System Identification of a Miniature Helicopter

Jared A. Grauer,∗ Joseph K. Conroy,† James E. Hubbard Jr.,‡ and Darryll J. Pines§

Department of Aerospace Engineering, University of Maryland, College Park, MD, 20742

Micro air vehicles are typically designed for mission profiles including surveillance andreconnaissance, and are envisioned to have a large degree of autonomy. Helicopters pro-vide a useful vehicle design as they may carry visual sensors, maneuver through clutteredenvironments, and hover. Knowledge of the vehicle dynamics facilitates the use of model-based state estimation and control techniques, which can be used to improve sensor mea-surements, augment pilot handling qualities, and improve autonomous flight performance.Towards that goal this work presents the identification of a linear model for a miniatureelectric helicopter in hovering flight. The model structure is built upon first principlemodeling, previous work, and the statistical contribution of candidate regressors to themodel accuracy. Parameter estimates and error bounds are estimated using maximumlikelihood methods in both the time and frequency domains, and resulting models are val-idated by comparing simulated outputs to measured flight data. Results show that theidentified models have a Eigenstructure consistent with and predictive capabilities similarto a previously identified model which employed the frequency response method.

Nomenclature

A, B, C, D linear system matricesA, B main rotor stability derivativesa, b main rotor flapping anglese unit vectorg gravitational accelerationi, j time domain data indicesJ(θ) parameter optimization costj imaginary numberk frequency domain data indexL, M , N moment stability derivativesN number of data samplesp, q, r rotational velocity projectionsR noise covariance matrixSνν residual autospectral densitys(θ) sample standard errort timeu input vectoru, v, w translational velocity projectionsX model regressor matrixx state vectorT , X, Y , Z force stability derivativesx, y, z inertial positiony model output vector

z measurement vectorδ pilot control inputζ, ω modal damping ratio and frequencyθ model parameter vectorν model residual vectorτ main rotor time constantφ, θ, ψ Euler anglesΩ main rotor rotational frequency

Subscriptsdir directionallat lateral cycliclon longitudinal cyclicthr throttlex, y, z orthonormal frame projections

SuperscriptsT transpose† complex conjugate transpose˜ frequency domain quantityˆ estimated value¯ sample mean˙ time derivative

∗Graduate Student, Department of Aerospace Engineering, Member AIAA.†Graduate Student, Department of Aerospace Engineering, Member AIAA.‡Langley Distinguished Professor, Department of Aerospace Engineering, Associate Fellow AIAA§Professor, Department of Aerospace Engineering, Member AIAA.

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AIAA Atmospheric Flight Mechanics Conference and Exhibit18 - 21 August 2008, Honolulu, Hawaii

AIAA 2008-6898

Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

I. Introduction

In recent years there has been a large effort to develop and mature micro air vehicle (MAV) technologyand vehicle platforms. Originally the designation MAV applied to aircraft with dimensions less than 0.15

m, but has since been used to describe many aircraft of slightly larger size. Micro air vehicles are typicallydesigned for mission profiles such as reconnaissance and surveillance, where their small size allows themto maneuver into remote locations while their visual and acoustic signatures deter detection. Miniaturehelicopters in particular provide a unique choice for MAV designs because of their maneuverability, as wellas their ability both to fly slowly and to hover. These traits make helicopters essential when attemptingto fly in close, crowded environments while streaming visual information to a ground station. Autonomousflight is not only desired, but also necessitated by these missions due to the fast and unstable dynamicsof the aircraft, which are exacerbated by transmission delays of sensor data and control inputs. Linear,time-invariant, state space models of the dynamics about trimmed flight conditions of the canonical form

x = Ax + Buy = Cx + Du

(1)

permit the use of a broad set of mathematical analysis and synthesis techniques which enable autonomousflight. For example a model of the helicopter dynamics in hovering flight may be used to improve theorientation tracking performance of the vehicle, which may result in higher quality camera images of amoving target. A model of this form is suitable for direct implementation in a Kalman filter to provideimproved state estimates in the presence of noise and disturbances.1 For piloted flight, this model can beused with classical control techniques to augment the handling qualities so that it is easier for a pilot tofly.2 For autonomous flight, modern control techniques such as LQG/LTR, H∞, and µ-synthesis may beemployed to design optimal and robust control laws to improve flight performance.3,4

In comparison with conventional fixed-wing aircraft, obtaining a state space model of the form of Equa-tion 1 for rotary-wing aircraft is difficult.5,6 Similar to fixed-wing counterparts, the fuselage of a rotary-wingvehicle is typically modeled using a single rigid body, the dynamics of which are well understood. However,the rotors on a helicopter are themselves dynamical systems which couple with the rigid body dynamicsand the surrounding flow field, introducing complex and unsteady aerodynamics which manifest in part asrotor/wake and rotor/fuselage interactions.7 The rotor dynamics also exhibit a large degree of inter-axiscouplings and higher harmonic responses which compete with requirements for an accurate, low order modelfor hardware implementation. First principles modeling offers insight into the dynamics, but often timesrequires numerous and drastic simplifications to obtain usable models. Additionally the hover dynamics of ahelicopter are unstable, requiring feedback to decrease the pilot workload, which in turn masks the naturaldynamics of the system and introduces collinearity into the flight data. The identification of miniaturehelicopters are further complicated by a number of obstacles. Onboard sensors are typically based on microelectromechanical systems (MEMS) and are of poorer quality than those used in larger aircraft, having astrong temperature dependence, noisy readings, and calibration biases which exhibit random walk. Smallervehicle sizes reduce the mass properties of the system which increase the sensitivity to wind gusts andsensor noise within the feedback loops. Miniature helicopters are commonly flown indoors for this reason,which limits the space in which helicopters may fly and thus hinders the identification of the low frequencydynamics.

