A method for optimizing terminal state of impact with small celestial bodies based on UT transformation

Abstract

The application discloses a UT transformation-based impact small celestial body orbit terminal state optimization method and belongs to the technical field of aerospace. The application realizes the method as follows: positions and velocities of the earth and small celestial bodies are obtained according to ephemeris and kinematic equations of the small celestial bodies, and the velocity variation of the small celestial body after the impact of a probe is defined as a constant, so that two angles of the velocity change direction of the small celestial body after the impact of the probe are taken as impact design variables; UT transformation is used to perform error evolution of the small celestial body, so that the space-time distribution probability of the target small celestial body in a future period of time is obtained; the space-time distribution probability of the small celestial body is calculated, the instantaneous impact probability under the given impact design is calculated based on a Legendre-Gauss quadrature method, and the overall impact probability of the small celestial body is obtained by integral based on a Newton-Cotes quadrature method; and based on the impact probability calculation method, the impact design variables are optimized in combination with a global optimization algorithm, so that the impact probability of the small celestial body under the optimized variables is the lowest.

Description

Technical Field

[0001] This invention relates to a method for deflecting a probe's orbit around a small celestial body to reduce the probability of an impact on Earth, and particularly to a method for optimizing the terminal state of an impacting small celestial body's orbit based on UT transformation. It belongs to the field of aerospace technology and is applicable to optimizing a probe's defense strategy against asteroid impacts. Background Technology

[0002] Numerous near-Earth asteroids exist in space, intersecting with Earth's orbit. Impacts with Earth often cause disasters such as explosions, earthquakes, and tsunamis, and can also lead to global cooling due to dust dispersion. Asteroid impacts have triggered more than ten mass extinction events of varying degrees, making near-Earth asteroid impacts a major long-term potential threat to humanity. Therefore, research on asteroid defense strategies is urgently needed and has long-term strategic significance. Currently, kinetic energy deflection technology, which uses probes to impact hazardous asteroids and deflect them from their current orbits, is considered a relatively feasible defense response. However, the space environment of asteroids is complex, and their orbits and velocities are difficult to predict accurately. Current traditional methods cannot determine the optimal impact mode for the probe. Therefore, it is necessary to optimize the probe impact design by calculating collision probabilities with uncertainties to maximize the kinetic energy deflection effect.

[0003] Currently, among the methods for calculating the probability of small celestial body impacts, the prior art [1] (see Foster, JL, and Estes, HS. Parametric Analysis of Orbital Debris Collision Probability and Maneuver Rate for Space Vehicles[J]. NASA TR-JSC-25898, Aug. 1992.) is based on linear relative motion, constant Gaussian relative position uncertainty, and zero relative velocity uncertainty, and has studied short-term collision scenarios. However, this method is not applicable to low relative velocity and long-term collision scenarios.

[0004] Prior technology [2] (see Carpenter J R. Non-parametric collision probability for low-velocity encounters[C] / / 2007AAS / AIAA Space Flight MechanicsMeeting.2007(AAS-07-201).) uses Monte Carlo simulation to calculate the collision probability, but its computational cost is very high and it is difficult to obtain accurate collision probability values. Summary of the Invention

[0005] The technical problem to be solved by the UT transform-based method for optimizing the terminal state of an impactor asteroid orbit is to use the change angle of the probe's impact asteroid orbit velocity as the design variable, employ the spherical surface integral method of UT transform and Gaussian mixture calculation to calculate the asteroid impact probability, and combine it with a global optimization algorithm to optimize the terminal state of the impactor asteroid orbit, thereby optimizing the probe's impact on the asteroid, reducing the probability of the asteroid hitting Earth, and avoiding the devastating disaster caused by the asteroid impact on Earth.

[0006] The objective of this invention is achieved through the following technical solution.

