Hima, Bestaoui 1 Trim trajectories MOTION GENERATION ON TRIM TRAJECTORIES FOR AN AUTONOMOUS UNDERACTUATED AIRSHIP Salim HIMA, Yasmina BESTAOUI Laboratoire des Systèmes Complexes, CNRS FRE 2492 Université d’Evry Val d’Essonne, 38 rue du pelvoux, 91020 Evry, France tel : (33) 169-47-75-19; fax : (33) 169-47-75-99 E-mail : hima@cemif.univ-evry.fr and bestaoui@cemif.univ-evry.fr Abstract : A blimp is a small airship that has no metal framework and collapses when deflated. In the first part of this paper, dynamic modeling of small autonomous non rigid airships is presented, using the Newton-Euler approach. This study discusses the motion in 6 degrees of freedom since 6 independent coordinates are necessary to determine the position and orientation of this vehicle. Euler angles are used in the formulation of this model. In the second part of the paper, path planning is introduced. Motion generation for trim trajectories is presented. This motion generation takes into account the dynamic model presented in the first part. Key-words : Autonomous Airship, Trajectory planning, Underactuated systems, Nonholonomic systems actuation) that influence the response of the blimp to the controls and external 1. Introduction atmospheric disturbances. The blimp is a member of the family of under-actuated Since their renaissance in early 1980’s, systems because it has fewer inputs than airships have been increasingly considered degrees of freedom. In some studies such for varied tasks such as transportation, as [FOS96, HYG00, KHO99, ZHA99], surveillance, freight carrier, advertising, motion is referenced to a system of monitoring, research, and military roles. orthogonal body axes fixed in the airship, More recently, attention has been given to with the origin at the center of volume the use of unmanned airships as aerial assumed to coincide with the gross center inspection platforms, with a very important of buoyancy. The model used was written application area in environmental, originally for a buoyant underwater vehicle biodiversity, and climatological research [FOS96, ZIA98]. It was modified later to and monitoring [CAM99, KHO99, take into account the specificity of the PAI99]. The first objective of this paper is airship [HYG00, KHO99, ZHA99]. In to present a model of a small autonomous [BES01], the origin of the body fixed blimp : kinematics and dynamics. For frame is the center of gravity. kinematics, Euler angles are presented. For The second objective of this paper is to dynamics, a mathematical description of a generate a desired flight path and motion to dirigible flight must contain the necessary be followed by the airship. A mission starts information about aerodynamic, structural with take-off from the platform where the and other internal dynamic effects (engine, 4th International Airship Convention and Exhibition
Hima, Bestaoui 2 Trim trajectories mast that holds the mooring device of the variables are very close to each other in blimp is mounted. Typically, flight their nature [ZEF99]. The usual minimal operation modes can be defined as : take- representation of orientation is given by a off, cruise, turn, landing, hover…[BES01, set of three Euler angles, assembled with CAM99, PAI99, ZHA99]. After the user the three position coordinates allow the has defined the goal tasks, the path description of the situation of a rigid generator then determines a path for the body. A 3*3 direction cosine matrix (of vehicle that is a trajectory in space. In this Euler rotations) is used to describe the paper, the trajectories considered are orientation of the body (achieved by 3 trimming or equilibrium trajectories. The successive rotations) with respect to some general condition for trim requires that the fixed frame reference. rate of change of the magnitude of the Two reference frames are considered in velocity vector is identically zero, in the the derivation of the kinematics and body fixed frame. In this paper we propose dynamics equations of motion. These are some motion generation on trim helices to the Earth fixed frame and the body R f be followed by the airship. fixed frame (figure 1). The position R m and orientation of the vehicle should be 2. AIRSHIP DYNAMIC MODELING described relative to the inertial reference frame while the linear and angular 2.1. Kinematics. velocities of the vehicle should be A general spatial displacement of a rigid expressed in the body-fixed coordinate body consists of a finite rotation about a system. This formulation has been first spatial axis and a finite translation along used for underwater vehicles [FOS96, some vector. The rotational and ZIA98]. translational axes in general need not be In this paper, the origin C of coincides R related to each other. It is often easiest to m with the center of volume of the vehicle. describe a spatial displacement as a ( ) combination of a rotation and a Its axes are the principal x y z v v v translation motions, where the two axes axes of symmetry when available. They are not related. However, the combined must form a right handed orthogonal effect of the two partial transformations normed frame. (i.e rotation, translation about their The position and the orientation of the respective axes) can be expressed as an vehicle C in can be respectively R equivalent unique screw displacement, f described by : where the rotational and translational φ ⎛ ⎞ ⎛ ⎞ axes in fact coincide. The concept of a x ⎜ ⎟ ⎜ ⎟ screw thus represents an ideal η = η 2 = θ eq 1 ⎜ ⎟ ⎜ ⎟ y 1 mathematical tool to analyze spatial ⎜ ⎟ ⎜ ⎟ ψ ⎝ ⎠ ⎝ ⎠ z transformation [ZEF99]. The finite with φ roll, θ pitch and ψ yaw angles. rotation of a rigid body does not obey to The orientation matrix R is given by: the laws of vector addition (in particular commutativity) and as a result the angular ⎛ − + + ⎞ c c s c c s s s s c s c ψ θ ψ φ ψ θ φ ψ φ ψ θ φ velocity of the body cannot be integrated ⎜ ⎟ eq 2 = + − + R s c c c s s s c s s s c ⎜ ψ θ ψ φ ψ θ φ ψ φ ψ θ φ ⎟ to give the attitude of the body. There are ⎜ ⎟ − ⎝ ⎠ s c s c c θ θ φ θ φ many ways to describe finite rotations. Direction cosines, Rodrigues – ( ) ( ) = θ = θ Where cos and sin c θ s θ Hamilton’s (quaternions) variables R ∈ ( 3 ) denotes the orthogonal rotation SO [FOS96], Euler parameters [WEN91], matrix that specifies the orientation of the Euler angles [BES01], can serve as airship frame relative to the inertial examples. Some of these groups of 4th International Airship Convention and Exhibition
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