A multi-stage trajectory uncertainty propagation method for high-speed flying objects

Abstract

The invention discloses a method for propagating uncertainty of a multi-stage trajectory of a high-speed flying body, which belongs to the technical field of research on trajectories of high-speed flying bodies. Firstly, a multi-stage trajectory model is established based on a three-degree-of-freedom center-of-mass motion model of a high-speed flying body; secondly, uncertainty factors in an actual flight environment of the high-speed flying body are taken into consideration in the multi-stage trajectory model, so as to construct a multi-stage uncertainty trajectory model; then, a Latin hypercube sampling strategy is adopted to design an uncertainty quantization propagation algorithm based on a combination of non-embedded polynomial chaos expansion and a random response surface method, so as to transform a random dynamic system into a high-order deterministic dynamic system; then, an overdetermined set of equations is solved by least quadratic regression; a fourth-order Runge-Kutta method is adopted to solve a regression coefficient matrix; finally, under the action of a specified optimal control quantity, a multi-stage trajectory output response statistical moment is obtained, so as to obtain a propagation mechanism of uncertainty in the random dynamic system of a high-speed flying body.

Description

Technical Field

[0001] The invention belongs to the technical field of high-speed flying body trajectory research, and in particular relates to a high-speed flying body multi-stage trajectory uncertainty propagation method. Background Art

[0002] The design of high-speed flying object's flight trajectory plays an important role in the research of flying objects, and its design directly affects the final mission execution effect. In the actual flight process of high-speed flying objects, there are certain uncertainties in the parameters such as dynamic model, aerodynamics, and environment, and the optimization design variables, objective functions, and constraints are closely related to these parameters.

[0003] At present, in the field of high-speed flying body trajectory research, most of the existing research results focus on deterministic systems and single-stage trajectories, and the uncertainty and multi-stage trajectories of high-speed flying bodies in actual flight environments are not fully considered. The design of idealized trajectory is very likely to lead to deviations in optimization targets, design variables exceeding the range, and constraint conditions breaking through boundaries, which seriously affect the overall mission effect of high-speed flying bodies. For the study of trajectories of high-speed flying bodies in uncertain flight environments, there are linear covariance analysis methods based on trajectory optimization, genetic algorithms, Gaussian pseudo-spectral methods, and Monte Carlo shooting methods based on uncertainty quantification propagation analysis methods, but there are few studies on multi-stage trajectories. This invention will study the uncertainty propagation law of multi-stage trajectories of high-speed flying bodies to provide a basis for subsequent multi-stage trajectory optimization work. Summary of the invention

[0004] The purpose of the present invention is to provide a method for propagating uncertainty in multi-stage trajectories of a high-speed flying body, so as to solve the problem that the existing trajectory research technology of the high-speed flying body does not adequately consider the multi-stage trajectories and uncertainties.

[0005] To achieve the above object, the present invention provides a method for propagating uncertainty of a multi-stage trajectory of a high-speed flying object, comprising the following steps:

[0006] Step 1: Based on the three-degree-of-freedom motion model of the high-speed flying body, a multi-stage trajectory model is established, wherein the trajectory model satisfies the constraint conditions;

[0007] Step 2: Taking the uncertainty factors in the actual flight environment of the high-speed flying object into consideration in the multi-stage trajectory model, a multi-stage uncertainty trajectory model of the high-speed flying object is constructed;

[0008] Step 3: Using the Latin hypercube sampling strategy, an uncertainty quantification propagation algorithm based on the combination of non-embedded polynomial chaos expansion and random response surface method is designed to transform the random dynamic system into a high-order deterministic dynamic system;

[0009] Step 4: The transformed high-order deterministic dynamic system is a set of overdetermined equations, which are solved by least quadratic regression; the fourth-order Runge-Kutta method is used to solve the regression coefficient matrix;

[0010] Step 5: Under the action of the specified optimal control quantity, the optimal control quantity is brought into the deterministic dynamic system model after dimension expansion to obtain the statistical moment of the multi-stage trajectory output response, and then the propagation method of uncertainty factors in the random dynamic system is obtained.

