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Chapter 4

Forces and Moments

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 1

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Architecture

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 2

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Equations of Motion from Chap 3

System of 12 first-order ODE’s

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 3

\newcommand{\sth}{\text{s}_\theta}

\newcommand{\cth}{\text{c}_\theta}

\newcommand{\sph}{\text{s}_\phi}

\newcommand{\cph}{\text{c}_\phi}

\newcommand{\sps}{\text{s}_\psi}

\newcommand{\cps}{\text{c}_\psi}

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

\newcommand{\Vbf}{\mathbf{V}}

\newcommand{\defeq}{\stackrel{\triangle}{=}}

The combined equations of motion from Chapter 3 are

\begin{align*}

\begin{pmatrix} \dot{p}_n \\ \dot{p}_e \\ \dot{p}_d \end{pmatrix}

&= \begin{pmatrix} \cth \cps & \sph \sth \cps - \cph \sps

& \cph \sth \cps + \sph \sps \\

\cth \sps & \sph \sth \sps + \cph \cps

& \cph \sth \sps - \sph \cps \\

-\sth & \sph \cth & \cph \cth

\end{pmatrix}

\begin{pmatrix} u \\ v \\ w \end{pmatrix}

\label{eq:kin-eom-xyh} \\

\begin{pmatrix} \dot{u} \\ \dot{v} \\ \dot{w} \end{pmatrix}

&= \begin{pmatrix} rv-qw \\ pw-ru \\ qu-pv \end{pmatrix} +

\frac{1}{\mass} \begin{pmatrix} f_x \\ f_y \\ f_z \end{pmatrix},

\\

\begin{pmatrix} \dot{\phi} \\ \dot{\theta} \\ \dot{\psi} \end{pmatrix}

& = \begin{pmatrix}

1 & \sin\phi\tan\theta & \cos\phi\tan\theta \\

0 & \cos\phi & -\sin\phi \\

0 & \frac{\sin\phi}{\cos\theta} & \frac{\cos\phi}{\cos\theta}

\end{pmatrix}

\begin{pmatrix} p \\ q \\ r \end{pmatrix} \\

\begin{pmatrix} \dot{p} \\ \dot{q} \\ \dot{r} \end{pmatrix}

&= \begin{pmatrix}

\Gamma_1 pq - \Gamma_2 qr \\

\Gamma_5 pr - \Gamma_6 (p^2-r^2) \\

\Gamma_7 pq - \Gamma_1 qr

\end{pmatrix}

+\begin{pmatrix}

\Gamma_3 l + \Gamma_4 n \\

\frac{1}{J_y} m \\

\Gamma_4 l + \Gamma_8 n

\end{pmatrix}

\end{align*}

--

The objective of this chapter is to show how to compute the force vector

\[

\mathbf{f}^b = \begin{pmatrix} f_x \\ f_y \\ f_z \end{pmatrix}

\]

and the moment vector

\[

\mathbf{m}^b = \begin{pmatrix} \ell \\ m \\ n \end{pmatrix}.

\]

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External Forces and Moments

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

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Gravity Force

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 5

\newcommand{\sth}{\text{s}_\theta}

\newcommand{\cth}{\text{c}_\theta}

\newcommand{\sph}{\text{s}_\phi}

\newcommand{\cph}{\text{c}_\phi}

\newcommand{\sps}{\text{s}_\psi}

\newcommand{\cps}{\text{c}_\psi}

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

\newcommand{\Vbf}{\mathbf{V}}

\newcommand{\defeq}{\stackrel{\triangle}{=}}

�

\noindent The gravity vector expressed in the vehicle frame is

\[

\mathbf{f}_g^v = \begin{pmatrix} 0 \\ 0 \\ \mass g \end{pmatrix}.

