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Aerospace Science and Technology ••• (••••) •••–•••

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Aerospace Science and Technology

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Aerodynamic parameter identification for symmetric projectiles:An improved gradient based method

Bradley T. Burchett 1

Department of Mechanical Engineering, Rose–Hulman Institute of Technology, Terre Haute, IN 47803, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 8 November 2012Received in revised form 3 July 2013Accepted 23 July 2013Available online xxxx

Keywords:ProjectileEstimationLinear theory

In order to estimate the aerodynamic coefficients of a symmetric projectile from free flight range data, analgorithm using traditional linear theory with non-linear programming is investigated. The linear theorysolution is reformulated resulting in matrix expressions of improved compactness and simplicity, whichprovide predictions of yaw, pitch, crossrange, and altitude as functions of downrange travel. The newsolution is easily differentiated with respect to aerodynamic coefficients such that a gradient basedoutput error estimation algorithm naturally follows. Numerical results are presented and compared toa previous method that required finite differencing for gradient estimates.

© 2013 Elsevier Masson SAS. All rights reserved.

1. Introduction

Estimation of aerodynamic coefficients from free flight rangedata is a well established field of study [6,7,16,18,20,24]. In gen-eral, the method is preferred over wind tunnel methods sincemeasurements are taken of actual flight with minimal disturbance,and full scale projectiles are observed.

Early studies [20] focused on transforming yaw card data andspark range images to positions and Euler angles such that linearpredictions could be directly fit to the observed trajectory. Morerecently, advances in optimization provide new avenues to matchnot only linear, but also non-linear trajectory predictions to therange data [2,6], and attempts have been made to expand capabil-ities to include projectiles which are aerodynamically asymmetric[11].

Projectile linear theory provides an essential element of manyestimation routines by allowing rapid prediction of free flight tra-jectories. Recent advances in linear theory have explored the effectof divert events [5,13,14], high gun elevations [15], and low launchvelocities [22]. Linear theory has also been used in several modelpredictive control schemes [4,12,21,22].

In this effort, noise-free actual trajectory data are assumed tobe available whether measured by yaw cards or spark range cam-eras. This work incorporates for the first time, a faster, more com-pact projectile linear theory solution including closed-form deriva-tives of trajectory predictions with respect to unknown aerody-namic parameters. Additionally this paper demonstrates the useof the Marquardt algorithm [17] for rapid non-linear program-

E-mail address: burchett@rose-hulman.edu.1 Associate Professor.

1270-9638/$ – see front matter © 2013 Elsevier Masson SAS. All rights reserved.http://dx.doi.org/10.1016/j.ast.2013.07.010

ming convergence of single and multiple launch data sets betweenwhich several, but not all aerodynamic parameters and launchconditions are shared. The algorithm is exercised in several casestudies to find the smallest set of measurements for satisfactoryparameter estimation, and the optimal placement of yaw cards orspark range stations. The range data are generated by a 6 DOFnon-linear simulation described in Section 2. The linear model,closed-form solution, and derivatives are presented in Section 3.Section 4 presents the Levenberg–Marquardt algorithm and Section5 compares the results for this algorithm when analytic derivativesare used as opposed to finite difference estimates.

2. Projectile dynamic model

The non-linear trajectory simulation used in this study is astandard six-degree-of-freedom model typically used in flight dy-namic modeling of projectiles. A schematic of the projectile con-figuration is shown in Figs. 1 and 2. The six degrees of freedomare the three inertial components of the position vector from aninertial frame to the projectile mass center and the three standardEuler orientation angles. The equations of motion are provided inEqs. (1)–(4) [4].

⎧⎨⎩

xyz

⎫⎬⎭ =

⎡⎣ cθ cψ sφsθ cψ − cφsψ cφsθ cψ + sφsψ

cθ sψ sφsθ sψ + cφcψ cφsθ sψ − sφcψ

−sθ sφcθ cφcθ

⎤⎦

⎧⎨⎩

uvw

⎫⎬⎭ (1)

⎧⎨⎩

φ

θ

ψ

⎫⎬⎭ =

⎡⎣ 1 sφtθ cφtθ

0 cφ −sφ

0 s /c c /c

⎤⎦

⎧⎨⎩

pqr

⎫⎬⎭ (2)

φ θ φ θ

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Nomenclature

L, M, N total external applied moment on the projectile aboutthe mass center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ft-lb