[0011] Preferably, the three-degree-of-freedom motion model based on the high-speed flying body is as follows:

[0012] ;

[0013] In the formula, is the flight speed of the projectile, To boost the average thrust of the rocket, is the mass of the glide projectile, is the acceleration due to gravity, is the ballistic inclination angle, is the ballistic deflection angle, is the aerodynamic drag, is the aerodynamic lift, is the aerodynamic lateral force, is the position of the guided projectile in the ground coordinate system, To balance the angle of attack, To balance the sideslip angle, is the mass consumption of the projectile per unit time, and its value is determined by the performance of the booster rocket. The expressions of aerodynamic drag, aerodynamic lift and aerodynamic lateral force are as follows:

[0014] ;

[0015] In the formula, is the atmospheric density, is the reference area of ​​the high-speed flying body, is the induction coefficient, is the zero lift drag coefficient, is the lift coefficient derivative, is the derivative of the lateral force coefficient, whose value can be obtained by interpolation from the wind tunnel test data. The state vector is expressed as , the control vector is expressed as , the parameter vector is expressed as .

[0016] Preferably, the multi-stage trajectory model of a high-speed flying body includes four stages, and the starting time of each stage is , the end time of each segment is , as follows:

[0017] Take-off phase: It starts at the launch position and ends at the booster rocket ignition position; the booster rocket needs to ignite some time after the projectile leaves the barrel. The time interval of this trajectory is ;

[0018] Boosting phase: starts from the booster rocket ignition position and ends at the ignition cut-off position; the boosting duration is a fixed value, and the time interval is ;

[0019] Climbing phase: starts from the closed position of the booster rocket and ends at the glide start control point. In the actual flight environment of the projectile, due to the existence of random disturbances, the projectile needs climbing time to restore its own balance after the rocket boost. The climbing time interval is ;

[0020] Gliding stage: It starts from the glide start point and ends at the landing. Its time interval is .

[0021] Preferably, the constraints in step 1 include take-off and climb phase constraints, boost phase constraints, glide phase constraints, time and state constraints, and the specific expressions are as follows:

[0022] The constraints for takeoff and climb are:

[0023] ;

[0024] The boosting phase constraints are:

[0025] ;

[0026] In the formula, is the mean thrust of the rocket engine, is the charge amount, Working hours for booster rockets;

[0027] The glide segment constraint is

[0028] ;

[0029] In the formula, , are the maximum allowable balanced angle of attack and the maximum balanced sideslip angle respectively;

[0030] The time and state constraints are

[0031] ;

[0032] In the formula, represents the final state of the nth stage, Represents the initial state of the n+1th stage.

[0033] Preferably, the uncertainty factors in step 2 include the uncertainty of state parameters and the uncertainty of model parameters; the uncertainty of state parameters is described as the uncertainty of the 6-dimensional initial flight state, specifically including the flight speed, trajectory inclination, trajectory deflection, and the particle position in the three-dimensional space of the high-speed flying body; the uncertainty of model parameters is described as a 5-dimensional uncertainty vector, specifically including the uncertainty of the average thrust of the booster rocket, the uncertainty of the meteorological environment, the uncertainty of the lift coefficient, the uncertainty of the drag coefficient, and the uncertainty of the lateral force coefficient; wherein the uncertainty of the meteorological environment is different from the uncertainty of the other four model parameters, and the uncertainty of the meteorological environment is specifically reflected in the atmospheric density deviation, temperature deviation and wind interference in the real flight environment of the high-speed flying body, the temperature deviation is attributed to the aerodynamic deviation, and the wind interference is attributed to the model deviation, so the uncertainty of the meteorological environment is the air density deviation, and the ratio of its standard deviation to the air density reference value at different altitudes is:

[0034] ;

[0035] Where y represents the flying height of the high-speed flying object.

