Improved theory of generalized meteo-ballistic weighting factor functions and their use

Vladimir CECH*，Jiri JEVICKY

aDepartment of Weapons and Ammunition，University of Defence，Kounicova 65，Brno 662 10，Czech Republic

bOprox，Inc.，Kulkova 8，Brno 615 00，Czech Republic

cDepartment of Mathematics and Physics，University of Defence，Kounicova 65，Brno 662 10，Czech Republic

Improved theory of generalized meteo-ballistic weighting factor functions and their use

Vladimir CECHa，b，*，Jiri JEVICKYc

aDepartment of Weapons and Ammunition，University of Defence，Kounicova 65，Brno 662 10，Czech Republic

bOprox，Inc.，Kulkova 8，Brno 615 00，Czech Republic

cDepartment of Mathematics and Physics，University of Defence，Kounicova 65，Brno 662 10，Czech Republic

It follows from the analysis of artillery f i re errors that approximately two-thirds of the inaccuracy of indirect artillery f i re is caused by inaccuracies in the determination of the meteo parameters included in f i re error budget model.Trajectories calculated under non-standard conditions are considered to be perturbed.The tools utilized for the analysis of perturbed trajectories are weighting factor functions （WFFs）which are a special kind of sensitivity functions.WFFs are used for calculation of meteo ballistic elements μB（ballistic wind wB，density ρB，virtual temperature τB，pressure pB）as well.We have found that the existing theory of WFF calculation has several signif i cant shortcomings.The aim of the article is to present a new，improved theory of generalizedWFFs that eliminates the def i ciencies found.Using this theory will improve methods for designing f i ring tables，f i re control systems algorithms，and meteo message generation algorithms.

Exterior ballistic；Non-standard projectile trajectory；Perturbation；Meteoconditions；Sensitivity function；（Generalized）weighting （factor）function（curve）；（Generalized）effect function；Norm effect

1.Introduction

1.1.Motivation

It follows from the analysis of artillery f i re errors，e.g.［1，2］，that approximately two-thirds of the inaccuracy of indirect artillery f i re is caused by inaccuracies in the determination of meteo parameters included in the error budget model ［1］.Consequently，it is always important to pay close attention to the problems of including the actual meteo parameters in ballistic calculations ［3］.The following meteo parameters μ are primarily utilized:Wind vector w，air pressure p，virtual temperature τ，and density ρ ［2-6］.

This paper deals only with problems relating to unguided projectiles without propulsion system for the sake of lucidity of the solved problems.

1.2.Weighting functions-basic information

The most important information about the inf l uence of meteo parameters （and not only them）on the trajectory of an unguided projectile is included in the relevant weight or weighting functions ［2，7-10］.

List of notation

μ met parameter （element）μ（y） real or measured magnitude of met parameter μ in height y r（μ） weighting factor function （curve，WFF）QP，QCP effect function μSTD（h） met parameter standard course with the height h Δμ（y） absolute deviation of met element μ in height y δμ（y） relative deviation of met element μ in height y ΔμB absolute ballistic deviation of ballistic element μBδμB relative ballistic deviation of ballistic element μB

The basis for the derivation of the weighting functions is perturbation theory ［11］.

We are interested in the exercise of the perturbation theory in dynamical systems theory，primarily in the control theory of dynamical systems ［12，13］.It is exercised especially in the exploration of stability and sensitivity ［12，13］.The mostwidespread variant of the perturbation theory is the simplest one-the f i rst-order perturbation theory.Its most important basis is the linearization of all requisite non-linear functions and equations ［11-13］.Unless otherwise specif i ed，the following information refers to this theory.

A special subset of controlled systems is comprised of aerospace vehicles，i.e.，aircrafts，space vehicles，rockets，space shuttles，guided missiles and spinning and non-spinning“unguided”projectiles with/without terminal guidance and Magnus rotors ［4-6，14］.

The state equations of non-linear dynamical systems have then the form

where x is state variable vector，u is input control variable vector，d is input disturbance variable vector，α is parameters vector，y is output variable vector.

The perturbation theory is used for transformation of these equations into their linearized form （f i rst example）［5-9］.The linearized state equations have for example the form

For the analysis of dynamical systems，it is interesting to observe changes of the system properties while some parameters α（t）change；the parameters are then often denoted as inf l uence quantities.We speak of differential sensitivity analysis of the control system or of sensitivity of a system to parameter variations.The perturbation theory is used again for linearization of Eq. （2）relative to parameters α.We obtain a sensitivity model of the （linearized）dynamical system，for instance in the form ［12，13，15，16］

where

are the absolute sensitivity functions.The absolute sensitivity functions of the output variables ηiare especially important for the practice.Non-dimensional Bode sensitivity functions are often used ［12，13，15，16］.

The perturbation theory is used in this second case for f i nding linearized relationsbetween changesofsystem parameters Δα and corresponding changes of the output variables Δy，which are represented by the sensitivity functions ηi（ηi≈ Δy/Δαi，i=1，2，...，n）and which can be expressed consecutively through the use of the corresponding transfer functions ［12，13］.

Standard test functions for the control variables u（t）and the disturbance variables d（t）are used for the analysis of properties of the systems that are described by Eqs.（1），（2）and （3）.The unit impulse is usually used，and also the unit step，the function sine and/or cosine，etc.［12，13］.

