# Thrust-Based Stabilization and Guidance for Airships without Thrust-Vectoring

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## Abstract

**:**

## 1. Introduction

## 2. Flight Dynamics Model of the Airship

## 3. Airship Stabilization

#### 3.1. Stabilization on an Airship Featuring Movable Aerodynamic Surfaces

#### 3.2. Thrust-Based Stabilization

## 4. Airship Guidance

#### 4.1. Guidance in the Longitudinal Plane

- a distance between the airship $\mathit{CB}$ and the beam, computed in the vertical plane containing the beam, and in a perpendicular direction with respect to the beam. Analytically, considering the unit vector ${\mathit{d}}_{V}$ in Figure 5, which is defined as normal to the rectilinear beam going from ${\mathit{P}}_{\mathbf{1}}$ to ${\mathit{P}}_{\mathbf{2}}$ and contained in the vertical plane where the target beam is, this distance is easily measured as ${e}_{V}^{disp}={\mathit{d}}_{V}\xb7\left({\mathit{x}}_{\mathit{CB}}-{\mathit{x}}_{{\mathit{P}}_{\mathbf{2}}}\right)$, where clearly the reference for this distance is taken as null. By definition, ${e}_{V}^{disp}$ will be positive when the airship is above the track, and negative vice versa.
- the error ${e}_{V}^{vel}$ between the velocity component of the reference point ${\mathit{v}}_{\mathit{CB}}$ along the same cross-beam vertical unit vector ${\mathit{d}}_{V}$ just introduced and a reference value ${v}_{v}^{*}$ of the cross-beam speed. This reference value is obtained as a bounded linear function of the cross-beam displacement error ${e}_{V}^{disp}$, such that$${v}_{v}^{*}=\left\{\begin{array}{cccc}{v}_{V}^{*,top},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\hfill & \hfill \phantom{\rule{0.166667em}{0ex}}& \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{e}_{V}^{disp}<& {e}_{V}^{disp,bot}\hfill \\ \frac{{e}_{V}^{disp}}{{e}_{V}^{disp,bot}}\xb7{v}_{V}^{*,top},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\hfill & \hfill {e}_{V}^{disp,bot}& \le {e}_{V}^{disp}<& 0\hfill \\ \frac{{e}_{V}^{disp}}{{e}_{V}^{disp,top}}\xb7{v}_{V}^{*,bot},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\hfill & \hfill 0& \le {e}_{V}^{disp}<& {e}_{V}^{disp,top}\hfill \\ {v}_{V}^{*,bot},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\hfill & \hfill \phantom{\rule{0.166667em}{0ex}}& \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{e}_{V}^{disp}\ge & {e}_{V}^{disp,top}\hfill \end{array}.\right.$$

#### 4.2. Lateral-Directional Guidance

#### 4.3. Thrust-Based Guidance: Navigation of a Four-Thrusters Airship with No Aerodynamic and Thrust Vector Control

## 5. Application Results

`SILCROAD`(Simulation Library for Craft-Object Advanced Dynamics), a novel object-oriented library developed in

`Matlab (R2019b)`

^{®}at the Department of Aerospace Science and Technology (DAER), Politecnico di Milano, to accurately simulate or co-simulate (in the case of interacting objects in the same scenario) the non-linear response of several machine types (generically named craft), subject to aerodynamic, gravity, buoyancy, and thrust forcing terms. Among the sub-classes in the library is the airship class, which was employed for the fully non-linear simulations presented in this work. This highly customizable library lends itself to multiple uses, including the systematic design, implementation, and testing of control laws applied for instance to the dynamic models of winged aircraft and airships (as well as submarines or torpedoes).

`SILCROAD`library, reproduces the layout obtained from a previous study [10], which emerged from an optimal approach in the arrangement of thrusters on board, based on an energy measure of performance, including control use and response to disturbance. Concerning aerodynamics, the envelope shape and size are the same as the Lotte airship, whereas the entire area of the tail surfaces has been considered fixed, thus removing the control degrees of freedom for elevons or rudder motion. The rear thruster of the baseline airship has been removed as well, and replaced by four thrusters arranged as in Figure 3. The mass of the actuation systems and rear thruster has been estimated and removed, whereas the mass for the four new thrusters has been added. Correspondingly, a slight forward shift of the $\mathit{CG}$ (by 5.9 mm) was observed, as well as an increase in mass of 3 kg [10], with respect to the original Lotte design.

