aerodynamic parameter estimation from flight data applying … · 2016-01-09 · extended kalman...

Aerospace Science and Technology 14 (2010) 106–117

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Aerospace Science and Technology

www.elsevier.com/locate/aescte

Aerodynamic parameter estimation from flight data applying extended andunscented Kalman filter

Girish Chowdhary a,∗,1, Ravindra Jategaonkar b,2

a Georgia Institute of Technology, 270 Ferst Drive, Atlanta, GA 30332, United Statesb The German Aerospace Center (DLR), Institute of Flight Systems, Lillienthalplatz 7, Germany 38108

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

Article history:Received 28 October 2008Accepted 13 October 2009Available online 27 October 2009

Keywords:System identificationKalman filteringRecursive parameter estimation

Aerodynamic parameter estimation is an integral part of aerospace system design and life cycle process.Recent advances in computational power have allowed the use of online parameter estimation techniquesin varied applications such as reconfigurable or adaptive control, system health monitoring, and faulttolerant control. The combined problem of state and parameter identification leads to a nonlinear filteringproblem; furthermore, many aerospace systems are characterized by nonlinear models as well as noisyand biased sensor measurements. Extended Kalman filter (EKF) is a commonly used algorithm forrecursive parameter identification due to its excellent filtering properties and is based on a first orderapproximation of the system dynamics. Recently, the unscented Kalman filter (UKF) has been proposedas a theoretically better alternative to the EKF in the field of nonlinear filtering and has received greatattention in navigation, parameter estimation, and dual estimation problems. However, the use of UKF asa recursive parameter estimation tool for aerodynamic modeling is relatively unexplored. In this paper wecompare the performance of three recursive parameter estimation algorithms for aerodynamic parameterestimation of two aircraft from real flight data. We consider the EKF, the simplified version of theUKF and the augmented version of the UKF. The aircraft under consideration are a fixed wing aircraft(HFB-320) and a rotary wing UAV (ARTIS). The results indicate that although the UKF shows a slightimprovement in some cases, the performance of the three algorithms remains comparable.

© 2009 Elsevier Masson SAS. All rights reserved.

1. Introduction

Parameter estimation and system identification techniques al-low the engineer to form a mathematical model of a system usingmeasured data. The aerospace industry has placed great empha-sis on system identification of various flight platforms since theresulting mathematical models are useful in the system designand management process, especially for the purpose of develop-ing elaborate simulation environments and control systems design.System identification techniques can roughly be classified in twogroups: offline techniques and online techniques [11]. Offline sys-tem identification techniques tend to depend on iterative meth-ods that exploit the advantage of having a complete set of dataavailable for processing, whereas online or recursive system iden-tification techniques must use the data as it becomes available.Recursive system identification is a valuable tool in the designof adaptive control laws [3,5,22], health monitoring algorithms,and the design of fault tolerant systems [20]. Increasing availabil-

* Corresponding author.E-mail address: [email protected] (G. Chowdhary).

1 Graduate research assistant.2 Senior scientist.

1270-9638/$ – see front matter © 2009 Elsevier Masson SAS. All rights reserved.doi:10.1016/j.ast.2009.10.003

ity of onboard computational power indicates further emergenceof applications employing recursive parameter identification algo-rithms.

Recursive system identification techniques handle flight dataas it is measured through onboard sensors and estimate the re-quired aerodynamic derivatives in real-time. Measured flight datacan contain considerable amount of noise, furthermore there mightbe biases and unobserved states in the system model whichmust be estimated; hence filtering techniques are generally em-ployed. Fundamental to all stochastic filtering methods is a twostep Bayesian procedure [2] consisting of prediction, or time up-date; and correction, or measurement update. The Kalman fil-ter [16], which assumes Gaussian distribution for the uncertain-ties in system dynamics and utilizes the first two moments ofthe state vector (mean and covariance) in its update rule is anoptimal sequential linear estimator ideally suited for recursive im-plementations. However, most aerospace systems involve a nonlin-earity of some form, furthermore the method of parameter esti-mation through state augmentation renders the filtering problemnonlinear, and hence nonlinear filtering techniques must be em-ployed.

The most popular nonlinear filtering technique in the aerospaceindustry is the extended Kalman filter (EKF), which employs in-

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stantaneous linearization at each time step to approximate thenonlinearities. The EKF can be hard to tune and implement whendealing with significant nonlinearities and exhibits divergence inextreme cases. Despite its theoretical shortcomings, however, theEKF has been employed successfully in various aircraft aerody-namic parameter estimation problems [4,8,12,13,21].

The problems associated with the EKF are attributed to theapproximation introduced by the linearization [9]. The EKF approx-imates the mean and covariance using a first order approximationof the system dynamics. The unscented Kalman filter (UKF) over-comes these theoretical deficiencies by using a set of carefully se-lected sample points (also known as sigma points) to approximatethe probability distribution of the random variable. The parame-terized sets of sigma points are propagated through the nonlineartransformation and the mean and covariance of the transformedvariables is used to approximate the mean and covariance of thesample space. Proper implementations of the UKF are accurate atleast to the second order and do not require the computation ofJacobian or Hessian matrices. It is claimed that the computationalcomplexity of the UKF is comparable to the EKF [15,24].

