Abstract
This paper presents the nonlinear system identification of a slightly flexible 25 kg fixedwing UAV in the timedomain using a computationally efficient distributed aerodynamics model and a linear structural dynamics representation. The equations of motion are formed by making use of the free vibration modes of the structure and the mean axes formulation. The structural modes and mode shapes are determined from ground vibration tests. The distributed aerodynamics, accounting for elastic deformations, are modeled using a quasisteady stability and control derivative approach and by applying strip theory. Initial distributions for the derivatives are obtained from vortexlatticemethod calculations. For matching the model response to the measured response, parameters for scaling the initial derivative distributions are introduced. The flexible model is subsequently identified based on flight test data using the output error method in the timedomain and maximumlikelihood estimation. A good overall identification result is achieved with a close match of the fast aircraft dynamics. Finally, an evaluation is given on the suitability of the identified model for realtime simulation, loads’ estimation, and active load control law design.
1 Introduction
Main drivers in aircraft development, in addition to safety, relate to the need of an increased fuel efficiency to reduce operational cost and fulfill increasingly demanding environmental regulations. The efforts have led to constructing aircraft with lighter structures and higher aspect ratio wings which thus become more flexible. Due to the increase in flexibility, such wings are characterized by lower natural frequencies and exhibit increased inflight deformations. These can lead to a coupling of rigidbody motion and elastic deformation through the aerodynamic forces and moments [1, 2]. Flight control systems further complicate the interaction, possibly leading to degraded [3] or unstable [4] control performance. Therefore, these interactions need to be considered in the design models for flight control systems.
Depending on the characteristics of the aircraft, the purpose of application, or the availability of experimental data, different modeling frameworks are required. In the context of this work, the objective is the development of an aeroelastic model for a slightly flexible 25 kg fixedwing UAV that can be adapted based on flight test data using system identification techniques in the timedomain. The model is intended for realtime simulation, loads’ estimation, and active load control law design. Therefore, a model of moderate complexity with a limited number of state variables is required. Beyond that, it is desired to maintain a distributed aerodynamics model, such that local forces and moments can be calculated.
An overview of different aerodynamics modeling and system identification activities including applications for flexible aircraft is given in [5]. The modeling of flexible aircraft is suggested with an additional dynamicpressuredependent part associated with each stability and control derivative of a classical model structure. However, this approach is only valid provided that the rigidbody and structural frequencies are sufficiently separated. A unified framework for modeling flexible aircraft is presented in [6] and later in [7]. The equations of motion are derived by Lagrange’s equation and the principle of virtual work. Further, the authors make use of a modal representation of the structure and the mean axes constraints to minimize inertial coupling between the rigidbody and elastic degrees of freedom (assuming small deformations). The nonlinear equations of motion of the flexible aircraft then simplify and are presented in terms of the nonlinear equations of the rigidbody motion and additional linear differential equations for the modal deflections of the structure. The equations are solely coupled by the aerodynamic forces and moments. The modeling of aerodynamics is further proposed with a quasisteady stability and control derivative approach and by applying strip theory. Strip theory is a simple method often found in models concerned with the investigation of wake vortex encounter, namely aerodynamic interaction models (AIM) [8,9,10]. In strip theory, the lifting surfaces of the aircraft are divided into spanwise strips. Each strip is then treated as a twodimensional airfoil with its local geometric and aerodynamic characteristics. This approach allows for the resolution of local relative flow conditions and the resulting change in force and moment distributions. However, the effects of threedimensional finite span wing are neglected. Outside aerodynamic interaction models, the application of strip theory is, e.g., found in realtime fullenvelope aerodynamic models for small UAVs [11]. The modeling methodology of [6] was further adopted in later works by various authors, in [12] together with quasisteady strip aerodynamics, and in [13, 14] together with unsteady aerodynamics using modified strip theory [15, 16] to account for the effects of finite span and indicial functions [17, 18]. The work of [12, 13] both aimed at modeling only incremental aerodynamics due to elastic deformations, assuming the availability of a rigidbody flight mechanics and aerodynamics model. System identification techniques for flexible aircraft in combination with the framework of [6, 7] can be found in [19, 20] for the identification of a highperformance glider. Although keeping the quasisteady derivative approach for modeling the aerodynamic forces, moments, and generalized loads, the derivatives are not determined by applying strip theory, but directly estimated as parameters within the parameter estimation process. This modeling approach is a the straightforward extension of the traditional rigidbody approach to a flexible structure, which can be easily combined with system identification techniques. However, the distributed property of the aerodynamics model is lost. Other applications of system identification for flexible aircraft are found, e.g., in [21] for the inflight identification of structural modes using an eigensystem realization algorithm (ERA).
Given the review on the literature, the framework of [6, 7] is a powerful method for modeling slightly flexible aircraft. Further, strip theory is simple yet effective for modeling local aerodynamic forces and moments. The objective of this work is to maintain this strip aerodynamics model while allowing for the adaption of the model using system identification techniques in the timedomain. The paper is structured as follows. Section 2 introduces the slightly flexible unmanned test aircraft GFlights Dimona and summarizes the test activities that were performed. Section 3 describes the modeling of the test aircraft. Within this section, the equations of motion are developed using the free vibration modes of the structure and the mean axes formulation of [6]. Further, the modeling of quasisteady strip aerodynamics is explained. Finally, Sect. 4 presents the identification and evaluation of the flexible model.
2 Unmanned test aircraft GFlights Dimona
The slightly flexible test aircraft GFlights Dimona, depicted in Fig. 1, is an unmanned replica of the HK36 Super Dimona at a scale of 1:3. It is driven by an electrical motor with a maximum power of 4kW, and has a total mass of 25 kg and a length of 2.4 m. The original wings of the aircraft have been replaced by custom spar, rib, and foil manufactured wings with increased span and flexibility. The total wingspan of the aircraft is 5.4 m with a total wing surface area of 1.68 m\(^2\). The aircraft can either be controlled by a safety pilot via remote control or by a flight control computer. It is equipped with a total of four control surfaces at the trailing edge of each wing: inner/outer ailerons and inner/outer flaperons, which are used for lateral–directional control and for load control. The aircraft further comprises a rudder for lateral–directional control and two elevators for longitudinal control and load control.