\]

Expressed in the body frame we have

\begin{align*}

\mathbf{f}_g^b &= \mathcal{R}_{v}^{b} \begin{pmatrix} 0 \\ 0 \\ \mass g \end{pmatrix}

= \begin{pmatrix}

\cth \cps

& \cth \sps

& -\sth \\

\sph \sth \cps - \cph \sps

& \sph \sth \sps + \cph \cps

& \sph \cth \\

\cph \sth \cps + \sph \sps

& \cph \sth \sps - \sph \cps

& \cph \cth

\end{pmatrix}\begin{pmatrix} 0 \\ 0 \\ \mass g \end{pmatrix}

= \mass g \begin{pmatrix}

-\sin\theta \\ \cos\theta\sin\phi \\ \cos\theta\cos\phi

\end{pmatrix}

\end{align*}

�

\noindent For quaternions, we have

\begin{align*}

\mathbf{f}_g^b &=

\begin{pmatrix}

e_0^2+e_x^2-e_y^2-e_z^2 & 2(e_xe_y-e_0e_z) & 2(e_xe_z+e_0e_y) \\

2(e_xe_y+e_0e_z) & e_0^2-e_x^2+e_y^2-e_z^2 & 2(e_ye_z-e_0e_x) \\

2(e_xe_z-e_0e_y) & 2(e_ye_z+e_0e_x) & e_0^2-e_x^2-e_y^2+e_z^2

\end{pmatrix}

\begin{pmatrix} 0 \\ 0 \\ \mass g \end{pmatrix}

= \mass g \begin{pmatrix}

2(e_xe_z-e_ye_0) \\ 2(e_ye_z+e_xe_0) \\ e_z^2+e_0^2-e_x^2-e_y^2

\end{pmatrix}.

\end{align*}

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Airfoil Pressure Distribution

above static pressure

below static pressure

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 6

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Aerodynamic Approximation

Assumption:

Forces and moment act at aerodynamic center of wing

Approximately at quarter chord

Defined as point where moment is constant with angle of attack

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 7

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Control Surfaces - Conventional

Ailerons
Elevator
Rudder

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 8

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Control Surfaces – V-tail

Ailerons
Ruddervators

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

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Control Surfaces – Flying Wing

Elevons

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 10

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Aircraft Dynamics

Aircraft dynamics and aerodynamics are commonly separated into two groups:

Longitudinal

Up-down, pitch plane, pitching motions

Lateral-directional

Side-to-side, turning motions (roll and yaw)

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 11

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Longitudinal Aerodynamics

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 12

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Aerodynamic Approximation

stability derivatives

control derivative

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 13

\newcommand{\pde}[2]{\frac{\partial#1}{\partial#2}}

\newcommand{\defeq}{\stackrel{\triangle}{=}}

\noindent In the general nonlinear case we have

\[

F_{\text{lift}} \approx \frac{1}{2}\rho V_a^2 S C_L (\alpha,q,\delta_e)

\]

Expanding $C_L$ as a Taylor series and keeping only the first-order (linear) terms gives

\begin{align*}

F_{\text{lift}} &= \frac{1}{2}\rho V_a^2 S \left[ C_{L_0} + \pde{C_L}{\alpha}\alpha +

\pde{C_L}{q} q + \pde{C_L}{\delta_e}\delta_e \right] \\

&= \frac{1}{2}\rho V_a^2 S \left[ C_{L_0} + C_{L_{\alpha}}\alpha +

C_{L_{q}}\frac{c}{2V_a}q + C_{L_{\delta_e}}\delta_e \right]

\end{align*}

where the coefficients $C_{L_0}$, $C_{L_\alpha}\defeq\pde{C_L}{\alpha}$, $C_{L_q}\defeq\pde{C_L}{\frac{qc}{2V_a}}$, and $C_{L_{\delta_e}}\defeq\pde{C_L}{\delta_e}$

are dimensionless quantities

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Linear Aerodynamic Model

Linear aerodynamic model is valid for small angles of attack – flow remains attached over wing

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 14

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Nonlinear Aerodynamics – Stall

