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SYSTEM IDENTIFICATION OF PARAMETERIZED STATE-SPACE MODEL OF A SMALL SCALE UAV HELICOPTER∗ ISMAILA B. TIJANI, RINI AKMELIAWATI

† AND

ARI LEGOWO

Abstract. The success of model-based flight control law design for autonomous helicopter is largely dependent on the availability of reliable model of the system. Considering the complexity of the helicopter dynamics, and inherent difficulty involves with physical measurement of the system parameters, the grey modeling approach which involves the development of parameterized model from first principles and estimation of these parameters using system identification (sysID) technique has been proposed in the literatures. Prediction Error Modeling (PEM) algorithm has been identified as an effective system identification technique. However, application of this method to complex system like helicopter is not a trivial exercise due to inherent coupling in the system states and the challenges associated with parameter initialization in PEM algorithm. In this work, an effective procedure in application of PEM algorithm available in MATLAB toolbox is presented for small scale helicopter using real-time flight data. The approach was able to yield satisfactory model suitable for model-based flight control design. Key words. system identification, modeling, prediction-error modeling AMS subject classifications. 93B30, 97M10

1. Introduction. The need for simple and effective mathematical representation of a helicopter dynamics has been one of the major challenges in model-based flight control design for the system. However, modeling of helicopter dynamics is not a trivial task due to the system’s complex dynamics, nonlinearities, instability and high degree of coupling among its state variables. Several approaches have been adopted in helicopter modelling since the inception of research activities on small-scale helicopters in academic institutions in the early 1990’s [1]. The modeling approaches can be generally categorized into the following: first principle approach (white model), parametric approach based on system identification (grey model) and non-parametric approach (black box). For model-based control application, only the first two approaches yield the needed mathematical representation for controller design. The first principle approach to modeling involves derivation of mathematical model describing the system of interest by using the fundamental laws of mechanics and aerodynamics [2]. Full modeling includes flexibility of the rotors and fuselage, stabilizer effects, dynamics of the actuators and combustion engine[3], [4]. A rigidbody assumption imposed on the system is used as the starting point with the inputs are forces and torques applied at center of gravity (cg), and outputs are the linear position and velocity of the cg as well asthe rotation angles and angular velocities. This is then followed by augmentation with the system sub-dynamics such as rotorfuselage dynamics, flybar dynamics, flapping dynamics, stabilizer bar and etc. [2], [4]. This approach usually leads to a high-order nonlinear coupled model useful mostly for simulation purposes [3]. In addition, this approach requires a detailed knowledge of and experience with all the phenomena involves in rotorcraft system. These have been the major demerits associated with the use of purely first-principle model approach. Also, it has been observed, that however detailed the resulting model could be, a rigorous validation against flight data collected in real-time from the intending platform ∗ This research was supported by RMGS (Research Matching Grant Scheme), RMGS-09-02, Research Management Center, IIUM, Malaysia. † [email protected], Intelligent Mechatronics Research units (IMSRU), Department of Mechatronics Engineering, International Islamic University Malaysia (IIUM).

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Ismaila B. Tijani, Rini Akmeliawati, Ari Legowo

