Dynamic System Identification and Validation of a Quadrotor UAV

Transcription

1 Dynami System Identifiation and Validation of a Quadrotor UAV Min Goo Yoo PG Student, Department of Aerospae Engineering, Sejong University, Seoul, Republi of Korea mingooyoo@gmailom Sung Kyung Hong* Corresponding Author, Professor, Department of Aerospae Engineering, Sejong University, Seoul, Republi of Korea skhong@sejongakr Abstrat In this paper, a pratial system identifiation method is applied to obtain a linear time-invariant (LI) model of a quadrotor using input-output flight data as opposed to painstaking mathematial modeling tehniques he system response data from an Asending ehnologies Hummingbird quadrotor is aquired in a arefully devised experiment proedure and an LI model is then obtained using a predition error method with a blak-box model approah Fidelity verifiation shows that the proposed method provides a fairly aurate model of a quadrotor flying robot his aquisition of a high-fidelity system model of a quadrotor will be a ruial step for the suessful design of a high-performane flight ontrol system Keywords: System Identifiation, Validation, Predition Error Method, Blak-box Model, Unmanned Aerial Vehile (UAV) Introdution In reent years, the quadrotor flying robot, whih is lifted and propelled by four rotors, has beome a standard platform in experiments and appliations of mini unmanned aerial vehiles (mini-uav) he major advantage of this type of UAV is that it redues the mehanial omplexity inherent in heliopters and other vertial take-off and landing (VOL) airraft Diretional ontrol is produed by individually altering the speed of the four motors Colletive pith propellers are not required in a quadrotor design Moreover, the orresponding gyrosope moments generated by the motors ounterat eah other As a result of these advantages, there have been a number of ommerial [1, 2, 3, 4] and researh [5, 6, 7, 8] quadrotor platforms developed However, there has been relatively little development of aurate dynami models of quadrotors he aquisition of high-fidelity system models and ontrol tehniques based upon these models are ritial for preision ontrol and trajetory traking In general, however, it is a hallenging proess to identify quadrotor dynamis due to its multi-input multi-output (MIMO) harateristis, nonlinearity, severe noise and disturbane, and wide flight envelope Hene, attempts are often made to design flight ontrollers with a simple struture whih are tested by trial and error during atual flight his alternative approah would require only a basi understanding of the plant, but the results from this proess is not likely to represent the optimal tuning herefore, we adopt a pratial method to obtain a linear time-invariant (LI) model of a quadrotor using input-output atual flight data as opposed to painstaking mathematial modeling tehniques [9] A ommerial off-the-shelf quadrotor platform (Hummingbird) with an embedded innerloop attitude feedbak ontroller from Asending ehnologies In was seleted for this researh he system response data from the Hummingbird is aquired in a arefully devised experiment proedure, after whih an LI model is obtained by means of the predition error method with the blak-box model approah [10] We then arry out a validation proess to obtain a fairly aurate model of the quadrotor UAV his aquisition of a high-fidelity system model of a quadrotor will be a ruial step for the suessful design of mid-loop veloity feedbak ontroller and outerloop position ontrol methods in further studies, espeially those by the urrent authors his paper proeeds as follows he seond setion presents the dynami model derived for the Hummingbird quadrotor he basi idea is to ombine a simple physial point-mass model, to approximate the kinematis of the mini-uav, with a blak-box model, whih represents the dynamis of the attitude ontroller as well as the rotor and blade dynamis he third setion desribes the fidelity of the identified system model through a omparison with atual input-output flight data Finally, the onlusions are given in the final setion Identifiation of the Dynami Model he system identifiation of a dynami system onsists of several steps, as shown in Figure 1 Figure 1: Shemati flowhart of the system identifiation proess 1089