Despite these obstacles, system identification of rotorcraft has become a standardized procedure. Themajority of the identification work in the literature employs the frequency response method, implementedwith the software package CIFER

TM(Comprehensive Identification from Frequency Responses).8–10 Sensor

measurements are recorded as pilots command sinusoidal inputs with rich spectral content. Data are trans-formed into the frequency domain and experimental frequency responses are estimated between each inputand each output. First principles modeling is performed to give predicted frequency responses, and thenan optimization routine is used to determine the model parameters which most closely match the exper-imental and theoretical frequency responses. This method is commonly used with rotorcraft because thespectral view of the dynamics removes noise, illuminates modal resonances, and uses fewer data points thana time domain identification. This method has been applied to large scale vehicles,11–13 miniature unmannedaircraft,5,14,15 and small electric helicopters.16,17

There are however many other methods for performing system identification.18,19 Although relativelysparse in literature, system identification procedures utilizing maximum likelihood (ML) methods have been

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applied to rotorcraft with good results.20 This paper presents the identification using maximum likelihoodmethods of a linear hover model for a small electric helicopter, which has previously been identified byConroy et al.17 using the conventional frequency response method. A model structure is determined andparameter estimates are found using the equation error (EE) and output error (OE) simplifications of theML estimator in the time and frequency domains. The modal eigenvalues, damping ratios, and resonantfrequencies identified are compared with previous results, and the predictive capabilities of the models aretested. The majority of numerical routines used are found in a MATLAB R©21 software package called SIDPAC(System Identification Programs for Aircraft).22,23

II. Identification Method

The first step in the system identification of a vehicle is to collect flight data in an experiment wherethe vehicle is piloted within a specified flight envelope, having the dynamics of interest sufficiently excited.Before parameters, such as stability and control derivatives, can be estimated, model structure determinationmust be performed to choose a model size and structure. Using first principle derivations, intuition with thesystem, and observations from the flight data, a pool of candidate model regressors is formed. A variety ofstatistical metrics are then used to determine which regressors significantly contribute to the model fit of theflight data and should be retained in the model. In general an accurate model which is as small as possibleis desired because increasing the number of parameters reduces the accuracy of parameter estimates and thepredictive capability of the model.23,24 In this work the main statistical metrics used were

R2 =θT XT z−N z2

zT z−N z2(2)

PSE =1N

(z− y)T (z− y) + σ2max

( p

N

)(3)

rjz =Sjz√SjjSzz

. (4)

The metric R2 ∈ [0, 1] in Equation 2 is the coefficient of determination, which measures how well thepostulated model matches the data, and should be maximized for accuracy. Equation 3 is the predictedsquare error (PSE), where σmax is a constant weighting term and p is the number of regressors in the model.This metric rewards model accuracy and penalizes the number of regressors, and is used to safeguard againstusing insignificant regressors in the model. Equation 4 is the correlation between the jth regressor andthe measured output, where the S quantities represent auto- and cross-correlations for scaled and centeredtime histories of measurements and regressors. When a regressor has a large correlation with the data,it is included in the model structure. These metrics are monitored in an iterative process called stepwiseregression, where regressors are manually added and removed until the model is deemed sufficient.

With a fixed model structure, parameters and error bounds can be estimated. Maximum likelihoodestimators obtain optimal estimates by minimizing a negative log-likelihood function under the Fisher modelassumptions of time-invariant model parameters and zero-mean, normally distributed random measurementnoise. For an adequate model structure these estimators are unbiased, consistent, asymptotically normaland efficient estimators.23 Estimators minimize the cost functions

J(θ) =12

N∑i=1

ν(i)T R−1ν(i) (5)

J(θ) = NN−1∑k=0

ν†(k)S−1νν ν(k) (6)

formulated in the time and frequency domains, respectively, where ν are the residual differences betweenthe measured and predicted outputs. Parameter accuracy for maximum likelihood estimators are bestcharacterized by the Cramer-Rao bounds, which provide a theoretical lower limit on the errors.24,25 Timedomain analysis is intuitive, but standard errors must be corrected for non-white residuals. In addition tousing fewer data points and weighting the modal resonances more heavily, frequency domain analysis doesnot require error corrections, as only the spectral bands of interest are used in the modeling process. Two

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simplifications of the ML estimator are the equation error and output error methods. The equation errormethod assumes a linear state space model of the form of Equation 1, where the model parameters arethe stability and control derivatives, and where the states and inputs are measured without error.23 Theresiduals are written as

ν(i) = z(i)−Ax(i)−Bu(i)ν(k) = z(k)−Ax(k)−Bu(k)

(7)

which are the differences between the measured state derivatives and the model for those state derivatives,constructed using measurements for states, controls, and the estimated system matrices, specified in the timeand frequency domains, respectfully. Equation error is a linear formulation and has a non-iterative solution.The method is applied to the model in Equation 1 one row at a time, where the derivative term on the leftside of the equation is found by smoothly differentiating a measured quantity. Output error assumes a statespace formulation of the dynamics free of process noise. The residuals are written as

ν(i) = z(i)− y(i)ν(k) = z(k)− y(k)

(8)

which are the differences between the measured and predicted outputs for the time and frequency domains,respectfully. The output error method requires an iterative, nonlinear optimizer to obtain parameter esti-mates, where equation error results are typically used as initial estimates of the parameters.

III. Experimental Setup

III.A. Aircraft Description

The test aircraft used in this study is the commercially available HoneyBee helicopter by E-Sky, shownwith dimensions in Figure 1. The helicopter is an electric hobby aircraft, controlled by a nearby pilot using aradio transmitter, which sends the pilot joystick commands to an onboard receiver, which in turn sends thecommands to the actuators. The throttle input δthr ∈ [0, 1] commands the speed of a brushless DC motor,which through a gearbox controls the rotational speed of the main rotor and governs the amount of thrustproduced. This fixed-pitch configuration is a low cost alternative to more advanced assemblies found onlarger aircraft, where the main rotor spins at a constant speed while the pilot controls the collective angle ofattack of the rotor blades in order to achieve the desired thrust. On this helicopter the rotor spins clock-wisewhen viewed from above. Longitudinal and lateral cyclic inputs δlon, δlat ∈ [−1,+1] command servo motorswhich position pitch linkages in a Hiller style stabilizer mechanism to impart control forces and torques onthe aircraft. The directional control input δdir ∈ [0, 1] governs the speed of a second DC motor and tailrotor, which is employed directly to cancel the reaction torque caused by the main rotor, and differentiallygenerate a yawing torque. Although easily incorporated, actuator dynamics are not identified because theservo motors are lightly loaded and actuated at 50 Hz, the tail rotor has a quick response, and the pilotinput spectra are generally limited to 3 Hz.

exBeyB

ezB

Parameter Value UnitMass 390 g

Main Rotor Diameter 0.505 mTail Rotor Diameter 0.145 m

Length 0.781 mWidth 0.120 mHeight 0.700 m

Figure 1. Helicopter test aircraft and specifications.