#### 5.1. Stability Augmentation System: Response to Perturbation

#### 5.2. Performance of the Guidance Control System: Examples of Navigation

- following a straight climbing track, ascending 20 m over a horizontal length of 200 m
- following a 200 m long horizontal track, misaligned with respect to the initial velocity of the airship by 60 degree
- following a six-checkpoint path, with checkpoints forming a regular hexagon viewed from top (60 degree angles, side length of 200 m), and with an altitude of alternatively 0 m or 10 m above ground, starting from 0 m at the beginning of the circuit.

#### 5.2.1. Ascending Track

#### 5.2.2. Misaligned Horizontal Track

#### 5.2.3. Multi-Checkpoint Track with Variable Altitude

#### 5.2.4. Flight with Constant Wind

`SILCROAD`library with arbitrary (including stochastic) wind. Sample results with a constant wind of 3 m/s from the west will be shown here.

## 6. Conclusions and Outlook

`SILCROAD`library and simulation tool) showed that both airships with the respective controllers perform realistically in terms of control use, and with good accuracy in the required navigation tasks.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

$\mathcal{B}$ | Body reference (supposed centered in $\mathit{CB}$ in this paper) |

$\mathcal{I}$ | Inertia reference (ground in this paper) |

$\mathcal{V}$ | Volume of lifting gas on airship |

$\mathit{CB}$ | Center of buoyancy |

$\mathit{CG}$ | Center of gravity |

${\mathit{J}}_{\mathit{CB}}$ | Inertia tensor in $\mathit{CB}$ |

${\mathit{M}}_{\mathit{CB}}$ | Generalized mass matrix in $\mathit{CB}$ |

${\mathit{N}}_{\mathit{CB}}$ | Matrix of aerodynamic reaction forcing term |

${\mathit{S}}_{\mathit{CB}}$ | Static moment in $\mathit{CB}$ |

${\mathit{S}}_{321}^{\mathcal{B}}$ | Kinematic matrix for rotational rate |

${\mathit{U}}_{\mathit{CB}}$ | Sensitivity matrix of active aerodynamic force in $\mathit{CB}$ wrt. generalized velocity |

${\mathit{V}}_{\mathit{CB}}$ | Sensitivity matrix of active aerodynamic force in $\mathit{CB}$ wrt. controls |

${\mathit{V}}_{\mathit{CB}}^{a}$ | Sensitivity matrix of active aerodynamic force in $\mathit{CB}$ wrt. aerodynamic controls |

${\mathit{V}}_{\mathit{CB}}^{t}$ | Sensitivity matrix of active aerodynamic force in $\mathit{CB}$ wrt. thrust controls |

${\mathit{d}}_{L}$ | Lateral unit vector (for guidance control computations) |

${\mathit{d}}_{V}$ | Vertical unit vector (for guidance control computations) |

${\mathit{e}}_{321}^{\mathcal{B}}$ | Array of attitude angles |

$\mathit{f}$ | Force vector |

${\mathit{m}}_{\mathit{CB}}$ | Moment vector in $\mathit{CB}$ |

${\mathit{r}}_{\mathit{CG}}$ | Position of center of gravity from $\mathit{CB}$ |

${\mathit{r}}_{{{\mathit{P}}_{\mathit{T}}}_{i}}$ | Position of point of application of i-th thrust force from $\mathit{CB}$ |

${\mathit{s}}_{\mathit{CB}}$ | Generalized forcing term vector in $\mathit{CB}$ |

${\mathit{s}}_{\mathit{CB}}^{a}$ | Aerodynamic forcing term vector in $\mathit{CB}$ |

${\mathit{s}}_{\mathit{CB}}^{a,b}$ | Active component of aerodynamic forcing term vector in $\mathit{CB}$ |

${\mathit{s}}_{\mathit{CB}}^{a,m}$ | Reactive component of aerodynamic forcing term vector in $\mathit{CB}$ |

