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# Investigation into the time varying mass effect on airship controller performance

Jayme Waishek, Vinay Kumar, Atilla Dogan, Yasmina Bestaoui

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Jayme Waishek, Vinay Kumar, Atilla Dogan, Yasmina Bestaoui. Investigation into the time varying mass effect on airship controller performance. AIAA Atmospheric Flight Mechanics Conference, Aug 2009, Chicago, United States. 10.2514/6.2009-6150. [hal-00869620](https://hal.science/hal-00869620)

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Distributed under a Creative Commons Attribution 4.0 International License
Investigation into the Time Varying Mass Effect on
Airship Controller Performance

Jayme Waishek *, Vinay Kumar †, and Atilla Dogan ‡

Department of Mechanical and Aerospace Engineering

University of Texas at Arlington, TX 76019

Yasmina Bestaoui §

Laboratoire IBISC, Universit d'Evry, 91025 EVRY, France

This paper addresses the significance of airship time varying inertia properties due to fuel consumption on the performance of an airship controller which is developed under the assumption that these properties are insignificant. A previously developed set of nonlinear, 6-DOF airship equations of motion which include the effect of time-varying mass and inertia properties associated with fuel consumption, as well as wind effect, are used. A proportional-integral controller is implemented to compute the three longitudinal control variables (main thrust, main thrust deflection, and elevator deflection) to control the altitude, pitch angle, and speed (relative to air or to ground). The simulation results obtained are both with and without the time varying mass terms in the equations of motion, in both steady cruise and prevailing wind conditions, for demonstration of the differences between the capabilities of a controller developed for a mass invariant aircraft when used in a "real" airship experience fuel burn-off.

# I. Introduction

Since the resurgence of lighter-than-air vehicles (a.k.a. airships, dirigibles) in the early 1990's, they have been increasingly considered for varied tasks involving transportation, surveillance, freighting, advertising, monitoring, research, and military roles. They are safe, cost-effective, durable, environmentally benign, and simple to operate, which makes them suitable for a wide range of applications and attractive to many varied customers.

The method by which airships achieve levitation is the employment of a lifting gas (i.e. helium or hot air). The only solid parts of the vehicle are the gondola, the propellers (a pair of propellers mounted at the gondola and a tail rotor with horizontal axis of rotation) and the tail fins. The envelope holds the helium that makes the airship lighter than air. In addition to the lift provided by helium, moving airships derive aerodynamic lift from the shape of the envelope as well as the elevator, if applicable. The airship ellipsoid shape exhibits a highly aerodynamic profile with good resistance to aerostatic pressures. Like the helicopter, airships possess different properties depending on their flight modes. The approach and landing, ground maneuvering, and masting phases of flight demand the highest degree of control precision, and yet these critical flight phases generally exhibit lack of controllability.¹ The control system must be robust enough to accommodate the airship's susceptibility to gusts, thermals, and mass variance, as well as its inherent aerodynamic instability and loss of aerodynamic control at low airspeeds. The problems associated with gust and mass variance sensitivity and aerodynamic instability are fundamental to ellipsoidal airships.

Airships offer many controller development challenges. One characteristic of airships that forces novel controller development techniques is their generally underactuated natures. In particular, airships are subject to control variable saturation in situations exhibiting reduced aerodynamic forces (e.g. low speed cruise or

<footnote>*Student Member AIAA, Graduate Student.</footnote>

<footnote>†Student Member AIAA, Graduate Student.</footnote>

<footnote>‡Senior Member AIAA, Assistant Professor.</footnote>

<footnote>§Member AIAA, Associate Professor.</footnote>
hover) and in aircraft configurations containing main propeller vectoring for the provision of both forward and vertical forces.² The vertical axis has proven particularly difficult to control while attempting station keeping because of factors including the gradual depletion of fuel mass and the gradual increase in the internal helium temperature during outdoor operation,³ even using a combination of ballonet volume control and vertical thrust vectoring. Additionally, airships are prone to lateral underactuation which some authors address by always adjusting the airship heading into the wind² using heading and path tracking control⁴ with rudder deflections and a small tail thruster.

