A rapid compensation method for launch data of long-range rockets under influence of the Earth’s disturbing gravity field

Baolin MA,Hongbo ZHANG,Wei ZHENG,Jie WU

College of Aerospace Science and Engineering,National University of Defense Technology,Changsha 410073,China

A rapid compensation method for launch data of long-range rockets under influence of the Earth’s disturbing gravity field

Baolin MA*,Hongbo ZHANG,Wei ZHENG,Jie WU

College of Aerospace Science and Engineering,National University of Defense Technology,Changsha 410073,China

Available online 20 April 2017

*Corresponding author.

E-mail address:feizhuxiongmao@126.com(B.MA).

Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.cja.2017.04.001

1000-9361©2017 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Regarding the rapid compensation of the influence of the Earth’s disturbing gravity field upon trajectory calculation,the key point lies in how to derive the analytical solutions to the partial derivatives of the state of burnout point with respect to the launch data.In view of this,this paper mainly expounds on two issues:one is based on the approximate analytical solution to the motion equation for the vacuum flight section of a long-range rocket,deriving the analytical solutions to the partial derivatives of the state of burnout point with respect to the changing rate of the finalstage pitch program;the other is based on the initial positioning and orientation error propagation mechanism,proposing the analytical calculation formula for the partial derivatives of the state of burnout point with respect to the launch azimuth.The calculation results of correction data are simulated and verified under different circumstances.The simulation results are as follows:(1)the accuracy of approximation between the analytical solutions and the results attained via the difference method is higher than 90%,and the ratio of calculation time between them is lower than 0.2%,thus demonstrating the accuracy of calculation of data corrections and advantages in calculation speed;(2)after the analytical solutions are compensated,the longitudinal landing deviation of the rocket is less than 20 m and the lateral landing deviation of the rocket is less than 10 m,demonstrating that the corrected data can meet the requirements for the hit accuracy of a long-range rocket.

©2017 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Analytical solution;

Earth’s disturbing gravity fi eld;

Launch data;

Partial derivative compensation method;

Rapid compensation

1.Introduction

With the intensive increase of future space missions,for longrange rockets,it is the main trend of the future development to have fast response and high hit accuracy at the same time.1–4It is of great practical significance to launch a rapid response in emergency,1such as space rescue,space interception,and ballistic missile launch mission.Future space shuttle missions will more demand high accuracy.The launch vehicle will be required to perform the mission at the first lap after the launch,and CEP for ballistic missile can be controlled in 100 m,1,4otherwise the significance of rapid response to launch will be lost.5

The disturbing gravitational potential is the difference between the actual Earth gravitational potential and the normal Earth gravitational potential.The force field formed by the disturbing gravitational potential is called disturbing gravity field.6In order to simplify calculation,the long-range rocket guidance system usually uses the normal Earth represented by the Earth reference ellipsoid as the reference condition to determine the directional data and calculate the gravity of the Earth.This model simplification will inevitably introduce errors such as trajectory calculation error and guidance calculation error.The results show that the impact of the disturbing gravity field on the hit accuracy can reach up to 1000 meters for a long-range ballistic missile with a range of more than 10000 km,7which has been an important error source in guidance method error.Therefore,it is necessary to compensate for the influence of the disturbing gravity field in order to improve the accuracy of long-range rockets.

In consideration of the current performance of the missile onboard computer,the compensation method of the disturbing gravity field in the space field can be divided into two categories.One is using online compensation method,which reconstructs the disturbing gravity,calculates the disturbing gravity in real time in flight,and compensates it by controlling the thrust direction of the missile.This method is simple in principle and good in real time,and it will not increase the computation time of the pre-launch compensation scheme.However,the limitation of this method is that it needs the fast calculation method of the disturbing gravity.7The other is using ground data of compensation.The partial derivative method to calculate the trajectory correction is more common in the space mission.7–9Compared with the online compensation method,the partial derivative compensation method can compensate the influence of the disturbing gravity field based on the existing trajectory calculation method,which is simple and reliable.However,the disadvantage of this method is that computational burden is high and the preparation time is long.For example,when the Richard algorithm is used in the difference method for calculating partial derivatives,9each calculation of a partial derivative needs to calculate trajectory 4 times.For the fast response task on the condition of maneuver,the calculation of partial derivatives by using the difference method will affect the preparation time seriously.In order to shorten the preparation time of the partial derivative compensation method,two preconditions are needed:one is calculating the landing error caused by disturbing gravity field fast;The other is establishing the mapping relationship between the landing error and trajectory data correction quickly and accurately.In Ref.7,by using the state-space perturbation method,the analytical solution to the landing deviation when the influence of disturbing gravity field was taken into account is derived.By using the neural network method,high-accuracy calculation for the trajectory constraint function is presented,which rapidly and effectively derives the landing deviation resulting from the disturbing gravity.10Therefore,based on the influence of the disturbing gravity field obtained by analytical form,the derivation of analytic solution of partial derivative is particularly urgent in engineering.

