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

Christian ForsterLuca CarloneFrank DellaertDavide Scaramuzza

article2015IEEE Transactions on robotics1,355 citations2017 King-Sun Fu Transactions on Robotics Best Paper Award

Develops an inertial measurement preintegration formulation on the rotation group with closed-form bias updates, allowing real-time visual-inertial odometry to achieve high accuracy within factor graph optimization frameworks.

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Autonomous navigation and motion tracking in environments without GPS rely heavily on combining monocular cameras with inertial measurement units (IMUs), a technique known as visual-inertial odometry (VIO). However, existing systems face an ongoing trade-off between estimation accuracy and real-time computation: traditional filtering methods are fast but suffer from accumulated linearization errors, while full optimization methods are highly accurate but computationally overwhelmed by the high rate of IMU data and the growing trajectory history.

The main objective of the article is to demonstrate an efficient and accurate VIO framework that achieves real-time, maximum a posteriori motion estimation. It evaluates a mathematical formulation for preintegrating high-rate inertial measurements directly on the rotation manifold and combining them with landmark-free visual factors in an incremental optimization pipeline.

The researchers developed a theoretical model that combines multiple IMU measurements between keyframes into single relative motion constraints on the 3D rotation manifold, avoiding repeated integrations when sensor biases are updated. They integrated this preintegrated IMU model into a factor graph using a structureless vision approach—which analytically eliminates 3D landmark points—and solved the resulting optimization incrementally using the iSAM2 algorithm. The system was validated using Monte Carlo simulations over 50 runs alongside real-world evaluations on a 430-meter indoor trajectory with motion-capture ground truth and multi-floor outdoor datasets, benchmarking against established methods such as OKVIS, MSCKF, and Google Tango.

The findings confirm substantial performance and accuracy gains over existing approaches. First, in real-world indoor testing, the proposed method accumulated only 0.3 meters of average drift over 360 meters of traveled distance, outperforming both OKVIS and MSCKF, which each accumulated roughly 0.7 meters (a drift reduction of more than 50%). Second, the framework achieved steady real-time execution on standard hardware, taking roughly 10 milliseconds for backend state updates and 3 milliseconds for visual feature tracking. Third, outdoor trials showed superior end-to-end accuracy, recording 1.0 meter and 0.5 meter drift on respective paths compared to 2.2 meters and 1.4 meters for Google Tango. Finally, statistical consistency evaluations demonstrated that the estimator properly tracks sensor biases and avoids overconfidence along unobservable degrees of freedom, completely removing the mathematical singularities and coordinate-dependent errors inherent to older Euler-angle preintegration methods.

These results show that engineering teams do not need to sacrifice trajectory accuracy to achieve real-time tracking on computationally constrained mobile platforms, such as drones or augmented reality devices. By avoiding redundant recalculations and landmark state explosion, the approach lowers computing overhead and minimizes risk of catastrophic tracking drift in GPS-denied environments.

Decision-makers and engineering leads should consider adopting on-manifold IMU preintegration and structureless vision factors for future autonomous navigation stacks, leveraging the authors' open-source implementation within the GTSAM toolbox. Future deployment planning should evaluate integrating loop-closure detection to eliminate long-term residual drift and explore higher-order integration schemes if deploying with lower-frequency IMUs.

Confidence in the system's performance is high based on the consistency of simulated Monte Carlo tests and physical validation against ground truth. Key operational considerations include sensitivity to initial sensor calibration (such as camera-IMU extrinsic alignment and time synchronization) and the assumption of sufficient visual feature tracking from the front-end pipeline, which may degrade in textureless or highly dynamic environments.

Georgia Institute of Technology Massachusetts Institute of Technology University of Zurich factor graphs imu inertial measurements visual-inertial odometry on-manifold preintegration visual-inertial SLAM

arXiv: 1512.02363 gtborg/gtsam

Paper: ORB-SLAM: A Versatile and Accurate Monocular SLAM System, Raul Mur-Artal et al. (2015). Introduces the keyframe-based tracking and local optimization architecture that underpins modern real-time visual estimation frameworks.
Paper: MonoSLAM: Real-Time Single Camera SLAM, Andrew J. Davison et al. (2007). Establishes real-time probabilistic visual SLAM using recursive filtering, providing fundamental context for subsequent graph-based and optimization-driven VIO methods.
Paper: An Introduction to the Kalman Filter, Greg Welch et al. (1995). Presents foundational state estimation and covariance propagation principles essential for formulating inertial sensor noise models and kinematic state propagation.
Paper: Estimating uncertain spatial relationships in robotics, Randall Smith et al. (1986). Develops the foundational probabilistic representation of uncertain spatial coordinate transformations used across robotics state estimation.

