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On-Manifold Preintegration for Real-Time Visual-Inertial Odometry

Christian Forster, Luca Carlone, Frank Dellaert, Davide Scaramuzza

arXiv:1512.02363v3cs.RO

TL;DR

Visual-inertial odometry can achieve accurate estimates with nonlinear optimization, but real-time optimization becomes infeasible as trajectories grow and high-rate inertial data add variables. This paper develops SO(3)-aware IMU preintegration into single motion constraints and reports full smoothing at 100 Hz with superior accuracy to competitive state-of-the-art approaches.

  • Problem

    Real-time full-smoothing VIO faces growing optimization costs as trajectories and maps expand, while high-rate IMU constraints require including states at high rate.

    Source: C. Full Smoothing · p. 3, §VI · p. 7

  • Method

    The paper develops SO(3)-aware preintegration that models many IMU measurements as a single motion constraint without committing to a linearization point during integration.

    Source: B. Uncertainty Description in SO(3) · p. 5, §IX · p. 16

  • Results

    100 Hz full smoothing achieves superior accuracy to competitive state-of-the-art filtering and optimization approaches.

    Source: §I · p. 3

  • Takeaways & Limitations

    Incremental smoothing achieves practically the same accuracy as batch estimation while preserving real-time capability.

    Source: 1) Pose Estimation Accuracy and Timing · p. 11

Abstract

from arXiv · show

Current approaches for visual-inertial odometry (VIO) are able to attain highly accurate state estimation via nonlinear optimization. However, real-time optimization quickly becomes infeasible as the trajectory grows over time, this problem is further emphasized by the fact that inertial measurements come at high rate, hence leading to fast growth of the number of variables in the optimization. In this paper, we address this issue by preintegrating inertial measurements between selected keyframes into single relative motion constraints. Our first contribution is a \emph{preintegration theory} that properly addresses the manifold structure of the rotation group. We formally discuss the generative measurement model as well as the nature of the rotation noise and derive the expression for the \emph{maximum a posteriori} state estimator. Our theoretical development enables the computation of all necessary Jacobians for the optimization and a-posteriori bias correction in analytic form. The second contribution is to show that the preintegrated IMU model can be seamlessly integrated into a visual-inertial pipeline under the unifying framework of factor graphs. This enables the application of incremental-smoothing algorithms and the use of a \emph{structureless} model for visual measurements, which avoids optimizing over the 3D points, further accelerating the computation. We perform an extensive evaluation of our monocular \VIO pipeline on real and simulated datasets. The results confirm that our modelling effort leads to accurate state estimation in real-time, outperforming state-of-the-art approaches.

I. INTRODUCTION … A. Notions of Riemannian geometry

The paper develops manifold-aware IMU preintegration and integrates it with incremental factor-graph smoothing for visual-inertial odometry. It reports real-time full smoothing at 100 Hz with superior accuracy to competitive filtering and optimization approaches.

I. INTRODUCTION

  • The method develops SO(3)-aware IMU preintegration and integrates it into incremental factor-graph smoothing with structureless visual measurements.Its theory formalizes rotation noise and the MAP estimator, derives analytic Jacobians for residuals, noise propagation, and a-posteriori bias correction, and avoids rotation-representation singularities. The structureless model eliminates 3D-point variables and permits visual measurements to be processed without delay and relinearized multiple times; the paper also provides implementation derivations and reports an extension of prior work with new experiments and a GTSAM 4.0 release.Source: §I · pp. 2–3
  • The implementation performs full smoothing at 100 Hz and achieves superior accuracy to competitive state-of-the-art filtering and optimization approaches.The proposed preintegration was integrated into a state-of-the-art VIO pipeline and tested on real and simulated datasets.Source: §I · pp. 2–3

II. RELATED WORK

  • VIO approaches differ in estimated pose history, uncertainty representation, and measurement relinearization frequency; iterated EKF equations can be equivalent to Gauss–Newton smoothing.Full smoothers estimate the complete pose history, fixed-lag smoothers a recent window, and filters only the latest state; both latter approaches marginalize older states into a Gaussian prior. EKFs use covariance matrices, while information methods use information matrices or their square roots [3, 8].Source: §II · p. 3

A. Filtering

  • Filtering is efficient but can accumulate linearization and marginalization errors, become overconfident and inconsistent, and require delayed landmark processing in structureless approaches.EKF complexity grows quadratically with landmark count, leading to typical tracking of about 20 landmarks for real-time operation. The VIO problem has four unobservable directions; erroneous linearization can add spurious yaw information.Source: A. Filtering · p. 3

