C.A. Rabbath, D. Corriveau

Defence R&D Canada, Valcartier Research Centre, 2459 De La Bravoure Rd., Quebec QC G3J 1X5, Canada

Keywords:Aerodynamic coefficients Piecewise polynomial functions Cubic splines Curve fitting Piecewise linear functions Piecewise cubic Hermite interpolating polynomial Projectile modelling and simulation Fire control inputs Precision Ballistic computer software

ABSTRACT Modelling and simulation of projectile flight is at the core of ballistic computer software and is essential to the study of performance of rifles and projectiles in various engagement conditions. An effective and representative numerical model of projectile flight requires a relatively good approximation of the aerodynamics. The aerodynamic coefficients of the projectile model should be described as a series of piecewise polynomial functions of the Mach number that ideally meet the following conditions:they are continuous, differentiable at least once, and have a relatively low degree. The paper provides the steps needed to generate such piecewise polynomial functions using readily available tools,and then compares Piecewise Cubic Hermite Interpolating Polynomial(PCHIP),cubic splines,and piecewise linear functions,and their variant,as potential curve fitting methods to approximate the aerodynamics of a generic small arms projectile. A key contribution of the paper is the application of PCHIP to the approximation of projectile aerodynamics,and its evaluation against a set of criteria.Finally,the paper provides a baseline assessment of the impact of the polynomial functions on flight trajectory predictions obtained with 6-degree-of-freedom simulations of a generic projectile.Crown Copyright ? 2019 Production and hosting by Elsevier B.V. on behalf of China Ordnance Society.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/).

1. Introduction

When used with small arms (SA), ballistic computer software(BCS) has the potential to improve the probability of hit for the marksmen,infantrymen,and snipers[1].The BCS is simply a handheld computer device that outputs gunnery solutions, typically in terms of rifle elevation and azimuth, for given engagement conditions. For its operation, the BCS runs a numerical model of the projectile,and simulates the trajectory of the projectile.Obviously,to be of any value, the numerical model must behave as closely as possible to the real projectile. Given meteorological conditions,shooter-target range and elevations, geolocation coordinates, and projectile characteristics,including its aerodynamics,the numerical model estimates the projectile flight. Hence one is able to determine appropriate elevation and azimuth for precision fires, or evaluate the precision characteristics of a projectile from modelling and simulation (M&S).

A model of the projectile implemented in the BCS is useful in military operations and in training. A model is also essential to carry out M&S studies of rifle-projectile performance under various operating and engagement conditions. An effective and representative numerical model of projectile flight is typically sought.At the very core of the numerical model, one finds the aerodynamics of the unguided body flying in the earth atmosphere. The aerodynamic characteristics of a projectile and the muzzle velocity are the so-called fire control inputs (FCI) to the BCS. According to Ref. [2], the FCI take the form of aerodynamic coefficients as functions of the Mach number,and muzzle velocity as a function of cartridge temperature. This paper focuses on the former. The projectile model requires, to a minimum, the values for the following eight aerodynamic coefficients [2,3]: the spin damping moment,the Magnus force,the two pitching moments,the lift and cubic lift forces, the zero yaw axial force,and the yaw drag force. It is internationally agreed through NATO Standard AOP-65 [2] that the aerodynamic coefficients should be described as series of polynomial functions of the Mach number M. These functions, also named piecewise polynomial functions, or PPF [4], should be continuous, preferably differentiable at least once (for third and fourth degree polynomials),and of degree smaller than or equal to four[2].Through modelling or trials,one may first obtain estimates of the eight aerodynamic coefficients at a finite,discrete set of Mach numbers.Then,as a second step,one may calculate the PPF for each aerodynamic coefficient by means of a curve fitting process where one seeks to obtain a curve that goes through the known discrete points in the plane of aerodynamic coefficient (ordinate) vs Mach number (abscissa). Several questions may then arise. For example,which PPF should be used? Are there curve fitting techniques that are more appropriate than others to represent the aerodynamics of the projectile over the entire set of Mach numbers? How can one measure the effectiveness of the PPF in the context of projectile flight? What is the impact of a particular PPF method on the projectile flight?And,what other considerations have an effect on the flight of the bullet?These are the types of questions that the paper seeks to address.

