Linear Aerodynamic Model Identification of a Flapping Wing MAV Based on Flight Test Data

Transcription

1 273 Linear Aerodynamic Model Identification of a Flapping Wing MAV Based on Flight Test Data J.V. Caetano *a, C.C. de Visser, G.C.H.E. de Croon, B. Remes, C. de Wagter, J. Verboom and M. Mulder Control and Simulation Section, Faculty of Aerospace Engineering, Delft University of Technology, The Netherlands. ABSTRACT This paper presents an approach to the system identification of the Delfly II Flapping Wing Micro Air Vehicle (FWMAV) using flight test data. It aims at providing simple FWMAV aerodynamic models that can be used in simulations as well as in nonlinear flight control systems. The undertaken methodology builds on normal aircraft system identification methods and extends these with techniques that are specific to FWMAV model identification. The entire aircraft model identification cycle is discussed covering the set-up and automatic execution of the flight test experiments, the aircraft states, the aerodynamic forces and moments reconstruction, the aerodynamic model structure selection, the parameter estimation and finally, the model validation. In particular, a motion capturing facility was used to record the flapper s position in time and from there compute the states and aerodynamic forces and moments that acted on it, assuming flapaveraged dynamics and linear aerodynamic model structures. It is shown that the approach leads to aerodynamic models that can predict the aerodynamic forces with high accuracy. Despite less accurate, the predictions of the aerodynamic moments still follow the general trend of the measured moments. Dynamic simulations based on the identified aerodynamic models show flight trajectories that closely match the ground truth spanning a number of flapping cycles. Finally, the dimensional aerodynamic forces and moments coefficients of two of the identified aerodynamic models are presented. NOMENCLATURE AFRL Air Force Research Laboratory CG Center of Gravity EOM Equations of Motion FWMAV Flapping Wing Micro Air Vehicle RC Radio Control µaviari Micro Air Vehicles Integration and Application Institute cov() covariance Rot(var) Rotation matrix of variable var Rot bi Rotation matrix from inertial to body frame Variables (L, M, N ) Aerodynamic Moments in the body axes (X, Y, Z ) Aerodynamic Forces in the body axes (ṗ, q, r ) Angular Accelerations in the body axes (u, v,w ) Linear Accelerations in the body axes (p, q, r) Angular Velocities in the body axes (u, v, w) Linear velocities in the body axes * corresponding author: j.v.caetano@tudelft.nl a Air Force Research Laboratory, Portuguese Air Force Academy, Sintra, Portugal Volume 5 Number 4 23

2 274 Linear Aerodynamic Model Identification of a Flapping Wing MAV Based on Flight Test Data C F C Fs F i N i R S V Affine Coefficient for a specific aerodynamic force or moment Coefficient for state s Aerodynamic Force or Moment i Total number of state observations Regression matrix States vector Total velocity Greek τ External tracking system recording frequency (φ, θ, ψ) Euler Angles (φ r, θ p, ψ y ) Attitude Angles for (roll,picth,yaw) α Angle of attack β Sideslip angle δ e Elevator input angle δ f Flapping frequency input δ r Rudder input angle ρ Pearson s correlation coefficient σ Variance Difference between two consecutive time-steps Vectors W LR Left-to-Right wing vector that connects the wing markers (i, j, k ) Unit vectors in the (x, y, z) axes (x, y, z ) Orthonormal axes Subscripts I Inertial Frame b Body Frame t time instant. INTRODUCTION It is envisioned that FWMAVs will occupy an existing gap in conventional fixed and rotary wing aircraft applications thanks to their agility, broad flight envelope, interesting behavior near obstacles and promising properties at very low Reynolds numbers. Especially for nano FWMAVs, for which flapping is believed to yield most advantages [, 3, 8, 25], the small size and mass heavily restrict the range of sensors and processing onboard. With a total take-off weight of only a few grams, attitude determination and flight control are still active areas of research [22, 32]. In this respect, a distinction has to be made between tailless and tailed designs. Tailless designs, which in general mimic insects, offer the possibility of performing highly dynamic maneuvers. However, these configurations require active stabilization, as they are passively unstable. So far, only two such configurations have been able show actual flight ability: the Nano Hummingbird [22] and the Robobee [24]. Tailed designs, which are similar to the configurations found in birds, benefit from being passively stable and do not require active stabilization. Examples of tailed FWMAVs include de Croon et al. (29)[8] and Baek (2)[3]. The advantage of these configurations is that the research can focus on higher-level problems and implementations, such as altitude control [2, ] or obstacle avoidance [9,, 3]. However, tailed designs are more sensitive to external perturbations and are hard to maintain stable throughout a broad flight envelope, i.e., from hover to fast forward flight. Hence, active control is still needed to further expand the flapper s flight envelope and fully explore its capabilities. The aforementioned approaches rely on manually tuned PID-controllers for flight control. However, the flight dynamics of FWMAV s are in general highly nonlinear across their flight envelope, requiring extensive gain scheduling schemes for full envelope control. The main reason for the current absence of model-based nonlinear control is the difficulty of designing a reliable model for FWMAVs. This derives from the not yet fully understood unsteady aerodynamics [7, 29, 28] associated with flapping wing flight. Furthermore the flapping of the wings International Journal of Micro Air Vehicles