2.1 Test instrumentation
The aircraft is equipped with the standard test instrumentation of a rigidbody aircraft and an additional test instrumentation integrated into the structure of fuselage, wings, and empennage. The standard test instrumentation comprises several sensors, computers, and radio equipment. An industrygrade highprecision inertial navigation platform (INS) supported by dual antennas is used for the measurement of GPS position, attitude, velocities, and accelerations. An increased position accuracy of 0.02 m is achieved with a third antenna on ground providing differential GPS correction. Airspeed, angle of attack, angle of sideslip, static air pressure, and air temperature are measured by three air data systems (ADS). They each consist of a fiveholeprobe with an additional inertial measurement unit (IMU) and temperature sensor for measurement correction. The airdata systems are located at the left (ADS1) and the right (ADS2) wing as well as at the vertical tail (ADS3). The air data systems were developed inhouse and calibrated in wind tunnel experiments [22]. The standard test instrumentation is completed by a data recorder computer and separate realtime flight control computer. The flight control computer serves as the main host for GNC applications and issues control outputs to the servo actuators. Further, it supports direct code deployment from Matlab/Simulink. Data distribution between the sensors and computers is implemented via Ethernetnetwork and Controller Area Network (CAN). Table 1 lists the available measurement parameters of the standard test instrumentation.
The additional test instrumentation is distributed at various sections along the fuselage, wings, and empennage to measure structural dynamics and loads. It comprises several IMUs and strain gauges (shear, bending, and torsion) which are concentrated in load measurement stations (LMS). Figure 2 displays the distribution of the sensors along the test aircraft. Since the measurement of loads and structural dynamics for wings and empannage is of prime importance, most sensors are located in these areas.
2.2 Ground vibration test and modal analysis
Considering the slight flexibility of the test aircraft, the assumption of linear structural dynamics is reasonable. Therefore, it was decided to represent the elastic deformation of the structure by a set of free vibration modes and mode shapes. To determine the structural modes of the aircraft, a ground vibration test (GVT) and experimental modal analysis was performed at the Technische Universität Berlin. Hammer and shaker inputs were used to excite the structural vibrations of the aircraft. The first seven structural modes up to the first symmetric wing torsion mode were identified and considered for the structural dynamics model. Details on the measurement setup, test execution, and results are presented in [23]. The structural mode shapes were subsequently determined from a finiteelement (FE) model adapted to the GVT data. Table 2 lists the seven structural modes with modal frequency \(\omega _{n}\) and modal damping \(\xi \). Figure 3 further visualizes the modes. Especially, the first symmetric wing bending mode exhibits a low natural frequency. Considering the servo actuator bandwidth of 6.74 Hz, it is the mode likely to be excited by control action or atmospheric disturbance.
2.3 Flight tests
A comprehensive flight test campaign was performed to acquire flight data for system identification. The campaign included a total of seven flights with 109 maneuvers performed. All maneuvers were executed with the help of an inhouse developed autopilot for aerodynamic parameter identification [24], which was adapted based on an a priori estimation of the rigidbody dynamics. Details on the overall concept and main components of the test setup are given in [25]. A variety of classical maneuvers specifically designed for the identification of the rigidbody dynamics were flown [26]. They comprised multiple longitudinal maneuvers including 3211 maneuvers, pulse maneuvers and level deceleration maneuvers. Further, different lateral–directional maneuvers were performed including banktobank maneuvers and rudder doublets. A full description of the maneuver characteristics is presented in [24]. The longitudinal maneuvers were executed by the elevators, while the lateral–directional maneuvers were executed by the ailerons, flaperons, and rudder. To identify the individual control effectiveness of ailerons and flaperons, four different control configurations were varied within the banktobank maneuvers: execution of the maneuvers by (1) inner and outer ailerons, (2) only inner ailerons, (3) only outer ailerons, and (4) inner and outer flaperons. The maneuvers were ultimately repeated around different trim velocities in the range of \(V_{TAS} =\) 18–30 m/s.
3 Flexible aircraft model
This section describes the flexible aircraft model. The modeling methodology is based on the framework of [6] and strip theory. The equations of motion are developed by representing the structural dynamics with free vibration modes and making use of the mean axes constraints. The aerodynamic strip forces and moments are modeled using a quasisteady stability and control derivative approach.
3.1 Equations of motion
The position \({\varvec{r}}\) of an arbitrary mass element \(\rho \textrm{d}V\) of an elastic body can be expressed in an inertial reference frame \(O_{I}\) (\(x_{I},y_{I},z_{I}\)) in terms of its relative position \({\varvec{p}}\) to a bodyreference frame \(O_{B^{R}}\) (x, y, z) and the position \({\varvec{r}}_{0}\) of the origin of \(O_{B^{R}}\) (see Fig. 4). The specified positions of the mass element in the inertial and bodyreference frames are then related by the expression \({\varvec{r}} = {\varvec{r}}_{0}+{\varvec{p}}\). The bodyreference axes \(O_{B^{R}}\) may rotate with an angular velocity \(\varvec{\omega }\) and their orientation may be defined by an arbitrary Euler angle sequence. The axes move with the body but are not necessarily attached to a material point.
Assume that a modal description of the structure is available (e.g., from ground vibration tests and finiteelement model), such that the elastic deformation \({\varvec{d}}\) at a point (x, y, z) can be expressed in terms of mode shapes \(\varvec{\Phi }_{j}(x,y,z)\) and the generalized displacement coordinates \(\eta _{j}(t)\) by
where \(n_{e}\) denotes the number of elastic modes of the unconstrained, undamped vibration problem. Then, the position of the mass element relative to \(O_{B^{R}}\) can be separated into its undeformed part \({\varvec{s}}(x,y,z)\) (rigid body) and its deformation part \({\varvec{d}}(x,y,z,t)\). The bodyreference frame \(O_{B^{R}}\) may be used to develop the equations of motion of the unconstrained elastic body, assuming that each mass element is treated as a point mass. When doing so, inertial coupling can occur between the rigid degrees of freedom and the elastic degrees of freedom. However, it is found that with a suitable choice of \(O_{B^{R}}\) satisfying the mean axes constraints and the origin of \(O_{B^{R}}\) located at the instantaneous center of mass, the inertial coupling can be neglected [6, 27].