B. Fixed-lag Smoothing

  • Fixed-lag smoothing relinearizes recent measurements and can handle outliers, but marginalization creates dense priors and retains some filtering problems, including inconsistency and linearization-error buildup.Some measurements may be dropped to preserve sparsity.Source: B. Fixed-lag Smoothing · p. 3

C. Full Smoothing

  • Full smoothing is highly accurate but becomes infeasible as trajectories and maps grow, while high-rate IMU data make adding a state per measurement impractical.Incremental methods such as iSAM2 update affected variables; IMU preintegration combines measurements between frames to avoid repeated integration as state estimates change. The paper extends preintegration with a formal SO(3) rotation-noise treatment, a complete MAP estimator, and analytic Jacobians, reporting improved accuracy and robustness over that proposal.Source: C. Full Smoothing · pp. 3–4

A. Notions of Riemannian geometry

  • The preliminaries review manifold-valued MAP optimization, focusing on SO(3), SE(3), uncertain rotations, and optimization on manifolds.The geometry subsection defines SO(3) rotations and its tangent Lie algebra so(3), identified with R^3 through the hat and vee operators; SE(3) represents rigid motions as a rotation and translation with a semidirect-product group operation. Exponential and logarithm maps connect rotation vectors and matrices, while the right Jacobian relates additive tangent perturbations to multiplicative rotation increments.Source: §III · p. 4, A. Notions of Riemannian geometry · pp. 4–5

B. Uncertainty Description in SO(3)

The section models rotational uncertainty as a Gaussian perturbation in the tangent space mapped onto SO(3), then derives a likelihood based on geodesic distance weighted by inverse covariance.

B. Uncertainty Description in SO(3)

  • A rotation is modeled by mapping a zero-mean Gaussian tangent-space perturbation with covariance Σ onto SO(3) through the exponential map.The noise-free rotation is the mean.Source: B. Uncertainty Description in SO(3) · p. 5
  • Changing variables with ϵ = Log(R^-1˜R) yields an SO(3) density whose normalization includes the determinant of the right Jacobian.The logarithm reparameterization applies for perturbations with norm below π.Source: B. Uncertainty Description in SO(3) · p. 5
  • The negative log-likelihood is the squared geodesic distance between measured and predicted rotations, weighted by Σ^-1.For small covariances, the normalization factor is approximated as constant; the log-map reparameterization clips probability tails outside the radius-π ball.Source: B. Uncertainty Description in SO(3) · p. 5

C. Gauss-Newton Method on Manifold

The section extends Gauss–Newton to manifold-valued variables by optimizing local tangent-space increments and retracting updates onto the manifold. This avoids over-parameterization and updates that leave the manifold.

C. Gauss-Newton Method on Manifold

  • Directly approximating the objective in manifold coordinates can over-parameterize variables, making normal equations under-determined, and produce solutions outside the manifold.For example, a rotation matrix uses nine elements although a 3D rotation is defined by a vector in R3.Source: C. Gauss-Newton Method on Manifold · p. 5
  • The lift-solve-retract scheme optimizes a local quadratic in the tangent space, then maps the resulting increment back to the manifold using a retraction.Lifting gives increments of suitable dimension, such as δx ∈ R3 for SO(3); the framework also generalizes to trust-region methods.Source: C. Gauss-Newton Method on Manifold · pp. 5–6
  • This manifold optimization framework provides a unifying generalization of the error state model used in aerospace filtering and robotics optimization.Source: C. Gauss-Newton Method on Manifold · p. 6
  • For SE(3), the chosen retraction is R_T(δφ, δp) = (R Exp(δφ), p + R δp); this choice avoids computing the exponential map for SE(3).The exponential map is a possible retraction, but may be computationally inconvenient.Source: C. Gauss-Newton Method on Manifold · p. 6

IV. MAXIMUM A POSTERIORI VISUAL-INERTIAL STATE ESTIMATION … V. IMU MODEL AND MOTION INTEGRATION

The paper formulates visual-inertial state estimation as a factor-graph MAP problem and models motion by integrating IMU measurements between keyframes. Its integration scheme is practical with a high-rate IMU, though its constant-orientation approximation is not exact during rotation.