The UKF is based on the concept of unscented transforms (UT)introduced by Julier and Uhlmann [15]. In Ref. [14] Julier andUhlmann demonstrate the feasibility and advantages of employingthe UT in recursive filtering through the UKF for higher order non-linear filtering problems such as reentry vehicle tracking. Their re-sults indicate that UKF produces consistent estimates in situationswhere the EKF tends to produce inconsistent estimates. Wendelet al. [26] compare the performance of sigma point Kalman fil-ter (SPKF) with EKF for the nonlinear problem of tightly coupledGPS/INS integration. They report that owing to the moderate non-linearities in their implementation of GPS-INS navigation equationsboth the EKF and the SPKF perform closely for all practical appli-cations and recommend that modification of existing EKF basednavigation systems may not result in significant performance im-provement. Crassidis and Markley [6] consider the UKF for space-craft attitude estimation problems and report that for realisticconditions their UKF implementation consistently outperforms theEKF especially when large initialization errors are present. Brunkeand Campbell [1] derive the elegant and robust square root sigmapoint filter and note that the SPKF is more amenable to onlineimplementation. Brunke has also used the SPKF for aerodynamicparameter estimation in presence of a 50% stabilator failure. Wanand Van der Merwe study its extension to the problem of param-eter estimation for selecting the weights of neural networks [23,25], the authors maintain that UKF consistently outperforms theEKF with comparable level of complexity.

Nonlinear dynamics of aerospace vehicles and the presence ofconsiderable noise and biases in measurements demand that anonlinear filtering algorithm be used. Traditionally, the EKF hasbeen used for recursive parameter estimation purposes. In this pa-per we compare three recursive nonlinear filtering algorithms foraerodynamic parameter estimation from real flight data. The algo-rithms considered are, the EKF, the simplified version of the UKFwhich assumes additive white noise, and the complete version ofthe UKF (referred to as augmented UKF), which caters to the gen-eral case of noise entering the system nonlinearly. We analyzethe feasibility and possible advantages of using the UKF for air-craft system identification from real flight data by comparing theresults of UKF parameter estimation runs with results of EKF recur-sive estimation runs and offline estimation runs for two differenttypes of aircraft. The two types of aircraft considered are a fixedwing platform (HFB 320) with nonlinear aerodynamic model anda miniature autonomous rotorcraft (ARTIS [7,18]) with linearizedstate model.

2. Recursive parameter estimation for aircraft

Estimation of parameters through the filtering approach is anindirect procedure, consisting of transforming the parameter esti-mation problem into a state estimation problem. This is done byaugmenting the system state vector by artificially defining the un-known parameters as additional state variables. It is important tonote that such a formulation will render the problem effectivelynonlinear regardless whether a linear estimation model is used orotherwise. This nonlinearity manifests itself from state products.

To be as close to reality as possible a continuous estimationmodel is used, while to facilitate real-time application the mea-surements are recorded at discrete time steps and a discrete imple-mentation of filtering algorithm is used. Such an approach whichemploys a continuous estimation model for prediction and a dis-crete filtering algorithm is known as the continuous-discrete fil-tering problem. The system dynamics are represented in genericcontinuous state space form along with the discrete measurementequation in Eq. (1),

x(t) = f[x(t), u(t),β

] + F w(t), x(t0) = x0

y(t) = g[x(t), u(t),β

]z(k) = y(k) + G v(k), k = 1, . . . , N (1)

where x is the state vector with initial value x0 at time t0, u isthe input vector, y is the observation vector, Θ is the vector ofunknown system parameters, f and g are the general nonlinearreal-valued functions, z is the measurement vector sampled at Ndiscrete time steps with a fixed sampling time of �t and k is thediscrete time index. The measurement noise vector v is assumed tobe sequence of independent zero mean white Gaussian noise. Thematrices F and G represent the additive state and measurementnoise matrices and are considered to be time invariant.

The unknown parameter vector Θ consists of system parame-ters β , the measurement biases �z, and the trim estimates formu-lated as input biases �ui and is represented in Eq. (2). It may notbe possible to estimate all the components of �z and �u sincethey could be linearly dependent or highly correlated [12]

Θ T = {βT ;�zT ;�uT }

(2)

We consider the constant system parameters Θ as output of anauxiliary dynamic system presented in Eq. (3),

Θ = 0 (3)

The augmented state vector is then defined in Eq. (4) as:

xa =[

xΘ

](4)

The extended system is represented in Eq. (5) as:

xa(t) = fa[xa(t), u(t)

] + Fa Wa(t)

=[

f [x(t), u(t),β]0

]+

[F 00 0

][w(t)

0

]y(t) = ga

[xa(t), u(t)

]z(k) = y(k) + G v(k) (5)

where the augmented variables are denoted by subscript “a”. Thethree algorithms that we consider are explained in this section.

2.1. The extended Kalman filter

For the augmented system of Eq. (5) the EKF consisting of anextrapolation (prediction) and an update step is summarized be-low [10,12]. In what follows we use the notation “tilde” ( ˜ ) and

108 G. Chowdhary, R. Jategaonkar / Aerospace Science and Technology 14 (2010) 106–117

“hat” ( ˆ ) to denote the predicted and corrected variables respec-tively.

