LMKF SLAM: Linear Reformulation of SLAM

• The paper introduces LMKF SLAM, which transforms the nonlinear SLAM state-space into an exactly linear model by leveraging direct heading measurements and Cartesian reformation for improved filter optimality.
• The methodology replaces the EKF’s iterative linearization with a measured input approach, minimizing linearization errors and enhancing convergence properties under standard observability assumptions.
• Empirical results show that LMKF SLAM achieves significantly lower RMSE, faster execution times, and greater robustness to sensor uncertainties compared to traditional EKF-based methods.

Searching arXiv for the specified paper and closely related SLAM work. Searching arXiv for LMKF SLAM and related EKF/KF SLAM formulations. LMKF SLAM is a simultaneous localization and mapping formulation introduced by transforming a nonlinear SLAM state-space model into a linear model and then applying the original Kalman filter rather than an extended Kalman filter. The method is motivated by a central failure mode of EKF-based SLAM: divergence induced by the nonlinearity of the motion and observation models and by linearization error. In the reported formulation, a simple compass provides a direct noisy measurement of the heading, which allows the heading to be treated as a measured input rather than a state variable; combined with an effective transformation of range-bearing observations into Cartesian relative coordinates, this yields a linear-Gaussian model for planar SLAM with static point landmarks (Bahreinian et al., 26 Jun 2026).

1. Problem setting and motivation

The work is situated in the standard mobile-robot SLAM setting in which the robot must estimate its own pose while simultaneously estimating landmark positions. The paper identifies EKF-based SLAM as a major and common class of algorithms for this task, and it states that one of their main problems is divergence when the robot’s motion and/or observation models are strongly nonlinear (Bahreinian et al., 26 Jun 2026).

The mechanism of divergence is specified in terms of repeated local linearization. Every EKF update requires linearizing the true nonlinear models around the current estimate. According to the reported analysis, linearization errors accumulate, inflate the Kalman gain, and lead to over-confidence, described as covariance collapse, and eventually to wrong state updates. The paper further emphasizes that bearing-only measurements and heading estimation errors introduce extra degrees of unobservability, making the filter inconsistent unless very small angular uncertainties are assumed.

Against this background, LMKF SLAM is framed as a linear-model alternative. A fully linear state-space model avoids any Taylor-series approximations in the filter equations. The claimed consequences are exact optimality of the Kalman filter with no linearization bias, provable convergence of landmark and robot estimates under standard detectability and observability assumptions, and reduced computational cost because no Jacobian of the state equations is required at each time step. This suggests that the method is not merely a numerical modification of EKF-SLAM, but a reformulation designed to remove the specific source of EKF inconsistency.

2. Nonlinear SLAM model from which LMKF is derived

The starting point is a nonlinear planar vehicle model. Let the robot pose at time $k$ be

$x_k^r = [x_k,\; y_k,\; \theta_k]^T.$

The robot measures instantaneous forward speed $v_k$ and steering angle $G_k$, and it observes $I_k$ landmarks whose true positions are $(x_k^i, y_k^i)$. The standard continuous-time discretized motion model is given as

$$\begin{bmatrix} x_{k+1}\ y_{k+1}\ \theta_{k+1} \end{bmatrix} = \begin{bmatrix} x_k + v_k d_t \cos(\theta_k + G_k)\ y_k + v_k d_t \sin(\theta_k + G_k)\ \theta_k + \dfrac{v_k d_t \sin(G_k)}{D} \end{bmatrix} + w_k,$$

where $d_t$ is the sampling interval, $D$ the wheelbase, and $w_k$ zero-mean process noise (Bahreinian et al., 26 Jun 2026).

The landmark observation model is the standard nonlinear range-bearing form:

$x_k^r = [x_k,\; y_k,\; \theta_k]^T.$0

Within this formulation, the robot heading $x_k^r = [x_k,\; y_k,\; \theta_k]^T.$1 is both dynamically evolved and embedded in the bearing observation, which is precisely the coupling that makes the overall state-space model nonlinear.

This nonlinear structure is the baseline against which LMKF SLAM is defined. A plausible implication is that the method is best understood as a model transformation strategy: it preserves the original SLAM objective while altering the representation so that linear Kalman filtering becomes admissible.

