HAPD aircraft Uncertain Norm-Bounded Mathematical model

TL;DR

This paper develops an uncertain norm-bounded mathematical model for the HAPD UAV, capturing its nonlinear dynamics within specified airspeed and altitude bounds to aid in performance analysis.

Contribution

It introduces a linear state space model that accurately describes the UAV's nonlinear behavior within defined operational limits, enhancing modeling precision.

Findings

Model effectively captures UAV dynamics within bounds

Provides a basis for control and performance analysis

Enhances understanding of UAV behavior in specified envelope

Abstract

In this paper an uncertain norm-bounded mathematical model for the UAV High Altitude Performance Demonstrator (HAPD) designed by Italian Aerospace Research Center (CIRA) is carried out. The linear state space description aims to describe the non-linear aircraft dynamic inside the operating envelope characterized by the following bounds: true air speed between 17 m/s and 23 m/s and altitude from 300 m to 700 m.

Tables

Table 1. Table 1: HAPD Main Parameters

Names	Values	Units
Wing Area (S)	13.5	$m^2$
Wing Span ($b$)	16.55	$m$
Mean Chord ($c$)	0.557	$m$
Mass (M)	184.4	$kg$
Moment of Inertia $I_x$	$1.997\cdot {10}^3$	$kg\cdot m^2$
Moment of Inertia $I_y$	$258.6$	$kg\cdot m^2$
Moment of Inertia $I_z$	$2.196\cdot {10}^3$	$kg\cdot m^2$
Product of Inertia $I_{xz}$	$-66.3$	$kg\cdot m^2$
Ailerons Slew Rates	$\pm 200$	$deg/s$
Elevators Slew Rates	$\pm 200$	$deg/s$
Rudders Slew Rates	$\pm 200$	$deg/s$
Ailerons deflections	$\pm 25$	$deg$
Elevators deflections	$\pm 25$	$deg$
Rudders deflections	$\pm 25$	$deg$

Equations

$$M\dot{V}=T\cos \alpha \cos \beta -D+M{g}_1$$

$$VM\dot{\beta }=-T\cos \alpha \sin \beta +Y-MVr+M{g}_2$$

$$MV\cos \beta \dot{\alpha }=-T\sin \alpha -L+MVq+M{g}_3$$

$$I_x\dot{p}-I_{xz}\dot{r}=\overline{L}+qr(I_y-I_z)+pq{I}_{xz}$$

$$I_y\dot{q}=\overline{M}+rp(I_z-I_x)+(r^2-p^2)I_{xz}$$

$$-I_{xz}\dot{p}+I_z\dot{r}=\overline{N}+pq(I_x-I_y)-qr{I}_{xz}$$

$$\dot{\varphi }=p+q\tan \theta \sin \varphi +r\tan \theta \cos \varphi$$

$$\dot{\theta }=q\cos \varphi -r\sin \varphi$$

$$\begin{array}[]{*{20}{l}}g_{1}=g(-\cos\alpha\cos\beta\sin\theta+\sin\beta\sin\phi\cos\theta+\sin\alpha\cos\beta\cos\phi\cos\theta)\\ g_{2}=g(\cos\alpha\sin\beta\sin\theta+\cos\beta\sin\phi\cos\theta-\sin\alpha\sin\beta\cos\phi\cos\theta)\\ g_{3}=g(\sin\alpha\sin\theta+\cos\alpha\cos\phi\cos\theta),\end{array}$$

$$M_{{\eta }_i}\ddot{{\eta }_i}+{\zeta }_{{\eta }_i}\dot{{\eta }_i}+M_{{\eta }_i}{\omega }_{{\eta }_i}{\eta }_i=Q_{{\eta }_i},i=1,\dots ,n_a,$$

$$\begin{array}[]{ll}L=&\!\!\!\!\!\!\frac{\rho V_{TAS}^{2}S}{2}\left[C_{L}\left(\!\alpha,\beta,p,q,r,\delta_{sup}\!\right)\!+\!\displaystyle\sum_{i=1}^{n_{a}}C_{L_{\eta_{i}}}\eta_{i}+\!\displaystyle\sum_{i=1}^{n_{a}}C_{L_{\dot{\eta}_{i}}}\dot{\eta}_{i}\right]\end{array}$$

$$\begin{array}[]{ll}D=&\!\!\!\!\!\!\frac{\rho V_{TAS}^{2}S}{2}\left[C_{D}\left(\!\alpha,\beta,p,q,r,\delta_{sup}\!\right)\!+\!\displaystyle\sum_{i=1}^{n_{a}}C_{D_{\eta_{i}}}\eta_{i}+\!\displaystyle\sum_{i=1}^{n_{a}}C_{D_{\dot{\eta}_{i}}}\dot{\eta}_{i}\right]\end{array}$$