Such a procedure is not suff i cient for analyses of movements of aerospace vehicles，so it is customary to use reference trajectories and maneuvers，respectively，which represent the typical maneuvers of a given type of aerospace vehicle［4-6，14］.

Moreover，it is necessary to differentiate whether reference maneuvers are pursued under standard conditions or perturbed conditions.

Standard conditions are def i ned contractually and determine the standard/normal values of the parameters respectively αSTD（t），for instance，parameters of the standard atmosphere are considered.The reference maneuvers under standard conditions are utilized for the basic analysis of aerospace vehicle properties，Eqs.（1）or （2）are used withal （d（t）=0）.

Thereferencemaneuversunderperturbed conditions（d（t）≠ 0，respectively α（t）= αSTD（t）+ Δα（t））serve for consequential analyses of stability or robustness of f l ight control；Eq.（3）are used together with Eqs.（1）or （2）.

The reference maneuver under standard conditions in the exterior ballistics of unguided projectiles is represented just by the standard projectile trajectory，and the reference maneuver under perturbed conditions is identical to the relevant perturbed projectile trajectory.

As mentioned above，corresponding sets of transfer functions are referred to Eqs.（2）and （3）；their equivalent in the time domain is the convolution operation represented by the convolution integral.Two functions f and g f i gure in the convolution integral.The functions f and g have a special signif i cance in the control theory of dynamical systems.The function f represents a generalized input variable zm（t），m=1，2，... （respectively uj（t），j=1，2，...and dk（t），k=1，2，...and Δαi（t），i=1，2，...）and the function gm，l（t）is the weighting function that corresponds with the relevant transfer function.The integral value then corresponds to the system response yl，l=1，2，...，to the excitation by the input variable ［12，13］.

The weighting functions gml（t）are impulse-response functions ［12，13］，i.e.responses of the dynamical system to the special excitation by impulse function zm（t）=zm0·δ（t-tp），where zm0is the excitation amplitude and δ（t-tp）is the Dirac delta function.The weighting function then has the form

where

tpis the moment of the impulse occurrence，γml（t-tp）is the normed form of the weighting function and Mmnis the relevant norm.

We have now presented all of the common information from the control theory of dynamical systems necessary to understand the importance of weighting functions in exterior ballistics.

The perturbation theory was already used-in a simple form -at the start of the 20th century in exterior ballistics，e.g.［17］. The equation system corresponding to Eq.（3）was f i rst derived during the First World War.These problems are often presented with the name Theory of trajectory （differential）corrections. The starting points were the motion equations of a projectile as a mass point（3 DoF-degree of freedom），which is an analogy to the Eq. （1）.The convolutory integral usage started in this period and therefore the usage of the needful weighting functions started too.Corresponding models were not published until after the war，starting in 1919，for instance ［7］.

We have adduced more information about the overall progress in perturbation theory utilization in exterior ballistics in our article ［3］.Only complementary information will be introduce here.

The control theory started to form at the end of the1930s and was not developed in full until the 1950s，so the procedures introduced into exterior ballistics had been formed almost 30 to 40 years prior.It should be no surprise，then，that the weighting functions were introduced differently.

The weighting （factor）function （WFF）rmlwas introduced into the exterior ballistics as the normed step response function［2，3，9，10，18］，followed by the response of the dynamical system to the special excitation by the step function zm（t）=zm0·H（t-tp），where zm0is the excitation amplitude and H（t-tp）is the Heaviside step function.Then，the non-normed weighting function has the form

where tpis the moment of the leap/perturbation，rml（t-tp）is the normed weighting （factor）function （curve）WFF ［2，3，7］，Nmlis the relevant norm，σml=+1 or-1 is the contractual sign-see sections 2.4，2.5.4.

The non-normed weighting function or perturbation functions Rml（t-tp）were named the effect functions （curves）-EFs originally ［2，8-10，19］.

It follows from the properties of the Dirac impulse function，the Heaviside step function and from system linearity that conversion relations among weighting functions are ［13］

We did not f i nd an explanation for Eq.（5）and its links with Eq. （4）def i ned by Eq. （6）in an explicit form in the available literature，but all the authors implicitly assume its validity［2，7，9，10，18］.Without appreciating the validity of this equation，the relation between modern control theory and the traditional theory of exterior ballistics，including the relevant weighting functions （WFFs），is not clear.

In the initial period，the following WFFs were introduced for meteo parameters:rwxfor the range wind，rwzfor the cross wind and rρfor the air density e.g.［7］.In this period，it was still assumed that the drag coeff i cient cD（vair）depends only on the air speed vair.It was not until the 1920s，especially in connection with the publication of the drag coeff i cient cD（M）=cD（vair，true）by Dupuis law，the respect for the dependence of the drag coeff i cient on the Mach number and on so-called f i ctive or true air speed-TAS begin.Therefore，a WFF was introduced；we named it rτ/ρ，because it exists only in pairs with WFF rρ.In our article ［3］，we explained that there are other combinations of WFFs，see also ［2，8，20］，and we refer to the problem in this contribution as well.The achieved f i ndings are published，for example，in ［9，10，19］.

Further development of this in the 1950s is documented，for example，in ［8，20］.

The development from the 1960s to the present can be considered paradoxical.Methods based on the theory of perturbations have been further developed and they are widely used in control theory，for example ［4-6，12，13，16］，whereas their use in exterior ballistics has declined.The status can be demonstrated by the content of important publications from this period.

