International Journal of Scientific & Engineering Research, Volume 5, Issue 2, February ISSN

Transcription

1 International Jornal of Scientific & Engineering Research, Volme 5, Isse, Febrary ISSN Qadrotor Comprehensive Identification from Freqency Responses Abbakar Srajo Imam, Robert Bicker Abstract Design and development of a qadrotor model-based flight control system entails the se of the vehicle's dynamic model. It is qite challenging to se the physical laws and first principle-based approaches to model the qadrotor dynamics as they are highly nonlinear, characterized by copled rotor-airframe interaction. However, system identification modeling method provides a less challenging approach to modeling the dynamics of highly non-linear systems sch as a qadrotor. This paper presents the freqency-domain system identification procedre for the extraction of linear models that correspond to the hover flight operating conditions of a qadrotor. Freqency response identification is a versatile procedre for rapidly and efficiently extracting accrate dynamic models of aerial vehicles from the measred response to control inpts. Dring the extraction of the qadrotor's model, flight test manoevres were sed to excite the variables of concern for flight dynamics and control by adopting a systematic selection procedre of the model strctre for the parameterized transfer-fnction model and the state-space model. The techniqe provides models that best characterized the vehicle's measred responses to the controls commands, and can be sed in the design of a flight control system. Index Terms Dynamic model, flight control system, freqency-domain system identification, flight dynamic and control, excitation and measred responses, qadrotor. INTRODUCTION Recently, the se of small-scale rotorcraft nmanned aerial vehicles (UAVs) for srveillance and monitoring tasks is becoming collected throgh identification method in [], after which two control laws were designed for the vehicle attitde stabilization. attractive. Amongst In [], system identification method was applied to the varios configrations of the small-scale rotorcraft, the se of a qadrotor gained more examine a high-bandwidth rotorcraft flight control system prominence, particlarly in the research commnity [], [], design. In the stdy, flight test and modeling reqirements [3], [4], [5]. A qadrotor is a small responsive for-rotor vehicle controlled by the rotational speed of its rotors. It is com- rotor helicopter. A systematic way is adopted in this stdy to were illstrated sing flight test data from a BO-5 hingeless pact in design with the ability to carry a high payload. derive a qadrotor dynamics models sing the freqencydomain system identification method. Once the models are The dynamics of rotorcraft is sbstantially more complex determined, a single-inpt-single-otpt (SISO) and mltipleinpt-mltiple-otpt than that of a fixed-wing aircraft [6], the complexity increases as the vehicle become smaller. The high non-linear natre of a (MIMO) control loops can be designed and implemented on a qadrotor qadrotor makes difficlt the se of physical law and first principle-based approach to model its dynamics. The qadrotor dynamics is characterized by the copled rotor-airframe dynamics. Hence, system identification method is needed to SYSTEM IDENTIFICATION CONCEPT System identification is the procedre for deriving a model the dynamics of non-linear systems sch as a qadrotor, and the procedre is condcted in either time or fre- mathematical model of a system based on experimental data of the system s control inpts and measred otpts. qency domain. A nmber of stdies have reported the se of system identification procedre to identify the dynamics of The procedre involves derivation of a mathematical model rotorcraft [7], [8], [9], []. For instance, a method for system identification sing Neral Networks was proposed in [7], based on experimental data of the vehicle's control in- pts and measred otpts; it also provides an excellent where inpt-otpt data was provided from nonlinear simlation of X-Cell 6 small-scale helicopter, and the data was sed tool for improving mathematical models sed for rotorcraft to train the mlti-layer perceptron combined with NNARXM flight control system design. System identification method time regression inpt vector to learn nonlinear behavior of the can be sed for derivation of both parametric and nonparametric vehicle. A rotorcraft system response data was acqired in models: examples of nonparametric models in- careflly devised experiment procedre in [8], and a time domain system identification method was applied in extracting a clde implse and freqency response models, and examples of parametric models are transfer fnction and state linear time-invariant system model. The acqired model was sed to design a feedback controller consisting of inner-loop space models. The nonparametric models are directly derived attitde feedback controller, mid-loop velocity feedback controller sing experimental data and provide an inpt and oter-loop position controller, when implemented otpt (I/O) description of the system. These model types on the Berkeley RUAV, the controllers showed remarkable are based on collections of data and do not reqire any hovering performance. Similarly, parametric and nonparametric models for a rotorcraft were identified sing data knowledge of the system strctre. However, the challenge 4