${\mathit{s}}_{\mathit{CB}}^{b}$ | Buoyancy forcing term vector in $\mathit{CB}$ |

${\mathit{s}}_{\mathit{CB}}^{g}$ | Gravity forcing term vector in $\mathit{CB}$ |

${\mathit{s}}_{\mathit{CB}}^{t}$ | Thrust forcing term vector in $\mathit{CB}$ |

$\mathit{u}$ | Array of controls |

${\mathit{u}}^{a}$ | Array of aerodynamic controls |

${\mathit{u}}^{t}$ | Array of thrust controls |

${\mathit{v}}_{\mathit{CB}}$ | Velocity vector of $\mathit{CB}$ |

${\mathit{w}}_{\mathit{CB}}$ | Generalized velocity vector of $\mathit{CB}$ |

${\mathit{x}}^{AC}$ | State array of flying craft |

${\mathit{x}}_{\mathit{CB}}$ | Position vector of $\mathit{CB}$ from the origin of reference $\mathcal{I}$ |

${\mathit{x}}_{\mathit{P}}$ | Position vector of navigation checkpoint from the origin of reference $\mathcal{I}$ |

${\mathit{\omega}}_{\mathcal{B}/\mathcal{I}}$ | Rotational speed of body reference wrt. inertial reference |

${\tilde{K}}_{i}$ | Modulating function of thrust vs. thrust setting for i-th thruster |

${L}_{\mathit{CB}}^{a,b}$ | First (roll) component of aerodynamic active moment in $\mathit{CB}$, in body ref. |

${M}_{\mathit{CB}}^{a,b}$ | Second (pitch) component of aerodynamic active moment in $\mathit{CB}$, in body ref. |

${N}_{\mathit{CB}}^{a,b}$ | Third (yaw) component of aerodynamic active moment in $\mathit{CB}$, in body ref. |

$RC$ | Rate of climb of airship |

${T}^{act}$, ${T}^{lp}$, ${T}^{thr}$, ${T}^{wo}$ | Time constants (actuator, low-pass, thruster, washout) |

${\tilde{T}}_{i}$ | Nominal thrust intensity for i-th thruster |

U | First (longitudinal) component of ${\mathit{v}}_{\mathit{CB}}$ in body ref. |

V | Second (lateral) component of ${\mathit{v}}_{\mathit{CB}}$ in body ref. |

W | Third (vertical) component of ${\mathit{v}}_{\mathit{CB}}$ in body ref. |

${X}^{a,b}$ | First (longitudinal) component of aerodynamic active force in body ref. |

${Y}^{a,b}$ | Second (lateral) component of aerodynamic active force in body ref. |

${Z}^{a,b}$ | Third (vertical) component of aerodynamic active force in body ref. |

${e}_{L}^{disp}$ | Lateral error on position (for guidance) |

${e}_{L}^{vel}$ | Lateral error on velocity (for guidance) |

${e}_{V}^{disp}$ | Vertical error on position (for guidance) |

${e}_{V}^{vel}$ | Vertical error on velocity (for guidance) |

${h}_{1,2}$, ${h}_{3,4}$ | Logical functions on four-thrusters configuration |

${k}_{I}$ | Integral gain |

${k}_{P}$ | Proportional gain |

m | Mass of airship |

p | First component of ${\mathit{\omega}}_{\mathcal{B}/\mathcal{I}}$ in body ref. |

q | Second component of ${\mathit{\omega}}_{\mathcal{B}/\mathcal{I}}$ in body ref. |

r | Third component of ${\mathit{\omega}}_{\mathcal{B}/\mathcal{I}}$ in body ref. |