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 15

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Nonlinear Lift Model

linear model

nonlinear model

flat-plate model

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 16

17 of 44

Nonlinear Aerodynamic Model

linear model

flat-plate model

blending function

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 17

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Blending Function

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 18

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Nonlinear Aerodynamic Model

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 19

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Drag vs. Angle of Attack

linear model incorrect for

negative angles of attack

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 20

Induced drag intuitive explanations:

Wingtip vortices change surface pressure distributions on wind to increase in direction of drag
Because local relative wind is canted downward due to contributions of wingtip vortices, the lift vector is tilted back and contributes to a force parallel with V_inf that is felt as drag
Wingtip vortices contain certain amount of kinetic energy that must come from somewhere – comes from the propulsion system. Extra power provided is overcoming the induced drag.

----------

\begin{align*}

C_{D}(\alpha) &= \underbrace{C_{D_p}}_{\text{parasitic drag}} + \underbrace{\frac{(C_{L_0} + C_{L_\alpha} \alpha)^2}{\pi e AR}}_{\text{induced drag}} \\

&= \left(C_{D_p} + \frac{C_{L_0}^2}{\pi e AR}\right)

+ \left(\frac{2C_{L_0}C_{L_\alpha}}{\pi e AR}\right)\alpha

+ \left(\frac{C_{L_\alpha}^2}{\pi e AR}\right)\alpha^2,

\end{align*}

where $0.8\leq e \leq 1.0$ is the Oswald efficiency factor.

�

\vspace{0.5cm}

�

{\bf Parasitic drag:} profile drag of wing, friction and pressure drag of tail surfaces, fuselage, landing gear, etc.

�

\vspace{0.5cm}

�

{\bf Induced drag:} drag due to lift

$e$ is the Oswald efficiency factor

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Linear Lift and Drag Models

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 21

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Longitudinal Forces – Body Frame

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 22

\begin{pmatrix} f_x \\ f_z \end{pmatrix}

&= \begin{pmatrix}

\cos\alpha & -\sin\alpha \\

\sin\alpha & \cos\alpha

\end{pmatrix}

\begin{pmatrix}

-F_{\textrm{drag}} \\ -F_{\textrm{lift}}

\end{pmatrix} \\

&= \frac{1}{2}\rho V_a^2 S \begin{pmatrix}

\left[ -C_{D}(\alpha)\cos\alpha + C_{L}(\alpha)\sin\alpha \right]

\\ \quad

+\left[ -C_{D_q}\cos\alpha + C_{L_q}\sin\alpha

\right]\frac{c}{2V_a}q

+\left[ -C_{D_{\delta_e}}\cos\alpha + C_{L_{\delta_e}}\sin\alpha \right] \delta_e

\\

---

\\

\left[ -C_{D}(\alpha)\sin\alpha - C_{L}(\alpha)\cos\alpha \right]

\\\quad

+\left[ -C_{D_q}\sin\alpha - C_{L_q}\cos\alpha

\right]\frac{c}{2V_a}q

+\left[ -C_{D_{\delta_e}}\sin\alpha - C_{L_{\delta_e}}\cos\alpha \right] \delta_e

\end{pmatrix}

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Pitching Moment

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 23

24 of 44

Lateral Aerodynamics

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 24

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Aerodynamic Coefficients

Static stability derivatives describe spring behavior of aerodynamics

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 25

\noindent

$C_{m_{\alpha}}$, $C_{\ell_{\beta}}$, $C_{n_\beta}$, $C_{m_q}$, $C_{\ell_p}$, $C_{n_r}$ are called the {\em stability derivatives} because their values determine the stability of the aircraft.

\vspace{0.75cm}

\underline{Static Stability Derivatives}

\begin{itemize}

\item $C_{m_{\alpha}}$ - longitudinal static stability derivative. Must be $\leq 0$ for stability: increase in $\alpha$ causes and downward pitching moment.

