# Nominal Trajectories of an autonomous under-actuated airship

Yasmina Bestaoui

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## To cite this version:

Yasmina Bestaoui. Nominal Trajectories of an autonomous under-actuated airship. International Journal of Control, Automation and Systems, 2006, 4 (4), pp.395-404. hal-00342778

HAL Id: hal-00342778

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# Nominal Trajectories of an Autonomous Under-actuated Airship

Yasmina Bestaoui

**Abstract:** The objective of this paper is to generate a desired flight path to be followed by an autonomous airship. The space is supposed without obstacles. As there are six degrees of freedom and only three inputs for the LSC AS200 airship, three equality constraints appear due to the under-actuation.

**Keywords:** Autonomous airship, trajectory generation, under-actuation.

## 1. INTRODUCTION

Lighter than air vehicles suit a wide range of applications, ranging from advertising, aerial photography and aerial inspection platforms, with a very important application in the areas of environmental, biodiversity, and climatological research and monitoring. Airships offer the advantage of quiet hover with noise levels much lower than helicopters. Unmanned remotely-operated airships have already proved themselves as camera and TV platforms and for specialized scientific tasks. An actual trend is toward autonomous airships.

Fig. 1 presents the AS200 airship. It is basically a large gas balloon. Its shape is maintained by its internal overpressure. The only solid parts are the gondola, a tail rotor with horizontal axis of rotation, the main rotor with a varying tilt angle and the tail fins. It is actually a remotely piloted airship designed for remote sensing, being transformed to be fully autonomous.

A basic problem to be solved by autonomous vehicles is the problem of trajectory generation. Trajectory generation means the generation and execution of a plan for moving from one location to another location in space to accomplish a desired task. The motion generation module generates a nominal state space trajectory and a nominal control input. Trajectory prediction consists in computing reference values to be given to the controller.

The problem of finding the time-optimal trajectory for a fully actuated robot manipulator along a specified path is a classical one in robotics. This problem has been solved by algorithms proposed by [1-5]. The algorithms find the minimum time scaling

of the path with respect to the actuator constraints: dynamic ones [1,4] and technological ones [2,6]. The problem of finding a smooth interpolating curve is well understood in Euclidean spaces, but it is not so clear in curved spaces [7]. When planning Cartesian trajectories, it is usually possible to characterize the performance of different trajectories [2,4,6,7-17].

In [18], in the first part, the trajectories considered are trim trajectories. The general condition for trim requires that the rate of change of the magnitude of the velocity vector is identically zero, in the body fixed frame. The trim problem is generally formulated as a set of non linear algebraic equations. The solution trajectories are helices with constant curvature and torsion. In the second part, variable curvature helices are investigated. In [9], a 3rd order expansion is used for transition maneuvering between two helices. [19] investigate optimal trajectory planning for hot air balloons in linear wind fields. The objective function to be minimized is fuel consumption with respect to free end states.

This article is concerned with methods of computing a trajectory in space that describes the desired motion. The contribution of this paper is the characterization of trajectories, considering the under-actuation. This paper consists of 6 sections. Section 2

<figure><img src="image_3.png" /><figcaption>Fig. 1. The AS200 airship.</figcaption></figure>

Manuscript received May 11, 2005; revised April 5, 2006; accepted May 10, 2006. Recommended by Editor Keum-Shik Hong.

Yasmina Bestaoui is with the Laboratoire IBISC CNRS FRE2873, University of Evry, 38 rue du Pelvoux, 91025 EVRY, France (e-mail: bestaoui@iup.univ-evry.fr).
presents the kinematics while the following one
introduces the mechanical system. Section 4
introduces the relationship between trajectory
generation algorithms and under-actuation. Simulation
results are discussed in Section 5 and finally some
concluding remarks are given in section 6.

## 2. KINEMATICS

Consider a rigid body moving in free space. Assume any inertial reference frame {F} fixed in space and a frame {M} fixed to the body at the center of gravity. At each instant, the configuration (position and orientation) of the rigid body can be described by a homogeneous transformation matrix corresponding to the displacement from frame {F} to frame {M}.

The origin C of {M} coincides with the center of gravity of the vehicle. Its axes ($X_c$, $Y_c$, $Z_c$) are the principal axes of symmetry when available. They must form a right handed orthogonal normed frame. The position of the vehicle C in {F} can be described by $\eta_1 = (x \ y \ z)^T$ while the orientation is given by $\eta_2 = (\phi \ \theta \ \psi)^T$ with $\phi$ Roll, $\theta$ pitch and $\psi$ Yaw angles. The orientation matrix R is given by

$$R = \begin{bmatrix} c\psi c\theta & -s\psi c\phi + c\psi s\theta s\phi & s\psi s\phi + c\psi s\theta c\phi \\ s\psi c\theta & c\psi c\phi + s\psi s\theta s\phi & -c\psi s\phi + s\psi s\theta c\phi \\ -s\theta & c\theta s\phi & c\theta c\phi \end{bmatrix}, (1)$$

where $c\theta = \cos(\theta)$ and $s\theta = \sin(\theta)$. This description is valid in the region $-\frac{\pi}{2} < \theta < \frac{\pi}{2}$. A singularity of this transformation exists for: $\theta = \frac{\pi}{2} \pm k\pi, k \in \mathbb{Z}$.