[0007] This invention discloses a method for optimizing the terminal state of an impactor's orbit around a small celestial body based on the U-shaped transform (UT). The method obtains the positions and velocities of Earth and the asteroid from ephemeris tables and the asteroid's kinematic equations, and defines the magnitude of the asteroid's velocity change after impact as constant. The method uses the two angles at which the velocity changes direction after impact as impact design variables. Based on the UT transform, the method performs asteroid error evolution to obtain the spatiotemporal probability distribution of the target asteroid over a future period. Based on the calculated spatiotemporal probability distribution, the instantaneous impact probability under a given impact design is calculated using the Legendre-Gauss quadrature method, and the overall impact probability of the asteroid is obtained by integration using the Newton-Cotes quadrature method. Based on the impact probability calculation method, combined with a global optimization algorithm, the impact design variables are optimized to minimize the asteroid impact probability under the optimized variables. This invention is used to optimize probe impacts on asteroids, reduce the probability of an asteroid impact on Earth, and avoid devastating disasters caused by asteroid impacts.

[0008] This invention discloses a method for optimizing the terminal state of impacting a small celestial body orbits based on UT transformation, comprising the following steps:

[0009] Step 1: Query Earth's position and velocity using the Earth's ephemeris, and solve for the target asteroid's position and velocity using the asteroid's kinematic equations. The magnitude of the velocity change dv of the probe impacting the asteroid is constant; the direction angles θ and α of the velocity change vector after the probe impacts the asteroid are used as impact design variables.

[0010] First, based on the Earth's ephemeris, the Earth's position coordinates and velocity for a future period are obtained from the table. Then, based on the given parameters of the target asteroid, the kinematic equations (1) of the asteroid in the equatorial coordinate system are solved to obtain the asteroid's position coordinates and velocity for a future period in the equatorial coordinate system, and the standard deviations of the target asteroid's position and velocity are given. The asteroid parameters include its semi-major axis, eccentricity, orbital inclination, ascending node, perigee argument, and mean perigee at the given initial ephemeris time, as well as the initial ephemeris time.

[0011] Kinematic equations of asteroids in equatorial coordinates:

[0012]

[0013] U s It is the gravitational constant of an asteroid.

[0014] The magnitude of the velocity change dv from the probe impacting the asteroid is constant. The direction angles θ and α of the velocity change vector after the probe's impact are used as the impact design variables. The velocity increment Δv = dv[cosθcosα,sinθcosα,sinα] is added to the asteroid's original velocity. T The average initial velocity of the subsequent asteroid evolution was obtained.

[0015] Step 2: Based on the initial position and average velocity of the asteroid obtained in Step 1 And the given covariance P x The future spatiotemporal distribution probability of the asteroid orbit is obtained by constructing a Sigma point set based on the 5th order CKF and then performing a nonlinear transformation of the Sigma point set based on the UT transformation.

[0016] Based on the asteroid's initial position and average velocity Covariance P x Using 5th-order CKF sampling, construct the Sigma point set:

[0017]

[0018] Where λ is the scale parameter and n is the dimension of variable x.

[0019] Based on the dynamic equation (1) of the asteroid in the equatorial coordinate system, a nonlinear transformation is performed on the Sigma point set to approximate the possible future position and velocity of the asteroid:

[0020] Y i =f(χ) i ), i = 0, 1, ..., 2n (3)

[0021] For the transformed Sigma point set Y i After weighting, we obtain the spatiotemporal distribution probability of the asteroid over a future period of time:

[0022]

[0023] Among them W i (c) and W i (m) Weights:

[0024]

[0025]

[0026]

[0027] Step 3: Based on the spatiotemporal distribution probability of the asteroid orbit calculated in Step 2, using the hard sphere model and the surface integral method of Gaussian mixture calculation, the surface integral of the sphere is simplified to a summation calculation using the Legendre-Gauss quadrature method. The instantaneous collision probability under the direction angle θ and α of the velocity change vector after the given probe impacts the asteroid is calculated. The instantaneous collision probability is then integrated over time using the Newton-Cotes quadrature method to obtain the overall impact probability of the asteroid.

