Batch Estimation of a Steady, Uniform, Flow-Field from Ground Velocity and Heading Measurements

TL;DR

The paper addresses estimating a steady, uniform flow-field from noisy ground velocity and heading measurements during circular or large-heading maneuvers. It introduces four batch estimators: two curve-fitting methods (a circle fit using $\{\dot x,\dot y\}$ and a quadratic fit to $\{v_g,\psi\}$) and two optimization-based methods (least-squares using $\{\dot x,\dot y,\psi\}$ and $\{v_g,\psi\}$) with explicit constraints and analytical gradients/Hessians. A Monte Carlo study shows that the optimization with $\{\dot x,\dot y,\psi\}$ data yields the lowest estimation errors, while the other approaches exhibit larger errors depending on heading-range and noise; experimental results with a Bluefin-21 corroborate the feasibility and align the estimated current directions with NOAA buoy tidal trends. The work demonstrates that flow-field estimation can be performed without requiring prior knowledge of the vehicle’s flow-relative speed and offers practical, implementable methods for onboard current estimation to support mission planning and path optimization.

Abstract

This paper presents three batch estimation methods that use noisy ground velocity and heading measurements from a vehicle executing a circular orbit (or similar large heading change maneuver) to estimate the speed and direction of a steady, uniform, flow-field. The methods are based on a simple kinematic model of the vehicle's motion and use curve-fitting or nonlinear least-square optimization. A Monte Carlo simulation with randomized flow conditions is used to evaluate the batch estimation methods while varying the measurement noise of the data and the interval of unique heading traversed during the maneuver. The methods are also compared using experimental data obtained with a Bluefin-21 unmanned underwater vehicle performing a series of circular orbit maneuvers over a five hour period in a tide-driven flow.

Figures (9)

• Figure 1: Left: The velocity triangle. The dashed circle represents the set of attainable ground velocities for a fixed magnitude $||{\bm v}_{\rm rel}||$. The center of the circle is defined by the flow velocity ${\bm w}$. Right: At least three unique ground velocity vectors are required to determine the dashed-line circle.
• Figure 2: Example of a circle fit (blue line) to a noisy $(\dot x, \dot y)$ dataset of $N = 100$ points (black dots) generated for parameters $(v, w, \theta) =$(3 m/s, 1 m/s, 90 deg.) with sensor noise standard deviation $\sigma = 0.1$. The red x marker indicates the true center point $(w_x, w_y)$. Using the circular curve-fit method the estimated values are $(\hat{v}, \hat{w}, \hat{\theta})=$ (3.00 m/s, 0.991 m/s, 90.10 deg.).
• Figure 3: Example of a quadratic curve fit (red lines) to $(v_{\rm g}, \psi)$ data (corresponding to the data in Fig. \ref{['eq:circle_example']}). The noise-free curve $v_{\rm g}(\psi)$ is shown as a black line. Using the quadratic curve-fit method the estimated values are $(\hat{v}, \hat{w}, \hat{\theta}) =$ (2.98 m/s, 0.994 m/s., 92.6 deg).
• Figure 4: Top: A surface plot of the cost function \ref{['eq:cost_xyh']} over two slices of the $(v, w, \theta)$ parameter space for the noisy $(\dot x, \dot y, \psi)$ data corresponding to Fig. \ref{['eq:circle_example']}. The left panel is a slice of the cost function at the true value $\theta$, and the right panel is a slice at the true value $v$. The shaded areas represent the parameter space that does not satisfy the constraints. The true $(v,w, \theta)$ point is indicated by a white x marker, and the optimizer iterations, projected on to each plane, are shown in magenta. Bottom: Visualization of the same optimization using data generated for a maneuver with a smaller heading angle change of $\Delta \psi = \pi$.
• Figure 5: Top: A surface plot of the cost function \ref{['eq:cost_vg']} for the dataset in Fig. \ref{['fig:qf']} with optimization iterations shown in magenta. Bottom: A similar plot for a dataset for a smaller change in heading $\Delta \psi = \pi$. Refer to the Fig. \ref{['fig:opt_XYP']} caption for further details.