3 J.V. Caetano, C.C. de Visser, G.C.H.E. de Croon, B. Remes, C. de Wagter, J. Verboom 275 and M. Mulder complicates the model structure as their added inertia effects contribute to the dynamics of the ornithopter[27]. A simple approach was taken by Dietl and Garcia [4, 6] that modelled the dynamics of a flapping wing aircraft using the general aircraft equations of motion and resulted in a simulated linear time-invariant alternative for the attitude control. This has its advantages due to simplicity since, on the other hand, a complete modeling approach [5, 27] typically results in a complex nonlinear timevariant multi-body representation of the ornithopter that is not suitable of being used onboard. On the experimental side, Grauer et al. [2, 2] succeeded in devising the kinematic equations of motion and was able to determine the Lift and Drag aerodynamic coefficients acting on a two wing birdlike ornithopter using experimental flight data. The previously mentioned theoretical approaches lack, however, a practical validation in the form of flight testing [5, 27, 5, 4, 6], as the models are limited to computational environments. Moreover, the experimental studies [2, 2] lack the repeatability of the data because the platform was not able to hover or stay aloft for enough time to perform specific flight test maneuvers that are necessary to perform system identification. More recently Caetano et al. [6] have been able to program a FWMAV to perform automatic maneuvers for system identification purposes and calculate the aerodynamic forces and moments that act on the FWMAV, using flight path reconstruction techniques and general aircraft equations of motion [7]. This study aims at bridging the gap between theoretical and experimental approaches and presents a set of benchmark linear aerodynamic models for the Delfly II FWMAV that were devised by applying aircraft system identification techniques and further extending them to flapping wing platforms. These linear aerodynamic models can be used to simulate Defly flight and form the basis of a future full flight envelope nonlinear control system for the Delfly. Our approach for the development of the aerodynamic models is presented in Figure. In particular, a high-fidelity sub-millimeter resolution external tracking system was used to directly measure the position and attitude of the ornithopter. The flight states were then reconstructed and used to calculate the aerodynamic forces and moments that acted on the flapper, under the assumptions of a rigid body and constant inertia properties (Section 2). Two linear aerodynamics models were devised and compared: () a full model that incorporates state variables reconstructed from the tracking system and (2) a reduced model that only includes state variables that may be measured or calculated directly from the existing onboard sensors (Section 3). The models are verified and validated using verification maneuvers and dynamic simulation in Section 4. The final conclusions are drawn in Section 5. Figure : Current contribution flowchart with associated framework. 2. SYSTEM OVERVIEW This section presents the Delfly II FWMAV and the experimental set-up that was used for the identification flight tests. 2. Delfly II Micro Air Vehicle The Delfly II [2] is a bio-inspired ornithopter, configured with 4 flapping wings and an inverted T tail. It weighs 7g and is capable of performing hovered flight as well as transitions to a maximum Volume 5 Number 4 23