A simplified model of ballistic missile motion considering the geophysical perturbation and positioning error propagation model can be used to solve the problem of the shutdown point error of the active disturbing gravity and positioning errors.3Based on the above two models,this paper proposes a fast calculation method for the corrections of data through deducting the partial derivatives of rocket stage pitch program change rate and launching azimuth for shutdown point state.Although there have been some methods which use partial derivative obtained by the difference method to compensate rocket launch data,the compensation method which uses analytical solutions of the partial derivative has not been reported in the open literature.In addition,the earth’s gravity field falls under the category of geodesy,11–14and the compensation of rocket data is the focus of research in flight dynamics and guidance,15–18so this paper pertains to an interdisciplinary research.Therefore,the research herein is of great significance to improve the hit accuracy of rockets and shorten the preparation time of rocket launching under the influence of the Earth’s disturbing gravity field.

This paper is organized as follows:Section 2 introduces the influence mechanism of the disturbing gravity field and the calculation method of the data correction;Section 3 reduces analytical solution of the partial derivative of the shutdown point state to the launch data;Section 4 simulates and verifies the accuracy and rapidness of the calculation of data corrections;Section 5 summarizes the work.

2.Influence of disturbing gravity field and calculation method of data corrections

As shown in Fig.1,the standard trajectory is obtained by the iteration of the ballistic model under the standard trajectory.The corresponding launch data are the basic data.The socalled standard conditions are that the Earth is generally regarded as the standard ellipsoid without considering the impact of the disturbing gravity field.According to the centroid motion equation of the powered-flight phase6

and the centroid motion equation of the unpowered-flight phase6

wherevandrrefer to the velocity and position in the launching inertial system respectively,grefers to the gravitational acceleration,andWrefers to the apparent acceleration,it can be seen that the value of the gravity termgin the motion equation of the powered and unpowered phase is different under the influence of disturbing gravity field.In the actual earth gravity,if the basic data are regarded as launch data,earth gravity model will make the flight trajectory deviation from the target point and produce landing error,and the corresponding trajectory is called disturbed trajectory.

As shown in Fig.2,the purpose of the trajectory data compensation is that the trajectory of the corrected data(basic elements+corrections)can also pass through the target accurately and the corresponding trajectory is a corrected trajectory under the influence of disturbing gravity field.

For a solid rocket,the launch data mainly includes the changing rateof the final-stage pitch program and launch azimuthA0.6Fig.3 shows the flowchart on calculation of data corrections when the basic data are known.

The calculation procedure is as follows:

Step 1.Determine a standard trajectory by using basic data

Step 2.According to the standard ballistic model and Earth’s gravity model,calculate the disturbing gravity of the flight trajectory.

Step 3.According to the basic data,determine the actual flight trajectory under the action of disturbing gravity.

Step 4.Calculate the influencing quantities ΔL and ΔH upon hit accuracy:

whereLandHrefer to the actual range and cross range of the rocket respectively when the disturbing gravity is taken into account,andL*refers to the range of the standard trajectory.Note that the superscript ʻ*’stands for the situation without considering the impact of the disturbing gravity field.

Step 5.Based on the standard trajectory determined at Step 1,calculate the partial derivatives(∂L/∂A0,∂L/∂˙φ,∂H/∂A0and∂H/∂)of the rangeLand cross rangeHwith respect to the launch data.

Step 6.According to the partial derivative relationship,7calculate the data corrections ΔA0and Δ˙φ:

Step 7.According to the basic dataand)and data corrections(ΔA0and Δ),determine the corrected launch data(A0and):

Step 8.According to A0and˙φ,calculate the landing point,and then determine the data corrections if the landing point meets the requirements for hit accuracy.