CN A normal force aerodynamic coefficientC X0 axial force aerodynamic coefficientCL P roll rate damping moment aerodynamic coefficientCLD D fin rolling moment aerodynamic coefficientCM A pitch moment due to AOA aerodynamic coefficientCM Q pitch rate damping moment aerodynamic coefficientD projectile characteristic lengthI identity matrixIxx, I yy roll and pitch inertia expressed in the projectile

reference frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sl-ft2

J Jacobian matrixm projectile mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . slp,q, r angular velocity vector components expressed in the

fixed plane reference frame . . . . . . . . . . . . . . . . . . . . . . rad/sS LC G stationline of the projectile c.g. location. . . . . . . . . . . . . ftS LC M stationline of the projectile Magnus c.p. location . . . ftS LC P stationline of the projectile c.p. location . . . . . . . . . . . . ftT residual vectoru, v, w translation velocity components of the projectile

center of mass resolved in the fixed plane referenceframe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ft/s

V magnitude of the mass center velocity . . . . . . . . . . . . ft/s

X, Y , Z total external applied force on the projectileexpressed in the body reference frame . . . . . . . . . . . . . lb

x, y, z position vector components of the projectile masscenter expressed in the inertial reference frame . . . . ft

Greek

Φ position state vector dynamics matrixΓ position states forcing functionχ velocity states initial condition vectorΔ1 factor in position states forced solutionη linear model velocity state vector {v w q r}T

Ψ combination matrixΛ combination variable in the roll rate solutionρ air density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sl/ft3

ψ,θ,φ Euler yaw, pitch, and roll angles . . . . . . . . . . . . . . . . . . . radϑ vector of unknown model parameters {CN A, C X0,

CL P , CLD D , S LC P , CM Q , p0,q0, r0}T

Ξ velocity state vector dynamics matrixξ linear model position state vector {y z θ ψ}T

Subscript

0 initial condition or previous statei, j row i, column j of matrixp particular solution

⎧⎨⎩

uvw

⎫⎬⎭ =

⎧⎨⎩

X/mY /mZ/m

⎫⎬⎭ −

⎡⎣ 0 −r q

r 0 −p−q p 0

⎤⎦

⎧⎨⎩

uvw

⎫⎬⎭ (3)

⎧⎨⎩

pqr

⎫⎬⎭ = [I]−1

⎡⎣

⎧⎨⎩

LMN

⎫⎬⎭ −

⎡⎣ 0 −r q

r 0 −p−q p 0

⎤⎦ [I]

⎧⎨⎩

pqr

⎫⎬⎭

⎤⎦ (4)

In Eqs. (1) and (2), the standard shorthand notation for trigono-metric functions is used: sin(α) ≡ sα , cos(α) ≡ cα , and tan(α) ≡tα . The force appearing in Eq. (3) contains contributions fromweight W , body aerodynamics A, and Magnus M aerodynamicforce:⎧⎨⎩

XYZ

⎫⎬⎭ =

⎧⎨⎩

XW

Y W

ZW

⎫⎬⎭ +

⎧⎨⎩

X A

Y A

Z A

⎫⎬⎭ +

⎧⎨⎩

XM

Y M

Z M

⎫⎬⎭ (5)

Gliding flight is assumed in this study. The dynamic equations areexpressed in a body-fixed reference frame, thus, all forces actingon the body are expressed in the projectile reference frame. Theprojectile weight force is shown in Eq. (6):⎧⎨⎩

XW

Y W

ZW

⎫⎬⎭ = mg

⎧⎨⎩

−sθ

sφcθ

cφcθ

⎫⎬⎭ (6)

whereas the aerodynamic force acting at the center of pressure ofthe projectile is given by Eq. (7):⎧⎨⎩

X A

Y A

Z A

⎫⎬⎭ = −π

8ρV 2 D2

⎧⎨⎩

C X0 + C X2(v2 + w2)/V 2

CN A v/VCN A w/V

⎫⎬⎭ (7)

and the Magnus force acting at the Magnus force center of pres-sure is [9,10]:⎧⎨⎩

XM

Y M

Z

⎫⎬⎭ = π

16ρD3

⎧⎨⎩

0pCY P A w−pC v

⎫⎬⎭

M Y P A

Fig. 1. Schematic of a fin-stabilized projectile, position coordinates.

The applied moments about the projectile mass center containcontributions from steady aerodynamics (SA), and unsteady aero-dynamics (UA).

⎧⎨⎩

LMN

⎫⎬⎭ =

⎧⎨⎩

LS A

M S A

N S A

⎫⎬⎭ +

⎧⎨⎩

LU A

MU A

NU A

⎫⎬⎭ (8)

The moment components due to steady aerodynamic forces arecomputed with a cross product between the distance vector fromthe mass center of the projectile to the location of the specificforce and the force itself. The moment due to the Magnus force iscomputed by a cross product of the distance vector from the masscenter to the Magnus force center of pressure. The unsteady bodyaerodynamic moment provides a damping source for projectile an-gular motion and is given by Eq. (9):

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Fig. 2. Schematic of a fin-stabilized projectile, attitude coordinates.

