Step 1: Identify Forces and Moments

• Buoyancy and Gravity: Understand how buoyancy and gravitational forces act on the AUV, considering its weight and displaced volume.
• Hydrodynamic Forces: Consider drag, lift, and added mass effects due to the AUV’s motion through water.
• Propulsive Forces: Identify forces generated by propellers or thrusters.
• Control Surfaces: If your AUV has control surfaces (like rudders), understand how they generate forces and moments.
• Disturbance Forces: Consider external disturbances like currents or contact with objects.

Step 2: Define the State Variables

• Position and Orientation: Define variables for the AUV’s position (x, y, z) and orientation (roll, pitch, yaw) in the global frame.
• Linear and Angular Velocities: Define variables for linear $(v_x, v_y, v_z)$ and angular $(ω_{roll}, ω_{pitch}, ω_{yaw})$ velocities in the body frame.

Step 3: Apply Newton-Euler Equations

• Linear Motion: Apply Newton’s second law to relate the sum of external forces (F) to the linear motion of the AUV: $m\dot{v} + (m-A)\dot{v}_{rel} = F$ Where:
  • m is the AUV’s mass.
  • v is the linear velocity.
  • $v_{rel}$ is the velocity relative to the water.
  • A is the added mass matrix.
    • Definition:
      • The added mass matrix represents the additional forces that an AUV must overcome due to the acceleration of the surrounding fluid as it moves.
      • It accounts for the inertia of the water that is displaced by the AUV as it accelerates in the fluid.
    • Characteristics:
      • Non-Diagonal Elements: Can capture the coupling between translational and rotational motions.
      • Symmetry: Typically, the added mass matrix is symmetric.
      • Dependence: Can depend on the AUV’s shape, size, and hydrodynamic properties.
    • Ex:
      • An added mass matrix might look like this for a 6-DOF (Degrees of Freedom) AUV: $A = \begin{bmatrix} X_{\dot{u}} & 0 & 0 & 0 & 0 & 0 \\ 0 & Y_{\dot{v}} & 0 & 0 & 0 & 0 \\ 0 & 0 & Z_{\dot{w}} & 0 & 0 & 0 \\ 0 & 0 & 0 & K_{\dot{p}} & 0 & 0 \\ 0 & 0 & 0 & 0 & M_{\dot{q}} & 0 \\ 0 & 0 & 0 & 0 & 0 & N_{\dot{r}} \end{bmatrix}$
        • $X_{\dot{u}}, Y_{\dot{v}}, Z_{\dot{w}}$ are the added mass coefficients in translational motion along the body-fixed x, y, and z axes, respectively.
        • $K_{\dot{p}}, M_{\dot{q}}, N_{\dot{r}}$ are the added mass coefficients in rotational motion about the body-fixed x, y, and z axes, respectively.
• Angular Motion: Apply Euler’s rotational motion equations to relate the sum of external moments (M) to the angular motion of the AUV: $I\dot{ω} + ω × (Iω) = M$ Where:
  • ω is the angular velocity.
  • I is the inertia matrix.
    • Definition:
      • The inertia matrix represents the distribution of the AUV’s mass with respect to its axes and thus how it resists changes in rotational motion.
      • It accounts for the resistance to changes in angular velocity due to the AUV's mass distribution.
    • Characteristics:
      • Diagonal Elements: Represent the moments of inertia about the principal axes.
      • Off-Diagonal Elements: Represent the products of inertia, capturing how rotations about one axis influence rotations about another.
      • Symmetry: The inertia matrix is symmetric.
    • Ex:
      • An inertia matrix might look like this: $I = \begin{bmatrix} I_{xx} & -I_{xy} & -I_{xz} \\ -I_{yx} & I_{yy} & -I_{yz} \\ -I_{zx} & -I_{zy} & I_{zz} \end{bmatrix}$
        • $I_{xx}, I_{yy}, I_{zz}$ are the moments of inertia about the body-fixed x, y, and z axes, respectively.
        • $I_{xy}, I_{xz}, I_{yx}, I_{yz}, I_{zx}, I_{zy}$ are the products of inertia.
• Estimating A, I:
  • Estimation: Added mass and inertia matrices can be estimated through hydrodynamic analysis, computational fluid dynamics simulations, or experimental measurements.
  • Complexity: In some cases, simplifications or assumptions (like assuming the AUV is symmetric) might be made to reduce the complexity of these matrices.

Step 4: Hydrodynamic Modeling

• Drag: Model drag forces and moments using empirical or theoretical hydrodynamic coefficients.
• Added Mass: Consider the added mass effects due to the acceleration of water around the AUV.
• Lift: If applicable, model lift forces generated by control surfaces or the AUV’s body.

Step 5: Propulsion Modeling

• Thrusters: Model the forces and moments generated by the AUV’s thrusters or propellers.
• Control Surfaces: If applicable, model the forces and moments generated by control surfaces like rudders or elevators.