4 276 Linear Aerodynamic Model Identification of a Flapping Wing MAV Based on Flight Test Data forward flight speed of 8 m/s and vice-versa. The version that was used in the flight tests was equipped with a Radio Control (RC) receiver for manual operation and a programmable autopilot that is capable of performing stable automatic flight at a trimmed configuration. A full description of the Delfly can be found in de Croon et al.[8] and Caetano et al.[6]. 2.2 Experimental Set-up The flight tests were conducted at the United States Air Force Research Laboratory (AFRL) Micro Air Vehicles Integration and Application Institute (µaviari), where the test chamber was equipped with a tracking system capable of recording the position of small reflective markers at a frequency of τ = 2Hz. The Delfly s position and attitude was tracked using eight markers placed on its structure, as presented in Figure 2. A set of longitudinal (elevator) inputs was designed and pre-programmed into its autopilot to assure a full dynamic response over all frequencies these were designed as steps, doublets and triplets with a reference time of 3 of a second. Furthermore, this method assured that the inputs were performed automatically guaranteeing the desired duration and flight regimes. More considerations about the markers positions and the maneuver input design can be found in Caetano et al. [6, 7]. 3. SYSTEM IDENTIFICATION FRAMEWORK This section presents the system identification framework that was used to go from the tracking system data to the linear models for the aerodynamic forces and moments. 3. Flight Path Reconstruction The first step in the system identification framework is the reconstruction of the Delfly s states from the recorded position data in time. This process is called flight path reconstruction [26] and aims at reconstructing the states that cannot be measured directly, such as the angle of attack, or to refine the accuracy of sensor measured states. 3.. Reference Frames In order to obtain the Delfly s states, each of the markers coordinates has to be described in the Delfly s body frame. The tracking system s reference frame will be addressed as being the inertial frame (subscript I) with the z I axis defined as vertical and positive up and the x I and y I axes in the horizontal plane creating a right-handed orthonormal frame with z I. The body reference frame (subscript b) was defined using an intermediate marker reference frame: the x b axis is positive forward; the z b is defined by the cross product of the unit vector of x b with the unit vector that is defined by the left-to-right wing marker vector (W LR), thus zb = normalized (i b W LR). The yb axis was then defined by the cross product of z b x b. The origin of the body frame is at the Delfly s Center of Gravity (CG). Figure 3 presents both the inertial (x I, y I, z I ) and body frames ( x b, y b, z b ). The rotation matrices between the frames were obtained using a direct cosine matrix. Eq. represents the rotation matrix from the inertial frame to the body frame, in the way that a vector can be calculated in the body frame by multiplying the rotation matrix Rot bi with that vector s coordinates written in the inertial frame. The Euler angles can be determined for each time step by corresponding the widely used 32 rotation matrix [4] to the markers coordinates in the inertial frame. The 32 rotation follows the yaw (ψ) pitch (θ) roll (φ) sequence. One can easily obtain Rot bi as a function of (φ, θ, ψ) by multiplying Rot (φ) Rot (θ) Rot (ψ). The result is presented in Eq. 2 and the (φ, θ, ψ) angles can be determined by relating the matrices entries in Eqs. and 2. () (2) International Journal of Micro Air Vehicles