Extrapolation:

xa(k) = xa(k − 1) +t(k)∫

t(k−1)

fa[xa(t), u(k)

]dt (6)

Pa(k) ≈ Φa(k) Pa(k − 1)ΦTa (k) + �t.Fa F T

a (7)

with the initial conditions

xa(1) = xa0, Pa(1) = Pa0 (8)

u denotes average or interpolated values of the inputs betweenpoints k − 1 and k, and Φa = e Aa�t denotes the discrete time statetransition matrix from the augmented system with

Aa(k) = ∂ fa

∂xa

∣∣∣∣xa=xa(k−1)

=[

∂ f∂x

∂ f∂Θ

0 0

]xa=xa(k−1)

(9)

Eq. (9) denotes the linearization process for the state matrix. Thelinearization can be carried out using a numerical implementationof the central difference formula around the last best state esti-mate at each discrete time step [10,12].

Update:

y(k) = ga[xa(k), u(k)

]Ka(k) = Pa(k)C T

a (k)[Ca(k) Pa(k)C T

a (k) + GGT ]−1

xa(k) = xa(k) + Ka(k)[z(k) − y(k)

]Pa(k) = [

I − Ka(k)Ca(k)]

Pa(k)

= [I − Ka(k)Ca(k)

]Pa(k)

[I − Ka(k)Ca(k)

]T

+ Ka(k)GGT K Ta (k) (10)

where Ca(k) is the linearized measurement matrix as representedin Eq. (11), as before the linearization can be carried out using thecentral difference formula

Ca(k) = ∂ ga

∂xa

∣∣∣∣xa=xa(k)

=[

∂ g

∂x

∂ g

∂Θ

]xa=xa(k)

(11)

The Runge–Kutta algorithm can be used for integration inEq. (6), the state transition matrix Φa can be approximated usingPadè approximation [10] or Taylor series expansion. Pa(0) repre-sents the confidence in the initial state estimates and must bespecified a priori. In the absence of any a priori knowledge ofthe parameter values it is common to assume high values for thePa(0) matrix [12]. The value of FFT and GGT i.e. the process andmeasurement noise covariance matrices must also be specified apriory. The measurement noise covariance matrix GGT can be cal-ibrated using laboratory measurements of sensors to ensure goodnoise filtering. The process noise covariance matrix FFT is howevermore difficult to determine and a trial and error method is em-ployed if no other suitable method is found. We do not considerthe problem of adaptive filtering in this paper, which attempts toadapt to the unknown noise characteristics [9].

The propagation of the covariance matrix P in Eq. (7) is a lin-ear approximation for a small �t . It ignores the higher order termsof the Taylor series expansion for propagating the covariance ma-trix. These ignored terms may contain higher order effects thataffect the performance of the filter [14]. This makes the EKF anon-optimal approximation of the KF based on a first order ap-proximation of nonlinear dynamic system. The EKF exhibits betterperformance if the system under consideration is close to linear,whereas strong system nonlinearities and wrong values of noisestatistics may result in biased estimates or in worst case lead todivergence [17].

2.2. The unscented Kalman filter: Augmented case

Various techniques have been proposed to address the inaccura-cies arising from the fundamental first order approximation inher-ent to the EKF implementation. The continuous-discrete mixed ap-proach which uses nonlinear model for propagation of the systemstates and a linearized model for propagation of the error covari-ance mitigates the problem somewhat, however the fundamentalproblem of not accounting for the nonlinear transformations thatthe random variable undergo remains, affecting the accuracy. Othertechniques, such as the iterated Kalman filter, or second and higherorder filters are possible and have shown good estimation errorreduction in specific application areas; but these are found to bemuch more involved. Julier and Uhlmann [14,15] introduced theconcept of unscented transforms and extended it to the problemof recursive estimation. The resulting filter is known as the un-scented Kalman filter (UKF).

The UKF is based on the idea that it is easier to approximatea probability distribution than to approximate an arbitrary nonlineartransformation [14]. The actual algorithm is based on propagatingcarefully selected finite set of points, called sigma points, throughthe system nonlinear dynamics, and then approximating the firsttwo moments of the distribution (mean and covariance) througha suitable method; such as weighted sample mean and covariancecalculations [14,23]. The flaw in the EKF that results from propa-gating the mean and covariance through linear approximations ofthe nonlinear transformation is thus eliminated in the UKF, lead-ing to theoretically better performance of the UKF. Furthermore,the UKF implementation does not need the calculation of any Ja-cobian or Hessian matrices, which not only results in considerablesimplicity in implementation, but also makes the UKF suitable forreal-time applications and applications involving non-differentiablefunctions. The accuracy of the UKF can be compared to that of thesecond order EKF, the computational order is comparable to theEKF [14,23].