The kinematics of the airship can be expressed in the following way:

$$\begin{pmatrix} \dot{\eta}_1 \\ \dot{\eta}_2 \end{pmatrix} = \begin{pmatrix} R & 0_{3*3} \\ 0_{3*3} & J(\eta_2) \end{pmatrix} \begin{pmatrix} V \\ \Omega \end{pmatrix}, \quad (2)$$

where

$$J(\eta_2) = \begin{pmatrix} 1 & s\phi \cdot \tan\theta & c\phi \cdot \tan\theta \\ 0 & c\phi & -s\phi \\ 0 & s\phi/c\theta & c\phi/c\theta \end{pmatrix}, \quad (3)$$

and

$$V = \begin{pmatrix} u \\ v \\ w \end{pmatrix}, \quad \omega = \begin{pmatrix} p \\ q \\ r \end{pmatrix}. \quad (4)$$

<figure><img src="image_4.png" /><figcaption>Fig. 2. The airship frames (cg: center of gravity, cv: center of volume).</figcaption></figure>

$\omega$ physically corresponds to the angular velocity of the rigid body, while $V$ is the linear velocity of the origin C of the frame {M}.

## 3. MECHANICAL SYSTEM

### 3.1. Dynamics

In this section, analytic expressions for the forces and moments of a system with added mass and inertia such as an airship are introduced [5,20-22]. An airship is a lighter than air vehicle using a lifting gas (helium in this particular case). We will make in the sequel some simplifying assumptions: the earth fixed frame is inertial, the gravitational field is constant, the airship is well inflated, the density of air is uniform. In [23], we considered the case of an airship with small deformations analyzed via the Updated Lagrangian Method.

The translational part being separated from the rotational part [20], the dynamic equations (Euler – Poincaré) are given by:

$$M \dot{V} = -\omega * MV - b(.) + f(u), \quad (5)$$

$$J \dot{\omega} = -\omega * J\omega - V * MV - \beta(.) + \tau(u),$$

where

$$b(.) = R^T e_3(mg - B) + D_l V, \quad (6)$$

$$\beta(.) = (R^T e_3 * \overline{BG})B + D_\omega \omega. \quad (7)$$

For a system with added masses, the term $V * MV$ is non zero. M and J are respectively the vehicle's mass and rotational tensors and $\tau$, $\beta$ and $f$, b represent respectively the control and non-conservative torques and forces in body axes. m is the mass of the airship, propellers and actuators. The tensor M includes both the airship's actual mass as well as the virtual mass elements associated with the dynamics of buoyant vehicles. The tensor J includes both the airship's
actual inertias as well as the virtual inertia elements associated with the dynamics of buoyant vehicles. As the airship displays a very large volume, its added masses and inertias become very significant [21]. We will assume that the added mass coefficients are constant. They can be estimated from the inertia ratios and the airship weight and dimension parameters. Added mass should be understood as pressure-induced forces and moments due to a harmonic motion of the body which are proportional to the acceleration of the body [21]. $D_v$ is the 3×3 aerodynamics forces diagonal matrix while $D_Ω$ is the 3×3 aerodynamics moments diagonal matrix. $e_3 = (0 0 1)^T$ a unit vector.

The 3×1 buoyancy force vector $Be_3$ with:

$$B = \rho \Delta g, \qquad (8)$$

where $Δ$ is the volume of the envelope, $ρ$ is the difference between the density of the ambient atmosphere $ρ_{air}$ and the density of the helium $ρ_{helium}$ in the envelope, g is the constant gravity acceleration.

$\overline{BG} = (x_b \ y_b \ z_b)$ represents the position of the center of buoyancy with respect to the body fixed frame.

