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AQUA-SLAM: Tightly-Coupled Underwater Acoustic-Visual-Inertial SLAM with Sensor Calibration

Shida Xu, Kaicheng Zhang, Sen Wang

arXiv:2503.11420v1cs.ROcs.CV

TL;DR

Underwater visual SLAM is challenged by degraded visibility and prolonged visual disruptions that visual-inertial methods do not fully address. AQUA-SLAM tightly fuses DVL, stereo camera, and IMU data in graph optimization and adds online sensor calibration. The reported tank results show improved translational and rotational accuracy across WaveTank sequences.

  • Problem

    Underwater visual SLAM faces limited visibility and long visual disruptions, while visual-inertial systems remain vulnerable to difficult underwater conditions.

    Source: §I · p. 1

  • Method

    AQUA-SLAM tightly fuses DVL, stereo-camera, and IMU measurements in graph optimization and provides online extrinsic and DVL-transducer calibration.

    Source: §I · p. 2

  • Results

    The proposed method outperformed alternatives across all WaveTank sequences in translational and rotational accuracy.

    Source: 1) Quantitative Evaluation · p. 11

  • Takeaways & Limitations

    The calibration techniques support precise sensor calibration even without a dedicated underwater facility.

    Source: §VIII · p. 17

Abstract

from arXiv · show

Underwater environments pose significant challenges for visual Simultaneous Localization and Mapping (SLAM) systems due to limited visibility, inadequate illumination, and sporadic loss of structural features in images. Addressing these challenges, this paper introduces a novel, tightly-coupled Acoustic-Visual-Inertial SLAM approach, termed AQUA-SLAM, to fuse a Doppler Velocity Log (DVL), a stereo camera, and an Inertial Measurement Unit (IMU) within a graph optimization framework. Moreover, we propose an efficient sensor calibration technique, encompassing multi-sensor extrinsic calibration (among the DVL, camera and IMU) and DVL transducer misalignment calibration, with a fast linear approximation procedure for real-time online execution. The proposed methods are extensively evaluated in a tank environment with ground truth, and validated for offshore applications in the North Sea. The results demonstrate that our method surpasses current state-of-the-art underwater and visual-inertial SLAM systems in terms of localization accuracy and robustness. The proposed system will be made open-source for the community.

I. INTRODUCTION

Underwater visual SLAM faces degraded imagery and prolonged visual disruptions that visual-inertial systems do not fully resolve. AQUA-SLAM combines DVL, camera, and IMU graph optimization with online calibration, addressing gaps in earlier underwater systems.

I. INTRODUCTION

  • Limited visibility, poor lighting, marine snow, and intermittent visual tracking challenge underwater visual SLAM; visual-inertial systems help with transient disruptions but remain vulnerable to longer degradation.IMU initialization also depends on visual conditions and sufficiently excited motion, both difficult underwater.Source: §I · pp. 1–2

II. RELATED WORK

  • Earlier DVL systems used odometry, loose fusion, filtering, or surface-focused formulations; the cited approaches did not jointly address sensor calibration.The paper positions rigorous DVL modeling and tightly coupled graph fusion as previously unexplored for underwater SLAM.Source: §I · p. 2, 1) Methods Using DVL · p. 3

A. Contributions

  • AQUA-SLAM tightly fuses DVL, camera, and IMU in graph optimization, alongside online extrinsic and DVL-transducer misalignment calibration with a rapid linear approximation.The paper reports tank experiments and North Sea validation, with performance surpassing state-of-the-art underwater and visual-inertial SLAM systems.Source: A. Contributions · p. 2

1) Methods Using DVL:

  • Prior DVL integrations included loose camera–gyroscope fusion without accelerometers, 2D surface-vehicle systems, filtering, and visual-DVL odometry without IMU integration.These designs had limitations including unbounded roll and pitch, surface-specific assumptions, or reduced orientation robustness under challenging vision.Source: 1) Methods Using DVL · p. 3

2) Methods Without Using DVL:

  • The reviewed sonar-visual-inertial systems and related approaches did not investigate incorporating DVL data.The cited work includes tightly coupled sonar, camera, and IMU systems, as well as a strategy that switches to a model-based approach after failure detection.Source: 2) Methods Without Using DVL · p. 3

B. Underwater Vision Based Extrinsic Calibration

The paper motivates natural-feature, online extrinsic calibration for underwater camera–DVL fusion and addresses DVL transducer misalignment. It also establishes the coordinate-frame, state, and measurement definitions used by the system.