The mean axes, first introduced by [28], define a body frame for which the relative translational and angular momenta, due to the elastic deformation, are zero for all time \(t \ge 0\). Combined with the modal description of the structure and only assuming small deformations (i.e., deformation and deformation rate are colinear), the constraints can be expressed by [6, 7]
It can be shown that for the constraints to be satisfied, the elastic degrees of freedom must be orthogonal to the translational and rotational rigidbody degrees of freedom, respectively [6, 29]. These conditions are guaranteed when the elastic deformations are represented by free vibration modes and the bodyreference frame \(O_{B^{R}}\) is located at the instantaneous center of mass. The decoupled equations of motion of the elastic body are then given by [6, 13]
The first two equations are formally equivalent to the standard nonlinear rigidbody flight dynamic equations for the translational and rotational degrees of freedom. Within these equations, \({\varvec{V}} \vert _{B^{R}}\) and \(\varvec{\omega } \vert _{B^{R}}\) denote the translational and angular velocity vectors of the bodyreference axes \(O_{B^{R}}\) with respect to the inertial reference axes \(O_{I}\), expressed in bodyreference coordinates. The gravity acceleration vector \({\varvec{G}} \vert _{I}\) is expressed in \(O_{I}\) coordinates and transformed to \(O_{B^{R}}\) with the transformation matrix \({\varvec{T}}_{B^{R}I}\). The aircraft mass is m and the inertial tensor \({\varvec{J}}\). Note that with the assumption of small deformations, \({\varvec{J}}\) is considered constant. The vectors \({\varvec{F}}^{ext} \vert _{B^{R}}\) and \({\varvec{M}}^{ext} \vert _{B^{R}}\) denote the sum of external forces and moments expressed in coordinates of \(O_{B^{R}}\). Only aerodynamic forces are considered in this work. The last Eq. 5 represents the \(n_{e}\) linear differential equations for the structural dynamics in generalized coordinates. Within these equations, \(\eta _{j}\) denotes the generalized displacement, \(\omega _{n,j}\) the undamped natural frequency, \(\xi _{j}\) the modal damping ratio, \(\mu _{j}\) the generalized mass, and \(Q_{\eta _{j}}\) the generalized forces of each mode. Given the nonlinear rigidbody equations of motion and the linear equations of motion of the structure decoupled as presented, a coupling is solely due to \({\varvec{F}}^{ext} \vert _{B^{R}}\), \({\varvec{M}}^{ext} \vert _{B^{R}}\), and \(Q_{\eta _{j}}\).
3.2 Quasisteady strip aerodynamics
The aerodynamic forces and moments of the test aircraft are modeled by applying strip theory. All effective lifting surfaces, i.e., wings, horizontal tail, and vertical tail, are divided into a finite number of spanwise strips. Each strip is then treated as a twodimensional airfoil with its own geometric and aerodynamic characteristics. The aerodynamic characteristics are modeled by means of quasisteady parameters. This implies that the resultant aerodynamic forces and moments at every time instant have reached their steadystate values and are only dependent on the instantaneous configuration and local relative flow. The aerodynamic forces and moments of the fuselage are expressed by additional quasisteady parameters acting on the center of mass.
Consider the aircraft with its lifting surfaces divided into a finite number of \(n_{s}\) spanwise strips, each with local width d\(y_{i}\), local chord \(c_{i}\), and local surface area \(S_{i}\). Each strip is assigned a structure support point (SP) at its centerline for which the instantaneous deformation (elastic translational deformation and elastic angular deformation) is known from the superposition of structural mode shapes and generalized displacements coordinates. The aircraft can then be treated as a discrete structure of interconnected points, as shown in Fig. 5, which mark the elastic axes of the lifting surfaces.
For the modeling of aerodynamic forces and moments, each strip is further assigned two aerodynamic control points along its centerline, i.e., a neutral point (NP), which is assumed at the 25\(\%\)chord position and a zero pressure point (PP0) which is assumed at the 50\(\%\)chord position. In strip theory, the strips themselves are assumed nondeformable. Therefore, their motion can be described similar to the motion of a flat plate, as indicated in Fig. 5 for the ith strip.
The instantaneous position of each strip’s structure support point, neutral point, and zero pressure point can be described relative to a global bodyfixed frame \(O_{B^{G}}\). The frame \(O_{B^{G}}\) is aligned with the bodyreference axes \(O_{B^{R}}\) (origin at the center of mass) but is translated by a vector \({\varvec{b}}_{cm}\), considered constant due to the assumption of small deformations [7]. Further, a local stripfixed frame \(O_{B^{S}}\) is introduced which defines the instantaneous translation and orientation of the strip’s 25\(\%\)chord line. Its origin is located at the neutral point and its orientation relative to \(O_{B^{G}}\) is defined by three rotations. The three rotation angles are the rigid and elastic dihedral angles (\(\nu _{0,i}\), \(\nu _{e,i}\)), the rigid and elastic twist angles (\(\varepsilon _{0,i}\), \(\varepsilon _{e,i}\)), and the rigid and elastic sweep angles (\(\varphi _{25c,i}\), \(\varphi _{e,i}\)). The rotation from \(O_{B^{G}}\) to \(O_{B^{S}}\) is given by the transformation matrix \({\varvec{T}}_{B^{S}B^{G}}\). With the alignment of \(O_{B^{G}}\) and \(O_{B^{R}}\), the transformation matrix \({\varvec{T}}_{B^{S}B^{G}}\) equals \({\varvec{T}}_{B^{S}B^{R}}\).
Let \({\varvec{b}}_{NP,i}\) denote the instantaneous position of the strip’s neutral point relative to the global bodyfixed axes \(O_{B^{G}}\). Further define \({\varvec{b}}_{r,NP,i}\) as the rigidbody part of the position vector and
as the elastic translational deformation and elastic angular deformation vectors of the ith structure support point along and about the bodyreference axes, respectively. In the equations, \(\varvec{\Phi }_{\text {trans},j}\) contains the mode shapes for elastic translational deformation and \(\varvec{\Phi }_{\text {ang},j}\) the mode shapes for elastic angular deformation. Then, under the assumption of no chordwise deformation of the strip, the position \({\varvec{b}}_{NP,i}\) relative to the bodyfixed axes \(O_{B^{G}}\) is calculated by
with \({\varvec{b}}_{r,SP,i}\) denoting the rigidbody position vector of the ith strip structure support point and \({\varvec{T}}_{\varphi ,i}\) denoting the transformation matrix of the rotation about \(O_{B^{R}}\) from the undeformed strip to \(O_{B^{S}}\) with the angles given by the elastic angular deformation in Eq. 7. Note that \({\varvec{b}}_{NP,i}\) is indicated time varying to distinguish between the nontimevarying terms in the equation. This notation is omitted for simplification in the following. Similar expressions can be derived for the position vectors of structure support point and zero pressure point.
The aerodynamic characteristics of the strips are modeled with quasisteady parameters. In this sense, each strip is assigned a local nondimensional lift coefficient \(C_{L,i}\) and a local nondimensional drag coefficient \(C_{D,i}\), where each coefficient itself is formed by stability and control derivatives normalized by \({}^{S_{i}}/_{S_{\text {ref}}}\)