The purpose of this paper will be to simulate and compare the performance of an airship controller when the airship is affected by the time varying mass effect of fuel burn-off. A simulation environment with airship equations of motion derived both with and without the time-varying mass terms present will be used. The airship simulation results will be compared to determine the true significance of the time varying inertial properties on the airship dynamic behavior. Additional simulation cases are presented to show the response of the airship to a long-term wind encounter, as modeled by the equations previously derived. The equations of motion simulated herein were derived⁵ from the more general set of equations of motion for an aircraft having both time varying mass properties and experiencing a wind effect, which were previously derived for the receiver aircraft in an aerial refueling scenario.⁶,⁷

## II. General System Modeling

The airship equations of motion were derived⁵ by defining the airship body frame in such a way that the motion of the airship Center of Mass (CM) due to fuel burn-off could be accounted for. This was accomplished by defining the origin of the airship's body frame ($B_A$-frame) at the location of its Center of Volume (CV), with a vector denoted $\rho_{CM}$ from the origin of the body frame to the location of rigid body empty CM.

The airship equations of motion were derived⁵ from previously developed⁶,⁷ general equations of motion in matrix form for an aircraft experiencing mass transfer and wind effect, derived relative to the tanker's body frame. The results are presented below and are implemented in a simulation environment in the state-space forms presented in the reference⁵ containing the full dynamics derivation. The translational dynamics equation is

$$ (M+m)\dot{U}_A = (M+m)[\mathbf{S}(\omega_{B_A})U_A - \mathbf{R}_{B_A I}\dot{\mathbf{W}}] + \mathbf{R}_{B_A I} F - M\mathbf{S}^2(\omega_{B_A})\rho_{CM} - M\mathbf{S}(\rho_{CM})\dot{\omega}_{B_A} \\ - \sum_{j=1}^{k} \dot{m}_j [-\mathbf{S}(\omega_{B_A}) \rho_{m_j} + \dot{\rho}_{m_j}] \\ - \sum_{j=1}^{k} m_j \{\mathbf{S}(\rho_{m_j}) \dot{\omega}_{B_A} + \mathbf{S}^2(\omega_{B_A}) \rho_{m_j} + \ddot{\rho}_{m_j}\} \\ - 2\sum_{j=1}^{k} m_j [-\mathbf{S}(\omega_{B_A}) \dot{\rho}_{m_j}] - \mathbf{S}(\omega_{B_A}) \sum_{l=1}^{p} \dot{m}_l \rho_{c_l} \quad (1) $$

where $\omega_{B_A}$ is the angular velocity vector of the airship relative to the inertial frame, $\mathbf{S}(\omega_{B_A})$ is the skew-symmetric form of $\omega_{B_A}$, $M$ is the weight of the airship minus fuel, and $m$ is the fuel weight, which is time varying due to fuel burn. In Eq. (1), $m_j$ represents the mass of fuel in the $j$th fuel tank, $\rho_{m_j}$ represents the position of the fuel center of mass in the $j$th fuel tank, $\dot{m}_l$ is the fuel burn rate of engine $l$, and $\rho_{c_l}$ is the position of engine $l$ in the body frame. In the derivation of the above equation, the fuel velocity at the fuel port relative to the $B_A$ frame is assumed to be zero.

The resultant force acting on the airship, ($\underline{F}$, in Eq. (1)) is considered to be the sum of the gravity force vector $\underline{F}_G$, the aerodynamic force vector $\underline{A}_A$, and the propulsive force vector $\underline{P}_A$. Thus, $F$ (representation in the inertial frame), in terms of $F_G$ (representation in the inertial frame), $A_A$ (representation in the body frame), and $P_A$ (representation in the body frame), is

$$ F = F_G + \mathbf{R}_{B_A I}^T A_A + \mathbf{R}_{B_A I}^T P_A \quad (2) $$
In general, the representation of the force vector $\underline{F}_G$ is

$$F_G = \begin{bmatrix} 0 \\ 0 \\ ((M+m) - (\rho_\Delta V))g \end{bmatrix} \qquad (3)$$

where $(M+m)g$ is the force caused by the total airship weight and $-\rho_\Delta Vg$ is the buoyant force of the airship as described by Archimedes' Principle. Herein, $\rho_\Delta$ is the difference between the density of the air and of the helium within the airship envelope, and $V$ is the envelope volume.

