Modelling of Underwater Robots
In this Chapter the mathematical model of UVMSs is derived. Modeling of rigid bodies moving in a fluid or underwater manipulators has been studied in literature by, among others, [1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 ], where a deeper discussi
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Modelling of Underwater Robots
“We have Einstein’s space, de Sitter’s spaces, expanding universes, contract- ing universes, vibrating universes, mysterious universes. In fact the pure mathematician may create universes just by writing down an equation, and indeed, if he is an individualist he can have an universe of his own”. J.J. Thomson, around 1919.
2.1 Introduction In this chapter the mathematical model of UVMSs is derived. Modeling of rigid bodies moving in a fluid or underwater manipulators has been studied in literature by, among others, [1–12], where a deeper discussion of specific aspects can be found. In [13], the model of two UVMSs holding the same rigid object is derived. A short introduction to underwater vehicles, without manipulators, thus, is given by [14], while deep discussion may be found in [15–17].
2.2 Rigid Body’s Kinematics A rigid body is completely described by its position and orientation with respect to a reference frame Σi , O − xyz that it is supposed to be earth-fixed and inertial. Let define η1 ∈ R3 as ⎡ ⎤ x η 1 = ⎣y ⎦ , z
G. Antonelli, Underwater Robots, Springer Tracts in Advanced Robotics 96, DOI: 10.1007/978-3-319-02877-4_2, © Springer International Publishing Switzerland 2014
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2 Modelling of Underwater Robots
the vector of the body position coordinates in a earth-fixed reference frame. The vector η˙ 1 is the corresponding time derivative (expressed in the earth-fixed frame). If one defines ⎡ ⎤ u ν1 = ⎣ v ⎦ w as the linear velocity of the origin of the body-fixed frame Σb , Ob − xb yb zb with respect to the origin of the earth-fixed frame expressed in the body-fixed frame (from now on: body-fixed linear velocity) the following relation between the defined linear velocities holds: (2.1) ν 1 = RBI η˙ 1 , where RBI is the rotation matrix expressing the transformation from the inertial frame to the body-fixed frame. In the following, two different attitude representations will be introduced: Euler angles and Euler parameters or quaternion. In marine terminology is common the use of Euler angles while several control strategies use the quaternion in order to avoid the representation singularities that might arise by the use of Euler angles.
2.2.1 Attitude Representation by Euler Angles Let define η2 ∈ R3 as
⎡ ⎤ φ η2 = ⎣ θ ⎦ ψ
the vector of body Euler-angle coordinates in a earth-fixed reference frame. In the nautical field those are commonly named roll, pitch and yaw angles and corresponds to the elementary rotation around x, y and z in fixed frame [18]. The vector η˙ 2 is the corresponding time derivative (expressed in the inertial frame). Let define ⎡ ⎤ p ν 2 = ⎣q ⎦ r as the angular velocity of the body-fixed frame with respect to the earth-fixed frame expressed in the body-fixed frame (from now on: body-fixed angular velocity). The vector η˙ 2 does not have a physical interpretation and it is related to the body-fixed angular velocity by a proper Jacobian matrix: ν 2 = J k,o (η2 )η˙ 2 .
(2.2)
2.2 Rigid Body’s Kinematics
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The matrix J k,o ∈ R3×3 can be express
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