In the equations, \(C_{L_{0},i}\) and \(C_{D_{0},i}\) are the strip’s zero lift and drag coefficient, \(C_{L_{\alpha }, \varphi = 0,i}\) indicates the lift curve slope of the unswept strip, \(\alpha _{\text {eff},i}\) is the strip’s local effective angle of attack, \(k_{i}\) is the strip’s kfactor of induced drag, and \(C_{L_{\delta _{c,n}},i}\) and \(\delta _{c}\) are the \(n_{c}\) control derivatives and deflections of the available control surfaces, respectively. Steady stall effects are included in terms of Kirchhoff’s theory of flow separation and an approximation of the steady flowseparation point \(X_{0,i}\) based on hyperbolic tangent [26]. For vertical stabilizer strips, the nondimensional lift coefficient is interpreted as a side force coefficient \(C_{Y,i}\) with the effective angle \(\beta _{\text {eff},i}\). All terms are assumed to act on the strip’s neutral point, except for \(C_{L_{0},i}\) and \(C_{L_{\delta _{c,n}},i} \cdot \delta _{c,n}\), which are assumed to act on the strip’s zero pressure point and a variable point along the centerline as a function of the deflection [30], respectively. They are ultimately transferred to the strip’s neutral point. The additional moments caused thereby are expressed in terms of nondimensional moment coefficients \(C_{l,i}\) (roll), \(C_{m,i}\) (pitch), and \(C_{n,i}\) (yaw). Nondimensionality is achieved through dividing the moments by \({\overline{s}}\) (roll, yaw) and \({\overline{c}}\) (pitch), respectively. Finally, the aerodynamic characteristics of the fuselage are modeled by additional nondimensional force and moment coefficients with classical 1point stability derivatives.
To calculate the effective angles in Eq. 9, the local relative flow at each strip has to be determined. This is achieved by summing all individual flow components at \(O_{B^{S}}\), i.e., the linear aerodynamic velocity of \(O_{B^{R}}\), plus additional terms due to rotation about \(O_{B^{R}}\), and the velocity of the structure
The induced velocity due to elastic angular deformation about the structure support point is neglected. In Eq. 11, \({\varvec{V}}_{A} \vert _{B^{R}}\) and \(\varvec{\omega }_{A} \vert _{B^{R}}\) denote the linear and angular aerodynamic velocity vector at \(O_{B^{R}}\), respectively. Downwash effects are taken into account at tailplane strips by the induced downwash angle \(\epsilon _{T,i}\). It is proportional to the angle of attack at \(O_{B^{R}}\) and flaperon deflections, delayed by \(\Delta t_{\epsilon _{T},i}\), the time the flow requires to reach the tailplane strips [31]
Herein, \({}^{\partial \epsilon _{T}}/_{\partial \alpha }\) and \({}^{\partial \epsilon _{T}}/_{\partial \delta _{f,n}}\) denote the partial derivatives of the downwash angle to the angle of attack at \(O_{B^{R}}\) and \(n_{f}\) flaperon deflections. In this work, the calculation of \(\Delta t_{\epsilon _{T},i}\) is simplified by taking the mean distance between wing and horizontal or vertical tail strip neutral points along \(x_{B^{R}}\), divided by the airspeed \(V_{A}\) at \(O_{B^{R}}\). The downwash angles are then used to correct the local relative flow at the tailplane strips. The local relative flow is originally defined in the local strip aerodynamic frame \(O_{A^{S}}\), but as given in Eq. 11, is already expressed in components along \(O_{B^{R}}\). Then, the angle of attack \(\alpha _{i}\) and angle of sideslip \(\beta _{i}\) of the local relative flow define the orientation of \(O_{A^{S}}\) relative to \(O_{B^{R}}\). The local relative flow at the ith strip is illustrated in Fig. 6.
With the transformation matrix \({\varvec{T}}_{B^{S}B^{R}}\), the local relative flow from Eq. 11 can be expressed in \(O_{B^{S}}\) coordinates. It is then straight forward to calculate the effective angles needed for the calculation of the aerodynamic force coefficient in Eq. 9 from the relative flow components \(u_{A,i}\), \(v_{A,i}\), and \(w_{A,i}\), by
Indicated in Fig. 6, the orientation of the force and moment coefficients is either given along the coordinates of \(O_{A^{S}}\) (defined by \(\alpha _{i}\) and \(\beta _{i}\)) or along the normal flow component \({\varvec{V}}_{N,i}\), but overall rotated about the bodyreference axis \(x_{B^{R}}\) by the rigid and elastic dihedral angle.^{Footnote 1} They can be expressed in components of the bodyreference axes \(O_{B^{R}}\) through a sequence of rotations involving these angles and are summarized to coefficient vectors for the aerodynamic forces \({\varvec{C}}^{A,F}_{i} \vert _{B^{R}}\) and moments \({\varvec{C}}^{A,M}_{i} \vert _{B^{R}}\), respectively. The resulting forces and moments at either the neutral point (strips) or the center of mass (fuselage) are then given by
with the diagonal matrix \({\varvec{D}}\), which has the main diagonal elements \(\left\{ {\overline{s}},{\overline{c}},{\overline{s}} \right\} \), and the effective dynamic pressures of the normal flow for wing and horizontal tail strips (Eq. 16a) and vertical tail strips (Eq. 16b) as
The forces and moments of all strips are ultimately transferred to the center of mass and summed up for the equations of motion of the rigidbody degrees of freedom. The generalized forces \(Q_{\eta _{j}}\) for the equations of the structural dynamics can be obtained by first transferring all strip forces and moments to their respective structure support points. Then, the resulting strip forces and moments are collected in vector form and are subsequently transformed with the transposed mode shapes
4 Flexible model identification and evaluation
This section is concerned with the identification of the flexible aircraft model and the evaluation of the results. Prior to the actual parameter estimation, a simulation model is assembled and implemented in Matlab/Simulink. Given the equations of motion of the flexible aircraft and the quasisteady strip aerodynamics model, the model is combined with additional models from the inhouse library FLYSim. These include an earth and atmosphere model, a wind and turbulence model, and an actuator model for the representation of servo and control surface dynamics. Moreover, a propulsion model and landing gear model are added. Subsequently, a suitable set of maneuvers is selected. This is accomplished by postprocessing the gathered flight test data and transferring all the available measurement parameters listed in Table 1 to the center of mass. Moreover, the load measurement parameters are corrected by the dead weight of the structure. The selection of maneuvers is based on different criteria, such as precision of maneuver execution, dynamic pressure variations, and atmospheric disturbance.
4.1 Parameter estimation
The estimation of model parameters is performed with the inhouse tool DAVIS, using the output error method in the timedomain, as depicted in Fig. 7, and maximumlikelihood estimation [26]. Based on an initial a priori estimation of the model parameters \(\varvec{\Theta }_{0}\), the simulation model is excited by the same inputs \({\varvec{u}}(t)\) as in the maneuvers of the flight tests (see Sect. 2.3) and the simulated model response \(\varvec{{\hat{y}}}(t)\) is compared to the measured aircraft system response \({\varvec{y}}(t)\). Only measurement noise \({\varvec{v}}(t)\) is considered to corrupt the deterministic system output \({\varvec{y}}_{d}(t)\). Applying a maximumlikelihood function, the response error \({\varvec{y}}(t)  \varvec{{\hat{y}}}(t)\) is minimized in an iterative optimization process through adaption of the model parameters by the parameter change \(\Delta \varvec{\Theta }\). To this end, sensitivities \(^{\partial \varvec{{\hat{y}}}}/_{\partial \varvec{\Theta }}\) of the model output with respect to the model parameters are calculated.