The rotational motion of the airship is analyzed with respect to the inertial frame in a manner similar to the translational motion. The rotational kinematics equations is determined to be

$$\mathbf{R}_{\mathbf{B}_A \mathbf{I}} \dot{\mathbf{R}}_{\mathbf{B}_A \mathbf{I}}^{\mathrm{T}} = -\mathbf{S}(\omega_{\mathbf{B}_A}) \qquad (4)$$

where $\omega_{\mathbf{B}_A}$ is the representation of the angular velocity vector of the airship relative to the inertial frame and can be expanded as

$$\omega_{\mathbf{B}_A} = [p \ q \ r]^T \qquad (5)$$

and matrix $\mathbf{S}(\omega_{\mathbf{B}_A})$ in Eq. (4) is obtained based on the skew-symmetric matrix operation.

The derivation of the equation of rotational dynamics of the airship including the effect of total mass change, change in the position of the airship's CM and the wind effect, in matrix form, is likewise accomplished by simplifying the more general previously developed$^{6,7}$ equation to be written relative to the inertial frame. The more general equation was derived in a manner which parallels the translational dynamics approach. By this approach, the airship rotational dynamics equation is determined to be

$$\begin{aligned} \underline{\mathbf{I}}_{\underline{\mathbf{t}}} \dot{\omega}_{\mathbf{B}_A} = & M_{\mathbf{B}_A} + \mathbf{S}(\omega_{\mathbf{B}_A}) \underline{\mathbf{I}}_{\underline{\mathbf{M}}} (\omega_{\mathbf{B}_A}) + M \mathbf{S}(\rho_{\mathrm{CM}}) [-\mathbf{S}(\omega_{\mathbf{B}_A}) U_A + \dot{U}_A + \mathbf{R}_{\mathbf{B}_A \mathbf{I}} \dot{\mathbf{W}}] \\ & + \sum_{j=1}^{k} \mathbf{S}(\rho_{\mathbf{m}_j}) [m_j \omega_{\mathbf{B}_A}^T \rho_{\mathbf{m}_j} \omega_{\mathbf{B}_A} + m_j \ddot{\rho}_{\mathbf{m}_j} + \dot{m}_j \dot{\rho}_{\mathbf{m}_j}] \\ & + \left[ \sum_{j=1}^{k} \mathbf{S}(\rho_{\mathbf{m}_j}) m_j \right] \left\{ -\mathbf{S}(\omega_{\mathbf{B}_A}) U_A + \dot{U}_A + \mathbf{R}_{\mathbf{B}_A \mathbf{I}} \dot{\mathbf{W}} \right\} \\ & + \left[ \sum_{j=1}^{k} \mathbf{S}(\rho_{\mathbf{m}_j}) \dot{m}_j \right] (U_A + \mathbf{R}_{\mathbf{B}_A \mathbf{I}} W) \\ & - 2 \sum_{j=1}^{k} m_j \left[ (\rho_{\mathbf{m}_j}^T \dot{\rho}_{\mathbf{m}_j}) \mathbf{I}_{3 \times 3} - \dot{\rho}_{\mathbf{m}_j} \rho_{\mathbf{m}_j}^T \right] \omega_{\mathbf{B}_A} \\ & - \sum_{j=1}^{k} \dot{m}_j \left[ (\rho_{\mathbf{m}_j}^T \rho_{\mathbf{m}_j}) \mathbf{I}_{3 \times 3} - \rho_{\mathbf{m}_j} \rho_{\mathbf{m}_j}^T \right] \omega_{\mathbf{B}_A} \\ & + \sum_{l=1}^{p} \dot{m}_l \left[ (\rho_{\mathrm{c}_l}^T \rho_{\mathrm{c}_l}) \mathbf{I}_{3 \times 3} - \rho_{\mathrm{c}_l} \rho_{\mathrm{c}_l}^T \right] \omega_{\mathbf{B}_A} \\ & - \left[ \sum_{l=1}^{p} \dot{m}_l \mathbf{S}(\rho_{\mathrm{c}_l}) \right] (U_A + \mathbf{R}_{\mathbf{B}_A \mathbf{I}} W) \end{aligned} \qquad (6)$$