⎧⎨⎩

LU A

MU A

NU A

⎫⎬⎭ = π

8ρV 2 D3

⎧⎪⎨⎪⎩

CLD D + pDCL P2V

qDCM Q2V

rDCM Q2V

⎫⎪⎬⎪⎭ (9)

The center of pressure location and all aerodynamic coefficients(C X0, C X2, CN A, CLD D , CL P , and CM Q ) depend upon local Machnumber and are computed during simulation using linear inter-polation.

The dynamic equations given in Eqs. (1)–(4) are numericallyintegrated forward in time using a fourth-order, fixed-step Runge–Kutta algorithm to generate virtual yaw card data.

3. Projectile linear theory trajectory solution

The six-degree-of-freedom projectile model just shown consistsof 12 highly non-linear differential equations that are not directlyamenable to a closed-form solution. Closed-form solutions of theprojectile trajectory are possible by applying projectile linear the-ory. Such solutions provide accurate prediction of the projectilestate history, and because they are algebraic in terms of the down-range distance, also lead to analytic derivatives of pitch, yaw, alti-tude, and swerve with respect to aerodynamic parameters. Projec-tile linear theory is used to rapidly compute projectile trajectories,to reduce aerodynamic range data, and to establish stability criteriafor both fin- and spin-stabilized projectiles. It is generally acceptedas an accurate dynamic model for a wide class of fin- and spin-stabilized projectiles. The following simplifying assumptions leadto the linear theory solution;

• Change of variables from fixed plane, station line velocity, u,to total velocity, V .

• Change of variables from time, t , to dimensionless arc length,s. The dimensionless arc length, as defined by Murphy [20] isgiven below and has units of calibers of travel:

s = 1

D

t∫0

V dτ

such that dotted terms in the sequel refer to time deriva-tives (d(.)/dt), and primed terms denote arc length derivatives(d(.)/ds). First derivatives are related by ζ = (V /D)ζ ′ .

• Euler pitch and yaw angles are small.• Aerodynamic angles of attack are small.• The projectile is mass balanced such that the center of gravity

lies on the rotational axis of symmetry.

• The projectile is aerodynamically symmetric. Note that this as-sumption restricts the algorithm as presented to identifyingprojectiles with axisymmetric aerodynamics. For instance, themodel cannot capture the effect of two forward canards, how-ever it will match the dynamics of three or more mounted inan axisymmetric fashion.

• A flat fire trajectory assumption is invoked such that the forceof gravity is neglected in the total velocity equation. Gravity isincluded in the epicyclic yawing and pitching equations, andthe swerve equations.

• The quantities V , and φ are large compared to θ,ψ,q, r, v , andw , such that products of small quantities and their derivativesare negligible.

The resulting equations are:

x′ = D (10)

φ′ = D

Vp (11)⎧⎪⎪⎨

⎪⎪⎩y′z′θ ′ψ ′

⎫⎪⎪⎬⎪⎪⎭ = Φ

⎧⎪⎪⎨⎪⎪⎩

yzθ

ψ

⎫⎪⎪⎬⎪⎪⎭ + D

VI

⎧⎪⎪⎨⎪⎪⎩

vwqr

⎫⎪⎪⎬⎪⎪⎭ (12)

where

Φ =

⎡⎢⎢⎣

0 0 0 D0 0 −D 00 0 0 00 0 0 0

⎤⎥⎥⎦

V ′ = −[ρS D

2m

]C X0 V (13)

p′ = ρS D3CL P

4Ixxp + ρS D2 V

2IxxCLD D (14)

The matrix equation for epicyclic pitching and yawing is:⎧⎪⎪⎨⎪⎪⎩

v ′w ′q′r′

⎫⎪⎪⎬⎪⎪⎭ = Ξ

⎧⎪⎪⎨⎪⎪⎩

vwqr

⎫⎪⎪⎬⎪⎪⎭ +

⎧⎪⎪⎨⎪⎪⎩

0G00

⎫⎪⎪⎬⎪⎪⎭ g (15)

where

Ξ =

⎡⎢⎢⎣

−A 0 0 −D0 −A D 0BD

CD E −F

−CD

BD F E

⎤⎥⎥⎦ (16)

and

A = ρS D

2mCN A (17)

C = ρS D2

2I yyCM A (18)

E = ρS D3

4I yyCM Q (19)

F = D

V 0

Ixx p

I yy(20)

G = D

V 0(21)

CM A = (S LC P − S LC G)CN A (22)

B is the Magnus moment term whose coefficient is shown as aproduct of CN P A and the distance from S LC G to S LC M [10,19].

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B = ρ

2

S

I yy

D3

V(S LC M − S LC G)CN P A p

and D is the projectile characteristic length (or diameter).

