A task success calculation method based on circular probability error
By establishing a two-dimensional joint probability density function and a circular probability error model for the landing point of an aircraft, the problem of quantifying the landing point error of an aircraft was solved, enabling accurate assessment of mission success and optimization of multi-aircraft collaborative missions.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-02-15
- Publication Date
- 2026-06-16
AI Technical Summary
Existing technologies struggle to accurately quantify the systematic and stochastic characteristics of aircraft landing point errors, leading to inaccurate mission success assessments.
A circular probability error-based approach is adopted. By establishing a two-dimensional joint probability density function of the longitudinal and lateral deviations of the aircraft landing point, the circular probability error is calculated. Combined with the radius of influence, a mission success probability model is established, and the calculation formula is improved by considering the target movement characteristics.
It enables accurate quantification of aircraft precision, provides precise calculation of mission success, especially for the evaluation of moving targets, and provides a reference for optimized launch strategies for multi-aircraft collaborative missions.
Smart Images
Figure CN122220673A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to a method for calculating the success rate of a task based on circular probability error, belonging to the field of computer simulation technology. Background Technology
[0002] Accuracy is a measure of how close a measurement result is to the true value; it is a statistical concept. To accurately determine the success probability of an aircraft's mission, theoretically, a large number of repeatable tests are needed. However, in practice, only a limited number of tests can be conducted, requiring analysis of limited experimental data to determine the success probability of the aircraft's mission.
[0003] Depending on the target, different concepts are used to describe the precision of an aircraft performing a mission. Generally, for fixed targets, such as ground buildings, precision is used to describe the accuracy of the mission. For moving targets, probability is used, referring to the probability that the aircraft will land in a specific area centered on the target.
[0004] Extensive engineering practice has shown that as the number of repeated launches increases, the geometric position of the impact point dispersion center of the same type of aircraft tends to stabilize. The deviation of the impact point dispersion center from the target point reflects the systematic error component (longitudinal and lateral) of the impact point error, describing the accuracy of the aircraft. Obviously, the smaller the impact point dispersion center is from the target point, the higher the accuracy. The distance of the impact point position from the impact point dispersion center reflects the random characteristics of the impact point deviation, describing the launch density. Obviously, the smaller the impact point dispersion, the lower the launch density. Accuracy and density together constitute the precision of the aircraft. High precision reflects the combined degree of systematic and random errors in the aircraft's impact point deviation. Summary of the Invention
[0005] This invention addresses the problem of quantifying the systematic and random characteristics of aircraft landing point errors, as well as the difficulty in accurately assessing overall mission success, by proposing a mission success calculation method based on circular probability error.
[0006] The technical solution adopted by the present invention to solve the above problems is as follows: The present invention includes the following steps: Step 1: Based on the longitudinal and lateral deviations of the aircraft's landing point, establish a two-dimensional joint probability density function for the aircraft's landing point distribution; Step 2: Determine the circular probability error of the aircraft landing point dispersion based on the two-dimensional joint probability density function; Step 3: Based on the circular probability error and the influence radius of the spacecraft, establish a probability calculation model for the success of the spacecraft mission; Step 4: Obtain the success probability of the target mission of the aircraft based on the aircraft mission success probability calculation model.
[0007] Furthermore, step 1 specifically includes: Based on the latitude and longitude coordinates of the launch point, theoretical landing point, and actual landing point of the aircraft, the longitudinal and lateral deviations of the aircraft landing point are calculated using the spherical trigonometry formula. Based on the statistical characteristics of longitudinal and lateral deviations, a joint probability density function following a two-dimensional normal distribution is established. The joint probability density function includes the mean longitudinal deviation, the mean lateral deviation, the standard deviation longitudinally, the standard deviation laterally, and the longitudinal and lateral correlation coefficients.
[0008] Furthermore, step 2 specifically includes: The longitudinal and lateral standard deviations are obtained based on the two-dimensional joint probability density function. Based on the longitudinal and lateral standard deviations, an integral equation for the circular probability error is established, where the integral equation indicates that the probability of the aircraft landing point within a circular domain with the target point as the center and the radius as the circular probability error is equal to a preset threshold. The circular probability error is obtained by numerically solving the integral equation.