As the mass properties of the rotors are small relative to the fuselage and the rotational frequencies aremuch faster than those of interest, the helicopter is modeled as a single rigid body with additional states to

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capture the rotor dynamics and the rotor/fuselage interactions. An inertial frame KI is placed at a fixedpoint on the surface of the Earth with the exI axis pointing north, the eyI axis pointing east, and the ezI

axis pointing down. A second frame KB is fixed at the center of mass of the vehicle with the exB axispointing out the nose, the eyB axis pointing laterally out the starboard side, and the ezB axis pointing downout of the bottom. The position of the aircraft is given in terms of rectilinear coordinates x, y, and z in theinertial coordinate frame. The orientation of the body frame relative to the inertial frame is parameterizedby the Euler angles φ, θ, and ψ. The translational velocity and rotational velocity are expressed by thebody-fixed projections u, v, w, and p, q, r, respectfully. Sign conventions of the control inputs are assignedsuch that positive perturbations to the throttle setting, longitudinal cyclic, lateral cyclic, and directionalsetting primarily induce a negative heave velocity, a negative pitch rate, a negative roll rate, and a positiveyaw rate, respectfully.

III.B. Measurements and Signal Processing

The helicopter is fitted with a custom avionics package,26 shown with dimensions in Figure 2. Theavionics are controlled by an 8-bit microprocessor running at 40 MHz, which samples sensors at 250 Hz,issues actuator commands at 50 Hz, and communicates with a ground station computer over two bluetoothchannels at 250 Hz. Measurement packets are timestamped using a running 16-bit integer counter, which areconverted to a time measurement in post processing. Pilot inputs are received over the bluetooth connectionin the form of 16-bit numbers, as measured by the transmitter unit. The avionics package is outfitted withmagnetometers for orientation information, gyroscopes for body fixed rotation rates, accelerometers for linearaccelerations, a sonar sensor for altitude, and an optic flow sensor for translational velocities. Additionallythe avionics package implements a minimal amount of onboard mixing to decrease the pilot workload duringflight tests: the throttle setting is fed forward and the yaw rate measurement fed back with proportionalgains to the directional control input to balance the main rotor reaction torque and slow the yaw dynamics.

Parameter Value UnitMass 30 g

Supply Voltage 7.4 VLength 0.090 mWidth 0.060 mHeight 0.015 m

Figure 2. Avionics package and specifications.

For larger aircraft, position and orientation information can be typically obtained using GPS data andinertial measurements. However for miniature helicopters flying indoors, GPS is prohibitive and does nothave spatial or temporal resolution fine enough for system identification. Instead a visual tracking system27

is used where during flight seven cameras track the three dimensional position of numerous retro-reflectivemarkers placed on the vehicle, as shown in Figure 3. Using a model of the helicopter geometry and thelocations of the markers on it, least-squares estimates for the position of the center of mass, orientation ofthe body frame, and azimuth angle of the main rotor are estimated at 350 Hz. Position and orientationdata are first low pass filtered to 10 Hz using a fixed-weight smoothing method,23 as this cutoff frequencyis above the expected rigid body modes and below the 50 Hz 2/rev vibrations caused by the rotor bladespassing over the fuselage. Inertial velocities and the Euler rates are computed by differentiating a movingpolynomial fit to the data.23,28 The body fixed translational and rotational velocities were computed vis u

v

w

=

cos θ cosψ cos θ sinψ − sin θsinφ sin θ cosψ − cosφ sinψ sinφ sin θ sinψ + cosφ cosψ sinψ cos θcosφ sin θ cosψ + sinφ sinψ cosφ sin θ sinψ − sinφ cosψ cosφ cos θ

x

y

z

(9)

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p

q

r

=

1 0 − sin θ0 cosφ sinφ cos θ0 − sinφ cosφ cos θ

φ

θ

ψ

(10)

which rotates the inertial derivatives to the body frame. Low pass smoothing is then applied to the deriv-atives. Body fixed accelerations are obtained by smoothly differentiating the velocities and smoothing thederivatives. The speed of the main rotor is found in a similar manner by differentiating and smoothing therotor azimuth angle. With the exception of the directional control, the pilot inputs were not filtered becausethe identification methods assume perfect knowledge of the inputs. While the onboard avionics provide manyof the measurements, it was decided to derive all of the measurements from the visual data for several reasons.The inclusion of the avionics data would require corrections for offset, alignment, and calibration errors ofeach measurement to ensure the data are kinematically consistent, whereas rotating and differentiating thevisual data enforces compatibility. The visual data is very accurate and of low variance, whereas the avionicsmeasurements are very noisy. Errors are introduced to the accelerometer measurements by correcting themeasurements to the aircraft center of mass and by removing the accelerations due to gravity. Additionallystructural vibrations are registered by the MEMS sensors, whereas the least-squares fit to the markers withthe visual system attenuates local vibratory disturbances.

(a) visual image (b) processed image

Figure 3. Visual tracking system configuration.