2 International Jornal of Scientific & Engineering Research, Volme 5, Isse, Febrary ISSN of the system identification procedre is to derive a parametric model of a system. The first step towards the extraction of a parametric model is the derivation of a parameterized model, which will serve as a logical gess of the actal system model. The se of an optimization algorithm determines the parameters of the model that minimize the error between the actal system responses and the model responses. Estimates of those characteristics may be obtained by analysis of the nonparametric model combined with information obtained by the first principles approach. The system identification procedre is an iterative process. Depending on the identification reslts, the parameterized model may be refined in terms of order and strctre ntil a satisfactory identification error is achieved. When the parameterized model is known, the system identification method redces to the parameter estimation problem []. A Key application of rotorcraft system identification reslts inclde piloted simlation models, comparison of wind tnnel test verss flight measrements, validation and improvement of physic based simlation models, flight control system development and validation There are nmeros methodologies for system identification techniqes which are well described in [3], [4]. A major classification amongst these methodologies depends on whether the compared responses are considered in the time or freqency domain. The similarities between freqency-domain and time-domain methods are; in both, good reslts depend on proper excitation of key dynamic modes; mltiple inpts shold not be flly correlated; both can be sed for parametric model identification that can be verified in the time domain. The major differences between the two are; the initial data for freqency-domain method consists of freqency responses derived from time-history data, while the time-domain method initial data consists of time history data. In addition, time domain method provides both linear and nonlinear models, whereas freqency domain method provides only a linearized characterization of the system, and a describing fnction for a nonlinear model. There exist none independent metrics to assess system excitation and linearity in time-domain method, whilst, in freqency-domain method there are a nmber of metrics sch as coherence fnction, Cramer-Rao ineqality and cost fnction. control inpts. Flight test manoevres are sed to excite the variables of concern for flight dynamics and control, or strctral stability. Typical excitations sed in system identification are freqency sweeps and doblets. The techniqes provide a model that best characterizes the vehicle's measred responses to controls commands [5], sch as (i) freqency response model, (ii) transfer fnction model and (iii) state space model. 3. Freqency response model This is a nonparametric model which represents the otpt/inpt amplitde ratio and phase shift of a system in an effective format, sch as a Bode plot. It can be regarded as a data crve identified from the flight test data, which represent the ratio of the response per nit of control inpt as a fnction of control inpt freqency. The freqency response is obtained sing the fast Forier transform and associated windowing techniqes. 3. Transfer fnction model This model provides a closed-form eqation that is a good representation of a freqency response data. The model is of the form: m m τs ( bs + bs bm ) e T(s) = () n n s _ as an The vales of the nmerator coefficients (b, b,... bm) and denominator coefficients (a...an) are determined sing system identification procedre. The transfer fnction is a parametric model comprising a limited set of characteristics parameters. 3.3 State-space model This can is the parametric model of the complete differential eqation of motion that describes the MIMO behavior of the vehicle. Eqation () represents the linear differential eqation for a small pertrbation abot a trim flight condition in statespace.. x () = Ax + B( t τ ) Where the control vector is composed of the control inpts and the vector of the vehicle states x comprises the response qantities (speed, anglar rates, and attitdes angles). The time-delay vector τ allows for a separate time-delay vale for each control. However, the set of available flight-test measrement y is composed of a sbset of the states and given by: y = Cx + D( t τ ) (3) The vales of the matrices A, B, C, D and the vector τ are determined sing the system identifications procedre. 3 ROTORCRAFT SYSTEM IDENTIFICATION System identification as applied to a qadrotor is a versatile procedre for rapidly and efficiently extracting accrate dynamic models of rotorcraft from the measred response to 4 4 FREQUENCY RESPONSE SYSTEM IDENTIFICATION Based on [] and [5] freqency domain identification is an ideal way for extracting linear rotorcraft models of high accracy. One of the main advantages of this approach is

3 International Jornal of Scientific & Engineering Research, Volme 5, Isse, Febrary ISSN METRICS FOR DECIDING MODEL ACCURACY the se of actal flight data for deriving and validating the model. Additionally, freqency domain identification has a coherent flow of the design steps starting from the inpt otpt characterization of the vehicle (nonparametric modeling), contining with the extraction of the state space model (parametric modeling) conclding with validating the predicted model in the time domain. This method is classified as an otpt-error method where the fitting error is defined between the actal flight data freqency responses and the freqency responses predicted by the model. To highlight this, sppose the vehicle is excited with a sine-wave inpt x(t) of amplitde A and freqency f in hertz, then: x( t) Asin(πft) = (4) When the transient response has decayed, the system otpt y(t) will also be a sine wave of the same freqency f, bt with an associated amplitde B and a phase shift ϕ : G yy ( f ) = Y ( f ) (9) T y ( t) = B sin(π ft + ϕ) (5) Finally, a rogh estimated of the cross spectrm or cross PSD displays the distribtion of the prodct of inpt mltiplied by This implies that for a linear time-invariant (LTI) system, otpt or inpt-otpt power transfer as a fnction of freqency, and is given by: a constant sine-wave periodic inpt reslts in a constant sine-wave otpt at the same freqency f, referred to as the first harmonic freqency. It is therefore important to note, G yy ( f ) = Y ( f ) * Y ( f ) () for linear systems, the higher harmonics of the response are T not considered, as the time fnction is the same for the inpt and otpt. For these systems, the focs is on the amplitde A and B, phase shiftϕ. The parameter vales A, Note that, the cross spectrm is a complex-valed fnction and ths conveys inpt=otpt phase information. B and ϕ in this case, can be obtained from the time-history plots or calclated nmerically sing the Forier series for only the first harmonic terms [6]. The freqency response fnction H(f) is a complexvaled fnction defined by the data crves for the magnification and phase shift at each freqency f given by: B( f ) H ( f ) = (6) A( f ) H ( f ) = Phase Shift = ϕ( f ) < (7) The freqency response can be obtained experimentally by exciting the system with discrete sine-wave inpts. 