3.Rapid calculation method for compensation of trajectory data

According to Eq.(4),in order to determine the data corrections,it is necessary to calculate the partial derivative matrixCand the landing deviations(ΔLand ΔH)resulting from the Earth’s disturbing gravity.The analytical solutions to the landing deviations resulting from the disturbing gravity can be obtained through the initial positioning and orientation error propagation model and state-space perturbation method.Therefore,the key issue is how to solve the partial derivative matrixCrapidly.The partial derivative matrixCcan be expanded into an expression regarding the error coefficients matrix ［∂L/∂X，∂H/∂X］Tand the partial derivative matrix［∂X/∂A0，∂X/∂］of the state of burnout pointX= ［vx（tk），vy（tk），vz（tk），rx（tk），ry（tk），rz（tk）］Twith respect to the launch data:

Therefore,the issue can also be translated into a problem whether the error coefficients(∂L/∂Xand ∂H/∂X)and the partial derivatives(∂X/∂A0and∂X/∂)of the state of burnout point with respect to the launch data can be obtained in an analytical form.

An error coefficients refers to the magnitude of the range deviation and lateral deviation arising from any unit deviation of the burnout point parameters.6The error coefficients∂L/∂Xand ∂H/∂Xcan be expressed as follows:

Therefore,the error coefficients can be obtained in a rapid calculation method.The key of the problem is whether the analytical solution of the partial derivative of the state error of burnout point with respect to the launch data can be deduced.

3.1.Partial derivative of state error of burnout point with respect to changing rate of final-stage pitch program

A long-range rocket is usually equipped with a multistage boost power plant.It is assumed that the rocket follows the fixed flight procedure in several front stages.9,18Then,this paper focuses the research on the vacuum flight(usually corresponding to the final stage of the rocket)of the powered- flight phase.In the launching inertial system,the final-stage motion equation is the same as Eq.(1).We assume that the transformation matrix from the body- fixed reference framebto the launching inertial reference frame I is

where φ（t）refers to the pitch program angle of the rocket at timet.

The component form of the thrust accelerationain theIframe is as follows:

wheremtrefers to the rocket mass at timet;Prefers to the thrust force.

After the rocket takes off,the time-varying law of the flight program angle is manifested as a piecewise linear form or a quadratic parabola.20For convenience,the final-stage pitch program angle of the powered-flight phase in the changing section is expressed as follows:

where φ0refers to the pitch program angle before the change,t0andt1refer to the start time and end time of the change of the final-stage pitch program angle respectively,and Δt=t-t0refers to the changed time quantum of the finalstage pitch program at timet.

In addition,we assume that the final-stage thrust forcePand second consumption˙mare constant values,and the rocket mass at the timetismt=mt0-˙mΔt.Eq.(9)can be converted into the following equation:

Values are assigned to makePa=Pcosφ0/mt0,Pb=Psinφ0/mt0andM=˙m/mt0.Then,in the launching inertial reference frame of the change section of the finalstage pitch program,the rocket motion equation is as follows7:

Likewise,the relational expression between the residual integral term in Eq. (16) and the sine integral

Therefore,the analytical solution to the motion state of the pitching section in the final-stage boost phase of the long-range rocket can be expressed as follows:

where

After deriving˙φ value in Eq.(18),we obtain the partial derivative of the state of burnout point with respect to the changing rate of the pitch program of the pitching section:

where

3.2.Partial derivative of state of burnout point with respect to launch azimuth

In Ref.7,the state errors of burnout point arising from the initial positioning and orientation errors are mainly classified into two terms:geometrical term ΔX1and initial value term ΔX2.Thus,the partial derivative matrix of the state errors of burnout point with respect to the launch azimuthA0can be decomposed into the partial derivatives of the two terms with respect to the launch azimuth:

3.2.1.Partial derivative

Considering that the orientation error is a small quantity,the transformation matrixfrom the platform coordinate system to the launching inertial system can be expressed as follows:7

and Ωx,Ωyand Ωzrefer to the three error angles between thePframe and theIframe.

The launch azimuthA0does not affect the initial position,so the geometrical errors ΔX1in the state errors of burnout point are only generated byThere exists the following relational expression between ΔX1and the standard state of burnout point

is the transformation matrix from the launching inertial coordinate frame to the geocentric coordinate frame.

Substituting Eq.(26)into Eq.(25)yields

Accordingly,the partial derivative of the geometrical error with respect to the launch azimuth is as follows:

Combining Eq.(27)with Eq.(29)gives

3.2.2.Partial derivativeof initial value term

We assume that the velocity of the carrier in the launch coordinate frame is 0 at the time of missile launch.Then,the initial velocity in the launching inertial frame is as follows7:

where ωeandR0，Irefer to the Earth’s rotational angular velocity and the position vector of the launch point in the launching inertial frame respectively.According to Eq.(31),v0is a function of ωeandR0，I.