Paper: VINS-Mono: A Robust and Versatile Monocular Visual-Inertial State Estimator, Tong Qin et al. (2017). Directly employs on-manifold IMU preintegration within a complete sliding-window monocular visual-inertial state estimator.
Paper: ORB-SLAM3: An Accurate Open-Source Library for Visual, Visual–Inertial, and Multimap SLAM, Carlos Campos et al. (2020). Incorporates IMU preintegration and maximum-a-posteriori optimization into a comprehensive visual-inertial SLAM library with multi-map capabilities.
Paper: LIO-SAM: Tightly-coupled Lidar Inertial Odometry via Smoothing and Mapping, Tixiao Shan et al. (2020). Extends factor-graph IMU preintegration to tightly coupled lidar-inertial odometry and mapping.
Paper: Past, Present, and Future of Simultaneous Localization and Mapping: Toward the Robust-Perception Age, Cesar Cadena et al. (2016). Surveys the evolution of factor-graph-based SLAM and reviews modern visual-inertial navigation methods within the broader robust perception landscape.

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Abstract

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.

Topics

factor graphs imu inertial measurements jacobians odometry optimization optimization problems estimation method

Keywords

visual-inertial odometry on-manifold preintegration visual-inertial SLAM preintegrated rotation measurement bundle adjustment Riemannian manifolds motion estimation inertial navigation incremental smoothing rigid body transformations

I Introduction
II Related Work
II-A Filtering
II-B Fixed-lag Smoothing
II-C Full Smoothing
III Preliminaries
III-A Notions of Riemannian geometry
III-B Uncertainty Description in SO⁡(3)\mathrm{SO}(3)
III-C Gauss-Newton Method on Manifold
IV Maximum a Posteriori Visual-Inertial State Estimation
IV-A The State
IV-B The Measurements
IV-C Factor Graphs and MAP Estimation
V IMU Model and Motion Integration
VI IMU Preintegration on Manifold
VI-A Preintegrated IMU Measurements
VI-B Noise Propagation
VI-C Incorporating Bias Updates
VI-D Preintegrated IMU Factors
VI-E Bias Model
VII Structureless Vision Factors
VIII Experimental Analysis
VIII-A Simulation Experiments
VIII-A1 Pose Estimation Accuracy and Timing
VIII-A2 Consistency
VIII-A3 Bias Estimation Accuracy
VIII-A4 First-Order Bias Correction
VIII-A5 Advantages over the Euler-angle-based formulation
VIII-B Real Experiments
VIII-B1 Implementation
VIII-B2 Indoor Experiments
VIII-B3 Outdoor Experiments
IX Conclusion
IX-A Iterative Noise Propagation
IX-B Bias Correction via First-Order Updates
IX-C Jacobians of Residual Errors
IX-C1 Jacobians of 𝐫Δ𝐩ij\mathbf{r}_{\Delta\mathbf{p}_{ij}}
IX-C2 Jacobians of 𝐫Δ𝐯ij\mathbf{r}_{\Delta\mathbf{v}_{ij}}
IX-C3 Jacobians of 𝐫Δ𝚁ij\mathbf{r}_{\Delta\mathtt{R}_{ij}}
IX-D Structureless Vision Factors: Null Space Projection
IX-E Rotation Rate Integration Using Euler Angles
References

Knowls

Knowl 1 — On-Manifold IMU Preintegration Model
model/method
In visual-inertial state estimation, an Inertial Measurement Unit (IMU) outputs high-rate angular velocity $\tilde{\omega}(t) = \omega(t) + b^g(t) + \eta^g(t)$ and linear acceleration $\tilde{a}(t) = R(t)^T(a(t) - g) + b^a(t) + \eta^a(t)$ , where $\omega(t) \in \mathbb{R}^3$ is instantaneous angular velocity, $a(t) \in \mathbb{R}^3$ is sensor acceleration, $g \in \mathbb{R}^3$ is the gravity vector in world coordinates, $R(t) \in SO(3)$ is the orientation of the sensor frame relative to the world frame, $b^g(t), b^a(t) \in \mathbb{R}^3$ are sensor biases, and $\eta^g(t), \eta^a(t)$ are additive white Gaussian noise processes.