No word about perturbation problems can be found in the key books ［21，22］.McCoy ［23］only pays attention to the problem of variable wind in two pages.Other authors，e.g. ［24-26］，clarify these problems through oversimplif i cation and without a more detailedexplanationofWFFproblems.Thebook ［2］dealswiththe problems of WFFs in the most detail，but a sensitivity model analogous to the Eq. （3）is not presented，unlike ［27］.There are very few articles that deal with the given problems，for example［18，28-30］and our contributions ［3，31，32］.

The question arises as to why this development has occurred. We do not know the answer.We try only to present the following hypothesis，for which we will use the following proverb:“They throw the baby out with the bathwater”.What is the“water” and what is the “child”？The “water”is the numerical algorithms for quick calculations of perturbed trajectories，and the “child” is the WFFs.

Till the early 1960s the main endeavor of publications about perturbations of trajectories focused on f i nding the most effective algorithms to solve perturbed trajectories.This problem became uninteresting after the massive arrival of digital computers.As a result，perturbation theory was quickly abandoned，and it was forgotten that the possibility of calculating WFFs was also lost.

1.3.The main objectives of the contribution

The weighting factorfunctions （WFFs） are special representants of sensitivity functions-see Eq.（3）and （6），and should be primarily considered as a post-processing tool.They allow for the compression of useful information very effectively and also allow for the display of it in synoptic graphs.Our main goal is a return to the use of WFFs （sensitivity functions）in the exterior ballistics.

We expect from this to

·streamline the teaching of exterior ballistics as a result of increasing its lucidity，

·improve the suggestive power of the published outputs from research in problems of the sensitivity analysis of projectile trajectories.

The consequential aim is to contribute to the improvement of methods for making f i ring tables，algorithms of f i re control systems，and methods for the preparation of documents for processing meteorological measurements and the subsequent generation of meteorological messages.

For the performance of the aims introduced，we present an improved theory of generalized meteo-ballistic weighting factor functions.The core of the theory is created by the publications of V.Cech ［33，34］.Moreover，selected problems are f i nished in this contribution.

1.4.Perturbations versus correction of projectile trajectories

Trajectory perturbation follows logic （Eqs.（4）and （5））-the primary change zm0of any of the parameters/input variables leads to the trajectory perturbation，which is a change of the output variables vector ΔyPm.

The theory of trajectory （differential）correction traditionally stems from a request［9，10，24，25，27，35-37］so the change of control variables Δu，which also leads to the change of output variables Δy（Δu）= ΔyC，compensates for the effect of perturbation ΔyP，i.e.，it is valid

so in the traditional notation and for the meteo-ballistic parameters ［3］we will present the most frequent case for range correction （ΔyCl= ΔX）［3］

where

ΔμB= μB- μSTD-absolute ballistic deviation of ballistic element μB，see below，δμB= ΔμB/μSTD-relative ballistic deviation of ballistic element μB，see below，ΔμBN，δμBN-constant norm values of the absolute and relative ballistic deviation that are presented in tabular f i ring tables，QxA，QxR-corresponding （unity）correction factors for range （x）

that are presented in tabular f i ring tables.Indices A （absolute）or R （relative）inform us of the fact that what enters into the calculations is the absolute deviation ΔμBor the relative deviation δμBof the ballistic elements.Next time-if no ambiguities will be possible-we will omit these indices.

The notation Q（μ|μ0）means ［3］that the correction factor Q（μ）is calculated under the assumption that Q（μ0）is used as a second correction factor and，for example，their common range correction ΔXcom= ΔXμ+ ΔXμ0.

It follows for the correction coeff i cient QxAfrom Eqs. （8）and （5）

where zm0= ΔμB0is the constant value by which the perturbation has been calculated. For example， ΔμB0= ±25 m/s is recommended ［37］for the range wind （ΔμB=wx）.For the f i ring tables by NATO methodology ［21，35，36］this is ΔμBN=1 knot=0.514 444 m/s.For the f i ring tables by the Soviet methodology ［2，24-27］it is ΔμBN=10 m/s.For the choice of（σml·Nml），see sections 2.4 and 2.5.4.

Equations for the relative values of the correction coeff icients QxRare determined by analogy，zm0= δμB0and δμBN.For example，this is recommended for the air density ρ ［9，37］δμB0= δρ = ±0.1 and δμBN=1% （NATO）， 10% （Soviet methodology）.

Relations analogous to Eq.（8），Eq.（9）can be derived for the azimuth （unity）corrections （ΔyCl= ΔZ，（QzA，QzR）），time of f l ight （ΔyCl= ΔtPI），etc.-see sections 2.2 and 2.3.

The linearization is understood in two different ways.The traditional way ［4-6，11-13，16］is based on the Taylor series of the function at the working point，and then there is a numerical estimate of the partial derivatives of the ηi（ηi≈ Δy/Δαi，i=1，2，...，n），which f i gured in Eq. （3）［8-10，19，24-27］.

The second way is more general.Two linear approximations at the working point-from the left and from the right-are numerically estimated ［35-37］.The aim is to extend the interval during which the linear approximation is suff i ciently accurate. An alternative to this procedure is the use of second-order perturbation theory ［8，11］.

For more information on the linearization of functions，see［38］.