\item $C_{\ell_\beta}$ - roll static stability derivative. Associated with dihedral in wings. Must be $\leq 0$ for stability: positive roll $\phi$ causes a restoring moment.

\item $C_{n_\beta}$ - yaw static stability derivative. Weathercock stability derivative. Influenced by design of tail. Causes airframe to align with the wind vector. Must be $\geq 0$ for stability: cocks airframe into wind driving $\beta$ to zero.

\end{itemize}

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Longitudinal Static Stability Derivative

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 26

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Roll Static Stability Derivative

(view from the tail)

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 27

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Yaw Static Stability Derivative

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 28

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Aerodynamic Coefficients

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 29

Q: If we linearize the equations of motion to be in the form xdot = Ax + Bu, where do the stability derivatives show up? Control derivatives?

�

----------------------

�

\underline{Dynamic Stability Derivatives}

\begin{itemize}

\item $C_{m_q}$, $C_{\ell_p}$, $C_{n_r}$ are known as the pitch damping derivative, roll damping derivative, and yaw damping derivative, respectively. They quantify the level of damping associated with angular motion of the airframe.

\end{itemize}

�

\vspace{0.5cm}

{\color{cyan} \large Dynamic stability derivatives describe damping behavior of aerodynamics}

\vspace{0.5cm}

�

\underline{Control Derivatives}

\begin{itemize}

\item $C_{m_{\delta_e}}$, $C_{\ell_{\delta_a}}$, and $C_{n_{\delta_r}}$ are the primary control derivatives and quantify the effect on the control surfaces on their primary intended axes of influence.

\item $C_{\ell_{\delta_r}}$ and $C_{n_{\delta_a}}$ are the cross-control derivatives.

\end{itemize}

30 of 44

Propeller Thrust and Torque

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 30

31 of 44

Propeller Thrust and Torque

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 31

32 of 44

Propeller Thrust and Torque

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 32

33 of 44

Propeller Thrust and Torque

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 33

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Propeller Thrust and Torque

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 34

Draw the two torque curves – show their intersection as the operating speed.

\noindent For a given voltage input to our motor, we want to know the propeller speed and how much thrust the propeller produces. This will depend on the airspeed of the aircraft (and other things). \\

�

\noindent Propeller torque can be expressed as a function of propeller speed as:

\[

Q_p(\Omega_p, C_Q) = \frac{\rho D^5}{4\pi^2}\Omega_p^2 C_Q

\]

�

\noindent Motor torque can also be expressed as a function of motor speed:

\[

Q_m = K_Q\left[ \frac{1}{R}\left(V_{in} - K_V \Omega \right) - i_0 \right],

\]

�

\noindent We know that the propeller torque and the motor torque are equivalent (they are connected by a shaft that we are assuming is rigid), so we can find the operating speed of the motor/prop combination by equating the torque expressions above and solving for $\Omega = \Omega_p$.

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Propeller Thrust and Torque

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 35

36 of 44

Propeller Thrust and Torque: Summary

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 36

Given $\delta_t\in[0,1]$, solve for motor input voltage: $V_{in} = V_{max}\delta_t$

�

\vspace{0.5cm}

�

Solve for rotor angular speed as the positive root of:

\[

\left(\frac{\rho D^5}{(2\pi)^2}C_{Q0}\right) \Omega^2 +

\left( \frac{\rho D^4}{2\pi}C_{Q1} V_a +\frac{K_Q K_V}{R} \right) \Omega %\\ &

+ \left( \rho D^3 C_{Q2} V_a^2 - \frac{K_Q}{R} V_{in} + K_Q i_0 \right) = 0

\]