To avoid reintegrating measurements when the state estimate at keyframe $i$ changes during nonlinear optimization, inertial measurements between consecutive keyframes $i$ and $j$ at discrete times $k \in \{i, \dots, j-1\}$ with time step $\Delta t$ are integrated into relative motion increments that are independent of the initial pose $(R_i, p_i) \in SE(3)$ and velocity $v_i \in \mathbb{R}^3$ :

$\Delta R_{ij} \doteq R_i^T R_j = \prod_{k=i}^{j-1} \text{Exp}\left( (\tilde{\omega}_k - b_i^g - \eta_k^{gd}) \Delta t \right)$

$\Delta v_{ij} \doteq R_i^T (v_j - v_i - g \Delta t_{ij}) = \sum_{k=i}^{j-1} \Delta R_{ik} (\tilde{a}_k - b_i^a - \eta_k^{ad}) \Delta t$

$\Delta p_{ij} \doteq R_i^T \left( p_j - p_i - v_i \Delta t_{ij} - \frac{1}{2} g \Delta t_{ij}^2 \right) = \sum_{k=i}^{j-1} \left( \Delta v_{ik} \Delta t + \frac{1}{2} \Delta R_{ik} (\tilde{a}_k - b_i^a - \eta_k^{ad}) \Delta t^2 \right)$

where $\Delta t_{ij} \doteq \sum_{k=i}^{j-1} \Delta t$ , $\Delta R_{ik} \doteq R_i^T R_k$ , $\Delta v_{ik} \doteq R_i^T (v_k - v_i - g \Delta t_{ik})$ , and $\eta_k^{gd}, \eta_k^{ad}$ are discrete-time noise terms. The operator $\text{Exp}(\phi) \doteq \exp(\phi^\wedge)$ denotes the exponential map from $\mathbb{R}^3$ to $SO(3)$ using the skew-symmetric matrix hat operator $(\cdot)^\wedge$ .
Knowl 2 — Preintegrated IMU Measurement Model and Residual Errors
equation
Given keyframe states $x_i = [R_i, p_i, v_i, b_i]$ and $x_j = [R_j, p_j, v_j, b_j]$ , where $(R, p) \in SE(3)$ , $v \in \mathbb{R}^3$ , and $b = [b^g, b^a] \in \mathbb{R}^6$ , the preintegrated IMU measurements computed at a linearization bias estimate $\bar{b}_i = [\bar{b}_i^g, \bar{b}_i^a]$ are defined as:

$\Delta \tilde{R}_{ij} \doteq \prod_{k=i}^{j-1} \text{Exp}\left( (\tilde{\omega}_k - \bar{b}_i^g) \Delta t \right)$

$\Delta \tilde{v}_{ij} \doteq \sum_{k=i}^{j-1} \Delta \tilde{R}_{ik} (\tilde{a}_k - \bar{b}_i^a) \Delta t$

$\Delta \tilde{p}_{ij} \doteq \sum_{k=i}^{j-1} \left( \Delta \tilde{v}_{ik} \Delta t + \frac{1}{2} \Delta \tilde{R}_{ik} (\tilde{a}_k - \bar{b}_i^a) \Delta t^2 \right)$

The generative measurement model relating the preintegrated measurements to the states and random noise vector $\eta_{ij}^\Delta = [\delta \phi_{ij}^T, \delta v_{ij}^T, \delta p_{ij}^T]^T \sim \mathcal{N}(0_{9\times 1}, \Sigma_{ij})$ is:

$\Delta \tilde{R}_{ij} = R_i^T R_j \text{Exp}(\delta \phi_{ij})$

$\Delta \tilde{v}_{ij} = R_i^T (v_j - v_i - g \Delta t_{ij}) + \delta v_{ij}$

$\Delta \tilde{p}_{ij} = R_i^T \left( p_j - p_i - v_i \Delta t_{ij} - \frac{1}{2} g \Delta t_{ij}^2 \right) + \delta p_{ij}$

Incorporating first-order bias updates $\delta b_i = [\delta b_i^g, \delta b_i^a]$ with respect to $\bar{b}_i$ , the preintegrated IMU factor residual error $r_{\mathcal{I}_{ij}} = [r_{\Delta R_{ij}}^T, r_{\Delta v_{ij}}^T, r_{\Delta p_{ij}}^T]^T \in \mathbb{R}^9$ is given by:

$r_{\Delta R_{ij}} = \text{Log}\left( \left( \Delta \tilde{R}_{ij}(\bar{b}_i^g) \text{Exp}\left( \frac{\partial \Delta \bar{R}_{ij}}{\partial b^g} \delta b_i^g \right) \right)^T R_i^T R_j \right)$

$r_{\Delta v_{ij}} = R_i^T (v_j - v_i - g \Delta t_{ij}) - \left[ \Delta \tilde{v}_{ij}(\bar{b}_i) + \frac{\partial \Delta \bar{v}_{ij}}{\partial b^g} \delta b_i^g + \frac{\partial \Delta \bar{v}_{ij}}{\partial b^a} \delta b_i^a \right]$