Suitable model parameters \(\Theta _{i}\) to be estimated need to be defined. Since the structural dynamics model was identified from GVT data, only parameters of the strip aerodynamics model are estimated. Initial distributions of the normalized stability and control derivatives are obtained from threedimensional vortexlatticemethod calculations of the aircraft in the open source software tool xflr5. The aircraft is modeled with its rigid shape lifting surfaces but without fuselage as advised by xflr5. The influence of lifting surface thickness is neglected and treated as an uncertainty. To this end, all lifting surfaces are divided into spanwise strips based on geometric properties and the resolution of local flow effects. A total number of 48 wing strips, 8 horizontal tail strips, and 5 vertical tail strips are considered. Subsequently, the locations of the strips are used to define associated structure support points as described in Sect. 3.2 and determine the structural mode shapes from the adapted FEmodel. The resulting distributions are shown in Fig. 8.
According to the aerodynamic parametrization of the nondimensional force coefficients, separate distributions are obtained for zero coefficients, stability derivatives, and control derivatives. The associated control surface deflections are \(\delta _{a,\text {in}}\) and \(\delta _{a,\text {out}}\) for inner and outer ailerons, \(\delta _{f,\text {in}}\) and \(\delta _{f,\text {out}}\) for inner and outer flaperons, \(\delta _{r}\) for the rudder, and \(\delta _{e}\) for the elevators. Each wing control surface is assumed to only influence the lift of all strips on the side of the wing it is attached to. Parameters for linearly scaling the initial distributions are introduced and defined as estimation parameters. In this way, the initial distributions can be adapted in the estimation process with a limited number of estimation parameters. On the other hand, the achievable solution is constrained by the qualitative shapes of the initial distributions. Separate scaling parameters are defined for each lifting surface, i.e., wings, horizontal tail, and vertical tail. Scaling parameters for the control derivative distributions of opposite control surfaces are either paired or grouped based on the control allocations used in the flight tests (indicated in Fig. 8 with the color code). The additional stability derivatives of the fuselage and downwash parameters are treated as direct estimation parameters with no effect on the distributions. Stall parameters and kfactors of induced drag are determined separately from flight data and the vortexlatticemethod calculations, respectively, and kept fixed for the estimation. Measurement outputs of the rigidbody motion (center of mass) are selected and set as criteria for the parameter estimation
Herein, \(V_{TAS}\), \(\alpha \), and \(\beta \) are the true airspeed, angle of attack, and angle of sideslip, p, q, and r the roll, pitch, and yaw rates, \(\Phi \), \(\Theta \), and \(\Psi \) are the Euler angles for roll, pitch, and yaw, and u, v, and w are the translational velocities expressed on \(O_{B^{R}}\) coordinates.
The values and relative standard deviations of the final estimated distribution scaling parameters are listed in Table 3. A value close to 1 indicates an estimation result close to the initial distributions from xflr5. This is found for the parameter \(k\_C_{Y_{\beta }}\) of the vertical tail, the parameters \(k\_C_{L_{\alpha },\text {wing}}\) and \(k\_C_{L_{\alpha },\text {htp}}\) of the wing and the horizontal tail, and the parameter \(k\_C_{L_{\delta _{a,\text {out}}}}\) of the outer ailerons. Least agreement with the initial distributions from xflr5 are found for the parameters \(k\_C_{D_{0}}\) of all lifting surfaces, \(k\_C_{L_{0},\text {wing}}\) of the wing, and \(k\_C_{L_{\delta _{e}}}\) of the elevators. The result of \(k\_C_{D_{0}}\) being significantly underestimated by xflr5 is expected considering the interpolation of viscous drag from twodimensional polar data associated with the calculation method. Low relative standard deviations are achieved for all estimated distribution scaling parameters.
Table 4 further lists the values and relative standard deviations of the final estimated additional parameters of the fuselage and downwash. Only a few parameters for the most dominant fuselage effects are incorporated in the final parameter set. An increased relative standard deviation is only observed for the parameter \(C_{Y_{\beta ,\text {fuse}}}\) of the fuselage.
Matching plots of the identification result are shown in Fig. 9 for longitudinal and lateral maneuvers with measurement parameters at the center of mass. A good overall identification result is achieved for all maneuvers. Especially, the fast aircraft dynamics are matched well, which is important for control law design. Good matches are also achieved for the translational rigidbody accelerations (\(a_{x}\), \(a_{y}\), \(a_{z}\)), which are directly related to the external forces acting on the aircraft. Note that gravitational accelerations are not measured by the sensors. Some model deficiencies are found for the measurement parameters true airspeed \(V_{TAS}\) and pitch angle \(\Theta \), which indicate slight deviations of the model’s pitching moment to the measured aircraft’s pitching moment. Unexpected model behavior is also observed in the last part of the level deceleration maneuver. However, judging from the control action of ailerons and rudder, an undetected asymmetric atmospheric disturbance is suspected to influence the maneuver.
The identification result is further analyzed by means of Theil’s inequality coefficient (TIC) [32]. First introduced in the field of economics, the coefficient provides a measure of the accuracy of a prediction from a model with its value ranging from zero to unity. Due to its simplicity, it is often adopted for system identification (e.g., [20]). The Theil’s inequality coefficient \(U_{i}\) is defined as
with N denoting the number of samples, and \(y_{i}(t_{k})\) and \({\hat{y}}_{i}(t_{k}))\), as previously defined, denoting the measured aircraft system response and simulated model response, respectively. However, in the calculation of the coefficient, these outputs are implemented as the variations from the initial condition \(y_{i}(t_{\text {init}})\) [33]. In the present case of a sequence of multiple maneuvers, the initial condition of each individual maneuver has to be used. Table 5 lists the Theil’s inequality coefficient for all measurement parameters of the identification result in Fig. 9. A value of zero corresponds to a perfect model prediction, whereas values close to unity indicate significantly different responses. Values below 0.3 are said to indicate a good agreement of measured and simulated responses [34]. As can be seen from Table 5, this is the case for all measurement parameters. The differences in TIC values further confirm the visual impression of the identification result in Fig. 9.
4.2 Evaluation of structural load distribution