The control input vector consists of the pilot joystick inputs, and the measurement vector consists of theroll angle, pitch angle, translational velocities, rotational velocities, and rotor speed. The position and yawangle are not included because in hover these variables do not significantly impact the dynamics. The inputand measurement vectors are written as

u =[δthr δlon δlat δdir

]T

(11)

z =[φ θ u v w p q r Ω

]T

(12)

where these quantities indicate deviations away from the trimmed values. Measurement statistics are pro-vided in Table 1, where the units specified are used consistently throughout this work. Resolution for thetime measurement is a function of the microprocessor oscillator frequency, the counter roll over frequency,and the number of bits used to store the number. The control input resolution is a function of the number ofbits used in the analog to digital converter and the voltage range specified. Estimates of the position resolu-tion were determined by the smallest differences measured while the vehicle was stationary on the ground.Resolutions for the remaining measurements were not estimated since these measurements are generated inpost processing. Variances were estimated by computing the sample variance of a segment of data where thehelicopter was stationary on the ground. As gyroscopes are noisy and are fed back to the directional input,variances for the yaw angle and yaw rate were relatively large and increased the estimated variances on theposition and rotational velocity measurements. Measurements available from a representative segment offlight data are shown in Figure 4 for illustration.

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Table 1. Measurement specifications.

Measurement Sensor Resolution Variance UnitTime t Avionics 25.600 · 10−6 - s

Control Input δthr, δlon, δlat, δdir Avionics 0.0015 0.2426 · 10−3 normPosition x, y, z Visual 8.9072 · 10−6 0.6128 · 10−3 m

Orientation φ, θ, ψ Visual - 0.0078 radTranslational Velocity u, v, w Visual - 0.2510 · 10−3 m/s

Rotational Velocity p, q, r Visual - 0.0012 rad/sRotor Speed Ω Visual - 0.2143 rad/s

0 5 10−0.05

0

0.05

δth

r

0 5 10−0.5

0

0.5

δlon

0 5 10−1

0

1δ

lat

0 5 10−0.5

0

0.5

δ dir

0 5 10−1

0

1

x

0 5 10−1

−0.5

0

y

0 5 10−0.5

0

0.5

z

0 5 10−0.5

0

0.5

φ

0 5 10−0.5

0

0.5

θ

0 5 10−2

0

2

ψ

0 5 10−1

0

1

u

0 5 10−1

0

1

v

0 5 10−1

0

1

w

1

0 5 10−1

0

1

p

0 5 10−2

0

2

q

time (s)

0 5 10−2

0

2

r

1

time (s)0 5 10

−20

0

20

Ω

time (s)

Figure 4. Measurements available from a representative segment of flight data.

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IV. Results

IV.A. Model Structure Determination

The state space model structure postulated is

φ

θ

u

v

w

p

q

r

Ωa

b

=

0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 1 0 0 0 00 −g Xu 0 0 0 0 0 0 0 0g 0 0 Yv 0 0 0 0 0 0 00 0 0 0 Zw 0 0 Zr ZΩ 0 00 0 0 Lv 0 0 0 0 0 0 Lb

0 0 Mu 0 0 0 0 0 0 Ma 00 0 0 Nv Nw 0 0 Nr NΩ 0 00 0 0 0 0 0 0 0 TΩ 0 00 0 0 0 0 0 −1 0 0 − 1

τAb

τ

0 0 0 0 0 −1 0 0 0 Ba

τ − 1τ

φ

θ

u

v

w

p

q

r

Ωa

b

+

0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0

Nthr 0 0 Ndir

Tthr 0 0 00 Alon

τAlat

τ 00 Blon

τBlat

τ 0

δthr

δlon

δlat

δdir

(13)where the model parameters are the stability and control derivatives. The C and D matrices consist ofones and zeros so that the system output vector pertains to the same quantities as the measurement vectorspecified in Equation 12. The dynamics for the roll and pitch angles are analytical linearizations of therotational kinematics about the trimmed hover condition.29 The dynamics for the roll and pitch rates aresimplified from the hybrid model first proposed by Tischler and Cauffman11 and later reduced as first ordertip path plane equations by Mettler.6 This model structure is able to capture the majority of the rotordynamics and rotor/fuselage interactions using the states a and b, which represent the longitudinal andlateral flapping angles of the lumped rotor and stabilizer system. Stability derivatives Ma and Lb quantifythe rotor torques on the fuselage, Mu and Lv represent speed derivatives, Ab and Ba capture cross-axisinteractions in the rotor dynamics, τ is the rotor time constant, and Alon, Alat, Blon, Blat are the gains fromthe pilot inputs to the flapping dynamics. Separating the rotor and stabilizer dynamics as in Mettler6 wasattempted, but increased the model order and complexity with only a marginal increase in accuracy.

The structure of the remaining dynamics was determined using stepwise regression, where the regressormatrix consisted of the measurements and control inputs given in Equations 11 and 12. A flight test lastingroughly 6.0 minutes was performed, in which 5.5 minutes of data were suitable for identifying a linear modelabout the hovering flight condition. For each dynamic variable in the model, a segment of flight data rangingbetween 10 and 20 seconds was used where the dynamics were excited. Coefficients of determination arepresented for the time domain fits, however data segments were analyzed in both the time domain andfrequency domains and resulted in the same model structure. The longitudinal velocity and lateral velocitydynamics have fits of 0.96 and 0.77 using regressors θ, u and φ, v, respectfully. The derivatives Xθ andYφ represent the linearized effect of gravity. Both derivatives resulted in estimates of gravity with color-corrected standard errors within the value of 9.81 m/s2, and hence this value was fixed for the remainder ofthe identification work. The derivatives Xu and Yv represent aerodynamic damping terms. The derivativesXa and Yb are often used to represent the longitudinal and lateral forces generated by the rotor,6 but werenot included since gravity and the damping terms capture the majority of the dynamics. Additionally Ydir

is often used to capture the lateral force generated by the tail rotor, but this regressor does not significantlyimprove the model fit to the data. The heave dynamics have a 0.85 model fit using regressors w, r, and Ω. Thederivative Zw is an aerodynamic drag damping term, whereas Zr and ZΩ describes thrust generation fromthe main rotor due to yaw rate and rotor speed perturbations. The altitude measurement was also significant,possibly indicating ground effect, but is omitted because it would provide only a modest improvement tothe model accuracy at the expense of increasing the model order. The yaw rate dynamics have a 0.76 fitusing regressors v, w, r, Ω, δthr, and δdir. The speed derivative Nv models the drag on the vertical tail andthe aerodynamics of the tail rotor, Nw is the additional reaction torque generated by heave motion, and Nr

represents both aerodynamic damping from rotation and the yaw rate feedback. Additionally Nthr capturesthe feedforward mixing, Ndir describes the gain associated with the pilot stick, and NΩ models the remainingvariations in yaw rate due to the reaction torque generated by the main rotor. The rotor speed dynamicsare modeled as a first order lag with a fit of 0.66 using regressors Ω and δthr. The derivative TΩ represents

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the rotational inertia and aerodynamic drag of the rotor, as well as the DC motor dynamics. The term Tthr

is the gain from the pilot joystick input. These two regressors captured the trends well; regressors r and wwere significant, but were omitted because the Eigenstructure of the rotor speed dynamics would have beenaltered in an non-intuitive manner. The altitude variable was not significant in the rotor speed dynamics.