$r_{\Delta p_{ij}} = R_i^T \left( p_j - p_i - v_i \Delta t_{ij} - \frac{1}{2} g \Delta t_{ij}^2 \right) - \left[ \Delta \tilde{p}_{ij}(\bar{b}_i) + \frac{\partial \Delta \bar{p}_{ij}}{\partial b^g} \delta b_i^g + \frac{\partial \Delta \bar{p}_{ij}}{\partial b^a} \delta b_i^a \right]$

where $\text{Log}(\cdot) \doteq (\log(\cdot))^\vee$ maps an element in $SO(3)$ to its tangent vector in $\mathbb{R}^3$ .
Knowl 3 — Iterative Covariance Propagation for Preintegrated IMU Measurements
equation
The preintegrated measurement noise vector $\eta_{ij}^\Delta = [\delta \phi_{ij}^T, \delta v_{ij}^T, \delta p_{ij}^T]^T \in \mathbb{R}^9$ is propagated in discrete time from the raw discrete IMU noise vector $\eta_k^d = [(\eta_k^{gd})^T, (\eta_k^{ad})^T]^T \sim \mathcal{N}(0_{6\times 1}, \Sigma_\eta)$ with covariance $\Sigma_\eta = \frac{1}{\Delta t} \text{diag}(\text{Cov}(\eta^g), \text{Cov}(\eta^a))$ .

The linear recurrence relation for the noise vector is:

$\eta_{ij}^\Delta = A_{j-1} \eta_{ij-1}^\Delta + B_{j-1} \eta_{j-1}^d$

with transition matrices:

$A_{j-1} = \begin{bmatrix} \Delta \tilde{R}_{j-1 j}^T & 0_{3\times 3} & 0_{3\times 3} \\ -\Delta \tilde{R}_{ij-1} (\tilde{a}_{j-1} - \bar{b}_i^a)^\wedge \Delta t & I_{3\times 3} & 0_{3\times 3} \\ -\frac{1}{2} \Delta \tilde{R}_{ij-1} (\tilde{a}_{j-1} - \bar{b}_i^a)^\wedge \Delta t^2 & \Delta t I_{3\times 3} & I_{3\times 3} \end{bmatrix}$

$B_{j-1} = \begin{bmatrix} J_r^{j-1} \Delta t & 0_{3\times 3} \\ 0_{3\times 3} & \Delta \tilde{R}_{ij-1} \Delta t \\ 0_{3\times 3} & \frac{1}{2} \Delta \tilde{R}_{ij-1} \Delta t^2 \end{bmatrix}$

where $J_r^{j-1} \doteq J_r((\tilde{\omega}_{j-1} - \bar{b}_i^g)\Delta t)$ is the right Jacobian of $SO(3)$ defined by:

$J_r(\phi) = I - \frac{1 - \cos \|\phi\|}{\|\phi\|^2} \phi^\wedge + \frac{\|\phi\| - \sin \|\phi\|}{\|\phi\|^3} (\phi^\wedge)^2$

The preintegrated measurement covariance $\Sigma_{ij} \in \mathbb{R}^{9\times 9}$ is computed recursively starting from $\Sigma_{ii} = 0_{9\times 9}$ via:

$\Sigma_{ij} = A_{j-1} \Sigma_{ij-1} A_{j-1}^T + B_{j-1} \Sigma_\eta B_{j-1}^T$
Knowl 4 — First-Order A-Posteriori Bias Correction for Preintegrated Measurements
model/method
When optimization updates the bias estimate from $\bar{b}_i = [\bar{b}_i^g, \bar{b}_i^a]$ to $\bar{b}_i + \delta b_i$ , repeating numerical integration across all intermediate IMU measurements is avoided by applying a first-order Taylor expansion on $SO(3)$ and $\mathbb{R}^3$ :

$\Delta \tilde{R}_{ij}(b_i^g) \approx \Delta \tilde{R}_{ij}(\bar{b}_i^g) \text{Exp}\left( \frac{\partial \Delta \bar{R}_{ij}}{\partial b^g} \delta b_i^g \right)$

$\Delta \tilde{v}_{ij}(b_i^g, b_i^a) \approx \Delta \tilde{v}_{ij}(\bar{b}_i^g, \bar{b}_i^a) + \frac{\partial \Delta \bar{v}_{ij}}{\partial b^g} \delta b_i^g + \frac{\partial \Delta \bar{v}_{ij}}{\partial b^a} \delta b_i^a$

$\Delta \tilde{p}_{ij}(b_i^g, b_i^a) \approx \Delta \tilde{p}_{ij}(\bar{b}_i^g, \bar{b}_i^a) + \frac{\partial \Delta \bar{p}_{ij}}{\partial b^g} \delta b_i^g + \frac{\partial \Delta \bar{p}_{ij}}{\partial b^a} \delta b_i^a$

The sensitivity Jacobians are computed incrementally during the preintegration step:

$\frac{\partial \Delta \bar{R}_{ij}}{\partial b^g} = - \sum_{k=i}^{j-1} \Delta \tilde{R}_{k+1 j}(\bar{b}_i)^T J_r^k \Delta t$

