UUV Navigation and State Estimation

Sensor roles, estimator integrity, calibration, and error-budget discipline for navigation without continuous GNSS.

Robotics and Controls · Underwater Systems

Underwater navigation is an observability and error-growth problem. The Global Navigation Satellite System (GNSS) usually provides a surface boundary condition, not a continuous underwater measurement. The system must limit drift, detect bad aiding, and expose when the estimate is no longer fit for the mission.

Sensor roles and failure modes

Sensor Primary observation Main limitation
inertial measurement unit (IMU) angular rate and specific force bias and noise integrate into drift
pressure sensor depth from ambient pressure offset, density, temperature, and port dynamics
magnetometer local field direction vehicle interference and local anomalies
Doppler velocity log (DVL) velocity relative to bottom or a water layer range, bottom type, correlation, geometry, and sound speed
ultra-short baseline (USBL) acoustic position relative to a topside array latency, multipath, calibration, geometry, support asset
long baseline (LBL) position from deployed transponders deployment, survey, and calibration burden
altimeter range to bottom beam geometry and bottom return
sonar or camera terrain or feature-relative constraints visibility, environment, and compute dependence
GNSS global position at the surface unavailable when submerged

WHOI’s Sentry vehicle lists INS, DVL, depth, USBL, and LBL navigation sources. The useful lesson is architectural: capable deep-water navigation layers independent measurements instead of treating one sensor as truth.

Keep the estimator state observable

A starting inertial state may include position $p$, velocity $v$, attitude $q$, gyro bias $b_g$, and accelerometer bias $b_a$:

\[x = \begin{bmatrix} p & v & q & b_g & b_a \end{bmatrix}\]

Add states only when the available motion and measurements can identify them. Candidates include:

  • DVL scale, alignment, and lever-arm error,
  • pressure-sensor bias,
  • local current velocity,
  • sound-speed correction,
  • sensor clock offset.

An unobservable state can absorb real errors while producing plausible covariance.

First-order drift checks

For a constant uncorrected accelerometer bias $b_a$, the velocity and position errors grow approximately as:

\[\Delta v(t) = b_a t\] \[\Delta p(t) = \frac{1}{2}b_a t^2\]

A 1 mg bias is about $9.81\times10^{-3}\ \text{m/s}^2$. If it remains uncorrected for 60 s, the first-order errors are approximately:

\[\Delta v = 0.59\ \text{m/s}, \qquad \Delta p = 17.7\ \text{m}\]

Likewise, a constant gyro bias $b_g$ produces a first-order attitude error $\Delta\theta \approx b_g t$. These are sanity checks, not complete inertial-error models. Gravity projection, attitude coupling, scale error, vibration, and estimator feedback can dominate.

A DVL scale error also accumulates with distance. A 0.2% uncompensated scale error over a 10 km bottom-track path corresponds to roughly 20 m of along-track error before other sources are included.

Illustrative mission sensitivity

For a path length $L$ and a small heading error $\delta\psi$ in radians, cross-track error is approximately:

\[e_\perp \approx L\delta\psi\]

For a 10 km leg:

Error source Assumption First-order position effect
DVL scale 0.2% uncompensated 20.0 m along track
Heading $0.1^\circ = 1.745$ mrad 17.5 m cross track
Initial surface fix 2 m horizontal 2 m initial offset
Acoustic aiding 5 m representative fix uncertainty geometry and latency dependent

The DVL and heading terms alone form an orthogonal error magnitude of approximately:

\[\sqrt{20.0^2 + 17.5^2} \approx 26.6\ \text{m}\]

If the terminal-error requirement is 30 m, little margin remains for initial position, lever arms, timing, outages, current, and acoustic-fix error. This sensitivity calculation suggests specific design work: improve DVL scale calibration, constrain heading error, shorten unaided legs, or add aiding.

Do not combine systematic terms by root-sum-square merely because the arithmetic is convenient. State whether each term is a bias bound, standard uncertainty, random walk, or observed repeatability before selecting a combination rule.