The UKF algorithm requires the definition of 2na + 1 sigmapoints, where na is the total number of states to be estimated,consisting of augmented system states with the unknown systemparameters as well as the process and measurement noise distur-bances. Each sigma point consists of a vector, one of the sigma vec-tors is the expected value of the augmented state vector and theremaining 2na points are obtained from the columns of the matrixsquare root ±(γ P ) for i = 1,2, . . . ,na , where P is the covariancematrix of the augmented state vector xa , and γ = α2(na + κ) − na ,λ = α2(

√na + κ) − na are scaling parameters. The constant α de-

termines the spread of the sigma points around xa and is usuallyset to small positive values less than one (typically in the range0.001 to 1) whereas κ is the secondary scaling parameter usuallyset to zero or 3 − n. When κ is set to zero, the sigma points andtheir weights are related to the dimension of the state vector di-rectly (n), when κ is set to κ = 3 − n, the fourth order momentinformation is mostly captured in the true Gaussian case [14]. Theweighted mean and covariance of the sigma points are captured inthe following manner:

n∑i=1

wixi ≈ x

n∑i=1

wmi (xi − x)(xi − x)T ≈ P xx (12)

Here the weights W are calculated as:

W m0 = λ/(na + λ)

W c0 = λ/(na + λ) + (

1 − α2 + β)

W m = W c = 1/{

2(na + λ)}, i = 1,2, . . . ,2na (13)
i i


where the subscript ‘0’ corresponds to the estimated states andi = 1,2, . . . ,2na the other sigma points; the superscripts ‘m’ and ‘c’indicate weights for the computation of mean and covariance re-spectively. The constant β is used to incorporate prior knowledgeof the distribution of x in the computation of weights for covari-ances W c

0; the optimum value is β = 2 for Gaussian distribution.The scale parameters may be tuned to match the specific problem;some guidelines to choose them are provided in [14].

To illustrate the UKF procedure, we consider a discrete timerepresentation widely used in the applications of the UKF. For thediscrete time nonlinear state space model be given by:

xk+1 = fd(xk,Θk, uk, wk) (14)

yk = gd(xk,Θk, uk, vk) (15)

where xk is the (nx × 1) state vector, Θk the (nq × 1) vector ofunknown parameters, uk the (nu × 1) vector of exogenous inputs,yk the (ny × 1) model output vector, fd and gd the correspondingstate and output functions. In a general case, the process noise wkand the measurement noise vk are assumed to enter the modelnonlinearly as implied by Eqs. (14) and (15) and are (nw × 1) and(nv × 1) size vectors respectively. The measurement vector is de-noted by zk . The augmented state vector of the size (na × 1) isgiven by:

xak = [

xTk Θ T

k wTk vT

k

]T(16)

where the superscript ‘a’ denotes the augmented state vector,na = nxp + nw + nv and nxp = nx + nq . We consider cases wherenw = nxp and nv = ny . The noise processes wk and vk are assumedto be zero mean, and with covariance matrices Q and R respec-tively. The starting values of the augmented state vector and itscovariance are given by:

xa0 = E

{xa

0

} = E{[

xT0 Θ T wT

0 vT0

]}= [

xT0 Θ T

0 0 0]

(17)

P a0 = E

{(xa

0 − xa0

)(xa

0 − xa0

)T } =

⎡⎢⎢⎣

P x0 0 0 00 PΘ0 0 00 0 Q 00 0 0 R

⎤⎥⎥⎦ (18)

Following the guidelines already discussed, we choose the con-stants α, β , and the scaling parameters κ , γ and λ, and computethe set of weights W m

i and W ci for i = 0,1, . . . ,2na . The UKF recur-

sive estimation algorithm follows the standard Bayesian solutionto continuous-discrete filtering problem consisting of a predictionand an update step [2]. The prediction step predicts the probabil-ity density at time step tk and the measurement step calculates theposterior probability density from the predicted probability densityfrom the prediction step and the measurement y at time tk .

The standard UKF algorithm can be summarized as follows [24]:

1) Set discrete time point k = 1 and build up the augmented statevector xa

k and the corresponding covariance matrix P ak for the

initial values according to Eqs. (17) and (18) respectively.2) Calculate the (2na + 1) sigma points

X ak = [

xak xa

k − γ

√P a

k xak + γ

√P a

k

](19)

3) Prediction (time update):

χak+1 = fd

[χ x

k , uk, χw

k

](20)

xak+1 =

2na∑W m

i χai,k+1 (21)

i=0

Pk+1 =2na∑i=0

W ci

[X x

i,k+1 − xk+1][

X xi,k+1 − xk+1

]T(22)

4) Measurement update:

Yk+1 = gd[

X xk+1, uk+1, X v

k+1

](23)

yk+1 =2na∑i=0

W mi Yi,k+1 (24)

P y yk+1=

2na∑i=0

W ci [Yi,k+1 − yk+1][Yi,k+1 − yk+1]T (25)

P x yk+1=

2na∑i=0

W ci

[X x

i,k+1 − xk+1][Yi,k+1 − yk+1]T (26)

Kk+1 = P xyk+1P−1

y yk+1(27)

xk+1 = xk+1 + Kk+1(zk+1 − yk+1) (28)

Pk+1 = Pk+1 − Kk+1 P y yk+1K T

k+1 (29)

5) Increment k and return to step 2 to continue to next timepoint.