2 he first step is the design of an experiment whih will aquire the input/output data over a time interval Aquiring data is not trivial and an be very laborious and expensive his involves areful planning of the inputs to be applied suh that suffiient information about the system dynamis is obtained he seond step defines the struture of the system, for example, the type and the order of the differential equation relating the input to the output he third step is identifiation/estimation, whih involves determining the numerial values of the strutural parameters, whih minimize the error between the system to be identified, and its model he estimation method that is used in this researh is the predition error method he final step, validation, onsists of relating the system to the identified model responses in the time or frequeny domain to instill onfidene in the obtained model [11] If the identified model is not aepted, a more omplex model struture must be onsidered, its parameters estimated, and the new model validated [12] A Experimental setup he aerial platform (Hummingbird) used in this study is a quadrotor developed by Asending ehnologies GmbH [13] he main feature of this platform is that it is equipped with on-board 1 khz flight eletronis, whih stabilizes the three rotational degrees of freedom with PD ontrollers hus, no diret input signals to the rotors are needed; instead, the desired tilt angle, yaw rate, and total thrust an diretly be transmitted able 1 provides the most important fats about the Hummingbird quadrotor able 1: ehnial Data of the Hummingbird Quadrotor Mass Payload apaity otal diameter Max flight time Cruise Speed 0550 kg 0200 kg 0600 m 20 min 36 km/h On-board Sensors MEMS inertial sensors, GPS, ompass Safe landing mode (If radio ontrol link is los Figure 2: Experimental setup for the identifiation B Charaterization of the Model Struture Due to the small size and symmetry of the quadrotor, it an be modeled as a point-mass system [16] Aording to the priniple of linear momentum (Newton's Seond Law) and the assumption that the mass m of the quadrotor is onstant, the hange of the momentum is as follows, m r I ( F i ), (1) I i where r is the aeleration of eah of the Cartesian axes at the enter of gravity, F i represents any outer fore ating on the point-mass, and I represents the inertial frame (see Figure 3) he equation of motion of the quadrotor is as follows: R IB r I RIB (,, ) m 0 0 (2) g (,, ) os os sin sin os os sin os sin os sin sin os os sin sin sin os os os sin sin sin os sin sin os os os Here, represents the total thrust pointing in the opposite diretion of the quadrotor s down axis, and g denotes the aeleration of gravity In equation (3), RIB(,, ) is the transformation matrix from the body frame (B) to the inertial frame (I), where,,and are the Euler angles (3) o ensure the quality of the data, the data-olletion experiment has to be arefully designed [14] hus, input signals to the quadrotor are the previously designed step signal, whih annot be too long onsidering the experimental flying spae he quadrotor is manually ontrolled near hover flight in an indoor spae to avoid disturbanes suh as wind A real-time omputer (RC) from dspace GmbH [15] is used to log the flight data and input ommands from the RC transmitter with a sampling frequeny of 10Hz he experimental setup for this identifiation proess is illustrated in Figure 2 Figure 3: Representation of the quadrotor system, introduing fores and oordinate systems 1090

3 With the on-board ontroller for attitude stabilization, the unknown parameters in equation (2), speifially the Euler angles and the total thrust, are the output variables of the attitude ontroller Given the absene of an exat desription of the attitude ontroller and given that the ontrol parameters are unknown, the total dynamis of the attitude ontroller, inluding the rotor and blade dynamis, are identified in the modeling task For this purpose, a linear blak-box modeling approah is used [17] Figure 4: Blak-box model of the attitude ontroller of the quadrotor As illustrated in Figure 4, the four deoupled subsystems used to model the roll, pith, yaw and thrust dynamis are approximated with a first-order system For the roll and pith dynamis, its input variables are the desired roll and pith angles, and its output variables are the urrent roll and pith angles For the yaw dynamis, in ontrast to the roll and pith dynamis he desired yaw angle rate is the input to the subsystem, and the urrent yaw rate is its output variable For the thrust model, the aeleration of the z axis of body frame is used as the output variable he predition error method uses an iterative nonlinear leastsquares algorithm to minimize the ost funtion, whih onsists of the weighted sum of the square of the errors he ost funtion is defined as follows, N 2 VN ( Gˆ, Hˆ ) ( (6) i1 Here, denotes the error between the estimated and the measured values For linear systems, the error an be expressed as ˆ 1 ( y( yˆ( H( ) ( y( Gˆ( ) u( ) (7) In this equation, is equal to the white noise e if the output is indeed generated by the model and for some sequenes u ( and y ( [18] o apply this parameter identifiation method, the software pakage MALAB was used [19] he MALAB ommand pem omputes the predition error estimate of a general linear system o inrease the auray of the estimation, the algorithm was applied several times using the prior estimation as the new initial guess hrough the abovementioned proess of identifiation, the unknown parameters of eah dynami model were determined Below Figure 6 shows an example of the roll model for the parameter seletion proess he parameter of the highest fitting rate among the results for eah experiment is seleted C Parameter Identifiation and Seletion he predition error method is adopted as the identifiation algorithm he system identifiation features are symbolially indiated in Figure 5 Figure 6: he fitting rate result (roll motion) Figure 5: he overall identifiation proedure With the model being defined as a linear time-invariant model, the relationship between the input and output is as follows: yˆ ( Gˆ( ) u( Hˆ ( ) e( (5) Here, e ( is the white noise (unpreditable), Ĥ denotes the harateristis of the noise, Ĝ is the dynami model of the system, the related input is u (, and the predited output is y ˆ( In the blak-box model approah, Ĝ and Ĥ are rational transfer funtions in the shift (delay) operator q and the unknown parameter θ able 2 presents the predited models and ratio of the standard deviation of eah parameter from eah experiment he units of ommand for all transfer funtions are shown in digital values (0~4095), as obtained from RC transmitter, and the outputs are in degrees or g units Deoupled Subsystem Roll Pith Yaw hrust able 2: Results of the Identifiation Proess ransfer funtion of the seleted model s 1 Standard deviation Standard deviation of the parameter of the parameter K s s e s

