---
title: Error-State Iterated Kalman Filter (ESIKF)
url: https://www.emergentmind.com/topics/error-state-iterated-kalman-filter-esikf
type: topic
---

# Error-State Iterated Kalman Filter (ESIKF)

The Error-State Iterated Kalman Filter (ESIKF) is a state estimation methodology that integrates the error-state representation of nonlinear systems with iterated measurement updates, producing a robust estimator particularly well-suited for high-dimensional, strongly nonlinear, and manifold-valued state spaces such as those encountered in inertial navigation, SLAM, and tightly coupled sensor fusion. ESIKF combines the strengths of the Error-State Kalman Filter (ESKF) and the Iterated Extended Kalman Filter (IEKF), leveraging the local linearity of error states and iterative re-linearization to deliver high accuracy even in the presence of highly nonlinear observations or rapid system dynamics [2406.06427][2509.15673][2307.09237][1711.02508].

## 1. Formalism and State Structure

The ESIKF represents the system state as a nominal reference $\hat{x}$ on a (possibly manifold) state space $M$, with local deviations encoded by a small error-state vector $\delta x$ in the tangent space at $\hat{x}$. The true state $x_{true}$ is thus parameterized as
$$
x_{true} = \hat{x} \boxplus \delta x, \quad \|\delta x\| \ll 1
$$
where $\boxplus$ denotes the manifold-lift operation. For systems on Lie groups, such as $SE(3)$ or $SO(3)$ (e.g., for IMU-driven navigation), orientation components are updated via group operations (typically, quaternion multiplication with small perturbations mapped through the exponential map).

In typical visual-inertial-odometry systems, as in Omni-LIVO, the state is formulated as
$$
x_k = [q_{WB},\, v_{WB},\, p_{WB},\, b_g,\, b_a]
$$
with $q_{WB} \in SO(3)$ a body-to-world quaternion, $v_{WB}, p_{WB} \in \mathbb{R}^3$ velocity and position, and $b_g, b_a$ gyroscope and accelerometer biases [2509.15673][1711.02508].

## 2. Error-State Dynamics and Propagation

The ESIKF adopts the error-state approach for filtering, modeling process (prediction) dynamics in the tangent space about the nominal trajectory. The propagation step consists of:

- **Nominal-state propagation**: The predicted nominal state is advanced according to the noise-free process model:
    $$
    \hat{x}_k^- = f(\hat{x}_{k-1}^+, u_{k-1}, 0)
    $$
- **Error-state covariance propagation**: Linearize the process model about the current nominal state,
    $$
    F_{k-1} = \left. \frac{\partial (f(\hat{x}_{k-1}^+\boxplus \epsilon, u_{k-1}, 0)\boxminus f(\hat{x}_{k-1}^+, u_{k-1}, 0))}{\partial \epsilon} \right|_{\epsilon=0}
    $$
    $$
    P_k^- = F_{k-1} P_{k-1}^+ F_{k-1}^\top + G_{k-1} Q_{k-1} G_{k-1}^\top
    $$
For IMU-driven systems, the continuous-time propagation employs quaternion differential equations and bias random-walk models, with discretization for real-time implementation [2509.15673][1711.02508].

## 3. Iterated Measurement Update

Unlike the standard ESKF, which linearizes the measurement model only once about the prior, the ESIKF applies a Gauss–Newton-style iterative refinement within each measurement update, reducing linearization error.

- **Residual and Jacobian computation**: At each iteration $\ell$, compute the measurement model residual and its Jacobian with respect to the error state at the current estimate $x_{k|k-1} \boxplus \delta x^{(\ell-1)}$.
- **Normal equations**: Solve for the error-state update:
    $$
    \delta x^{(\ell)} = \delta x^{(\ell-1)} + K^{(\ell)} \big[r^{(\ell)} - H^{(\ell)} \delta x^{(\ell-1)}\big]
    $$
    where $K^{(\ell)}$ is the Kalman gain computed with the linearized Jacobian and measurement covariance.
- **State correction**: Upon convergence, inject the total increment into the nominal state:
    $$
    x_{k|k} = x_{k|k-1} \boxplus \delta x^{(L)}
    $$
    Covariance is updated via the first-order or Joseph form, ensuring positive-definiteness and correct uncertainty propagation [2406.06427][2509.15673][2307.09237][1711.02508].

This iterative approach offers increased accuracy in regimes with high nonlinearity, such as vision-based measurements or aggressive vehicle maneuvers.

## 4. Adaptive Multi-Modal Fusion and Covariance Weighting

In systems with heterogeneous measurements (e.g., multi-camera, LiDAR), ESIKF supports the joint fusion of diverse sensor modalities, balancing their contributions by adaptive covariance weighting. In Omni-LIVO:

- The measurement covariance $R_{\text{multi}}$ is constructed as a block-diagonal matrix with scaling factors per modality or per camera:
    $$
    R_{\text{multi}} = \mathrm{diag}(\alpha_1 R_{\text{base}}^{(1)}, \ldots, \alpha_C R_{\text{base}}^{(C)}, \alpha_{\text{cross}} R_{\text{cross}})
    $$
- Scaling factors $\alpha_i$ are computed online as a function of per-view tracking statistics, such as patch coverage and average residuals, down-weighting unreliable measurements and dynamically prioritizing robust sensor streams [2509.15673].

