1 Split-KalmanNet A Robust Model-Based Deep Learning Approach for SLAM

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Split-KalmanNet: A Robust Model-Based Deep

Learning Approach for SLAM

Geon Choi, Graduate Student Member, IEEE, Jeonghun Park, Member, IEEE, Nir Shlezinger, Member, IEEE,

Yonina C. Eldar Fellow, IEEE and Namyoon Lee, Senior Member, IEEE

Abstract—Simultaneous localization and mapping (SLAM) is

a method that constructs a map of an unknown environment and

localizes the position of a moving agent on the map simultane-

ously. Extended Kalman ﬁlter (EKF) has been widely adopted

as a low complexity solution for online SLAM, which relies on a

motion and measurement model of the moving agent. In practice,

however, acquiring precise information about these models is

very challenging, and the model mismatch effect causes severe

performance loss in SLAM. In this paper, inspired by the recently

proposed KalmanNet, we present a robust EKF algorithm using

the power of deep learning for online SLAM, referred to as

Split-KalmanNet. The key idea of Split-KalmanNet is to compute

the Kalman gain using the Jacobian matrix of a measurement

function and two recurrent neural networks (RNNs). The two

RNNs independently learn the covariance matrices for a prior

state estimate and the innovation from data. The proposed split

structure in the computation of the Kalman gain allows to

compensate for state and measurement model mismatch effects

independently. Numerical simulation results verify that Split-

KalmanNet outperforms the traditional EKF and the state-of-the-

art KalmanNet algorithm in various model mismatch scenarios.

Index Terms—Kalman ﬁlter, model-based deep learning, si-

multaneous localization and mapping (SLAM).

I. INTRODUCTION

Simultaneous localization and mapping (SLAM) is a

method that jointly estimates the location of a moving agent

and the map of an unknown environment [1]. This method

was initially studied in the context of autonomous control of

robots, and has been extended to various applications, includ-

ing unmanned aerial vehicles (UAV), self-driving cars, and

augmented reality (AR) [2]–[4]. However, performing precise

SLAM is very challenging in practice. The main difﬁculty

arises from the fact that building a map of the unknown

environment requires perfect knowledge of the position infor-

mation of a moving agent. Furthermore, accurate localization

also requires exact map information. To apply SLAM, three

models are needed: i) a motion model of a moving agent, ii)

an inverse measurement model that determines the locations

G. Choi is with the Department of Electrical Engineering, POSTECH,

Pohang, Gyeongbuk, 37673 Korea (e-mail: simon03062@postech.ac.kr).

J. Park is with the School of Electronics Engineering, College of IT

Engineering, Kyungpook National University, Daegu 41566, South Korea (e-

mail: jeonghun.park@knu.ac.kr).

N. Shlezinger is with the School of Electrical and Computer Engineer-

ing, Ben-Gurion University of the Negev, Beer-Sheva, Israel (e-mail: nir-

shl@bgu.ac.il).

Y. C. Eldar is with the Math and CS Faculty, Weizmann Institute of Science,

Rehovot, Israel (e-mail: yonina.eldar@weizmann.ac.il).

N. Lee is with the School of Electrical Engineering, Korea University, Seoul

02841, South Korea (e-mail: namyoon@korea.ac.kr).

of landmarks (or features) from measurement data, and iii)

a measurement model that predicts the agent’s location from

estimated landmark positions.

Standard approaches for SLAM are based on the Kalman

ﬁlter (KF), particle ﬁlter, and graph-based methods [5]–[7].

In particular, the extended Kalman ﬁlter (EKF) is the most

popular model-based algorithm for online SLAM [5]. The

EKF method exploits various model knowledge obtained from

the underlying physical dynamics in SLAM. Speciﬁcally, it

alternately performs i) state update according to a predeﬁned

motion model of the agent and ii) measurement update ac-

cording to a measurement model that acquires information for

the relative ranges and angles from the agent to the landmarks

in each time slot. The EKF is mathematically tractable and

can achieve optimal estimation under some mild conditions

(e.g., linear and Gaussian process). This model-based online

optimization method, however, is vulnerable to the physical

and statistical model mismatch effects, and it may diverge

when these effects are pronounced [8].

Recently, a model-based DNN-aided Kalman ﬁlter algo-

rithm called KalmanNet was proposed in [9]. The idea of

KalmanNet is to learn the Kalman gain matrix using model-

less deep neural networks (DNNs) [10]. The learned Kalman

gain matrix is then plugged into the model-based state and

measurement updates of the KF algorithm. Thanks to the

power of deep learning, the trained Kalman gain matrix is

relatively robust to model mismatch effects compared to the

traditional KF. When optimizing the Kalman gain with the

DNN, however, KalmanNet does not completely exploit the

model-based structure in computing the Kalman gain matrix.

In particular, when state and measurement model mismatch

effects coexist, the model mismatch effects can be coupled,

leading to performance loss in the state estimation.

In this paper, we present a new model-based deep learning

algorithm for SLAM referred to as Split-KalmanNet. The main

idea of Split-KalmanNet is to independently train the covari-

ance matrices for a prior state estimate and for the innovation

using two DNNs in parallel. Then, the Kalman gain matrix is

obtained by combining the learned covariance matrices with

the knowledge of the Jacobian matrix of the measurement

function according to the model-based Kalman gain update

rule. Our split learning structure makes the algorithm robust

to both the state and measurement model mismatch effects

because it mutually decouples these effects by independently

optimizing the two DNNs. From simulations, we show that

the proposed Split-KalmanNet provides a considerable gain

in terms of mean-squared error (MSE) and standard deviation

arXiv:2210.09636v1 [eess.SP] 18 Oct 2022

compared to existing model-based EKF and KalmanNet under

both the state and measurement model mismatch effects.