The residuals are also analyzed to evaluate the postulated model structure. Representative results areshown in Figure 5 for the dynamics of the longitudinal velocity using the equation error method in thetime domain. For clarity, results are decimated from 350 Hz to 35 Hz. Figure 5(a) shows the measuredaccelerations and the model output, which match well. The residuals and the 95% confidence bound areshown below in the Figure 5(c). There is deterministic content left in the residuals, but the regressors θ andu account for the majority of the dynamics and the confidence intervals are roughly 25% of the accelerationmagnitudes measured in the data. The cross plot is shown in Figure 5(b) graphs the model outputs againstthe model residuals and indicates neither unmodeled nonlinearities, nor non-stationary variances in theresiduals. Figure 5(d) shows the autocorrelation of the residuals, which resembles white noise in that thereis a strong peak at the zero lag index, but the remaining content shows that the residuals are indeed coloredand standard errors of parameter estimates computed using conventional calculations must be corrected.

0 1000 2000 3000 4000 5000−1.5

−1

−0.5

0

0.5

1

1.5

sample index

outp

ut

datamodel

(a) equation error model fit

−1.5 −1 −0.5 0 0.5 1 1.5−0.5

0

0.5

model output

resi

dual

data95% confidence

(b) residuals

0 1000 2000 3000 4000 5000−0.5

0

0.5

sample index

resi

dual

data95% confidence

(c) residuals

−5000 0 5000−5

0

5

10

15

lag index

Rvv

⋅ 10

−3

autocorrelation2−σ confidence

(d) residual autocorrelation

Figure 5. Residual diagnostic plots for the longitudinal velocity dynamics model structure using the equation errormethod in the time domain.

IV.B. Parameter Estimation and Modal Characterization

Parameter estimates are given in Table 2. The same segments of flight data used for model structuredetermination are used for the equation error analysis, whereas the entire 5.5 minutes of flight data areused for the output error analysis. Frequency domain identification used the same data as the time domainidentification. A high accuracy chirp-z transform30 is employed to obtain frequency domain data in terms ofFourier coefficients spaced at 0.01 Hz increments between 0.10 Hz and 5.00 Hz. The chosen spectral grid wasfiner and the range was extended farther than suggested by conventional rules of thumb23 because rotorcrafthave more rigid body modes than standard fixed-wing aircraft.

The equation error model fits are shown in Figure 6, where again the time domain results are decimatedto 35 Hz for clarity. As the rotor flapping angles were not measured, the dynamics of the roll rate, pitchrate, longitudinal flapping angle, and lateral flapping angle can not be identified using equation error. Thedynamics of the longitudinal and heave velocities match very well in both the time and frequency domains.The lateral velocity dynamics under predict the frequency content above 4 Hz, which under predict the

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Table 2. Parameter estimates and color-corrected standard errors.

Parameter Time EE Frequency EE Time OE Frequency OEθ θ ± s(θ) θ ± s(θ) θ ± s(θ) θ ± s(θ)Xu −0.2223± 0.0554 −0.3218± 0.0004 −0.8073± 0.3219 −0.8159± 0.0007Yv −0.9180± 0.1817 −0.2119± 0.0026 −0.6907± 0.5667 −0.8733± 0.0045Zw −0.7659± 0.2063 −0.6304± 0.0038 −2.1845± 0.2399 −1.9198± 0.1372Zr −0.0885± 0.0542 −0.1066± 0.0011 +0.7370± 0.1288 +1.7203± 0.1491ZΩ −0.1354± 0.0087 −0.1525± 0.0000 −0.1761± 0.0135 −0.3228± 0.0162Lv - - −60.563± 2.6544 −29.251± 2.2306Lb - - +3369.9± 40.543 +2128.3± 162.58Mu - - +7.1191± 0.3663 +7.1093± 0.1538Ma - - +306.97± 4.0491 +309.24± 2.0355Nv −0.5420± 1.2283 −1.3089± 0.3577 −0.1843± 0.1266 −1.0943± 0.5484Nw −1.8458± 1.3561 −0.5023± 0.2023 −2.2427± 0.7279 −5.3766± 0.6652Nr −10.743± 0.8740 −5.0806± 0.2323 −14.862± 0.6269 −12.185± 0.9581NΩ −0.1682± 0.0658 −0.2025± 0.0023 −0.1096± 0.0587 −0.0780± 0.0872Nthr −140.22± 15.183 −36.491± 11.813 −70.458± 10.341 −3.1788± 16.226Ndir +138.96± 8.6589 +72.258± 27.829 +142.15± 5.1011 +123.07± 8.7711TΩ −2.0603± 0.2583 −1.4781± 0.0288 −2.9584± 0.0823 −2.3144± 0.0797Tthr 809.50± 51.282 +644.96± 11.041 +784.31± 18.670 +760.89± 18.818τ - - +0.1260± 0.0040 +0.1342± 0.0022Ab - - −1.1992± 0.1079 −2.3003± 0.0882Alon - - −0.1524± 0.0046 −0.1760± 0.0026Alat - - −0.0559± 0.0034 −0.0357± 0.0019Ba - - +0.5838± 0.0290 +0.7179± 0.0164Blon - - +0.0741± 0.0044 +0.0883± 0.0024Blat - - −0.0868± 0.0067 −0.1151± 0.0030

peaks of the fast oscillations in the time domain. The yaw rate dynamics under predict the frequencycontent between 1 Hz and 3.5 Hz, which under predict the time domain peaks, but fits the data well. Therotor speed dynamics under predict the high frequency content, but match the data well with only a singleregressor.