$$\begin{array}[]{ll}Y=&\!\!\!\!\!\!\frac{\rho V_{TAS}^{2}S}{2}\left[C_{Y}\left(\!\alpha,\beta,p,q,r,\delta_{sup}\!\right)\!+\!\displaystyle\sum_{i=1}^{n_{a}}C_{Y_{\eta_{i}}}\eta_{i}+\!\displaystyle\sum_{i=1}^{n_{a}}C_{Y_{\dot{\eta}_{i}}}\dot{\eta}_{i}\right]\end{array}$$

$$\begin{array}[]{ll}\bar{L}=&\!\!\!\!\!\!\frac{\rho V_{TAS}^{2}Sb}{2}\left[C_{l}\left(\!\alpha,\beta,p,q,r,\delta_{sup}\!\right)\!+\!\displaystyle\sum_{i=1}^{n_{a}}C_{l_{\eta_{i}}}\eta_{i}+\!\displaystyle\sum_{i=1}^{n_{a}}C_{l_{\dot{\eta}_{i}}}\dot{\eta}_{i}\right]\end{array}$$

$$\begin{array}[]{ll}\bar{M}=&\!\!\!\!\!\!\frac{\rho V_{TAS}^{2}Sc}{2}\left[C_{m}\left(\!\alpha,\beta,p,q,r,\delta_{sup}\!\right)\!+\!\displaystyle\sum_{i=1}^{n_{a}}C_{m_{\eta_{i}}}\eta_{i}+\!\displaystyle\sum_{i=1}^{n_{a}}C_{m_{\dot{\eta}_{i}}}\dot{\eta}_{i}\right]\end{array}$$

$$\begin{array}[]{ll}\bar{N}=&\!\!\!\!\!\!\frac{\rho V_{TAS}^{2}Sb}{2}\left[C_{n}\left(\!\alpha,\beta,p,q,r,\delta_{sup}\!\right)\!+\!\displaystyle\sum_{i=1}^{n_{a}}C_{n_{\eta_{i}}}\eta_{i}+\!\displaystyle\sum_{i=1}^{n_{a}}C_{n_{\dot{\eta}_{i}}}\dot{\eta}_{i}\right]\end{array}$$

$$\begin{array}[]{ll}\bar{Q}_{\eta_{i}}=&\!\!\!\!\!\!\frac{\rho V_{TAS}^{2}S}{2}\left[C_{0}^{i}+C_{\alpha}^{i}\alpha+C_{\beta}^{i}\beta+C_{p}^{i}p+C_{q}^{i}q+C_{r}^{i}r+C_{\delta_{sup}}^{i}\delta_{sup}\right.\\ &\left.+\displaystyle\sum_{j=1}^{n_{a}}C^{ij}_{\eta_{j}}\eta_{j}+\displaystyle\sum_{j=1}^{n}C^{ij}_{\dot{\eta}_{j}}\dot{\eta}_{j}\!\right]\!\end{array}$$

$$\left\{\begin{array}[]{lcl}x(t+1)&=&\Phi\,x(t)+G\,u(t)+B_{p}\,p(t)\\ y(t)&=&C\,x(t)\\ q(t)&=&C_{q}\,x(t)+D_{q}\,u(t)\\ p(t)&=&\Delta(t)\,q(t)\end{array}\right.$$

$$x(t)=[V,\alpha ,\beta ,p,q,r,\varphi ,\theta ,{\eta }_s,{\dot{\eta }}_s,{\eta }_a,{\dot{\eta }}_a{]}^T,u(t)=[{\delta }_{sup}T{]}^T$$

$$\delta_{sup}=\left[\begin{array}[]{l}ElevatorIB-DX\\ ElevatorIB-SX\\ ElevatorMID-DX\\ ElevatorMID-SX\\ ElevatorOB-DX\\ ElevatorOB-SX\\ AileronIB-DX\\ AileronIB-SX\\ AileronOB-DX\\ AileronOB-SX\\ RudderSUP\\ RudderINF\end{array}\right]$$

$$y(t)=[V,\alpha ,\beta ,p,q,r,\varphi ,\theta {]}^T$$

$$\sum _{j=0}^{t}p(j{)}^{T}p(j)\le \sum _{j=0}^{t}q(j{)}^{T}q(j),\forall t\ge 0$$

Full text

1 Description of the UAV mathematical model

The HAPD shown in Fig.1, is an over-actuated UAV equipped with the following redundant aerodynamic control surfaces: three pairs of elevators, namely inboard (IB), middle (MID) and outboard (OB), two pairs of ailerons, namely inboard and outboard, and two rudders, namely upper (SUP) and lower (INF). The thrust is generated by eight independent, electrically powered, propellers. In view of the light weight and the high wing aspect ratio, the HAPD flexibility dynamics have to be taken into account by increasing the number of states with respect to classical aircraft rigid body mathematical models, see e. g. [1].

