3: Given the initial state error of the tracking spacecraft and the maximum value of the spacecraft's small thrust control, obtain the initial state error set and control set, and divide the collision warning area set for the target spacecraft;
[0085] The reachable set is a region that evolves over time, and initial errors also affect the propagation of the reachable set. Mahalanobis distance is chosen. The error ellipsoid, the initial covariance matrix of the error ellipsoid is... Using a polymorphic cell, the ellipsoid is represented as a set. For input control, the maximum permissible acceleration centered at the origin is used in the acceleration space. A sphere with radius r can represent the set of control inputs that a spacecraft can apply at any given time. Based on the relative motion requirements of the spacecraft, a minimum warning distance of 500 m is selected, and a BOX no-fly zone is demarcated for the target spacecraft. This zone is represented by a set. .
[0086] Step 4: Map the initial error set through the state transition tensor, while considering the effect of the control set, obtain the reachable set of the tracked spacecraft at each time point, and determine the set relationship between the set and the collision warning area set in the position space. Based on the relationship between the sets, determine whether to issue a collision warning.
[0087] Figure 2 A schematic diagram illustrating the propagation of spacecraft reachable sets using multi-cell computing.
[0088] Step three proposes that there is a deviation in the initial state of the tracked spacecraft. Substituting this deviation into the augmented differential equation proposed in step one, and integrating, the deviation of the system state variables at any given time is:
[0089]
[0090] The formula With a Taylor expansion to second order, the deviations of the various state variables can be re-expressed as:
[0091]
[0092] In the formula, Represents the order of the tensor, when When the state transition tensor degenerates into the state transition matrix; Indicates the first The deviation of each state variable at the initial time; Indicates the first The state transition tensor of order 1 has a dimension of 0. ,in Let be the dimension of the system state variables. For the first The dimension of the augmented state.
[0093] The formula Represented using a polymorphic cell set, for any time... The deviation set at the next moment can be represented as:
[0094]
[0095] In the formula, Tensor mapping operations on sets; ,in, for The state deviation set of the spacecraft at any given moment for Spacecraft control set This represents the Cartesian product operation of two sets.
[0096] Translate the obtained deviation set center to The nominal state of relative motion of the spacecraft at a given moment ,get The reachable set of time:
[0097]
[0098] In the formula, This represents a translation operation to the center of the set.
[0099] Through Japanese style By iterating continuously, the reachable set of the spacecraft at any given time can be calculated; then, the reachable set of the spacecraft at each time point can be compared with the no-fly zone mentioned in step three. Determine the positional relationship:
[0100]
[0101] In the formula: A function to determine the relationship between sets;
[0102] Since the reachable set boundary is the farthest distance that a spacecraft can reach by continuously applying control, if the judgment result returns 0, there is no risk of collision between the two spacecraft at the current moment, and no collision warning is triggered. The spacecraft continues to propagate the reachable set along the nominal orbit. If the judgment result returns 1, there is a risk of collision between the spacecraft at the current moment, and the current position of the nominal orbit of the spacecraft is recorded as the warning position. Figure 4 This diagram illustrates the relationship between the spacecraft reachability set and the collision warning region at the moment of a collision warning. At the 59th integral duration, the spacecraft reachability set and the collision warning region intersect, and the warning moment is recorded. The status of the tracking spacecraft at that moment is At this moment, the tracking spacecraft needs to begin collision avoidance planning to prevent accidental collisions between spacecraft.
[0103] 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 collision warning method for relative motion of a three-body spacecraft based on multi-cell reachable sets, characterized in that: Includes the following steps: Step 1: Establish the dynamic equations of the tracking spacecraft based on the relative motion relationships of the spacecraft. Given the initial state of the tracking spacecraft, obtain the nominal orbit by integrating the equations. Step 2: Based on the nominal orbital state obtained in Step 1, calculate the state transition tensor of the dynamic system at the corresponding integration step size; Step 3: Given the initial state error of the tracking spacecraft and the maximum value of the spacecraft's small thrust control, obtain the initial state error set and control set, and divide the collision warning area set for the target spacecraft; Step 4: Map the initial error set through the state transition tensor, while considering the effect of the control set, to obtain the reachable set of the tracked spacecraft at each time point, and determine the set relationship between it and the collision warning area set in Step 3. Based on the relationship between the sets, determine whether to issue a collision warning.