4 In the freqency response system identification techniqe, the following independent metrics are sed to measre model accracy in terms of system excitation, data qality and system response linearity: 4. Spectral Fnction The prodcts of the Forier transform comptation are the Forier coefficients of the inpt X(f) excitation and otpt Y(f) response. This leads to the definition of the three spectral fnctions (i) inpt spectrm, (ii) otpt spectrm and (iii) cross spectrm. Based on [7], a rogh estimate of the inpt atospectrm is determined from the Forier coefficients by: G xx ( f ) = X ( f ) (8) T Similarly, a rogh estimate of the otpt ato-spectrm or inpt PSD displays the distribtion of the otpt sqared or response power as fnction of freqency given by: 4. Coherence fnction The coherence fnction is an important prodct of the smooth spectral fnctions. It can be interpreted as the fraction of the otpt spectrm that is linearly attribtable to the inpt spectrm at a certain freqency. ( f ) xy = G xx G xy ( f ) ( f ) G yy γ () ( f ) The coherence fnction is a normalized metric having a vale ranging from zero to nity. It is an indicator of the linearity between the inpt and otpt. A vale of the coherence fnction close to nity indicates that the otpt is significantly linearly correlated with the inpt of the system. In practical applications, there are several reasons for a low vale of the coherence fnction [8]. Following a simple gide, the coherence fnction can be sed to effectively and rapidly examine the accracy of freqency-response identification []. Gener-

4 International Jornal of Scientific & Engineering Research, Volme 5, Isse, Febrary ISSN ally, if the coherence fnction satisfies Eqation (), and is not oscillating, then the freqency response can be said to have acceptable accracy [5]. γ xy.6 () After some preprocessing to eliminate the noise and other types of inconsistencies in the time domain otpt data, the second phase comptes the inpt otpt freqency responses sing a Fast Forier Transform. This phase of the process establishes the nonparametric model of the vehicle. The design of the parameterized linear state space model follows sing information from the physical laws and the nonparametric modeling phase. 4.3 Craner-Rao ineqality The Cramer-Rao ineqality is another reliable measre of parameter accracy in the freqency-response identification method. The ineqality establishes the Cramer-Rao bonds (CR) as the minimm expected standard deviation in a parameter estimate obtained from many repeated manoevres. The Cramer-Rao bond is given by: σ CR (3) Relative vales of the Cramer-Rao bonds associated with the identification parameters are of key significance for refining the model strctre. High vales of Cramer-Rao bonds for individal parameter sggest indicate poorly identified parameters and sggest these parameters to be removed or fixed in the model strctre. Fig.. Flowchat for freqency response system identification. 4.4 Cost fnction The qadratic factor J referred to as cost fnction is also sefl in deciding an acceptable level of model accracy. For flight dynamic modeling, a cost fnction of J, generally represents an acceptable level of accracy, whereas a cost fnction of J 5 is expected to prodce an exact match of the flight data. After some preprocessing to eliminate the noise and other types of inconsistencies in the time domain otpt data, the second phase comptes the inpt otpt freqency responses sing a Fast Forier Transform. This phase of the process establishes the nonparametric model of the vehicle. The design of the parameterized linear state space model follows sing information from the physical laws and the nonparametric modeling phase. 5 FREQUENCY RESPONSE SYSTEM IDENTIFICATION PROCEDURE Fig. illstrates the seqence of a freqency response identification procedre, in which the initial step is the excitation of the vehicle sing specially designed inpt signals, sch as a freqency sweep, to excite the vehicle dynamics over a desired freqency range. The choice of the desired freqency range has an important role in the identification process and has to be wide enogh in order to captre all the dynamic effects of interest (i.e., airframe and rotor dynamics). After some preprocessing to eliminate the noise and other types of inconsistencies in the time domain otpt data, the second phase comptes the inpt otpt freqency responses sing a Fast Forier Transform. This phase of the process establishes the nonparametric model of the vehicle. The design of the parameterized linear state space model follows sing information from the physical laws and the nonparametric modeling phase. The freqency domain identification method is only sitable for the derivation of a linear parametric model. Althogh the rotorcraft dynamics are nonlinear, arond certain trimmed flight conditions, the nonlinearities from the eqations of motion and aerodynamics are relatively mild. When this is the case, a linearized model is sfficient to accrately predict the vehicle's response. Usally, the validity of the linearized model is adeqate over a wide area of the flight envelope arond the trim point. However, a single linear model in most cases is not sfficient to represent globally the flight envelope. Therefore, different models are reqired for each operating condition. The final step of the identification procedre is the validation of the model. This step takes place in the time domain, with different flight data from the identification procedre. For the same inpt seqence, the vehicle responses from the flight data are compared with the predicted vales of the model, obtained by integration of the state space model. However, if the validation portion of the problem is not satisfactory the parametric modeling setp shold be modified and the procedre repeated. 4 6 SOFTWARE FOR FREQUENCY RESPONSE METHODS A nmber of software packages can be sed for rotorcraft freqency-response identification. Amongst the poplar