B. Underwater Vision Based Extrinsic Calibration

  • Natural scene features support automatic online calibration integrated with SLAM, avoiding the fiducial panel and pre-setup required by a prior camera–DVL approach.The prior method relied on a marinized panel of fiducial markers and pre-setup, which the paper says was impractical or time-consuming offshore.Source: B. Underwater Vision Based Extrinsic… · p. 3

C. DVL Calibration

  • Earlier DVL calibration work focused on SINS-DVL systems, and the cited methods did not address transducer misalignment, which can considerably affect velocity measurements and sensor fusion,,.The cited approaches used EKF, particle swarm optimization, or observability-based trajectory design.Source: C. DVL Calibration · p. 3

III. PRELIMINARIES

  • The paper specifies scalar, vector, matrix, coordinate-frame, and set notation used throughout.Scalars use lowercase italics, vectors lowercase bold Roman, matrices uppercase bold Roman, frames typewriter, and sets calligraphic typeface.Source: §III · p. 3

A. Riemannian Geometry Representation

  • SO(3) denotes the group of 3D rotations, while its tangent space so(3) is represented by skew-symmetric matrices formed from vectors with the hat operator.The paper also introduces SE(3) for 3D rigid motion.Source: A. Riemannian Geometry Representation · p. 4

B. Definition of Coordinate Frames

  • The work uses static world, IMU, camera, and DVL frames; the world z-axis follows gravity, and a frame subscript marks a time-dependent keyframe frame.For example, C_i denotes camera frame C at keyframe i.Source: B. Definition of Coordinate Frames · p. 4

C. State Estimation

  • Each keyframe state includes camera pose, DVL-frame linear velocity, and IMU gyroscope and accelerometer biases.The historical state set also includes visible landmarks, fixed sensor extrinsics, and the initial IMU orientation for gravity.Source: 1) State Definition · p. 4

1) State Definition:

  • The state uses fixed IMU–DVL and DVL–camera extrinsics; the IMU–camera extrinsic is derived by composing these transformations rather than estimated separately.The coordinate transformation convention defines a transform from frame B to frame A and its translation in frame A.Source: 1) State Definition · p. 4, A. Riemannian Geometry Representation · p. 4

2) Measurement Definitions:

  • DVL data are decoded from acoustic signals to estimate velocity, IMU data provide angular velocity and acceleration, and stereo images supply landmarks.The measurement set collects these DVL, IMU, and camera measurements through keyframe n.Source: 2) Measurement Definitions · p. 4

3) Maximum a Posteriori (MAP) Estimation:

The system estimates states from DVL, IMU, and camera measurements using a Gaussian-noise least-squares formulation. Its DVL model derives transducer radial velocities from the 3D velocity and uses them to recover the DVL velocity measurement.

3) Maximum a Posteriori (MAP) Estimation:

  • Gaussian measurement-noise assumptions recast maximum a posteriori state estimation as nonlinear least squares, with prior, DVL, IMU, and camera residuals weighted by covariance.The sensor residuals measure discrepancies between observed and predicted measurements from their respective models.Source: 3) Maximum a Posteriori (MAP) Estimation · p. 4

A. DVL Measurement Model and Its Residuals

  • The DVL measures seabed-relative velocity through four transducers that emit and receive acoustic signals reflected from the seabed.Their Doppler shifts yield one-dimensional radial velocities, which can be aggregated into 3D velocity.Source: A. DVL Measurement Model and Its… · pp. 4–5

1) Velocity Measurement from an Individual Transducer:

  • A transducer’s radial velocity is modeled as its orientation-dependent projection of the DVL-frame 3D velocity, with Gaussian noise.The radial measurement follows the Doppler-shift relation, with water sound speed and transmitted and received frequencies determining the velocity.Source: 1) Velocity Measurement from an… · p. 5

2) DVL Velocity Measurement and Its Model:

  • Four transducer radial velocities form an overdetermined linear system whose closed-form solution estimates the DVL-frame 3D velocity.Each transducer’s projection vector encodes its orientation; the model derives the vectors from the transducer angles and combines them across the four measurements.Source: 2) DVL Velocity Measurement and Its… · p. 5
  • The DVL velocity measurement model uses the estimated DVL-frame velocity to constrain motion between keyframes.The section assumes the extrinsic parameters are known, with their estimation handled by the calibration method described elsewhere in the paper.Source: §IV · p. 4, 2) DVL Velocity Measurement and Its… · p. 5

3) DVL Translation Measurement Model:

DVL translation between keyframes is formed by integrating DVL velocities using gyroscope orientation increments. The paper pre-integrates this contribution to avoid repeated integration during optimization and expresses DVL constraints as velocity and translation residuals.

