Table of Contents
1. Rigid Body Kinematics
1.1 Reference Frames
UUV motion is described using two primary reference frames:
- Earth-Fixed Frame (NED): North-East-Down coordinate system
- Body-Fixed Frame: Origin at vehicle center of buoyancy
1.2 State Vector
The complete state of a 6-DOF underwater vehicle:
\[ \boldsymbol{\eta} = [x, y, z, \phi, \theta, \psi]^T \] \[ \boldsymbol{\nu} = [u, v, w, p, q, r]^T \]
Where η represents position/orientation in NED frame and ν represents velocities in body frame.
1.3 Transformation Matrix
The rotation matrix from body to NED frame using Euler angles:
\[ R(\phi, \theta, \psi) = R_z(\psi) R_y(\theta) R_x(\phi) \]
2. Hydrodynamic Forces and Moments
2.1 Equations of Motion
The 6-DOF equations of motion in matrix form:
\[ M\dot{\nu} + C(\nu)\nu + D(\nu)\nu + g(\eta) = \tau \]
Where:
- M: Inertia matrix (rigid body + added mass)
- C(ν): Coriolis and centripetal matrix
- D(ν): Damping matrix (linear + quadratic)
- g(η): Restoring forces (gravity and buoyancy)
- τ: Control inputs (forces and moments)
2.2 Added Mass
Added mass coefficients for an ellipsoid body:
\[ M_A = -\text{diag}(X_{\dot{u}}, Y_{\dot{v}}, Z_{\dot{w}}, K_{\dot{p}}, M_{\dot{q}}, N_{\dot{r}}) \]
2.3 Drag Forces
Quadratic drag model:
\[ F_D = -\frac{1}{2} \rho C_D A |V| V \]
3. Control System Design
3.1 PID Control
Standard PID controller formulation:
\[ u(t) = K_p e(t) + K_i \int_0^t e(\tau)d\tau + K_d \frac{de(t)}{dt} \]
3.2 Sliding Mode Control
Sliding surface definition:
\[ s = \dot{e} + \lambda e \] \[ u = u_{eq} - K \cdot \text{sgn}(s) \]
3.3 Model Predictive Control
MPC optimization problem:
\[ \min_u \sum_{k=0}^{N-1} \|x_k - x_{ref}\|_Q^2 + \|u_k\|_R^2 \]
5. Underwater Acoustics
5.1 Sound Propagation
Sound speed in seawater (UNESCO equation):
\[ c = 1449.2 + 4.6T - 0.055T^2 + 1.34(S-35) + 0.016D \]
5.2 Transmission Loss
Thorp absorption coefficient:
\[ TL = 20\log_{10}(r) + \alpha r \]
6. Sensor Fusion
6.1 IMU Integration
Strapdown inertial navigation equations for attitude and velocity propagation from accelerometer and gyroscope measurements.
6.2 DVL Processing
Doppler Velocity Log provides bottom-track or water-track velocity measurements with typical accuracy of 0.1% of measured velocity.
6.3 Multi-Sensor Fusion
Complementary filter combining IMU high-frequency data with DVL low-frequency corrections:
\[ \hat{v} = \alpha \cdot v_{IMU} + (1-\alpha) \cdot v_{DVL} \]
References
- Fossen, T.I. (2011). Handbook of Marine Craft Hydrodynamics and Motion Control. Wiley.
- Antonelli, G. (2014). Underwater Robots. Springer.
- Leonard, J.J., & Bahr, A. (2016). Autonomous Underwater Vehicle Navigation. Springer Handbook of Ocean Engineering.