CN112861249A - Gliding trajectory degradation solution along with speed change based on attack angle and resistance acceleration - Google Patents
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Abstract
The invention belongs to the technical field of guidance, and relates to a gliding trajectory degradation solution along with speed change based on an attack angle and a resistance acceleration. The gliding trajectory degradation solution along with the change of the speed based on the attack angle and the resistance acceleration comprises the following steps: establishing a dynamic model of the gliding aircraft; establishing an approximate equation of a dynamic model of the gliding aircraft containing three-dimensional motion; and solving the analytic solution of the first approximation equation. Compared with the traditional method, the method has the advantages that the method takes the attack angle and the resistance acceleration as control variables, establishes the three-dimensional gliding aircraft dynamics approximate equation, obviously improves the precision, can quickly solve the large-range and three-dimensional maneuvering trajectory, and can meet the requirements of on-line trajectory planning and quick guidance solving of the missile-borne computer in engineering realization.
Description
Technical Field
The invention belongs to the technical field of guidance, and particularly relates to a gliding trajectory degradation along with speed change based on an attack angle and a resistance acceleration.
Background
The guidance planning of the hypersonic trajectory is complex and the calculation amount is large, the traditional numerical calculation method for solving the differential equation is long in calculation time, and the requirements for online trajectory planning and guidance quick calculation are difficult to meet. Because ballistic planning needs to execute a large number of iterative operations, the analytic solutions have great value in the aspect of online ballistic planning, but most of glide ballistic analytic solutions in the prior art are difficult to meet the engineering application requirements of online ballistic planning and guidance quick solving due to the defects in the aspects of calculation speed and precision when processing three-dimensional maneuvering ballistics with large-scale transverse motion, so that a price-reducing solution which can quickly solve the large-scale three-dimensional maneuvering ballistics and can meet the engineering application requirements in precision is necessary to be researched.
Disclosure of Invention
The invention aims to provide a gliding trajectory degradation solution of an attack angle and a resistance acceleration along with the change of speed, the attack angle and the resistance acceleration are used as control variables, the requirements of high-speed gliding trajectory rapid guidance solution and online trajectory planning are met, a large-range three-dimensional maneuvering trajectory can be rapidly solved compared with a traditional analysis solution, and the calculation speed and the calculation precision can meet the engineering application requirements of online guidance rapid solution.
The technical scheme adopted by the invention is as follows:
1. the gliding trajectory degradation solution along with the change of the speed based on the attack angle and the resistance acceleration is characterized in that:
the method comprises the following steps:
step 1: establishing a dynamics model of a gliding aircraft with a velocity v as an independent variable
Defining: initial longitude, initial latitude and initial azimuth angle of gliding aircraft in geocentric coordinate system are respectively lambda0、φ0And alpha0;
Defining: the offset geocentric coordinate system rotates the geocentric coordinate system around the z, y and x axes by a rotating angle lambda respectively0、-φ0And alpha0Obtaining a coordinate system;
defining: the longitude, the latitude and the heading angle of the gliding aircraft under the offset geocentric coordinate system are respectively lambda, phi and psi;
introducing dimensionless heightsWhereinre=Re+Hm,HmIs the average flying height R of the gliding aircraft in the gliding flight sectioneIs the radius of the earth, reThe distance between the center of mass of the aircraft and the geocentric; v. the0And vfThe speeds of the aircraft at the glide starting point and the glide ending point respectively;
establishing a dynamic model of the gliding aircraft with the speed v as an independent variable:
wherein A isDFor resisting acceleration, LDyIs the longitudinal lift-to-drag ratio, LDzIs the transverse lift-drag ratio, omegaeIs the angular velocity of rotation of the earth, mu is the gravitational constant of the earth;
Cσ≈2vωe(sinφ-sinψtanθcosφ)
Cθ≈-2vωecosφcosψ