where $\underline{\mathbf{I}}_{\underline{\mathbf{M}}}$ is the inertia matrix of the rigid (time-invariant mass) portion of the airship with respect to the origin of the $\mathbf{B}_A$ frame and $\underline{\mathbf{I}}_{\underline{\mathbf{t}}}$ is the total inertia matrix including the effect of the fuel in each fuel tank.

The total moment of the external forces about the origin of $\mathbf{B}_A$-frame can be written as

$$\underline{M}_{\mathbf{B}_A} = \underline{M}_G + \underline{M}_A + \underline{M}_P \qquad (7)$$

where $\underline{M}_G$, $\underline{M}_A$, and $\underline{M}_P$ are the moment vectors due to gravity, aerodynamics, and propulsion, respectively. Note that the moment due to gravity is not zero since the origin of $\mathbf{B}_A$-frame is at the CV of the airship
rather than its CM. Therefore, the gravitational moment is due to the rigid and fuel masses, as well as the buoyant force in the event that its line of action does not pass through the $B_A$-frame origin, such that

$$ \underline{M}_G = \sum_{j=1}^{k} (\underline{\rho}_{m_j} \times m_j \underline{G}) + \underline{\rho}_B \times \underline{B} + \underline{\rho}_{CM} M \times \underline{G} \quad (8) $$

where $G$ is the vector of the gravitational acceleration and $B$ is the vector of the buoyant force. Let $G$ be the representation of the $G$ vector in the inertial frame. Thus, $G = [0 \ 0 \ g]$ where $g$ is the magnitude of the acceleration due to gravity. Additionally let $B$ be the representation of the $B$ vector in the inertial frame. Thus, $B = [0 \ 0 \ -\rho V g]$. Equation (8) is written in matrix form as

$$ M_G = - \sum_{j=1}^{k} m_j \mathbf{S}(\rho_{m_j}) \mathbf{R}_{B_A} \mathbf{I} G + \mathbf{S}(\rho_B) \mathbf{R}_{B_A} \mathbf{I} B + M \mathbf{S}(\rho_{CM}) \mathbf{R}_{B_A} \mathbf{I} G \quad (9) $$

The aerodynamic and propulsion moment vectors are expanded in Section III.

The "added masses and inertias" associated with the displacement of the fluid medium through which the airship travels are detailed in the reference⁵ containing the original airship dynamics equations derivation.

### III. AS500 Airship Model with PI Controller

This section shows how the physical attributes of the airship fuel tanks and fuel consumption are incorporated as mathematical quantities in the general equations of motion, as well as the implemented controller created simply for a mass invariant airship. The simulation is based on the Airspeed Airship AS500, the aircraft data for which can be found in the original dynamics derivation.⁵

<figure><img src="image_3.png" /><figcaption>Figure 1. The Airspeed Airship AS-500</figcaption></figure>

The airship contains a single fuel tank located at (0.95, 0, 1.17) with respect to the $B_A$ frame origin, fuel flow from which is denoted $\dot{m}_1 = \dot{m}$. This tank fuels the main and tail engines simultaneously at a rate that is a linear function of each engine's throttle. Therefore, the sum of the mass consumed by the main and tail engines, $\sum_{l=1}^{p} \dot{m}_l$, is equal to the total mass change of the single fuel tank. The actual mass of $m_1$, the fuel amount left in the fuel tank, at any instant of time can be computed by integrating $\dot{m}_1$, a negative number, over time. With the main propellers providing full thrust, generating such a fuel burn rate, it takes the fuel tank 10300 seconds to empty.