There are very few published papers and reports that study the approximation of projectile aerodynamic coefficients as functions of the Mach number.There is therefore very little known.In Ref.[5],spark-range trials are carried out to obtain aerodynamic coefficients at a discrete set of Mach numbers. Reduction and fitting methods are applied to those data to generate the continuous curves of aerodynamic coefficient versus Mach number. However,there is no explicit comparison of methods, discussion on the mathematical process involved in the curve fit,or assessment of the impact of the curve fit on the projectile flight prediction. Polynomials are studied in the context of drag aerodynamic coefficient expansion in Ref. [6]. Various expansion techniques are evaluated.Splines are used in the expansion of the coefficients.Yet,the results in Ref.[6]are not directly comparable with those presented in this paper. At least, reference [6] demonstrates one potential use of polynomials in the context of aerodynamic coefficient representation. The NATO standard presented in Ref. [2] prioritizes the methods with which the aerodynamic coefficients of a projectile may be obtained, and states the conditions on the polynomials,defined over the domain of Mach numbers, to achieve an appropriate curve fit. In Ref. [2], however, there is no discussion of the possible PPF that may be used for the curve fit.

There is limited literature on projectile M&S and the use of various PPF to express the aerodynamic coefficients as curves or functions of the Mach number.BALCO,which is a six/seven-degreeof-freedom (6/7-DOF) projectile trajectory M&S software developed by the French-German Research Institute of Saint-Louis, is a shareable software within NATO that enables using various PPF for the aerodynamic coefficients[1,7].Any polynomial function may be used to represent the aerodynamic coefficients in terms of Mach numbers.As far as we are aware,there is no known result obtained with BALCO that illustrate the performance of different PPF for the approximation of projectile aerodynamics.

The contribution of the paper is threefold. First, the paper provides the steps needed to generate piecewise polynomial functions using readily available tools, and then compares Piecewise Cubic Hermite Interpolating Polynomial (PCHIP), cubic splines, and piecewise linear functions (PLF), and their variant, as potential curve fitting methods to approximate the aerodynamics of a generic small arms projectile as functions of the Mach number.Second, the paper proposes PCHIP to approximate the projectile aerodynamics, and validates its performance against a set of criteria, and against other PPF/PLF. Third, the paper assesses the impact of PPF,and the numerical truncation of the coefficients,on a generic small arms projectile flight prediction using a 6-DOF model implemented in BALCO [7].

The paper is divided as follows.Section 2 presents the baseline,generic projectile model used in the study. Importantly, Section 2 gives the starting point of the study: the values for the generic projectile aerodynamic coefficients for a discrete set of Mach numbers. Section 3 presents the piecewise polynomial functions(PPF) and PLF for the approximation of the aerodynamics of the projectile as functions defined over the range of Mach numbers,including the proposed PCHIP function.The comparison of the PPF and PLF is done in Section 4.The section contains the definition of the metrics,and the approximations obtained in the domain of the Mach numbers and in terms of the flight of the generic projectile for the PPF and PLF studied in this paper.Finally,the conclusions on the performances obtained with the various PPF and PLF are stated in Section 5.

2. Baseline projectile model

This section presents the basics of the aerodynamic model of the projectile used in the analysis.

2.1. Description

The aerodynamics and flight behavior studied in this paper are those of a generic small arms projectile. The approaches and methods proposed in this paper are not connected to a particular projectile. Hence, the content of the paper is applicable to various projectile types.The aerodynamics of the baseline projectile model are the starting point of the study on approximation methods.The aerodynamics may contain errors with respect to true values, but this is outside the scope of the paper.The key aspect is that all the approximation methods are applied to the same baseline model.

The baseline model could correspond to 7.62 mm, 5.56 mm,0.338 caliber, and other SA projectiles. There is a certain body of knowledge in the literature on the means of obtaining aerodynamic tables, and basic tabular aerodynamics data for the 5.56 mm projectile, for example. Readers are referred to Refs. [8,9] for the specifics of the 5.56 mm projectile,which could be helpful to generate a model for this particular ammunition.Alternatively one may use software such as PRODAS (Projectile Rocket Ordnance Design &Analysis System)[10]to generate the aerodynamic coefficients at a discrete set of Mach numbers.PRODAS is a graphical user interface(GUI)-based projectile M&S environment.