The aerodynamic force can be resolved into two component forces, one parallel and the other perpendicular to the direction of motion. Lift is the component of the aerodynamic force perpendicular to the direction of motion and drag is the component opposite to the direction of motion. As the airship is a slow moving object in the air, we can assume a linear relationship between the speed and the drag.

$$D_v = \operatorname{diag}(-X_u \ -Y_v \ -Z_w), \qquad (9)$$

$$D_Ω = \operatorname{diag}(-L_p \ -M_q \ -N_r). \qquad (10)$$

The gravitational force vector is given by the difference between the airship weight and the buoyancy acting upwards on it:

$$R^T e_3 (mg - B) = \begin{pmatrix} (mg - B)s\theta \\ -(mg - B)c\theta.s\phi \\ -(mg - B)c\theta.c\phi \end{pmatrix}. \qquad (11)$$

The gravitational and buoyant moments are given by:

$$\left( R^T e_3 * \overline{BG} \right) B = B \begin{pmatrix} z_b c\theta s\phi - y_b c\theta c\phi \\ x_b c\theta c\phi + z_b s\theta \\ -y_b s\theta - x_b c\theta s\phi \end{pmatrix}. \qquad (12)$$

If a system is fully actuated, it can be steered along any given curve on the configuration manifold, i.e it is controllable. This is not true in general for an underactuated system. However, an underactuated system can be locally controllable if it enjoys the property of nonholonomy. The existence of nonholonomic const-

rints translates into the fact that the system can be locally steered along a manifold of dimension larger than the number of independent control inputs [3,13-15,18,21]. However, practical methods have only been found for simple underactuated systems.

## 3.2. Propulsion

An airship is propelled by thrust. Propellers are designed to exert thrust to drive the airship forward. The most popular propulsion system layout for pressurized non rigid airships is twin ducted propellers mounted either side of the envelope bottom. Another one exists in the tail for torque correction and attitude control. The airship AS200 is an underactuated system with two types of control in a low velocity flight: forces generated by thrusters and angular inputs controlling the direction of the thrusters ($γ$ is the tilt angle of the propellers)

$$F_1 = \begin{pmatrix} T_M \sin \gamma \\ 0 \\ T_M \cos \gamma \end{pmatrix}, \quad F_2 = \begin{pmatrix} 0 \\ T_T \\ 0 \end{pmatrix}, \qquad (13)$$

where $T_M$ and $T_T$ represent respectively the main and tail thrusters.

Thus in building the non linear six degrees of freedom mathematical model, the additional following assumptions are made:

$$\overline{P_1 G} = \begin{pmatrix} 0 \\ 0 \\ P_1^3 \end{pmatrix}, \quad \overline{P_2 G} = \begin{pmatrix} P_2^1 \\ 0 \\ 0 \end{pmatrix}. \qquad (14)$$

If we consider the plane XZ as a plane of symmetry, the mass and inertia matrices can be written as:

$$M = \begin{pmatrix} m + X_u & 0 & X_z \\ 0 & m + Y_v & 0 \\ Z_x & 0 & m + Z_w \end{pmatrix}, \qquad (15)$$

$$J = \begin{pmatrix} I_x + L_p & 0 & -I_{xz} \\ 0 & I_y + M_q & 0 \\ -I_{xz} & 0 & I_z + N_r \end{pmatrix}. \qquad (16)$$

If the center of gravity sits below the center of buoyancy, then $\overline{BG} = (0 \ 0 \ z_b)^T$. It is important to gain insight into the geometric structure of the equations since this knowledge is useful in motion planning and control.

# 4. TRAJECTORY GENERATION AND UNDERACTUATION

In this paragraph, nominal trajectories are characterized, the model is supposed perfect and any
perturbation such as wind or sensor disturbance is
neglected. The three equality constraints deriving
from the under-actuation are sought. Considering the
dynamics of the airship and its propulsion, the
following dynamics equations can be written:

$$
f(u) = M \dot{V} + \omega^* M V + b(.) = \begin{pmatrix} T_M \sin \gamma \\ T_T \\ T_M \cos \gamma \end{pmatrix}, \quad (17)
$$

$$
\tau(u) = J \dot{\omega} + \omega^* J \omega + V^* M V + \beta(.)
= \begin{pmatrix} 0 \\ P_1^3 T_M \sin \gamma \\ P_2^1 T_T \end{pmatrix} = \begin{pmatrix} 0 \\ P_1^3 f_1(u) \\ P_2^1 f_2(u) \end{pmatrix}, \quad (18)
$$

where the kinematics are:

$$
\begin{align}
p &= \dot{\phi} - \dot{\psi} S\theta, \notag \\
q &= \dot{\theta} C\phi + \dot{\psi} S\phi C\theta, \tag{19} \\
r &= -\dot{\theta} S\phi + \dot{\psi} C\phi C\theta. \notag
\end{align}
$$

**First equality constraint:** The roll moment being
zero (see (18)), $\tau_1 = 0$ gives

$$
J_{11} \left( \ddot{\phi} - \ddot{\psi} S\theta - \dot{\psi} \dot{\theta} C\theta \right) \\
+ (J_{33} - J_{22}) \left( \dot{\theta} C\phi + \dot{\psi} S\phi C\theta \right) \left( -\dot{\theta} S\phi + \dot{\psi} C\phi C\theta \right) \\
+ D_p \left( \dot{\phi} - \dot{\psi} S\theta \right) + z_b B C\theta S\phi = 0. \quad (20)
$$