step 2: approximate equation for establishing dynamics model of gliding aircraft based on Newton iteration method
Let xvWhen v, the variable of formula (1) is replaced and rewritten to y' (x)v)=f(xv,y),
remember yiIs y' (x)v)=f(xvY) and setting the i +1 th iteration value as yi+1=yi+δyiTo, forA first order Taylor approximation is performed on theta, phi, psi, lambda to expand the function y' (x)v)=f(xvY) is constructed as a newton iteration in function space, i.e.:
wherein
knowing the value y of the ith iterationiThen, the solution y of the (i + 1) th step can be obtained by using the formula (2)i+1;
Defining an equation formed by using the formula (2) at the Nth time as an approximate equation of the formula (1) at the Nth time, and solving the approximate equation into an approximate solution of the Nth time; definition ofθ0,φ0,ψ0,λ0Is a zero-order approximate solution of formula (1);θ1,φ1,ψ1,λ1is a first order approximation solution of formula (1);
when the offset geocentric system equation is adopted, the gliding aircraft approximately flies to the target along the equator of the offset geocentric coordinate system, and the gliding flight section has phi approximately equal to 0, theta approximately equal to 0, psi approximately equal to 0,The characteristic of (1) is that the zero-order approximate solution is:
substituting the equation (3) into the equation (2) to obtain a first order approximation equation of the glide flight segment as follows:
wherein
As can be seen from the equation (4), the Jacobian matrix has the characteristic of block sparsity, and theta can be solved first1Andand phi1And psi1Then solve for λ1;
Due to phi1And psi1Is the solution of the lateral motion variable,and theta1Is a solution of a longitudinal motion variable, phi1And psi1To pairAnd theta1Insensitive, equation (4) is further simplified as:
step 3: establishing the relation between the three-dimensional section and the motion state
Using angle of attack and resistance acceleration as control variables, order
Wherein, c0,c1,c2Are all controlled variable resistance acceleration ADThe design parameters of (1); α is the angle of attack, and f (v) is the angle of attack control function designed with velocity as the independent variable.
Defining: the flight height of the gliding aircraft is h, and the first derivative of the aerodynamic drag acceleration and the aerodynamic drag speed with respect to the energy is obtained by:
finishing to obtain:
wherein: cDIs the drag coefficient of the gliding aircraft, hsThe coefficients of the exponential formula being a function of atmospheric density;
neglecting the effect of the drag coefficient derivative, the approximate altitude of the gliding aircraft can be found:
and (3) solving a second derivative related to energy for the resistance acceleration to obtain a longitudinal lift-drag ratio:
LDy=a(AD″-b)
wherein the content of the first and second substances,
and (3) utilizing a pneumatic coefficient table or a pneumatic interpolation function to obtain a transverse lift-drag ratio:
wherein:
step 4: analytic solution to solve first order approximation equation
Substituting formula (6) for formula (3) according to the definition of the offset geocentric coordinate system, and calculating from v0By v, we get:
the first two formulas of the combined vertical type (5) can obtain
Order to
Left-multiplying equation (7) by M (v, v)0) After arrangement, the differential multiplication rule is applied reversely, and then integration and arrangement are carried out to obtain:
using M (v, v)0) Solving its inverse matrix [ M (v, v)0)]-1Formula (8) is substituted forFinish to obtain phi1And psi1The analytical formula (2):
lagrange interpolation polynomial in interval [ v ] by taking root of Legendre of degree n as sampling nodef,v0]Approximate λ (v, v) on segment0)、cos(λ(xv,v0))m3(xv)、sin(λ(xv,v0))m3(xv) Obtaining:
substituting formula (9) to obtain phi1And psi1Analytic solution of (2):
wherein: c. Cp0(k)、cp1(k)、cp2(k)Is the coefficient of the interpolation polynomial;
by calculated phi1And psi1Substituting formula (5) to obtain λ1Analytic solution of (2):
the invention has the beneficial effects that:
the invention establishes a gliding trajectory degradation solution of the attack angle and the resistance acceleration along with the change of the speed, realizes the rapid calculation of the high-altitude gliding three-dimensional trajectory, has good calculation speed and precision in the aspect of calculating the large-range three-dimensional maneuvering trajectory compared with the traditional analytic solution, and can meet the application requirement of the online guidance planning engineering of the large-range three-dimensional gliding aircraft.
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Detailed Description
The technical scheme of the invention is further specifically described by combining specific embodiments.