1.5.Explicit versus implicit algorithm

What has been mentioned up to now relates to the explicit algorithm for calculations of the sensitivity functions and WFFs.A pre-requisite is to build Eqs.（2）and （3）in a form that corresponds to the analyzed model of the dynamical system［4-10，19，27］.

The advantage of the explicit algorithm is the possibility to study the structure of the links in the state-space model（matrices A，B，C，D and their derivatives）.In fact，the real models are so complicated that they are confusing，and therefore several simpler，partial models are derived from them so that these new models will already be clear and appropriate for analysis［4-6］.

As mentioned in ［9］，M.Garnier published in 1929 the WFF calculation method，which can be described as the implicit algorithm.Linearized Eqs.（2）and （3）do not have to be derived at all.The original non-linear system Eq.（1）is suff i cient for the work.The basis of the algorithm is the def i nition of the partially perturbed functions-see section 2.2.The algorithm is extremely convenient for programming on a digital computer. In the following text we describe the essence of the method.The linearity can be assumed implicitly and will appear in the form of relations not used until post-processing.The disadvantage of the algorithm is that the structure of dependences in the perturbation model is not obvious.

2.Improved theory of generalized weighting factor functions

Pieces of work made by V.Cech form the core of the theory［33，34］.

2.1.Ballistic atmosphere models

Atmospheric conditions have to be known in advance，at the time of planning the shot.This means that appropriate measurements have to be made in advance and consequently the results of these measurements need to be extrapolated in time and space，i.e.to the points the projectile will f l y through.For data extrapolation it is necessary to choose hypotheses about their future changes.The methodology of measurements of the required magnitudes，along with the algorithms of their processing and extrapolation，form the core of the ballistic models of the atmosphere ［2，9，39-41］.With respect to the aim of the article，we are going to def i ne and describe f i ve groups of models.The f i rst group of models serves for practical calculations of f i ring data.The remaining four groups of models are used for theoretical analyses and tabular and graphical f i ring tables.

2.1.1.Current atmosphere models

Current atmospheric models have the following basic features.They are based on currently-measured data，from which the noise and relatively quick trend components are eliminated. The data are exported to the users in the form of meteorological messages.There are three subsets of them.

In the f i rst group no concrete information about the weapon or the projectile planned for shooting is entered，i.e.，the data are universally usable by any weapon system.This especially embodies METCMQ meteo messages according to NATO methodology ［35，42，43］and METEO-11 （“Meteoaverage”）according to Soviet methodology ［2，3，24，26，28，41］.

The second group is represented by meteo message METBKQ according to NATO methodology ［35，40，43，44］.The data are modif i ed by means of weighting factors-WFs，deduced from particular WFFs.The applied WFs ［44］and WFFs are accurately valid only for a totally specif i c gun，projectile，charge and quadrant elevation.For other guns the data stated in METBKQ are valid only approximately.

The third group of models is represented by meteo message METGM ［35，43，45，46］.This modern method is based on complex modeling of the development of the meteorological situation and custom sending of the meteo messages ［35］.

2.1.2.Standard atmosphere models

In practice a number of general standard （normal，etalon）atmospheres are used.The most important is the International StandardAtmosphere （ISA）according to ISO 2533.In exterior ballistics a number of different standard atmospheres have also been used ［8］-not only generally，but also special ones.For our purposes we will mention only two of them:Ventcel's atmosphere （also Artillery Normal Atmosphere-ANA）and ICAO standard atmosphere.Ventcel's atmosphere has been used for the majority of calculations of f i ring tables according to Soviet methodology ［2，24-27，47］.ICAO standard atmosphere is being used for f i ring table calculations according to NATO methodology ［4，12，21，23，37，40，43，48］.

Figures about standard atmospheric parameters are indicated depending either on geometric altitude above mean sea level（MSL）H or geopotential altitude above MSL hgeopot.In the following text we will use only the altitude H.

This deals mainly with the following set of functions μSTD（H）= （τSTD（H）， pSTD（H）， ρSTD（H）， aSTD（H）， gSTD（H）），namely:virtual temperature，air pressure，air density，speed of sound and gravity acceleration+gN=9.806 65 m/s2-normal gravity acceleration.

2.1.3.Standard meteo-ballistic atmosphere model

In practice，a single model of meteo-ballistic atmosphere is used ［2，40，41］.This model serves for evaluating measurements and processing METBKQ meteo messages according to NATO methodology ［35，42，43］and METEO-11 （“Meteo-average”）according to Soviet methodology ［2，3，24，26-28，41］.It is a selected standard atmosphere that is vertically shifted according to the relation

where

hMDPis the altitude above MSL of a meteorological station（Meteorological Datum Plane-MDP），h is an altitude above MSL of the atmospheric layer measured［2，3，40，44］，yZis a superelevation of the atmospheric layer measured above MDP.

This transformation is illogical in principle and causes a number of complications when using the meteo messages METBKQ and METEO-11 ［2］in practice.

2.1.4.Standard f i ring table atmosphere models

In practice，a series of standard f i ring table atmosphere models is used.We are concerned especially about those models based on Ventcel's atmosphere （also ANA）and ICAO standard atmosphere.

The following relation is used for conversion of coordinates

where

hGis an altitude above MSL of the origin of a ballistic coordinate system （x，y，z），y is the height of the projectile trajectory above the level（x，y，z），y=0.

The altitude above MSL hGcan reach 10 000 m or more while shooting or bombing from an airplane.