The steps in the design of a projectile model in PRODAS are shown in Fig. 1. The first step is to gather information on the ammunition.2D and 3D drawings may come from laboratory work on actual ammunitions, such as cutting in halves the body and obtaining geometries,engineering drawings of the ammunition,or simply from pictures of the ammunition. The user enters material properties,such as density,for the different parts and components,and the dimensions of the projectile. The user uses the model editor. PRODAS then outputs a number of calculated ammunition characteristics, matching in part with existing database values,including center of mass, moments of inertia, aerodynamic coefficients versus Mach number, and spin rate.

The aerodynamics module of PRODAS is used to predict or estimate the aerodynamic coefficients and stability derivatives that are necessary to conduct the stability analysis, to compute the projectile trajectories,and to produce the firing tables.Based on the configuration developed in the model editor module of PRODAS,and with the function of each model component defined by the designer,an aerodynamically equivalent configuration is generated to obtain the aerodynamic coefficient predictions at a discrete set of Mach numbers.

A CAD drawing of a generic SA projectile for use with PRODAS is shown in Fig. 2.

Fig.1. Baseline model development in PRODAS.

Fig. 2. Projectile CAD drawing for use with PRODAS.

2.2. Aerodynamic coefficients at discrete set of Mach numbers

The main aerodynamic coefficients are as follows [3]:

·CD0 is the zero-yaw axial force coefficient,

·CD2 is the yaw drag coefficient,

·CLa is the linear lift force coefficient,

·CLa3 is the cubic lift force coefficient,

·CYpa is the Magnus force coefficient,

·Cma is the pitching moment coefficient,

·Cma3 is the degree 3 pitching moment coefficient, and

·Clp is the spin damping moment coefficient.

Table 1 contains the values for the eight aerodynamic coefficients of the generic projectile at a discrete set of Mach numbers as obtained by means of PRODAS and using typical geometrical and mass properties from Refs. [8,9]. There are seven non-zero aerodynamic coefficients. Cma3 is irrelevant for the remainder of the paper.

3. Aerodynamic coefficients as functions of Mach number

This section presents the piecewise polynomial functions (PPF)and piecewise linear function(PLF),including the proposed PCHIP,to represent the aerodynamic coefficients of the projectile as functions of the Mach number.

Table 1 Aerodynamic coefficients at discrete set of Mach numbers for the projectile conceptualized in this paper.

3.1. Mathematical problem

The mathematical problem is defined as follows: To express each aerodynamic coefficient C as a continuous function of Mach number M such that the function passes through the values of C at the finite set M*given in Table 1(the set of 30 Mach numbers in the first column from the left), where

From Ref.[2],two important properties of the function C?f(M)are sought: (1) the function is continuous, so that the function is differentiable at every point M,and the limit both from the left and from the right at any given point is equal to the value of the function at the same point,and this is true for every point in the admissible domain;and(2)the function passes through the available values of C for the points M2M* (columns 2 to 9 of Table 1).

Furthermore, aerodynamic coefficient functions for fire control solutions must comply with the following five requirements(R1 to R5) stated in the NATO Standard [2]:

·R1: The aerodynamic coefficients C are functions of Mach number M, so C?f(M) where f( ) is some function, as already indicated,

·R2: The functions are consecutive polynomials of degree up to four, with the polynomial coefficients assumed to be real numbers,

·R3:The functions are defined over regions,or intervals,of Mach number, from a minimum to a maximum Mach number value,

·R4:The upper limit of the highest interval should be higher than the Mach number corresponding to the maximum possible muzzle velocity,

·R5: The series of polynomials must be continuous and preferably differentiable at the connecting break points when using high-degree (third or fourth degree)polynomials in M.

It should be noted that there are no requirements on the number of intervals and on the length of each interval.