**Second equality constraint:** $P_1^3 f_1 = \tau_2$ gives

$$
\begin{align*}
J_{22} \chi + (M_{11} - M_{33})uw & \\
& + (J_{11} - J_{33}) \left( \dot{\phi} - \dot{\psi} \dot{S} \theta \right) \left( -\dot{\theta} S \phi + \dot{\psi} C \dot{\phi} C \theta \right) \\
& + D_q \left( \dot{\theta} C \phi + \dot{\psi} S \dot{\phi} C \theta \right) - z_b B S \theta + P_1^3 M_{11} \dot{u} \\
& + P_1^3 M_{22} w \left( \dot{\theta} C \phi + \dot{\psi} S \dot{\phi} C \theta \right) \\
& - P_1^3 M_{22} v \left( -\dot{\theta} S \phi + \dot{\psi} C \dot{\phi} C \theta \right) \\
& + P_1^3 D_u u + P_1^3 (mg - B) S \theta = 0, \tag{21}
\end{align*}
$$

where

$$
\chi = \ddot{\theta} C\phi - \dot{\theta}\dot{\phi} S\phi + \ddot{\psi} S\phi C\theta + \dot{\psi}\dot{\phi} C\phi C\theta - \dot{\psi}\dot{\theta} S\phi S\theta.
$$

**Third equality constraint:** $P_2^1 f_2 = \tau_3$ gives

$$
\begin{align}
& J_{33}\chi' + (M_{22} - M_{11})uv \nonumber \\
& \quad + (J_{22} - J_{11})\left(\dot{\phi} - \dot{\psi}\dot{S}\theta\right)\left(\dot{\theta}C\phi + \dot{\psi}S\dot{\phi}C\theta\right) \tag{22} \\
& \quad + N_r\left(-\dot{\theta}S\phi + \dot{\psi}C\dot{\phi}C\theta\right) + R_1^2 M_{22}\dot{v} \nonumber \\
& \quad - P_1^2 M_{22}w\left(\dot{\phi} - \dot{\psi}\dot{S}\theta\right) + R_1^2 M_{11}u\left(-\dot{\theta}S\phi + \dot{\psi}C\dot{\phi}C\theta\right) \nonumber \\
& \quad + P_1^2 D_v v - R_1^2(mg - B)S\phi C\theta = 0, \nonumber
\end{align}
$$

where

$$
\chi' = -\ddot{\theta} S \phi - \dot{\theta} \dot{\phi} C \phi + \ddot{\psi} C \phi C \theta - \psi \dot{\phi} S \phi C \theta - \dot{\psi} \dot{\theta} C \phi S \theta.
$$

The following approach is considered. The
variations of the roll and pitch angles as well as the
longitudinal velocity are imposed and the influence of
the under-actuation on the variations of the yaw angle,
the lateral and vertical velocities are studied.

The first equality constraint (20) is equivalent to the
resolution of an ordinary differential equation of the
form

$$
a(t) \ddot{\psi} + b(t) \left(\dot{\psi}\right)^2 + c(t) \dot{\psi} + d(t) = 0, \quad (23)
$$

where

$$
\begin{align}
a(t) &= J_{11}(S\theta), \notag \\
b(t) &= (J_{33} - J_{22})(C\phi S\phi C^2\theta), \notag \\
c(t) &= -S\theta D_p + 2(J_{33} - J_{22})C\theta C^2\phi \notag \\
&\quad - (J_{11} + J_{33} - J_{22})\dot{\theta}C\theta, \tag{24}
\end{align}
$$

$$
d(t) = -z_b B C \theta S \phi - J_{11} \ddot{\phi} - D_p \dot{\phi} \\
\qquad + (J_{33} - J_{22}) \dot{\theta}^2 S \phi C \phi.
$$

If $\Xi(t) = \dot{\psi}(t)$ then the non autonomous generalized
logistic equation must be solved:

$$
a(t) \dot{\Xi}(t) + b(t) (\Xi(t))^2 + c(t) \Xi(t) + d(t) = 0. \quad (25)
$$

The third equality constraint (22) can be written as:

$$
w(t) = \alpha_0 + \alpha_1 u + \alpha_2 v + \alpha_3 u v + \alpha_4 \dot{v}, \quad (26)
$$

where

$$
\alpha_0 = \frac{\alpha'_0 (J_{22} - J_{11}) - P_1^2 (mg - B) C\theta S\phi + D_r \left( \dot{\psi} C\theta C\phi - \dot{\theta} S\phi \right)}{-P_1^2 M_{22} \left( \dot{\psi} S\theta - \dot{\phi} \right)}
$$
$$
\begin{align*}
& J_{33} \left( \ddot{\psi} C \theta C \phi - \ddot{\theta} S \phi - \dot{\psi} \dot{\theta} S \theta C \phi - \dot{\psi} \dot{\phi} C \theta S \phi - \dot{\theta} \dot{\phi} C \phi \right) \\
& \qquad + \frac{R_1^2 M_{22} \left( \dot{\psi} S \theta - \dot{\phi} \right)}{-R_1^2 M_{22} \left( \dot{\psi} S \theta - \dot{\phi} \right)} \\
& \alpha_1 = - \frac{\left( \dot{\psi} C \theta C \phi - \dot{\theta} S \phi \right) M_{11}}{\left( \dot{\psi} S \theta - \dot{\phi} \right) M_{22}},
\end{align*}
$$