Contractual-f i ring table values of the altitude hFT（hG=hFT）are chosen for setting up the f i ring tables.

According to Soviet methodology ［2，24-27，47］，hFT=0，500，1000，1500，2000，2500，3000 m above MSL.Firing tables set for hFT＞ 0 m are denoted as Mountain Firing Tables.This system has a convenient accuracy for approx.95%of the continent's surface ［2，33］.

Swiss methodology used the implicit def i nition of f i ring table altitudes hFTsuch as to def i ne the table's standard densities ρSTD（hFT）=1208，1150，1100，...，900 g/m3［9］.NATO methodology ［35，36］presupposes only hFT=0 m above MSL.Sowhen shooting in the mountains，signif i cant errors of shooting appear ［10］.This system has a convenient accuracy for not more than approx.50%of the continent's surface ［33］.

2.1.5.Perturbed f i ring table atmosphere models

For each standard atmosphere and each altitude hGat least three perturbed table atmosphere models exist ［8，9，20］.

In the interest of maximal simplif i cation of the computational algorithm，we will introduce a set of artif i ciallyconstructed relations in the model ［34］.

Perturbed magnitudes will be indicated by μP（H，t）= （τP（H，t），pP（H，t），ρP（H，t），aP（H，t），gP（H，t））.These are generalized input step functions-Eq.（5）.We will not consider the perturbation of the acceleration of gravity （gP（H，t）=gSTD（H））.

We consider wind vector w0= （wx，wy，wz）0to be a disturbance input variables belonging to vector d （Eqs. （1），（2）and（3））which are also perturbation constants （zm0）-see Eqs. （5）and （9）.

We will implement three perturbed virtual temperatures ［34］

where

（δτ0，Δτ0i）are perturbation constants （zm0）-see Eqs.（5）and （9），εP（t）is perturbation function.For the basic perturbation algorithm it is always εP（t）=H（t-tP）-see Eq. （5）. Perturbed hypsometric equation has the form ［34］

where hB（H，t）is perturbed pressure scale height ［m］

where

rDAis the gas constant of dry air ［J/（kg·K）］，τB，P（H，t）is the barometric average virtual temperature of the interval 〈0，H〉［K］

We will implement perturbed relative function of pressures［34］

and perturbed atmospheric pressure

where

where

（δp0，Δp0i）are perturbation constants （zm0）-see Eqs.（5）and （9）. Next we def i ne the perturbed relative function of virtual temperatures ［34］

and the perturbed relative function of air density ［34］

where

then perturbed air density （the f i rst output of the model）is def i ned by relation ［34］

where

where

（δρ0，Δρ0i）are perturbation constants （zm0）-see relations （5）and （9）.

We def i ne two perturbed speeds of sound ［34］

where

κMAis the adiabatic index of moist air.It approximately holds that ［8］κMA≈ κDA，κDAis the adiabatic index of dry air.

The vector of the projectile towards the air （air speed）-see Eq.（5）-is where

v= （vx，vy，vz）is a vector of the projectile towards the Earth（ground speed），then we def i ne two perturbed Mach numbers（second output of the model）［34］

For drag function φ（M）as def i ned in ［8］on page 95，and also［20］，its perturbed form can be used ［34］

where

cD（MP3（H，t））is a drag coeff i cient ［-］，which is used in all models.

Table 1Input data for perturbation models.

It is appropriate to supplement the model with a perturbed ballistic coeff i cient ［8，34］.

The model can be included in projectile trajectory models with 3 DoF，5 DoF or 6 DoF ［21，23，27，45］.

We get the （non-perturbed）standard table atmosphere model by setting all the perturbation constants as equal to zero-see Table 1 （Option 1）.

In practice，basically three models of perturbed table atmosphere are used ［8］-see Table 1 （Option 2，3，4）.

The f i rst model （Option 2）is the most widespread［8，10，19，20，29］.It is a prerequisite for f i ring table formations according to the NATO methodology ［35-37］and generating meteo messages analogical to METBKQ ［40，43，44］.For generating the WFF，rρδρ0≠ 0 and δτ0=wx0=wz0=0 is chosen. For generating the WFF rτ/ρis chosen δτ0≠ 0 and δρ0=wx0=wz0=0.For generating the WFF rwxwx0≠ 0 and δρ0= δτ0=wz0=0 is chosen.For generating the WFF rwzwz0≠ 0 and δρ0= δτ0=wx0=0 is chosen.For research use，different combinations of nonzero values of δρ0，δτ0，wx0，wz0can be chosen.

Even in the 1940s and 1950s it was discovered ［10］that this f i rst model has a very inappropriate course of the WFF rτ/ρ.That is why an alternative solution has been sought.For example，in ［20］the second model has been analyzed （Option 3）.It has been discovered that using drag function φ（M）-see Eq. （27）-instead of drag coeff i cient cD（M）brings signif i cant improvement.The new WFF rτ/phas a much more favorable course in the majority of cases in comparison with the original WFF rτ/ρ.Our brief comment can be found in ［32］. For generating the WFF rτ/pδτ01= δτ02= δτ0≠ 0 and δp0=wx0=wz0=0 is chosen.For generating the WFF rpδp0≠ 0 and δτ0=wx0=wz0=0 is chosen.