is required in establishing the effectiveness of the model [4], [5]. All these have necessitated parametric modeling approach based on integration of mathematical model with system identification (sysID) methods in developing a simplified, linear model suitable for control system design. In this approach, a simplified linear model is usually extracted by linearizing the resulting nonlinear model from first principle approach method around a particular operating point. Then with appropriate input/output experimental flight data collected at the operating point, a frequency/time domain system identification technique is then applied to obtain the parameters of the linearized model. Apart from extraction of linearized model of the system from detailed nonlinear mathematical representation, determination of the linearized model’s parameters using sysID approach has been a topical issue in the literatures [6], [7], [8]. Due to the uniqueness of the helicopter system, the basic sysID procedures of : data collection, data pre-processing, model structure formulation and parameter estimation algorithm have received special attention and treatment to overcome the challenges posed by the complexity of the helicopter system. A guide on aircraft system identification with a custom software package written in MATLAB m-files known as SIDPAC (System Identification Programs for Aircraft) is reported in [9]. Mettler and Kanade [3] reported the sysID approach in developing a parameterized model of Carnegie Mellon’s Yamaha R50 using frequency domain method with customized software CIFER (Comprehensive Identification from Frequency Responses). The detailed hybrid of first-principle approach with sysID in helicopter model development represents the core contribution of the work. The extraction of single input-single output (SISO) model using this approach was reported in [7]. To obtain data, the helicopter movement was restricted with the use of stands to 1-DOF. An integrated modeling based on first principle and sysID using customized software MOSCA (modeling for flight simulation and control analysis) was used in extracting a high order linear model of R50 Yamaha with nonlinear simulation model of the helicopter at Carnegie Melon University [8]. The use of prediction-error modeling (PEM) technique has also been reported as effective tool in UAV helicopter model estimation [10]. The availability of this tool in MATLAB software package has also been an added advantage in UAV helicopter design and development. However, the PEM algorithm is highly sensitive to parameter initialization, and this becomes more difficult due to the helicopter system complexity as a MIMO system with cross-coupling among the system states. This work as part of an ongoing research on autonomous UAV helicopter development for disaster monitoring in our research unit IMSRU, presents an effective approach in application of PEM algorithm for estimation and analysis of linearized state-space model of a small scale UAV helicopter system. Apart from breaking down of the identification process into simple sub-systems involve in helicopter dynamics as suggested by Mettler using CIFER [3], a simple MATLAB algorithm is developed to iteratively initialize the system parameters, compare the performance of the resulting models, and select the best model using a simple statistical comparison for single output and pareto selection for multi-outputs sub-systems. The rest of the paper is organized as follows. Section 2 presents the system description and mathematical modeling. The system identification process is described in section 3 followed by results and discussion in section 4. The paper is concluded in section 5 with further study. 2. System Description and Mathematical modeling. A 50 class Hirobo RC helicopter (Hirobo SDX50) is adopted as the UAV helicopter platform. The helicopter

System Identification of Parameterized State-Space Model of a Small Scale UAV Helicopter

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Table 2.1 General specifications of Hirobo SDX50 helicopter Specifications Full length of fuselage Full width of fuselage Height Main rotor diameter

Values 1220 mm 186 mm 395mm 1348 mm

Specifications Tail rotor diameter Gear ratio Dry weight Engine

Values 258 mm 8.7 : 1 : 4.71 3400 g 50 class

is shown in Fig. 2.1, and its general specifications of are given in Table 2.1. Currently the system has been modified and instrumented with an embedded data-acquisition system [11]. The onboard system comprises of microchip dspic33fj256m710 as core processor, and IMU/AHRS VN100 by vectornav, GPS, ultrasonic and barometric sensors. The system produces the measurement of the vehicle state variables require for identification process. The pilot PWM control signals from the RC receiver are captured using the input capture I/O features of the onboard processor.

Fig. 2.1. Hirobo SDX 50.

2.1. Parameterized State-space Model. The translational and rotational motion of the vehicle referred to an axes fixed at the center of mass of the craft are deT T scribed by three linear velocities v¯ = [u, v, w] , and three angular rates ω ¯ = [p, q, r] . The complete system model is generally developed from rigid body assumption augmented with a simplified rotor dynamics and rotor blade representation [2]. Based on rigid body assumption and application of law of conservation of linear and angular momentum, the Euler-Newton equations of motion are: (2.1)

X

d(¯ v) , F¯ = m d(t)

X

ω) ¯ = I¯d(¯ M d(t)