Requirements R2 and R3 indicate that a single polynomial function that covers the entire Mach number domain is not acceptable.A set of polynomial functions defined over intervals,or sub-regions,of the entire set of M are preferred.The reason for such requirements is briefly explained. It is true that, in theory, there exists a unique polynomial function of degree n-1 that passes through the entire set of known n points at which the values of a real function are defined [11]. Yet, in practice, for relatively large values for n,(1) the numerical computations become difficult, and(2) when a polynomial function is actually obtained, the fluctuations associated with that function are typically large and unacceptable. One may further reduce the number of points through which a single polynomial function should pass.So,one may select a reduced set of salient(M,C)pairs,but then the single polynomial curve may not necessarily go through the original 30 pairs, in addition to still offering large fluctuations in-between the discrete set of (M,C) pairs. Thus, such a function is not pursued further in this paper for the obvious reasons. For R5, please note that differentiability at the connecting points is preferred, but is not mandatory, for relatively high-degree polynomials.

From Table 1, n ? 30, and there is the information on the aerodynamic coefficients at those 30 M values. Associate an index to each known Mach number in an ordered sequence from the smallest to the largest,again using the entries of Table 1. Let M1?0:01, M2? 0:4, M3? 0:6, and so on up to M30? 8. Then, Mjand Mjt1,; are known as consecutive Mach numbers in the set M*.

Define a fourth-degree polynomial expressed in terms of M2M*valid over a portion of the range of Mach numbers with the general form,from Ref. [2]:

Define the unit step function 1ewT as

Then, the overall expression for C ?CeMT can be written as a sum of the piecewise components CieMT as follows

In other words, for values of M in ?Mi; Mit1T,CeMT?CieMT;: In terms of the computations involved,for fourth-degree polynomials, there are 5 unknowns to calculate per interval, 29 intervals per aerodynamic coefficient, and 8 aerodynamic coefficients according to Table 1.

To meet the continuity requirement(R5),the PPF must be such that there is no discontinuity at the known Mach numbers in the set M*, namely, at the Mi,.

Table 2 Index and Mach number interval.

3.2. Piecewise linear functions

One particular type of PPF is the baseline piecewise linear function,or PLF.PLF comprises segments of straight lines that meet at a finite number of points. For any given ith interval, the linear function is simply given as

In Eq. (5), the polynomial coefficients A0 and A1 are real numbers. For each Mach number interval and aerodynamic coefficient, there is a pair eA0;A1T.

The approach to obtain the coefficients of the PLF is straightforward.Consider CieMT at M ?Miand at M ?Mit1.Then,one may write

and

One wants to solve for A0 and A1 at the ith interval.From Eq.(6),

Substituting the expression for A0 given in Eq. (8) into Eq. (7),one obtains value for the first- and second-order derivatives of C with respect to M,and so one may observe a relatively smooth curve for a given aerodynamic coefficient over the entire set of Mach numbers,

·The spline polynomial coefficients are obtained by solving a system of equations with a number of unknowns (polynomial coefficients), with constraints or conditions at the knots (value of functions and derivatives), and at the boundaries (first and last points).

The cubic splines are calculated with the MATLAB? built-in function spline (arg1,arg2), where arg1 is the vector containing the set of discrete values for M, and arg2 is the vector of aerodynamic coefficients of interest at the discrete set of M. The cubic spline piecewise polynomials have the general form given as

The script for the cubic splines is given in Table 4.

The coefficients a0;i; a1;i,a2;i,a3;iare calculated for every interval i;: It should be noted that the polynomials obtained with the function unmkpp(as shown in Table 4)are in the form of variable D. Thus a conversion to third-degree polynomials in M is needed. Such subtlety makes a difference. The polynomials in M are obtained as follows. The cubic spline polynomials in D obtained with function unmkpp may be expressed as

Replacing A1 in Eq. (8) with the expression Eq. (9), one may solve for A0:

It is important to note that in theory CieMit1T does not exist given our definition of the intervals. It is the value of CieMT as M approaches Mit1in the interval?Mi;Mit1T that is defined.

The PLF coefficients are given in Table 3. The PLF curves are shown in Fig.3.The Cma3 curve is not shown for obvious reason.It should be noted that the slope of the curve for a given aerodynamic coefficient may experience a sudden change at the points where the linear functions meet. Requirement R5 is therefore not met de facto for the PLF.