[0009] Furthermore, the integral equation for the circular probability error is established, including: Determine whether the longitudinal standard deviation and the transverse standard deviation are equal. When the longitudinal standard deviation and the transverse standard deviation are equal and the systematic deviation is zero, determine the analytical relationship between the circular probability error and the standard deviation based on the characteristics of the Rayleigh distribution, and obtain the integral equation of the original probability error, where the circular probability error is equal to a preset multiple of the standard deviation.
[0010] Furthermore, step 3 specifically includes: Based on the Rayleigh distribution parameters of the uncompleted mission of the spacecraft, a deviation probability density function is established. Based on the influence radius and deviation probability density function of the aircraft, a mission success probability calculation model is established. The mission success probability calculation model is a double integral model in polar coordinates, where the integration domain of the double integral model is a circular domain centered on the target point and with the influence radius as the radius. The mission success probability of a single aircraft is obtained by solving the double integral model.
[0011] Furthermore, a probability calculation model for the success of spacecraft missions is established, including: Obtain the displacement of the mission target during the flight of the aircraft, wherein the displacement follows a Rice distribution; Based on the statistical characteristics of displacement and the aircraft's miss distance, an equivalent total error model is established, where the equivalent total error is the square root of the sum of the variances of displacement and miss distance. Based on the equivalent total error and the aircraft's influence radius, the probability of mission success is obtained.
[0012] Furthermore, step 4 specifically includes: A binomial distribution model is established based on the mission success probability of a single aircraft and the preset number of aircraft to be launched; Based on the binomial distribution model, the optimal launch strategy for the mission is determined, and the probability that at least one aircraft will complete the mission when multiple aircraft are performing the mission is calculated, where the probability is a function of the number of aircraft launched and the probability of a single aircraft completing the mission. Based on the probability that at least one aircraft completes the mission, obtain the combined mission success probability under the condition that multiple aircraft perform the mission.
[0013] Furthermore, determining the optimal spacecraft launch strategy for the mission includes: Based on the probability of a single aircraft completing a mission and the preset number of aircraft to be launched, a binomial distribution model is established. According to the binomial distribution model and the mission success probability model of a single aircraft, the average loss of the mission mode in which all aircraft jointly execute missions and the mission mode in which a single aircraft continuously executes missions are calculated respectively. Based on the comparison results of the average losses, the optimal aircraft launch strategy for the mission is determined.
[0014] The beneficial effects of this invention are: 1. This invention completes an unbiased estimation method for the mean and variance of the longitudinal and lateral deviations of the landing point of an aircraft based on the normal distribution, determines the calculation model of the circular probability error (including polar coordinate simplification and numerical integration), realizes the statistical quantification of the accuracy of the aircraft using finite simulation test data, can accurately distinguish the accuracy and density of the aircraft, and provides a theoretical basis for the accuracy detection of the aircraft.
[0015] 2. This invention improves the formula for calculating the success probability of a mission by considering the target's movement characteristics. It treats the target's movement distance as independent random noise and equates it to the total system error, thus solving the problem that the Rice distribution has no analytical solution. This enables accurate calculation of the success probability of a mission with a moving target and establishes a quantitative judgment standard for the probability of mission failure and the influence range of the aircraft.