$$
\alpha_2 = \frac{-D_v}{\left( \dot{\psi} S\theta - \dot{\phi} \right) M_{22}}, \qquad (27)
$$

$$
\alpha_3 = \frac{M_{11} - M_{22}}{P_1^2 M_{22} \left( \dot{\psi} S\theta - \dot{\phi} \right)},
$$

$$
\alpha_4 = \frac{-1}{\left( \dot{\psi} S\theta - \dot{\phi} \right)},
$$

and

$$
\alpha'_{0} = \left( -\dot{\psi}^2 S\theta C\theta S\phi - \dot{\psi}\dot{\theta} S\theta C\phi + \dot{\psi}\dot{\phi} C\theta S\phi + \dot{\theta}\dot{\phi} C\phi \right).
$$

The second equality constraint (21) gives:

$$
\begin{equation}
\begin{split}
& \beta_0 + \beta_1 u + \beta_2 u^2 + \beta_3 uv + \beta_4 u^2 v + \beta_5 u \dot{v} \\
& \qquad + \beta_6 v + \beta_7 \dot{v} + \beta_8 \dot{u} = 0,
\end{split}
\tag{28}
\end{equation}
$$

where

$$
\begin{align}
\beta_0 &= \beta'_0 + \beta'_3 \alpha_0, & \beta_1 &= \beta'_1 + \alpha_1 \beta'_3 + \alpha_0 \beta'_5, \notag \\
\beta_2 &= \beta'_5 \alpha_1, & \beta_3 &= \beta'_5 \alpha_2 + \alpha_3 \beta'_3, \notag \\
\beta_4 &= \beta'_5 \alpha_3, & \beta_5 &= \beta'_5 \alpha_4, \tag{29} \\
\beta_6 &= \beta'_2 + \alpha_2 \beta'_3, & \beta_7 &= \alpha_4 \beta'_3, \notag \\
\beta_8 &= \beta'_4, \notag
\end{align}
$$

$$
\begin{align*}
\beta_6 &= \beta'_2 + \alpha_2 \beta'_3, & \beta_7 &= \alpha_4 \beta'_3, \\
\beta_8 &= \beta'_4,
\end{align*}
$$

$$
\begin{align*}
\beta'_0 &= J_{22} \beta''_0 + (J_{11} - J_{33}) \beta'''_0 \\
&\quad + D_q \left( \dot{\psi} C \theta S \phi + \dot{\theta} C \phi \right) \\
&\quad - B z_b S \theta + P_1^3 (m g - B) S \theta, \\
\beta''_0 &= \ddot{\psi} C \theta S \phi + \ddot{\theta} C \phi + \dot{\psi} \dot{\phi} C \theta C \phi \\
&\quad - \dot{\psi} \dot{\theta} S \theta S \phi - \dot{\theta} \dot{\phi} S \phi,
\end{align*}
$$

$$
\begin{align*}
\beta'''_0 &= -\dot{\theta}\dot{\phi}S\phi + \dot{\psi}\dot{\phi}C\phi C\theta \\
&\quad + \dot{\psi}\dot{\theta}S\theta S\phi - \dot{\psi}^2 C\phi C\theta S\theta,
\end{align*}
$$

$$
\begin{align*}
\beta'_1 &= P_1^3 X_u, \\
\beta'_2 &= -P_1^3 M_{22} \left( \dot{\psi} C \theta C \phi - \dot{\theta} S \phi \right), \\
\beta'_3 &= P_1^3 M_{22} \left( \dot{\psi} C \theta S \phi + \dot{\theta} C \phi \right), \\
\beta'_4 &= P_1^3 M_{11}, \\
\beta'_5 &= (M_{11} - M_{22}).
\end{align*}
$$

4.1. Roll and pitch angles constant
The differential equation

$$
a \ddot{\psi} + b \left( \dot{\psi} \right)^2 + c \dot{\psi} + d = 0, \quad (30)
$$

where

$$
\begin{align}
a &= -J_{11}(S\theta), \notag \\
b &= (J_{33} - J_{22})(S\phi C\theta)(C\phi C\theta), \tag{31} \\
c &= D_p(-S\theta), \notag \\
d &= -z_b B C \theta S \phi \notag
\end{align}
$$

admits an analytic general solution.

By using the method of separation of variables and
integration. by partial fractions, in the constant
coefficient case, logistic equation can be solved and
the behavior of all solutions is analyzed [24-27].