In a final step, the identified model is evaluated with regard to its capability to estimate distributed structural loads and its use in active load control law design. To this end, structural loads measured at wing load measurement stations are compared to the model outputs. The outputs are constructed as part of a separate loads model and capture the internal loads acting on the strain gauges of the load measurement stations. Aerodynamic forces and moments as well as inertial forces at local mass points (MP) are considered. The mass points are defined based on the locations of the strain gauges and a detailed CAD model. The aerodynamic strip forces and moments are allocated and transferred to these mass points considering their elastic points of attack. Elastic position vectors of the mass points can be obtained similar to Eq. 8 and an interpolation of mode shapes \(\varvec{\Phi }_{j}(x,y,z)\) at the mass point positions, assuming a rigid connection between mass point and elastic axis. Next, inertial forces are added. Local accelerations are derived based on [35] which include rigidbody accelerations and the effects of elastic deformation on angular rates and translational acceleration. The total forces and moments at the mass points \({\varvec{F}}_{\text {MP}_{i}} = [F_{x} \; F_{y} \; F_{z} \; M_{x} \; M_{y} \; M_{z}]^{T}\) are then used to calculate the internal loads \({\varvec{F}}_{\text {SG}_{i}} = [Q_{x} \; Q_{y} \; Q_{z} \; M_{x} \; M_{y} \; M_{z}]^{T}\) by transferring them to the strain gauge positions. Figure 10 exemplifies the internal loads at \(\text {SG}_{10}\) on the right wing.
For the single pair of \(\text {MP}_{10}\) and \(\text {SG}_{10}\) with individual distance \([\Delta x, \Delta y, \Delta z]\) from mass point to strain gauge along the axes of \(O_{B^{R}}\), the internal loads at \(\text {SG}_{10}\) due to the forces and moments at \(\text {MP}_{10}\) can be derived according to [36, 37] as
Similarly, the internal loads from other pairs of mass points and strain gauges can be calculated, each by applying the respective distance from mass point to strain gauge. For the total internal loads at all strain gauges on the right wing, a total transformation matrix can be constructed with each of the transformation matrices corresponding to single pairs of mass points and strain gauges, as shown in Eq. 20, to
As seen in the equation, the forces and moments at all mass points \(\text {MP}_{6}\) to \(\text {MP}_{10}\) contribute to the total internal loads at the innermost strain gauge \(\text {SG}_{6}\).
With Eq. 22, the internal loads at the strain gauges are given in components of \(O_{B^{R}}\) which is compliant with the calibration axes of the sensors on the GFlights Dimona. However, if the calibration is given in the sensor coordinate systems \(O_{B^{SG_{i}}}\), in a final step, the internal loads need to be transformed into the sensor coordinate systems with the transformation matrices \({\varvec{T}}_{B^{SG_{i}} B^{R}}\). The rotation angles are then given by the sum of rigidbody and elastic angular deformation angles at the sensor positions relative to \(O_{B^{R}}\). The latter can be determined through an interpolation of mode shapes for elastic angular deformation \(\varvec{\Phi }_{\text {ang},j}(x,y,z)\) at the sensor positions and generalized displacement coordinates \(\eta _{j}(t)\). The same overall approach is used to calculate the internal loads at the strain gauges on the left wing and empennage.
Figure 11 compares the shear forces \(Q_{z}\) measured at middle and inner load measurement stations along the wing (see Fig. 2) to the respective outputs of the structural loads’ model for the same maneuver sequence.
Close matches are found for all shear forces for the majority of maneuvers. This result indicates a plausible representation of the actual load distribution along the wing. Moreover, the close match of shear forces at inner load measurement stations \(Q_{z,\text {LMS5}}\) (left wing) and \(Q_{z,\text {LMS6}}\) (right wing) is in good agreement with the identification result of \(a_{z}\), since both stations capture the majority of aerodynamic loads. An unexpected model behavior is solely found for the inner load measurement stations during flaperon banktobank maneuvers, where the effects of local load increase/decrease due to flaperon deflection and the resulting rotation of the aircraft are not represented correctly. It is suspected that either the negligence of a dedicated fuselage model within xflr5 and therewith negligence of wing–fuselage interaction effects on the initial derivative distributions, or the load application on the inner load measurement stations is a possible cause of the deviation.
Threedimensional effects on the induced angle of attack were investigated in [38] with a 1 g and 2.5g wind tunnel wing shape and were found to cause deviations in the sectional aerodynamic loads if not properly modeled. The effects were observed to be approximately proportional to the wing’s elastic twist. With the initial derivative distributions being derived from threedimensional vortex lattice method calculations of the rigid aircraft shape only, these effects are not accounted for and thus add to the uncertainty of the results. However, considering the moderate magnitudes of elastic twist experienced during the maneuvers and the plausible load distributions found from the evaluation of shear forces in Fig. 11, this uncertainty is considered small. If more aggressive maneuvers or gust encounters with larger elastic deformations and increased elastic mode participation are to be investigated in the future, the influence of the threedimensional effects on the induced angle of attack is expected to increase. In this case, even unsteady aerodynamic effects can become important for slightly flexible aircraft as represented by the GFlights Dimona [13].
Based on these results, the identified model is capable of computing realistic distributed aerodynamic forces and moments and thus is well suited for state of the art modelbased load estimation techniques [36]. This aspect combined with the simple model structure also provides a suitable basis for realtime simulation and the design of active load control laws.
5 Conclusion
A controloriented modeling framework for slightly flexible aircraft suitable for parameter identification in the timedomain was presented. It combines linear structural dynamics and a distributed quasisteady aerodynamics model using strip theory. The applicability of the modeling framework was demonstrated for a slightly flexible 25 kg fixedwing UAV. Free vibration modes and mode shapes of the structure were obtained from ground vibration tests. Initial distributions for the stability and control derivatives of the strip aerodynamics were derived from threedimensional vortexlatticemethod steadyflow calculations. They were subsequently adapted based on flight test data using the output error method in the timedomain and maximumlikelihood estimation. The use of scaling parameters for adapting the initial distributions is demonstrated to be an efficient approach for achieving a limited number of estimation parameters. A good overall identification result and close match of the fast aircraft dynamics was achieved. Good agreement with the initial derivative distributions was found for the lift curve slope and angle of sideslipdependent stability derivative distributions, while viscous drag effects were significantly underestimated by the vortexlatticemethod calculations. Further, the evaluation of the structural load distribution measured along the wing demonstrated the capability of the model to compute realistic distributed aerodynamic forces and moments. The resultant computationally efficient model provides a suitable basis for realtime simulation, loads’ estimation, and active load control law design. The latter highly benefit from the capability of the strip aerodynamic model structure to yield distributed forces and moments. Possible future work includes the improvement of the derivative distributions based on highfidelity numerical or experimental data.
Availability of data and materials
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Notes
Assuming no elastic sweep angle \(\varphi _{e,i} = 0\). If \(\varphi _{e,i} \ne 0\), a further rotation is given by this angle.
Abbreviations
 \(a_{x},a_{y},a_{z}\) :