5 International Jornal of Scientific & Engineering Research, Volme 5, Isse, Febrary ISSN ones inclde: MATLAB/SIMULINK, NI Labview and CIFER (Comprehensive Identification from Freqency Responses). MATLAB and LabView are generalized packages which for solving varios engineering problems. However, CIFER software package was developed at Ames research centre primarily for the task of aircraft and rotorcraft freqency response identification from flight-test data, and hence it is well sited for application in this stdy. The program is composed of six tility packages that interact with a sophisticated database of freqency responses. The importance of a well organized and flexible database system is very crcial in a large scale MIMO identification procedre of an air vehicle. The CIFER package is designed to cover all the intermediate steps necessary for the development of air vehicles parametric modeling. The key characteristic of CIFER is its ability to generate and analyse high qality freqency responses for MIMO systems, by sing Discrete Forier Transform (DFT) and windowing algorithms [9]. Fig.. CIFERs software and database components. 6. Data flow in CIFER Fig. depicts how the CIFER software package is linked to a relational database tility to facilitate the comptation of the freqency response identification method of Fig. 3. The six core programs within the CIFER perform the following processes: data conditioning and performing FFTs, FRESPID; mlti-inpt conditioning, MISOSA; window combination, COMPOSITE; transfer-fnction model identification, NAVFIT; state-space model identification, DERIVID and model verification, VERIFY. The package also has tilities that allow interfacing to many standard data formats, inclding MATLAB, Excel, ASCII comma and tab delimited, among others. Freqency responses generated dring any session of the system 4 identification procedre are stored and cataloged in a dedicated database folder. The database entry that is available to all CIFER programs and tilities contains all the information abot how the procedre was carried ot. In addition, the database can be shared by mltiple sers of CIFER and mltiple databases can be combined or compressed. 6. Overview of CIFER package The CIFER software facilitates the se of freqency domain analysis of flight data to achieve a nmber of objectives, inclding handling qality analysis and specification compli- Fig. 3. CIFER software components. ance, vibration analysis, and identification of linearized models. The package contains varios tilities that can be sed As depicted in Fig 3, time-history data enters the system at interactively as shown in Fig.. two points. The first set of time-history is processed into freqency responses at the beginning of the CIFER procedre. At the end of the procedre, a second set of time-history data from inpts dissimilar from those sed for the identification is then sed for the model verification. Three programs are rn seqentially to generate the MIMO freqency-response database. Beginning with the time-history data, FREPID comptes SISO freqency response for a range of spectral windows sing a chirp z-transform. The reslts are then written into the freqency-response database. Next, MISOSA reads in the SISO data from the freqency-response database and conditions these responses for the effect of mltiple, partially correlated controls that might have been present in the same manoevre record. Again, the reslts are written back to the database. Finally, COMPOSITE performs optimization across the mltiple spectral windows to achieve a final freqency-response database with excellent resoltion, broad bandwidth and low random error. Two programs spport parametric model identification. NAVFIT is sed to identify a pole-zero transfer-fnction model that best fits a selected SISO freqency-response. DERIVID is sed to identify a complete generic state-space model strctre that best fits the MIMO freqency-response database. Lastly, VERIFY is sed to check the identified model's timedomain response based on time-history data from manoevres different from those sed for the identification. An important tility program is the smoothing from aircraft

6 International Jornal of Scientific & Engineering Research, Volme 5, Isse, Febrary-4 8 ISSN kinematics (SMACK), althogh not part of CIFER software package, is sed for processing of time-history data before the identification proper. Table smmarizes the fnctions of the CIFER components. Table Smmary of the CIFER components fnctions Serial Component Fnction Time history data Inpt to FRESPID and SMACK SMACK Data consistency 3 FRESPID Freqency-response identification 4 MISOSA Mlti-inpt conditioning 5 COMPOSITE Window combination 6 NAVFIT Transfer-fnction identification 7 DERIVID State-space model identification 8 VERIFY State-space model verification 7 QUADROTOR IDENTIFICATION This section presents the general reqirements and procedre that leads to the comprehensive freqency response system identification of a qadrotor sing the CIFER software package. The description of the vehicle platform sed has been presented in [5]. 7. Model strctre determination An important and challenging aspect in parametric model identification process is the proper selection of the different aspects of model strctre that depends on many factors, critical amongst inclde (i) the ltimate application of the model; (ii) selection of the inpt-to-otpt variable pair; (iii) selection of freqency range of the fit; (iv) selection of the order of nmerator (n) and denominator (m); (v) inclsion of the eqivalent time delay τ and (vi) fixing or freeing specific coefficients in the fitting process. 