For $\phi = 0$;

$$
\psi(t) = \psi_0 e^{-\frac{L_p t}{I_x}}. \tag{32}
$$

For $\theta = 0$;

$$
\psi(t) = t \sqrt{\frac{B z_b}{C \phi (I_z - I_y)}} + \psi_0. \quad (33)
$$

For the particular case, where $\dot{\psi}$ is constant,
classical trim trajectories are encountered.

4.1.1 $\dot{\psi}$ is constant:

$$
\left(
\begin{array}{l}
\phi = \text{cst} = \phi_0 \\
\theta = \text{cst} = \theta_0 \\
\psi = \dot{\psi}_0 t
\end{array}
\right).
\qquad (34)
$$

The first equality constraint becomes a second order
polynomial equation:

$$
b \left( \dot{\psi} \right)^2 + c \dot{\psi} + d = 0. \qquad (35)
$$
The third equality constraint gives

$$w(t) = \alpha_0 + \alpha_1 u + \alpha_2 v + \alpha_3 uv + \alpha_4 \dot{v}, \quad (36)$$

where

$$\begin{align}
\alpha_0 &= \frac{\dot{\psi}^2 S\theta C\theta S\phi (J_{22} - J_{11}) + D_r \dot{\psi} C\theta C\phi}{R_1^2 \dot{\psi} S\theta M_{22}} \nonumber \\
&\quad + \frac{-R_1^2 (mb - B)C\theta S\phi + J_{33} \ddot{\psi} C\theta C\phi}{P_1^2 \dot{\psi} S\theta M_{22}}, \nonumber \\
\alpha_1 &= -\frac{C\theta C\phi M_{11}}{S\theta M_{22}}, \nonumber \\
\alpha_2 &= \frac{D_v}{M_{22} \dot{\psi} S\theta}, \nonumber \\
\alpha_3 &= \frac{M_{11} - M_{22}}{P_1^2 M_{22} \dot{\psi} S\theta}, \nonumber \\
\alpha_4 &= \frac{1}{\dot{\psi} S\theta}, \tag{37}
\end{align}$$

while the second equality constraint gives

$$v = \frac{\beta_0 + \beta_1 u + \beta_2 u^2}{\beta_6 + \beta_3 u + \beta_4 u^2} \quad (38)$$

for $\dot{u} = \dot{v} = 0$,
otherwise

$$\beta_0 + \beta_1 u + \beta_2 u^2 + \beta_3 uv + \beta_4 u^2 v + \beta_5 u \dot{v} + \beta_6 v + \beta_7 \dot{v} + \beta_8 \dot{u} = 0 \quad (39)$$

with the parameters $\beta_i$ given in (29)-(31).

When $\dot{u} = \dot{v} = \dot{w} = 0$, we retrieve the classical kinematics equations of the trim trajectories

$$\dot{x} = a_x \cos(\dot{\psi}_0 t) + b_x \sin(\dot{\psi}_0 t), \quad (40)$$

$$\dot{y} = a_y \cos(\dot{\psi}_0 t) + b_y \sin(\dot{\psi}_0 t),$$

$$\dot{z} = -\sin(\theta_0)u_0 + \cos(\theta_0)\sin(\phi_0)v_0 + \cos(\theta_0)\cos(\phi_0)w_0,$$

where

$$\begin{align*}
a_x &= \cos(\theta_0)u_0 + \sin(\theta_0)\sin(\phi_0)v_0 + \sin(\theta_0)\cos(\phi_0)w_0, \\
b_y &= a_x,
\end{align*}$$

$$b_x = -\cos(\phi_0)v_0 + \sin(\phi_0)w_0, \quad (41)$$

$$a_y = -b_x.$$

Integrating these equations, we obtain

$$\begin{align}
x &= \frac{a_x}{\dot{\psi}_0} \sin(\frac{\dot{\psi}_0}{V_e} s) - \frac{b_x}{\dot{\psi}_0} \cos(\frac{\dot{\psi}_0}{V_e} s), \nonumber \\
y &= -\frac{b_x}{\dot{\psi}_0} \sin(\frac{\dot{\psi}_0}{V_e} s) - \frac{a_x}{\dot{\psi}_0} \cos(\frac{\dot{\psi}_0}{V_e} s), \tag{42} \\
z &= \frac{z}{V_e} s, \nonumber
\end{align}$$

where $s$ represents the curvilinear abscissa and we suppose a uniform motion such that

$$s = V_e t = t \sqrt{u_0^2 + v_0^2 + w_0^2}. \quad (43)$$

The trajectories represented by these equations (44) are classical helices rotated around the vertical axe by an angle of $\arctan\left(\frac{b_x}{a_x}\right)$ and have a constant curvature and torsion

$$k = \frac{V^2 \sqrt{A_x^2 + B_x^2}}{\sqrt{V^2 (A_x^2 + B_x^2) + C_x^2}}, \quad (44)$$

$$\tau = \frac{C_x V}{\sqrt{V^2 (A_x^2 + B_x^2) + C_x^2}}, \quad (45)$$

where

$$A_x = \frac{a_x}{\dot{\psi}_0}, \quad B_x = \frac{b_x}{\dot{\psi}_0}, \quad (46)$$

$$C_x = \frac{z_0}{V_e}, \quad V = \frac{\dot{\psi}_0}{V_e}. \quad (47)$$

The trajectory is shown in Fig. 3.