T ¯ = [Mx , My , Mz ]T are vectors of external forces and where F¯ = [Fx , Fy , Fz ] and M moment acting on the vehicle, m is the helicopter mass, and I¯ is the moment of inertial about the reference axes. The forces and moments are produced by: the main motor and tail rotor which are the control forces and moments, while other sources are the gravitational forces, aerodynamic forces produced by the fuselage, horizontal and vertical fins. Let the superscripts; mr, tr, f us, hf, and vf represents components of main rotor, tail rotor, fuselage, horizontal fin and vertical fin respectively, the summation of forces and moments can be written as:

(2.2)

F¯ = F¯ mr + F¯ f us + F¯ hf + F¯ vf ,

¯ =M ¯ mr + M ¯ f us + M ¯ hf + M ¯ vf M

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Ismaila B. Tijani, Rini Akmeliawati, Ari Legowo

Using kinematics principles of moving reference frames [2], the six nonlinear differential equations of motion describing the rigid-body dynamics of the helicopter with respect to body coordinate system are: 1 X 1 X Fx , v˙ = pw − ru + Fyg + Fy u˙ = rv − qw + Fxg + m m X X 1 w˙ = qu − pv + Fzg + Fz , p˙ = qr (Iyy − Izz ) /Ixx + Mx m X X (2.3) q˙ = pr (Izz − Ixx ) /Iyy + My , r˙ = pq (Ixx − Iyy ) /Izz + Mz Fxg , Fyg , Fzg are the gravitational forces in x, y, and z body axes, respectively. These forces can be expressed in inertial reference frame using Euler angles: φ-roll (about x-axis), θ-pitch (about y-axis), and ψ-yaw(about z-axis) as: Fxg = −g sin θ, Fyg = g cos θ sin φ, Fzg = g cos θ cos φ. The angular velocity of the vehicle is related to the time rate of change of Euler angles through the following relation:     φ˙ − ψ˙ sin θ p  q  =  θ˙ cos φ + ψ˙ sin φ cos θ  (2.4) r ψ˙ cos φ cos θ − θ˙ sin φ Re-arranging (2.4) gives the attitudes components: φ˙ = p + (q sin φ + r cos φ) tan θ θ˙ = q cos θ + r sin φ ψ˙ = (q sin φ + r cos φ) sec θ

(2.5)

Equation (2.3) - (2.5) describe the motion and orientation of the craft. This is expressed in compact form as x˙ = f (x, δ) where x is vector of vehicle states given as:x = T T [u, v, w, φ, θ, ψ, p, q, r] and δ is the vector of control inputs δ = [δlat , δlon , δcol , δped ] . δlat and δlon are the lateral and longitudinal cyclic rotor controls, respectively. δcol is the collective pitch control and δped tail rotor pedal control input. The simplified rotor dynamics and rotor blade representation yields the simplified coupled first-order tip-path plane equations of rotor-fuselage dynamics that have been used successfully in the literature [2],[4]: 1 ∂a a + τc τc ∂µmr 1 ∂b b b˙ = − − τc τc ∂µmr

a˙ = − (2.6)

ua 1 ∂a wa 1 + − q + Aδlon δlon ΩR τc ∂µzmr ΩR τc va 1 − p + Bδlat δlat ΩR τc

where a˙ and b˙ are rotor flapping states, τc is motor time constant,Ω rotor head speed, µzmr is normal component of airflow,R is rotor diameter, Aδlon and Bδlat are the effective gains from δlon and δlat , respectively, and [ua , va , wa ] is wind speed vector along x, y, z axes. A linearized state-space model of the system is extracted from the nonlinear equations (2.3) to (2.6) by linearization about an equilibrium state (e.g. hovering/cruise) using small perturbation theory [4]. At hovering point, only the partial derivatives of the forces and moments significantly exist, and are expressed as: (2.7) ∂Fx/y/z =

