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Chapter 9

Nonlinear Design Models

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 1

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Architecture

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 2

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Simplifying Dynamic Models

When designing higher level autopilot functions, we need models that are easier to analyze and simulate
Models must capture the essential behavior of system
We will derive reduced-order, reduced-complexity models suitable for design of higher-level guidance strategies
Two types of guidance models:

Kinematic – utilize kinematic relationships, do not consider aerodynamics, forces directly
Dynamic – apply force balance relations to point-mass models

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 3

4 of 25

Autopilot Models (transfer function)

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 4

\begin{minipage}{3.0in}

Airspeed hold and roll hold loops can be modeled as:

\[

V_a(s) = \frac{{\color{magenta}b_{V_a}}}{s+{\color{magenta}b_{V_a}}}{\color{blue}V_a^c}(s) \qquad\qquad

\phi(s) = \frac{{\color{magenta}b_{\phi}}}{s+{\color{magenta}b_{\phi}}}{\color{blue}\phi^c}(s)

\]

Altitude and course hold loops can be modeled as:

\[

h(s) = \frac{{\color{magenta}b_{\dot{h}}}s+{\color{magenta}b_h}}{s^2 +

{\color{magenta}b_{\dot{h}}}s + {\color{magenta}b_h}}{\color{blue}h^c}(s) \qquad\qquad

\chi(s) = \frac{{\color{magenta}b_{\dot{\chi}}}s + {\color{magenta}b_{\chi}}}

{s^2 + {\color{magenta}b_{\dot{\chi}}}s + {\color{magenta}b_{\chi}}}{\color{blue}\chi^c}(s)

\]

Alternatively, the heading hold loop can be modeled as:

\[

\psi(s) = \frac{{\color{magenta}b_{\dot{\psi}}}s + {\color{magenta}b_{\psi}}}

{s^2 + {\color{magenta}b_{\dot{\psi}}}s + {\color{magenta}b_{\psi}}}{\color{blue}\psi^c}(s)

\]

Flight-path angle and load factor loops can be modeled as:

\[

\gamma(s) = \frac{{\color{magenta}b_{\gamma}}}{s+{\color{magenta}b_{\gamma}}}{\color{blue}\gamma^c}(s) \qquad\qquad

n_{\mathit{lf}}(s) = \frac{{\color{magenta}b_n}}{s+{\color{magenta}b_n}}{\color{blue}n_{\mathit{lf}}^c}(s)

\]

\end{minipage}

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Autopilot Models

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 5

\begin{minipage}{3.0in}

Airspeed hold and roll hold loops can be modeled as:

\begin{align*}

\dot{V}_a &= {\color{magenta}b_{V_a}}({\color{blue}V_a^c} - V_a) \\

\dot{\phi} &= {\color{magenta}b_{\phi}}({\color{blue}\phi^c} - \phi)

\end{align*}

Altitude and course hold loops can be modeled as:

\begin{align*}

\ddot{h} &= {\color{magenta}b_{\dot{h}}}(\dot{h}^c - \dot{h}) + {\color{magenta}b_h}({\color{blue}h^c}-h) \\

\ddot{\chi} &= {\color{magenta}b_{\dot{\chi}}}(\dot{\chi}^c-\dot{\chi}) + {\color{magenta}b_{\chi}}({\color{blue}\chi^c} - \chi)

\end{align*}

Alternatively, the heading hold loop can be modeled as:

\[

\ddot{\psi} = {\color{magenta}b_{\dot{\psi}}}(\dot{\psi}^c-\dot{\psi}) + {\color{magenta}b_{\psi}}({\color{blue}\psi^c} - \psi)

\]

Flight-path angle and load factor loops can be modeled as:

\begin{align*}

\dot{\gamma} &= {\color{magenta}b_{\gamma}}({\color{blue}\gamma^c} - \gamma) \\

\dot{n}_{\mathit{lf}} &= {\color{magenta}b_n} ({\color{blue}n_{\mathit{lf}}^c} - n_{\mathit{lf}})

\end{align*}

\end{minipage}

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Kinematic Model of Controlled Flight

Flight path projected onto ground

horizontal component

of groundspeed vector

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 6

7 of 25

Kinematic Model of Controlled Flight

Flight path projected onto ground

horizontal component

of groundspeed vector

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 7

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Coordinated Turn

Differentiate both sides of

vector wind-triangle equation (2.9). Solve resulting messy matrix

equation.