The output error model fits are shown in Figure 7. As the hover model is unstable, the parameteroptimization was performed on the data in 11 second intervals to keep the model outputs from divergingfrom the measurements. Equation error parameter estimates were used as initial guesses for the first interval,after which parameter estimates and Cramer-Rao bound matrices were used recursively as initial estimates insubsequent data intervals. Initial estimates for the rotor flapping derivatives were taken from Conroy et al.,17

as these could not be predicted with equation error. The numerical optimization routine typically convergedwithin 14 iterations on the 11 second intervals in the time domain analysis, and required 93 iterations toconverge in the frequency domain analysis. The fit results shown in Figure 7(a) are for a representativeinterval of flight data, and are representative of other data intervals. Again the time domain results aredecimated to 35 Hz for clarity. Generally the model outputs match the measurements well. Lateral velocityestimates slightly under predict the response and heave velocity estimates slightly over predict the response.Frequency content around 2.5 Hz is under predicted in the roll rate, but the time domain data match verywell. Frequency content in the pitch rate match well, but the time domain data has errors. To improve thefit to the roll and pitch rate measurements, the model structure of Equation 13 was modified as per Mettler6

to include more parameters and to separate the rotor and stabilizer dynamics, but these augmentations wereabandoned as model accuracy did not significantly improve. Model outputs of the longitudinal velocity, yawrate, and rotor speed match very well in the time and frequency domains.

Generally the estimates for the parameters and errors are reasonable. For each method, the time domain

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Figure 6. Measured and predicted state derivatives using the equation error method.

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Figure 7. Measured and predicted outputs using the output error method.

results are for the most part consistent with the 95% confidence bounds of the output error results. However,the equation error results have some inconsistencies with the output error results. Errors bounds are similar,however the frequency domain analysis estimated smaller error bounds than the time domain analysis. Sinceequation error could only predict a subset of the parameters and because the estimates were often differentfrom the output error estimates, only the parameters estimated with the output error method are consideredfor the remainder of this study. The two output error models predict different values for the derivatives

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Lv and Lb, which primarily influence the modeled roll rate dynamics. The differences in the Zw, Nr, andTΩ derivatives most greatly influence the differences between the heave velocity, yaw rate, and rotor speedoutputs. Estimates for the yawing moment derivatives varied and had large standard errors due to thecollinearity in the data introduced by the feedback and feedforward terms. The aerodynamic damping termsXu, Yv, Zw, Nr and TΩ all have the correct sign, indicating that they stabilize the respective responses. Thespeed derivatives Lv and Mu have the correct sign and exhibit the expected stabilizing dihedral effect. Thedirectional speed derivative Nv produces an unexpected destabilizing effect, unlike the nominal stabilizingweathervane behavior created by the tail rotor and tail fin. This parameter has a large standard error, whichagain is attributed to the data collinearity in the yaw dynamics.

The modal characteristics of the models identified using output error in the time and frequency domainsare described by the Eigenvalues, given in Table 3, and the Eigenvectors, illustrated in Figure 8. TheEigenvectors are normalized and the significant directions in the state space are graphed on a polar plot foreach mode. While directions are similar between the time domain and frequency domain models, resultsin Figure 8 are shown only for the time domain model. Generally the poles match well between the timedomain and the frequency domain analyses. The first mode is the lateral flapping mode, which is a fast andlightly damped oscillatory mode. The main contributor to this mode is the roll rate, although the bank angleand the pitch rate are also present. The second mode is the longitudinal flapping mode, which is a sloweroscillatory mode with more damping. The main contributors are the pitch and roll rates, although the pitchangle is also present. The yaw subsidence mode is a fast first order mode consisting of the yaw rate and asmall portion in the direction of the heave velocity. The fourth and fifth modes are the unstable and stablephugoid modes, respectively. These are slower oscillatory modes, where the translational velocities, roll andpitch rates, and roll and pitch angles are involved. The sixth mode is the first order heave subsidence mode,which combines the heave motion with yaw rate. The seventh mode is the first order rotor speed subsidencemode, which combines the rotor speed with the heave velocity.

Table 3. Eigenvalues and modal characteristics.

Mode Time OE Frequency OE# Description Eigenvalue ζ ω (rad/s) Eigenvalue ζ ω (rad/s)1 lateral flapping −3.94± 58.3j 0.07 58.4 −3.84± 47.1j 0.08 47.32 longitudinal flapping −3.80± 16.9j 0.22 17.3 −3.42± 16.7j 0.20 17.13 yaw subsidence -14.7 1.00 14.7 -11.2 1.00 -11.24 unstable phugoid +0.05± 1.27j - 1.27 +0.15± 1.21 - 1.225 stable phugoid −1.00± 1.31j 0.61 1.65 −1.20± 1.26 0.69 1.746 heave subsidence -2.32 1.00 2.32 -2.31 1.00 2.317 rotor speed subsidence -2.96 1.00 2.96 -2.92 1.00 2.92

Although a simpler model structure was used, similar modal characteristics were found by Conroy etal.17 The pole locations for the output error models and the model identified by Conroy et al. are displayedin Figure 9. In general the dynamics identified using maximum likelihood methods were faster than thoseidentified with the frequency response method. Specifically, the lateral flapping mode was less damped andoccurred at a lower frequency, while the longitudinal flapping mode was more damped and also occurred ata lower frequency. The yaw subsidence mode was considerably slower. The phugoid modes had a similardamping values, but the ML results occurred at higher frequencies. The heave subsidence and rotor speedsubsidence modes were very similar.

IV.C. Model Validation

To validate the predictive capability of the models a second flight test about hovering flight was conducted.Magnitudes for the measurements and pilot inputs are of approximately the same magnitude as those usedin the identification data. Three portions of the flight test are analyzed. Figure 10 shows the primaryresponses due to longitudinal cyclic excitation, which are predicted well by the models. The ML modelstend to follow the means of the bank angle and roll rate while the frequency response model over predictstheir amplitudes. The pitch angle, translational velocity, and pitch rate are the primary dynamics involvedand are captured very well. Figure 11 shows the main responses to a lateral cyclic input. All of the models

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Figure 8. Graphical representation of Eigenvectors.