[0036] Preferably, the high-speed flying body multi-stage uncertainty trajectory model constructed in step 2 is specifically:

[0037] ;

[0038] in, is the actual state parameter, is the model parameter, the superscript “~” indicates the nominal state parameter, and Each element in represents the corresponding initial state and initial model parameters, and the operator " ” is the Hadamard product, and Each element in is a random number with mean 0 and variance 1. and The elements in the formula represent the deviation of each parameter from the nominal value, which is expressed as the state parameter and model parameter. The ratio to the nominal value of the corresponding parameter defines the degree of deviation of each parameter from the nominal value. is the standard deviation corresponding to each parameter;

[0039] The uncertainty in atmospheric density is expressed as:

[0040] ;

[0041] in, is the reference value of atmospheric density, represents random numbers that follow a standard normal distribution, Indicates the ratio of the air density to the reference value at different altitudes;

[0042] set up ,make The random uncertainty variables of the state parameters and model parameters of the three-degree-of-freedom motion model of the high-speed flying body in step 1 are represented. The three-degree-of-freedom uncertainty dynamic model of the high-speed flying body is represented as a set of random ordinary differential equations:

[0043] .

[0044] Preferably, in step 3, a Latin hypercube sampling strategy is adopted to design an uncertainty quantization propagation algorithm based on a combination of non-embedded polynomial chaos expansion and random response surface method, and the specific process of converting the random dynamic system into a high-order deterministic dynamic system is as follows:

[0045] S31, let the number of sample points selected be H, and select samples twice the number of unknown chaotic polynomial coefficients in the standard random space (i.e. ) can ensure the accuracy of the solution and transform the selected H sample points from the standard random N space to the original random x space. The specific expression is as follows:

[0046] ;

[0047] In the formula, the superscript "S" indicates that the indicated quantity is a sample point, and the subscript jh Indicates j The h-th dimension sample point among the sample points, For the conversion process, Represents the original random space j The first of the sample points h dimensional sample points, Indicates standard random N Space j The first of the sample points h dimensional sample points, for random numbers of normal distribution type, the expression of the conversion process is:

[0048] ;

[0049] In the formula, and are the h-th dimension random input variables The mean and variance of The true response model is called to obtain the true response value at each sample point;

[0050] S32. According to the generalized polynomial chaos expansion theory, the random ordinary differential equations are approximated into polynomial form, as follows:

[0051] ;

[0052] In the formula, is a polynomial basis of multidimensional random variables, yes The coefficient vector of yes The order of yes dimensional standard random variable, the number of PCE coefficients for , Indicates the expansion order. The number of PCE coefficients increases with the order of the PCE model. and random variables Dimension The specific functional relationship is:

[0053] ;

[0054] In the formula, is the maximum expansion order of the polynomial;

[0055] S33, the polynomial form of the stochastic ordinary differential equations of S32 is transformed into a high-order deterministic dynamic system in the stochastic dynamic system:

[0056] ;

[0057] In the formula, is the PCE coefficient of the state variable, To control the amount, is the PCE coefficient of the model parameters, is the true response value of H sample points brought into the original 6-dimensional system, is the transfer matrix of the transformed deterministic dynamic system, , The symbol represents the Kronecker product, is an overdetermined system of equations, represents the intermediate variable, , .

[0058] Preferably, the specific process of step 4 is as follows:

[0059] S41, solving the overdetermined equations obtained in step 3 by the least quadratic regression method:

[0060] ;

[0061] In the formula, is the residual of the jth sample point;

[0062] S42. According to the principle of minimum residual sum of squares, we can get:

[0063] ;

[0064] In the formula, is the coefficient regression matrix, and the fourth-order Runge-Kutta method is used to solve the above formula to obtain the variation law of all PCE coefficients with time t.

[0065] Preferably, the optimal control quantity in step 5 is a variation array consisting of the values ​​of the flight attack angle and sideslip angle of the high-speed flying body that change with time.