[0016] 3. This invention provides the theoretical basis for multi-aircraft collaborative mission execution, derives the success probability formula and average loss calculation model for multi-aircraft collaborative mission execution and sequential mission execution, and clarifies that the collaborative mission execution mode has a higher first mission success rate and lower mission loss, providing a quantitative reference for the formulation of aircraft launch strategies. Attached Figure Description
[0017] Figure 1 This is a flowchart illustrating a method for calculating task success based on circular probability error. Figure 2 This is a schematic diagram of the circular probability error. Figure 3The landing point is scattered in a circle and A diagram illustrating the relationship between circles; Figure 4 This is a schematic diagram showing the probability of task completion and the integral region. Detailed Implementation
[0018] The accuracy and density of impact point dispersion can be mathematically described using the mean and standard deviation of a normal distribution. Therefore, aircraft accuracy can also be characterized by the mean and standard deviation of the longitudinal and lateral deviations of the impact points. For the longitudinal and lateral deviations of the impact points, two-dimensional random variables can be used. As a measure of aircraft accuracy, its probability density function is shown in the following formula, with mean... , Describe the accuracy standard deviations of the longitudinal and lateral dispersion of the landing points, respectively. , : Describes the density of the vertical and horizontal distribution of the landing points, respectively.
[0019] When the longitudinal and lateral deviations of the landing point are considered independent, i.e., the correlation coefficient... , The probability density function is: (1); Furthermore, when considering a systematic bias of zero, i.e. At that time, The probability density functions are as follows: (2); When the standard deviations of the longitudinal and transverse axes are equal, that is... That is, the landing points are distributed in a circle, then The probability density function is: (3); mean , This describes a systematic error in the longitudinal and lateral deviations of the landing point. In engineering practice, the systematic deviation of an aircraft's accuracy is generally not too large. In fact, if the systematic error is too large, it is usually considered that there is a defect or error in the design, production, or use of the aircraft, and the cause must be found and corrected to reduce the systematic error.
[0020] The normal distribution is the most widely used distribution type in nature, and its study is relatively thorough. It is a random variable that follows a normal distribution. It is a normal distribution independent samples, with a mean of _ . The variance is Then the estimates of the mean and variance are: (4); in, The sample mean. The above estimates of the mean and variance are unbiased estimates, representing the sample variance. In fact, random variables... mean It remains a random variable and follows a normal distribution. Regarding the mean... ,have (5); That is, the sample mean Follow the mean variance is The variance follows a normal distribution. It can be seen that after repeated measurements, the variance decreased, becoming only a fraction of the original variance. This is also why repeated measurements can improve measurement accuracy.
[0021] When the sample variance is unknown, the sample mean is... With sample variance The following relationships exist: (6); That is, obeying the degree of freedom is of Distribution. Sample variance and The following relationship exists: (7); That is, obeying the degree of freedom is The chi-square distribution is used. Interval estimates of the mean and variance can be calculated based on this distribution.
[0022] S1: Calculation of circular probability error; Circular probability error (CEP) refers to the error in a circle centered at the target point. Draw a circle with radius such that the probability of the aircraft landing within that circle is 50%. What is the radius of this circle? That is called When the longitudinal and lateral deviations of the landing point are independent, The radius can be expressed as: (8); in, This represents the radius of the integration domain in the above double integral. The circular probability error can be expressed as... Figure 2 express.
[0023] The integral represented by equation (8) does not have a universal analytical solution. Numerical integration methods are generally required to find analytical and approximate solutions in some special cases. This invention addresses the case where the systematic deviation of the landing point is not considered, and the landing point exhibits circular dispersion characteristics (i.e., ...). , Let's discuss this further.
[0024] According to equation (8), we have: (9); Using polar coordinates, let , We can obtain: (10); Right now for: .
[0025] At this point, the dispersion circle of the landing point is... The relationship of circles is as follows Figure 3 As shown.
[0026] At this time The relationship between the radius of a circle and the linear probability error is as follows: (11); At this point, the probability errors of the longitudinal and transverse lines are equal, that is... .
[0027] S2: Calculation of the probability of mission completion for the aircraft; The probability of an aircraft successfully completing its mission is equal to the integral of the longitudinal and lateral probability density functions of the landing point over the target area. Assume the planar target area is... According to equation (1), the success probability is... )for: (12); Theoretically, planar target area It can be a closed region of any shape, for example Figure 3 The gray area in the diagram represents the target region, which can be described using a polygon, and its boundaries are... The success probability of a task is calculated by integrating the probability density function of the landing point distribution over the target area.