7. Parameterized transfer fnction model A transfer-fnction model is the linear inpt-to-otpt description of a dynamic system; it represents the simplest form of parametric model that can be extracted from the nmerical freqencyresponse database. The transfer-fnction model is sfficient for describing the majority of the qadrotor dynamics, inclding handling qality analysis, rotors and airframe models and flight control system design. In transfer-fnction modeling, the system to be modeled is treated like a black box with no attempt to represent the actal dynamics of the vehicle. Transfer-fnction models are composed of a nmerator and denominator polynomial in the Laplace variable s, with an eqivalent time delay to accont for additional nmodeled high freqency dynamics and transport delays in the system. However, despite this simplification, the transfer-fnction model can provide a remarkably accrate representation of system response behavior and has the form of Eqation (). The order of the nmerator and denominator orders are selected in sch that a good fit of the freqency-response data in the freqency range of interest is achieved Qadrotor state-space model Determination of the parameterized model is one of the critical aspects in the freqency domain identification method. The challenge here is deciding on which stability derivatives shold be inclded in the development of the parameterized model. To simplify the identification task, the linear parameterized model sed for parameter identification of the qadrotor is based on Mettler s model (with some modifications) described in [], [], [] for the Carnegie Mellon s Yamaha R-5 and MIT s X-Cell-6. The strctre of the parameterized model proposed by Mettler has already been sccessflly sed for the parametric identification of several helicopters of different sizes and specifications [3] and [4]. The ability of this model strctre to establish a generic soltion to the small-scale rotorcraft identification problem is based on two important factors: a) that the Mettler s parameterized model provides a physically meaningfl representation of the system dynamics. All stability derivatives inclded in this model are related to kinematic and aerodynamic effects of the airframe and the rotor systems, and b) the ability to represent the many cross-copling effects that dominate the rotorcraft motion. This allows for the integration of the rotors model with the linearized eqations of motion. The modifications made to the proposed parameterized model is de to the absence of a stabilizer bar on qadrotor, which provides additional damping to the pitch and roll rates. However, this fnction on the qadrotor is addressed by proportional reglation of the rotor speeds.. x = Fx + G( t τ ). y H x + H The qadrotor physical model strctre represents direct implementation of the eqations of motion of the vehicle. In a system identification procedre, the choice of model strctre depends on those points highlighted previosly. Hence, Eqations () and (3) can be written in the state-space form as: M (4) = (5) x The matrices M, F, G and vector τ contain model parameters to be identified as well as the known model parameters and constants. Time delays are sometimes inclded to accont for nmodeled system dynamics. A measrement vector y is inclded to accont for the difficlty associated with directly measring the state x. The matrices H and H are composed of known constants, sch as gravity, nit conversion, kinematics, etc. However, once the identification parameters are determined, Eqations (4) and (5) can easily be expressed in the conventional state-spaceform (Eqations and 3): A = M F (6) B = M G (7) C = H + HM F (8) D = H M G (9) 7.3 State and control variables There are nine states and for control variables in the qadrotor's 6-DOF eqations describing its airframe motion, centre of mass

7 International Jornal of Scientific & Engineering Research, Volme 5, Isse, Febrary-4 8 ISSN (CM) and body rotation, given by: x [ v w p q r φ θ ψ ] T = () ] T = [ 3 4 () Where v B = [, v, w] T and ω B =[p, q, r] T denote the linear and anglar velocities components of the vehicle relative to the bodyfixed frame. 7.4 Otpt vector Dring the vehicle's take-off, the otpt vector consists of the qadrotor's linear and anglar velocities, and a Z, the linear acceleration along z-axis. However, at hover flight, acceleration along the x and y axes eqals zero, and the vector is redced to: y ] T = [ v w p q r a z () 7.5 Parameterized state-space model The qadrotor parameterized state-space model represents the linearized dynamics of the pertrbed states and control inpts of the helicopter from a trimmed reference flight condition. The trim operating condition considered is the hover mode. Althogh the parameterized model is associated with the pertrbed vales of the states and inpts, the linear state-space parameterized model is given by: A = Lθ L φ g L ψ 4 = Y L N Y L N Y L L N 4 Y 4 Y 4 L 4 N 4 B (4) The nknown coefficient to be identified in matrices A and B of the parameterized model strctre are the conventional stability and control derivatives, which reslt from Taylorseries representation of the vehicle's aerodynamics, composed of stability and control derivatives of the vehicle, and are a complex combination of the vehicle geometric parameters, aerodynamic parameters and inertia parameters. These derivatives also represent the complex combination of the qadrotor geometric and inertial parameters. 8 THE IDENTIFICATION PROCESS The identification procedre for the qadrotor starts with the collection of the experimental time domain flight data. For each flight data record, the qadrotor was set to hover, and a piloted freqency sweep excitation signal was applied to the for-control variables (,, 3, 4) one after the other. The bandwidth of the excitation signal ranges between. rad/sec, whilst the freqency sweep was exected by the primary inpt of interest, the secondary inpts were kept ncorrelated to the main inpt maintaining the vehicle near the reference operating point. For each control inpt, for records have been collected; the minimm and Nθ maximm freqency of the excitation sweeps and the dration of N φ (3) the flight records for each control inpt are shown in Table. Nψ Y θ Table Yφ Smmary of the CIFER components fnctions Y Control channel ω ψ min (rad/sec) ω max (rad/sec). 3 4 The variables collected for the identification process were the Eler angles θ, ϕ, ψ; anglar velocities p, q, r and body frame acceleration a Z as well as the linear velocity w. For translation, the body frame accelerations were selected instead of the velocity measrements, as these provide a more symmetrical response arond the trim vale, facilitating the calclations of the respective FFTs. After the collection of the time domain experimental