3) DVL Translation Measurement Model:

  • DVL translation between keyframes aggregates per-interval velocity integrations, assuming velocity remains constant between consecutive DVL measurements and using gyroscope measurements during integration.The integrated translations are combined to obtain the displacement between keyframes.Source: 3) DVL Translation Measurement Model · p. 5

DCDpDC + RDC(RT

4) Residual Derivation:

  • The DVL residual combines velocity constraints at both keyframes with a relative-translation constraint between them.The velocity residuals constrain endpoint velocities, while the translation residual constrains their relative displacement.Source: 4) Residual Derivation · p. 6

B. IMU Measurement Model and Its Residuals

The IMU model uses gyroscope measurements to estimate orientation and accelerometer measurements to estimate velocity and position between keyframes. Pre-integration reformulates these measurements to avoid repeated integration during optimization.

1) Gyroscope Model:

  • Gyroscope measurements with white noise and random-walk bias are integrated to compute the IMU orientation at a later keyframe.The gyroscope bias at the initial keyframe is included in the model.Source: 1) Gyroscope Model · p. 6

2) Accelerometer Model:

  • Accelerometer measurements, affected by gravity, white noise, and random-walk bias, provide velocity and position increments between keyframes.The model reformulates the accelerometer residuals with pre-integration to avoid repeated integrations during optimization.Source: 2) Accelerometer Model · p. 6
  • IMU pre-integration reformulates the gyroscope and accelerometer residual calculations to avoid repeatedly integrating measurements during optimization.Repeated integration of the original residual formulation could require multiple integrations per optimization iteration and considerable time.Source: 2) Accelerometer Model · p. 6
  • The accelerometer velocity and translation models use pre-integrated measurements and corresponding noise terms to represent relative increments between keyframes.The section identifies separate velocity and translation pre-integrations for the accelerometer measurements.Source: 2) Accelerometer Model · pp. 6–7

3) Residual Definition:

The section defines IMU residuals for rotation, velocity, and translation. The rotation residual uses the logarithm map from SO(3) to vectorized so(3).

3) Residual Definition:

  • The IMU residuals comprise rotation, velocity, and translation components.The supplied passages introduce each component separately.Source: 3) Residual Definition · p. 7
  • The IMU rotation residual uses the logarithm map from SO(3) to vectorized so(3).The passage defines Log(·) as that map.Source: 3) Residual Definition · p. 7
  • The IMU velocity residual and translation residual are introduced as separate terms.The passages refer to their derivations from equations (15) and (16), respectively.Source: 3) Residual Definition · p. 7

C. Camera Measurement Model and Its Residuals

The section introduces the camera measurement model and a staged calibration procedure for estimating DVL, camera, and IMU extrinsics. The procedure starts with vision-only bundle adjustment, then progressively adds sensor constraints and parameters.

C. Camera Measurement Model and Its Residuals

  • Camera measurements use stereo image pairs to extract features and estimate 3D landmarks, with the camera model projecting landmarks into pixel coordinates.The residual is formulated over landmarks visible in each keyframe under a Gaussian measurement-noise assumption.Source: C. Camera Measurement Model and Its… · p. 7

V. ONLINE SENSOR CALIBRATION

  • Extrinsic calibration addresses errors in camera–DVL–IMU transformations that are difficult to measure accurately by hand and may vary over time.The paper proposes online calibration to address both extrinsic errors and DVL transducer misalignment.Source: §V · p. 7

A. Extrinsic Calibration of DVL, Camera and IMU Sensors

  • Jointly optimizing SLAM states and unknown extrinsics is described as impracticable because it is sensitive to initialization and prone to local minima.The proposed alternative is a systematic coarse-to-fine calibration procedure.Source: A. Extrinsic Calibration of DVL,… · p. 7

1) Vision-only Bundle Adjustment:

  • Vision-only bundle adjustment first estimates camera poses and 3D landmarks from a short image sequence, typically 10–100 keyframes.Stereo provides metric scale; successful calibration requires reasonable image quality, sufficient visual texture, and preferably observable motion.Source: 1) Vision-only Bundle Adjustment · p. 7

