4-DOF Maneuvering Motion

Consider the motion of a body, for example a ship, that is forced to move with constant velocity V and a constant course angle ψ. This type of motion has four degrees of freedom. For a ship, the free degrees of freedom would be sway, heave, roll, and pitch motions, which can be used as independent variables.

The independent coordinates can be written as:

Figure 1. EQUATION_DISPLAY
q=(yzϕθ)
(4915)

The body position r and the orientation T can then be written as:

Figure 2. EQUATION_DISPLAY
r=(rxryrz)=(−y⋅sin⁡ψ+V⋅t⋅cos⁡ψy⋅cos⁡ψ+V⋅t⋅sin⁡ψz)
(4916)

and

Figure 3. EQUATION_DISPLAY
T = ( cos ⁡ θ ⁢ cos ⁡ ψ sin ⁡ φ ⁢ sin ⁡ θ ⁢ cos ⁡ ψ − cos φ ⁢ sin ψ ⁡ cos ⁡ φ ⁢ sin ⁡ θ ⁢ cos ⁡ ψ + sin ⁡ φ ⁢ sin ⁡ ψ cos ⁡ θ ⁢ sin ⁡ ψ sin ⁡ φ ⁢ sin ⁡ θ ⁢ ⁡ sin ψ + cos ⁡ φ ⁢ cos ⁡ ψ cos ⁡ φ ⁢ sin ⁡ θ ⁢ sin ⁡ ψ − sin ⁡ φ ⁢ cos ⁡ ψ − sin ⁡ θ sin ⁡ φ ⁢ cos ⁡ θ cos ⁡ φ ⁢ cos ⁡ θ )
(4917)

For the 4-DOF maneuvering motion, the Jacobian matrices for translation and rotation are:

Figure 4. EQUATION_DISPLAY
JT=(−sin⁡ψ000cos⁡ψ0000100)
(4918)
Figure 5. EQUATION_DISPLAY
JR=(00cos⁡ψ⁢cos⁡θ−sin⁡ψ00sin⁡ψ⁢cos⁡θcos⁡ψ00−sin⁡θ0)
(4919)
For the 4-DOF maneuvering motion, the remaining quantities in the general constrained motion equation (Eqn. (4909)) reduce to:
Figure 6. EQUATION_DISPLAY
LT=0LR=(000−φ⁢˙cos⁡ψ⁢sin⁡θ000−φ⁢˙sin⁡ψ⁢sin⁡θ000−φ⁢˙⁢cos⁡θ)v'=(V⋅cos⁡ψV⋅sin⁡ψ0)ω'=0v˙'=0ω˙'=0
(4920)