1. The gliding trajectory degradation solution along with the change of the speed based on the attack angle and the resistance acceleration is characterized in that:
the method comprises the following steps:
step 1: establishing a dynamics model of a gliding aircraft with a velocity v as an independent variable
Defining: initial longitude, initial latitude and initial azimuth angle of gliding aircraft in geocentric coordinate system are respectively lambda0、φ0And alpha0;
Defining: the offset geocentric coordinate system rotates the geocentric coordinate system around the z, y and x axes by a rotating angle lambda respectively0、-φ0And alpha0Obtaining a coordinate system;
defining: the longitude, the latitude and the heading angle of the gliding aircraft under the offset geocentric coordinate system are respectively lambda, phi and psi;
taking into account that the flying height during gliding flight is a small quantity relative to the radius of the earth, dimensionless heights are introducedWhereinre=Re+Hm,HmFor the average flight height, R, of the gliding aircraft in the gliding flight sectioneIs the radius of the earth, reThe distance between the center of mass of the aircraft and the geocentric; v. the0And vfThe speeds of the aircraft at the glide starting point and the glide ending point respectively;
the gliding flight section is that after the gliding aircraft enters again, the gliding aircraft has a small ballistic inclination angle and a long distanceA flight segment away from the landing area toward the land. Establishing a dynamics model of the gliding aircraft with the velocity v as an independent variable (introducing dimensionless altitude into a classical model of the hypersonic dynamics)To obtain):
wherein A isDFor resisting acceleration, LDyIs the longitudinal lift-to-drag ratio, LDzIs the transverse lift-drag ratio, omegaeIs the angular velocity of rotation of the earth, mu is the gravitational constant of the earth;
Cσ≈2vωe(sinφ-sinψtanθcosφ)
Cθ≈-2vωecosφcosψ
step 2: approximate equation for establishing dynamics model of gliding aircraft based on Newton iteration method
Let xvWhen v, the variable of formula (1) is replaced and rewritten to y' (x)v)=f(xv,y),
and (3) a Newton iteration method is applied to approximate the multi-dimensional nonlinear trajectory equation into a plurality of low-dimensional variable coefficient linear differential equations so as to realize high-precision rapid calculation of the flight trajectory. Remember yiIs y' (x)v)=f(xvY) and setting the i +1 th iteration value as yi+1=yi+δyi. To solve equation (2), a zero-order approximation solution of equation (1) needs to be provided before starting the iterative computation. The closer the zero-order solution is to the true solution, the number of iterations required to achieve the same accuracy requirementThe less. For the problem of aircraft guidance, a larger ballistic estimation error is usually allowed, if a zero-time solution is closer to a true solution, the requirement can be met by one-time solution, and the method is suitable for the problem of aircraft guidanceA first order Taylor approximation is performed on theta, phi, psi, lambda to expand the function y' (x)v)=f(xvY) is constructed as a newton iteration in function space, i.e.:
wherein
knowing the value y of the ith iterationiThen, the solution y of the (i + 1) th step can be obtained by using the formula (2)i+1;
Defining an equation formed by using the formula (2) at the Nth time as an approximate equation of the formula (1) at the Nth time, and solving the approximate equation into an approximate solution of the Nth time; definition ofθ0,φ0,ψ0,λ0Is a zero-order approximate solution of formula (1);θ1,φ1,ψ1,λ1is a first order approximation solution of formula (1);
c used for calculating the Jacobian matrix in order to obtain the Jacobian matrix which is accurate and has the blocking sparse characteristic of decoupling the equationθ、CσIn the expression, only ω is ignorede 2And J2, trying to retain ωeThe first order item of (2). When the offset geocentric system equation is adopted, the gliding aircraft approximately flies to the target along the equator of the offset geocentric coordinate system, and the gliding flight section has phi ≈ 0, theta ≈ 0, psi ≈ 0,The characteristic of (1) is that the zero-order approximate solution is:
substituting the equation (3) into the equation (2) to obtain a first order approximation equation of the glide flight segment as follows:
wherein
Equation (4) retains the basic characteristics of equation (1) and contains a first term of the earth rotation rate, so that the accuracy of the 1-time approximate solution is obviously improved compared with the accuracy of the 0-time approximate solution, and the 1-time approximate solution is used for guidance calculation to obtain better guidance performance. If desired, equation (2) can be used multiple times to obtain a more accurate solution.