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 8

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Accelerating Climb

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 9

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Kinematic Guidance Models

Several guidance models can be derived, with varying levels of fidelity

Choice of model depends on application

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 10

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Kinematic Guidance Model - #1a

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 11

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Kinematic Guidance Model - #1b

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 12

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Kinematic Guidance Models - #2a

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 13

\begin{minipage}{5.0in}

\noindent

A more accurate model is to command the roll angle and use the coordinated turn model for $\chi$:

\begin{align*}

\dot{p}_n &= V_a \cos{\color{red}\psi} + {\color{blue}w_n} \\

\dot{p}_e &= V_a \sin{\color{red}\psi} + {\color{blue}w_e} \\

\dot\chi &= \frac{g}{{\color{red}V_g}}\tan\phi \cos(\chi-{\color{red}\psi})\\

\ddot{h} &= b_{\dot{h}}(\dot{h}^c - \dot{h}) + b_h({\color{blue}h^c}-h) \\

\dot{V_a} &= b_{V_a} ({\color{blue}V_a^c} - V_a) \\

\dot\phi &= b_{\phi} ({\color{blue}\phi^c}-\phi)

\end{align*}

where $\psi$ and $V_g$ are given by (equations (2.10) and (2.12))

\begin{align*}

{\color{red}\psi} &= \chi - \sin^{-1}\left(\frac{1}{V_a}\begin{pmatrix}w_n \\ w_e\end{pmatrix}^{\top}\begin{pmatrix}-\sin\chi \\ \cos\chi\end{pmatrix}\right) \\

{\color{red}V_g} &= w_n\cos\chi+w_e\sin\chi + \sqrt{(w_n\cos\chi+w_e\sin\chi)^2+V_a^2-w_n^2-w_e^2}

\end{align*}

\end{minipage}

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Kinematic Guidance Models - #2b

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 14

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Kinematic Guidance Models - #3

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 15

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Kinematic Guidance Models - #4

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 16

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Dubins Airplane Model

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 17

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Dubins Airplane Model (simplified)

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 18

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Dynamic Guidance Models

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 19

20 of 25

Dynamic Guidance Models

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 20

21 of 25

Dynamic Guidance Models

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 21

22 of 25

Dynamic Guidance Models

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 22

23 of 25

Dynamic Guidance Models

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 23

\begin{minipage}{3.0in}

\newcommand{\mass}{\mathsf{m}}

�

Given:

\begin{align*}

\mathbf{F}^{v2}

&= \begin{pmatrix}

-D + T \cos\alpha\cos\beta \\

-(Y+T \cos\alpha\sin\beta)\cos\phi - (L+T\sin\alpha)\sin\phi \\

(Y+T \cos\alpha\sin\beta)\sin\phi - (L+T\sin\alpha)\cos\phi

\end{pmatrix}.

\end{align*}

�

\vspace{1cm}

�

Longitudinal direction:

\begin{align*}

\mass V_g\dot{\gamma} &= - (Y+T \cos\alpha\sin\beta)\sin\phi + (L+T\sin\alpha)\cos\phi - \mass g\cos\gamma \\

\implies

\dot{\gamma} &= \frac{1}{\mass V}\left[(L+T\sin\alpha)\cos\phi - (Y+T \cos\alpha\sin\beta)\sin\phi - \mass g\cos\gamma \right].