${v}_{\mathit{CB}}^{*}$ | Velocity set-point (scalar) |

${v}_{l}^{*}$ | Lateral velocity set-point |

${v}_{v}^{*}$ | Vertical velocity set-point |

$\beta $ | Airship sideslip angle |

${\delta}_{e}$ | Elevator deflection |

${\delta}_{a}$ | Aileron deflection |

${\delta}_{r}$ | Rudder deflection |

${\delta}_{{T}_{i}}$ | Thrust setting of i-th thruster |

$\vartheta $ | Pitch attitude angle |

${\lambda}_{i}$ | Tilt of i-th thrust line |

$\rho $ | Density of air |

${\sigma}_{i}$ | Lateral misalignment of i-th thrust line |

$\phi $ | Roll attitude angle |

$\psi $ | Yaw attitude angle |

${(\xb7)}^{\mathcal{B}}$ | Representation in body components |

${(\xb7)}^{\mathcal{I}}$ | Representation in inertial components |

${(\xb7)}^{bf}$ | Bandwidth-filtered signal |

${(\xb7)}^{coord}$ | Turn coordination input signal |

${(\xb7)}^{g}$ | Guidance input signal |

${(\xb7)}^{pilot}$ | Pilot/Auto-pilot input signal |

${(\xb7)}^{req}$ | Required command value set as input to actuator |

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**Figure 2.**Proposed stabilization control for an airship with deflectable tail control surfaces. Cyan: pitch-rate damper. Purple: roll-rate damper. Green: yaw-rate damper.

**Figure 3.**Conceptual layout of a four-thrusters airship with no movable surfaces and no thrust vector control. (

**Left**) three-quarters view. (

**Right**) view from the port side of the airship.

**Figure 5.**Sketch of beam-tracking measurements in the longitudinal plane. (

**Left**) definition of ${e}_{V}^{disp}$. (

**Right**) computation of ${\mathit{d}}_{V}\xb7{\mathit{v}}_{\mathit{CB}}$.

**Figure 6.**Proposed guidance control for the baseline airship with deflectable aerodynamic surfaces and a single thruster. Red: airspeed tracking. Brown: longitudinal beam-tracking. Grey: lateral beam-tracking with turn coordination.

**Figure 7.**Sketch of beam-tracking measurements for lateral guidance. (

**Left**) definition of ${e}_{L}^{disp}$. (

**Right**) computation of ${\mathit{d}}_{L}\xb7{\mathit{v}}_{\mathit{CB}}$.

**Figure 8.**Proposed guidance control for the four-thrusters concept. Same color codes as for Figure 6.

**Figure 9.**Time response of baseline airship with stabilizing control, in trimmed horizontal flight at a ground speed of 8 m/s, to an initial perturbation of $\Delta W=\Delta V=$ 0.5 m/s. (

**Top-left**) and (

**bottom-left**) longitudinal and lateral-directional states. (

**Bottom-right**) controls. Black dash-dotted lines: trimmed value.

**Figure 10.**Time response of four-thrusters airship with thrust-based stabilizing control, in trimmed horizontal flight at a ground speed of 8 m/s, to an initial perturbation of $\Delta W=\Delta V=$ 0.5 m/s. (

**Top-left**) and (

**bottom-left**) longitudinal and lateral-directional states. (

**Bottom-right**) controls. Black dash-dotted lines: trimmed value.

**Figure 11.**Trajectory and control behavior for an ascending track, as flown by the stabilized baseline airship with the proposed guidance control system. Ground speed setting 6 m/s. (

**Top-left**) three-quarter view of the 3D trajectory from the northeast. (

**Bottom-left**) view of the trajectory from the east. Black dash-dotted lines: target track. Black spheres: target checkpoint capture volume. (

**Bottom-right**) time histories of controls. Black dash-dotted lines: trimmed values.

**Figure 12.**Trajectory and control behavior for an ascending track, as flown by the thrust-controlled, four-thrusters stabilized platform, with the proposed guidance control system. Ground speed setting 6 m/s. (

**Top-left**) three-quarter view of the 3D trajectory from the northeast. (

**Bottom-left**) view of the trajectory from the east. Black dash-dotted lines: target track. Black spheres: target checkpoint capture volume. (

**Bottom-right**) time histories of controls. Black dash-dotted lines: trimmed values.

**Figure 13.**Trajectory and control behavior for an initial misalignment $\Delta {\psi}_{0}=$ 60 degree with respect to a horizontal track, as flown by the stabilized baseline airship with the proposed guidance control system, at 10 m/s of ground speed. (