3.1. Linear model closed-form solution

The total velocity solution is

V (s) = V 0 exp

(−ρS D

2mC X0s

)The roll rate solution is

p(s) = p0Λ + 2V 0CLD D

DCL Pexp

(−ρS DC X0

2ms

)(Λ − 1) (23)

where

Λ = exp

(ρS D3CL P

4Ixxs

)(24)

The solution to Eq. (15) is the sum of a particular solution dueto the gravity constant, and a homogeneous solution [5]. The par-ticular solution is given by setting the derivatives equal to zero andsolving the resulting algebraic equation:⎧⎪⎪⎨⎪⎪⎩

v p

w p

qp

rp

⎫⎪⎪⎬⎪⎪⎭ = −Ξ−1

⎧⎪⎪⎨⎪⎪⎩

0G00

⎫⎪⎪⎬⎪⎪⎭ g (25)

resulting in:⎧⎪⎪⎨⎪⎪⎩

v p

w p

qp

rp

⎫⎪⎪⎬⎪⎪⎭ = Gg

det(Ξ)

⎧⎪⎪⎨⎪⎪⎩

−F C + B EEC + A F 2 + AE2 + B F

−(C AE + C2 + B A F + B2)/D(C F − B E)A/D

⎫⎪⎪⎬⎪⎪⎭ (26)

where:

det(Ξ) = A2 F 2 + A2 E2 + 2AEC + C2 + 2B A F + B2 (27)

This particular solution is then subtracted from the initial condi-tions prior to solving for the homogeneous response:

χ = η0 − ηp (28)

The homogeneous response for epicyclic pitching and yawing maybe found using the matrix exponential, then adding the particularsolution to yield the total solution as

η = eΞ sχ + ηp (29)

where the epicyclic velocity states are gathered into the vector η ={v w q r}T .

For the crossrange, altitude, pitch, and yaw state vector, ξ ={y, z, θ,ψ}T , the particular solution cannot be found by matrix in-version since Φ is clearly singular. Also, the epicyclic pitching andyawing states (v, w,q, r) serve as a time varying forcing functionfor y, z, θ , and ψ . Thus, a general solution to forced linear systemsattributed to Athans et al. [1] is invoked. For the linear system

x′(s) = Φx(s) + Γ (s) (30)

The total solution is given as

x(s) = eΦsx0 + eΦs

s∫0

e−ΦτΓ (τ )dτ (31)

The integration is handled well by a method from Van Loan [3,23].In the most compact form, suppose

Ψ =[

Φ Γ0 R

](32)

Then

eΨ s =[

Ω1 Δ10 Ω2

](33)

and

Ω1 = eΦs (34)

Δ1 = Ω1

s∫0

e−Φτ Γ eRτ dτ (35)

Thus setting R = 0 and

Γ = D

Vη

evolution of the position state vector is computed by

ξ = Ω1ξ0 + Δ1 (36)

Note that in Eq. (32), Γ is a constant. Thus Eq. (36) is used re-cursively, treating η as a constant each time and averaging it withthe previous velocity state for improved accuracy. Practically, Γ iscomputed simply by

Γ = D

2V(η + η0)

in the actual algorithm.

3.2. Linear model closed-form derivatives

Analytic derivatives of the position state vector ξ wrt anyaerodynamic parameter ϑi may then be found by differentiatingEq. (36) resulting in

∂ξ

∂ϑi= Ω1

∂ξ0

∂ϑi+ ∂Δ1

∂ϑi(37)

Since Ω1 does not vary with the aerodynamic parameters, the po-tential term involving derivatives of Ω1 does not appear. Thus inaddition to previous calculations, only the forced part of the posi-tion state vector solution needs to be differentiated

∂Δ1

∂ϑi=

s∫0

e−Φτ ∂Γ

∂ϑidτ (38)

which can be found from once again invoking the Van Loan for-mula with

∂

∂ϑiΨ =

[Φ ∂

∂ϑiΓ

0 0

](39)

such that

exp

(s

∂

∂ϑiΨ

)=

[Ω3

∂Δ1∂ϑi

0 Ω4

](40)

And ∂Γ /∂ϑi is found from differentiating the closed-form solutionfor the velocity state vector η = {v w q r}T . That is(

V

D

)∂

∂ϑiΓ = ∂

∂ϑiη = s

∂Ξ

∂ϑieΞ sχ + eΞ s ∂χ

∂ϑi+ ∂

∂ϑiηp (41)

The last two terms are found by realizing that

∂

∂ϑi

⎧⎪⎪⎨⎪⎪⎩

v p

w p

qp

r

⎫⎪⎪⎬⎪⎪⎭ = − ∂

∂ϑiΞ−1

⎧⎪⎪⎨⎪⎪⎩

0G00

⎫⎪⎪⎬⎪⎪⎭ g (42)

p

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where

∂Ξ−1

∂ϑi= −Ξ−1 ∂Ξ

∂ϑiΞ−1

such that

∂

∂ϑiηp = −Ξ−1 ∂Ξ

∂ϑiηp (43)

and substituting Eqs. (43) and (25) into Eq. (41) the following isobtained.(

V

D

)∂

∂ϑiΓ = s

∂Ξ

∂ϑiη + (

eΞ sΞ−1 − sI − Ξ−1) ∂Ξ

∂ϑiηp

+ eΞ s ∂

∂ϑiη0 (44)