IV. MAXIMUM A POSTERIORI VISUAL-INERTIAL STATE ESTIMATION

  • The VIO setup estimates keyframe poses for an IMU-equipped monocular-camera system, assuming a known fixed camera-to-IMU transformation and front-end landmark observations.The IMU frame coincides with the tracked body frame, and a front-end selects the keyframes.Source: §IV · p. 6

A. The State

  • The estimated keyframe state comprises IMU orientation, position, velocity, and biases; the implementation excludes 3D landmarks using a structureless approach.The approach can also estimate landmarks and camera intrinsic and extrinsic calibration parameters.Source: A. The State · p. 6

B. The Measurements

  • Camera measurements record observed landmarks at keyframes, while each interval between consecutive keyframes can contain from a few to hundreds of IMU measurements.The available measurements are the inputs to the estimation problem.Source: B. The Measurements · p. 6

C. Factor Graphs and MAP Estimation

  • Under zero-mean Gaussian noise, MAP estimation minimizes a sum of squared measurement residuals, represented by a factor graph with visual and inertial factors.The graph summarizes multiple IMU measurements in a preintegrated factor and uses structureless vision factors to constrain keyframes observing the same landmark.Source: C. Factor Graphs and MAP Estimation · pp. 6–7, Figure 3 · p. 7

V. IMU MODEL AND MOTION INTEGRATION

  • IMU measurements combine additive white noise with slowly varying sensor bias, and the motion model infers pose and velocity while neglecting Earth’s rotation.The paper treats the world frame as inertial under this assumption.Source: §V · p. 7
  • The adopted numerical integration assumes constant orientation between measurements, so it is inexact during nonzero rotation; high-rate IMU sampling mitigates this approximation.The authors note higher-order numerical integration as an option for slower IMU rates.Source: §V · p. 7

VI. IMU PREINTEGRATION ON MANIFOLD · A. Preintegrated IMU Measurements

The paper replaces high-rate IMU state constraints with preintegrated measurements between keyframes, extending earlier Euler-angle preintegration to SO(3). It rewrites the resulting compound measurement model to isolate noise and enable straightforward likelihood construction.

VI. IMU PREINTEGRATION ON MANIFOLD

  • Preintegrating IMU measurements between keyframes replaces constraints at every IMU sample with a single compound motion measurement, extending from Euler angles to SO(3).Without preintegration, each high-rate measurement would require adding states to the factor graph.Source: §VI · pp. 7–8
  • The direct integration must be repeated when the initial keyframe’s linearization point changes; the method defines relative increments to avoid that dependence.The velocity and position increments are defined to remove dependence on the initial pose, velocity, and gravity, so they can be computed from intervening IMU measurements.Source: §VI · p. 8
  • In Section A, the measurement-model derivation assumes the bias at the initial keyframe is known and remains constant between keyframes.The paper defers avoiding reintegration when the bias estimate changes to Section VI-C.Source: A. Preintegrated IMU Measurements · p. 8, §VI · p. 8

A. Preintegrated IMU Measurements

  • The derivation isolates noise in the rotation increment using a first-order approximation that assumes rotation noise is small.The resulting preintegrated rotation measurement has an associated rotation-noise term.Source: A. Preintegrated IMU Measurements · p. 8
  • The paper substitutes the noise-separated increments into the original model to obtain a compound preintegrated measurement model with random noise.The derivation uses first-order approximations and drops higher-order noise terms when forming the velocity and position measurements.Source: A. Preintegrated IMU Measurements · pp. 8–9
  • The rewritten model makes likelihood construction straightforward: zero-mean additive Gaussian noise gives quadratic negative log-likelihoods for velocity and position, while rotation noise is analyzed subsequently.The rotation likelihood follows when its noise is zero-mean Gaussian.Source: A. Preintegrated IMU Measurements · p. 9

B. Noise Propagation

The section derives the preintegrated measurement-noise covariance, which weights terms in the MAP estimator, and shows that the preintegrated rotation, velocity, and position noises are zero-mean Gaussian to first order. It also describes linear and iterative covariance propagation from IMU noise statistics.