3.3. Cubic splines

Consider the general PPF form given in Eq.(2)and Eq.(4).Cubic splines for the aerodynamic coefficients are a series of third-degree piecewise polynomials in M that satisfy the following properties from M ?0:01 to M ?8 [4,11,12]:

·The curves are continuous,

·The first and second derivatives at the nodes where two piecewise polynomials meet (typically called knots) are continuous, and so at a knot, adjacent splines have the same

Table 5 presents an excerpt of the MATLAB?script to implement Eq.(14).Such a script should be run after the one shown in Table 4.

For brevity, the cubic spline polynomial coefficients of Eq. (11),where index i corresponds to the first column of each table,are only given for representative coefficients CD0, CD2 and CLa in Table 6,Table 7,and Table 8.It is straightforward to calculate the polynomial coefficients for the other aerodynamic coefficients. The resulting cubic spline PPF curves and the discrete values of C at the finite set M*are shown in Fig. 4.

3.4. PCHIP

Piecewise Cubic Hermite Interpolating Polynomial,or PCHIP,is a third-degree PPF with the following characteristics [11,13,14]:

·it is a shape-preserving piecewise cubic Hermite interpolation,

Table 3 Polynomial coefficients of the PLF.

·it is different from cubic splines in the sense that it seeks to match only the first-order derivatives at the data points with those of the intervals before and after (a characteristic of Hermite interpolation),

·its minimum matches the minimum of the data,

·it is monotonic over intervals where the data are monotonic,

·it exhibits generally less overshoot/undershoot than cubic splines.

The aforementioned characteristics make PCHIP an attractive method when one seeks a curve fit for all of the aerodynamic data points. In particular, it is known that in general PCHIP curves present no peaks and valleys.The curves are also differentiable at every point along M. In general, cubic splines are smoother than PCHIP curves, however.

The PCHIP piecewise polynomials have the general form given in Eq. (11). The PCHIP curves are calculated with MATLAB? using built-in functions. Table 9 presents an excerpt of the MATLAB?script written to obtain the PCHIP coefficients. The excerpt shown in Table 9 pertains to CD0. Similar scripts are used for the other aerodynamic coefficients, but are not shown for brevity.

The CD0,CD2 and CLa PCHIP PPF coefficients of Eq.(11)are given in Table 10,Table 11,and Table 12.As in the case of the cubic spline polynomial coefficients, index i (interval number) corresponds to the first column of each table. The PCHIP PPF curves and the discrete values of C at the finite set M*are shown in Fig. 5.

3.5. Variants

There exist three variants to calculate PPF/PLF: (1) apply the methods described in Sections 3.2 to 3.4 on a reduced number of intervals(here there are 29 intervals,and 30 discrete aerodynamic coefficients), (2) modify the degree of the polynomials, and (3)truncate or round the polynomial coefficients of the PPF/PLF.

Variant 1(V1):To reduce the number of intervals,one may carry out a visual inspection of each of the 7 non-zero aerodynamic coefficients as a function of M,and use only a subset of the data points to generate the curves. The data points selected are typically inflection points and start/end points of the overall curve. It is clear that the resulting curve may not pass through the n?30 points of the original data set.Such curve is useful for comparison purposes:to assess the impact of reducing the number of intervals on the curve fitting performance.

Variant 2 (V2): To select and then force the degree of the polynomials,which is not a straightforward process,is not pursued further in this paper, as it is expected to require further mathematical development.

Variant 3 (V3): To truncate or round the calculated polynomial coefficients is readily achievable. Such a variant is useful to illustrate, albeit in a limited fashion, the effect of limited/constrained computing power on the approximation of the aerodynamics, and the impact of a simplification of the polynomial coefficient representation.Limited computing power may arise from the use of lowcost, fixed-point arithmetic processors.

4. Comparison of PPF and PLF

This section contains the definition of the metrics, and the approximations obtained in the domain of the Mach numbers and in terms of projectile flight for the PPF and PLF studied in this paper.The PPF and PLF are compared qualitatively and quantitatively.