In Eqs. (19)–(29), X a = [(X x)T (X w)T (X v )T ] denotes thesigma points, X x corresponds to the system states and unknownparameters, ‘ ˜ ’ (tilde) the predicted values, and ‘ ˆ ’ (hat) the cor-rected values. The presented UKF algorithm is for the general dis-crete time case of noise entering nonlinearly into the system dy-namics and is referred to as the augmented case [27]. A continuousestimation model is used since aircraft dynamics evolve naturallyover time. Hence, similar to the presented EKF implementation, weuse numerical integration methods to propagate the sigma pointsthrough the continuous nonlinear equations instead of Eq. (20):

xak+1 = xa

k(k) +tk+1∫tk

f[χa(t), u(k)

]dt (30)

Although the UKF bears some resemblance to particle filter al-gorithms, the main difference between UKF or SPKF algorithms andparticle filter algorithms is that the sigma points are not drawn atrandom, but are carefully chosen to capture specific properties ofthe state variable distribution.

2.3. The unscented Kalman filter: Simplified case

For the continuous representation of nonlinear system dynam-ics with additive noise of Eq. (1) the UKF steps are very similar tothose of Eqs. (19)–(29), but for the two modifications:

1) Propagation of sigma points through numerical integration,and

2) Simplification resulting from additive noise disturbances.

As in the case of EKF or the augmented UKF, we use numeri-cal integration methods (Eq. (30)) to propagate the sigma pointsthrough the continuous nonlinear equations instead of Eq. (20).For additive zero-mean noise disturbances the augmentation of thestate vector through noise vectors is not necessary. Hence the aug-mented state vector is given by xa

k = [xTk Θ T

k ]T , as in the case ofEKF. Consequently, the initial covariance matrix P a

0 correspondsto that of the states and system parameters only, and the totalnumber of states to be estimated is na = nxp . The respective noisecovariances Q and R are simply added to the right-hand sides ofEqs. (22) and (25):


Pk+1 =2na∑i=0

W ci

[χ x

i,k+1 − xk+1][

χ xi,k+1 − xk+1

]T + Q (31)

P y yk+1=

2na∑i=0

W ci [Yi,k+1 − yk+1][Yi,k+1 − yk+1]T + R (32)

The reduced size of the augmented state vector and sigmapoints leads to a reduction in computational load. By comparingthe UKF steps given above with those of the EKF in the previoussection, it is seen that the major changes pertain to computing themean and covariances of states and outputs; whereas the measure-ment update is basically similar. It is also apparent that the first orsecond-order approximations of the system dynamics are not re-quired.

Fig. 1. The DLR HFB-320 research aircraft.

3. Application to flight vehicles: Fixed wing platform

Two different types of aircraft are considered. The first aircraftconsidered is a fixed wing research aircraft HFB-320 (Fig. 1).

We estimate the lift, drag, and pitching moment coefficientsof the research aircraft HFB-320. Flight tests were carried out toexcite the longitudinal motion through a multi-step elevator in-put resulting in short period motion and a pulse input leading tophugoid motion. The following model is used to estimate the non-dimensional derivatives:

State equations:

V = − qS

mC D + g sin(α − θ) + Fe

mcos(α + σT )

α = − qS

mVCL + q + g

Vcos(α − θ) − Fe

mVsin(α + σT )

θ = q

q = qSc

I yCm + Fe

I y(�tx sinσT + �tz cosσT ) (33)

where the lift, drag, and pitching moment coefficients are modeledas:

C D = C D0 + C D VV

V 0+ C Dαα

CL = CL0 + CLVV

V 0+ CLαα

Cm = Cm0 + CmVV + Cmαα + Cmq

qc + Cmδeδe (34)
V 0 2V 0
Fig. 2. Time history for HBF 320.


Fig. 3. Performance of recursive parameter estimation methods for aerodynamic parameter identification of the research aircraft HFB-320.

Observation equations:

Vm = V

αm = α

θm = θ

qm = q

qm = qSc

I yCm + Fe

I y(�tx sinσT + �tz cosσT )

axm = qS

mC X + Fe

mcosσT

azm = qS

mC Z + Fe

msinσT (35)

where the longitudinal and vertical force coefficients C X and C Z

are given by:

C X = CL sinα − C D cosα

C Z = −CL cosα − C D sinα (36)

Here V is the true airspeed, α is the angle of attack, θ is the pitchattitude, q is the pitch rate, δe is the elevator deflection, Fe is thethrust, σT is the inclination angle of the engines, q (= 1/2ρV 2) isthe dynamic pressure, m is the mass, S is the wing area, c is thewing chord, I y is the moment of inertia, and ρ is the density ofair. The subscript m on the right-hand side of Eq. (35) denotesthe measured quantities. These system equations in terms of vari-ables in the stability axes (V ,α) contain not only the commontrigonometric and multiplicative nonlinearities, but in addition, thevariable dynamic pressure q (= 1/2ρV 2), which multiplies all of

the aerodynamic derivatives, introduces an additional nonlinear-ity. Furthermore, inversion of the state variable V leads to furthernonlinearities in the state equation for α. The unknown parame-ter vector Θ consisting of the non-dimensional derivatives is givenby:

Θ = [C D0 C D V C Dα CL0 CLV CLα Cm0 CmV Cmα Cmq Cmδe]T (37)

The recorded flight data has been used without any smooth-ing or filtering; the angle of attack was calibrated using flight pathreconstruction techniques [11], the thrust was computed prior toparameter estimation and used in the estimation process as an in-put variable.