[0066] Preferably, the response statistical moment in step 5 is the corresponding expectation and variance. According to the calculated expectation and variance, the statistical law of each state quantity in the original dynamic system can be obtained. The specific expression is as follows:

[0067] ;

[0068] In the formula, E is the operator for solving the expectation, and is the intermediate variable, It means to find the average value of the state variables during the motion of a high-speed flying body. It means to find the variance of the state variables in the process of high-speed flying body. , All are intermediate variables.

[0069] Therefore, the present invention adopts the above-mentioned method for propagating uncertainty in multi-stage trajectories of high-speed flying bodies, defines a corresponding multi-stage trajectory dynamics model, solves the problem of quantitative propagation of multi-stage trajectories considering uncertainty factors, and the resulting method and system solve the problem of subsequent multi-stage trajectory optimization.

[0070] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] Figure 1 It is an overall flow chart of a method for propagating uncertainty in multi-stage trajectories of a high-speed flying body according to the present invention. DETAILED DESCRIPTION

[0072] The following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the invention claimed for protection, but merely represents selected embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0073] See also Figure 1 , a high-speed flying body multi-stage trajectory uncertainty propagation method, comprising the following steps:

[0074] Step 1: Based on the three-degree-of-freedom motion model of the high-speed flying body, a multi-stage trajectory model is established, wherein the trajectory model satisfies the constraint conditions;

[0075] The three-degree-of-freedom motion model based on high-speed flying body is as follows:

[0076] ;

[0077] In the formula, is the flight speed of the projectile, To boost the average thrust of the rocket, is the mass of the glide projectile, is the acceleration due to gravity, is the ballistic inclination angle, is the ballistic deflection angle, is the aerodynamic drag, is the aerodynamic lift, is the aerodynamic lateral force, is the position of the guided projectile in the ground coordinate system, To balance the angle of attack, To balance the sideslip angle, is the mass consumption of the projectile per unit time, and its value is determined by the performance of the booster rocket. The expressions of aerodynamic drag, aerodynamic lift and aerodynamic lateral force are as follows:

[0078] ;

[0079] In the formula, is the atmospheric density, is the reference area of ​​the high-speed flying body, is the induction coefficient, is the zero lift drag coefficient, is the lift coefficient derivative, is the derivative of the lateral force coefficient, whose value can be obtained by interpolation from the wind tunnel test data. The state vector is expressed as , the control vector is expressed as , the parameter vector is expressed as .

[0080] The multi-stage trajectory model of a high-speed flying body consists of four stages, and the starting time of each stage is , the end time of each segment is , as follows:

[0081] Take-off phase: It starts at the launch position and ends at the booster rocket ignition position; the booster rocket needs to ignite some time after the projectile leaves the barrel. The time interval of this trajectory is ;

[0082] Boosting phase: starts from the booster rocket ignition position and ends at the ignition cut-off position; the boosting duration is a fixed value, and the time interval is ;

[0083] Climbing phase: starts from the closed position of the booster rocket and ends at the glide start control point. In the actual flight environment of the projectile, due to the existence of random disturbances, the projectile needs climbing time to restore its own balance after the rocket boost. The climbing time interval is ;

[0084] Gliding stage: It starts from the glide start point and ends at the landing. Its time interval is .

[0085] The constraints include take-off and climb phase constraints, boost phase constraints, glide phase constraints, time and state constraints. The specific expressions are as follows:

[0086] The constraints for takeoff and climb are:

[0087] ;

[0088] The boosting phase constraints are:

[0089] ;

[0090] In the formula, is the mean thrust of the rocket engine, is the charge amount, Working hours for booster rockets;

[0091] The glide segment constraint is

[0092] ;

[0093] In the formula, , are the maximum allowable balanced angle of attack and maximum balanced sideslip angle respectively;

[0094] The time and state constraints are

[0095] ;

[0096] In the formula, represents the final state of the nth stage, Represents the initial state of the n+1th stage.