Derivatives of Ξ with respect to aerodynamic parameters arefairly straightforward as shown in Appendix A, but result in sparsematrices, so it is fortunate that no inversion of the derivative ma-trix is required. The total velocity V in Eq. (44) is treated as aconstant for all derivatives except C X0. For C X0, the variation oftotal velocity V due to drag must be considered. Thus the deriva-tive of Γ is found as

∂Γ

∂C X0= ∂Γ

∂V

∂V

∂C X0+ D

V

∂

∂C X0η

where

∂Γ

∂V

∂V

∂C X0= − D

V 2η

(−ρS D

2ms

)V 0 exp

(−ρS D

2mC X0s

)Derivatives with respect to the unknown initial angular rates arefairly straightforward. The initial roll rate influences F since F =f (p).

∂ F

∂ p0= ∂ F

∂ p

∂ p

∂ p0= ∂ F

∂ pΛ

Initial roll rate is thus the only angular rate that influences Ξ as

∂Ξ4,3

∂ p0= −∂Ξ3,4

∂ p0= ∂ F

∂ p0

and ∂ F/∂ p is shown in Appendix A. Initial pitch and yaw ratesinfluence the initial values of the η vector as

∂η

∂q0

∣∣∣∣s=0

= eΞ {0 0 1 0}T

∂η

∂r0

∣∣∣∣s=0

= eΞ {0 0 0 1}T

This influence must then be propagated along the trajectory, there-fore ∂η/∂q0 and ∂η/∂r0 are treated as co-states which are updatedby

∂η

∂q0

∣∣∣∣s=k

= eΞ ∂η

∂q0

∣∣∣∣s=k−1

∂η

∂r0

∣∣∣∣s=k

= eΞ ∂η

∂r0

∣∣∣∣s=k−1

Derivatives of the position states wrt q0 and r0 are then computedby respective instances of Eq. (37) where

∂Γ

∂q0= D

V 0

∂η

∂q0

∣∣∣∣s=k

∂Γ

∂r0= D

V 0

∂η

∂r0

∣∣∣∣s=k

Practical calculation of Ω1, Δ1, and their derivatives are discussedin the sequel.

3.3. Some notes on practical computation

The matrix exponential of Eqs. (33) and (40) will need to beevaluated once for each trajectory prediction step, and once foreach step of each of the derivative predictions respectively. In orderto reduce computational burden, use of the series expansion

eΨ s = I + Ψ s + 1

2!Ψ2s2 + 1

3!Ψ3s3 + · · ·

is explored and due to the sparsity of Ψ , Ψ n = 0 ∀n � 3. Thus theseries expansion for eΨ s converges in just three terms, that is:

eΨ s = I + Ψ s + 1

2!Ψ2s2

Since Φ is invariant with respect to aerodynamic parameters, Ψ ispartitioned into variant and invariant parts in hopes of speedingup computation of the derivatives of Δ1. This results in

eΨ s =[

I 00 1

]+

[Φ Γ0 0

]s + 1

2![

Φ2 ΦΓ0 0

]s2

and since Φ2 = 0, Ω1 and Δ1 are computed directly by

Ω1 = I + Φs (45)

Δ1 = Γ s + 1

2!ΦΓ s2 (46)

Thus derivatives, of the position states are found simply byevaluating Eq. (47).

∂Δ1

∂ϑi= ∂Γ

∂ϑis + 1

2!Φ∂Γ

∂ϑis2 (47)

The linear theory solutions are solved recursively such that{v0, w0,q0, r0} represents the solution at the last sample and{v, w,q, r} and {v p, w p,qp, rp} are the respective quantities at thecurrent sample.

4. Gradient based parameter identification

The gradient based method assumes that yaw card data areavailable at 24 stations over a short downrange interval. The Tran-sonic Experimental Facility (TEF) at ARL currently has 25 stations[8], and the Aerodynamic Experimental Facility (AEF), a smallerscale range, has 39 stations [18], so 24 is not considered unrealis-tic. Nine parameters are to be estimated including the three initialbody angular rates (p0, q0 and r0) and six major aerodynamic pa-rameters (including S LC P ) for a brief interval of flight where Machnumber varies only slightly. The initial and final total velocities areassumed to be known from chronometer measurements. Launchposition is defined to be the origin, and launch Euler angles areassumed to be zero. A maximum of four measurements are madefrom each yaw card—altitude, crossrange, yaw, and pitch—for a to-tal of 96 measurements.