[0028] The spatiotemporal distribution probability obtained in step two is decomposed according to position and velocity:

[0029]

[0030] Based on the hard-sphere model, the instantaneous collision probability of an asteroid is obtained by using Gaussian mixture calculations to address the uncertainties in the orbits of celestial bodies in space:

[0031]

[0032] Where R is the radius of the hard sphere model. This is the normal vector for the area element of the hard sphere model. Parameters for:

[0033]

[0034] in and It is given by the following formula:

[0035]

[0036] Gaussian probability density function P g The definition of is:

[0037]

[0038] To calculate the surface integral of equation (6), the Legendre-Gauss quadrature method is used to transform the integral into a summation calculation:

[0039]

[0040] The Gaussian point x in the quadrature formula (10) k It is a zero of the Legendre polynomial, Ak These are the relevant weighting coefficients. The Legendre polynomial is defined by the following recurrence relation:

[0041]

[0042] The zeros and weights of the N Legendre polynomials are calculated using recursion and the Newton-Raphson method. Substituting equation (10) into equation (6) yields the summation form of the instantaneous collision probability formula:

[0043]

[0044] Integrating equation (12) over time with a fixed time interval, and using the Newton-Cotes quadrature method, the overall collision probability is obtained:

[0045]

[0046] Where n is the number of time steps in the division.

[0047] Step 4: Based on the spatiotemporal distribution probability of the asteroid's orbit predicted in Steps 1 to 3, and combined with a global optimization algorithm, while keeping the magnitude of the probe's velocity dv when it impacts the asteroid constant, optimize the design variables θ and α of the velocity change vector after the probe's impact with the asteroid. The optimized result is the impact probability P of the asteroid with Earth. c The minimum post-impact velocity change angles θ and α are found to minimize the overall impact probability P of the asteroid at the optimized angles θ and α. c Minimum.

[0048] It also includes step five: based on the speed and angle of the probe impacting the asteroid obtained in step four, improve the probe's ability to deflect the asteroid by impacting it with kinetic energy, implement planetary defense, and prevent the asteroid from hitting the Earth.

[0049] Beneficial effects:

[0050] 1. This invention discloses a method for optimizing the final state of an impacted asteroid orbit based on the UT transform. The method obtains the positions and velocities of Earth and the asteroid from ephemeris and the asteroid's kinematic equations, and defines that the magnitude of the asteroid's velocity change after the probe impact remains constant. The method uses two angles representing the change in velocity direction after the probe impacts the asteroid as impact design variables. Based on the UT transform, the method performs asteroid error evolution to obtain the spatiotemporal distribution probability of the target asteroid over a future period. Based on the calculated spatiotemporal distribution probability of the asteroid, the method calculates the asteroid impact probability using discrete Legendre-Gauss and Newton-Cotes quadrature methods. The method simplifies the general spherical surface integral form into a summation form, making it easier to calculate and solve. This saves on the cost of optimizing the final state of an impacted asteroid orbit while maintaining integration accuracy.

[0051] 2. The present invention discloses a method for optimizing the orbital terminal state of impacting asteroids based on UT transformation. It combines a global optimization algorithm with the calculation of the collision probability between asteroids and Earth, and designs an optimization algorithm for the angle of the probe impacting the asteroid. By optimizing the velocity angle of the probe impacting the asteroid, the probability of the asteroid hitting Earth is reduced, and the ability of the probe to deflect the asteroid by kinetic energy impact is improved. Attached Figure Description

[0052] Figure 1 A flowchart of a method for optimizing the terminal state of an impacting asteroid based on UT transformation according to the present invention;

[0053] Figure 2 A graph showing the Earth's position and velocity changes over the next 60 days;

[0054] Figure 3 Spatiotemporal distribution probability-mean plot of 2011AG5 over the next 60 days;

[0055] Figure 4 Spatiotemporal distribution probability-covariance plot of 2011AG5 over the next 60 days;

[0056] Figure 5 A graph showing the change in instantaneous collision probability over time;

[0057] Figure 6 Graph showing the change in impact probability over time. Detailed Implementation

[0058] 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.

[0059] 2011AG5 is a near-Earth asteroid classified as a potentially hazardous object. This example uses this asteroid as the target for a probe collision and employs the method proposed in this invention to calculate the collision angle that minimizes the probability of a collision with Earth.