[0097] Step 2: Consider the uncertainty factors in the actual flight environment of the high-speed flying body into the multi-stage trajectory model, and construct a multi-stage uncertainty trajectory model for the high-speed flying body; wherein, the uncertainty factors include the uncertainty of state parameters and the uncertainty of model parameters; the uncertainty of state parameters is described as the uncertainty of the 6-dimensional initial flight state, specifically including the flight speed, trajectory inclination, trajectory deflection, and the particle position in the three-dimensional space of the high-speed flying body; the uncertainty of model parameters is described as a 5-dimensional uncertainty vector, specifically including the uncertainty of the average thrust of the booster rocket, the uncertainty of the meteorological environment, the uncertainty of the lift coefficient, the uncertainty of the drag coefficient, and the uncertainty of the lateral force coefficient; wherein the uncertainty of the meteorological environment is different from the uncertainty of the other four model parameters, and the uncertainty of the meteorological environment is specifically reflected in the atmospheric density deviation, temperature deviation and wind interference in the actual flight environment of the high-speed flying body. The temperature deviation is attributed to the aerodynamic deviation, and the wind interference is attributed to the model deviation. Therefore, the uncertainty of the meteorological environment is the air density deviation, and the ratio of its standard deviation to the air density reference value at different altitudes is:

[0098] ;

[0099] Where y represents the flying height of the high-speed flying object.

[0100] The multi-stage uncertainty trajectory model of high-speed flying objects is constructed as follows:

[0101] ;

[0102] in, is the actual state parameter, is the model parameter, the superscript “~” indicates the nominal state parameter, and Each element in represents the corresponding initial state and initial model parameters, and the operator " ” is the Hadamard product, and Each element in is a random number with mean 0 and variance 1. and The elements in the formula represent the deviation of each parameter from the nominal value, which is expressed as the state parameter and model parameter. The ratio to the nominal value of the corresponding parameter defines the degree of deviation of each parameter from the nominal value. is the standard deviation corresponding to each parameter;

[0103] The uncertainty in atmospheric density is expressed as:

[0104] ;

[0105] in, is the reference value of atmospheric density, represents random numbers that follow a standard normal distribution, Indicates the ratio of the air density to the reference value at different altitudes;

[0106] set up ,make The random uncertainty variables of the state parameters and model parameters of the three-degree-of-freedom motion model of the high-speed flying body in step 1 are represented. The three-degree-of-freedom uncertainty dynamic model of the high-speed flying body is represented as a set of random ordinary differential equations:

[0107] .

[0108] Step 3: Using the Latin hypercube sampling strategy, design an uncertainty quantification propagation algorithm based on the combination of non-embedded polynomial chaos expansion and random response surface method to transform the random dynamic system into a high-order deterministic dynamic system; the specific process is as follows:

[0109] S31, let the number of sample points selected be H, and select samples twice the number of unknown chaotic polynomial coefficients in the standard random space (i.e. ) can ensure the accuracy of the solution and transform the selected H sample points from the standard random N space to the original random x space. The specific expression is as follows:

[0110] ;

[0111] In the formula, the superscript "S" indicates that the indicated quantity is a sample point, and the subscript jh Indicates j The h-th dimension sample point among the sample points, For the conversion process, Represents the original random space j The first of the sample points h dimensional sample points, Indicates standard random N Space j The first of the sample points h dimensional sample points, for random numbers of normal distribution type, the expression of the conversion process is:

[0112] ;

[0113] In the formula, and are the h-th dimension random input variables The mean and variance of The true response model is called to obtain the true response value at each sample point;

[0114] S32. According to the generalized polynomial chaos expansion theory, the random ordinary differential equations are approximated into polynomial form, as follows:

[0115] ;

[0116] In the formula, is a polynomial basis of multidimensional random variables, yes The coefficient vector of yes The order of yes dimensional standard random variable, the number of PCE coefficients for , Indicates the expansion order. The number of PCE coefficients increases with the order of the PCE model. and random variables Dimension The specific functional relationship is:

[0117] ;

[0118] In the formula, is the maximum expansion order of the polynomial;

[0119] S33, the polynomial form of the stochastic ordinary differential equations of S32 is transformed into a high-order deterministic dynamic system in the stochastic dynamic system:

[0120] ;

[0121] In the formula, is the PCE coefficient of the state variable, To control the amount, is the PCE coefficient of the model parameters, is the true response value of H sample points brought into the original 6-dimensional system, is the transfer matrix of the transformed deterministic dynamic system, , The symbol represents the Kronecker product, is an overdetermined system of equations, represents the intermediate variable, , .