3. Effective transformation and the linear state-space model

The key idea is to equip the robot with a compass giving a direct noisy measurement of the heading $x_k^r = [x_k,\; y_k,\; \theta_k]^T.$2, so that $x_k^r = [x_k,\; y_k,\; \theta_k]^T.$3 can be treated as a measured input rather than a state variable. The transformed inputs are defined as

$x_k^r = [x_k,\; y_k,\; \theta_k]^T.$4

Using these quantities, the planar motion is rewritten purely in the $x_k^r = [x_k,\; y_k,\; \theta_k]^T.$5 subspace and linearized in the sensor noises:

$x_k^r = [x_k,\; y_k,\; \theta_k]^T.$6

where

$x_k^r = [x_k,\; y_k,\; \theta_k]^T.$7

Because landmarks are static, the augmented state becomes

$x_k^r = [x_k,\; y_k,\; \theta_k]^T.$8

The observation transformation is defined by the nonlinear mapping $x_k^r = [x_k,\; y_k,\; \theta_k]^T.$9:

$v_k$0

Linearizing in the measurement noises yields

$v_k$1

where $v_k$2 are zero-mean Gaussian, $v_k$3 the Jacobians, and $v_k$4 picks out the robot and $v_k$5-th landmark coordinates (Bahreinian et al., 26 Jun 2026).

In full state form, the transformed model is

$v_k$6

with

$v_k$7

For all visible landmarks,

$v_k$8

The paper characterizes this transformed model as exactly linear-Gaussian. In context, the central significance of LMKF SLAM lies here: the state-transition and observation operators used by the filter are linear in the augmented state, so the estimator no longer relies on the EKF’s repeated local Jacobian linearizations of the original nonlinear state equations.

4. Kalman filtering procedure

Once the transformed model is obtained, LMKF SLAM applies the standard discrete-time Kalman filter. The prediction equations are

$v_k$9

with

$G_k$0

The update equations are

$G_k$1

$G_k$2

The algorithm summary is explicit. Initialization uses $G_k$3 and $G_k$4. At each time $G_k$5, the robot reads the compass $G_k$6, odometry $G_k$7, and optionally $G_k$8; computes $G_k$9, $I_k$0, and $I_k$1; performs the prediction step; and then, for each observed landmark $I_k$2, computes $I_k$3, the Jacobian $I_k$4, and the observation matrix $I_k$5. The update may be implemented sequentially:

$I_k$6

followed by

$I_k$7

The output is $I_k$8 and $I_k$9 (Bahreinian et al., 26 Jun 2026).

The importance of this procedure is twofold. First, it uses the original Kalman filter rather than an extended variant. Second, the sequential update is reported to further accelerate execution, which aligns with the paper’s broader claim of lower computational complexity relative to EKF-based alternatives.

5. Reported empirical behavior

The reported comparative analysis states that LMKF SLAM is significantly superior to state-of-the-art methods, especially EKF-based SLAMs, in accuracy, convergence, and computational complexity (Bahreinian et al., 26 Jun 2026). The specific baselines named in the experiments are EKF, UnFS-SLAM, and ICKF.

The main quantitative results are summarized below.

Setting	LMKF	Comparators
Simulated 200×200 m², closed-loop path (1.35 km, 139 landmarks) and open-loop path (0.58 km, 63 landmarks)	Robot RMSE $(x_k^i, y_k^i)$0–$(x_k^i, y_k^i)$1 m	EKF $(x_k^i, y_k^i)$2 m, UnFS $(x_k^i, y_k^i)$3 m, ICKF $(x_k^i, y_k^i)$4 m
Same simulated setting	Landmark RMSE similarly halved	Relative to the compared methods
Victoria Park real dataset (2D LiDAR + GPS + odometry)	Vehicle RMSE $(x_k^i, y_k^i)$5 m, MAE $(x_k^i, y_k^i)$6 m	EKF RMSE $(x_k^i, y_k^i)$7 m, MAE $(x_k^i, y_k^i)$8 m
Victoria Park real dataset, 620 updates	Execution time $(x_k^i, y_k^i)$9 s	EKF $$\begin{bmatrix} x_{k+1}\ y_{k+1}\ \theta_{k+1} \end{bmatrix} = \begin{bmatrix} x_k + v_k d_t \cos(\theta_k + G_k)\ y_k + v_k d_t \sin(\theta_k + G_k)\ \theta_k + \dfrac{v_k d_t \sin(G_k)}{D} \end{bmatrix} + w_k,$$0 s, ICKF $$\begin{bmatrix} x_{k+1}\ y_{k+1}\ \theta_{k+1} \end{bmatrix} = \begin{bmatrix} x_k + v_k d_t \cos(\theta_k + G_k)\ y_k + v_k d_t \sin(\theta_k + G_k)\ \theta_k + \dfrac{v_k d_t \sin(G_k)}{D} \end{bmatrix} + w_k,$$1 s, UnFS $$\begin{bmatrix} x_{k+1}\ y_{k+1}\ \theta_{k+1} \end{bmatrix} = \begin{bmatrix} x_k + v_k d_t \cos(\theta_k + G_k)\ y_k + v_k d_t \sin(\theta_k + G_k)\ \theta_k + \dfrac{v_k d_t \sin(G_k)}{D} \end{bmatrix} + w_k,$$2 s