The flight measured and model estimated responses are shownin Fig. 2. Fig. 3 compares the performance of the three recursiveparameter estimation methods for the purpose of aerodynamic pa-rameter estimation. It is clearly seen that the parameter estimatesof all these methods are in close vicinity of one another. Further-more, excellent agreement is seen between the recursive param-eter estimation method and the offline filter error method (FEM)estimates of the parameters.

The excellent performance of all the methods and their closeagreement can largely be attributed to an accurate mathematicalmodel of the system under consideration.

Table 1 compares the numerical values of the estimates of theparameters arrived at with the different methods. All numericalvalues are in excellent agreement. The design parameters for theUKF implementation are as follow, α is a small positive value inthe range of 0.001 to 1, κ = 0 or 3 − ne , β = 2.


Table 1Comparison of parameter estimates for the HFB-320.

Parameter Filter errormethod (FEM)

RPE methods

EKF UKF UKFaug

C D0 0.123 0.124 0.123 0.124(2.45)* (2.50) (2.64) (2.55)

C D V −0.0645 −0.0652 −0.0642 −0.0653(3.95) (4.01) (4.27) (4.1)

C Dα 0.320 0.319 0.319 0.316(2.26) (2.33) (2.40) (2.37)

CL0 −0.0929 −0.0853 −0.087 −0.099(21.1) (23.5) (22.9) (20.0)

CLV 0.149 0.144 0.147 0.157(11.1) (11.7) (11.4) (10.6)

CLα 4.328 4.303 4.289 4.303(1.08) (1.14) (1.14) (1.13)

Cm0 0.112 0.112 0.112 0.115(3.27) (4.28) (4.29) (3.40)

CmV 0.0039 0.0046 0.0045 0.0022(82.1) (90.5) (92.2) (154)

Cmα −0.968 −0.971 −0.970 −0.983(1.12) (1.54) (1.54) (1.24)

Cmq −34.710 −34.937 −35.363 −35.098(2.27) (2.85) (2.82) (2.27)

Cmδe −1.529 −1.533 −1.539 −1.552(1.27) (1.65) (1.65) (1.31)

* The values in parenthesis denote standard deviation values in percent.

4. Application to flight vehicles: Rotary wing platform

The second aircraft considered in this paper is a rotary wing air-craft: ARTIS [5,7,18] (Autonomous Rotorcraft Testbed for IntelligentSystems) which is an Unmanned Aerial Vehicle (UAV). The ARTIS(Fig. 4) is based on a commercially available miniature rotorcraftflight platform that is often used for acrobatic flight by pilots ofmodel helicopters. The baseline vehicle is a modified Benda Gene-sis 1800 helicopter. Due to the extreme maneuverability and a veryhigh thrust-to-weight ratio of the flight platform, the instrumentedUAV is able to fly even the most dynamic and aggressive ma-neuvers. A multi-sensor navigation system based on off the shelfsensors has been developed for ARTIS which provides optimizednavigation solutions using sensor fusion techniques. Detailed infor-mation on the modular avionics suite can be found in Refs. [3,18].

A linear estimation model is used which is valid for the ro-torcrafts hover domain. Sequences of frequency sweep inputs areused for the purpose of flight data collection as these provide avery rich set of estimation data for the inherently unstable rotor-craft platform [18,19].

The system state and observation equations are modeled incontinuous time domain as:

x(t) = Ax(t) + B(u(t) + �utrim

), x(t0) = x0

y = Cx(t) + �z (38)

where the system matrix A, the input matrix B and the measure-ment matrix C are functions of parameters to be identified β andβ = [βA;βB ;βC ]. Extended states �utrim and �z represent the re-spective input and measurement biases to be identified. The statevector x consists of u, v and w which are the rotorcraft veloci-ties in body x, y and z axes respectively, p, q and r which arethe rotorcraft angular rates around the body x, y and z axes re-spectively. φ and θ which are the Euler angles as well as a andb which are respectively rotorcraft internal states for longitudinaland lateral rotor flapping angles. Sensor measurements are avail-able for all the velocities, angular rates and Euler angles, howeverthey may contain noise and sensor biases. The internal states aand b are not measured and must be estimated. It is importantto note that the system model is time variant in the true senseas well as state dependent. Hence the parameters as well as the

Fig. 4. The ARTIS VTOL UAV.

trim and output bias values are not constant, rather, are func-tions of time and the current state of the rotorcraft. However,for the purpose of this paper we will consider a time invari-ant model trimmed around the hover domain. For linear repre-sentation of rotorcraft systems we use a measurement matrix Cequating measurements directly to states; hence simplifying theestimation problem by reducing the number of parameters to beestimated.