∂Fx/y/z ∂Fx/y/z ∂Mx/y/z ∂Mx/y/z ∂x + ∂δ, ∂Mx/y/z = ∂x + ∂δ ∂x ∂δ ∂x ∂δ

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∂Fx ∂u ∂δlat x and so on, the parameterized state space of Let ∂F ∂u = Fx , ∂δlat = Fx the system comprises of matrices A with stability derivatives and B with control derivatives is given by (2.8) containing the most significant terms with respect to each states. Additional rate feedback term, rrf b is added to account for the effect of the tail gyro, [3]:   ∂u  u˙ Fx 0 0 0 0 0 0 −g Fx∂a 0 0 ∂v ∂b  v˙   0 F 0 0 0 0 g 0 0 F 0 y y    ∂w ∂r ∂a ∂b  w˙   0 0 F 0 0 F 0 0 F F 0 z z z z    ∂a ∂b  p˙   Mx∂u Mx∂v 0 0 0 0 0 0 M M 0 x x    ∂a ∂b  q˙   My∂u My∂v 0 0 0 0 0 0 M M 0 y y    ∂rrf b  r˙  =  0 ∂v ∂w ∂p ∂r Mz Mz Mz 0 Mz 0 0 0 0 Mz     φ˙   0 0 0 1 0 0 0 0 0 0 0     θ˙   0 0 0 0 1 0 0 0 0 0 0     a˙   0 0 0 0 −1 0 0 0 Aa 0 0     b˙   0 0 0 −1 0 0 0 0 0 Bb 0 r˙rf b 0 0 0 0 0 Kr 0 0 0 0 Krf b     0 0 0 0 u δ  v   0 0 0 Fy ped      δ   w   0 0 Fz col 0       p   0  0 0 0  δlat      q   0 0 0 0   δlon     δ    r + 0 (2.8) 0 Mzδcol Mz ped    δcol       φ   0 0 0 0  δped      θ   0 0 0 0       a   0 0 0 0       b   A Aδlon 0 0 δlat rrf b Bδlat Bδlon 0 0

(2.9)

rrf b = Kr r + Krf b rrf b

3. System Identification Process. System identification is generally an iterative process that starts from design of experiment for data collection, and ends only when a satisfactory model is obtained for an intended application. This section presents step by step procedure proposed for the estimation of the unknown parameters of the state-space model (2.8). 3.1. Flight data collection and processing. In order to obtain reliable realtime data for the parameter estimation, the instrumented helicopter system described in 2 was used for the flight experiment. The pilot excites each of the system channels (roll, pitch, yaw and heave) with a sinusoidal input of varying frequency while keeping the system at desired hovering operating point as much as possible. Due to difficulty in maintaining the system at the operating for long time as a result of wind disturbance and system inherent instability, the experiments are repeated several times to obtain sufficient data. The necessary states data are logged in real-time through a wireless transmission to a ground computer station. The collected data are separated into individual channel as roll, pitch, yaw and heave data corresponding to lateral cyclic, longitudinal cyclic, pedal and collective inputs excitation respectively. Since the linear model describes the change in input signals and the change in output signals, the data

                 

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are ’detrend’ by removing means, offsets and linear trend that may be present in the data. Then a low pass filter with 10Hz cut-off frequency was used to remove high frequency noise in the data. The quality of the collected data is examined by computing coherence factor which indicates how well an input corresponds to an output at each frequency. A coherence value of 0.8 and above is considered as indication of good data quality, [3]. For input x and output y, the coherence is given as ratio of the power spectral density Pxx and Pyy of x and y and the cross power spectral density Py of x and y: 2

(3.1)

Cxy =

|Pxy (f )| Pxx (f )Pyy (f )

In this study, a coherence of well above 0.8 has been obtained for all the data. However, due to the space constraint, the coherence plots cannot be presented here. 3.2. Parameters Estimation Procedures. Giving a parameterized function f Z k , ρ and a experimental data set Z N of length N , the goal in PEM-based sysID [12] is to determine the set of parameters ρb that minimizes the distance (error) between the predicted outputs and the measured outputs: ρbN = arg minρ VN (ρ) (3.2)