In addition to point metrics, the comparative analysis emphasizes several qualitative properties. The convergence region is described as wider because exact linearity guarantees convergence under standard Kalman-filter assumptions, whereas EKF needs very small angular noise for consistency. The computational advantage is attributed to the absence of online Jacobians of the state equations, with approximately $$\begin{bmatrix} x_{k+1}\ y_{k+1}\ \theta_{k+1} \end{bmatrix} = \begin{bmatrix} x_k + v_k d_t \cos(\theta_k + G_k)\ y_k + v_k d_t \sin(\theta_k + G_k)\ \theta_k + \dfrac{v_k d_t \sin(G_k)}{D} \end{bmatrix} + w_k,$$3–$$\begin{bmatrix} x_{k+1}\ y_{k+1}\ \theta_{k+1} \end{bmatrix} = \begin{bmatrix} x_k + v_k d_t \cos(\theta_k + G_k)\ y_k + v_k d_t \sin(\theta_k + G_k)\ \theta_k + \dfrac{v_k d_t \sin(G_k)}{D} \end{bmatrix} + w_k,$$4 CPU savings versus EKF; the sequential update is reported to further accelerate execution. Stability under sensor uncertainty is also highlighted: the RMSE of LMKF is almost invariant to increases in speed uncertainty $$\begin{bmatrix} x_{k+1}\ y_{k+1}\ \theta_{k+1} \end{bmatrix} = \begin{bmatrix} x_k + v_k d_t \cos(\theta_k + G_k)\ y_k + v_k d_t \sin(\theta_k + G_k)\ \theta_k + \dfrac{v_k d_t \sin(G_k)}{D} \end{bmatrix} + w_k,$$5 or heading noise $$\begin{bmatrix} x_{k+1}\ y_{k+1}\ \theta_{k+1} \end{bmatrix} = \begin{bmatrix} x_k + v_k d_t \cos(\theta_k + G_k)\ y_k + v_k d_t \sin(\theta_k + G_k)\ \theta_k + \dfrac{v_k d_t \sin(G_k)}{D} \end{bmatrix} + w_k,$$6, while EKF/UnFS error explodes as $$\begin{bmatrix} x_{k+1}\ y_{k+1}\ \theta_{k+1} \end{bmatrix} = \begin{bmatrix} x_k + v_k d_t \cos(\theta_k + G_k)\ y_k + v_k d_t \sin(\theta_k + G_k)\ \theta_k + \dfrac{v_k d_t \sin(G_k)}{D} \end{bmatrix} + w_k,$$7 or $$\begin{bmatrix} x_{k+1}\ y_{k+1}\ \theta_{k+1} \end{bmatrix} = \begin{bmatrix} x_k + v_k d_t \cos(\theta_k + G_k)\ y_k + v_k d_t \sin(\theta_k + G_k)\ \theta_k + \dfrac{v_k d_t \sin(G_k)}{D} \end{bmatrix} + w_k,$$8 grows.

These reported results are significant because they tie the model transformation directly to observable estimator behavior. In the paper’s presentation, the gains are not limited to nominal accuracy; they extend to convergence properties, runtime, and robustness to sensor uncertainty.

6. Scope, requirements, and implications

The formulation has explicit requirements. It requires a reliable heading sensor, stated as an electronic compass or IMU, to remove $$\begin{bmatrix} x_{k+1}\ y_{k+1}\ \theta_{k+1} \end{bmatrix} = \begin{bmatrix} x_k + v_k d_t \cos(\theta_k + G_k)\ y_k + v_k d_t \sin(\theta_k + G_k)\ \theta_k + \dfrac{v_k d_t \sin(G_k)}{D} \end{bmatrix} + w_k,$$9 from the state. It also assumes static, point-landmarks and planar motion (Bahreinian et al., 26 Jun 2026). These are not peripheral engineering details; they define the regime in which the linear reformulation is available.

A common misunderstanding would be to read LMKF SLAM as a universal elimination of SLAM nonlinearity. The reported formulation is narrower: by “measuring” the robot heading with a simple compass and transforming range/bearing into Cartesian relative coordinates, one obtains an exactly linear state-space model. The linearity therefore depends on the sensing setup and on the geometric assumptions of the map representation. This suggests that the method is best matched to 2D platforms with dependable heading sensing and landmark-based perception.

The paper also identifies several potential extensions. These include 3D SLAM via analogous roll/pitch/yaw measurements and point-cloud transforms, integration with map-parametrized landmarks such as lines and planes using the same linearization principle, and fusion with batch graph-SLAM or smoothing for global consistency checks. These directions indicate that LMKF SLAM may be interpreted not only as a specific filter instance, but also as a design principle: restructure the estimation problem so that the dominant nonlinear coupling is absorbed into measured inputs and transformed observations, leaving a linear state-space model for inference.