[0122] Step 4: The transformed high-order deterministic dynamic system is a set of overdetermined equations, which are solved by least quadratic regression. The fourth-order Runge-Kutta method is used to solve the regression coefficient matrix. The specific process is as follows:

[0123] S41, solving the overdetermined equations obtained in step 3 by the least quadratic regression method:

[0124] ;

[0125] In the formula, is the residual of the jth sample point;

[0126] S42. According to the principle of minimum residual sum of squares, we can get:

[0127] ;

[0128] In the formula, is the coefficient regression matrix, and the fourth-order Runge-Kutta method is used to solve the above formula to obtain the variation law of all PCE coefficients with time t.

[0129] Step 5. Under the action of the specified optimal control quantity, the optimal control quantity is brought into the deterministic dynamic system model after dimension expansion to obtain the multi-stage trajectory output response statistical moment, and then obtain the propagation method of uncertainty factors in the random dynamic system; wherein, the optimal control quantity is a change array composed of the values ​​of the flight angle of attack and sideslip angle of the high-speed flying body changing with time; the response statistical moment is the corresponding expectation and variance, and the statistical law of each state quantity in the original dynamic system can be obtained according to the calculated expectation and variance. The specific expression is as follows:

[0130] ;

[0131] In the formula, E is the operator for solving the expectation, and is the intermediate variable, It means to find the average value of the state variables during the motion of a high-speed flying body. It means to find the variance of the state variables in the process of high-speed flying body. , All are intermediate variables.

[0132] Therefore, the present invention adopts the above-mentioned method for propagating uncertainty in the multi-stage trajectory of a high-speed flying body, including a three-degree-of-freedom center of mass motion model based on the high-speed flying body, defining a four-stage trajectory model based on the motion characteristics of the high-speed flying body, and giving constraints to the motion state of each stage; constructing a dynamic model that considers the uncertainty of the initial state parameters of the multi-stage trajectory and the uncertainty of the model parameters; adopting a combination of non-embedded polynomial chaos expansion and random response surface method to transform the random dynamic system into a high-order deterministic dynamic system, and transforming it into a set of overdetermined equations; solving the overdetermined equations by least quadratic regression; using the fourth-order Runge-Kutta method to solve the regression coefficient matrix; and under the action of the specified optimal control quantity, bringing it into the expanded-dimensional dynamic system model to carry out the uncertainty propagation of the multi-stage trajectory of the high-speed flying body.

[0133] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solution of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solution to deviate from the spirit and scope of the technical solution of the present invention.