Treat DVL mode and quality as measurements

Bottom track estimates velocity relative to the seafloor when beam returns are valid. Water track estimates velocity relative to scatterers in a selected water layer. It is not ground velocity unless the water velocity is independently known.

Log the vector with its beam validity, correlation, altitude or range, error velocity, operating mode, timestamps, and sound-speed inputs. Rejecting only on whether a vector was emitted is too weak. Teledyne’s Navigator documentation describes separate bottom and water-reference operation, correlation thresholds, altitude search, and installation calibration.

Calibrate the installed system

  • Characterize IMU axes, scale, bias, temperature response, and mounting.
  • Measure magnetometer hard-iron and soft-iron effects with the vehicle assembled and energized.
  • Calibrate DVL rotation, lever arm, scale, and latency against an independent reference.
  • Establish pressure offset, temperature behavior, port response, and density assumptions.
  • Measure transducer lever arms and acoustic timing.
  • Maintain one controlled set of sensor-to-body transforms and timestamp conventions.

Repeat magnetic characterization across thruster commands, payload states, and power modes. A clean bench calibration does not represent the powered vehicle.

Bound timing and frame errors

A measurement belongs to the state at its acquisition time, not when a host process receives it. For a vehicle moving at speed $\lVert v\rVert$ with timestamp error $\Delta t$, the first-order position inconsistency is:

\[e_t \approx \lVert v\rVert |\Delta t|\]

At 2 m/s, a 100 ms timing error creates a 0.20 m position inconsistency before estimator dynamics, acoustic travel time, or processing delay are considered. Rotation also moves an offset sensor. For lever arm $r$ and angular rate $\lVert\omega\rVert$:

\[e_r \approx r\lVert\omega\rVert|\Delta t|\]

A sensor 1 m from the vehicle origin during a $30^\circ$/s turn accumulates about 52 mm of timing-induced displacement over 100 ms.

Define and version:

  • body, sensor, navigation, geodetic, and acoustic-array frames,
  • NED or ENU world convention and positive rotation convention,
  • transform direction, quaternion ordering, and units,
  • lever arms and mounting rotations with uncertainty,
  • acquisition, transmit, receive, and host timestamps,
  • clock source, synchronization method, drift bound, and holdover behavior.

Test timestamp integrity by injecting known delays and clock steps. Log both source and receive times so transport delay is observable. Delayed acoustic fixes should update the state at their measurement epoch or use a justified delayed-measurement method, rather than being treated as current position.

Expose estimator integrity

Log and monitor:

  • innovations and normalized residuals,
  • covariance or another defensible uncertainty representation,
  • measurement acceptance and rejection,
  • filter resets and mode changes,
  • age of the last valid aiding source,
  • disagreement between redundant depth, altitude, or velocity sources,
  • time synchronization health.

Mission logic should act on navigation quality, not only whether the estimator process is running. Define degraded modes before sea trials.

Allocate the error budget

Work backward from the maximum acceptable position and heading error. Include:

  • initial alignment and surface-fix uncertainty,
  • inertial bias and scale error,
  • DVL error and outage duration,
  • acoustic-fix uncertainty and latency,
  • sensor extrinsic and timing error,
  • current-model error,
  • final surfacing and recovery localization.

Use the budget to set sensor, calibration, aiding-frequency, and mission-duration requirements.

Test progression

  1. Static sensor, thermal, and clock characterization.
  2. Known-motion or rate-table testing where available.
  3. Surface trajectory with independent ground truth.
  4. Tank testing for integration and failure handling, with explicit geometry limits.
  5. Open-water runs with independent acoustic or surveyed truth.
  6. Deliberate aiding dropouts, rejected measurements, clock errors, and recovery.
  7. Mission-duration drift and end-to-end recovery tests.

UUV Underwater Communications and Networking, Measurement and Instrumentation, and Design Verification and Test.

Sources