Translational accelerations expressed in \(O_{B^{R}}\) coordinates, m/s\(^2\)
 \({\varvec{b}}_{cm}\) :

Position vector of \(O_{B^{G}}\) relative to \(O_{B^{R}}\) to, m
 \({\varvec{b}}_{NP}\) :

Position vector of neutral point relative to \(O_{B^{G}}\), m
 \({\varvec{b}}_{r,NP}\) :

Rigidbody position vector of neutral point relative to \(O_{B^{G}}\), m
 \({\varvec{b}}_{r,SP}\) :

Rigidbody position vector of structure support point relative to \(O_{B^{G}}\), m
 \({\varvec{C}}^{A,F}\) :

Coefficient vector of aerodynamic forces
 \({\varvec{C}}^{A,M}\) :

Coefficient vector of aerodynamic moments
 \(C_{D}\) :

Drag coefficient
 \(C_{D_{0}}\) :

Zero drag coefficient
 \(C_{l}\) :

Roll moment coefficient
 \(C_{l_{0}}\) :

Zero roll moment coefficient
 \(C_{L}\) :

Lift coefficient
 \(C_{L_{0}}\) :

Zero lift coefficient
 \(C_{L_{\alpha ,\varphi = 0}}\) :

Lift curve slope of unswept wing, 1/rad
 \(C_{L_{\delta _{c}}}\) :

Lift stability derivative of control surfaces, 1/rad
 \(C_{m}\) :

Pitch moment coefficient
 \(C_{m_{0}}\) :

Zero pitch moment coefficient
 \(C_{n}\) :

Yaw moment coefficient
 \(C_{Y}\) :

Side force coefficient
 \(C_{Y_{\beta }}\) :

Side force stability derivative, 1/rad
 \(C_{Y_{\delta _{r}}}\) :

Side force rudder control derivative, 1/rad
 \({\overline{c}}\) :

Wing aerodynamic mean chord, m
 \(c_{i}\) :

Local strip chord, m
 \({\varvec{D}}\) :

Diagonal matrix with entries \(\left\{ {\overline{s}},{\overline{c}},{\overline{s}} \right\} \), m
 \({\varvec{d}}\) :

Elastic (translational) deformation vector in \(O_{B^{R}}\) coordinates, m
 \(\textrm{d}V\) :

Volume element, m\(^{3}\)
 y :

Local strip width, m
 \({\varvec{F}}^{A}\) :

Vector of aerodynamic forces, N
 \({\varvec{F}}^{ext}\) :

Vector of external forces, N
 \({\varvec{F}}_{\text {MP}}\) :

Vector of forces and moments at mass point, N and N\(\cdot \)m
 \({\varvec{F}}_{\text {SG}}\) :

Vector of internal loads at strain gauge, N and N\(\cdot \)m
 \({\varvec{G}}\) :

Gravity acceleration vector, m/s\(^2\)
 h :

Altitude, m
 \({\varvec{J}}\) :

Inertia tensor, kg\(\cdot \)m\(^2\)
 k :

Kfactor of induced drag
 \(k\_C\) :

Scaling parameter for derivative distribution
 \({\varvec{M}}^{A}\) :

Vector of aerodynamic moments, N\(\cdot \)m
 \({\varvec{M}}^{ext}\) :

Vector of external moments, N\(\cdot \)m
 m :

Aircraft mass, kg
 N :

Number of samples
 \(n_{e}\) :

Number of elastic modes
 \(n_{f}\) :

Number of flaperons
 \(n_{s}\) :

Number of strips
 \(O_{A^{S}}\) :

Local aerodynamic strip frame
 \(O_{B^{G}}\) :

Global bodyfixed frame
 \(O_{B^{R}}\) :

Bodyreference frame (center of mass)
 \(O_{B^{S}}\) :

Local stripfixed frame
 \(O_{B^{SG}}\) :

Strain gauge sensor frame
 \(O_{I}\) :

Inertial reference frame
 \({\varvec{p}}\) :

Position vector of mass element relative to \(O_{B^{R}}\), m
 p, q, r :

Angular rates (roll, pitch, yaw), rad/s
 \(p_{s}\) :

Static pressure, Pa
 \({\overline{q}}\) :

Dynamic pressure, Pa
 \({\overline{q}}_{A}\) :

Dynamic pressure of relative flow, Pa
 \({\overline{q}}_{N}\) :

Dynamic pressure of normal flow, Pa
 \(Q_{\eta _{j}}\) :

Generalized forces of jth mode, N\(\cdot \)m
 \({\varvec{r}}\) :

Position vector of mass element relative to \(O_{I}\), m
 \({\varvec{r}}_{0}\) :