[0028] like Figure 4 As shown, for space targets, whether the mission has been accomplished is generally measured by the probability of failure. For a specific aircraft target, given the radius of influence... The number of failures during a particular mission by the aircraft was Then: (13); If the spacecraft fails to complete its mission, it is considered to have accomplished its mission; otherwise, it is considered to have failed.
[0029] Simplify equation (12) using polar coordinates (i.e., let...) After that, we can obtain: (14); in, Indicates the radius of influence of the aircraft. This represents the standard deviation of the distance between the actual landing point of the aircraft and the target.
[0030] Considering that the target will continue to move during the flight of the aircraft when it strikes the target, equation (14) needs to be improved by taking the distance the target moves during the flight of the aircraft as... ( (It follows a Rice distribution), and its probability density function is: (15); Since equation (15) cannot yield a definite analytical solution, this invention will... Treating it as an independent random noise, the total system error can therefore be equivalent to... Therefore, the formula for the final success probability of the spacecraft mission can be expressed as: (16); S3: Calculation of the success probability of multi-aircraft collaborative mission execution; In real-world scenarios, the success rate of a single-engine aircraft mission is generally low, and the completion of the target mission cannot be guaranteed. In such cases, multiple aircraft can be used to perform the same mission to increase the success rate.
[0031] S301: Success rate of multiple aircraft performing missions independently and continuously; Multiple identical aircraft continuously execute missions against the same target until the mission is completed, at which point launches cease. This method depends on the success probability of a single-engine aircraft's mission. Let's assume the success probability of a single-engine flight is... Until the During the next launch The probability of completing the task is: (17); Based on the above task execution method, the first Completing the target task this time means that the previous The mission was not completed in the first launch. Therefore, the average loss required to complete at least one mission is: (18); make , , Then we have: (19); By summing the numbers, we get: (20); because ,so It is a convergent geometric sequence whose sum converges to: (twenty one); but The sum of the sequence is: (twenty two); According to formula (331): (twenty three); Therefore, it can be assumed that the average loss required for the aircraft to complete the mission at least once is It should be noted that formula (23) gives the average loss, which is a theoretical value and not the actual loss. For example, if the probability of a single aircraft completing a mission is 0.5, then on average, two aircraft are needed to complete one mission.
[0032] 302: Success rate of completing a mission through coordinated launch of multiple spacecraft Assuming a collaboration Launch similar types of aircraft Each spacecraft is launched independently, and the success rate is 100% each time. If using random variables express The number of aircraft that completed the mission among the launched aircraft, then It follows the binomial distribution shown in formula (24).
[0033] (twenty four); but At least complete the launch of the aircraft The probability of secondary hot tasks for: (25); when Sometimes, (26); Therefore, a coordinated launch When launching the same type of aircraft, the average number of aircraft required to complete the target mission is: (27); Comparing formulas (24) and (27), we can see that: (28); because Therefore, the following formula always holds true: (29); In other words, the average loss during a coordinated launch is always greater than that of consecutive independent launches. However, when the success probability of an independent launch is low, the probability of a coordinated launch successfully completing the mission on the first attempt increases significantly. For example, when... When launching three aircraft in a single launch, according to formula (26), the success probability is 0.784, which significantly increases the probability of completing the first mission in multiple coordinated launches. However, launching aircraft independently requires determining whether the mission is completed after the previous launch before deciding whether to launch the next aircraft. This method has a longer execution time and lower efficiency.
[0034] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make some modifications or alterations to the above-disclosed technical content to create equivalent embodiments without departing from the scope of the present invention. Any simple modifications, equivalent substitutions, and improvements made to the above embodiments without departing from the scope of the present invention, based on the technical essence of the present invention and within the spirit and principles of the present invention, shall still fall within the protection scope of the present invention.
Claims
1. A method for calculating task success based on circular probability error, characterized in that, include: Step 1: Based on the longitudinal and lateral deviations of the aircraft's landing point, establish a two-dimensional joint probability density function for the aircraft's landing point distribution; Step 2: Determine the circular probability error of the aircraft landing point dispersion based on the two-dimensional joint probability density function; Step 3: Based on the circular probability error and the influence radius of the aircraft, establish a probability calculation model for the success of the aircraft mission; Step 4: Obtain the success probability of the target mission of the aircraft based on the aircraft mission success probability calculation model.