The sensor measurements are corrupt due to sensor inducednoise and vibrations of the rotorcraft frame. Furthermore the mea-surements may contain a considerable amount of process noisewhich among other factors, is also accounted to windy condi-tions on the day of the flight test. Apart from these 7 parameters,4 input trims (one on each input) two biases (on phi and thetaEuler angles) are also estimated. We pay special attention to theperformance of the filter in estimating states, filtering noise, andestimating parameters in order to asses the feasibility of the filterto be used in recursive state and parameter estimation applica-tions [5]. The estimates acquired from recursive parameter meth-ods are compared to offline estimation-runs undertaken using theoutput error method for the same set of data.

The rotorcraft state and input matrices are given as:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

uvpqabwrφ

θ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Xu 0 0 0 Xa 0 0 0 0 −g0 Y v 0 0 0 Yb 0 0 g 0Lu Lv 0 0 0 Lb Lw 0 0 0Mu Mv 0 0 Ma 0 Mw 0 0 0

0 0 0 −1 −1τ f

Abτ f

0 0 0 0

0 0 −1 0 Baτ f

−1τ f

0 0 0 0

0 0 0 0 Za Zb Z w Zr 0 00 Nv Np 0 0 Nb Nw Nr 0 00 0 1 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

×

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

uvpqabwrφ

θ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 00 0 Yped 00 0 0 00 0 0 Mcol

Alatτ f

Alonτ f

0 0

Blatτ f

Blonτ f

0 0

0 0 0 Zcol0 0 Nped Ncol0 0 0 00 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎨⎪⎪⎩

δlatδlonδpedδcol

⎫⎪⎪⎬⎪⎪⎭


Fig. 5. Recorded pilot inputs for ARTIS Sys-ID maneuvers.

Fig. 6. EKF histories of output variables, measured and estimated.


Fig. 7. S-UKF time histories of output variables, measured and estimated.

Measurement equation:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

uvwpqrφ

θ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 00 0 0 0 0 0 1 0 0 00 0 1 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 00 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

uvpqabwrφ

θ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

The unknown parameter vector Θ consists of system parame-ters β , the measurement biases �z and the trim estimates formu-lated as input biases �u.3

Θ T = {βt;�zT ;�uT }

(39)

The parameters to be estimated are:Lb: Effect of lateral rotor flapping angle on the lateral moment.Ma: Effect of longitudinal rotor flapping angle on longitudinal

moment.Za: Effect of longitudinal rotor flapping angle on vertical force.Zb: Effect of lateral rotor flapping angle on vertical force.Nb: Effect of lateral rotor flapping angle on yawing moment.

3 It may not be possible to estimate all the components of �z and �u since theycould be linearly dependent or highly correlated [18].

Alon: Effect of longitudinal input on lateral rotor flapping angle.Blat: Effect of lateral input on longitudinal rotor flapping angle.The recorded pilot input for the identification maneuvers per-

formed are shown in Fig. 5, the maneuvers consist of variablefrequency input sweeps on the lateral cyclic and the longitudinalcyclic inputs and are executed manually through remote control bya trained pilot.

The EKF state filtering performance is shown in Fig. 6, it can beseen that the EKF output matches well with the measured outputs;furthermore the measurement noise is fairly well filtered by theEKF. The S-UKF performs equally well in terms of state filtering, thestate filtering performance of S-UKF is seen in Fig. 7. The perfor-mance of the EKF and the S-UKF is almost identical; this stressesthe importance of the additive white noise assumption. The aug-mented UKF, which caters to the case of noise entering nonlinearlyin the system, performs marginally better in filtering out the noisein the w (velocity along the body Z axis), the state filtering per-formance in other channels is almost identical to the other twomethods, indicating that additive white noise assumption does notresult in great loss of accuracy in the state estimation problem.The state filtering performance of the augmented UKF is shown inFig. 8. Different tuning of the Q matrix, which accounts for rela-tive process noise covariance, was required for the augmented UKFalgorithm compared to that of the EKF and S-UKF algorithms. TheR matrix tuning, accounting for relative measurement noise covari-ance, was based on empirical methods and were kept constant forall three algorithms.


Fig. 8. UKFaug: Time histories of output variables, measured and estimated.

Fig. 9. Comparison of recursive parameter estimation algorithms for the ARTIS VTOL UAV.


Table 2Comparison of parameter estimates for the ARTIS VTOL UAV.

Parameter Output errormethod (FEM)

RPE methods

EKF UKF UKFaug

Lb 351.294 352.37 353.65 320.84(0.31)* (3.1) (3.07) (3.07)

Ma 119.2 123.38 123.48 111.55(0.34) (5.83) (6.0) (7.64)

Za 23.284 5.03 12.18 19.24(8.70) (36.78) (15.70) (81.49)

Zb −61.484 4.83 1.03 −9.08(7.66) (32.05) (151.19) (138.36)

Nb 94.049 72.67 80.88 80.70(1.93) (3.64) (3.12) (5.55)

Alon −2.626 −2.81 −2.71 −2.89(0.28) (12.79) (13.29) (22.47)

Blat −2.357 −2.97 −2.82 −2.75(0.39) (8.04) (8.52) (21.30)

* The values in parenthesis denote standard deviation values in percent.

