10.3 Autonomous Vehicles

Step 1: Position Update from Velocity Measurements

Dead reckoning integrates velocity measurements (from DVL and IMU) over time to estimate position. In a body-fixed frame rotated to Earth coordinates by heading $\psi$:

$$\begin{pmatrix} x(t) \\ y(t) \end{pmatrix} = \begin{pmatrix} x_0 \\ y_0 \end{pmatrix} + \int_0^t \begin{pmatrix} \cos\psi & -\sin\psi \\ \sin\psi & \cos\psi \end{pmatrix} \begin{pmatrix} u_b(\tau) \\ v_b(\tau) \end{pmatrix} d\tau$$

Step 2: DVL Velocity Error Model

The DVL measures velocity relative to the seabed with a scale factor error $\epsilon_s$ and a white noise component $\sigma_v$. The measured velocity is:

$$\hat{v} = v_{\text{true}}(1 + \epsilon_s) + n_v, \quad n_v \sim \mathcal{N}(0, \sigma_v^2)$$

Step 3: Position Error Growth

The position error has two components: a linearly growing term from the DVL scale factor error and a random-walk term from velocity noise. Integrating the errors over time $t$:

$$\sigma_{\text{pos}}^2(t) = (\epsilon_s \bar{v} t)^2 + \sigma_v^2 t \, \Delta t_{\text{sample}}$$

Step 4: Heading Error Contribution

Gyroscope drift introduces a heading error $\delta\psi$ that grows with time, causing a cross-track position error proportional to $v \cdot t \cdot \delta\psi$. Combined with the along-track DVL error, the total position uncertainty is:

$$\sigma_{\text{total}}(t) = \sqrt{(\epsilon_s \bar{v} t)^2 + \sigma_v^2 t \Delta t + (\bar{v} \dot{\psi}_{\text{drift}} t^2/2)^2}$$

Step 5: Acoustic Fix Reset

When an acoustic fix (USBL or LBL) is obtained, the position estimate is updated using a Kalman filter. The error resets to the acoustic positioning accuracy $\sigma_{\text{acoustic}}$ and begins growing again. Optimal fix intervals balance navigation accuracy against acoustic communication costs.

Step 1: Total Power Decomposition

The total power consumption of an AUV comprises propulsion power (speed-dependent) and hotel load (constant power for sensors, computers, and communication):

$$P_{\text{total}}(v) = P_{\text{prop}}(v) + P_{\text{hotel}}$$

Step 2: Propulsion Power from Drag

Propulsion power equals drag force times velocity, divided by propulsion efficiency $\eta$. At low Reynolds numbers typical of AUVs, drag is dominated by skin friction proportional to $v^2$:

$$P_{\text{prop}} = \frac{D \cdot v}{\eta} = \frac{\frac{1}{2}\rho C_D A v^2 \cdot v}{\eta} = \frac{\rho C_D A v^3}{2\eta}$$

Step 3: Range as a Function of Speed

Range $R$ is the distance covered before the battery (energy $E$) is exhausted. Since endurance is $T = E/P_{\text{total}}$ and distance is $R = vT$:

$$R(v) = \frac{E \cdot v}{P_{\text{hotel}} + \frac{\rho C_D A}{2\eta} v^3}$$

Step 4: Optimal Speed for Maximum Range

Setting $dR/dv = 0$ and solving, the optimal transit speed occurs when propulsion power equals half the hotel load:

$$\frac{dR}{dv} = 0 \;\Rightarrow\; P_{\text{prop}}(v_{\text{opt}}) = \frac{P_{\text{hotel}}}{2} \;\Rightarrow\; v_{\text{opt}} = \left(\frac{P_{\text{hotel}} \eta}{\rho C_D A}\right)^{1/3}$$

Step 5: Numerical Example

For a typical survey AUV: $P_{\text{hotel}} = 30$ W, $C_D = 0.01$, $A = 0.1$ m$^2$, $\eta = 0.5$, $E = 10$ MJ. The optimal speed is $v_{\text{opt}} \approx 1.5$ m/s, giving a maximum range of $\sim 150$ km and an endurance of $\sim 28$ hours.