[0060] like Figure 1 As shown in the figure, this example discloses a method for optimizing the terminal state of an impacted asteroid orbit based on UT transformation. The specific implementation steps are as follows:

[0061] Step 1: Query Earth's position and velocity using the Earth's ephemeris, and solve for the target asteroid's position and velocity using the asteroid's kinematic equations. The magnitude of the velocity change dv of the probe impacting the asteroid is constant; the direction angles θ and α of the velocity change vector after the probe impacts the asteroid are used as impact design variables.

[0062] First, based on the Earth's ephemeris, the Earth's position coordinates and velocity for a future period are obtained from the table. Then, based on the given parameters of the target asteroid, the kinematic equations (1) of the asteroid in the equatorial coordinate system are solved to obtain the asteroid's position coordinates and velocity for a future period in the equatorial coordinate system, and the standard deviations of the target asteroid's position and velocity are given. The asteroid parameters include its semi-major axis, eccentricity, orbital inclination, ascending node, perigee argument, and mean perigee at the given initial ephemeris time, as well as the initial ephemeris time.

[0063] Kinematic equations of asteroids in equatorial coordinates:

[0064]

[0065] U s It is the gravitational constant of an asteroid.

[0066] The magnitude of the velocity change dv from the probe impacting the asteroid is constant. The direction angles θ and α of the velocity change vector after the probe's impact are used as the impact design variables. The velocity increment Δv = dv[cosθcosα,sinθcosα,sinα] is added to the asteroid's original velocity. T The average initial velocity of the subsequent asteroid evolution was obtained.

[0067] The density of asteroid 2011 AG5 is approximately 5.518 × 10⁻⁶. 3 kg·m 3 The mass is 7.9280 × 10⁻⁶. 9 kg. The semi-major axis, eccentricity, orbital inclination, ascending node, argument of perigee, and mean perigee at a given initial ephemeris time of 2011AG5 are 1.43078 AU, 0.390306, 3.682, 135.66, 53.543, and 247.757, respectively. The initial ephemeris time is 00:00 on January 1, 2023. Calculate the current position of the asteroid r = [-41.211 + 171.32 + -6.0316] × 10⁻⁶. 6 The probe's initial position and average velocity are given. The velocity of the probe upon impact with the asteroid is v = [-24.5710 -16.6778 + 1.8727] km / s. The magnitude of the velocity of the probe upon impact with the asteroid is fixed at dv = 0.3391 m / s. The initial velocity change angle after the initial impact is [θα] = [-1.3077 -0.4336]. Covariance P x =diag[500km 500km 500km 1m / s 1m / s 1m / s]. Calculate the Earth's position and velocity changes over the next 60 days using equation (1), such as... Figure 2As shown.

[0068] Step 2: Based on the initial position and average velocity of the asteroid obtained in Step 1 And the given covariance P x The future spatiotemporal distribution probability of the asteroid orbit is obtained by constructing a Sigma point set based on the 5th order CKF and then performing a nonlinear transformation of the Sigma point set based on the UT transformation.

[0069] First, based on the asteroid's initial position and average velocity... Covariance P x Using 5th-order CKF sampling, construct the Sigma point set:

[0070]

[0071] Where λ is the scale parameter and n is the dimension of variable x.

[0072] Based on the dynamic equation (1) of the asteroid in the equatorial coordinate system, a nonlinear transformation is performed on the Sigma point set to approximate the possible future position and velocity of the asteroid:

[0073] Y i =f(χ) i ), i = 0, 1, ..., 2n (16)

[0074] For the transformed Sigma point set Y i After weighting, we obtain the spatiotemporal distribution probability of the asteroid over a future period of time:

[0075]

[0076] Among them W i (c) and W i (m) Weights:

[0077]

[0078]

[0079]

[0080] Based on the relevant parameters of asteroid 2011 AG5, the calculated spatiotemporal distribution probability is as follows: Figure 3 and Figure 4 As shown.