Fig. 9 compares the time histories of estimated parameters forthe three recursive estimation methods. It was seen that the pa-rameter estimates from all three methods were in close agree-ment with each other, as well as independent of the initial condi-tions. While the parameter estimates from the recursive estimationmethods are in the vicinity of offline estimates (estimated usingthe OEM method and presented in the graph with solid lines), theyare not in complete agreement with the OEM estimates. This is at-tributed to the following main reasons:

1. Presence of noise: The EKF, UKF, and the S-UKF can be tunedto account for both process and measurement noise which isamply present in this data. Whereas the OEM algorithm usedfor offline estimation did not offer any noise filtering capabili-ties. The effect of filtering is more obvious in the estimation ofthe parameters Za and Zb where accounting for process noiseresults in different parameter estimates to that of the OEMmethod.

2. State dependence of parameters: The rotorcraft is a nonlinear,time varying, and state dependent system. The OEM methodattempts to find the best match for parameters that is valid forthe complete set of data, while the recursive filtering methodstend to adapt to parameters based on the current informationin the data. In Fig. 9 it is observed that the recursive methodsare in close agreement of the OEM methods when rich datafor the parameter in question is available. For example: theparameter estimate of Nb is in closer agreement with OEMestimates for the first 25 seconds, as during this time the ex-citations in roll rate produce yaw excitations resulting in richdata for the estimation of Nb .

These explainable discrepancies in the parameter estimationserve well to point out the advantage of using recursive parameterestimation methods in aerodynamic parameter estimation of non-linear, state-dependent systems approximated by a linear model.Table 2 presents the numerical values of the parameters and thepercent standard deviation. In all three cases the trim values andthe output biases (not shown here) showed rapid convergence totheir approximate expected values (approximated from pilot feed-back).

The run time of our implementation of the S-UKF algorithmwas approximately thrice that of the EKF algorithm, while the runtime of the augmented UKF algorithm was approximately six timesthat of the EKF.

5. Conclusions

In this paper we compared three recursive parameter identifi-cation algorithms for nonlinear filtering of aircraft flight data usingthree nonlinear recursive parameter estimation algorithms basedon Kalman filters:

1) The EKF, employing a continuous estimation model with ad-ditive noise represented by state vector xa

k = [xTk Θ T

k ]T ; andusing a first order approximation for propagation of the co-variance matrix,

2) General or augmented case of the UKF employing a contin-uous estimation model with noise disturbance entering thesystem nonlinearly, represented by the augmented vector xa

k =[xT

k Θ Tk w T

k v Tk ]T , Eqs. (19)–(29), and

3) Special or simplified case of the UKF (S-UKF); employing acontinuous estimation model with additive noise, representedby state vector xa

k = [xTk Θ T

k ]T and to which the simplificationsin Eqs. (31) and (32) are used to calculate the covariances.

The results indicate that the UKF augmented recursive param-eter estimation method is the fastest in terms of time to conver-gence. However, it is also the costliest in terms of computationalpower. The simplified UKF case, which assumes additive uncor-related noise is equivalent in accuracy and filtering quality withthe EKF, and shows little improvement in time to convergence ascompared to the EKF. However, even the simplified UKF implemen-tation is computationally more costly than the EKF. The EKF, asexpected, maintains excellent filtering quality and decent time toconvergence, it is also the computationally least expensive.

For the flight parameter estimation of the HFB-320 researchaircraft with a nonlinear estimation model, no great difference be-tween the numerical values of the parameters was seen. This ismainly attributed an accurate estimation model. For the aerody-namic derivative estimation of the ARTIS VTOL UAV; which em-ploys a linear model, general agreement was seen between thenumeric results of the EKF and S-UKF recursive parameter esti-mation algorithms which assume additive white noise. However,the parameter estimates from the augmented UKF (which caters tothe general case of noise entering nonlinearly) algorithm showedslight deviation from those of the EKF and the S-UKF algorithms.This result highlights the impact of the assumption on how noiseenters the system for estimation problems with considerable mea-surement and process noise coupled with approximate estimationmodel. Furthermore, the results of the recursive estimation algo-rithms did not match well with the results of offline output errormethod results. This discrepancy is mostly attributed to the factthat the OEM method used for offline parameter estimation did notimplement filtering of measurement and process noise, and to thefact that the linear representation of a rotorcraft model is state-dependent. It is important to note that the filtering performanceof the three algorithms in state estimation was nearly identical forboth the linear and the nonlinear estimation models.

In conclusion, our results indicate that for flight system param-eter identification purposes, the EKF algorithm remains a viabletool, which consistently returns quality results and is the leastcostly in terms of computational demand. The UKF algorithm isa theoretically superior alternative to the EKF, however it maynot add much to the recursive system identification process if theidentification model and the identification data are not appropri-ate. The UKF is indeed superior in terms of time to convergence,and relative reliability of the estimates; however it is it is com-putationally more expensive. Hence, a need is not seen for thereplacement of existing EKF based recursive aerodynamic param-eter identification algorithms with the UKF. However, it is clearthat the UKF is a powerful system identification tool and should be


given due attention in the development of future applications in-volving recursive parameter identification algorithms. Furthermore,the UKF proves itself handy in validating the accuracy of existingEKF based algorithms, double checking the results of offline param-eter estimation methods, and improving the general reliability ofparameter estimates especially when the flight data contains con-siderable noise.

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http://cslu.cse.ogi.edu/nsel/ukf/

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