8 International Jornal of Scientific & Engineering Research, Volme 5, Isse, Febrary-4 8 ISSN data, flight records excited by the same primary control inpt were concatenated into a single record. The time domain experimental data was then entered in the CIFER software and processed sing the PRESPID, MISOSA and COMPOSITE to prodce a high qality MIMO freqency response database. This database comprises the conditioned freqency responses and partial coherences for each inpt otpt pair. After calclating the flight data freqency responses, the parametric models were extracted sing the NAVFIT modle to determine the model transfer-fnction parameters. The DERIVID modle was sed to extract and determine the state-space model and its parameters. In both cases, the model parameters were determined sch that the estimated freqency responses provide best fits to the flight data freqency responses. The first task exected in the parametric modeling process was the determination of the flight data freqency response inptotpt pairs, to be inclded in the identification process, followed by the determination of the freqency range of interest. For the qadrotor, the selected freqency responses and their corresponding ranges are depicted in Table 5.9. The coherence fnction ϒ has been sed as the criterion for the freqency response selection, for which the coherence fnction has vales greater than.6 over the desired freqency range of the model. The determination of the freqency response pairs to be inclded in the identification process was followed by extraction of the transfer-fnction model; which involved determining the strctre and order of the parameterized model, and followed by making initial gesses for the vales of the model parameters. CIFER ses an optimization algorithm which comptes the cost fnction J satisfying Eqation 5. for each inpt-otpt pair. The optimization algorithm is based on an iterative robst secant algorithm that redces the phase and magnitde error between the state space model and the flight data freqency responses. The exection of the optimization algorithm contines ntil the average of the selected freqency responses and cost fnctions are minimized. Similarly, the parametric state-space model extraction involved an iterative procedre sing DERIVID. The iteration rn ntil the most sitable stability and control derivatives of the state-space model were selected based on the three accracy metrics discssed previosly, namely: freqency responses cost fnctions; percentage of the Cramér Rao (CR) bond for each parameter; percentage of the insensitivity of each parameter with respect to the cost fnction. Parameters having high CR bond were dropped or fixed to a specific vale, high insensitive parameters have minimal or no effect on the comptation of the cost fnction and were dropped. 9 IDENTICATION RESULTS The final extracted model obtained from the procedre described in the previos section has an excellent average cost fnctions vale of and prodced physically reasonable vales for 4 the stability derivatives. The identified stability and control derivatives with their respective CR bond and insensitivity percentage for the qadrotor are depicted n Table 5.. For instance, the anglar body position damping parameters Y θ and Y ϕ exhibit negative (stable) vales is an indication that the vehicle has a good anglar position damping, whereas a positive Y ψ points to an nstable yaw mode. Table 3 Linear state space model identified parameters for matrix B Parameter Vale CR Insensitivity % Y 6.965E Y.97E Y E Y 4.36E L -.35E L 3.594E L 3.95E L 4.33E N.348E N.37E N 3.347E N 4.444E E E E E Some of the identified parameters exhibit high CR bonds and insensitivities (Table 5.), i.e., the anglar position derivatives of the roll and pitch L θ, L ϕ, and L ѱ can be dropped from the model withot sacrificing the accracy of the identification reslts. However, these derivatives are kept to maintain the final state space dynamics as close as possible to the parameterized model. According to [5], the large ncertainty of the specific stability derivatives reslts from the lack of low freqency excitation. The signs and magnitdes of the anglar position damping derivatives Y θ and Y ϕ, together with the low vale accracy metrics, indicate that these parameters are completely reliable. The most important parameters of the state space model are the control variable copling terms N i (in matrix B) presented in Table 4, and their vales indicate the qadrotor is a high maneverable and agile vehicle. 9. Altitde model The magnitde, phase, coherence and error plots of the qadrotor altitde model response obtained from the CIFER identification is depicted in Fig The model shows excellent coherence (ϒ.7) from rad/sec p to 9 rad/sec, with both magnitde and phase constant to within % over excitation freqency range; hence this indicates a good model. The experimental flight data is represented with a second order critically damp system (Eqation 5). Similarly, the magnitde of the identified model fits the experimental flight data from. rad/s to rad/s, while the phase fits from zero rad/s to. rad /s as depicted in Fig. 4.