VN (ρ) =

N X

` (),

 = y (k) − yb (k, ρ)

k=1

Using a quadratic criterion ` () = 21 2 for SISO model, and ` () = 21 T Λ−1  for MIMO/SIMO model. where Λ is a positive semi definite matrix that weighs the relative importance of the component of . The default value is an identity matrix indicating equal importance of all the outputs. For a discrete state space parameterized model with measurement noise v (k): x (k + 1) = A (k) x (k) + B (k) δ (k) (3.3)

y (k) = C (k) x (k) + D (k) δ (k) + v (k)

The predicted output is computed using: (3.4)

yb (k, ρ) = Wy (z) y (k) + Wδ (z) δ (k)

where Wy (z) and Wy (z) are stable predictor filters defined as −1

B (ρ)

−1

K (ρ)

Wy (z) = C (ρ) [zI − A (ρ) + K (ρ) C (ρ)] (3.5)

Wδ (z) = C (ρ) [zI − A (ρ) + K (ρ) C (ρ)]

where K (ρ) is Kalman filter gain obtained from solution of the Riccati equation [12]. The optimization process in the computation of yb (k, ρ) requires that the predictor filters be stable, and the process is usually computationally intensive with many local minima. These factors make PEM algorithm highly sensitive to initial parameter values. To overcome these challenges, two major steps are taken in the identification process. First, the model is broken down into sub-systems comprise the four dynamics of the system: roll, pitch, yaw and heave. The on-axes terms are identified first and

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followed by the cross-coupling terms. The main stages are: identification of roll and pitch rotor dynamics terms, followed by the lateral and longitudinal translational motion terms, heave and yaw dynamics terms, and lastly cross-coupling terms of lateral and longitudinal with heave and yaw dynamics. Second, a MATLAB routine is developed to iterate the parameter values initialization as follows: 1. Generate randomly a population size NP of initial parameters for each unknown parameters, given lower and upper bounds: (3.6)

Pi = Li + (Hi − Li ) ∗ rand (1, N P )

Pi , Li and Hi are population, lower and upper bounds for parameters i. 2. Determine if the initial parameters gives a stable predictor: if yes process to model estimation using MATLAB inbuilt PEM algorithm, else, set the prediction error as infinity 3. Compute the prediction errors for all the resulting models and compare the performance 4. Select the best model: for a single output sub-system, this is based on the model with the least prediction error using ’min’ operator, while for multioutputs sub-system, a pareto dominance selection approach usually applied in evolutionary Multiobjective optimization [13] is adopted to ensure optimal compromise between the outputs. Thus, the vector of the prediction errors is treated as a multi-objective function, and Definition (3.1) applies. 5. The parameters of the selected model in step 4 is then used as initial guess for further iteration 6. Validate the model with different set of data from the one used in the identifcation process Definition 3.1. given y and y ∗ as two vector objective functions with K objectives for solutions x and x∗ respectively, such that y = (f1 (x) , f2 (x) , ...fK (x)) and y ∗ = (f1 (x∗ ) , f2 (x∗ ) , ...fK (x∗ )), then vector y ∗ dominates y if and only if y ∗ is partially less than y, that is: if ∀k ∈ {1, 2, ...K} , y ∗ ≤ y ∧ ∃k ∈ {1, 2, ...K} : y ∗ < y. 4. Results and Discussion. The identification process discussed in section 3 was employed to estimate the unknown parameters set of the UAV helicopter statespace model of (2.8). For on-axes parameters estimation, the lower and upper bounds are set at −100 and 100 respectively, while for cross-coupling terms, the bounds are −10 and 10 for lower and upper bounds. This is informed by the fact that the effect of cross-coupling terms is small compare to the on-axes terms, hence with expected small parameter values compare to the on-axes terms. The comparison of the predicted model outputs with measured outputs together with prediction error plot using a separate set of data that are not used during the identification process is performed. The root mean square values of the prediction error for the roll, pitch and yaw responses are 0.005 rad/s, 0.007 rad/s and 0.003 rad/s, respectively. Those of the lateral, longitudinal and the heave velocity are 0.1 m/s, 0.15 m/s and 0.6 m/s, respectively. This is generally due to unstable characteristics of the translational motion of the system, and identification of the associated parameters has been the most challenging part of the system identification. Nevertheless, the model is able to follow closely the actual system responses. 5. Conclusion. An effective PEM-based identification procedure has been reported for estimation of a small scale UAV helicopter parameterized model. The