Position vector of origin of \(O_{B^{R}}\), m
 \(S_{i}\) :

Local strip surface area, m\(^{2}\)
 \(S_{\text {ref}}\) :

Wing reference surface area, m\(^{2}\)
 \({\overline{s}}\) :

Half wing span, m
 \({\varvec{s}}\) :

Undeformed position vector relative to \(O_{B^{R}}\), m
 T :

Air temperature, deg
 \({\varvec{T}}_{B^{R}I}\) :

Transformation matrix from \(O_{I}\) to \(O_{B^{R}}\)
 \({\varvec{T}}_{B^{S}B^{G}}\) :

Transformation matrix from \(O_{B^{G}}\) to \(O_{B^{S}}\)
 \({\varvec{T}}_{B^{S}B^{R}}\) :

Transformation matrix from \(O_{B^{R}}\) to \(O_{B^{S}}\)
 \({\varvec{T}}_{B^{SG}B^{R}}\) :

Transformation matrix from \(O_{B^{R}}\) to \(O_{B^{SG}}\)
 \({\varvec{T}}_{\text {SG}\text {MP}}\) :

Transformation matrix from mass point forces/moments to strain gauge internal loads
 \({\varvec{T}}_{\varphi }\) :

Transformation matrix from the undeformed strip to \(O_{B^{S}}\)
 \(t_{\text {init}}\) :

Time of initial maneuver condition, s
 \(U_{i}\) :

Theil’s inequality coefficient for ith output
 u, v, w :

Translational velocities expressed in \(O_{B^{R}}\) coordinates, m/s
 \(u_{I},v_{I},w_{I}\) :

Translational velocities expressed in \(O_{I}\) coordinates, m/s
 \(u_{A},v_{A},w_{A}\) :

Translational aerodynamic velocities, m/s
 \({\varvec{u}}\) :

Input vector
 \({\varvec{V}}\) :

Translational velocity vector of \(O_{B^{R}}\), m/s
 \(V_{A}\) :

Aerodynamic velocity at \(O_{B^{R}}\), m/s
 \({\varvec{V}}_{A,i}\) :

Local aerodynamic velocity vector, m/s
 \({\varvec{V}}_{N,i} \) :

Local normal flow vector, m/s
 \(V_{TAS}\) :

True airspeed, m/s
 \({\varvec{v}}\) :

Measurement noise vector
 \(X_{0}\) :

Steady flowseparation point
 \({\varvec{y}}\) :

System output vector
 \(\varvec{{\hat{y}}}\) :

Simulation model output vector
 \({\varvec{y}}_{d}\) :

Deterministic system output vector
 \(\alpha \) :

Angle of attack, rad
 \(\alpha _{\text {eff}}\) :

Effective angle of attack, rad
 \(\beta \) :

Angle of sideslip, rad
 \(\beta _{\text {eff}}\) :

Effective angle of sideslip, rad
 \(\Delta t_{\epsilon _{T}}\) :

Time delay of downwash at tailplane, s
 \(\delta _{a}\) :

Aileron deflection, rad
 \(\delta _{c}\) :

Control surface deflection, rad
 \(\delta _{e}\) :

Elevator deflection, rad
 \(\delta _{f}\) :

Flaperon deflection, rad
 \(\delta _{r}\) :

Rudder deflection, rad
 \(\epsilon _{b}\) :

Bending strain, 0.01\(\%\)
 \(\epsilon _{s}\) :

Shear strain, 0.01\(\%\)
 \(\epsilon _{t}\) :

Torsion strain, 0.01\(\%\)
 \(\epsilon _{T}\) :

Induced downwash angle, rad
 \(\varepsilon _{0},\varepsilon _{e}\) :

Rigid and elastic twist angle, rad
 \(\eta \) :

Generalized displacement coordinate
 \(\lambda \) :

Longitude, deg
 \(\mu \) :

Generalized mass, kg\(\cdot \)m\(^2\)
 \(\nu _{0}, \nu _{e}\) :

Rigid and elastic dihedral angle, rad
 \(\varvec{\omega }\) :

Angular velocity vector of \(O_{B^{R}}\), rad/s
 \(\varvec{\omega }_{A}\) :

Aerodynamic angular velocity vector of \(O_{B^{R}}\), rad/s
 \(\omega _{n}\) :

Undamped natural (modal) frequency, rad/s
 \(\rho \) :

Air density, kg\(\cdot \)m\(^{3}\)
 \(\xi \) :

Structural (modal) damping ratio
 \(\varphi \) :

Latitude, deg
 \(\varvec{\varphi }\) :

Elastic angular deformation vector relative to \(O_{B^{R}}\), rad
 \(\varphi _{e}\) :

Elastic sweep angle, rad
 \(\varphi _{25c}\) :

Rigid sweep angle at quarter chord line, rad
 \(\Phi , \Theta , \Psi \) :

Euler angles (roll, pitch, yaw), rad
 \(\varvec{\Phi }_{j}\) :

Mode shapes
 \(\varvec{\Phi }_{\text {ang},j}\) :

Mode shapes for elastic angular deformation
 \(\varvec{\Phi }_{\text {trans},j}\) :

Mode shapes for elastic translational deformation
 \(\varvec{\Theta }\) :

Estimation parameter vector
 \(\varvec{\Theta }_{0}\) :

Initial estimation parameter vector
 \(\Theta _{i}\) :

Estimation parameter
 \((\cdot ) \vert _{B^{R}}\) :

\((\cdot )\) expressed in \(O_{B^{R}}\)
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Acknowledgements
The authors would like to acknowledge Lothar Desel and Rasmus Köhler for their support during the flight test campaign. Special thanks is further devoted to Karsten Henning for his contribution to this work. Parts of this work have been presented previously at the 6th CEAS Conference on Guidance, Navigation and Control (EuroGNC) [39].
Funding
Open Access funding enabled and organized by Projekt DEAL. This work was funded by the research project “Advanced load analysis and observer methods for innovative aircraft configurations” (ELASTIK), which is supported by the German Federal Ministry for Economic Affairs and Climate Action in the national LuFo V3 program. Any opinions, findings, and conclusions expressed in this document are those of the authors and do not necessarily reflect the views of the other project partners.
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Herrmann, B., Theis, J. & Thielecke, F. Nonlinear system identification of a UAV model with distributed aerodynamics and flexible structure. CEAS Aeronaut J 14, 661–677 (2023). https://doi.org/10.1007/s1327202300674x
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DOI: https://doi.org/10.1007/s1327202300674x