**Top-left**) three-quarter view of the 3D trajectory from the northeast. (

**Bottom-left**) view of the trajectory from the top. Black dash-dotted lines: target track. Black spheres: target checkpoint capture volume. (

**Bottom-right**) time histories of controls. Black dash-dotted lines: trimmed values.

**Figure 14.**Trajectory and control behavior for an initial misalignment $\Delta {\psi}_{0}=$ 60 degree with respect to a horizontal track, as flown by the four-thrusters stabilized airship, with the thrust-based guidance control system, at 10 m/s of ground speed. (

**Top-left**) three-quarter view of the 3D trajectory from the northeast. (

**Bottom-left**) view of the trajectory from the top. Black dash-dotted lines: target track. Black spheres: target checkpoint capture volume. (

**Bottom-right**) time histories of controls. Black dash-dotted lines: trimmed values.

**Figure 15.**Trajectory and control behavior for a six-checkpoints circuit, flown by the stabilized baseline airship, at 8 m/s of ground speed. (

**Top-left**) three-quarter view of the 3D trajectory from the northeast. (

**Bottom-left**) view of the trajectory from the east. (

**Top-right**) view of the trajectory from the top. Black dash-dotted lines: target track. Black spheres: target checkpoint capture volume. (

**Bottom-right**) time histories of controls (a different color for each segment).

**Figure 16.**Trajectory and control behavior for a 6-checkpoints circuit, flown by the thrust controller on the stabilized four-thrusters concept airship, at 8 m/s of ground speed. (

**Top-left**) three-quarter view of the 3D trajectory from the northeast. (

**Bottom-left**) view of the trajectory from the east. (

**Top-right**) view of the trajectory from the top. Black dash-dotted lines: target track. Black spheres: target checkpoint capture volume. (

**Bottom-right**) time histories of controls (a color for each segment).

**Figure 17.**Trajectory views in the presence of a western wind of 3 m/s, considering the same target track and checkpoint of Figure 11 and Figure 12. Ground speed setting of 6 m/s. (

**Top**) baseline airship. (

**Bottom**) four-thrusters thrust-controlled airship. (

**Left**) top view. (

**Right**) view from the east. Black dash-dotted lines: target track. Black spheres: target checkpoint capture volume.

**Figure 18.**Trajectory views in the presence of a western wind of 3 m/s, considering the same 6-checkpoints circuit of Figure 15 and Figure 16. Ground speed setting of 6 m/s. (

**Top**) baseline airship. (

**Bottom**) four-thrusters thrust-controlled airship. (

**Left**) top view. (

**Right**) view from the east. Black dash-dotted lines: target track. Black spheres: target checkpoint capture volume.

Parameter | Baseline Airship | 4-Thrusters Airship Concept |
---|---|---|

Mass (kg) | 134.28 | 137.28 |

Envelope volume (m${}^{3}$) | 107.42 | 107.42 |

Overall length (m) | 16.0 | 16.0 |

Hor. disp. $\mathit{CG}$ from $\mathit{CB}$ (mm) | 0 | 5.9 |

Ver. disp. $\mathit{CG}$ from $\mathit{CB}$ (mm) | 455.0 | 455.0 |

Number of thrusters | 1 | 4 |

Nominal thrust (each unit) (N) | 500 | 250 |

Aerodynamic control | Rudder, elevons | (no aerodynamic control) |

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## Share and Cite

**MDPI and ACS Style**

Riboldi, C.E.D.; Rolando, A.
Thrust-Based Stabilization and Guidance for Airships without Thrust-Vectoring. *Aerospace* **2023**, *10*, 344.
https://doi.org/10.3390/aerospace10040344

**AMA Style**

Riboldi CED, Rolando A.
Thrust-Based Stabilization and Guidance for Airships without Thrust-Vectoring. *Aerospace*. 2023; 10(4):344.
https://doi.org/10.3390/aerospace10040344

**Chicago/Turabian Style**

Riboldi, Carlo E.D., and Alberto Rolando.
2023. "Thrust-Based Stabilization and Guidance for Airships without Thrust-Vectoring" *Aerospace* 10, no. 4: 344.
https://doi.org/10.3390/aerospace10040344