To determine the position vector of the fuel mass concentrated at the single tank, $\underline{\rho}_{m_1}$, it is assumed that fuel in each fuel tank is concentrated at its CM. In general, the initial location $\underline{\rho}_{m_j}(0)$ is considered to be the CM of the fuel in the fuel tank at the initial time. As the fuel tank mass changes, the vector $\underline{\rho}_{m_j}(t)$ always points at the CM of the fuel under the assumptions that the fuel tanks are rectangular and the fuel stays level within each tank. The position vectors $\underline{\rho}_{m_1}$ along with its first and second time derivatives are used in the airship's equations of motion.
The AS500 is an under-actuated system having the most popular propulsion system layout for pressurized non rigid airships (ref Fig. 1). The propulsion system consists of twin ducted main propellers mounted either side of the envelope bottom as well as a propeller in the tail for torque correction and attitude control. If the thrust provided by the main envelope propellers in the body frame is denoted as $T_M$ with a tilt angle of $\gamma$, and the thrust provided by the tail propeller in the body frame is denoted as $T_T$, then the representation of the force vector $\underline{P}_A$ as it appears in Eq. (2) is

$$P_A = P_{A_M} + P_{A_T} \qquad (10)$$

where $P_{A_M}$ is the propulsion force associated with the main propellers and $P_{A_T}$ is that associated with the tail propellers, which are written as

$$P_{A_M} = \begin{bmatrix} T_M \cos \gamma \\ 0 \\ T_M \sin \gamma \end{bmatrix} \qquad P_{A_T} = \begin{bmatrix} 0 \\ T_T \\ 0 \end{bmatrix} \qquad (11)$$

where $\gamma$ is the tilt angle of the main propellers, which tilt simultaneously. The moment due to propulsion as it appears in Eq (7) is

$$\underline{M}_P = \underline{\rho}_M \times \underline{P}_{A_M} + \underline{\rho}_T \times \underline{P}_{A_T} \qquad (12)$$

where $\underline{\rho}_M$ and $\underline{\rho}_T$ are the main propeller and tail propeller moment arms in the airship's body frame, respectively.

The AS500 is additionally controlled with elevator and rudder surfaces for improved controllability of the pitch and heading angles, respectively, at significant airspeeds. These surfaces are mounted at the rear of the airship (ref Fig. 1). The aerodynamic force and moment effect of the elevator and rudder can be seen in the equations for $A_X$, $A_Y$, $A_Z$, $M_L$, $M_M$, and $M_N$ (the aerodynamic force and moment vector components), repeated from the original derivation⁵,⁸ for reference below:

$$\begin{align}
A_X &= \frac{1}{2} \rho V_a^2 \left[ C_{X1} \cos^2 \alpha \cos^2 \beta + C_{X2} \sin(2\alpha) \sin(\alpha/2) \right] \nonumber \\
A_Y &= \frac{1}{2} \rho V_a^2 \left[ C_{Y1} \cos(\beta/2) \sin(2\beta) + C_{Y2} \sin(2\beta) + C_{Y3} \sin(\beta) \sin(|\beta|) + 2C_{Y4} \delta_r \right] \nonumber \\
A_Z &= \frac{1}{2} \rho V_a^2 \left[ C_{Z1} \cos(\alpha/2) \sin(2\alpha) + C_{Z2} \sin(2\alpha) + C_{Z3} \sin(\alpha) \sin(|\alpha|) + 2C_{Z4} \delta_e \right] \tag{13}
\end{align}$$

$$\begin{align}
M_L &= \frac{1}{2} \rho V_a^2 C_{L2} \sin(\beta) \sin(|\beta|) \nonumber \\
M_M &= \frac{1}{2} \rho V_a^2 \left[ C_{M1} \cos(\alpha/2) \sin(2\alpha) + C_{M2} \sin(2\alpha) + C_{M3} \sin(\alpha) \sin(|\alpha|) + 2C_{M4} \delta_e \right] \nonumber \\
M_N &= \frac{1}{2} \rho V_a^2 \left[ C_{N1} \cos(\beta/2) \sin(2\beta) + C_{N2} \sin(2\beta) + C_{N3} \sin(\beta) \sin(|\beta|) + 2C_{N4} \delta_r \right] \tag{14}
\end{align}$$