�

Find the advanced ratio $J=\frac{2\pi V_a}{\Omega_\text{op} D}$, and solve for the aerodynamic coefficients:

\begin{align*}

C_T ( J ) &\approx C_{T2} J^2 + C_{T1} J + C_{T0} \\

C_Q ( J ) &\approx C_{Q2} J^2 + C_{Q1} J + C_{Q0} \\

\end{align*}

�

Solve for thrust and torque:

\begin{align*}

T_p &= \rho \left(\frac{\Omega}{2\pi}\right)^2 D^4 C_T(J) \\

Q_p &= \rho \left(\frac{\Omega}{2\pi}\right)^2 D^5 C_Q(J),

\end{align*}

37 of 44

Propeller Thrust and Torque

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 37

\begin{lstlisting}[language=python]

# compute thrust and torque due to propeller (See addendum by McLain)

# map delta_t throttle command(0 to 1) into motor input voltage

V_in = MAV.V_max * delta_t

# Quadratic formula to solve for motor speed

a = MAV.C_Q0 * MAV.rho * np.power(MAV.D_prop, 5) \

/ ((2.*np.pi)**2)

b = (MAV.C_Q1 * MAV.rho * np.power(MAV.D_prop, 4)

/ (2.*np.pi)) * self._Va + KQ**2/MAV.R_motor

c = MAV.C_Q2 * MAV.rho * np.power(MAV.D_prop, 3) \

* self._Va**2 - (KQ / MAV.R_motor) * Volts + KQ * MAV.i0

# Consider only positive root

Omega_op = (-b + np.sqrt(b**2 - 4*a*c)) / (2.*a)

# compute advance ratio

J_op = 2 * np.pi * self._Va / (Omega_op * MAV.D_prop)

# compute non-dimensionalized coefficients of thrust and torque

C_T = MAV.C_T2 * J_op**2 + MAV.C_T1 * J_op + MAV.C_T0

C_Q = MAV.C_Q2 * J_op**2 + MAV.C_Q1 * J_op + MAV.C_Q0

# add thrust and torque due to propeller

n = Omega_op / (2 * np.pi)

fx += MAV.rho * n**2 * np.power(MAV.D_prop, 4) * C_T

Mx += -MAV.rho * n**2 * np.power(MAV.D_prop, 5) * C_Q

\end{lstlisting}

38 of 44

Wind Model

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 38

39 of 44

Wind Model

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 39

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Dryden Gust Model

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 40

41 of 44

Transfer Function Implementation

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 41

42 of 44

Adding in the Effects of Wind

Key concept:

Wind and turbulence affect airspeed,�angle of attack, and sideslip angle

It is through these parameters that�wind and atmospheric effects enter�the calculation of aerodynamic forces�and moments, and thereby affect�motion of aircraft

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 42

43 of 44

Summary

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 43

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

\begin{align*}

\begin{pmatrix} f_x \\ f_y \\ f_z \end{pmatrix}

= \begin{pmatrix}

-\mass g\sin\theta \\ \mass g\cos\theta\sin\phi \\ \mass g\cos\theta\cos\phi

\end{pmatrix}

&+ \frac{1}{2}\rho V_a^2 S \begin{pmatrix}

C_{X}(\alpha)

+ C_{X_{q}}(\alpha) \frac{c}{2V_a}q \\

C_{Y_0}

+ C_{Y_{\beta}} \beta

+ C_{Y_p} \frac{b}{2V_a}p

+ C_{Y_r} \frac{b}{2V_a}r \\

C_{Z}(\alpha)

+ C_{Z_q}(\alpha) \frac{c}{2V_a}q

\end{pmatrix}

\\

&+ \frac{1}{2}\rho V_a^2 S \begin{pmatrix}

C_{X_{\delta_e}}(\alpha) \delta_e \\

C_{Y_{\delta_a}} \delta_a + C_{Y_{\delta_r}} \delta_r \\

C_{Z_{\delta_e}}(\alpha) \delta_e

\end{pmatrix}

+ \begin{pmatrix}

T_p(\delta_t V_a) \\ 0 \\ 0

\end{pmatrix}

\end{align*}

44 of 44

Project 4

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012 Chapter 4: Slide 44