The third model （Option 4）practically agrees with the original model set by P.Langevin ［2，8，18，26-28］.It is used for setting up the f i ring tables according to the Soviet methodology and for meteo messages METEO-11.Based on our previous research，while the corresponding WFF rτ/p0has relatively the best features，nevertheless it does not comply under marginal conditions.The problem is not closed at all，so we plan to continue our research.For generating the WFF rτ/p0τ01= Δτ02= Δτ03= Δτ0≠ 0 and Δp0=wx0=wz0=0 is chosen.

2.2.Garnier's algorithm of the weighting factor function calculation

The essence of Garnier's method （see section 1.5）resides in the cyclic calculation of the partially-perturbed trajectories［9，33］.It derives from the properties of the Heaviside step function H（t-tP）-Eq. （5）.

The partially-perturbed trajectory is composed of two segments.The f i rst segment（H（t-tP）=0 for t

The parameter （the variable）is the start time of the perturbation tP，which is selected in steps from the interval〈0，tend〉.In the time t=tend，it is reached （x，y，z）end，（vx，vy，vz）end，etc.The meaning of the time tendwill be explained in the section 2.3.

The standard （unperturbed）trajectory is a special case of the partially-perturbed trajectory-the whole trajectory is unperturbed （tP≥ tend=tPI）.The point of impact/burst （the index PI）arises for the standard trajectory in the time t=tend=tPI- （x，y，z）PI，（vx，vy，vz）PI，etc.

The （full）perturbed trajectory is also a special case of the partially-perturbed trajectory-the whole trajectory is perturbed.It is valid tp=0.In the time t=tend，0，it is reached （x，y，z）end，0，（vx，vy，vz）end，0，etc.

The numerical calculations generate the set of partiallyperturbed trajectories that differ at chosen times tPi，i=0，1，2，...The times tPi∈ 〈0，tend〉are selected densely to make it possible to consequently express the courses of the WFFs.

Now it is possible to calculate perturbations of the elements in the point of calculations termination — point of impact/burst（t=tPI）for the perturbation of the meteo parameter μ

·range perturbation

·perturbation of height of impact/burst point

·azimuth perturbation

·time of f l ight perturbation

·perturbation of velocity horizontal component

etc.

These functions of times tPand tPIare the special cases of effect functions Rml（t-tP）-Eq.（5）.

From the practical point of view，the most signif i cant effect functions （EFs）are of the range，height，azimuth and time of f l ight（Eqs.（28）to （31））.We will use the special notation QP（μ，tP）=QP（tP）=QPfor these EFs.

2.3.Coordinate perturbation of the point of impact

There exist at least f i ve ways to def i ne the time tend，in which the calculation of the partially-perturbed trajectory is f i nished，and so we have minimally f i ve variants of perturbations of coordinates of the impact/burst point.

The f i rst variant is the simplest from the view of numerical calculation.We choose contractually tend=tPI，i.e.，the time of the calculation is always equal to the time of the projectile f l ight on the standard trajectory tPI.Consequently，the isochronous perturbations are considered and Δtt（μ，tP）=0.

The time tendis def i ned implicitly in the other four variants of perturbations.

In the second variant，the time tendis def i ned by the condition y（tend）=yPI，consequently ΔYy（μ，tp）=0.The iso-height of impact/burst perturbations takes effect in this case.Corresponding correction factors to ΔXy（μ，tP）and Δty（μ，tP）for tP=0 are usually included in the f i ring tables-Eqs. （8）and （9）［18，26，35-37，47］.Trivial approximate formulas exist for the conversion of isochronal perturbations into iso-height［7，8，10，19，27］.

In the third variant，the time tendis def i ned by the condition x（tend）=xPI，consequently ΔXx（μ，tP）=0.The iso-range of impact/burst perturbations takes effect in this case.Corresponding correction factors to ΔYx（μ，tP）and Δtx（μ，tP）for tP=0 are usually included in the f i ring tables-Eqs. （8）and （9）［18，26，35-37，47］.The perturbations Δty（μ，tP）and Δtx（μ，tP）for tP=0 are transformed into corrections of a fuze setting.Trivial approximate formulas exist for the conversion of isochronal perturbations into iso-range.

In the fourth variant，the time tendis def i ned by the condition ε（tend）= εPI，consequently Δεε（μ，tp）=0，where εPIis the angle of impact point site.In this case，the perturbations ΔDε（μ，tP）are calculated of the slant range D of the impact point and perturbations of the time Δtε（μ，tP）.The iso-angle of site perturbations takes effect in this case.

In the f i fth variant，the time tendis def i ned by the condition D（tend）=DPI，consequently ΔDD（μ，tP）=0，where DPIis the slant range of impact point.In this case，the perturbations ΔεD（μ，tP）are calculated of the angle of impact point site and perturbations of the time ΔtD（μ，tP）.The iso-slant range perturbations take effect in this case.

The total perturbation QPS（Δμ）of parameters of the point of impact/burst-e.g.the range ΔXP（Δμ），Eqs. （5）to （9）-under known （measured）courses of the absolute deviations Δμ（t）from the standard values （Δμ（t）= μmeasured（t）- μSTD（t））is given by the convolutory integral

and

where

（σQ·NQ）corresponds to Eq.（5），see sections 2.4 and 2.5.4，ΔμB0-see Eq.（9），ΔμBis the absolute ballistic deviation of the meteo ballistic element μ，rA（μ，tP）is the （normed）weighting factor function WFF，see Eqs. （5）and （6）［2，3，10］；the index A-see the commentary to the Eq.（8）.