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match the roll and pitch rates well, although again the frequency response method tends to over predict theamplitudes. While the models capture the orientation variables fairly well, the velocity variables are underpredicted, but in a similar manner between the models. Figure 12 shows the responses to excitation usingthe throttle and directional controls. All models capture the rotor speed measurement very well. The yawrate is generally matched well. The ML models capture the high frequency dynamics, but under predictedthe larger amplitudes. The frequency response model does not capture the high frequency responses, and asa result over predicted the responses at some points. The lateral velocity is captured fairly well by the MLmodels, but the frequency response model becomes unstable. The heave measurements were also capturedbetter by the ML models. Overall both models predicted well, although the more complex structure of themodel presented in this work allows for a better match to the flight data.

V. Conclusions and Future Work

This work presents the system identification of a linearized model describing the flight dynamics of aminiature electric helicopter in hovering flight. First principles, previous work, and statistical metrics appliedto flight data are used to postulate a model structure. Equation error and output error simplifications of

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the maximum likelihood estimator are used to estimate model parameters and their standard errors in boththe time and frequency domains. Equation error could not estimate all of the model parameters because theflapping angles were not measured. Parameter estimation in the time and frequency domains using outputerror resulted in two models, which have good prediction capability and are expected to be sufficient for usewith future observer and controller designs.

The good prediction capability of the models and the general consistency of the parameter estimatesis evidence that the modeling and identification process was a success. Furthermore the similarity in themodal characteristics identified to those found by Conroy et al.17 suggest that both works accounted forthe important dynamics exhibited by the helicopter. These two works also collectively show that the modelequations and simplifying assumptions used on larger rotary-wing aircraft are also appropriate for miniatureelectric helicopters. While models identified for larger aircraft can typically obtain higher coefficients ofdetermination,23 the relatively low model fits found in this study, which are attributed to unmodeled higherorder dynamics, were sufficient for capturing the dominant response characteristics. Filtering the measure-ments to a lower frequency would increase the fit values. It has been shown for a large transport aircraft thatgiven a sufficient amount of data and a fixed model structure, maximum likelihood and frequency responsemethods produce similar results;13 it would be interesting to similarly compare numerical results using thetwo identification methods for the miniature electric helicopter. While the maximum likelihood models pre-dict better, this result may be due only to the differences in the model structure. When H∞ and µ-synthesiscontrollers are used in the future, it is not unexpected that similar performance will be achieved since thereis inherent robustness in the control algorithms.

It is recommended that a variety of methods be used when performing system identification, as somemethods have particular strengths over others and the collective consistency of the estimate ensembles addsconfidence to the model. Equation error estimates required little computation time and were useful instepwise regression, where the model fit had to be continuously changed and evaluated. However equationerror could not estimate the parameters in the rotor dynamics, whereas the output error method couldestimate these parameters but required much more computation time. Analysis in the frequency domainwas much easier than in the time domain, as computations were more efficient and standard errors didnot require corrections for colored residuals. Additionally it is generally expected that frequency domainresults are superior, as the optimization routine weights the modal components heavier, whereas in the timedomain analysis the large amplitudes, which may represent nonlinear dynamics, are weighted heavier. While

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more intuitive, time domain analysis required much more computation time, as the entire measurementtime history was used and standard errors were corrected. Implementation of the output error method inthe time domain was particularly time consuming because the optimization was recursively performed ondata segments which were long enough to exhibit dynamical responses, but short enough to keep the modeloutputs from diverging from the measurements.

With the exception of the input vector, all signals used for the identification work in this paper werederived from the visual tracking system outputs, which consisted of least-squares fits for the position andorientation of the body frame, as well as the rotor azimuth angle. This choice simplified the data reductionprocess because the data did not require corrections for position, alignment, and calibration errors; theacceleration of gravity was automatically removed; the measurements were of higher quality and requiredless filtering; and kinematic compatibility was automatically enforced. The visual tracking system offersa non-intrusive method for providing high-fidelity measurements for systems which fly indoors, lack thepayload capacity for a full suite of avionics, or would otherwise require the reconstruction of measurements.Work has recently been performed with this system for the identification of samara flight dynamics31 andornithopter aerodynamics.32,33 Future work is planned for the identification of smaller vehicles and thevalidation of state estimation and sensor fusion algorithms using onboard avionics.

While good results were achieved, several portions of the system identification process could be refinedfor better results. The model fits of the lateral variables are in general not as good as the fits of thelongitudinal variables. As these dynamics have good excitation in the flight data, errors are attributed toeither environmental disturbances or a model deficiency. The lateral inertia is less than the longitudinalinertia, and so there is a heightened sensitivity to environmental disturbances and noise in the control loops.It would be interesting to include the nonlinear gravitational, Coriolis, and centripetal dynamics as in Bruceet al.,20 and measure the main rotor flapping angles and observe the effect on the model fidelity and theflight envelope for which the model predicts well. Additionally it would be interesting to see if the modelstructure for the roll rate dynamics could be improved to match the fidelity seen by the pitch rate modelstructure. The heave dynamics could also be improved to account for ground effect, since this conditionwould most likely be experienced during the conceived mission profiles. Alternatively the model could beidentified from flight data where the helicopter is not in ground effect, but this increases the pilot work loadand significantly limits the volume of the flight arena and hinders the identification of the low frequencydynamics. The fidelity of the yaw dynamics could also be improved. Feedback and feedforward terms weredeemed necessary to pilot maneuvers, and so either the mixing could be incorporated into the model as perMettler,6 or multivariate orthogonal functions23 could be used.

VI. Acknowledgements

The authors would like to thank the NASA Langley Research Center, the National Institute of Aerospace,and the University of Maryland for their support in this research. Flight tests were conducted at theUniversity of Maryland in the Autonomous Vehicle Laboratory. Many conversations with Eugene Morelliat the NASA Langley Research Center are acknowledged and greatly appreciated. Additionally the authorswould like to thank the members of the Morpheus Laboratory and Autonomous Vehicle Laboratory for theircontinued support and guidance.