5 J.V. Caetano, C.C. de Visser, G.C.H.E. de Croon, B. Remes, C. de Wagter, J. Verboom 277 and M. Mulder Figure 2: Markers positions on the Delfly. Figure 3: Inertial and body reference frames 3..2 State Reconstruction The Euler angle definition is limited to specific intervals [4]. For the Delfly case, in a near-hover configuration, an elevator input can induce the transition to an inverted flight regime, where z b is above the horizontal plane, pointing up. In this situation the ψ and φ have a sudden variation of approximately 8, as presented in Figure 6 due to the change in heading induced by the inverted flight. In order to keep the Euler relations valid to be used in the general aircraft equations of motion (EOM) [9, 7, 3], the Euler angles were converted to new attitude angles φ r (roll), θ p (pitch) and ψ y (yaw) that can count above 9 in pitch. This allows the dynamic equations to be defined for pitch angles in the interval] 2 π ; 3π [ not needing alternative formulations in terms of, e.g., quaternions. 2 The developed routine evaluated the change in the computed Euler angles: if the ψ and φ angles rotated of values close to 8 (thus too big to be caused by a turn) and the z b was pointing upwards in the inertial frame the θ p would be equal to (π + θ), while assuming ψ y and φ r to be equal to the de-rotated initial value that caused the detection of the inverted flight regime. The angular rates (p, q, r) were obtained using discrete attitude measurements that where then differentiated into turn rates using the tracking system sampling rate (τ), as indicated in Eq. 3. The linear velocities (u, v, w) and accelerations (u, v, w ), as well as the body angular accelerations (ṗ, q, ṙ) were obtained by differentiating the markers positions and body rates, respectively, with respect to time. Table presents all the states that were computed the variables follow the commonly accepted definition in aeronautical engineering [9]. (3) To decrease the influence of the high oscillatory modes induced by the flapping frequency, the states were filtered using a 3 rd order zero-phase lag Butterworth low-pass filter with cut-off frequency of 3Hz, thus conserving the information up to the second flapping frequency harmonic. Table : Reconstructed states, using the flight data and respective control surface inputs. Volume 5 Number 4 23

6 278 Linear Aerodynamic Model Identification of a Flapping Wing MAV Based on Flight Test Data The states evolutions around an elevator step input are presented in Figures 4 to 7. These are somewhat constant before the input, being dampened after the input ceases. The elevator input had a period of 3 of a second deflected upwards (Figure 4). Here the rudder oscillation (middle plot) is related to the tail s rotation and bending excited by the flapping, as no input over the rudder was performed by the autopilot. The abscissa axes are defined in seconds with its interval selected to present the FWMAV s response to a step input on the elevator. Figure 5 presents the evolution of the velocities in the body frame: the velocity component u decreases to a negative value as the Delfly loses lift and height. The w component varies due to the pitch angle oscillation when w goes to negative means the Delfly is flying in the opposite direction, in inverted flight. As expected, v does not suffer considerable variations during the longitudinal maneuver. Figure 6 presents the Euler (red) and the attitude (blue) angles variation around the maneuver. The angular velocities in the body frame are depicted in Figure 7 here one can see the considerably high oscillatory behavior of the Delfly caused by the flapping frequency due to its low mass and inertia. e ( ) 3 5 Inputs u (ms ).5 Velocities r ( ) 2 v (ms ) f (Hz) w (ms ) Figure 4: Control surface inputs: elevator (top) and rudder (middle) deflection angles, defined as positive down and left, respectively; flapping frequency (bottom), with step variations due to discrete time calculations. Figure 5: Velocities written in the body reference frame (i b, j b, k b ). The velocities oscillations due to the input are dampened in less than 5 seconds. 3.2 Aerodynamic Model Identification The second step in the system identification framework is aerodynamic model identification. This identification aims at creating a model that relates the control inputs to the resulting aerodynamic forces and moments. This way the aerodynamic models can then be used in model-based flight control systems. International Journal of Micro Air Vehicles