Analogous Eqs.to （33）and （34）are valid also for the known（measured）relative deviations δμ（t）（δμ（t）= Δμ（t）/μSTD（t））［2，3］-see Eq.（8）.

2.4.Weighting factor functions for f l at-f i re trajectories

Measured deviations are evaluated for the requirements of the f l at-f i re depending on the topographic range，i.e.，on the coordinate x，and so （Δμ（x），δμ（x））is used ［23，30］.As a consequence，Eqs.（33）and （34）must be modif i ed.

We will use the function tP=F（x），which is valid for the unperturbed （standard）trajectory；then it will be QP（μ，x）and r（μ，x）.Let us remind the reader that dx=vx·dtP，thus Eq.（34）will have the form

We choose （σQ·NQ）=QP（μ，x）for x=0 in all cases.

It is important to be aware that

is valid.

The WFFs for the range wind rwx（x）and for the cross wind rwz（x）are important only for the f l at-f i re from a practical point of view.

2.5.Weighting factor functions for common trajectories

2.5.1.Generalized two-branched effect function

For shooting at common trajectories，measured deviations（Δμ，δμ）are evaluated depending on coordinate y of the projectile trajectory，thus （Δμ（y），δμ（y））is used ［7-10，17-27，33，45，49，50］.Therefore it is necessary to modify Eqs.（33）and（34）again.

The meteo message de facto determines （Δμ（yzi），δμ（yzi），i=0，1，2，...）［2，3，18，31-35，40-46］.As a consequence it is necessary to transform the data of the meteo message at f i rst（discrete coordinates yzi，see the paragraph 2.1.3）into data dependent on the coordinate y ［2，24，26，34，35，40，41，43］.

We will use the function tP=F（y）valid for standard trajectory.It is a one-to-two function.Such an essential failure will be eliminated by deliberating the particular dependence separately for the ascending branch （AB）tP，AB=tP1（y）=FAB（y）and separately for the descending branch （DB）tP，DB=tP2（y）=FDB（y）.It holds that tP1（y）≤ tP2（y）.In consequence of this，it is essential to separately consider traditional effect functions （EFs）for the ascending QP，AB（μ，y）and descending QP，DB（μ，y）branch.Thenext step requires sorting the true projectile trajectories into four groups-Table 2.

Table 2Four groups of true projectile trajectories.

By means of the traditional method it was possible to calculate WFFs r（y）only in three special cases ［10］:The trajectory has either only an ascending （1st trajectory）［10］，or only a descending （4th trajectory）［10］branch，or the trajectory has both branches of equal height （3rd trajectory and R=xLP）［2，7-10，18-20，24，26］.

Our goal is to provide calculating WFFs r（y）for each trajectory-Table 2.In order to do so，generalized effect functions（EFs）QCP（tP），def i ned for basic projectile trajectory，is introduced ［33］.

The basic trajectory ［34］consists of true projectile trajectory and virtual sections that are chosen so that the height of the ascending branch will be the same as the descending branch.In virtual sections of the basic projectile trajectory all perturbations contractually equal zero.

The minimal basic trajectory ［34］is a basic trajectory that has the same origin point or end point，i.e.，point of impact，or both with a true trajectory-Table 2 and Fig.1.

In accordance with the def i nition of the basic trajectory，the derivative of generalized effect functions ［34］is valid

where

i=1，2，3，4 are indices of the true trajectories-Fig.1，Table 2，tPis the perturbation time for the basic trajectory.

The link between tPand tPiis apparent from Fig.1 and consequently we def i ne generalized effect functions ［34］-Fig.1

Traditional effect functions QP，AB（μ，y）and QP，DB（μ，y）will be implemented by their generalized varieties QCP，AB（μ，y）and QCP，DB（μ，y）which have-unlike the traditional ones-the same height［34］.

Next，by means of unifying QCP，AB（μ，y）and QCP，DB（μ，y）into one function we will create a generalized two-branched effect function （curve）QCP（μ，y）=QCP（y）-Fig.2 ［2，8-10，19，20，33，34］.

2.5.2.Generalized Garnier's effect function

Generalized Garnier's effect function QCG（y）-Fig.3 is calculated ［34］according to the same def i nitional relation as a traditional Garnier's effect function QG（y）［8，9，20，33，49］but differs in inputs （QCP，AB（μ，y），QCP，DB（μ，y））and （QP，AB（μ，y），QP，DB（μ，y））respectively

The value QCG（μ，y0）represents the cumulative effect of all perturbations in heights y ≥ y0.

2.5.3.Generalized Bliss'effect function

Generalized Bliss'effect function QCB（y）-Fig.4 is counted［34］according to the same def i nitional relation as the traditional Bliss'effect function QB（y）［2，7，8，10，18，19，24，28，33，43，44，47］，but differs in inputs QCG（μ，y）and QP（μ，y）respectively

Fig.1.Basic trajectory and its four subsets-true trajectories and their corresponding generalized effect functions QCP（tP）（illustrative example）［34］，Table 2.Oi-origin of the i-th trajectory，PIi-point of impact/burst of the i-th trajectory.

Fig.2.Generalized two-branched effect functions （curves）QCP（y）for four varieties of projectile trajectories （illustrative example follows up the Fig.1）［34］，Table 2.

Value QCB（μ，y0）represents the cumulative effect of all perturbations in heights y ≤ y0.