References

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Journal of Guidance, Control, and Dynamics, Vol. 19, No. 2, March-April 1996.5Theodore, C., Tischler, M., and Colbourne, J., “Rapid Frequnecy-Domain Modeling Methods for Unmanned Aerial

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ed., 2006.8Tischler, M. and Cauffman, M., “Comprehensive Identification from Frequency Responses, Vol.1 - Class Notes,” No.

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9Tischler, M. and Cauffman, M., “Comprehensive Identification from Frequency Responses, Vol.2 - User’s Manual,” No.10150, NASA CP, September 1994.

10Tischler, M. and Remple, R., Aircraft and Rotorcraft System Identification: Engineering Methods with Flight TestExamples, AIAA Education Series, American Institute of Aeronautics and Astronautics, 2006.

11Tischler, M. and Cauffman, M., “Frequency-Response Method for Rotorcraft System Identification: Flight Applicationsto BO-105 Coupled Rotor/Fuselage Dynamics,” Journal of the American Helicopter Society, Vol. 37, No. 3, 1992, pp. 3–17.

12Tischler, M., Leung, J., and Dugan, D., “Frequency-Domain Identification of XV-15 Tilt-Rotor Aircraft Dynamics inHovering Flight,” No. 83-2695 in AIAA 2nd Flight Testing Conference, American Institute of Aeronautics and Astronautics,Las Vegas, Nov 1983.

13Morelli, E., “Low-Order Equivalent System Identification for the Tu-144LL Supersonic Transport Aircraft,” Journal ofGuidance, Control, and Dynamics, Vol. 26, No. 2, March-April 2003, pp. 354–362.

14Mettler, B., Tischler, M., and Kanade, T., “System Identification of Small-Size Unmanned Helicopter Dynamics,” 55th

Forum of the American Helicopter Society, American Helicopter Society, Montreal, Canada, May 1999.15Mettler, B., Tischler, M., and Kanade, T., “System Identification Modeling of a Small-Scale Unmanned Rotorcraft for

Control Design,” Journal of the American Helicopter Society, Vol. 47, No. 1, January 2002, pp. 50–63.16Conroy, J. and Pines, D., “System Identification of a Miniature Electric Helicopter using MEMS Inertial, Optic Flow,

and Sonar Sensing,” Forum of the American Helicopter Society, American Helicopter Society, Virginia Beach, VA, May 2007.17Conroy, J., Humbert, S., and Pines, D., “System Identification of a Rotary Wing Micro Air Vehicle,” Journal of the

American Helicopter Society, 2008, under review.18Ljung, L., System Identification: Theory for the User , Information and Sciences Series, Prentice Hall, 2nd ed., 1999.19Hamel, P. and Jategaonkar, R., “Evolution of Flight Vehicle System Identification,” Journal of Aircraft , Vol. 33, No. 1,

January-February 1996, pp. 9–28.20Bruce, P., Silva, J., and Kellett, M., “Maximum Likelihood Identification of a Rotary-Wing RPV Simulation Model from

Flight-Test Data,” No. AIAA-98-4157 in AIAA Atmospheric Flight Mechanics Conference and Exhibit, American Institute forAeronautics and Astronautics, Boston, MA, 10-12 August 1998.

21“MATLAB,” www.themathworks.com.22Morelli, E., “System Identification Programs for Aircraft (SIDPAC),” No. AIAA-2002-4704 in AIAA Atmospheric Flight

Mechanics Conference and Exhibit, American Institute of Aeronautics and Astronautics, Monterey, California, 6-8 August 2002.23Klein, V. and Morelli, E., Aircraft System Idenfication: Theory and Practice, AIAA Education Series, American Institute

of Aeronautics and Astronautics, 2006.24Morelli, E. and Klein, V., “Accuracy of Aerodynamic Model Parameters Estimated from Flight Test Data,” Journal of

Guidance, Control, and Dynamics, Vol. 20, No. 1, 1997, pp. 74–80.25Maine, R., “The Theory and Practice of Estimating the Accuracy of Dynamic Flight-Determined Coefficients,” Tech.

Rep. NASA RP 1077, National Aeronautics and Space Administration, July 1981.26Conroy, J. and Pines, D., “Development of a Micro Air Vehicle Avionics Package for System Identification and Vehicle

Control Applications,” Unmanned Rotorcraft Specialist’s Meeting, American Helicopter Society, Chandler, AZ, 23-25 January2007.

27“Vicon Motion Systems,” www.vicon.com.28Morelli, E., “Practical Aspects of the Equation-Error Method for Aircraft Parameter Estimation,” No. AIAA-2006-6144 in

AIAA Atmospheric Flight Mechanics Conference and Exhibit, American Institute for Aeronautics and Astronautics, Keystone,Colorado, 21-24 2006.

29McRuer, D., Ashkenas, I., and Graham, D., Aircraft Dynamics and Automatic Control , Princeton University Press, 1973.30Morelli, E., “High Accuracy Evaluation of the Finite Fourier Transform using Sampled Data,” Tech. Rep. NASA TM

110340, National Aeronautics and Space Administration, 1997.31Ulrich, E. and Pines, D., “Planform Geometric Variation, and its Effect on the Autorotation Efficiency of a Mechanical

Samara,” 64th annual forum, American Helicopter Society, Montreal Canada, April 29 - May 1 2008.32Sitaraman, J., Roget, B., Harmon, R., Grauer, J., Conroy, J., Hubbard, J., and Humbert, S., “A Computational Study

of Flexible Wing Ornithopter Flight,” 26th AIAA Applied Aerodynamics Conference, American Institute of Aeronautics andAstronautics, Honolulu, Hawaii, 18-21 August 2008.

33Harmon, R., Grauer, J., Hubbard, J., and Humbert, S., “Experimental Determination of Ornithopter Membrane WingShapes Used for Simple Aerodynamic Modeling,” 26th AIAA Applied Aerodynamics Conference, American Institute of Aero-nautics and Astronautics, Honolulu, Hawaii, 18-21 August 2008.

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