Available online at www.sciencedirect.com Procedia Engineering 16 (2011 ) 137 143 International Workshop on Automobile, Power and Energy Engineering Identification of Lateral/Directional Model for a UAV Helicopter in Forward Flight Peng Liu *, Zhijun Meng, Zhe Wu School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China Abstract A lateral/directional state-space model of the Raptor 50 helicopter is identified in forward flight condition. A frequency sweep test is properly designed and executed to collect a well-suited time-history database. The measured accelerometers are transferred from the sensor location to the gravity, then, a complete set of non-parametric input-to-output frequency responses that fully characterizes the coupled helicopter dynamics are extracted from the frequency sweep test data. With the help of small perturbation theory, a linearized dynamic model for the stability and control derivative is extracted from linearization of a nonlinear mathematical model. A nonlinear search based on secant method is conducted for the linear lateral/directional state-space model that matches the frequency-response data set. The identified model is verified in the time domain, which indicates that the model matches the flight data well. 2010 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Society for Automobile, Power and Energy Engineering Open access under CC BY-NC-ND license. Keywords:Helicopter; Frequency identification; Lateral/directional model; Secant method; Forward flight. * Corresponding author. Tel.: +86-10-8233-8797; fax: +86-10-8231-5584. E-mail address: lppl2008@163.com. 1877-7058 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2011.08.1063 Open access under CC BY-NC-ND license.
138 Peng Liu et al. / Procedia Engineering 16 ( 2011 ) 137 143 1. Introduction There has been a significant growth in the use of UAV helicopters for a multitude of military and civilian applications over the last few years. The main research focus is on the development of automatic flight systems (AFCS). The AFCS design methodologies require a linear model that accurately represents the aircraft that is to be controlled [1]. Frequency domain system identification can be used to develop state-space models directly from flight data, and it has been proven to be highly accurate and efficient [2]. In frequency-domain system identification, the identification of the dynamic models in forward flight is more challengeable than in hover because it is more difficult to collect proper flight data [3]. The extraction and analysis of dynamic models first involves generating a set of nonparametric frequency responses from a set of time history flight data and then fitting transfer functions and higher-order state-space models to the frequency responses. The goal of the system identification process is to achieve the best possible fit between the flight data and a model that is consistent with the physical knowledge of the vehicle dynamics. 2. Development of the lateral/directional state-space model of helicopter in forward flight The Newton-Euler rigid body equations of motion for a generic helicopter have the axes fixed at the center of mass of the helicopter. With the help of small perturbation theory, a linearized lateral/directional dynamic model for the stability and control derivative is extracted from linearization of a non-linear mathematical model in forward flight [4]. x = Ax+ Bu y = Cx+ Du 1 where x is the state u is the control inputs x = [ v p r ] T u = [ ] T Lat Ped 2 The state matrix A is composed of stability derivatives and the control matrix B is composed of control derivatives. Some of these elements may be known from physical considerations, and the unknown parameters could be gotten by the frequency identification method. Yv Yp+ W0 Yr U0 gcosθ0 Y Y Lat Ped Lv Lp Lr 0 L L Lat Ped A = B = Nv Np Nr 0 N N Lat Ped 0 1 cosφ0tanθ0 0 0 0 3
Peng Liu et al. / Procedia Engineering 16 ( 2011 ) 137 143 139 3. Frequency sweep test for system identification 3.1. The design of the frequency sweep test The platform we used to do research on the frequency identification is Raptor 50 helicopter with equipments, which is shown in Fig. 1. A secure and easier way to collect flight data for the forward flight frequency sweeps is placing the pilot on a vehicle which is driving down a runway, the pilot is flying the helicopter parallel to and at the same forward speed as the helicopter. Fig. 1. Instrumented helicopter used for experiment The flight maneuver that is used for the identification of frequency responses is the frequency sweep. The maneuver starts with a low frequency sinusoidal shaped input and increases in frequency as time progress [5]. Fig. 2 give an example of a frequency sweep input and response. This maneuver provides excellent spectral content for purpose of system identification. Fig.2 Lateral frequency sweep and roll rate response time history
140 Peng Liu et al. / Procedia Engineering 16 ( 2011 ) 137 143 3.2. Data post processing The DMU that houses the three accelerometers has a slight offset from the vehicle center of gravity. Therefore, the measured accelerations have to be transferred from the sensor location to the gravity [6]. The accelerations measured are easily expressed in terms of the state and state rates with correction terms included to account for the offsets r s of the accelerometer package relative to the center of gravity by the following equation: 4 acg = ameas + rs + ( rs ) a ycg a is the part of the rs and a y 2 is the part of the ( r s ). The results of transferred lateral acceleration is shown in Fig. 3, where acceleration of gravity, y1 is the transferred lateral 1.2 The acceleration of Y axis( m/s 2 ) 1.0 0.8 0.6 0.4 0.2 0.0-0.2 a y a y1 a y2 4. Frequency identification for helicopter 4.1. State-space model identification method -0.4 0 20 40 60 80 100 120 t(s) Fig.3 The results of transferred lateral acceleration In the current frequency-response approach, stability-derivative identification is achieved directly through iterative multi-input/multi-output matching of the identified conditioned frequency responses with those of the following linear model: x = Ax+ Bu 5 y= Cx+ Du The elements of A B C D are the unknown stability and control derivatives. Taking the Laplace transfer of Eq. (5) results in the following transfer function [7]: T 1 () s = C ( s I A ) B + D 6