2. The method as described in claim 1, characterized in that: The tracking spacecraft dynamics equations described in step one are expressed as follows: In the formula, the augmented vector ; To track spacecraft state variables, Among them, tracking spacecraft position vector velocity vector ; The differential equations for relative motion without external acceleration; system control vector. , These are the control components for tracking the spacecraft in various directions under the relative motion coordinate system; , ; Expanding the components of the dynamic equations as follows: In the formula, The gravity parameters are in the Earth-Moon-lunar coordinate system; the target spacecraft's position vector. In the Earth-Moon-centered coordinate system, the positional relationship between the target spacecraft and the tracking spacecraft satisfies ,in To track the spacecraft's position vector relative to the moon; To track the distance between spacecraft and Earth; To track the distance between the spacecraft and the moon; The distance between the target spacecraft and Earth; The distance between the target spacecraft and the moon satisfies: 。 3. The method as described in claim 1, characterized in that: The arbitrary-order state transition tensor is represented as follows: In the formula: Let the order be the tensor. Represents the state components of the system; Indicates the first The system differential equations Find the partial derivative of the variable; It is the set of all non-empty combinations of variables. ; The differential equation of the first-order state transition tensor is: In the formula, yes The first-order state transition tensor of the system at each time step is the identity matrix at the initial time step; It is the Jacobian matrix of the relative motion augmented system; Indicates the first There are several state variables; the specific expression is: in, In the formula, Indicates the first Differential equations for each state quantity Find the first-order partial derivative. ; The second-order state transition tensor differential equation of the relative motion system is expressed as: In the formula, This is the index of the corresponding position in the matrix; Indicates the first The differential equation for the nth state variable is related to the nth Find the first-order partial derivatives of each state variable; Indicates the first The differential equation for the nth state variable is related to the nth and the Find the second-order partial derivatives of each state variable; express The first-order state transition tensor at time t, whose subscripts correspond to the positions in the matrix; express The second-order state transition tensor at time step 1. Note that initially, the second-order state transition tensor has nine 9s. A zero matrix of 9; From the formula In a relative motion system , , , , , The first-order partial derivatives are all constants, and their corresponding... All matrices are zero matrices; the rest are non-zero matrices. The matrix is represented as: In the formula, Indicates the first Differential equations for each state quantity and Find the second-order partial derivative. , .
4. The method as described in claim 3, characterized in that: 。 5. The method as described in claim 3, characterized in that: 。 6. The method as described in claim 3, characterized in that: 。 7. The method as described in claim 3, characterized in that: 。 8. The method as described in claim 1, characterized in that: The implementation method for step three is as follows: The initial state error range is selected using Mahalanobis distance. The error ellipsoid is represented as a set using a polycell. Any state variable inside the ellipsoid can be regarded as the initial deviation state of the spacecraft. For input control, a sphere centered at the origin and with the maximum permissible acceleration as its radius can be used in acceleration space to represent the set of control inputs that the spacecraft can apply at any given time. Based on the relative motion requirements of the spacecraft, a minimum warning distance is selected. A no-fly zone (BOX) is defined by a cuboid centered on the target spacecraft, with length equal to the minimum warning distance in each coordinate direction of the location space. This region is represented by a set. .
9. The method as described in claim 1, characterized in that: The implementation method for step four is as follows: Deviation of state variables at any given time: The formula With a Taylor expansion to second order, the deviations of the various state variables can be re-expressed as: In the formula, Represents the order of the tensor, when When the state transition tensor degenerates into the state transition matrix; Indicates the first The deviation of each state variable at the initial time; Indicates the first The state transition tensor of order 1 has a dimension of 0. ,in Let be the dimension of the system state variables. For the first The dimension of the augmented state; The formula Represented using a polymorphic cell set, for any time... The deviation set at the next moment can be represented as: In the formula, Tensor mapping operations on sets; ,in, for The state deviation set of the spacecraft at any given moment for Spacecraft control set This represents the Cartesian product operation between two sets; it shifts the center of the resulting deviation set to... The nominal state of relative motion of the spacecraft at a given moment ,get The reachable set of time: In the formula, This represents a translation operation to the center of the set. Through Japanese style By iterating continuously, the reachable set of the spacecraft at any given time can be calculated; then, the reachable set of the spacecraft at each time point can be compared with the no-fly zone mentioned in step three. Determine the positional relationship: In the formula: A function to determine the relationship between sets; Since the reachable set boundary is the farthest distance that a spacecraft can reach by continuously applying control, if the judgment result returns 0, there is no risk of collision between the two spacecraft at the current moment, and no collision warning is triggered. The spacecraft continues to propagate the reachable set along the nominal orbit. If the judgment result returns 1, there is a risk of collision between the spacecraft at the current moment, and the current position of the nominal orbit of the spacecraft is recorded as the warning position.