Thus, a Newton type search may be applied where the nor-mal equation is over-determined (nine unknowns and 96 measure-ments). The ‘extra’ measurements should insure that the Newtonstep is well conditioned for each iteration. The Jacobian matrixtakes the form:

J =

⎡⎢⎢⎢⎢⎢⎣

∂ y1∂ S LC P

· · · ∂ y1∂ p0

· · · ∂ y1∂r0

......

...∂z1

∂ S LC P· · · ∂z1

∂ p0· · · ∂z1

∂r0...

......

⎤⎥⎥⎥⎥⎥⎦

The 96 × 9 Jacobian matrix is formed directly from the deriva-tives found in Section 3.2.

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The difference between actual and predicted state history forthe nominal case is stored as the current iteration residual vec-tor T .

Given the rectangular J matrix, and residual vector T withequal row dimension, a second-order Newton method is approx-imated by the Levenberg–Marquardt algorithm [17]. The algorithmproceeds as follows.

1. Set Marquardt parameters μ = 0.01 and β = 10.2. Form the Jacobian matrix and residual vector from data and

predictions.3. Make a trial correction

ϑ i+1 = ϑ i − (J T J + μI

)−1J T T

4. Find the new T vector based on ϑ i+1.5. If the error (‖T ‖2) is reduced, set μ = μ/β and repeat steps 2

and following;If not, set μ = μβ and repeat steps 3 and following.

When μ is small, the algorithm approximates a second-orderNewton step. When μ gets significantly large, the iteration uses asmall steepest-descent step. The author suggests that for this ap-plication μ is limited such that when μ� 10, the new parametersare adopted whether or not the error is reduced.

In addition to handling many measurements with a small num-ber of adjustable parameters, the Levenberg–Marquardt algorithmcan handle multiple data sets that have a few common parame-ters. This way multiple trajectories for a common projectile canbe used such that aerodynamic coefficients are shared (assumingequal launch velocity) and initial angular rates are varied betweenthe trajectories. In this case, a composite Jacobian matrix can beformed from independently determined Jacobian matrices J 1 andJ 2. The combined Jacobian matrix takes the form

J =[

J 1:,1:6 J 1:,7:9 0

J 2:,1:6 0 J 2:,7:9

](48)

where the subscript (:,a:b) indicates all rows, columns athrough b. The combined residual is merely a vertical concate-nation of the two residual vectors ⇒ T = [T T

1 T T2 ]T .

For the Army–Navy finner, two configurations are tested—onewith fin cant and one without. This is assumed to vary only CLD D

while all other aero coefficients are held constant. If two initialconditions are tested, then a total of four test trajectories are avail-able. A combined Jacobian matrix can be formed to handle all fourdata sets simultaneously

J =

⎡⎢⎢⎢⎣

J 1:,1:4 J 1:,5:6 J 1:,7:9 0 0

J 2:,1:4 J 2:,5:6 0 J 2:,7:9 0

J 3:,1:4 0 J 3

:,6:9 0 J 3:,5

J 4:,1:4 0 J 4:,6 J 4:,7:9 J 4:,5

⎤⎥⎥⎥⎦ (49)

where J i:,5 corresponds to derivatives wrt CLD D and J i7:9 those wrt

initial conditions. The corresponding residual is a vertical concate-nation of all four corresponding residual vectors. Such an approachcan be used to quantify symmetric aerodynamic modifications toa projectile. For instance, trajectories for a rocket with and with-out four passive canards have been used to identify the changes inCN A and S LC P required to model the canards.

5. Results

The Marquardt algorithm was exercised on four sets of virtualrange data of the Army–Navy finner from two configurations and

Fig. 3. Army–Navy finner trajectory matching, with cant, initial condition 1, cross-range.

Fig. 4. Army–Navy finner trajectory matching, with cant, initial condition 1, altitude.

Fig. 5. Army–Navy finner trajectory matching, with cant, initial condition 1, yaw.

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Fig. 6. Army–Navy finner trajectory matching, with cant, initial condition 1, pitch.

two possible initial conditions as described in the previous sec-tion. Figs. 3–6 show the quality of trajectory matching for one ofthe four trajectories available using analytic estimates of the gra-dient. Fig. 7 shows a comparison of the cost function evolution forthe algorithm when using finite difference estimates as opposed toanalytic calculations for gradient information. The finite differencebased algorithm reduces the estimation error in fewer iterations,but stagnates after about 20 iterations. This stagnation is due tothe nature of finite differencing—a perturbation is chosen whichworks well when the gradients are steep, however as the iteratesapproach a local minima, the gradients become shallow, neededcorrections become small, and the previously chosen perturbationmay skip over the nearest minima, thus providing a gradient esti-mate that doesn’t even represent the correct search direction.

The algorithm with analytic gradient estimates is about sixtimes faster and continues to make improvements out to 50 it-erations. A vertical line is drawn at eight iterations to show theprogress of the finite difference algorithm after 770 s of CPU time.Note at this point the cost function is still greater than 10−2. Af-ter an equal amount of CPU time, the analytic method has reducedthe cost function to 3.3(10−3).