DenoteB0the geodetic latitude of launching point,and λ0the geodetic longitude of launching point.The projection of ωein the I frame is

and the transformation relationship betweenR0，IandR0，Eis

withMi［θ］（i=1，2，3）the direction cosine matrix that rotates θ aroundiaxis.

According to Eqs.(32)–(34),we know that ωeandR0，Iare the function ofA0.

Taking the derivate of ωeandR0，Iwith respect toA0and substituting the resulted derivate into Eq.(31),the initial value error can be obtained:

Therefore,the partial derivative of the initial value term with respect to the launch azimuth is as follows:

According to Eqs.(30)and(36),the partial derivative of the state of burnout point with respect to the launch azimuth can be obtained in an analytical form.

According to Eqs.(20)and(22),the partial derivatives of the state of burnout point with respect to the launch data can be obtained in an analytical form.

4.Simulation results

The simulation is intended to verify the following items:(1)Accuracy of the data corrections calculated by the analytical method;(2)Advantages of the analytical solutions in speed;(3)Compensation effect of the analytical solutions to data corrections upon hit accuracy.

We select three launch points with distinctive topographic conditions:(1)A plain(the altitude is less than 100 m,and the terrain slopes are gentle);(2)A general mountainous area(the altitude ranges from 1000 m to 2000 m,and the terrain is very complex);(3)A super mountainous area(the altitude is more than 2000 m,and the terrain is extremely complex).Then,based on the EGM-2008 model,the spherical harmonics method is used to calculate the Earth’s gravity during the flight of the rocket.Fig.4 shows the influence upon hit accuracy by the disturbing gravity across the entire flight process under omni-directional launch conditions.

We select one group of the results that influence the landing point most significantly among the three launching points.Table 1 describes the maximum landing deviation and the corresponding launch direction.The analytical method and difference method are used respectively to calculate data corrections,and the corrected data are used to calculate the landing point.The above two methods are compared using a Windows XP 3.2 GHz computer.Table 2 compares the data corrections,calculation time and corrected hit accuracy between the two methods.

The simulation results are as follows:(1)The accuracy of approximation between the data corrections calculated by the analytical method and difference method is higher than 90%,thus demonstrating the correctness of the analytical solutions to the data corrections obtained by the analytical method herein;(2)The ratio of calculation time between the two methods is lower than 0.2%,thus demonstrating that the analytical method has an obvious advantage in the rapidness of calculation,which is critical to the survival of long-range ballistic missiles.If the corrected data are used to calculate the trajectory,the longitudinal landing deviation is less than 20 m and the lateral landing deviation is less than 10 m,meeting the requirements for the hit accuracy of the long-range rocket.This proves that the proposed method can compensate the influence of the disturbing gravity effectively and quickly.

Table 1 One group of results that influence landing point most significantly among three launching points.

Table 2 Calculation results of data corrections and their compensative effect upon hit accuracy.

5.Conclusions

The Earth’s disturbing gravity field is a major error source for the trajectory calculation of long-range rockets.While the influence of disturbing gravity field is taken into account,calculating the corrections of trajectory data is of great significance for improving the hit accuracy of long-range rockets.The solutions to data corrections can be obtained according to the relationship between the partial derivative matrix and the influencing quantity of the Earth’s disturbing gravity field upon hit accuracy.On the premise that the influencing quantity of hit accuracy and error coefficients can be obtained in an analytical form,it is particularly urgent to derive the analytical solutions to the partial derivatives of the state of burnout point with respect to the launch data.

Based on the motion equation for the vacuum flight section of a long-range rocket,and the initial positioning and orientation error propagation mechanism(the influence of disturbing gravity field is taken into account),this paper derives the analytical solutions to the partial derivatives of the state errors of burnout point with respect to the launch data.The simulation results are as follows:(1)the accuracy of approximation between the analytical solutions to data corrections and the results attained via the trajectory difference method is higher than 90%,thus demonstrating the correctness of the calculation method;(2)the calculation time is shorter than that of the difference method,thus demonstrating the excellent calculation efficiency of the calculation method;(3)after the launch data is corrected,the longitudinal landing deviation is less than 20 m and the lateral landing deviation is less than 10 m,thus meeting the requirements for hit accuracy.Therefore,the proposed rapid calculation method for the launch data completely meets the requirements for rapid rocket launch and high hit accuracy when the influence of disturbing gravity field is taken into account.

The rapid calculation method herein is also of great reference significance to the research on the influence of other error terms upon trajectory data.In this paper,the fast calculation method of trajectory data can effectively improve the computational efficiency.This method can also be used as a technical reference for other engineering applications.

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10 August 2016;revised 8 February 2017;accepted 17 March 2017