9 International Jornal of Scientific & Engineering Research, Volme 5, Isse, Febrary-4 83 ISSN φ = s.4s _ 8.7 e + 5.8s +..45s (5) Fig. 6. Pitch attitde model (b) model verss flight data fit. Fig. 4. (a) Altitde model (b) model verss flight data fit. 9. Roll attitde model The magnitde, phase, coherence and error plots of the qadrotor roll attitde response obtained from the CIFER identification depicted in Fig. 5a, shows excellent coherence (ϒ.7) from rad/sec p to rad/sec, with both magnitde and phase constant to within % over excitation freqency range, hence indicates a good model. However, at.6 rad/s, there was a 9 degrees phase a rollover. The vehicle's roll response experimental flight data is represented with a second order critically damp system (Eqation 6). Similarly, the magnitde and phase of the identified model fits the experimental flight data from over the entire freqency range as depicted in Fig. 5b. θ ω.7 = s s + 5.7s _ s = e (6) 3 s + 6.s + 8. Fig. 5. Roll attitde model (b) model verss flight data fit. 9.3 Pitch attitde model Similar to the vehicle's roll attitde model, the magnitde, phase, coherence and error plots of the qadrotor pitch attitde model depicted in Fig. 6a, shows excellent coherence (ϒ.7) from rad/sec p to rad/sec, with both magnitde and phase constant to within % over excitation freqency range. Hence, this indicates a good model; however, at.6 rad/s, there was a 9 degrees phase a rollover. The vehicle's roll response experimental flight data is represented with a second order critically damp system (Eqation 7). The magnitde and phase of the identified model fits the experimental flight data from over the entire freqency range as depicted in Fig. 6b. ψ s 9.6 _ s = e 4 s (7) Yaw attitde model The magnitde, phase, coherence and error plots of the qadrotor yaw attitde model depicted in Fig. 7a shows poor coherence (ϒ <.6 ) at lower and pper freqency, a good coherence over the middle freqency (ϒ >.7) from.3 rad/sec p to.5 rad/sec having both magnitde and phase constant at within 5% over the freqency range. Hence, this indicates a good model. The vehicle's yaw response experimental flight data is represented with a second order critically damp system (Eqation 8). Similarly, the magnitde and phase of the identified model fits the experimental flight data from over the entire freqency range as depicted in Fig. 7b. 8 e.s (8) Even thogh the single axis transfer-fnctions can accrately model the on-axis anglar and vertical responses, the MIMO state-space models are needed to flly characterize the copled dynamics of the qadrotor. The 6-DOF hover state-space model generated in the system identification process exhibited copled dynamics between its responses to the inpts siganals. However, based on [5], the F and G matrices containing the stability and control derivatives were tned by CIFER sch that the model match those derived from the flight test data. The MIMO model shows a good agreement between the on-axis state-space model and the transfer-fnctions control and damping derivatives. The on-axis delays on the inpt control channels are as follows; a time delay of.54 rad/sec (Eqation 5) on the roll channel,.67 rad/sec (Eqation 6) on the pitch channel,.74 rad/sec (Eqation 7) on the yaw channel, and.4 rad/sec (Eqation 8) on the altitde channel. In the transfer fnction models, the gains are the control derivatives and the poles are the damping derivative. The slight change in vales occrs since the state-space model acconts for the simltaneos fit to the complete MIMO freqency responses. By and large, the models show excellent fit with the actal and predicted freqency reposnse which can be se for flight control system design. 5 CONCLUSION This paper has presented the freqency-domain system identification of a qadrotor. The hover flight condition of the vehicle was considered as the reference flight operating point in extracting the vehicle's parameterized model sing the CIFER software

10 International Jornal of Scientific & Engineering Research, Volme 5, Isse, Febrary-4 84 ISSN REFERENCES [] M. Ryll, H. H. Blthoff, and P. R. Giordano, Modeling and control of a qadrotor av with tilting propellers, in proceedings of the IEEE International Conference on Robotics and Atomation (ICRA), (Saint Pal, Minnesota, USA), pp , May. [] S. Bellens, J. D. Schtter, and H. Bryninckx, A hybrid pose / wrench control framework for qadrotor helicopters, in proceedings of the IEEE International Conference on Robotics and Atomation (ICRA), (Saint Pal, Minnesota, USA), pp , May. [3] L. Jn and L. Yntang, Dynamic analysis and pid control for a qadrotor, in proceedings of the IEEE International Conference on Mechatronics and Atomation, (China), pp , Agst. [4] D. M. Ly and H. Cheolken, Modeling and control of qadrotor mav sing vision-based measrement, in proceedings of the International Form of Strategic Technology (IFOST), (Ulsan, Soth Korea), pp , October. [5] F. Wang, B. Xian, G. Hang, and B. Zhao. Atonomos hovering control for a qadrotor nmanned aerial vehicle. In proceedings of the Control Conference (CCC), 3, pages 6 65, 3. [6] F. Wang, B. Xian, G. Hang, and B. Zhao. Atonomos hovering