$\frac{\partial \Delta \bar{v}_{ij}}{\partial b^a} = - \sum_{k=i}^{j-1} \Delta \bar{R}_{ik} \Delta t$

$\frac{\partial \Delta \bar{v}_{ij}}{\partial b^g} = - \sum_{k=i}^{j-1} \Delta \bar{R}_{ik} (\tilde{a}_k - \bar{b}_i^a)^\wedge \frac{\partial \Delta \bar{R}_{ik}}{\partial b^g} \Delta t$

$\frac{\partial \Delta \bar{p}_{ij}}{\partial b^a} = \sum_{k=i}^{j-1} \left( \frac{\partial \Delta \bar{v}_{ik}}{\partial b^a} \Delta t - \frac{1}{2} \Delta \bar{R}_{ik} \Delta t^2 \right)$

$\frac{\partial \Delta \bar{p}_{ij}}{\partial b^g} = \sum_{k=i}^{j-1} \left( \frac{\partial \Delta \bar{v}_{ik}}{\partial b^g} \Delta t - \frac{1}{2} \Delta \bar{R}_{ik} (\tilde{a}_k - \bar{b}_i^a)^\wedge \frac{\partial \Delta \bar{R}_{ik}}{\partial b^g} \Delta t^2 \right)$

where $J_r^k = J_r((\tilde{\omega}_k - \bar{b}_i^g)\Delta t)$ .
Knowl 5 — Analytic Jacobians of Preintegrated IMU Residual Errors
theoretical result
Gauss-Newton optimization on manifolds lifts the residual errors $r_{\mathcal{I}_{ij}} = [r_{\Delta R_{ij}}^T, r_{\Delta v_{ij}}^T, r_{\Delta p_{ij}}^T]^T$ using local retractions: $R_i \leftarrow R_i \text{Exp}(\delta \phi_i), \quad p_i \leftarrow p_i + R_i \delta p_i, \quad v_i \leftarrow v_i + \delta v_i, \quad \delta b_i \leftarrow \delta b_i + \tilde{\delta b}_i$ $R_j \leftarrow R_j \text{Exp}(\delta \phi_j), \quad p_j \leftarrow p_j + R_j \delta p_j, \quad v_j \leftarrow v_j + \delta v_j$

The analytic Jacobians of the preintegrated residuals with respect to these local perturbations are:

For rotation residual $r_{\Delta R_{ij}}$ : $\frac{\partial r_{\Delta R_{ij}}}{\partial \delta \phi_i} = -J_r^{-1}(r_{\Delta R}(R_i)) R_j^T R_i, \quad \frac{\partial r_{\Delta R_{ij}}}{\partial \delta \phi_j} = J_r^{-1}(r_{\Delta R}(R_j))$ $\frac{\partial r_{\Delta R_{ij}}}{\partial \tilde{\delta b}_i^g} = -J_r^{-1}(r_{\Delta R_{ij}}(\delta b_i^g)) \text{Exp}(r_{\Delta R_{ij}}(\delta b_i^g))^T J_r\left( \frac{\partial \Delta \bar{R}_{ij}}{\partial b^g} \delta b_i^g \right) \frac{\partial \Delta \bar{R}_{ij}}{\partial b^g}$ $\frac{\partial r_{\Delta R_{ij}}}{\partial \delta p_i} = \frac{\partial r_{\Delta R_{ij}}}{\partial \delta v_i} = \frac{\partial r_{\Delta R_{ij}}}{\partial \delta p_j} = \frac{\partial r_{\Delta R_{ij}}}{\partial \delta v_j} = \frac{\partial r_{\Delta R_{ij}}}{\partial \tilde{\delta b}_i^a} = 0_{3\times 3}$

For velocity residual $r_{\Delta v_{ij}}$ : $\frac{\partial r_{\Delta v_{ij}}}{\partial \delta \phi_i} = \left( R_i^T (v_j - v_i - g \Delta t_{ij}) \right)^\wedge, \quad \frac{\partial r_{\Delta v_{ij}}}{\partial \delta v_i} = -R_i^T, \quad \frac{\partial r_{\Delta v_{ij}}}{\partial \delta v_j} = R_i^T$ $\frac{\partial r_{\Delta v_{ij}}}{\partial \tilde{\delta b}_i^g} = -\frac{\partial \Delta \bar{v}_{ij}}{\partial b^g}, \quad \frac{\partial r_{\Delta v_{ij}}}{\partial \tilde{\delta b}_i^a} = -\frac{\partial \Delta \bar{v}_{ij}}{\partial b^a}$ $\frac{\partial r_{\Delta v_{ij}}}{\partial \delta \phi_j} = \frac{\partial r_{\Delta v_{ij}}}{\partial \delta p_i} = \frac{\partial r_{\Delta v_{ij}}}{\partial \delta p_j} = 0_{3\times 3}$