7 J.V. Caetano, C.C. de Visser, G.C.H.E. de Croon, B. Remes, C. de Wagter, J. Verboom 279 and M. Mulder ( ) 2 Euler and Attitude Angles r p (s ) 3.5 Angular Velocities ( ) 2 p q (s ) ( ) 2 y r (s ) Figure 6: Euler angles (red) and attitude angles (blue) around the maneuver Figure 7: Angular velocities in the body frame around the maneuver Dynamics Model Creating accurate full flight envelope aerodynamic models of aircraft is a highly challenging task, espe cially when the aircraft is subject to significant nonlinear aerodynamics. However, the nonlinear aerody namics can be approximated with linear models [23] for small excitations around a stationary trimmed trajectory. In this work ornithopter was flying in a stationary regime when the inputs were performed. Under these premises the linear modeling approach is valid. Within a well know constant static margin the Delfiy is an open-loop stable platform in slow forward flight. This allows for the control strategies to focus on flapping cycle averaged forces and moments. This way the dynamics of the platform were assumed to be described by the Newton-Euler equations of motion found in [23] or [7], under the assumptions of: () rigid body kinematics; (2) no flapping (the flapping is then modelled as a thrust force); (3) constant mass and no inertia changes due to flapping or bending; (4) fiat Earth; (5) no wind. Despite neglecting the flapping behavior and kinematics, these assumptions can be used because the wing phases and angles are neither used for stabilization nor for attitude control on the Delfiy Aerodynamic Model Structure Selection Several linear model structures of the form of Eq. 4 were devised. Here the left-hand side term F i represents the forces and moments obtained from the Newton-Euler equations of motion (mentioned in Section 3.2.); the first term on the right-hand side, C F, is the affine coefficient; S represents a state and C Fs, the state s coefficient or parameter for a given force or moment F i. The flight test data was divided into identification and validation sets. The identification set is used to estimate the aerodynamic parameters; the validation set is used to verify if the estimated model is able to represent the aerodynamic forces and moments. The full model structure was defined such that each of the aerodynamic forces (X, Y, Z) and moments (L, M, N) is a linear function of all the states, as in Eq. 4. The reduced model structure was defined such that it requires only states that are measurable using on-board sensors, with the goal of using it in a nonlinear flight controller. In particular, the coefficients of Eq. 4 were substituted in the reduced model equation, presented in Eq. 5. (4) Equation 4 Volume 5 Number 4 23

8 28 Linear Aerodynamic Model Identification of a Flapping Wing MAV Based on Flight Test Data (5) 4. RESULTS The Parameter Estimation and Validation boxes from the Figure will be addressed in the present section. 4. Parameter Estimation An ordinary least squares estimator was used to estimate the parameters of the linear models. Eq. 7 is an example for the estimation of the parameters of the aerodynamics for X, with the regression matrix R containing a total of N i observations: (7) Hence, the models parameters were estimated for a total of five longitudinal maneuvers that covered step, doublet and triplet inputs 2 on the elevator for near-hover and slow forward flight regimes (V.5m/s). It is interesting to notice that despite the different maneuvers, the estimated aerodynamic coefficients were found to have very similar values, with best results for the reduced model. The coefficient s average and standard deviation values for each model and force are presented in Table 2. Furthermore, the analysis of the covariance matrix showed that the coefficients with higher variances also had the higher standard deviations across the maneuvers. This proximity between the estimated coefficients allows for a single aerodynamic model to be used for similar flight regimes, with beneficial consequences in the control strategies. 4.2 Model Validation The models were validated using two different approaches: () by means of validation maneuvers in which the parameters that were estimated in the identification cycle were used to predict the aerodynamic forces and moments of other maneuvers; (2) using a dynamic simulation, where the identified aerodynamic models were used in a nonlinear dynamics simulator to reconstruct the Delfly s flight path and states and then compare them with the original flight path and states that were recorded in the flight tests Validation Maneuvers This subsection presents the aerodynamic forces and moments estimation results for both the full and the reduced models. For the sake of coherence and length, the estimated aerodynamic forces and moments will be presented for the test case already described in Figures 4 to 7. Hence, Figure 8 presents the graphical evolution of the filtered forces (8a) and moments (8b) for the validation part of the system identification cycle. The blue lines represent the forces and moments calculated using the aircraft EOM; the red lines depict the full linear model s evolution; whereas the green lines describe the reduced model s behavior. 2 the maneuver input design is fully described in Caetano et al.[6]. International Journal of Micro Air Vehicles