Claims

1. A method for propagating uncertainty in multi-stage trajectories of a high-speed flying object, characterized in that: The following steps are involved: Step 1: Based on the three-degree-of-freedom motion model of the high-speed flying body, a multi-stage trajectory model is established, wherein the trajectory model satisfies the constraint conditions; Step 2: Taking the uncertainty factors in the actual flight environment of the high-speed flying object into consideration in the multi-stage trajectory model, a multi-stage uncertainty trajectory model of the high-speed flying object is constructed; Step 3: Using the Latin hypercube sampling strategy, an uncertainty quantification propagation algorithm based on the combination of non-embedded polynomial chaos expansion and random response surface method is designed to transform the random dynamic system into a high-order deterministic dynamic system; Step 4: The transformed high-order deterministic dynamic system is a set of overdetermined equations, which are solved by least quadratic regression; the fourth-order Runge-Kutta method is used to solve the regression coefficient matrix; Step 5: Under the action of the specified optimal control quantity, the optimal control quantity is brought into the dimensionally expanded deterministic dynamic system model to obtain the statistical moment of the multi-stage trajectory output response, and then the propagation method of uncertainty factors in the random dynamic system is obtained; In step 3, the Latin hypercube sampling strategy is used to design an uncertainty quantification propagation algorithm based on the combination of non-embedded polynomial chaos expansion and random response surface method. The specific process of transforming the random dynamic system into a high-order deterministic dynamic system is as follows: S31, let the number of sample points selected be H, select samples twice the number of unknown chaotic polynomial coefficients in the standard random space, transform the selected H sample points from the standard random N space to the original random x Space, the specific expression is as follows: In the formula, the superscript "S" indicates that the indicated quantity is a sample point, and the subscript jh Indicates j The h-th dimension sample point among the sample points, For the conversion process, Represents the original random space j The first of the sample points h dimensional sample points, Indicates standard random N Space j The first of the sample points h dimensional sample points, for random numbers of normal distribution type, the expression of the conversion process is: In the formula, and are the h-th dimension random input variables The mean and variance of The true response model is called to obtain the true response value at each sample point; S32. According to the generalized polynomial chaos expansion theory, the random ordinary differential equations are approximated into polynomial form, as follows: In the formula, is a polynomial basis of multidimensional random variables, yes The coefficient vector of yes The order of yes dimensional standard random variable, the number of PCE coefficients for , Indicates the expansion order. The number of PCE coefficients increases with the order of the PCE model. and random variables Dimension The specific functional relationship is: In the formula, is the maximum expansion order of the polynomial; S33, the polynomial form of the stochastic ordinary differential equations of S32 is transformed into a high-order deterministic dynamic system in the stochastic dynamic system: In the formula, is the PCE coefficient of the state variable, To control the amount, is the PCE coefficient of the model parameters, is the true response value of H sample points brought into the original 6-dimensional system, is the transfer matrix of the transformed deterministic dynamic system, , The symbol represents the Kronecker product, is an overdetermined system of equations, represents the intermediate variable, , .

2. A high-speed flying object multi-stage trajectory uncertainty propagation method according to claim 1, characterized in that: The three-degree-of-freedom motion model based on high-speed flying body is as follows: In the formula, is the flight speed of the projectile, To boost the average thrust of the rocket, is the mass of the glide projectile, is the acceleration due to gravity, is the ballistic inclination angle, is the ballistic deflection angle, is the aerodynamic drag, is the aerodynamic lift, is the aerodynamic lateral force, is the position of the guided projectile in the ground coordinate system, To balance the angle of attack, To balance the sideslip angle, is the mass consumption of the projectile per unit time, and its value is determined by the performance of the booster rocket. The expressions of aerodynamic drag, aerodynamic lift and aerodynamic lateral force are as follows: In the formula, is the atmospheric density, is the reference area of ​​the high-speed flying body, is the induction coefficient, is the zero lift drag coefficient, is the lift coefficient derivative, is the derivative of the lateral force coefficient, whose value can be obtained by interpolation from the wind tunnel test data. The state vector is expressed as , the control vector is expressed as , the parameter vector is expressed as .

3. A high-speed flying object multi-stage trajectory uncertainty propagation method according to claim 2, characterized in that: The multi-stage trajectory model of a high-speed flying body consists of four stages, and the starting time of each stage is , the end time of each segment is , as follows: Take-off phase: It starts at the launch position and ends at the booster rocket ignition position; the booster rocket needs to ignite some time after the projectile leaves the barrel. The time interval of this trajectory is ; Boosting phase: starts from the booster rocket ignition position and ends at the ignition cut-off position; the boosting duration is a fixed value, and the time interval is ; Climbing phase: starts from the closed position of the booster rocket and ends at the glide start control point. In the actual flight environment of the projectile, due to the existence of random disturbances, the projectile needs climbing time to restore its own balance after the rocket boost. The climbing time interval is ; Gliding stage: It starts from the glide start point and ends at the landing. Its time interval is .