For position residual $r_{\Delta p_{ij}}$ : $\frac{\partial r_{\Delta p_{ij}}}{\partial \delta \phi_i} = \left( R_i^T \left( p_j - p_i - v_i \Delta t_{ij} - \frac{1}{2} g \Delta t_{ij}^2 \right) \right)^\wedge, \quad \frac{\partial r_{\Delta p_{ij}}}{\partial \delta p_i} = -I_{3\times 3}, \quad \frac{\partial r_{\Delta p_{ij}}}{\partial \delta p_j} = R_i^T R_j$ $\frac{\partial r_{\Delta p_{ij}}}{\partial \delta v_i} = -R_i^T \Delta t_{ij}, \quad \frac{\partial r_{\Delta p_{ij}}}{\partial \tilde{\delta b}_i^g} = -\frac{\partial \Delta \bar{p}_{ij}}{\partial b^g}, \quad \frac{\partial r_{\Delta p_{ij}}}{\partial \tilde{\delta b}_i^a} = -\frac{\partial \Delta \bar{p}_{ij}}{\partial b^a}$ $\frac{\partial r_{\Delta p_{ij}}}{\partial \delta \phi_j} = \frac{\partial r_{\Delta p_{ij}}}{\partial \delta v_j} = 0_{3\times 3}$

where the inverse right Jacobian is: $J_r^{-1}(\phi) = I + \frac{1}{2} \phi^\wedge + \left( \frac{1}{\|\phi\|^2} - \frac{1 + \cos \|\phi\|}{2 \|\phi\| \sin \|\phi\|} \right) (\phi^\wedge)^2$
Knowl 6 — Structureless Vision Factor via Null Space Projection
model/method
To eliminate the need to include 3D landmark positions $\rho_l \in \mathbb{R}^3$ ( $l = 1, \dots, L$ ) in the state vector, visual reprojection errors are marginalized out analytically at each Gauss-Newton iteration.

For a landmark $l$ observed in a set of $n_l$ keyframes $\mathcal{X}(l)$ with image measurements $z_{il} \in \mathbb{R}^2$ , the lifted and linearized reprojection error is written in stacked matrix form as:

$\| F_l \delta T_{\mathcal{X}(l)} + E_l \delta \rho_l - b_l \|^2$

where $\delta T_{\mathcal{X}(l)}$ stacks the pose perturbations $[\delta \phi_i^T, \delta p_i^T]^T \in \mathbb{R}^6$ for $i \in \mathcal{X}(l)$ , $E_l \in \mathbb{R}^{2n_l \times 3}$ is the stacked landmark Jacobian, $F_l \in \mathbb{R}^{2n_l \times 6n_l}$ is the stacked pose Jacobian, and $b_l \in \mathbb{R}^{2n_l}$ is the residual error vector normalized by the measurement noise standard deviation.

Minimizing this quadratic term with respect to the landmark position correction yields:

$\delta \rho_l^* = -(E_l^T E_l)^{-1} E_l^T (F_l \delta T_{\mathcal{X}(l)} - b_l)$

Substituting $\delta \rho_l^*$ back eliminates the landmark variable, producing the structureless vision factor:

$\| (E_l^\perp)^T (F_l \delta T_{\mathcal{X}(l)} - b_l) \|^2$

where $E_l^\perp \in \mathbb{R}^{2n_l \times (2n_l - 3)}$ is an orthonormal basis for the null space of $E_l$ computed via Singular Value Decomposition (SVD), satisfying $(E_l^\perp)^T E_l^\perp = I$ and $E_l^\perp (E_l^\perp)^T = I - E_l(E_l^T E_l)^{-1} E_l^T$ . This formulation avoids matrix inversions and reduces the optimization problem to only camera poses.
Knowl 7 — Maximum A Posteriori Visual-Inertial Odometry on Factor Graphs
model/method
The visual-inertial odometry problem is formulated as Maximum A Posteriori (MAP) estimation over the trajectory states $\mathcal{X}_k = \{x_i\}_{i \in \mathcal{K}_k}$ at keyframe times $\mathcal{K}_k$ , where each state is $x_i = [R_i, p_i, v_i, b_i]$ .