2) Initial Estimation of Extrinsic Parameters:

  • Initial extrinsic estimation uses fixed camera poses with DVL translation and IMU rotation residuals, assuming zero gyroscope bias.The optimization estimates extrinsics while mitigating the influence of IMU biases and gravity.Source: 2) Initial Estimation of Extrinsic… · p. 8

3) Refined Estimation of Extrinsic Parameters with Gyroscope Bias:

  • The next stage refines extrinsics by adding gyroscope biases as optimization variables while retaining the preceding residual terms.The biases are associated with keyframes in the image sequence.Source: 3) Refined Estimation of Extrinsic… · p. 8

4) Initialization of Gravity Direction:

The calibration procedure progressively adds IMU residuals and refines DVL transducer alignment. Linear approximations reduce repeated DVL translation pre-integration when parameter updates are small.

4) Initialization of Gravity Direction:

  • Gravity-direction initialization uses camera poses, extrinsics, and gyroscope bias before incorporating IMU velocity and translation residuals.The initialization optimizes the initial world-to-IMU rotation.Source: 4) Initialization of Gravity Direction · p. 8

5) Full Refinement:

  • Full refinement estimates extrinsics using the DVL translation residual and all three IMU residual terms, initialized from the preceding optimal values.Changing the DVL rotation extrinsic requires reintegrating DVL translation pre-integration, a computationally intensive operation that hinders online optimization.Source: 5) Full Refinement · p. 8

B. DVL Misalignment Calibration

  • DVL misalignment calibration treats each transducer’s tilt angles as unknown while assuming the extrinsics are known.The alignment parameters are estimated through a coarse-to-fine procedure.Source: B. DVL Misalignment Calibration · p. 8

1) Vision-only Bundle Adjustment:

2) Optimization of DVL Body Velocity:

  • The DVL body-velocity stage optimizes velocities using the DVL translation residual while treating its measurement prediction as constant.The variables are the DVL body velocities for the keyframes.Source: 2) Optimization of DVL Body Velocity · p. 8

3) Optimization of DVL Transducer Alignment Parameters:

  • Given the optimized DVL body velocities, the next stage estimates each transducer’s alignment parameters from its individual velocity measurements.The formulation indexes the measured velocity by transducer and keyframe.Source: 3) Optimization of DVL Transducer… · p. 8

C. Rapid Linear Approximation Iteration

  • A linear approximation accelerates DVL translation-residual minimization when incremental parameter updates are relatively small.It approximates the full integration of DVL translation pre-integration during an optimization iteration.Source: C. Rapid Linear Approximation Iteration · p. 8

1) Approximation Iteration for Extrinsic Calibration:

  • For extrinsic calibration, DVL translation pre-integration is linearized with respect to the DVL rotation extrinsic and used for sufficiently small updates.The update-to-integration mapping is applied below a magnitude threshold.Source: 1) Approximation Iteration for… · pp. 8–9

2) Approximation Iteration for Misalignment Calibration:

The implementation uses DVL and IMU information to support tracking and mapping when visual tracking is poor, and applies calibration over data from 10 keyframes. Linearization is used for misalignment calibration when velocity updates are small.

2) Approximation Iteration for Misalignment Calibration:

  • The misalignment-calibration approximation maps incremental DVL body-velocity updates to updated translation pre-integration and is used below a norm threshold.This linearization approximates the pre-integration during optimization.Source: 2) Approximation Iteration for… · p. 9

VI. SYSTEM IMPLEMENTATION

  • The system integrates acoustic-visual-inertial SLAM and online calibration into ORB-SLAM3 while retaining its three-thread design and Atlas multi-map strategy.The tracking, local mapping, and loop-closure modules are upgraded.Source: §VI · p. 9

A. Sensor Data Processing

  • Stereo images provide depth with absolute scale, while DVL measurements are pre-integrated and IMU data is processed through its rotation, velocity, and translation terms.These processed inputs supply the sensor measurements used by the system.Source: A. Sensor Data Processing · p. 9

B. Tracking

  • When frame-to-local tracking fails, DVL and IMU residuals support acoustic-inertial pose estimation in poor image quality.The system also adds acoustic-inertial constraints between consecutive sub-maps.Source: B. Tracking · p. 9