2. The method for calculating task success based on circular probability error according to claim 1, characterized in that, Step 1 specifically includes: Based on the latitude and longitude coordinates of the launch point, theoretical landing point, and actual landing point of the aircraft, the longitudinal and lateral deviations of the aircraft landing point are calculated using the spherical trigonometry formula. Based on the statistical characteristics of longitudinal and lateral deviations, a joint probability density function following a two-dimensional normal distribution is established. The joint probability density function includes the mean longitudinal deviation, the mean lateral deviation, the standard deviation longitudinally, the standard deviation laterally, and the longitudinal and lateral correlation coefficients.
3. The method for calculating task success based on circular probability error according to claim 1, characterized in that, Step 2 specifically includes: The longitudinal and lateral standard deviations are obtained based on the two-dimensional joint probability density function. Based on the longitudinal and lateral standard deviations, an integral equation for the circular probability error is established, where the integral equation indicates that the probability of the aircraft landing point within a circular domain with the target point as the center and the radius as the circular probability error is equal to a preset threshold. The circular probability error is obtained by numerically solving the integral equation.
4. The method for calculating task success based on circular probability error according to claim 3, characterized in that, Establish the integral equation for the circular probability error, including: Determine whether the longitudinal standard deviation and the transverse standard deviation are equal. When the longitudinal standard deviation and the transverse standard deviation are equal and the systematic deviation is zero, determine the analytical relationship between the circular probability error and the standard deviation based on the characteristics of the Rayleigh distribution, and obtain the integral equation of the original probability error, where the circular probability error is equal to a preset multiple of the standard deviation.
5. The method for calculating task success based on circular probability error according to claim 1, characterized in that, Step 3 specifically includes: Based on the Rayleigh distribution parameters of the uncompleted mission of the spacecraft, a deviation probability density function is established. Based on the influence radius and deviation probability density function of the aircraft, a mission success probability calculation model is established. The mission success probability calculation model is a double integral model in polar coordinates, where the integration domain of the double integral model is a circular domain centered on the target point and with the influence radius as the radius. The mission success probability of a single aircraft is obtained by solving the double integral model.
6. The method for calculating task success based on circular probability error according to claim 5, characterized in that, Establish a probability calculation model for the success of spacecraft missions, including: Obtain the displacement of the mission target during the flight of the aircraft, wherein the displacement follows a Rice distribution; Based on the statistical characteristics of displacement and the aircraft's miss distance, an equivalent total error model is established, where the equivalent total error is the square root of the sum of the variances of displacement and miss distance. Based on the equivalent total error and the aircraft's influence radius, the probability of mission success is obtained.
7. The method for calculating task success based on circular probability error according to claim 1, characterized in that, Step 4 specifically includes: A binomial distribution model is established based on the mission success probability of a single aircraft and the preset number of aircraft to be launched; Based on the binomial distribution model, the optimal launch strategy for the mission is determined, and the probability that at least one aircraft will complete the mission when multiple aircraft are performing the mission is calculated, where the probability is a function of the number of aircraft launched and the probability of a single aircraft completing the mission. Based on the probability that at least one aircraft completes the mission, obtain the combined mission success probability under the condition that multiple aircraft perform the mission.
8. The method for calculating task success based on circular probability error according to claim 7, characterized in that, Determine the optimal spacecraft launch strategy for the mission, including: Based on the probability of a single aircraft completing a mission and the preset number of aircraft to be launched, a binomial distribution model is established. According to the binomial distribution model and the mission success probability model of a single aircraft, the average loss of the mission mode in which all aircraft jointly execute missions and the mission mode in which a single aircraft continuously executes missions are calculated respectively. Based on the comparison results of the average losses, the optimal aircraft launch strategy for the mission is determined.