B. Noise Propagation

  • The preintegrated covariance Σij determines optimization weights in the MAP estimator, so accurately modeling it is important.The inverse noise covariance weights the terms in the optimization.Source: B. Noise Propagation · p. 9
  • The preintegrated rotation noise is zero-mean Gaussian to first order, as a linear combination of zero-mean gyro-noise terms.This puts the rotation measurement model in the same form as the paper’s earlier rotation-noise model.Source: B. Noise Propagation · p. 9
  • Velocity and position noises are also zero-mean Gaussian to first order because they are linear combinations of acceleration noise and preintegrated rotation noise.The relations used to derive their noise terms are valid up to first order.Source: B. Noise Propagation · p. 9
  • The covariance Σij can be computed by linear propagation from IMU noise statistics or updated iteratively as each new measurement arrives.The iterative form avoids recomputing the covariance from scratch and is more amenable to online inference.Source: B. Noise Propagation · p. 9

C. Incorporating Bias Updates … VII. STRUCTURELESS VISION FACTORS

The paper updates preintegrated IMU measurements for changing bias estimates and models bias evolution between keyframes, then incorporates IMU and structureless vision constraints into optimization. Vision landmarks are eliminated at each Gauss–Newton iteration while preserving the optimal MAP estimate.

C. Incorporating Bias Updates

  • A first-order bias update corrects preintegrated measurements directly on SO(3), using Jacobians computed at integration-time bias that remain constant and can be precomputed; this is similar to.Source: C. Incorporating Bias Updates · p. 9

D. Preintegrated IMU Factors

  • The preintegrated IMU residual incorporates bias updates and first-order zero-mean Gaussian measurement noise with covariance Σij; analytic residual Jacobians support Gauss–Newton linearization.Source: D. Preintegrated IMU Factors · p. 10

E. Bias Model

  • Biases are modeled as Brownian motion, or integrated white noise; discretization gives zero-mean bias noises with covariances proportional to keyframe interval Δtij, added as a factor between consecutive keyframes.Source: E. Bias Model · p. 10

VII. STRUCTURELESS VISION FACTORS

  • Landmarks are eliminated linearly at each Gauss–Newton iteration, avoiding landmark optimization while preserving the optimal MAP estimate.The landmark elimination is repeated during optimization.Source: §VII · p. 10
  • For each landmark, minimizing its linearized quadratic cost removes its perturbation, yielding one pose-only factor over the keyframes that observe it.The approach reduces pose-and-landmark factors to L pose-only factors and updates landmark positions by fast linear triangulation.Source: §VII · p. 10
  • Unlike MSCKF, the optimization-based approach permits repeated relinearization and incremental inclusion of measurements without delaying them until all observations of a landmark arrive.Source: §VII · p. 11

VIII. EXPERIMENTAL ANALYSIS · A. Simulation Experiments · 1) Pose Estimation Accuracy and Timing:

The experiments assess the proposed approach on simulated and real data, with simulation evaluating accuracy, speed, and consistency and real indoor and outdoor experiments comparing accuracy against the state of the art.

VIII. EXPERIMENTAL ANALYSIS

  • The analysis reports that the approach is accurate, fast, and consistent in simulation, and that it achieves superior accuracy in real indoor and outdoor experiments against the state of the art.Source: §VIII · p. 11

A. Simulation Experiments

  • The simulated camera follows a 120-meter circular trajectory of three-meter radius with sinusoidal vertical motion and observes at most 50 landmarks per frame.The camera runs at 2.5 Hz to simulate keyframes.Source: A. Simulation Experiments · p. 11
  • Simulated landmark measurements use 1-pixel isotropic Gaussian noise, while acceleration and gyroscope measurements derive from the trajectory and include white noise and slowly time-varying bias.The camera focal length is 315 pixels.Source: A. Simulation Experiments · p. 11
  • A 50-run Monte Carlo analysis computes MAP estimates from the paper’s IMU and vision models using iSAM2.Each run uses different realizations of process and measurement noise.Source: A. Simulation Experiments · p. 11

1) Pose Estimation Accuracy and Timing:

  • iSAM2 achieves accuracy practically the same as batch estimation while maintaining approximately 10 milliseconds per update; batch optimization cost quickly grows with trajectory length.In odometry, iSAM2 update time is approximately constant per frame.Source: Figure 6 · p. 11, 1) Pose Estimation Accuracy and Timing · p. 11

A. Simulation Experiments

  • iSAM2 uses Bayes-tree ordering to reduce fill-in, then incrementally updates the square-root information matrix and resolves variables affected by new measurements.This exploits the often-local effect of new measurements on the MAP estimate.Source: A. Simulation Experiments · p. 11

2) Consistency: … B. Real Experiments

The experiments report consistent VIO estimates and accurate bias tracking, and compare on-manifold preintegration with Euler-angle integration. They also report negligible first-order bias-correction errors and benchmark the integrated VIO pipeline against state-of-the-art methods.