The full negative log-posterior cost function optimized by the back-end is:

$\mathcal{X}_k^* = \arg\min_{\mathcal{X}_k} \left( \| r_0 \|_{\Sigma_0}^2 + \sum_{(i,j) \in \mathcal{K}_k} \| r_{\mathcal{I}_{ij}} \|_{\Sigma_{ij}}^2 + \sum_{(i,j) \in \mathcal{K}_k} \| r_{b_{ij}} \|_{\Sigma_{b_{ij}}}^2 + \sum_{l=1}^L \| (E_l^\perp)^T (F_l \delta T_{\mathcal{X}(l)} - b_l) \|^2 \right)$

where:
1. $\| r_0 \|_{\Sigma_0}^2$ is the prior factor on the initial state $x_0$ .
2. $\| r_{\mathcal{I}_{ij}} \|_{\Sigma_{ij}}^2$ is the preintegrated IMU factor between keyframes $i$ and $j$ weighted by covariance $\Sigma_{ij}$ .
3. $\| r_{b_{ij}} \|_{\Sigma_{b_{ij}}}^2 = \| b_j^g - b_i^g \|_{\Sigma_{bgd}}^2 + \| b_j^a - b_i^a \|_{\Sigma_{bad}}^2$ represents the random-walk IMU bias drift factors between keyframes, with covariances $\Sigma_{bgd} = \Delta t_{ij} \text{Cov}(\eta^{bg})$ and $\Sigma_{bad} = \Delta t_{ij} \text{Cov}(\eta^{ba})$ derived from continuous white noise densities $\eta^{bg}, \eta^{ba}$ .
4. The final term represents structureless vision factors constraining keyframes observing landmark $l$ .
Inference is executed incrementally using the iSAM2 algorithm over the Bayes tree, which updates only the subset of variables affected by new measurements.
Knowl 8 — SO(3) Manifold Preintegration Invariance and Singularity Avoidance over Euler Angles
theoretical result
Preintegration directly on the $SO(3)$ manifold overcomes three structural flaws inherent in Euler-angle-based IMU preintegration:
1. Integration Exactness: Integrating angular rates using the matrix exponential map on $SO(3)$ is exact for constant angular velocity across the interval $\Delta t$ . In contrast, Euler angle integration $\theta_{k+1} = \theta_k + [E'(\theta_k)]^{-1}(\tilde{\omega}_k - \eta_k^g)\Delta t$ (where $E'(\theta_k)$ is the conjugate Euler angle rate matrix) is only a first-order approximation whose error accumulates rapidly with large sampling periods $\Delta t$ or high angular rates $\tilde{\omega}$ .
2. Statistical Fairness and Coordinate Invariance: The negative log-likelihood on $SO(3)$ , $\mathcal{L}(R) = \frac{1}{2}\| \text{Log}(\tilde{R}^{-1} R) \|_\Sigma^2$ , is left-invariant under rigid body coordinate transformations (fair metric). The Euler angle likelihood $\mathcal{L}(\theta) = \frac{1}{2}\| \tilde{\theta} - \theta \|_\Sigma^2$ is not invariant, causing state estimation results to depend on the choice of the reference world frame.
3. Singularity Avoidance: Euler angle parametrizations (such as $ZYX$ ) suffer from gimbal lock singularities at pitch angles $\theta = \pm \pi/2 + n\pi$ ( $n \in \mathbb{Z}$ ). In the vicinity of these points, Euler angle noise propagation severely degrades, producing diverging Kullback-Leibler (KL) divergence against the ground-truth covariance. $SO(3)$ preintegration avoids representation singularities entirely.
Knowl 9 — Real-Time Benchmark Performance and Odometry Accuracy on Indoor Benchmark
empirical result
The proposed monocular VIO pipeline (combining the SVO front-end, on-manifold preintegrated IMU factors, structureless vision factors, and iSAM2 back-end) was evaluated on a 430 m indoor trajectory recorded with a forward-looking VI-Sensor (20 Hz camera, 800 Hz ADIS16448 IMU) and ground truth from a Vicon motion-capture system.
1. Accuracy vs. State-of-the-Art: Over trajectory segments of lengths $\{10, 40, 90, 160, 250, 360\}$ m, the proposed system achieved an average translation drift of 0.3 m over 360 m traveled distance, compared to 0.7 m average drift for both OKVIS (keyframe-based sliding-window optimization) and MSCKF (filtering). The approach exhibited significantly lower yaw drift while pitch and roll errors remained low and constant across all methods due to gravity observability.
2. Outdoor Trajectory Accuracy: On a 300 m outdoor loop around a building, the proposed system accumulated 1.0 m end-to-end drift, compared to 2.2 m drift for Google Tango Peanut (mapper v3.15). On a 160 m multi-floor trajectory across three building floors, the proposed system accumulated 0.5 m drift compared to 1.4 m for Google Tango.
3. Execution Runtime: The SVO front-end required ~3 ms per frame for feature tracking on an Intel i7 2.4 GHz laptop. The back-end iSAM2 incremental update optimized keyframes with 10 iterations in ~10 ms per keyframe on average.
Knowl 10 — Estimation Consistency and Observability-Preserving Covariance Bounds
empirical result
The statistical consistency of the on-manifold VIO estimator was verified using 50 Monte Carlo simulation runs over a 120 m 3D sinusoidal circular trajectory (corrupted with continuous gyro noise $\sigma^g = 0.0007\text{ rad}/(\text{s}\sqrt{\text{Hz}})$ , accel noise $\sigma^a = 0.019\text{ m}/(\text{s}^2\sqrt{\text{Hz}})$ , gyro bias noise $\sigma^{bg} = 0.0004\text{ rad}/(\text{s}^2\sqrt{\text{Hz}})$ , accel bias noise $\sigma^{ba} = 0.012\text{ m}/(\text{s}^3\sqrt{\text{Hz}})$ , and $\sigma_{px} = 1\text{ pixel}$ image noise).
1. NEES Hypothesis Test: The average Normalized Estimation Error Squared (NEES) across $N = 50$ runs is $\bar{\eta}_k = \frac{1}{N} \sum_{i=1}^N \epsilon_k^{(i)T} \hat{\Sigma}_k^{-1} \epsilon_k^{(i)}$ , where $\epsilon_k = [\text{Log}(\hat{R}_k^T R_k^{gt})^T, (\hat{R}_k^T(\hat{p}_k - p_k^{gt}))^T]^T \in \mathbb{R}^6$ . For significance level $\alpha = 2.5\%$ and $n = 6 \times 50 = 300$ degrees of freedom, the two-sided $\chi_{300}^2$ acceptance region is $\bar{\eta}_k \in [5.0, 7.0]$ . The proposed estimator's average pose NEES remained strictly within this interval and below 7.0 throughout the entire trajectory, demonstrating that the estimator does not become overconfident.
2. Observability Properties: The estimation uncertainty and errors for roll, pitch, and velocity remained strictly bounded due to gravity observability, whereas yaw orientation and 3D position errors grew slowly over time, matching the theoretical 4 unobservable degrees of freedom in monocular visual-inertial odometry without spurious information gain.