As detailed in [2], assume the following hypotheses: the inertia matrix $I$ does not depend on the aircraft elastic deformations; the linear elastic theory can be exploited to model the aero-elastic dynamics; aero-elastic modes are quasi-stationary. Starting from these premises, consider the polar form of the nonlinear equations of the 6DoF motion:

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where $T$ is the thrust, $V_{TAS}=\|V_{B}-V_{W}\|$ the true air speed,$V_{B}=(u_{B},v_{B},w_{B})^{T}$ the 6DoF linear velocity vector, $V_{W}=(u_{W},v_{W},w_{W})^{T}$ the atmospheric wind velocity vector, $V=\|V_{B}\|$, $\omega_{B}=(p,q,r)^{T}$ the rotational velocity vector, $\phi$ the roll angle, $\theta$ the pitch angle, $\alpha=\arctan\left(\frac{w_{B}-w_{w}}{u_{B}-u_{w}}\right)$ the angle of attack, $\beta=\arcsin\left(\frac{v_{B}-v_{w}}{V}\right)$ the sideslip angle, $I_{x}$, $I_{y}$, $I_{z}$, $I_{xz}$ the moments and products of inertia in body axes, $M$ the aircraft mass and

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with $g$ the gravity acceleration.

Notice that a formal definition of forces and moments involved in (1)-(6) requires the use of the flexibility dynamics which in turn prescribes the introduction of additional state variables for $n_{a}$ aero-elastic modes, namely the generalized state variables $\eta_{i}$ and $\dot{\eta}_{i},\,i=1,\ldots,n_{a}.$ These dynamics are modelled by means of interacting second order linear state space descriptions:

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where $M_{\eta_{i}}$ is the generalized mass of the $i-th$ aero-elastic mode, $\zeta_{\eta_{i}}$ the generalized damping coefficient, $\omega_{\eta_{i}}$ the generalized natural frequency and $Q_{\eta_{i}}$ the generalized force.

The evolution of the aerodynamic forces and moments acting on (1)-(6) and (10) depends on both the rigid body and the flexibility state variables:

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where the numerical values of the main involved variables are reported in Table 1. Moreover, $\delta_{sup}$ is a vector accounting for control surfaces deflections, $\rho$ the air density at the flying altitude and all the terms indexed by $C$ refer to adimensional aerodynamic coefficients arising from the common hypothesis that aerodynamic forces are affine functions of the motion variables and generalized states $\eta_{i}$ and $\dot{\eta}_{i}.$

Note also that the flexible UAV mathematical model requires the calculation of both generalized masses, damping coefficiens, natural frequencies and aerodynamic coefficients. The first are computed via finite structural elements methods, while aerodynamic coefficients are obtained by resorting to CFD (Computational Fluid Dynamics) calculations [3] and/or wind tunnel or flight tests.

2 Uncertain Norm-Bounded modeling

By considering only the two slower and most significant aero-elastic modes accounting for symmetrical ($\eta_{1}=\eta_{s}$) and asymmetrical ($\eta_{2}=\eta_{a}$) aircraft deformations, faster aero-elastic dynamics being considered instantaneous, the high nonlinear model description (1)-(17) has been recast into the following uncertain linear state space description with uncertainties (or perturbations) appearing in a feedback loop

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where $x\in\mathop{{\rm I}\mskip-4.0mu{\rm R}}\nolimits^{12}$ denoting the state, $u\in\mathop{{\rm I}\mskip-4.0mu{\rm R}}\nolimits^{12}$ the control input, $y\in\mathop{{\rm I}\mskip-4.0mu{\rm R}}\nolimits^{8}$ the output.

The input and state vectors definition are below reported

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Moreover, $p,q\in\mathop{{\rm I}\mskip-4.0mu{\rm R}}\nolimits^{12}$ are the additional variables accounting for the uncertainty. The uncertain operator $\Delta$ may represent either a memoryless, possibly time-varying, matrix with $\left\|\Delta(t)\right\|_{2}=\bar{\sigma}\left(\Delta(t)\right)\leq 1$ $\forall t\geq 0$, or a convolution operator with norm, induced by the truncated $\ell_{2}$-norm, less than 1 viz.

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For a more extensive discussion about this type of uncertainty see [4]. Such a representation (18) can be obtained by exploiting the fact that a suitable collection of linear models well approximates aircraft dynamics in wide regions of the flight envelope including steady state and transient flight conditions. Thus, a Polytopic Linear Differential Inclusion (PLDI) of the HAPD nonlinear model (1)-(10) can be obtained by deriving a convex outer approximation of regions covered by a set of $30$ linearized models around different operating flight conditions characterized by the following bounds on speed and altitude: true air speed between $17$ and $23$ m/s and altitude from $300$ m to $700$ m. PLDI is then approximated as a Norm-bound Linear Differential Inclusion (NDLI) by exploting the optimization procedure described in [4].