2) Consistency:

  • For generic motion, VIO has four unobservable degrees of freedom—global translation and yaw—and Fig. 8 reports errors within the corresponding 3σ bounds, indicating consistency.Roll and pitch uncertainty remain bounded because gravity direction is observable, while global yaw and position uncertainty grow over time.Source: 2) Consistency · p. 11, Figure 8 · p. 12

3) Bias Estimation Accuracy:

  • The estimated gyroscope and accelerometer biases correctly track the ground-truth biases in simulation.Because smoothing can revise the full bias-estimate history at each step, the estimates are visualized as multiple curves.Source: 3) Bias Estimation Accuracy · p. 12, Figure 10 · p. 13

4) First-Order Bias Correction:

  • First-order correction errors are reported as negligible, even for bias perturbations with magnitudes between 0.04 and 0.2.The correction avoids repeating preintegration when an older measurement's bias estimate changes; accuracy was tested by comparing the correction with reintegration.Source: 4) First-Order Bias Correction · p. 12, Figure 11 · p. 13

2) Consistency:

  • Figure 9 reports average NEES over 50 Monte Carlo runs for current pose, position, and rotation; the stated interval is [5.0, 7.0], with values above its upper bound indicating overconfidence.Source: 2) Consistency · p. 12, Figure 9 · p. 13

5) Advantages over the Euler-angle-based formulation:

  • Euler-angle integration errors accumulate quickly for large sampling intervals or angular rates, whereas the proposed rotation-manifold integration is exact regardless of their values.The comparison integrates randomly selected rotation axes with angular-rate magnitudes from 1 to 3 rad/s.Source: 5) Advantages over the… · pp. 12–13
  • Euler-angle likelihoods are not invariant to rigid-body transformations and can produce different estimates under different world-frame choices, whereas the SO(3) likelihood is left invariant.The paper reports experimental comparison of likelihood changes when the reference frame is rotated around randomly selected axes.Source: 5) Advantages over the… · p. 13
  • Near zyx Euler-angle gimbal-lock singularities, Euler-based noise propagation increasingly mismatches ground-truth covariance, while the on-manifold approach estimates covariance accurately independently of platform motion.The analysis varies maximum trajectory pitch and uses KL divergence to quantify covariance mismatch.Source: 5) Advantages over the… · p. 13

B. Real Experiments

  • The monocular VIO pipeline is reported as more accurate than state-of-the-art filtering and fixed-lag smoothing algorithms, while enabling fast inference in real-world problems.The section introduces indoor motion-capture and longer outdoor experiments but provides no benchmark values here.Source: B. Real Experiments · p. 13

1) Implementation: … IX. CONCLUSION

The paper combines manifold-aware IMU preintegration with a structureless visual-inertial pipeline and reports accurate real-time estimation in indoor and outdoor experiments.

1) Implementation:

  • SVO tracks salient image features and supplies triangulated landmarks to an iSAM2 back end, which optimizes selected keyframes in parallel.Gyroscope-derived rotation increments serve as priors for sparse image alignment; the approach does not marginalize past states and could be extended to loop closures.Source: 1) Implementation · pp. 13–14

2) Indoor Experiments:

  • The indoor trial reports 0.3m average drift over 360m, versus 0.7m for OKVIS and MSCKF; yaw drift was lower, while pitch and roll errors were similar.The methods use different feature-tracking front ends, so this comparison evaluates the complete systems rather than isolating preintegration.Source: Figure 15 · p. 14, 2) Indoor Experiments · p. 15
  • The indoor implementation averages 10ms per iSAM2 back-end update and about 3ms for SVO front-end frame processing.The back end uses 10 optimization iterations on a standard laptop and runs in parallel while optimizing keyframes.Source: Figure 16 · p. 14, 2) Indoor Experiments · p. 15
  • In the indoor run, estimation errors remained within the plotted 3σ uncertainty bounds.The plot is from a single run aligned to the Vicon trajectory at the first frame.Source: Figure 17 · p. 15, 2) Indoor Experiments · p. 15

3) Outdoor Experiments:

  • The first outdoor trial reports 1.5m end-to-end error for the proposed approach versus 2.2m for Tango; Fig. 18’s caption instead gives 1.0m versus 2.2m.The trajectory begins and ends at the same location; the paper notes that different sensors make the comparison qualitative.Source: Figure 18 · p. 15, 3) Outdoor Experiments · p. 16
  • On the three-floor outdoor trajectory, the proposed approach reports 0.5m end-to-end error versus Tango’s 1.4m.The comparison is qualitative because the systems use different sensors.Source: Figure 19 · p. 15, 3) Outdoor Experiments · p. 16

IX. CONCLUSION:

  • The proposed preintegration models many IMU measurements as one motion constraint without fixing an integration linearization point, extending preintegration and uncertainty propagation on SO(3).The conclusion contrasts global-frame integration methods.Source: §IX · p. 16
  • The VIO pipeline uses structureless visual factors to avoid optimizing 3D landmarks and iSAM2 for constant-time incremental smoothing; inference takes 10ms back end and 3ms front end.The paper reports greater accuracy than state-of-the-art filtering and optimization-based alternatives and releases the preintegration and structureless vision-factor code in GTSAM 4.0.Source: §IX · p. 16

APPENDIX · A. Iterative Noise Propagation · B. Bias Correction via First-Order Updates

The appendix gives an iterative procedure for propagating preintegrated noise covariance and derives first-order updates for preintegrated measurements when the bias estimate changes. It also states that the bias-update Jacobians can be computed incrementally as measurements arrive.

A. Iterative Noise Propagation

  • The preintegrated noise covariance can be computed iteratively, yielding simpler expressions that are more amenable to online inference.This appendix develops the computation discussed in Section VI-B.Source: A. Iterative Noise Propagation · p. 16
  • The rotation, velocity, and position preintegrated noise terms are each rewritten in iterative form by separating the last term of their sums.The position-noise expression is also related to the velocity-noise expression.Source: A. Iterative Noise Propagation · p. 16
  • A compact linear model of the preintegrated noise, together with the raw IMU measurement-noise covariance, enables iterative computation of the preintegrated measurement covariance.Source: A. Iterative Noise Propagation · p. 16

B. Bias Correction via First-Order Updates

  • Bias correction starts from preintegrated measurements computed at an existing bias estimate and considers a new estimate differing by a small correction.The appendix derives updates for the rotation, velocity, and position measurements.Source: B. Bias Correction via First-Order… · p. 17
  • For rotation, the new-bias preintegrated measurement is expressed as the old measurement plus a first-order correction, allowing an update without repeating integration.Source: B. Bias Correction via First-Order… · p. 17
  • The same derivation is applied to velocity and position, and the bias-update Jacobians can be computed incrementally as new measurements arrive.The appendix summarizes the Jacobians used for the a posteriori bias update.Source: B. Bias Correction via First-Order… · p. 17

C. Jacobians of Residual Errors

The section derives analytic Jacobians for the residual errors used in iterative optimization. It lifts the cost through state retractions so the residuals become functions on a vector space, where their Jacobians can be computed.

C. Jacobians of Residual Errors

  • Lifting the cost with state retractions makes the residual errors functions on a vector space, enabling Jacobian computation with respect to the retracted state variables.The retractions perturb rotations on the rotation group and update positions, velocities, and bias corrections.Source: C. Jacobians of Residual Errors · p. 17

1) Jacobians of r∆pij

2) Jacobians of r∆vij

3) Jacobians of r∆Rij

D. Structureless Vision Factors: Null Space Projection · E. Rotation Rate Integration Using Euler Angles

The section gives a more efficient null-space implementation of structureless vision factors and recalls Euler-angle integration of rotation rates with first-order covariance propagation.

D. Structureless Vision Factors: Null Space Projection

E. Rotation Rate Integration Using Euler Angles

  • Euler-angle integration uses the rotation-rate measurement and noise at time k to update the Euler-angle vector θ_k to θ_{k+1}.The update uses the conjugate Euler angle rate matrix E′(θ_k).Source: E. Rotation Rate Integration Using… · p. 18
  • The covariance of θ_{k+1} is approximated by first-order propagation.The propagation uses the Jacobian with respect to θ_k and the rotation-rate measurement-noise covariance Σ_η.Source: E. Rotation Rate Integration Using… · p. 18
  • The measurement-noise term in the covariance propagation uses B_k = −[E′(θ_k)]^−1Δt and the noise covariance Σ_η.The passage identifies Σ_η as the covariance of the rotation-rate measurement noise.Source: E. Rotation Rate Integration Using… · p. 18

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