control for a qadrotor nmanned aerial vehicle. In proceedings of the Control Conference (CCC), 3, pages 6 65, 3. [7] I. E. Ptro, A. Bdiyono, K. J. Yoon, and D. H. Kim. Modeling of nmanned small scale rotorcraft based on neral network identifica- package. The identification procedre started with the collection tion. In proceeding of the IEEE International Conference on Robotics and Biomimetics, ROBIO, 8, pages , 9. of the experimental time domain flight data. For each flight data record, the qadrotor was set to hover and a piloted freqency [8] D.H. Shim, Hyon Jin Kim, and S. Sastry. Control system design for rotorcraft-based nmanned aerial vehicles sing time-domain sweep excitation signal was applied to the for control variables, system identification. In proceedings of the IEEE International one after the other, with the bandwidth of the excitation signal Conference on Control Applications,, pages 88 83,. ranged between. rad/sec rad/sec. While the freqency [9] G. Zhengbang, F.Wei, G. Tongye, and L. Jn. Modeling and identification of a hovering sb-miniatre rotorcraft. In International sweep was exected by the primary inpt of interest, the secondary inpts were kept ncorrelated to maintain the vehicle near the Symposim on Intelligent In- 4 Bibliography formation Technology Application Workshops, IITAW, 8, pages 4 8, 8. reference operating point. A systematic selection procedre of the [] Y. S. Chang, B.I. Kim, and J. E. Keh. Rotorcraft dynamics model model strctre for the parameterized transfer-fnction model identification and hovering motion control simlation. In proceedings of the 34th IEEE Annal Conference on Indstrial Electronics and the state-space model was employed, with emphasis on the selection of stability and control derivatives inclded in the parameterized state-space model. The extraction of the vehicle s [] M. B. Tischler. System identification reqirements for high- (IECON), 8, pages , 8. transfer-fnction model involved the determination of the strctre and order of the parameterized model, and the determination bandwidth rotorcraft flight control system design. In American Control Conference, 99, pages 494 5, 99. [] I. A. Raptis and K. P. Valvanis. Linear and Nonlinear Control of of the vales of the model parameters sing an optimization algorithm. The optimization algorithm was based on an iterative ro- [3] L. L. Jng. System Identification: Theory for the User. Prentice Small-Scale Unmanned Helicopters. Springer,. bst secant algorithm that redced the phase and magnitde error Hall, New York 999. between the state space model and the flight data freqency responses, the iteration was exected ntil the average of the se- New York 989. [4] T. Soderstrom and P. Stoica. System Identification. Prentice Hall, lected freqency responses cost fnctions were minimized. Similarly, the parametric state-space model extraction involved an [5] M. B. Tischler and R. K. Remple. Aircraft and Rotorcraft System Identification. AIAA Edcation Series (AIAA, Washington), 6. [6] R. W. Ramirez. The FFT: Fndamentals and Concepts. Prentice iterative procedre ntil the most sitable stability and control Hall, Upper Saddle River, NJ, 985. derivatives of the state-space model were selected based on the [7] J. S. Bendat and A. G. Piersol. Engineering Apllications of Correlation and Spectral Analysis, nd Edition. Wiley, New York, 993. three accracy metrics namely: freqency responses cost fnctions; percentage of the Cramér Rao (CR) bond for each parameter; percentage of the insensitivity of each parameter with and Practice, AIAA Edcation Series. AIAA, Washington, 6. [8] V. Klein and E. A. Moreli. Aircraft System Identification Theory [9] B. T. Mark and K. R. Robert. Aircraft and Rotorcraft System Identication. AIAA, Virginia, USA, 6. respect to the cost fnction. The model of the qadrotor extracted possesses an excellent average cost fnctions vale of and [] B. Mettler, M. B. Tischler, and T. Kanade. System identification of prodced physically reasonable vales for the stability derivatives. Similarly, the MIMO model shows a reasonable agreement 55th Form of American Helicopter Society, May 999. small-size nmanned helicopter dynamics. In proceedings of the between the on-axis state-space model and the transfer-fnctions [] B. Mettler. Identification Modeling and Characteristics of Miniatre control and damping derivatives. Rotorcraft. Klwer Academic Pblishers, Norwel, 3. 4 [] B. Mettler, T. Kanade, and M. B. Tischler. System identification modeling of a model-scale helicopter, technical report. Technical report, Carnegie Mellon University,. [3] C. Gowei, B. M. Chen, K. Peng, M. Dong, and T. H. Lee. Modeling and control system design for a av helicopter. In proceedings of the 4th Mediterranean Conference on Control and Atomation, 6, MED, 6, pages 6, 6. [4] J. Gadewadikar, F. Lewis, K. Sbbarao, and B. Chen. Strctred h infinity command and control loop design for nmanned helicopters. Jornal of Gidance, Control and Dynamics, 3:93, 8. [5] A.S. Imam and R. Bicker. Design and constrction of a small-scale rotorcraft av system. International Jornal of Engineering Science and Innovative Technology (IJESIT), Vol., in Pres, 4.