Coverage note — None was omitted; all primary theoretical foundations (on-manifold preintegration, iterative noise propagation, a-posteriori bias corrections, residual Jacobians, structureless factors), graphical model formulations, and empirical evaluations are represented.

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Citation

MLA

Forster, C., et al. “On-Manifold Preintegration for Real-Time Visual--Inertial Odometry”. IEEE Transactions on Robotics, vol. 33, no. 1, 2017, pp. 1–1, https://doi.org/10.1109/TRO.2016.2597321.

APA

Forster, C., Carlone, L., Dellaert, F., & Scaramuzza, D. (2017). On-Manifold Preintegration for Real-Time Visual--Inertial Odometry. IEEE Transactions on Robotics, 33(1), 1–21. https://doi.org/10.1109/TRO.2016.2597321

Chicago

Forster, C., L. Carlone, F. Dellaert, and D. Scaramuzza. 2017. “On-Manifold Preintegration for Real-Time Visual--Inertial Odometry”. IEEE Transactions on Robotics 33 (1): 1–21. https://doi.org/10.1109/TRO.2016.2597321.

Harvard

Forster, C. et al. (2017) “On-Manifold Preintegration for Real-Time Visual--Inertial Odometry”, IEEE Transactions on Robotics, 33(1), pp. 1–21. Available at: https://doi.org/10.1109/TRO.2016.2597321.

Vancouver

1. Forster C, Carlone L, Dellaert F, Scaramuzza D (2017) On-Manifold Preintegration for Real-Time Visual--Inertial Odometry. IEEE Transactions on Robotics 33:1–21

BibTeX

@article{Forster_2017, title={On-Manifold Preintegration for Real-Time Visual--Inertial Odometry}, volume={33}, ISSN={1941-0468}, url={http://dx.doi.org/10.1109/TRO.2016.2597321}, DOI={10.1109/tro.2016.2597321}, number={1}, journal={IEEE Transactions on Robotics}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Forster, Christian and Carlone, Luca and Dellaert, Frank and Scaramuzza, Davide}, year={2017}, month=Feb, pages={1–21} }

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

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Knowls

Knowl 1 — On-Manifold IMU Preintegration Model

Knowl 2 — Preintegrated IMU Measurement Model and Residual Errors

Knowl 3 — Iterative Covariance Propagation for Preintegrated IMU Measurements

Knowl 4 — First-Order A-Posteriori Bias Correction for Preintegrated Measurements

Knowl 5 — Analytic Jacobians of Preintegrated IMU Residual Errors

Knowl 6 — Structureless Vision Factor via Null Space Projection

Knowl 7 — Maximum A Posteriori Visual-Inertial Odometry on Factor Graphs

Knowl 8 — SO(3) Manifold Preintegration Invariance and Singularity Avoidance over Euler Angles

Knowl 9 — Real-Time Benchmark Performance and Odometry Accuracy on Indoor Benchmark

Knowl 10 — Estimation Consistency and Observability-Preserving Covariance Bounds

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