where $\rho$ is the air density and $V_a$ is the body frame velocity of the airship determined by

$$V_a = \sqrt{u^2 + v^2 + w^2} \qquad (15)$$

Additionally, the angle of attack and sideslip angle are calculated by

$$\begin{align}
\alpha &= \tan^{-1} \frac{w}{u} \nonumber \\
\beta &= \tan^{-1} \frac{v \cos(\alpha)}{u} \tag{16}
\end{align}$$

and $\delta_e$ and $\delta_r$ are the elevator and rudder deflections, respectively. All coefficients $C_{X1}...C_{N4}$ are given in the original derivation.⁵

Control variable authority is limited with saturation values typical for the AS500 model airship. The main thrust, $T_M$, is limited to a range of 0 to 20 N. The tail thrust, $T_T$, is limited to a range of 0 to 10 N.
The main thrust deflection angle, $\gamma$, is limited to a range of $\pm 20$ degrees. The elevator and rudder deflection angles, $\delta_e$ and $\delta_r$ respectively, are each limited to a range of $\pm 30$ degrees.

Because the AS500 fuel tank is centered about the body x-axis and no unsteady or "slosh" effect is considered, only the longitudinal flight and controllability characteristics are significant in this investigation. A simple proportional-integral (PI) controller was developed to command the individual longitudinal control variables $T_M$ (main propeller thrust), $\gamma$ (main propeller horizontal deflection angle), and $\delta_e$ (elevator deflection angle) under the assumption that fuel mass was constant. This type of controller allows for stable, zero-error steady state response without requiring the derivatives of potentially fast-changing control variable values. The proportional and integral gains for each control variable were tuned by trial for acceptable overshoot and settling time for commanded airship speed, altitude, and pitch angle. The final controller configuration is

$$ 
\begin{aligned}
T_M(t) &= 0.8e_u(t) + 0.1 \int_0^t e_u(\tau)d\tau \\
\gamma(t) &= 1.2e_Z(t) + 0.1 \int_0^t e_Z(\tau)d\tau \\
\delta_e(t) &= -3e_\theta(t) - 1 \int_0^t e_\theta(\tau)d\tau
\end{aligned} 
\qquad (17) $$

where the error values are computed individually for each control variable. The value $e_u$ is the difference between the commanded value of the body frame x-direction velocity relative to the surrounding air, $u$, and the actual value of $u$. In a case where groundspeed (inertial frame x-direction speed) is controlled instead of $u$, the value $e_u$ is the difference between the commanded x-direction groundspeed and the actual x-direction groundspeed. The value $e_Z$ is the difference between the commanded inertial frame vertical position, $Z$, and the actual value of $Z$. The value of $e_\theta$ is the difference between the commanded body-frame pitch angle, $\theta$, and the actual value of $\theta$.

## IV. Simulation and Results

Simulation results are presented below for several cases. The airship is always initialized (time $t = 0$) at an altitude of 20 meters in a level hover with respect to the ground. Results are shown with the aircraft (i) thrusting to achieve and maintain a forward velocity both with and without mass variation due to fuel burn, and (ii) thrusting as required to maintain either zero forward airspeed or zero forward groundspeed when encountering a steady headwind.

Figures 2 through 5 illustrate the differences in aircraft behavior when simulated as a mass variant, fuel burning system ("Normal") versus when simulated as a mass invariant system ("No burn"). In both cases, the airship is commanded from hover to achieve a 10 m/s steady forward cruise while maintaining a 20-meter altitude with zero steady-state pitch angle. The difference between the cases is that the mass varying terms in Eqs. (1) and (6) are set to zero in the "No burn" scenario.