This mechanism enables ESIKF to automatically favor the most informative and consistent measurements during fusion, increasing estimator robustness across challenging multimodal scenarios.

## 5. Algorithmic Structure and Implementation Considerations

A canonical ESIKF iteration encompasses the following sequence:
1. Synchronize sensor input frames (IMU, camera(s), LiDAR).
2. Propagate nominal state and covariance.
3. Process sensor residuals, compute analytic Jacobians.
4. Initialize iterative update (with $\delta x^{(0)} = 0$).
5. For each iteration:
    - Re-linearize at the current state hypothesis.
    - Stack all residuals; update measurement covariance adaptively.
    - Compute Kalman gain; update the error-state increment.
    - Validate convergence (e.g., norm of increment below $\epsilon$, or iteration cap $\ell_{\max}$ reached).
6. Inject the converged error-state into the nominal state.
7. Normalize quaternions after updates, if the state includes orientation.
8. Reset and update covariance using a numerically robust scheme [2509.15673][2406.06427][2307.09237].

Tukey weighting or robust loss functions may be employed on visual residuals to mitigate the influence of outliers, as in Omni-LIVO. Analytical Jacobians are recommended for performance and stability. SLAM backends benefit from block-sparsity exploitation in covariance updates.

## 6. Comparative Properties and Performance

The ESIKF inherits and extends the advantages of both ESKF and IEKF while mitigating their respective weaknesses:

| Filter       | Update Domain   | Linearization Method        | Iteration | Key Attributes                      |
|--------------|----------------|----------------------------|-----------|-------------------------------------|
| ESKF         | Error-state    | Single (tangent space)     | No        | Minimal parameterization, efficient |
| IEKF         | Full state     | Iterated (full state)      | Yes       | Robust to nonlinearity, but higher computational cost, can suffer from parameterization singularities |
| ESIKF        | Error-state    | Iterated (tangent space)   | Yes       | Reduces linearization error, robust for strongly nonlinear measurements, preserves minimal parameterization, real-time feasible (2–8 iterations) |

Empirically, ESIKF demonstrates superior convergence and numerical stability under high measurement nonlinearity and longer update intervals. In Omni-LIVO, ESIKF enables the system to outperform previous tightly coupled odometry frameworks:
- Successfully processes all 14 sequences in challenging SLAM benchmarks where single-camera methods fail on up to half the cases.
- Delivers approximately 34% lower average RMSE versus FAST-LIVO2, maintaining sub-0.1m errors in degraded visual and dynamic environments.
- Achieves 35 ms processing per packet (four cameras), incurring only ~1.2x overhead relative to single-camera ESKF baselines, and generates 2–4x denser colored point clouds [2509.15673].

## 7. Numerical and Practical Issues

Several technical subtleties are crucial for reliable ESIKF deployment:

- **Convergence and complexity**: Each iteration is $\mathcal{O}(n^3)$, with $n$ the error-state dimension; total iterations are typically capped ($\ell_{\max}$ between 5–8) with a tight convergence threshold ($\epsilon \sim 10^{-5}$).
- **Manifold operations**: Consistent choice and implementation of $\boxplus$, $\boxminus$ is required for correct Jacobians, especially for rotation groups [1711.02508][2307.09237].
- **Covariance management**: Ensure positive-definite, symmetric $P$ at each update; Joseph form is preferred for numerical stability.
- **Quaternion normalization**: After updates in the orientation error-state, the nominal quaternion must be renormalized to avoid drift.
- **Block-sparsity**: Exploit block-diagonal structure in covariance and measurement models to accelerate high-dimensional updates, especially in multi-view or multi-sensor settings [2307.09237][2406.06427].
- **Robustification**: Incorporating robust error norms (e.g., Tukey loss) enhances resilience to outliers in sensor data [2509.15673].

These considerations are fundamental for real-time operation and reliable performance, particularly when fusing high-rate heterogeneous sensory streams in SLAM and odometry systems.

---

**References:**
- "Omni-LIVO: Robust RGB-Colored Multi-Camera Visual-Inertial-LiDAR Odometry via Photometric Migration and ESIKF Fusion" [2509.15673]
- "Notes on Kalman Filter (KF, EKF, ESKF, IEKF, IESKF)" [2406.06427]
- "A Quick Guide for the Iterated Extended Kalman Filter on Manifolds" [2307.09237]
- "Quaternion kinematics for the error-state Kalman filter" [1711.02508]

Source: https://www.emergentmind.com/topics/error-state-iterated-kalman-filter-esikf