Table 1 compares the accuracy of aerodynamic parameter esti-mates from each algorithm. The virtual range data were generatedusing aerodynamic parameters which vary as a function of Machnumber. The ‘Actual’ column gives the value averaged over theMach numbers encountered along a typical trajectory. Launch ve-locities are assumed to be identical for the four data sets, and forthe short gliding flight, Mach numbers are similar. Note that pa-rameters which most strongly affect epicyclic pitching and yawing

Fig. 7. Evolution of cost function for the algorithms considered.

are estimated to within ten percent of their actual value when us-ing analytic gradient estimates. Parameters that directly influenceroll and roll rate are poorly estimated, however this is most likelydue to not including any direct roll information in the cost func-tion. Roll rate influences the frequency and stability of epicyclicmodes, however with such an indirect influence, the interactionsbetween fin cant, roll damping constant, and initial roll rate maycancel each other out, resulting in poor estimates of all three. Inpractice, projectiles are often fitted with a roll pin such that sparkrange data offers estimates of roll angle. Including measurementsof roll would greatly enhance the accuracy of estimated p0, CLD D ,and CL P .

The algorithm was also tested with the TEF configuration, as-suming the first station is 200 calibers from launch. This configu-ration placed the final station at 7002 calibers instead of 4800 for24 equally spaced stations. For the TEF configuration, convergenceis slow, however after 100 iterations the 2-norm of the length 400residual vector is reduced to 3(10−3), which is comparable to per-formance using the default configuration.

In hopes of finding a range configuration that would provideadequate parameter estimates without the tedium of setting up25 yaw cards, of the expense of 25 spark range cameras, sev-eral strategies were devised. In each case, a subset of the original24 station measurements were used for estimation while all oth-ers were ignored. The ‘linear skip’ keeps cards n = 1,3,5, . . . inthe cost function. ‘Binary’ keeps cards n ∈ {1,2,4,8,16}. ‘Fibonac-ci’ keeps cards 1, 2, 3, 5, 8, 13, and 21. Recognizing that stationsfurther downrange should be given greater weight, a ‘reverse Fi-bonacci’ scheme where cards 4, 12, 17, 20, 22, 23, and 24 are kept,and a ‘reverse binary’ scheme where cards 9, 17, 21, 23, and 24are kept were also devised. Performance of the algorithm undervarious pruning strategies is shown in Fig. 8. The residual vector2-norms (‖T ‖2) are scaled such that initial error is equal regard-

Table 1Army–Navy finner aero identification results.

Parameter Actual Analytic estimate Percent difference FD estimate Percent difference

SLC P 0.295 0.296 0.34 0.257 12.9CN A 14 13.998 0.014 10.99 21.5CM Q −590 −535.8 9.19 −861.2 46.0C X0 0.65 0.654 0.62 0.6625 1.92CLD D1 0.0 0.005 N/A −0.0028 N/ACL P −8.75 −10.63 21.5 −6.72 23.1p0(1) −9.9804 −16.69 67.2 0.6069 93.9q0(1) −0.0286 −0.0326 14.0 −0.0399 39.4r0(1) −0.784 −0.798 1.79 −1.022 30.2p0(2) −4.9804 −25.54 413 −17.67 255q0(2) −0.286 −0.295 3.15 −0.375 30.9r0(2) −0.0784 −0.0803 2.42 −0.1035 31.95CLD D2 −0.2 −0.165 17.5 −0.141 29.5

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8 B.T. Burchett / Aerospace Science and Technology ••• (••••) •••–•••

Fig. 8. Investigating strategies to reduce the number of yaw card stations required.

Fig. 9. Investigating the number of yaw card stations required.

less of the dimension of T (which equals four times the number ofcards). Despite having the most sparse arrangement, the ‘reversebinary’ strategy shows the most promise.

Having determined that reverse binary gives the best perfor-mance with the least number of measurements, we further soughta minimum set of stations to render performance comparable tothe full set. Cards were added by keeping equal spacing in loga-rithmic space, i.e. the kept cards have numbers

25 − 2μ, μ ∈ {0,1,1 + mki}such that 1 +mkmax < log2 24 and ki = 1,2, . . . ,kmax where kmax ∈{3,7,10,13,16}. If duplicate numbers arise, the duplications aredeleted. Fig. 9 shows the result. Performance improves with theaddition of stations, but all reduced card sets stagnate nearly anorder of magnitude above the full data set final residual. In con-clusion, a minimum of 24 stations are needed for adequate esti-mation.