4.1. Qualitative and quantitative evaluations in the domain of Mach numbers

The simplest means of evaluating the PPF is a visual inspection of the curves and a comparison with the available values of C for the points M2M*. The functions produced are shown on the same plots in Fig. 6 and Fig. 7 for only two representative aerodynamic coefficients, for the sake of brevity: CD0 and CLa. The same observations can be made for the entire set of aerodynamic coefficients.To facilitate the visual inspection,key zones in the figures are magnified.

Fig. 3. PLF curves.

Table 4 Cubic spline script.

The curve labeled as“PLF reduced set”presented in the figures is a variant of the PLF (actually, V1). Seven Mach numbers and aerodynamic coefficient values are selected from the set of the original 30 values(given in Table 1).This selection of seven discrete values(and six intervals)is arbitrary.It is chosen for illustrative purposes,and to quantify the impact on the approximation of the aerodynamics of using a significantly reduced number of intervals. To reduce the number of intervals, one carries out a visual inspection of the original piecewise linear curve,and uses only a subset of the data points.The data points are selected such that the straight lines connecting these points stay relatively close to a maximum number of the n?30 points of the original data set.The selected data points are typically the major inflection points of the curve.This process is carried out for each aerodynamic coefficient, except Cma3. For brevity, and to illustrate the results, only the polynomial coefficients of CD0 are presented in Table 13.The resulting CD0 curve and the seven,selected discrete values are shown in Fig. 8.One may observe from Figs. 6,7, and 8 that

Table 5 Script for polynomial coefficient conversion.

·Cubic splines are the only curves that exhibit oscillations, and the oscillatory behavior is especially important at low and at high M, and in case the consecutive data points experience a significant change in slope,

·The reduced PLF, namely the PLF with connecting intervals at the selected subsets of M,provide the largest deviation from the data points among the PPF,

·PCHIP is smooth (at least differentiable once over all M) and non-oscillatory despite sharp turns, and has values relatively close to those of the PLF over the full set of data points,

·There is deterioration in the quality of the curve fit that follows from a reduction in the number of intervals used for the approximation.

4.1.1. Metrics

Several quantitative measures of performance, or metrics, may be used to evaluate the different PPF.Performance of a PPF is in the

Table 6 Cubic spline polynomial coefficients for CD0.

Table 7 Cubic spline polynomial coefficients for CD2.

sense of the closeness of the curve to the data points. The data points are the available values of C for the points M2M*(columns 2 to 9 of Table 1). Equivalently, performance of a PPF may be interpreted as the evaluation of the goodness of curve fitting from the knowledge of the discrete data points.

In this paper, two metrics are used:

Table 8 Cubic spline polynomial coefficients for CLa.

Fig. 4. Cubic spline curves.

·Maximum separation (MS) between any given curve and the reference PLF curve defined over the full set M.

·Area (A) between any given curve and the reference PLF curve defined over the full set M.

The reference PLF curve defined over the full set M serves as the yardstick curve against which the other curves are measured. The reference PLF curve is that obtained in Section 3.2.

The rationale is as follows: the reference PLF curve is the simplest possible curve that connects all the data points in a linear fashion,and has no overshoot/undershoot or oscillations.However,such a curve is not differentiable(at least once)everywhere.It doesnoes meet all the criteria given in Ref. [2]. In simple terms, at the points where successive intervals meet,namely at the data points,the piecewise linear curve may offer a sudden change in slope when approached from the left and from the right.As for PCHIP and cubic splines,they are differentiable with respect to M at least once everywhere.

Table 9 Part of the PCHIP script to calculate the CD0 polynomials.

Table 10 PCHIP polynomial coefficients for CD0.

Table 11 PCHIP polynomial coefficients for CD2.

Table 12 PCHIP polynomial coefficients for CLa.

For the evaluation in terms of MS and A,one divides the range of M from 0.01 to 8 into a larger set of data points available at integer multiples of T ? 0:001. One obtains the set Md?f0:01;0:011;0:012;…;8g One evaluates each PPF at those values of Md. For MS, one identifies the maximum distance between each PPF and the yardstick PLF curve defined over the full set M as evaluated over all Mach numbers in Md. For A, the area is approximated by a Riemann sum over all Mach numbers Md, as given by Eq. (15). In the equation, C is the curve being evaluated,and Cy is the yardstick PLF curve. Obviously, one looks for the smallest values of MS and A for the best performance with respect to the yardstick PLF curve defined over the full set M.