From the inertial-frame positions exhibited in Fig. 2, it can be seen that the "Normal" and the "No burn" cases both exhibit the same longitudinal and lateral travel. However, it is significant to note that the main thrust vectoring is unable to compensate for the fuel burn in the "Normal" case around 4940 seconds in the inertial z-direction. This corresponds to Fig. 5, where the thrust deflection angle, $\gamma$, can be seen to saturate at its maximum allowed angle of 20 degrees at this same time. Beyond this time, the mass loss caused by fuel burn cannot be compensated for by the controller designed for a mass invariant aircraft due to thrust deflection authority limits. As a result, the aircraft ascends. This behavior is especially significant when compared with the thrust deflection angle for the "No burn" mass-invariant case, where the angle is close to its lower limit constantly. The deflection angle saturation for the "Normal" case is additionally reflected in Fig. 3 as the vertical body velocity becomes negative (indicating ascension) at 4940 seconds.

It is also noteworthy to compare the other control variables as seen in Fig. 5. The commanded main thrust, $T_M$, can be seen to increase for the "Normal" case until the thrust vectoring deflection angle saturates, at which point it levels off. This corresponds to the thrust deflection angle deviation from the horizontal, which is greater in magnitude than for the "No burn" case. The aircraft therefore requires a greater commanded thrust to maintain the reference airspeed of 10 m/s, since the body x-component of the thrust is slightly lessened in the "Normal" case. The elevator deflection angle, $\delta_e$, is significantly different in the
<figure><img src="image_4.png" /><figcaption>Figure 2. Time history of the airship position during commanded 10 m/s cruise</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 3. Time history of the airship body axis velocity with respect to air during commanded 10 m/s cruise</figcaption></figure>

"Normal" and "No burn" cases because, in the "Normal" case, the mass of the single fuel tank is decreasing. Because the fuel tank is located forward of the body frame origin (the CV), this decreasing fuel mass results in less elevator angle required to maintain a zero-degree level pitch attitude, which is successfully accomplished as seen in Fig. 4. Therefore, while the constant mass airship requires a -18-degree elevator deflection for level pitch, the "Normal" fuel burn case requires only a -12-degree deflection after 5000 seconds at the
commanded speed.

<figure><img src="image_6.png" /><figcaption>Figure 4. Time history of the airship orientation during commanded 10 m/s cruise</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 5. Time history of the airship control variables during commanded 10 m/s cruise</figcaption></figure>

Figures 7 through 10 exhibit a comparison of cases in which the airship is commanded to maintain 0 m/s airspeed (case "u cont") or 0 m/s groundspeed (case "xdot cont") while encountering a steady 10 m/s headwind, initiated at time $t = 150$ seconds. The profile for the filtered step wind input is displayed in Fig. 6.

Figure 7 contains plots for the three dimensional inertial frame airship position during commanded 0
<figure><img src="image_8.png" /><figcaption>Figure 6. Time history of the x-direction wind input during commanded 0 m/s hover</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 7. Time history of the airship position during commanded 0 m/s hover</figcaption></figure>

m/s airspeed and groundspeed. It can be seen that, because the main thruster is limited to the bounds of 0 to +20 N, the thruster has no authority to counteract a headwind when airspeed (u) is controlled, and the airship is continually forced backwards. This is consistent with the "u cont" plot of airspeed in Fig. 8, where it can be seen that the headwind creates a constant positive airspeed. The airship control configuration cannot execute the commands from the PI controller for a speed error demanding a negative thrust. On a related note, it may also be seen in Fig. 7 that the y-direction position for the airspeed-controlled scenario is oscillatory, which is consistent with the yaw result of Fig. 9 for the same scenario. The common problem of underactuated airship controllability is well demonstrated here because, without thruster commands due to airspeed control, the airship is helpless when faced with wind gusts. The aircraft may be prone to enter the spin illustrated here or have other undesirable behavior if the wind excites the lateral dynamics due to rudder positioning or envelope aerodynamic characteristics. However, for the cases where groundspeed is controlled instead of airspeed, it can be seen that there are no x- or y-direction issues of note, and that the
controller is capable of maintaining a 0 m/s x-direction groundspeed such that the airspeed of the aircraft encountering steady wind is exactly the wind speed (Fig. 8).