The algorithm was also tested using only one data set, and per-formance compared to with trials using two and four data sets re-spectively. As shown in Fig. 10, the single and dual trajectory caseseventually stagnate to (‖T ‖2) values nearly an order of magnitudehigher than the final 2-norm for the full algorithm. In Fig. 10, thevalues of (‖T ‖2) are normalized between trials such that all tri-als have the same initial cost. This eliminates artificial scaling ofthe cost function due to the smaller row dimension of T for casesusing only one or two data sets. Thus having multiple data sets

Fig. 10. Variation of estimation performance depending on number of trajectoriesused.

Fig. 11. Variation of estimation performance depending on number of measurementsused.

from repeated launches of a common projectile greatly enhancesthe accuracy of aerodynamic parameter identification. The notionof multiple trajectories sharing common initial angular rates maybe questionable, however, unless the range is equipped to directlycontrol the launch condition.

Finally, individual channels of the position predictions (y, z,θ,ψ ) were ignored to determine whether all were essential toaccurate parameter estimation. Again, if the data were from yawcards, being able to ignore pitch and/or yaw would eliminate a te-dious manual step in the reduction process. Performance of thefull algorithm using all 96 × 4 measurements was compared withfour trials where one coordinate of the four is ignored or only72 × 4 measurements are used. The result is shown in Fig. 11,where it is readily apparent that all measurements are essentialto accurate parameter estimation. One might imply that yaw isthe least important, since it has the next lowest final estimationcost residual, and likewise crossrange is the most important. Alltrials with reduced measurements resulted in the algorithm stag-nating at or prior to 20 iterations, with cost functions of 7(10−3)

or higher.Thus, the algorithm requires a minimum of 24 stations, each

providing measurements of crossrange, altitude, pitch and yaw. Es-timation accuracy is best when at least four trajectories with atleast 96 measurements each are used.

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B.T. Burchett / Aerospace Science and Technology ••• (••••) •••–••• 9

6. Conclusions

Reformulating the closed-form solution to the projectile lineartheory equations in terms of explicit matrix exponentials providesexpressions that are more compact, and easier to differentiate withrespect to aerodynamic parameters and initial conditions. Deriva-tives with respect to six aerodynamic parameters and three angu-lar rates have been found in closed form and used to improve agradient based parameter estimation algorithm. The speed of thisalgorithm is improved by a factor of six. Estimates from this al-gorithm are improved such that five out of seven aerodynamicparameters and four out of six initial conditions are estimated towithin ten percent of actual. Four of the seven aerodynamic pa-rameters are estimated to within one percent of actual. Algorithmsusing reduced numbers of trajectories or reduced number of mea-surements within the full set of trajectories resulted in significantdegradation of the parameter estimates.

Appendix A

Here we present the derivatives of the velocity state dynam-ics matrix Ξ with respect to aerodynamic parameters. Since eachmatrix is sparse, only the non-zero terms are shown.

∂Ξ3,3

∂CM Q= ∂Ξ4,4

∂CM Q= ∂ E

∂CM Q= ρS D3

4I yy(50)

∂Ξ3,2

∂ S LC P= 1

D

∂C

∂ S LC P= − ∂Ξ4,1

∂ S LC P

∂C

∂ S LC P= ρS D2

2I yyCN A (51)

∂Ξ4,3

∂CLD D= − ∂Ξ3,4

∂CLD D= ∂ F

∂CLD D

∂Ξ4,3

∂C X0= −∂Ξ3,4

∂C X0= ∂ F

∂C X0

∂Ξ4,3

∂CL P= −∂Ξ3,4

∂CL P= ∂ F

∂CL P(52)

The derivatives of F with respect to aerodynamic parameters allarise from the fact that F is a function of p (F = f (p)). Theseterms are then easily found by applying the chain rule.

∂ F

∂Cγ= ∂ F

∂ p

∂ p

∂Cγ, γ ∈ {LD D, X0, L P }

and

∂ p

∂CLD D= 2V 0

DCL Pexp

(−ρS D

2mC X0s

)(Λ − 1)

∂ p

∂CL P= 2V 0CLD D

DCL Pexp

(−ρS D

2mC X0s

)(∂Λ

∂CL P− Λ − 1

CL P

)∂Λ

∂CL P=

(ρS D3

4Ixxs

)exp

(ρS D3CL P

4Ixxs

)∂ p

∂C X0=

(−ρS D

2ms

)(2V 0CLD D

DCL P

)exp

(−ρS D

2mC X0s

)(Λ − 1)

∂ F

∂ p= D

V 0

I X X

IY Y(53)

∂Ξ1,1

∂CN A= ∂Ξ2,2

∂CN A= − ∂ A

∂CN A

∂Ξ3,2 = −∂Ξ4,1 = 1 ∂C

∂CN A ∂CN A D ∂CN A

∂ A

∂CN A= ρS D

2m∂C

∂CN A= ρS D

2I yy(S LC P − S Lcg) (54)

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Further reading

[25] B.T. Burchett, Robust lateral pulse jet control of an atmospheric projectile, Ph.D.thesis, Oregon State University, 2001.