4.1.2. Results

The results of the calculations for MS and A are summarized in Table 14.Clearly,PCHIP exhibits the smallest metrics for all but one case tested (MS calculated for Clp). This is expected as PCHIP is known to offer fewer valleys and peaks than cubic splines,from the differentiability property of the polynomial curve fit. The PLF reduced set offers in general the worst performance as measured with MS and A.

4.1.3. Effect of truncation

Rounding or truncation of the PPF coefficients impacts on the value of C for any given M.This case corresponds to PPF variant 3,or V3, as defined in Section 3.5. There is no significant truncation or rounding error when the PPF are implemented on high-end computers with their representation and numerical computations carried out with a relatively large number of bits and floating-point arithmetic.There may be a problem when one is constrained to use a limited number of bits in the representation/computations of the PPF, and fixed-point arithmetic.

Fig. 5. PCHIP curves.

An example is provided to briefly examine the impact of truncating CD0 polynomial coefficients. Similar conclusions can be made for all the aerodynamic coefficients. Suppose that the PCHIP and cubic spline coefficients are truncated to a certain number of decimal places.Furthermore,assume that the computations are not constrained in any way, and hence are performed with floating point arithmetic as available on standard computers. Maximum errors when truncating the polynomial coefficients to one and two digits after the decimal point are given in Table 15.The CD0 curves obtained from the truncated polynomial coefficients are shown in Fig. 9 and Fig.10.

Two observations can be made from the results obtained with the truncated polynomial coefficients:

·When the error due to rounding/truncation is significant enough, the PPF curves obtained from the rounded/truncated polynomial coefficients, curves indicated as bCi; become discontinuous and may show bias,

·The maximum error in CD0 (namely,over all i)increases with the rounding/truncation error,and is in the same order of magnitude for all polynomials,with PCHIP having the largest error in CD0 in case of a truncation to 1 decimal place.

4.2. PPF and projectile trajectory

One may assess the impact of the PPF and PLF developed in the previous sections on projectile flight prediction using 6-degree-offreedom M&S with BALCO [1,7]. Any polynomial function may be used to represent the aerodynamic coefficients in terms of Mach numbers.Implementing in BALCO a PPF or a PLF for an aerodynamic coefficient is a straightforward process [7].

4.2.1. Approach for M&S tests

The M&S test procedure proposed in this paper is as follows:

·Step 1,Setup the scenarios at firing ranges of 600 m and 1000 m to capture in simulation the projectile trajectory over a relatively wide span of Mach numbers, using the main simulation parameters shown in Table 16, of course other ranges could be selected to expand the study,

Fig. 6. Superimposed CD0 PPF and PLF curves for visual inspection.

·Step 2,Develop a model in the BALCO[1,7]environment relying on the discrete aerodynamic coefficients of Table 1 and the scenario/simulation parameters produced in Step 1, and apply the PLF, the cubic spline PPF (with/without truncation to 2 decimal places), and the proposed PCHIP PPF (with/without truncation to 2 decimal places)presented in Sections 3.2-3.4 on the eight aerodynamic coefficients,actually the seven non-zero functions,

·Step 3, Run the BALCO simulations, and record the projectile speed, trajectory, and position at the target's downrange location for the cases modeled in Step 2.

The Quadrant elevation(QE)in Table 16 is obtained by trial and error. The process is as follows. With the BALCO input script that uses the PLF that connects all 30 discrete data points for each aerodynamic coefficient(as given in Table 1),the QE is set to a value selected by the designer. Then the BALCO simulation is run with this QE.If the projectile falls on the ground(or at a prescribed target height) at the desired range with this QE, the QE value becomes final,and is used for all the PPF/PLF.Otherwise,the QE is modified in the BALCO input file, and the simulation is run again with the updated QE value. The process is repeated until the projectile falls on the ground(or at target height)at the desired range.The muzzle velocity is set to 990 m/s(Mach number of approximately 2.9).The time step used for the solution to the ordinary differential equations iss except for the cubic spline PPF for the 1000 m range scenario which requires reducing the time step tos to obtain convergence of the numerical solution.