9 J.V. Caetano, C.C. de Visser, G.C.H.E. de Croon, B. Remes, C. de Wagter, J. Verboom 28 and M. Mulder Table 2: Average and standard deviation of the estimated coefficients for both the full and the reduced models for a total of 5 elevator input maneuvers. The C Fs form in Eq. 4 was replaced by a more general C s for convenience of representation. Both the full and the reduced models were able to predict all three aerodynamic forces (Figure 8a). The variation in the curves for the peak in the X force results from the fact that the X force that was computed from the EOM is highly affected by the vertical velocity component w, which goes to negative values when the Delfly flies inverted. As for the X force, the models are able to predict the Z force with a high accuracy. Moreover, the Y force does not vary considerably in the (longitudinal) maneuver, as no rudder input was present and, despite some punctual differences, it is also well predicted by both models. However, none of the models predict the aerodynamic moments evolution (Figure 8b) as good as they did for the forces. Nevertheless, the most important moment for longitudinal dynamics analysis, the pitching moment, is also most accurately predicted. Despite these less than optimal results, the predicted moments show a cycle averaged behavior that is similar to that of the moments calculated directly from the data using the EOM s. This points to a possible application for onboard control, using online filtering of the states or a more refined linear model. A quantitative measurement of the performance of the model is given by calculating Pearson s correlation coefficient between each of the model s estimations and the forces and moments that were computed from the EOM. The correlation coefficient captures how two signals vary with respect to their means and is defined as. The best performance would be the highest correlation of ρ =, while completely decorrelated signals would give ρ =. Table 3 shows the correlation coefficients for the full model and reduced model. It can be seen that the predicted forces X, Y, and Z are closely correlated to the calculated forces with ρ [.85,.99]. The moment predictions are still reasonably correlated for the full model (ρ [.39,.62]), but only slightly correlated for the reduced model ρ [.4,.43]. Table 3: Pearson s Correlation Coefficient between each of the linear models and the forces and moments that were calculated from the aircraft EOMs. Volume 5 Number 4 23

10 282 Linear Aerodynamic Model Identification of a Flapping Wing MAV Based on Flight Test Data Forces 2 x Moments.2.8 X (N).6.4 L(Nm) EOM Full Reduced EOM Full Reduced..2.5 Y (N) M(Nm) x.5 Z (N) N(Nm) Figure 8: Calculated (blue) versus estimated forces and moments for both the full (red) and reduced (green) models, around the elevator step input maneuver. The aerodynamic forces are very well estimated; the aerodynamic moments are still able to follow the calculated moments with a similar behavior, presenting the same evolution and tendency, despite punctual changes Dynamic Simulation The second model validation method involved the reconstruction of the omithopter s flight path. Here the identification coefficients (presented in Table 2) were used to calculate the aerodynamic forces and moments that were input in the nonlinear equations of motion to compute the new state derivatives, which were integrated resulting in the new states. The method is very well documented in other sources, i.e. Stevens(23, p.)[3] and uses the general aircraft equations of motion. The inertia properties that were used in the simulation can be found in Caetano et al. [6]. The control surface deflections that were recorded during the flight tests were used as inputs to the simulation. The initial time was selected to assure low inputs and small rates in the initial states, as well as to be coherent with the time-frame that was presented in Figures 4 to 8. The initial states and inputs are presented in Table 4. Despite the good predictions obtained for the aerodynamic forces and moments reconstruction, pre sented in Figure 9, the numerical results point to instability in the method for simulations that lasted longer.5 seconds. In particular, the full model is able to predict the Delfly s flight path for half a second but the reduced model starts diverging after the first flapping cycle. Figures and present the dynamic simulation results. The first (Figure ) presents the real tra jectory of the Delfly in the inertial frame versus the trajectories that were reconstructed using both the full and the reduced models, for two different lapsed times:.5 and.8 seconds. Here the difference in the trajectories of the reduced model (green) and real recorded one (blue) is clearly noticeable. Nevertheless, the full model is able to follow the trajectory with minor errors for around half a second, only diverging from that point on. The small differences in the aerodynamic forces and moments influence the states evolution, especially the angular velocities and, consequently, the Euler angles. Figure presents the first half of a second of the inputs and states evolution where the reconstructed states diverge away from the real states, especially at second 6. (the time at which the elevator input moves the platform away from the equilibrium). As before, the full model was able to follow the states evolution more closely than the reduced model. Keeping in mind that the motivation behind the current approach was the limited onboard computational resources and that the linear model was intended for onboard control purposes, this study points to the possibility of onboard nonlinear control of the Delfly (or a similar passively stable platform) using a linear aerodynamic model, given the f act that the states are being updated in real time to the controller. International Journal of Micro Air Vehicles