2.5.4.Generalized weighting factor functions

Generalized weighting factor functions WFFs are calculated by norming from generalized effect functions （curves）.

For generalized Garnier's weighting factor functions WFFs it holds that ［8，9，20，49］

Fig.3.Generalized Garnier's effect functions QCG（y）（illustrative example follows up the Fig.2）［34］，Eq.（38）.

Fig.4.Generalized Bliss'effect functions QCB（y）（illustrative example follows up the Fig.3）［33］，Eq.（39）.

Fig.5.Course of the generalized Bliss'WFF srCB（μ，η）（illustrative example follows up Fig.4）［34］，Eq.（41）.

where

（σml·Nml）= （σQ·NQ）-see the relations （5），（33），（42），（47），（48）and （49）.

For generalized Bliss'weighting factor functions WFFs it holds that ［2，3，7，8，10，18，19，23-28，31-34，41，44，50］

where

ymin-see Table 2，

ymax-see Table 2，

QCG（μ，ymin）=QCP（μ，ymin）=QCP（μ，tP）=QCP（tP）for tP=0-see

Figs.1-3，

rCG（ymax）=rCB（ymin）=0-see Figs.1-4.

Traditionally ［2，3，7，8，10，18，19，23，24，26，28，30-34，49，50］are chosen （for tP=0） Hereafter，if the following condition is valid

then it always holds that

According to NATO and Soviet methodologies using Bliss' WFFs rB（μ，y）are presupposed，so we will limit our following analysis only to generalized Bliss'WFFs rCB（μ，y）-Figs.5 and 6.If interchange is not possible，we will no longer mention index “CB”in description of WFFs.

For the graphic presentation of the WFFs course the coordinate y is also normed.In Figs.5 and 6 two of many possible varieties of norming are presented.

Fig.6.The course of generalized Bliss'WFF srCB（μ，η）（illustrative example follows up Fig.4）［34］，Eq.（41）.

Analogically for Eqs.（34）and （35），（dy=vy（tP）·tP）will apply［2，3，8，10，18，19，24，26，28，31-34］

It is important to realize that analogically，in Eq.（36）it holds that

In the summit of the trajectory （x，y）sit applies that vy（ys）=0，so it is necessary to use l'Hospital's rule for analyzing relations for trajectories from 2 to 4 ［34］-Table 2.

When calculating WFF rτ/ρ（section 2.1.5）Eq. （43）is not usually fulf i lled and instead it usually applies that ［3，10，20，31-34］

The relation mentioned above is described as a “norm effect”［10］.In extreme cases it is possible that also QCP（0）=0 ［10，20］. In these cases it is essential，or more precisely，necessary to choose norm NQin a different way than according to the traditional method given by Eq.（42）.Moreover it is necessary to add that the described complication can appear even while calculating other WFFs ［20，33，34］.

The authors of the book ［10］chose the norm NQas the total variation of the function QB（y）（in our case it is QCB（μ，y））.If this norm is being used，the WFFs are indicated as “normalized effect functions （curves）”.The introduced process is correct from the mathematical point of view，but it is not suitable in practice.

Based on the analyses of the problem，we propose the following norm ［34］

Fig.7.Two generalized effect functions QCP（tP）（illustrative example follows up Fig.1）［34］.Problem of the “norm-effect”，Eqs. （46）-（49）.

and at the same time

if it applies that QCP（tP）≠ 0 for tP=0 i.e.QCP（0）.

If QCP（0）=0 then a number of varieties how to choose σQexist.For example we can choose ［34］

where

In case that equality Eq. （43）holds，Eqs. （47）and （48）becomes consistent with the traditional Eq.（42）.

Extreme variety （QCP（0）=0）will be explained by means of Figs.7-9.

3.Conclusion

This article presents our newly-conceived theory of generalized meteo-ballistic weighting factor functions-WFF as a special kind of sensitivity functions-Fig.10.The limited extent of this contribution has allowed us only to indicate the applicational possibilities of the new theory.

In the publications that will follow we plan to especially apply the new theory to the problematics of calculating the reference height of the projectile trajectory YR［2，3，18，28，31，33，34］that will mainly demand a detailed analysis of WFFs for the virtual temperature ［20，32］.

Fig.8.Two generalized Garnier's effect functions QCG（y）（illustrative example follows up Figs.3 and 7）［34］.Problem of the “norm-effect”，Eqs.（46）-（49）.

Fig.9.Two generalized Bliss'WFFs rCB（μ，η）（illustrative example follows up Figs.5 and 8）［34］.Problem of the “norm-effect”，Eqs.（46）-（49）.

Fig.10.The simplif i ed f l owchart of improved algorithm of the weighting factor functions and corresponding norms calculation.

Acknowledgments

This work originated with the support of f i nancing from the Research Project for the Development of the Department of Weapons and Ammunition，F(xiàn)aculty of Military Technology，University of Defence，Brno，DZRO K-201.

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Received 15 September 2015；revised 10 January 2016；accepted 21 January 2016 Available online 2 March 2016

Peer review under responsibility of China Ordnance Society.

*Corresponding author.Tel.:+420 539 010 703.

E-mail address:cech-vladimir@volny.cz （V.CECH）.

http://dx.doi.org/10.1016/j.dt.2016.01.009

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? 2016 China Ordnance Society.Production and hosting by Elsevier B.V.All rights reserved.