<figure><img src="image_5.png" /><figcaption>Fig. 3. Trim trajectory.</figcaption></figure>
The trim condition can be a turning (about the vertical axis), descending or climbing (assuming constant air density and temperature), side-slipping maneuver at constant speed. More conventional flight conditions such as hover, cruise, auto-rotation or sustained turns are also trims.

### 4.1.2 $\dot{\psi}$ is not constant:

Fig. 4 shows the solution $\psi(t)$ of the differential in (20) while Fig. 5 shows its derivative $\dot{\psi}(t)$.

Even though there is a nonlinear variation of $\psi$ in the beginning of the simulation, the yaw angle has a quasilinear variation after a certain time.

A transitional behavior can be recognized before the yaw velocity attains a permanent (constant) value.

## 4.2. Roll and pitch angles linear functions of time

In this paragraph, the roll and pitch angles are assumed to have linear variations:

$$ \theta = \dot{\theta}_0 t + \theta_0, \quad \phi = \dot{\phi}_0 t + \phi_0. $$

When the coefficients of the non autonomous logistic equation are no longer constant, no explicit solutions can be found in general and the equilibrium point may become unstable [24]. For a study to be complete, the

<figure><img src="image_6.png" /><figcaption>Fig. 4. Yaw angle.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5. Yaw derivative.</figcaption></figure>

existence of stable periodic or stable bounded solutions is an essential part of qualitative theory of this differential equation, in order to determine non trivial solutions and study their behavior [24-27]. Nkashama [27] proved that the logistic equation with positive non autonomous bounded coefficients has exactly one bounded solutions that is positive and does not tend to the zero solution.

Solving the first equality constraint, the roll moment being null, $\forall t$, implies $L_p \dot{\phi}_0 = 0 \Rightarrow \dot{\phi}_0 = 0$. Rearranging the first equality constraint with this requirement gives: $\dot{\theta}_0 C \phi_0 S \phi_0 = 0$, three cases are possible

$$ \dot{\theta}_0 = 0 \text{ or } \phi_0 = 0 \text{ or } \phi_0 = \frac{\pi}{2}. $$

The first case: trim trajectories has already been studied in paragraph 4.1.1.

If the roll angle is null, the following differential equations must be solved:

$$ \ddot{\psi} + \dot{\psi} \left( a + b \frac{\dot{\theta}_0 C \theta}{S \theta} \right) = 0, $$

$$ \text{where} \quad a = L_p / (I_x), \\ b = -(I_z - I_y - I_z) / (I_x), \qquad (48) $$

the following derivative $\dot{\psi}(t)$ is obtained

$$ \dot{\psi}(t) = - \frac{\dot{\theta}_0 S \theta_0^b S \theta^{-b} e^{\frac{-a}{c} \dot{\theta}_0}}{-\cosh\left(\frac{a \theta_0}{\dot{\theta}_0}\right) + \sinh\left(\frac{a \theta_0}{\dot{\theta}_0}\right)}. \qquad (49) $$

The case $\phi_0 = \frac{\pi}{2}$ gives the following differential equations

$$ \ddot{\psi} + \dot{\psi} \left( a_1 + a_2 \frac{\dot{\theta}_0 C \theta}{S \theta} \right) + a_3 C \theta / S \theta = 0, $$

$$ \text{where} \quad a_1 = L_p / (I_x), \\ a_2 = (I_z - I_y + I_z) / (I_x), \qquad (50) \\ a_3 = B z_b / I_x. $$

The third equality constraint gives

$$ w = \delta_1 + \delta_2 u v + \delta_3 u + \delta_4 v + \delta_5 \dot{u}. \qquad (51) $$

where

$$ \delta_1 = \dot{\theta}_0 \frac{I_y + I_z - I_x}{P_2^1 M_y} - \frac{N_r C \theta}{P_2^1 S \theta M_y} - \frac{I_z \ddot{\psi} C \theta}{P_2^1 \dot{\psi} S \theta M_y}, $$
$$
\begin{align*}
\delta_2 &= -\frac{M_y - M_x}{P_2 \dot{\psi} S \theta M_y}, \\
\delta_3 &= -\frac{C \theta M_y}{S \theta M_y}, \\
\delta_4 &= -\frac{Y_v}{\dot{\psi} S \theta M_y}, \\
\delta_5 &= -\frac{1}{\dot{\psi} S \theta}. \tag{52}
\end{align*}
$$

Once the yaw angle is calculated, the linear and
angular velocities are determined as well as the 3D
path.