\end{align*}

�

\vspace{1cm}

�

Lateral direction:

\begin{align*}

\mass V_g\dot{\chi}\cos\gamma &= (Y+T \cos\alpha\sin\beta)\cos\phi + (L+T\sin\alpha)\sin\phi \\

\implies

\dot{\chi} &= \frac{1}{\mass V_g\cos\gamma}\left[(Y+T \cos\alpha\sin\beta)\cos\phi + (L+T\sin\alpha)\sin\phi\right].

\end{align*}

�

\end{minipage}

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Dynamic Guidance Models: Summary

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 24

\begin{minipage}{3.0in}

\newcommand{\mass}{\mathsf{m}}

�

\begin{align*}

\dot{p}_n &= V_g \cos\chi\cos\gamma \\

\dot{p}_e &= V_g \sin\chi\cos\gamma \\

\dot{h} &= V_g \sin\gamma \\

\dot{V}_g &= \frac{{\color{blue}T}}{\mass} \cos{\color{blue}\alpha}\cos{\color{blue}\beta} - \frac{D}{\mass} - g\sin\gamma \\

\dot{\chi} &= \frac{1}{\mass V_g\cos\gamma}\left[(Y+{\color{blue}T} \cos{\color{blue}\alpha}\sin{\color{blue}\beta})\cos{\color{blue}\phi} + (L+{\color{blue}T}\sin{\color{blue}\alpha})\sin{\color{blue}\phi}\right] \\

\dot{\gamma} &= \frac{1}{\mass V_g}\left[(L+{\color{blue}T}\sin{\color{blue}\alpha})\cos{\color{blue}\phi} - (Y+{\color{blue}T} \cos{\color{blue}\alpha}\sin{\color{blue}\beta})\sin{\color{blue}\phi} - \mass g\cos\gamma \right],

\end{align*}

where

\begin{align*}

L(\alpha) &= \frac{1}{2}\rho V_a^2 S \left(C_{L_0} + C_{L_\alpha}{\color{blue}\alpha}\right) \\

D(\alpha) &= \frac{1}{2}\rho V_a^2 S \left(C_{D_0} + C_{D_\alpha}{\color{blue}\alpha} + C_{D_{\alpha^2}}{\color{blue}\alpha}^2\right) \\

Y(\beta) &= \frac{1}{2}\rho V_a^2 S C_{Y_\beta} {\color{blue}\beta},

\end{align*}

\end{minipage}

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Dynamic Guidance Models: Summary (simplified)

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 9, Slide 25

\begin{minipage}{3.0in}

\newcommand{\mass}{\mathsf{m}}

When sideslip is regulated to zeros:

\begin{align*}

\dot{p}_n &= V_g \cos\chi\cos\gamma \\

\dot{p}_e &= V_g \sin\chi\cos\gamma \\

\dot{h} &= V_g \sin\gamma \\

\dot{V}_g &= \frac{{\color{blue}T}}{\mass} \cos{\color{blue}\alpha} - \frac{D}{\mass} - g\sin\gamma \\

\dot{\chi} &= \frac{1}{\mass V_g\cos\gamma}\left(L+{\color{blue}T}\sin{\color{blue}\alpha}\right)\sin{\color{blue}\phi} \\

\dot{\gamma} &= \frac{1}{\mass V_g}\left[(L+{\color{blue}T}\sin{\color{blue}\alpha})\cos{\color{blue}\phi} - \mass g\cos\gamma \right],

\end{align*}

where

\begin{align*}

L(\alpha) &= \frac{1}{2}\rho V_a^2 S \left(C_{L_0} + C_{L_\alpha}{\color{blue}\alpha}\right) \\

D(\alpha) &= \frac{1}{2}\rho V_a^2 S \left(C_{D_0} + C_{D_\alpha}{\color{blue}\alpha} + C_{D_{\alpha^2}}{\color{blue}\alpha}^2\right)

\end{align*}

\end{minipage}