C. Local BA

  • Local bundle adjustment fuses DVL, IMU, and camera data in a sliding window of up to 10 keyframes.It runs when a new keyframe is inserted.Source: C. Local BA · p. 9

D. Atlas

  • The Atlas module interlinks all sub-maps through acoustic-inertial constraints to maintain a consistent map.This differs from ORB-SLAM3’s active and inactive sub-map division.Source: D. Atlas · p. 9

E. Loop Closure

  • Loop closure corrects all sub-maps through acoustic-inertial constraints when sub-maps are merged.The paper contrasts this with ORB-SLAM3, which corrects only the active map and a sub-map being merged.Source: E. Loop Closure · p. 9

F. Sensor Calibration

A. Experiment Settings and Datasets

The evaluation uses tank and offshore datasets, compares AQUA-SLAM with underwater and visual-inertial systems, and finds better odometry accuracy across WaveTank sequences. Comparisons align pose frames and timestamps before scoring estimates against available ground truth.

1) WaveTank Dataset:

  • WaveTank data come from a 9×12 m tank, with AprilTags used to generate ground-truth poses rather than as inputs to evaluated SLAM systems.The BlueROV2 carries a DVL at 5 Hz, an IMU at 330 Hz, and stereo cameras at 20 Hz; eight sequences span differing conditions and trajectories.Source: 1) WaveTank Dataset · p. 10

2) Offshore Dataset:

  • The Offshore dataset was collected at a wind farm with a Saab Seaeye Falcon ROV, and its results are compared with COLMAP because open-sea ground truth is unavailable.Source: 2) Offshore Dataset · p. 10

B. Competing Methods

  • The comparison includes five state-of-the-art systems: two underwater SLAM approaches and three visual-inertial systems, configured largely with shared camera, IMU, and extrinsic parameters.Method-specific parameters use author-provided defaults; underwater and visual-inertial competitors operate in stereo-inertial mode where applicable.Source: B. Competing Methods · pp. 10–11

C. Evaluation Metrics and Preprocessing

1) Preprocessing:

  • Preprocessing converts poses to a common camera frame, matches them to ground truth by timestamp, and aligns each trajectory's first pose to the ground-truth origin.Source: 1) Preprocessing · p. 11

D. Evaluation on Odometry

1) Quantitative Evaluation:

  • DVL-based prior work improves translation over visual-inertial systems, but loose coupling and missing accelerometer compromise rotational accuracy; visual baselines drift or fail under degraded imagery.The paper attributes prior-work orientation drift partly to uncorrected gyroscope bias and unobservable absolute roll and pitch, and reports substantial errors or tracking failures among competing systems.Source: 1) Quantitative Evaluation · pp. 11–12
  • AQUA-SLAM outperforms the compared systems across WaveTank sequences in translational and rotational odometry accuracy.The evaluation reports average RMSE and standard deviation over 10 runs; the results are presented in Table III and Fig. 8.Source: 1) Quantitative Evaluation · p. 11, Figure 8 · p. 11, Table III · p. 12

2) Qualitative Evaluation on HalfTank Easy Sequence:

Across the tank sequences, AQUA-SLAM shows the least drift in odometry and generally outperforms the other systems in SLAM mode. Loop closure improves several systems, but challenging visual conditions still expose substantial errors and failures.

2) Qualitative Evaluation on HalfTank Easy Sequence:

  • On HalfTank Easy odometry, AQUA-SLAM has the least drift and the most accurate, comprehensive dense reconstruction; visual-inertial systems drift more, while ORB3 loses tracking.Its 6-DoF errors remain consistently low, whereas visual-inertial methods show drift on axes beyond roll and pitch under challenging visual conditions.Source: 2) Qualitative Evaluation on HalfTank… · pp. 12–13

1) Quantitative Evaluation:

  • AQUA-SLAM surpasses the other systems in most SLAM scenarios, although poor features or lighting still degrade visual-inertial methods despite loop closure.Results average RMSE and standard deviation over 10 runs; Basalt is slightly better on the SE sequence but drifts on harder sequences and often crashes.Source: Table IV · p. 13, 1) Quantitative Evaluation · p. 14

3) Qualitative Evaluation on WholeTank Hard Sequence:

  • On WholeTank Hard odometry, AQUA-SLAM has the least drift after the full-tank traversal; ORB3 and VINS show the largest overall drift.The prior method drifts mainly along z, while SVIN2 and Basalt show mainly translational drift.Source: 3) Qualitative Evaluation on WholeTank… · p. 14