11 J.V. Caetano, C.C. de Visser, G.C.H.E. de Croon, B. Remes, C. de Wagter, J. Verboom 283 and M. Mulder.2 Forces 2 x Moments X (N).5 L (Nm) Y (N) EOM Full Reduced M (Nm) EOM Full Reduced x Z (N) N (Nm) Figure 9: Reconstructed Forces and Moments that were used to compute the new states. Real Full Reduced Real Full Reduced Z (m) 2.39 Z (m) Y (m) X (m) Y (m) X (m).3 Figure : Reconstructed flight path using the fttllll (red) and reduced (green) models coefficients com pared to the original (blue) recorded path. The full model is able to predict the Delfly s flight path evolution for approximately half a second, whereas the reduced model starts diverging at the first iterations. Table 4: Initial states and inputs for the states reconstruction and dynamic simulation, at t = 5.85sec Volume 5 Number 4 23?2

12 284 Linear Aerodynamic Model Identification of a Flapping Wing MAV Based on Flight Test Data e ( ) 2 Inputs u (ms ).5 Velocities r ( ) 2 v (ms ).5 Real Full Reduced f (Hz) w (ms ) ( ) 5 Euler Angles p (s ) 3.5 Angular Velocities ( ) Real Full Reduced q (s ) 3.5 Real Full Reduced ( ) r (s ) Figure : Simulated states (in subfigures b to d) that were computed using the input sequence (in a) for full (red) and reduced (green) models versus the real (blue) captured states. 5. CONCLUSION System Identification techniques were used to create a nonlinear flight dynamics model of the Delfly II FWMAV. For this, two linear aerodynamic models were devised and tested in the framework. The more complex model used a total of states and 3 inputs, whereas the reduced model used only the states that could be measured by onboard sensors and 3 inputs. After the estimation of the models parameters it was seen that both of them were able to predict the aerodynamic forces with great accuracy, showing correlation factors that ranged from.85 to.99. The results were less optimal for the aerodynamic moments, where the correlations where smaller. However, both models were still able to follow the cycle average evolution described by the calculated aerodynamic moments. The models were then used to predict the new states of the ornithopter in a nonlinear dynamic simulator. This simulator used the real inputs sequence to predict the Delfly s new states, based on a real initial condition. The results showed that it was possible to reconstruct its flight path and attitude with considerable accuracy for the initial parts of the simulation. However, the small differences between the real and predicted aerodynamic forces and moments cause the nonlinear simulation to International Journal of Micro Air Vehicles