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proposed procedure yields a model with satisfactory performance for the intending flight controller design. The challenges of initial parameter values together with complexity of the system model are addressed with this procedure. As part of the ongoing work, this model is to be used for the design of flight control law for autonomous hovering/low speed control. Also, similar model is to be developed for cruise flight at trim conditions. REFERENCES [1] Volker Remu, Carsten Deeg, Marek Musial, Gunter Hommel, Manuel Bejar, Federico Cuesta, and An?bal Ollero, Autonomous Helicopters, A. Ollero and I. Maza (Eds.): Mult. Hetero. Unmanned Aerial Vehi., STAR 37, pp. 111-146, 2007. springerlink.com Springer-Verlag Berlin Heidelberg (2007) [2] G. D. Padfield, Helicopters Flight Dynamics: The Theory and Application of Flying Qualities and simulation Modeling, Blackwell Science LTD, (1996). [3] B. F. Mettler, Identification modeling and characteristics of miniature rotorcraft, Kluwer Academic Publisher, (2003). [4] A. Budiyono,, First Principle Approach to Modeling of Small Scale Helicopter, International Conference on Intelligent Unmanned System, ICIUS, Bali, October 24-25, (2007). [5] A. Budiyono, T. Sudiyanto,and H. Lesmana, Global Linear Modeling of Small Scale Helicopter, in Budiyono,A., B. Riyanto, E. Joelianto, (Eds), 2009, Intelligent Unmanned Systems: Theory and Applications, Studies on Computational Intelligence, Vol. 192, Springer, Berlin-Germany, pp.27-62, (2009). [6] C. John Morris, Michiel van Nieuwstadt and Pascale Bendotti, Identification and Control of a Model Helicopter in Hover, in proceeding of the American Control Conference Baltimore,Marvland, June (1994). [7] S. K. Kim and D. M. Tilbury, Mathematical modelling and experimental identification of a model helicopter, Journal of Robotic Systems, 21(3):95-116, (2004). [8] M. La Civita, W. C. Messner, and T. Kanade, Modeling of small-scale helicopters with integrated first principles and system-identification techniques, in proceedings ofthe 58th Forum of the American Helicopter Society, volume 2, pages 2505-2516. Montreal, Canada, June (2002). [9] Vladislav Klein and A. Eugene Morelli, Aircraft System Identification: Theory and Practice, AIAA Education Series by AIAA, (2006), 1st edition. [10] Hardian Reza Dharmayanda, Agus Budiyono and Taesam Kang, State space identification and implementation of H-infinity control design for small-scale helicopter, Aircraft Engineering and Aerospace Technology: An International Journal vl.82 pg.6 (2010) 340-352, Emerald Group Publishing Limited. [11] Ismaila B. Tijani, Akmeliawati, Ari Legowo, Mahmud Iwan, and A. G. Abdul Muthalif, Matlab-based low-cost autopilot for UAV helicopter deployment, submitted for 8th International Conference on Intelligent Unmanned Systems (ICIUS-2012), SIM university, Singapore. [12] L. Ljung, System Identification: Theory For the User, Prentice Hall, Upper Saddle River, NJ, USA, 2nd edition, (1999). [13] Xin-SheYang, Engineering Optimization:An Introduction with Metaheuristic Applications, Wiley, A John Wiley & Sons, Inc., Publication,(2010).