1) Quantitative Evaluation:

  • After loop closure, AQUA-SLAM retains superior accuracy across 6-DoF poses; other systems show varied corrections, while Basalt's false-positive closures cause trajectory errors.The prior method's roll and pitch errors remain uncorrected, and SVIN2 does not detect the loop.Source: 1) Quantitative Evaluation · p. 14

2) Qualitative Evaluation on HalfTank Easy Sequence:

  • In HalfTank Easy SLAM, AQUA-SLAM and the prior method align trajectories with ground truth and produce good maps, correcting odometry-mode map misalignment.ORB3 improves but still fails to localize along the tank wall; SVIN2 and VINS retain errors, while Basalt has trajectory and map errors.Source: 2) Qualitative Evaluation on HalfTank… · p. 15

3) Qualitative Evaluation on WholeTank Hard Sequence:

F. Validation in Real Offshore Environments

Offshore tests show consistent reconstruction despite poor imagery, while calibration and ablation studies assess parameter convergence and the roles of inertial terms. The section also reports a challenging initialization case and concludes with the system’s broader performance claims.

F. Validation in Real Offshore Environments

1) Extrinsic Calibration of DVL, IMU and Camera:

  • Extrinsic estimates gradually converged toward manual measurements, demonstrating calibration without prior parameter information.The study initialized the transformation as identity and received keyframes during optimization; small offsets were attributed as likely measurement errors.Source: 1) Extrinsic Calibration of DVL, IMU… · p. 16, Figure 20 · p. 17

2) DVL Misalignment Calibration:

  • DVL transducer alignment estimates converged toward the default setting, consistent with the sensor’s reported lack of substantial misalignment.The authors report this as evidence that the method can calibrate transducer alignment accurately.Source: 2) DVL Misalignment Calibration · p. 16, Figure 21 · p. 17

1) Effect of Bias Correction and Accelerometer Residual:

  • Disabling either accelerometer residuals or bias correction increased roll, pitch, and z drift; disabling both produced the highest error and performance similar to prior work.IMU residuals constrain roll and pitch through gravity, while roll drift maps to z drift because the BlueROV2 moves mainly along y.Source: 1) Effect of Bias Correction and… · p. 16, Figure 18 · p. 16

2) Effect of Sensor Calibration:

  • In UUV simulation, added noise to extrinsics and DVL transducer orientation was recalibrated, then calibrated and uncalibrated SLAM were compared across three sequences.The passage describes the evaluation setup; the supplied text does not state its outcome.Source: 2) Effect of Sensor Calibration · p. 16, Table V · p. 17

3) Challenging Cases:

  • No dataset failure was observed, but poor-visual, aggressive-motion initialization could leave IMU bias at zero and cause pose drift; favorable visuals and sufficient initialization motion are recommended.The reported challenging case produced a distorted sparse reconstruction and map misalignment.Source: 3) Challenging Cases · p. 17, Figure 19 · p. 17

VIII. CONCLUSIONS

  • The paper concludes that tightly coupled DVL-camera-IMU graph SLAM with rapid linearized calibration outperforms state-of-the-art underwater and visual-inertial SLAM methods.The reported evaluations were qualitative and quantitative in a tank and at an offshore site.Source: §VIII · p. 17

APPENDIX

The appendix derives the DVL translation measurement model and pre-integration expression, then outlines Jacobian derivations for extrinsic and misalignment calibration.

A. DVL Translation Measurement Model Derivation

DCDpDC + RDC(RT

  • The intermediate algebraic reformulation moves terms, applies a rotation substitution, and reaches the DVL translation measurement model defined in equation (8).Source: DCDpDC + RDC(RT · p. 18

B. DVL Pre-integration Derivation

  • The relative DVL translation increment is reformulated using a first-order rotation approximation and algebraic identities into the pre-integration quantities defined in equation (8).The steps expand the expression and apply u^v = -v^u.Source: B. DVL Pre-integration Derivation · p. 18

C. Jacobian of Extrinsic Calibration Approximation Iteration Derivation

  • The extrinsic-calibration Jacobian is rewritten in the form of equation (25) using a left-multiplication update and first-order approximation.The derivation also applies an algebraic identity to reformulate the expression.Source: C. Jacobian of Extrinsic Calibration… · p. 18

D. Jacobian of Misalignment Calibration Approximation Iteration Derivation

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