<figure><img src="image_10.png" /><figcaption>Figure 8. Time history of the airship body axis velocity with respect to air during commanded 0 m/s hover</figcaption></figure>

It can also be seen in Fig. 7 that, due to the same thrust vectoring saturation that occurred in the previously described commanded cruise case, the airship begins to climb near 5000 seconds when thrust vectoring authority has saturated due to fuel burn off as the airship becomes too light to maintain altitude. This saturation can be seen in Fig. 10, where the thrust deflection angle reaches its limit of 20 degrees correspondingly around 5000 seconds. Figure 10 likewise exhibits the main thrust required to maintain a 0 m/s groundspeed given the steady headwind, and also illustrates the fact that the thrust saturates at the minimum limit (zero thrust) because airspeed is always greater than the commanded value. Also shown are the elevator angles required to maintain a level pitch attitude (Fig. 9), which can be seen to be effective in both the airspeed-controlled and groundspeed-controlled cases as expected.
<figure><img src="image_11.png" /><figcaption>Figure 9. Time history of the airship orientation during commanded 0 m/s hover</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure 10. Time history of the airship control variables during commanded 0 m/s hover</figcaption></figure>

V. Conclusion and Future Work

A PI controller developed under the assumption that mass variation due to fuel burn is negligible is implemented with previously derived equations of motion having terms for time varying inertia properties due to fuel depletion and wind exposure. The controller was developed for acceptable controllability of
forward groundspeed or airspeed, pitch angle, and altitude using the available control variables of thrust, thrust deflection angle, and elevator deflection angle. This study shows that the depletion of fuel from an airship fuel tank in a realistic manner over the course of a flight scenario results in significantly altered control requirements, and that a controller developed under a constant mass assumption on a conventionally configured aircraft will be insufficient to maintain a flight profile. It is also found that controlling airspeed rather than groundspeed when attempting a hover in a headwind results in severely handicapped control-lability when compared with a scenario controlling groundspeed, and that the airship is prone to suffering from underactuation in such low-speed conditions. As future work, a controllability study in critical flight phases using additional lateral control will be studied. Additionally, the use of active ballonets to adjust the internal pressure for additional altitude control will be implemented.

## References

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<footnote n="3">³Miller, C., Sullivan, J., and McDonald, S., "High Altitude Airship Simulation Control and Low Altitude Flight Demonstration," proceeding of AIAA Infotech@Aerospace 2007 Conference and Exhibit, Rohnert Park, CA, May 2007, AIAA paper 2007-2766.</footnote>

<footnote n="4">⁴Kulczycki, E., Joshi, S., and Hess, R., "Towards Controller Design for Autonomous Airships Using SLC and LQR Methods," proceeding of AIAA Guidance Navigation, and Control Conference and Exhibit, Keystone, CO, Aug 2006, AIAA paper 2006-6778.</footnote>

<footnote n="5">⁵Waishek, J., Dogan, A., and Bestaoui, Y., "Investigation into the Time Varying Mass Effect on Airship Dynamics Response," proceeding of 47th AIAA Aerospace Sciences Meeting and Exhibit, Orlando, FL, Jan 5-9, 2009, AIAA paper 2009-735.</footnote>

<footnote n="6">⁶Tucker, J., Derivation of the Dynamics Equations of Receiver Aircraft in Aerial Refueling, Master's thesis, The University of Texas at Arlington, Arlington, TX, December 2007.</footnote>

<footnote n="7">⁷Tucker, J., Dogan, A., and Blake, W., "Derivation of the Dynamics Equations of Receiver Aircraft in Aerial Refueling," Journal of Guidance, Control, and Dynamics, Vol. 32 no. 2, 2009, pp. 586-598.</footnote>

<footnote n="8">⁸Mueller, J. and Paluszek, M., "Development of an Aerodynamic Model and Control Law Design for a High Altitude Airship," AIAA 3rd "Unmanned Unlimited" Technical Conference, Workshop and Exhibit, Chicago, IL, Sept 2004, AIAA paper 2004-6479.</footnote>