4 Eventually, the final system of the omplete model of the quadrotor an be illustrated as shown in Figure 7 he model is used to build a simulator where for example ontrol algorithms an be tested before they are implemented on an atual platform Figure 7: Completed dynami system of the quadrotor Dynami Model Validation We performed experiments four times near the hover ondition to validate the model, ie, to ompare the flight data with the response of the model simulated with the orresponding input data he results of the validation proess are presented in able 3 with the fitting rate, whih is the perentage of output variane between the simulated and the measured output Figures (8 ~ 11) show the results of the system validation proess As shown in the plots assoiated with the validation, the dynamial model whih evolved approximates the behavior of the atual plant of the quadrotor quite well Espeially for the roll, pith, and yaw dynamis, great results were ahieved; for the aeleration in the Z diretion, the obtained results are also satisfying able 3: Results of the Fitting Rate hrust Model Roll Model Pith Model Yaw Model est #1 [%] est #2 [%] est #3 [%] est #4 [%] Figure 9: he original and estimated responses (roll motion) Figure 10: he original and estimated responses (pith motion) Figure 11: he original and estimated responses (yaw motion) Figure 8: he original and estimated responses (z-axis motion) Conlusions In this paper, a pratial system identifiation method is applied to obtain a linear time-invariant (LI) model of a quadrotor UAV using input-output flight data as opposed to painstaking mathematial modeling tehniques he system response data from a ommerial off-the-shelf quadrotor platform is aquired in a arefully devised experiment proedure and an LI model is then obtained using the 1092

5 predition error method with the blak-box model approah Fidelity validation shows that the proposed method provides a fairly aurate model of the quadrotor flying robot his aquisition of a high-fidelity system model of the quadrotor will be a ruial step for the suessful design of a highperformane flight ontrol system Aknowledgments his work was supported by Institute for Information & ommuniations ehnology Promotion(IIP) grant funded by the Korea government(msip) (R , Development of Unmanned Aerial Vehile Sensor-based Smart Eye ehnology for Loal Disaster Monitoring and Situational Response) [15] dspace GmbH, MiroAutoBox, om [16] Ruesh, Dynamis Identifiation & Validation, and Position Control for a Quadrotor, Semester- hesis, Eidgenossishe ehnishe Hoshshule Zurih, 2010 [17] Lennart Ljung, System Identifiation, heory for the User, 2 nd edition Prentie Hall PR [18] omas MKelvey, Relations Between ime Domain and Frequeny Domain Predition Error Methods Control Systems, Robotis, and Automation, Vol V, Enylopedia of Life Support Systems (EOLSS), 2009 [19] Lennart Ljung, System Identifiation oolbox, MathWorks In, 1991 Referenes [1] Mirodrones GmbH, MD4-200 quadrotor heliopter, [2] Asending ehnologies GmbH, Ase hummingbird, [3] DraganFly-Innovations, DraganFlyer X4 (2009) [4] Aeryon Labs In, Aeryon Sout (2009) wwwaeryonom [5] M Valenti, B Bethke, G Fiore, J P How, E Feron, Indoor multivehile flight testbed for fault detetion, isolation, and reovery, in: Pro AIAA Guidane, Navigation, and Control Conf, Keystone, CO [6] M Ahtelik, A Bahrah, R He, S Prentie, N Roy, Autonomous navigation and exploration of a quadrotor heliopter in GPS-denied indoor environments, in: AUVSI First Symposium on Indoor Flight Issues, Mayaguez and Puerto Rio [7] N Mihael, J Fink, V Kumar, Cooperative manipulation and transportation with aerial robots, Robotis: Siene and Systems, (Seattle, WA) (2009) [8] S Lupashin, A Sherbak, R D'Andrea, A simple learning strategy for high-speed quadroopter multiflips (2010) [9] Min Goo Yoo, Sung Kyung Hong, System Identifiation of the Quadrotor Flying Robot in Hover using Predition Error Method, Eletrial and Eletroni Engineering 2015, Vol118, pp21-26 [10] Lennart Ljung, Predition Error Estimation Methods, Ciruits Systems Signal Proessing, VOL21, No1, pp11-23, 2002 [11] W Naeem, R Sutton and J Chudley, System Identifiation, Modeling and Control of an Autonomous Underwater Vehile IFAC, 2003 [12] orsten Soderstrom, Peter Stoia, System Identifiation, Prentie Hall [13] Ase Hummingbird with AutoPilot User s Manual, Asending ehnologies GmbH [14] Guowei Cai, Ben M Chen, Kemeo Peng, Miaobo Dong and ong H Lee, Modeling and Control of the Yaw Channel of a UAV Heliopter, IEEE ransations on Industrial Eletronis, Vol 55, No 9, September