As can be seen from the equation (4), the Jacobian matrix has the characteristic of block sparsity, and theta can be solved first1Andand phi1And psi1Then solve for λ1;
Due to theta1And psi1Is in the transverse directionThe solution of the motion variable is then determined,and theta1Is a solution of a longitudinal motion variable, phi1And psi1To pairAnd theta1Insensitive, equation (4) is further simplified as:
step 3: establishing the relation between the three-dimensional section and the motion state
On the premise of meeting the requirements of trajectory planning, the angle of attack and the resistance acceleration are taken as control variables to order
Wherein, c0,c1,c2Are all controlled variable resistance acceleration ADThe design parameters of (1); α is the angle of attack, and f (v) is the angle of attack control function designed with velocity as the independent variable.
Defining: the flight height of the gliding aircraft is h, and the first derivative of the aerodynamic drag acceleration and the aerodynamic drag speed with respect to the energy is obtained by:
finishing to obtain:
wherein: cDIs the drag coefficient of the gliding aircraft, hsThe coefficients of the exponential formula being a function of atmospheric density;
neglecting the effect of the drag coefficient derivative, the approximate altitude of the gliding aircraft can be found:
and (3) solving a second derivative related to energy for the resistance acceleration to obtain a longitudinal lift-drag ratio:
LDy=a(AD″-b)
wherein the content of the first and second substances,
and (3) utilizing a pneumatic coefficient table or a pneumatic interpolation function to obtain a transverse lift-drag ratio:
wherein:
step 4: analytic solution to solve first order approximation equation
Since the zeroth order generalized longitude equation is independent, first derivation is made here about λ0Is defined by the offset geocentric coordinate system, formula (6) is substituted for formula (3), and the formula (6) is derived from v0By v, we get:
to solve for phi1,ψ1The first two formulas of the combined vertical type (5) can be obtained
Order to
Left-multiplying equation (7) by M (v, v)0) After arrangement, the differential multiplication rule is applied reversely, and then integration and arrangement are carried out to obtain:
using M (v, v)0) Solving its inverse matrix [ M (v, v)0)]-1Formula (8) is substituted forFinish to obtain phi1And psi1The analytical formula (2):
lagrange interpolation polynomial in interval [ v ] by taking root of Legendre of degree n as sampling nodef,v0]Approximate λ (v, v) on segment0)、cos(λ(xv,v0))m3(xv)、sin(λ(xv,v0))m3(xv) Obtaining:
substituting formula (9) to obtain phi1And psi1Analytic solution of (2):
wherein: c. Cp0(k)、cp1(k)、cp2(k)Is the coefficient of the interpolation polynomial;
by calculated phi1And psi1Substituting formula (5) to obtain λ1Analytic solution of (2):
the above detailed description of the present invention is only used for illustrating the present invention and is not limited to the technical solutions described in the embodiments of the present invention, and it should be understood by those skilled in the art that the present invention can be modified or substituted equally to achieve the same technical effects; as long as the use requirements are met, the method is within the protection scope of the invention.
Claims (1)
1. The gliding trajectory degradation solution along with the change of the speed based on the attack angle and the resistance acceleration is characterized in that:
the method comprises the following steps:
step 1: establishing a dynamics model of a gliding aircraft with a velocity v as an independent variable
Defining: initial longitude, initial latitude and initial azimuth angle of gliding aircraft in geocentric coordinate system are respectively lambda0、φ0And alpha0;
Defining: the offset geocentric coordinate system rotates the geocentric coordinate system around the z, y and x axes by a rotating angle lambda respectively0、-φ0And alpha0Obtaining a coordinate system;
defining: the longitude, the latitude and the heading angle of the gliding aircraft under the offset geocentric coordinate system are respectively lambda, phi and psi;
introducing dimensionless heightsWhereinre=Re+Hm,HmFor the average flight height, R, of the gliding aircraft in the gliding flight sectioneIs the radius of the earth, reThe distance between the center of mass of the aircraft and the geocentric; v. the0And vfThe speeds of the aircraft at the glide starting point and the glide ending point respectively;
establishing a dynamic model of the gliding aircraft with the speed v as an independent variable:
wherein A isDFor resisting acceleration, LDyIs the longitudinal lift-to-drag ratio, LDzIs the transverse lift-drag ratio, omegaeIs the angular velocity of rotation of the earth, mu is the gravitational constant of the earth;
Cσ≈2vωe(sinφ-sinψtanθcosφ)
Cθ≈-2vωecosφcosψ
step 2: approximate equation for establishing dynamics model of gliding aircraft based on Newton iteration method
Let xvWhen v, the variable of formula (1) is replaced and rewritten to y' (x)v)=f(xv,y),
remember yiIs y' (x)v)=f(xvY) and setting the i +1 th iteration value as yi+1=yi+δyiTo, forA first order Taylor approximation is performed on theta, phi, psi, lambda to expand the function y' (x)v)=f(xvY) is constructed as a newton iteration in function space, i.e.:
yi+1(vini)=yini
wherein
knowing the value y of the ith iterationiThen, the solution y of the (i + 1) th step can be obtained by using the formula (2)i+1;
Defining an equation formed by using the formula (2) at the Nth time as an approximate equation of the formula (1) at the Nth time, and solving the approximate equation into an approximate solution of the Nth time; definition ofθ0,φ0,ψ0,λ0Is a zero-order approximate solution of formula (1);θ1,φ1,ψ1,λ1is a first order approximation solution of formula (1);
when the offset geocentric system equation is adopted, the gliding aircraft approximately flies to the target along the equator of the offset geocentric coordinate system, and the gliding flight section has phi approximately equal to 0, theta approximately equal to 0, psi approximately equal to 0,The characteristic of (1) is that the zero-order approximate solution is:
substituting the equation (3) into the equation (2) to obtain a first order approximation equation of the glide flight segment as follows:
wherein
As can be seen from the equation (4), the Jacobian matrix has the characteristic of block sparsity, and theta can be solved first1Andand phi1And psi1Then solve for λ1;
Due to phi1And psi1Is the solution of the lateral motion variable,and theta1Is a solution of a longitudinal motion variable, phi1And psi1To pairAnd theta1Insensitive, equation (4) is further simplified as:
step 3: establishing the relation between the three-dimensional section and the motion state
Using angle of attack and resistance acceleration as control variables, order
α=f(v)
Wherein, c0,c1,c2Are all controlled variable resistance acceleration ADThe design parameters of (1); α is the angle of attack, and f (v) is the angle of attack control function designed with velocity as the independent variable.
Defining: the flight height of the gliding aircraft is h, and the first derivative of the aerodynamic drag acceleration and the aerodynamic drag speed with respect to the energy is obtained by:
finishing to obtain:
wherein: cDIs the drag coefficient of the gliding aircraft, hsThe coefficients of the exponential formula being a function of atmospheric density;
neglecting the effect of the drag coefficient derivative, the approximate altitude of the gliding aircraft can be found:
and (3) solving a second derivative related to energy for the resistance acceleration to obtain a longitudinal lift-drag ratio:
LDy=a(AD″-b)
wherein the content of the first and second substances,
and (3) utilizing a pneumatic coefficient table or a pneumatic interpolation function to obtain a transverse lift-drag ratio:
wherein:
step 4: analytic solution to solve first order approximation equation
Substituting formula (6) for formula (3) according to the definition of the offset geocentric coordinate system, and calculating from v0By v, we get:
the first two formulas of the combined vertical type (5) can obtain
Order to
Left-multiplying equation (7) by M (v, v)0) After arrangement, the differential multiplication rule is applied reversely, and then integration and arrangement are carried out to obtain:
using M (v, v)0) Solving its inverse matrix [ M (v, v)0)]-1Formula (8) is substituted forFinish to obtain phi1And psi1The analytical formula (2):
lagrange interpolation polynomial in interval [ v ] by taking root of Legendre of degree n as sampling nodef,v0]Approximate λ (v, v) on segment0)、cos(λ(xv,v0))m3(xv)、sin(λ(xv,v0))m3(xv) Obtaining:
substituting formula (9) to obtain phi1And psi1Analytic solution of (2):
wherein: c. Cp0(k)、cp1(k)、cp2(k)Is the coefficient of the interpolation polynomial;
by calculated phi1And psi1Substituting formula (5) to obtain λ1Analytic solution of (2):
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CN107941087A (en) * | 2017-10-18 | 2018-04-20 | 北京航空航天大学 | A kind of superb steady gliding reentry guidance method of high lift-drag ratio based on resistance profiles |
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