[0081] Step 3: Based on the spatiotemporal distribution probability of the asteroid orbit calculated in Step 2, using the hard sphere model and the surface integral method of Gaussian mixture calculation, the surface integral of the sphere is simplified to a summation calculation using the Legendre-Gauss quadrature method. The instantaneous collision probability under the direction angle θ and α of the velocity change vector after the given probe impacts the asteroid is calculated. The instantaneous collision probability is then integrated over time using the Newton-Cotes quadrature method to obtain the overall impact probability of the asteroid.

[0082] The spatiotemporal distribution probability obtained in step two is decomposed according to position and velocity:

[0083]

[0084] Based on a general hard-sphere model, the instantaneous collision probability of an asteroid is obtained by using Gaussian mixture calculations to address the uncertainties in the orbits of celestial bodies in space:

[0085]

[0086] Where R is the radius of the hard sphere model. This is the normal vector for the area element of the hard sphere model. Parameters for:

[0087]

[0088] in and It is given by the following formula:

[0089]

[0090] Gaussian probability density function P g The definition of is:

[0091]

[0092] To calculate the surface integral of equation (6), the Legendre-Gauss quadrature method is used to transform the integral into a summation calculation:

[0093]

[0094] The Gaussian point x in the quadrature formula (10) k It is a zero of the Legendre polynomial, A k These are the relevant weighting coefficients. The Legendre polynomial is defined by the following recurrence relation:

[0095]

[0096] The zeros and weights of the N Legendre polynomials are calculated using recursion and the Newton-Raphson method. Substituting equation (10) into equation (6) yields the summation form of the instantaneous collision probability formula:

[0097]

[0098] Integrating equation (12) over time with a fixed time interval, and using the Newton-Cotes quadrature method, the overall collision probability can be obtained:

[0099]

[0100] Where n is the number of time steps in the division.

[0101] Substituting the spatiotemporal distribution probability of the asteroid obtained in step two over the next 60 days into the collision probability calculation in step three, the instantaneous collision probability P is obtained. c The curve of (t) changing with time is as follows: Figure 5 As shown, the impact probability P obtained by time integration c like Figure 6 As shown, P on day 60 c This means the probability of an asteroid hitting Earth is 0.0462%.

[0102] Step 4: Based on the impact probability calculation method described in Steps 1 to 3, and combined with a global optimization algorithm, while keeping the magnitude of the probe's velocity dv constant when it impacts the asteroid, optimize the design variables θ and α of the velocity change vector after the probe's impact with the asteroid. The optimized result is the impact probability P of the asteroid on Earth. c The minimum post-impact velocity change angles θ and α are found to minimize the overall impact probability P of the asteroid at the optimized angles θ and α. c Minimum.

[0103] Using the genetic algorithm in the global optimization algorithm for optimization design, the angle of velocity change of the probe after impacting the asteroid is obtained as [θα] = [-2.7534 -0.1008], and the probability of the asteroid impacting Earth is reduced to 0.03525%.

[0104] 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 optimizing the terminal state of an impacting asteroid's orbit based on UT transformation, characterized in that: Includes the following steps, Step 1: Query Earth's position and velocity using the Earth's ephemeris, and solve for the target asteroid's position and velocity using the asteroid's kinematic equations; the change in velocity of the probe impacting the asteroid. The magnitude is constant, and the direction and angle of the vector change based on the velocity of the probe after impacting the asteroid. and Design variables for impact; Step 2: Based on the initial position and average velocity of the asteroid obtained in Step 1 and the given covariance The future spatiotemporal distribution probability of the asteroid orbit is obtained by constructing a Sigma point set based on the 5th order CKF and then performing a nonlinear transformation of the Sigma point set based on the UT transformation. Step 3: Based on the spatiotemporal distribution probability of the asteroid orbit calculated in Step 2, using the hard sphere model and the Gaussian mixture method for surface integral calculation, the Legendre-Gauss quadrature method is used to simplify the surface integral to a summation calculation, thus calculating the direction angle of the velocity change vector after the given probe impacts the asteroid. and The instantaneous collision probability under the impact design variables is calculated, and the Newton-Cotes quadrature method is used to integrate the instantaneous collision probability over time to obtain the overall impact probability of the asteroid. Step 4: Based on the spatiotemporal distribution probability of the asteroid's orbit predicted in Steps 1 to 3, and combined with a global optimization algorithm, maintain the change in the probe's velocity when impacting the asteroid. The design variable whose magnitude remains constant is the vector that changes the velocity of the probe after it collides with the asteroid. and Optimization was performed to obtain the probability of an asteroid colliding with Earth. Minimal velocity change angle after impact and This makes the optimized angle and Below, the overall probability of an asteroid impact. Minimum, that is, optimizing the terminal state of the impacting small celestial body's orbit based on UT transformation.