4. A high-speed flying object multi-stage trajectory uncertainty propagation method according to claim 3, characterized in that: The constraints in step 1 include takeoff and climb constraints, boost constraints, glide constraints, time and state constraints. The specific expressions are as follows: The constraints for takeoff and climb are: The boosting phase constraints are: In the formula, is the mean thrust of the rocket engine, is the charge amount, Working hours for booster rockets; The glide segment constraint is In the formula, , are the maximum allowable balanced angle of attack and the maximum balanced sideslip angle respectively; The time and state constraints are In the formula, represents the final state of the nth stage, Represents the initial state of the n+1th stage.

5. A high-speed flying object multi-stage trajectory uncertainty propagation method according to claim 4, characterized in that: The uncertainty factors in step 2 include the uncertainty of state parameters and the uncertainty of model parameters; the uncertainty of state parameters is described as the uncertainty of the 6-dimensional initial flight state, including flight speed, trajectory inclination, trajectory deflection, and particle position in the three-dimensional space of the high-speed flying body; the uncertainty of model parameters is described as a 5-dimensional uncertainty vector, including the uncertainty of the average thrust of the booster rocket, the uncertainty of the meteorological environment, the uncertainty of the lift coefficient, the uncertainty of the drag coefficient, and the uncertainty of the lateral force coefficient; the uncertainty of the meteorological environment is different from the uncertainty of the other four model parameters. The uncertainty of the meteorological environment is specifically reflected in the atmospheric density deviation, temperature deviation and wind interference in the real flight environment of the high-speed flying body. The temperature deviation is attributed to the aerodynamic deviation, and the wind interference is attributed to the model deviation. Therefore, the uncertainty of the meteorological environment is the air density deviation, and the ratio of its standard deviation to the air density reference value at different altitudes is: Where y represents the flying height of the high-speed flying object.

6. A high-speed flying object multi-stage trajectory uncertainty propagation method according to claim 5, characterized in that: The multi-stage uncertainty trajectory model of the high-speed flying body constructed in step 2 is specifically: in, is the actual state parameter, is the model parameter, the superscript "~" indicates the nominal state parameter, and Each element in represents the corresponding initial state and initial model parameters, operator" ” is the Hadamard product, and Each element in is a random number with mean 0 and variance 1. and The elements in represent the degree of deviation of each parameter from the nominal value, with the state parameters and model parameters The ratio to the nominal value of the corresponding parameter defines the degree of deviation of each parameter from the nominal value. is the standard deviation corresponding to each parameter; The uncertainty in atmospheric density is expressed as: in, is the reference value of atmospheric density, represents random numbers that follow a standard normal distribution, Indicates the ratio of the air density to the reference value at different altitudes; set up ,make The random uncertainty variables of the state parameters and model parameters of the three-degree-of-freedom motion model of the high-speed flying body in step 1 are represented. The three-degree-of-freedom uncertainty dynamic model of the high-speed flying body is represented as a set of random ordinary differential equations: 。 7. A high-speed flying object multi-stage trajectory uncertainty propagation method according to claim 6, characterized in that: The specific process of step 4 is as follows: S41, solving the overdetermined equations obtained in step 3 by the least quadratic regression method: In the formula, is the residual of the jth sample point; S42. According to the principle of minimum residual sum of squares, we can get: In the formula, is the coefficient regression matrix, and the fourth-order Runge-Kutta method is used to solve the above formula to obtain the variation law of all PCE coefficients with time t.

8. A high-speed flying object multi-stage trajectory uncertainty propagation method according to claim 7, characterized in that: The optimal control quantity in step 5 is a variable array composed of the values ​​of the flight angle of attack and sideslip angle of the high-speed flying body changing with time.

9. A high-speed flying object multi-stage trajectory uncertainty propagation method according to claim 8, characterized in that: The response statistics in step 5 are the corresponding expectations and variances, and the specific expressions are as follows: In the formula, E is the operator for solving the expectation, and is the intermediate variable, It means to find the average value of the state variables during the motion of a high-speed flying body. It means to find the variance of the state variables in the process of high-speed flying body. , All are intermediate variables.

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