4.2.2. Results

The simulation results are shown in Fig.11,Fig.12,and Fig.13.In the figures,the unit is the meter.A similar drift trajectory profile is found at the 600 m and 1000 m ranges,and so only the 600 m range plot is provided for brevity.At 600 m,the projectile reaches a Mach number of approximately 1.05 for PCHIP, cubic spline and PLF. At 1000 m, the Mach number is about 0.72 for the same 3 approximation methods.

The following observations can be made on the M&S results for the 600 m and 1000 m ranges:

·Truncation of the polynomial coefficients to 2 decimal places introduces a significant error in the trajectory of the projectile with respect to the trajectory obtained without truncation(1 m at 1000 m range),

·PCHIP PPF,without coefficient truncation,produces a projectile trajectory that is relatively close to that obtained with PLF,with a maximum difference of only 5 cm between the trajectory resulting from cubic spline,PCHIP and PLF,hence the differences observed in the Mach domain between PCHIP and cubic splines do not translate into significant differences in bullet trajectory for the cases simulated.

Fig. 7. Superimposed CLa PPF and PLF curves for visual inspection.

5. Conclusions

The paper presents the steps needed to approximate the aerodynamic coefficients of a generic projectile model as a series of piecewise polynomial functions (PPF) of the Mach number. The functions are calculated with readily available numerical tools.The paper introduces the Piecewise Cubic Hermite Interpolating Polynomial(PCHIP)to approximate the aerodynamics,and validates its performance against a set of criteria, or metrics, and against other piecewise linear functions and cubic splines. The paper also investigates the effects of variants of the polynomial functions on curve fitting performance. Variants include a reduced number of intervals, and truncation or rounding of the polynomialcoefficients. Finally, the paper provides a basic assessment of the impact of the polynomial functions on a generic small arms projectile flight prediction using a 6-DOF model implemented in BALCO.

Table 13 CD0 polynomial coefficients for PLF reduced set.

Fig. 8. CD0 curve labeled as “PLF reduced set” with original and selected data points.

Table 14 MS and A values for aerodynamic coefficients CD0, CD2, CLa, CLa3, Cma, Clp, and CYPa.

Table 15 Maximum errors for two truncation cases.

Fig. 9. CD0 curves with and without truncation of the polynomial coefficients (2 decimal places truncation).

Fig.10. CD0 curves with and without truncation of the polynomial coefficients (1 decimal place truncation).

Table 16 Main parameters for the simulation scenarios implemented in BALCO.

Fig.11. Side view of trajectories for the 600 m firings.

It was found that a piecewise linear function (PLF)approximation to the aerodynamics of a generic projectile result in trajectories that are relatively close to those obtained with aerodynamics approximated with PCHIP and cubic splines.PLF is much simpler to implement than PCHIP and cubic splines. However,PCHIP provides a smoother curve than PLF in the domain of Mach numbers. PCHIP meets all of the NATO standards for third- and fourth-degree polynomials. When compared with cubic splines,PCHIP results in aerodynamic curves that are continuous in the Mach domain, and are less oscillatory. Reducing the number of intervals and truncating/rounding the polynomial coefficients of the PPF were shown to significantly deteriorate the approximation of the aerodynamics,and to adversely affect the numerical model of the bullet flight trajectory.Further work is needed to generalize the observations and conclusions stated in this paper, particularly for the analysis of the impact of PPF on bullet trajectory.The influence of engagement parameters and projectile initial conditions,such as muzzle velocity,on curve fit should be assessed.The specific ranges of Mach numbers for which there are significant differences in curve fit observed in the Mach domain should be further explored in terms of trajectory scenarios. Overall, the contribution of the paper is the first step toward developing knowledge in the use of polynomial functions to approximate small arms projectile aerodynamics.

Fig.12. Top view of trajectories for the 600 m firings.

Fig.13. Side view of trajectories for the 1000 m firings.