13 J.V. Caetano, C.C. de Visser, G.C.H.E. de Croon, B. Remes, C. de Wagter, J. Verboom 285 and M. Mulder diverge from the ground-truth within.5 seconds. These results point to the possible use of linear aerodynamic models for model based onboard flight control. However, for flight simulation purposes of longer periods of time the aerodynamic forces and moments have to be predicted more accurately. Furthermore, the model of the moments is not accurate enough to be used in nonlinear simulations. This is expected firstly because the angle rates suffer from noise amplification due to the time differentiation of the states and secondly because the moment estimation correlation coefficients are low, often even below.5. Concerning the forces estimate it can be seen that the assumption of no-flapping is likely to account for a lot of the remaining model errors. In this regard, future research will focus on assessing the influence of different linear models that use more flapping wing vehicle relevant states, i.e. the flapping phase. Future analyses will encompass the use of more advanced modeling methods, such as multivariate splines[]. Additionally, the modeling efforts will concentrate on creating global aerodynamic models which are valid on the full flight envelope. Finally, a nonlinear control system will be developed that can control the Delfly over its entire flight envelope. 6. ACKNOWLEDGEMENTS The authors would like to thank the United States AFRL µaviari for the use of their motion capturing facility. REFERENCES [] S. Ansari, R. Zbikowski, and K. Knowles. Aerodynamic modelling of insect-like flapping flight for micro air vehicles. Progress in Aerospace Sciences, 42(2):29 72, 26. [2] S. Baek and R. Fearing. Flight forces and altitude regulation of 2 gram i-bird. In IEEE RAS and EMBS International Conference on Biomedical Robotics and Biomechatronics (BioRob), pages , 2. [3] S. S. Baek. Autonomous ornithopter flight with sensor-based behavior. Univ. California, Berkeley, Tech. Rep. UCB/EECS-2-65, 2. [4] H. Baruh. Analytical Dynamics. McGraw-Hill Higher Education, 999. [5] M. A. Bolender. Rigid multi-body equations-of-motion for flapping wing mavs using kanes equations. In AIAA Guidance, Navigation, and Control Conference, August 29. [6] J. Caetano, C. C. de Visser, B. Remes, C. de Wagter, and M. Mulder. Controlled flight maneuvers of a flapping wing micro air vehicle: a step towards the delfly ii identification. In AIAA Atmospheric Flight Mechanics Conference, 23. [7] J Caetano, C. C. de Visser, B. Remes, C. de Wagter, and M. Mulder. Modeling a flappingwing mav: Flight path reconstruction of the delfly ii. AIAA, 23. [8] G. De Croon, K. De Clercq, R. Ruijsink, B. Remes, and C. De Wagter. Design, aerodynamics, and vision-based control of the delfly. International Journal of Micro Air Vehicles, (2):7 97, 29. [9] G. de Croon, E. de Weerdt, C. de Wagter, B. Remes, and R. Ruijsink. The appearance variation cue for obstacle avoidance. IEEE Transactions on Robotics, 28(2): , 22. [] G. de Croon, Groen, M., C. de Wagter, B. Remes, R. Ruijsink, and B. van Oudheusden. (accepted) design, aerodynamics, and autonomy of the delfly. In Bioinspiration and Biomimetics, 22. [] C. C. de Visser, Q. Chu, and J. A. Mulder. A new approach to linear regression with multivariate splines. Automatica, 42(2): , December 29. [2] Delfly Team. Delfly. accessed on January 5th, 23. [3] M. H. Dickinson and S. P. S. F. O. Lehmann. Wing rotation and the aerodynamic basis of insect flight. Science, 284:954 96, 999. [4] J. M. Dietl and E. Garcia. Stability in ornithopter longitudinal flight dynamics. Journal of Guidance, Control and Dynamics, 3:57 62, 28. [5] J. M. Dietl and E. Garcia. Ornithopter optimal trajectory control. Aerospace Science and Technology, In press, 23. [6] J. M. Dietl, T. Herrmann, G. Reich, and E. Garcia. Dynamic modeling, testing, and stability analysis of an ornithoptic blimp. Journal of Bionic Engineering, 8(8): , 2. Volume 5 Number 4 23

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