Peng Liu et al. / Procedia Engineering 16 ( 2011 ) 137 143 141 The unknown state-space model parameters are determined by minimizing J, a weighted function of the error between the identification frequency response and the model responses over a selected frequency range: n TF n 20 w ˆl l 2 l l 2 J= W Wg( Tc T ) + Wp( Tc T ) l= 1 n 1 Where 7 = magnitude (db) at each frequency ; = phase (deg) at each frequency ; n = number of frequency points (typically selected n = 20 ); 1 2 2 xy and n = starting and ending frequencies of fit; Wr = 1.58(1 e ) ; W 1.0 g = ; W p = 0.01745. The frequency ranges for the identification criterion are selected individually for each input/output pair according to their individual ranges of good coherence. The n p parameters to be identified in the model matrices A B are collected into an identification vector Θ = 1 2 np. A nonlinear search method based on secant method is much better suited for this application and is the method used in this paper [8]. The algorithm is sketched in Fig. 4. When a secant is projected through the function evaluations Fu ( 1) and Fu ( 2), it estimates a better approximation u. new F( u 2 ) F( u 1 ) u new u 1 u 2 Fig.4 Sketch of secant approximation However, the method can still fall if Fu ( ) has a local minimum which is not a solution, e.g., as in Fig. 4. In this event, the iteration procedure must be restarted from a different set of initial approximations. 4.2. Result of state-space model identification Comparisons of the identified model fit to the flight data are shown for the frequency responses in Fig. 5. These figures indicate that the model and the flight data are in good agreement.
142 Peng Liu et al. / Procedia Engineering 16 ( 2011 ) 137 143 Fig. 5. Comparison between the frequency response computed from the flight data (solid) and the identified model (dashed) The identification results are given as follows: 0.2903 2.6336 25.8835 32.174 0.0031 0.2192 0.0083 0.6742 0.0021 0 0.0753 0.0076 A = B = (8) 0.0078 0.2253 0.9764 0 0.0274 0.0225 0 1 0.08 0 0 0 The trim conditions in forward flight for the flight data are as follows: U0 = 32.2 ft/ s, W0 = 4.9 ft/ s, Θ 0 = 4.48deg, Φ 0 = 1.52 deg. 5. Model verification The identification model is verified in the time domain to ensure that it can accurately predict the aircraft dynamic response [9]. The pilot s inputs are used to excite the model, and the model responses and the aircraft responses are then compared. If the responses match, then the model has good predictive accuracy. The lateral verification results for forward flight are given in Fig. 6. As seen in the figure, the model matches the flight data well in the time domain.
Peng Liu et al. / Procedia Engineering 16 ( 2011 ) 137 143 143 Control Deflection Control Deflection Lat Ped p r a y Time (s) Fig.6 Lateral verification results 6. Conclusion A parameterized lateral/directional model of a helicopter in forward flight is developed and successfully identified using frequency domain identification method. The key results are: 1) A frequency sweep test is properly designed and executed to collect a well-suited time-history database in forward flight condition. 2) A complete set of non-parametric frequency responses derived from the flight data in forward flight condition that fully characterizes the coupled coaxial helicopter dynamics is extracted. 3) A nonlinear search based on secant method for the lateral/directional linear state-space model in forward flight that matches the frequency-response data set is conducted. 4) Time domain verification results showed that the model accurately predicts the response of the coaxial helicopter to control input. References [1] Mettler B. Modeling small-scale unmanned rotorcraft for advanced flight control design. Pittsburgh: Carnegie Mellon University, 2001:43-66. [2] Colin R. Theodore, Jason D. Colbourne. Rapid Frequency-Domain Modeling Methods for Unmanned Aerial Vehicle Flight Control Applications. Journal of Aircraft, Vol. 41, No.4, p. 735-743, August 2004. [3] Bhandari Subodh, Kowalchuk Scott. Six-DoF Dynamic Modeling and Flight Testing of a UAV Helicopter. AIAA Modeling and Simulation Technologies Conference and Exhibit, San Francisco, Caligornia, 2005. [4] Prouty R W. Helicopter Performance, Stability and Control. Krieger Publishing Company. Malabar, Florida, 2002. [5] Tischler,M.B., Remple,R.K. Aircraft and Rotorcraft System Identification: Engineering Methods With Flight Test Examples. AIAA, Aug, 2006. [6] Bhandari Subodh. Flight validated high-order models of UAV helicopter dynamics in hover and forward flight using analytical and parameter identification techniques. University of Kansas, 2004. [7] Tischler,M.B., Gauffman,M.G. Frequency- Method for Rotorcraft System Identification: Flight Application to BO 105 Coupled Rotor/Fuselage Dynamics. Journal of the American Helicopter Society, Vol. 37, No.3, Pgs 3-17, July 1992. [8] Burrows R. R., McDaniel,G.A.. A Method of Trajectory Analysis with Multi-Mission Capability and Guidance Application. AIAA Guidance, Control and Flight Dynamics Conference. p. 68-844, August, 1968. [9] Christina M. Ivler. Control System Development and Flight Test Experience with the MQ-8B Fire Scout Vertical Take-Off Unmanned Aerial Vehicle. Journal of the American Helicopter Society. May, 2007.