5. SIMULATION RESULTS

The lighter than air platform is the AS200 by
Airspeed Airships. It is a remotely piloted airship
designed for remote sensing. It is a non rigid 6m long,
1.4m diameter and 8.6m³ volume airship equipped
with two vectorable engines on the sides of the
gondola and 4 control surfaces at the stern. Envelope
pressure is maintained by air fed from the propellers
into the two ballonets located inside the central
portion of the hull. These ballonets are self regulating
and can be fed from either engine. The engines are
standard model aircraft type units. The propellers can
be rotated through 120 degrees. During flight the
rudders and elevators are used for all movements in
pitch and yaw.

In Table 1, five cases are presented for a normalized
simulation time = 1. For each case, four subplots are
presented: the first one presents the trajectory in space,
the second one the variation of the yaw angle $\psi$, the
linear velocities $v$ and $w$ and finally the angular
velocities $p, q, r$.

The 3D trajectory in Fig. 6 represents a straight line
as the angular velocity is null while the linear velocity
is constant. The yaw angle is constant. This 3D

<table><caption>Table 1. Five cases for a normalized simulation time
= 1.</caption>
  <thead>
    <tr>
      <th>Case</th>
      <th>1</th>
      <th>2</th>
      <th>3</th>
      <th>4</th>
      <th>5</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>&phi;<sub>0</sub></td>
      <td>0</td>
      <td>&pi;/12</td>
      <td>&pi;/12</td>
      <td>&pi;/12</td>
      <td>&pi;/12</td>
    </tr>
    <tr>
      <td>&phi;&#775;<sub>0</sub></td>
      <td>0</td>
      <td>0</td>
      <td>0</td>
      <td>0.1</td>
      <td>0.1</td>
    </tr>
    <tr>
      <td>&theta;<sub>0</sub></td>
      <td>0</td>
      <td>&pi;/6</td>
      <td>&pi;/6</td>
      <td>&pi;/6</td>
      <td>&pi;/6</td>
    </tr>
    <tr>
      <td>&theta;&#775;<sub>0</sub></td>
      <td>0</td>
      <td>0</td>
      <td>0.1</td>
      <td>0</td>
      <td>0.1</td>
    </tr>
    <tr>
      <td>u</td>
      <td>1</td>
      <td>1</td>
      <td>1</td>
      <td>1</td>
      <td>1</td>
    </tr>
    <tr>
      <td>u&#775;</td>
      <td>0</td>
      <td>0</td>
      <td>0</td>
      <td>0</td>
      <td>0.1</td>
    </tr>
    <tr>
      <td>Figure</td>
      <td>6</td>
      <td>7</td>
      <td>8</td>
      <td>9</td>
      <td>10</td>
    </tr>
  </tbody>
</table>

<figure><img src="image_8.png" /><figcaption>Fig. 6. Case #1.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7. Case # 2.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 8. Case # 3.</figcaption></figure>

trajectory in Fig. 7 represents a part of a helix with
constant curvature and torsion. The yaw angle has a
linear variation while the angular and linear velocities
are constant. As can be seen in Fig. 8, after a transient
phenomenon, in case #3 the yaw angle has a linear
variation and the path tends to a classical helix with
<figure><img src="image_11.png" /><figcaption>Fig. 9. Case # 4.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 10. Case # 5.</figcaption></figure>

constant curvature and torsion.

In case #4, Fig. 9 shows that the angular and linear velocities have a slight nonlinear variation. After a certain time, they tend to have permanent values. For case #5, when the derivative of the longitudinal velocity is non zero, the nonlinear phenomenon is amplified in Fig. 10.

## 6. CONCLUSION

This paper addresses the problem of characterizing continuous paths in 3D, taking into account the underactuation. Three differential algebraic equations must be solved as there is six degrees of freedom and three inputs. The constraints on the yaw angle is in the form of a generalized logistic equation while the others are differential algebraic equations in v and w, when the variations of the longitudinal velocity u, and the pitch and roll angles $θ, φ$ are imposed. The role of the trajectory generator is to generate a feasible time trajectory for the UAV. Once the path has been calculated in the Earth fixed frame, motion must be

investigated and reference trajectories determined taking into account actuators constraints. This is the subject of our actual research. This method can be suitable for precise flight path tracking tasks, such as in landing approach. As a further application of the trajectory determination, the prediction of a cone of feasible future positions can be determined to evaluate the influence of the different kinematic parameters on the future flight path.

This methodology can be applied to other types of UAV, taking into account their characteristics. For fixed wing aircraft or helicopter, the added mass and inertia are neglected.

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