2. The method for optimizing the terminal state of impacting a small celestial body orbit based on UT transformation as described in claim 1, characterized in that: The implementation method for step one is as follows: First, based on the Earth's ephemeris, the position coordinates and velocity of the Earth for a future period of time are obtained by looking up the table; based on the given target asteroid parameters, the kinematic equations (1) of the asteroid in the equatorial coordinate system are solved to obtain the position coordinates and velocity of the asteroid in the equatorial coordinate system for a future period of time, and the standard deviation of the target asteroid's position and velocity is given; the asteroid parameters include its orbital semi-major axis, eccentricity, orbital inclination, ascending node, perigee argument and the mean perigee angle at the given initial ephemeris time, as well as the initial ephemeris time; Kinematic equations of asteroids in equatorial coordinates: in It is the gravitational constant of the asteroid; Change in velocity of the probe impacting the asteroid The magnitude is constant, and the direction and angle of the vector change based on the velocity of the probe after impacting the asteroid. and The impact design variable is the velocity increment generated by the probe impacting the asteroid, added to the asteroid's own velocity. The average initial velocity of the subsequent asteroid evolution was obtained. .

3. The method for optimizing the terminal state of impacting a small celestial body orbit based on UT transformation as described in claim 2, characterized in that: The second step is implemented as follows: Based on the asteroid's initial position and average velocity Covariance Using 5th-order CKF sampling, construct the Sigma point set: in For scale parameters, For variables The dimension; Based on the dynamic equation (1) of the asteroid in the equatorial coordinate system, a nonlinear transformation is performed on the Sigma point set to approximate the possible future position and velocity of the asteroid: For the transformed Sigma point set After weighting, we obtain the spatiotemporal distribution probability of the asteroid over a future period of time: in and Weights: 。 4. The method for optimizing the terminal state of impacting a small celestial body orbit based on UT transformation as described in claim 3, characterized in that: The method for implementing step three is as follows: The spatiotemporal distribution probability obtained in step two is decomposed according to position and velocity: Based on the hard-sphere model, the instantaneous collision probability of an asteroid is obtained by using Gaussian mixture calculations to address the uncertainties in the orbits of celestial bodies in space: Where R is the radius of the hard sphere model; The normal vector of the area unit of the hard sphere model; parameters for: in and It is given by the following formula: Gaussian probability density function The definition of is: To calculate the surface integral of equation (6), the Legendre-Gauss quadrature method is used to transform the integral into a summation calculation: The Gaussian point of the quadrature formula (10) These are the zeros of the Legendre polynomial. These are the relevant weighting coefficients; the Legendre polynomial is defined by the following recurrence relation: The zeros and weights of the N Legendre polynomials are calculated using recursion and the Newton-Raphson method; substituting equation (10) into equation (6) yields the summation form of the instantaneous collision probability formula: Integrating equation (12) over time with a fixed time interval, and using the Newton-Cotes quadrature method, the overall collision probability is obtained: Where n is the number of time steps in the division.

5. A method for optimizing the terminal state of an impacting asteroid orbit based on UT transformation as described in claim 1, 2, 3, or 4, characterized in that: It also includes step five, which, based on the speed and angle of the probe impacting the asteroid obtained in step four, improves the probe's ability to deflect the asteroid by impacting it with kinetic energy, implements planetary defense, and avoids the asteroid impacting Earth.

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