Optimal rescue orbital elements online decision-making method based on RBFNN for launch vehicles under thrust drop fault
An optimal rescue orbital elements online decision-making method based on RBFNN for launch vehicles under thrust drop fault includes establishing the flight dynamic equations of launch vehicles in the second-stage ascending phase in the geocentric inertial coordinate system, to construct a series of optimization problems of maximum semi-major axis of circular orbit under the thrust drop fault. The method further includes using the adaptive pseudo-spectrum method to solve the optimization problems of maximum semi-major, and using the maximum and minimum method to normalize the sample data to [−1, 1], using the orthogonal least square method to select the data center of the radial basis function neural network (RBFNN), where the Gaussian function is selected as the radial basis function, and the RBFNN is trained offline to establish a nonlinear mapping relationship from the fault states to the optimal rescue orbital elements.
1. An optimal rescue orbital elements online decision-making method based on RBFNN for launch vehicles under thrust drop fault, comprising:
establishing dynamic equations of launch vehicles in a second-stage ascending phase in a geocentric inertial coordinate system; setting a boundary condition and a constraint condition with different fault times and thrust drop percentages, to construct a series of optimization problems of maximum semi-major axis of circular orbit under the thrust drop fault;
using an adaptive pseudo-spectral method (APM) to solve the optimization problems of maximum semi-major axis offline, to obtain a sample set of the optimal rescue orbital elements under different fault states, wherein input features of the sample set are the fault states, and the fault states include a time of a thrust fault, a percentage of thrust drop, a position, a speed, a mass and output features of the sample set are the rescue orbital elements, the rescue orbital elements include the semi-major axis, an inclination, and a longitude of an ascending node of a rescue orbit;
using a maximum-minimum method to normalize a sample data, and all data are normalized to [−1,1]; using an orthogonal least square method to select a data center of a radial basis function neural network (RBFNN), wherein a Gaussian basis function is selected as a radial basis function; the radial basis function neural network is trained offline to establish a nonlinear mapping relationship from the fault states to the optimal rescue orbital elements; and
transferring the well-trained radial basis function neural network to an actual flight; taking a fault state of the actual flight as an input of the radial basis function neural network, to determine the optimal rescue orbital elements online.
2. The method according to claim 1 , wherein, when constructing the optimization problems of maximum semi-major axis:
setting an X 1 axis to point a primary meridian at a time of launch in an equatorial plane, a Z 1 axis vertical to the equatorial plane and point to a North Pole, and a Y 1 axis to meet a right-hand rule; establishing the dynamic equations of the launch vehicles in the second-stage ascending phase in the geocentric inertial coordinate system as follows:
r
.
=
v
(
1
)
v
.
=
-
μ
r
3
r
+
(
1
-
η
)
T
nom
m
u
(
2
)
m
.
=
-
(
1
-
η
)
T
nom
g
0
I
sp
(
3
)
wherein r and v represent a position and a velocity vector of launch vehicles; μ=GM is an earth's gravitational constant; m represents a total mass of the launch vehicles, and I sp represents a specific impulse of an engine of the launch vehicles; μ=[u x , u y , u z ] T is a component of a thrust unit vector of the engine; when an engine fault occurs, a percentage of thrust drop is η; a thrust magnitude is (1−η)T nom , wherein T nom is a nominal thrust of the engine; in the case of a thrust drop fault, a specific impulse of the engine remains unchanged, a propellant consumption per second decreased η, and a total flight time exceeds a nominal flight time; assuming an engine thrust drop failure occurs at t 0 , so a constraint condition of a starting point is expressed as follow:
x ( t 0 )= x 0 (4)
wherein x 0 is a state of the starting point, a nonlinear relationship from a number of orbital elements to a terminal state is expressed as
[ a f ,e f ,i f ,Ω f ,ω f ] T =ψ( r ( t f ), v ( t f )) (5)
wherein t f is a terminal moment; a f , e f , i f , Ω f , ω f are orbital elements of a target orbit including a semi-major axis, an eccentricity, an inclination, a longitude of an ascending node, an argument perigee of a terminal point,
a total mass of the launch vehicle and payload after fuels are exhausted is expressed as m f , a radius of the earth is R 0 , and a minimum safe orbit height is defined as h safe , a terminal mass and a height meets:
m ( t f )≥ m f ,h safe ≤r ( t f )− R 0 (6)
when the thrust drop fault occurs, because an energy required to send a load into a circular orbit is less than an energy of an elliptical orbit under a same perigee height, so searching for a highest circular orbit within a current orbital plane as the optimal rescue orbital elements; describing, by a solution to the highest circular orbit under a thrust drop failure, a maximum optimization problem of the semi-major axis:
Objective Function: min J=−a ( t f )
Boundary Conditions: [ e f ,i f ,Ω f ] T =[0, i f res ,Ω f res ] T ,
x ( t 0 )= x 0 ,m ( t f )≥ m f ,h safe ≤r ( t f )− R 0
Dynamics Constraints: Eq. (1)-(3)
Control Constraints: ∥ u∥= 1; and
determining the orbital inclination i f res and the longitude of the ascending node Ω f res , according to the fault states of launch vehicles.