The paper is organized as follows. In Section II, we present

a system model and problem formulation for SLAM. Then, we

brieﬂy review conventional model-based EKF and KalmanNet

algorithms for SLAM in Section III. Section IV introduces

the proposed Split-KalmanNet algorithm. Section V shows

the effectiveness of the proposed method for SLAM via

simulations.

II. SYSTEM MODEL AND PROBLEM FORMULATION

In this section, we explain a SLAM problem. The goal of

SLAM is to jointly estimate the position of a moving agent and

landmarks using relative range and angle information between

the agent and landmarks. This SLAM system can be modeled

as a discrete-time nonlinear dynamical system comprised of

two models: i) a state evolution model and ii) a measurement

model. We shall brieﬂy explain these models according to [11,

Chapter 10].

State evolution model: We denote the position of a moving

agent and the moving-direction angle at time tby xR

t, yR

t

and θR

t. Further, we deﬁne the position of the mth landmark

as xL

m, yL

m. By concatenating all these variables, the state

vector at time tis deﬁned as

xt=xR

t, yR

t, θR

t, xL

1, yL

1, . . . , xL

M, yL

M>∈R3+2M.(1)

The agent is assumed to move in an unknown environment

according to a motion model with a constant velocity vR

tand

rotation angle dθR

t, namely,

t+1 =xR

t+vR

tcos θR

t+wx

t+1 =yR

t+vR

tsin θR

t+wy

θR

t+1 =θR

t+dθR

t+wθ

t,(2)

where wx

t,wy

t, and wθ

tdenote process noise in x-axis, y-axis,

and the rotation angle. The distributions of wx

t,wy

t, and wθ

are assumed to be independent zero-mean Gaussian. Using

(2), the state vector xtevolves to xt+1 at time t+ 1 with a

nonlinear mapping function f:R3+2M×R2→R3+2Mas

xt+1 =fxt, vR

t, dθR

t+wt,(3)

with process noise vector wt=wx

twy

twθ

t02M>.

The time-invariant covariance matrix of wtis deﬁned as

Q=Ewtw>

t.

Measurement model: For each time t, the moving agent is

assumed to acquire the range and angle measurements from

Mlandmarks. Let rm,t and φm,t be the range and angle

measurements from the mth landmark at time t. Then, these

measurements are given by

rm,t =qxL

m−xR

t2+yL

m−yR

t2,

φm,t = atan2 yL

m−yR

t, xL

m−xR

t−θR

t,(4)

for m∈[M]. Here, atan2 (y, x)denotes the angle between

the x-axis and the point (x, y)in radians in the Euclidean

plane. By concatenating all these measurements, we deﬁne a

measurement vector at time tas

h(xt)=[r1,t, φ1,t, . . . , rM,t, φM,t]>∈R2M,(5)

where h(·) : R3+2M→R2Mdenotes a nonlinear measure-

ment function from state vector xtto the 2Mmeasurements

in (4). As a result, the noisy measurement vector at time tis

given by

yt=h(xt) + vt,(6)

where vtdenotes the measurement noise vector at time t

whose distribution follows a zero-mean Gaussian. Under the

premise that the measurement noise is independent and identi-

cally distributed over time index and landmarks, the covariance

matrix is deﬁned as R=Evtv>

t=IM×M⊗DR

2×2. Here,

2×2is the noise covariance matrix of the range and angle

measurements.

Online SLAM optimization: The online SLAM optimiza-

tion problem is equivalent to the sequential minimum mean

squared error (MMSE) estimation problem for each time

t∈[T]with known the state transition function f(·)and

measurement function h(·):

arg min

Eh(xt−ˆ

xt)2|y1,...,yti.(7)

The EKF algorithm provides a recursive solution for this

problem. When implementing the EKF algorithm, it is required

to know both process and measurement noise distributions

accurately. For instance, under the zero-mean Gaussian noise

assumption, the precise knowledge of the covariance matrices

Qand Ris needed. In practice, however, acquiring perfect

knowledge of this statistical information is difﬁcult. This

model mismatch effect causes severe performance loss in

the EKF algorithm. Consequently, a robust sequential state

estimation algorithm is required for the model mismatch

problem.

III. CONVENTIONAL APPROACHES

In this section, we brieﬂy review the EKF and KalmanNet

algorithm in [9] to solve the online SLAM problem in (7).

A. Model-Based Extended Kalman Filter (MB-EKF)

The EKF algorithm performs i) state-update and ii) mea-

surement update recursively. To present the algorithm con-

cisely, we ﬁrst deﬁne some notations. Let a priori mean

and covariance of the state vector at time tbe ˆxt|t−1and

Σt|t−1=E(xt−ˆxt|t−1)(xt−ˆxt|t−1)>. We also denote

a posteriori mean and its covariance at time tby ˆxt|tand

Σt|t=E(xt−ˆxt|t)(xt−ˆxt|t)>.

State-update: The state-update step predicts a priori mean

and covariance of the state vector using a posteriori mean

ˆxt−1|t−1and covariance matrix Σt−1|t−1, which are obtained

in the previous measurement-update step:

ˆxt|t−1=fˆxt−1|t−1, vR

t, dθR

t,

Σt|t−1=Ft−1Σt−1|t−1F>

t−1+Q,(8)

where Ft=∂f

∂x|x=ˆxt|tis the Jacobian of f(·)evaluated at ˆxt|t.

Measurement-update: In the measurement-update step, a

posteriori mean ˆxt|tand covariance matrix Σt|tare computed

using a priori mean ˆxt|t−1and covariance matrix Σt|t−1

attained in the previous state-update step. In this ﬁltering step,

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1Split-KalmanNet:ARobustModel-BasedDeepLearningApproachforSLAMGeonChoi,GraduateStudentMember,IEEE,JeonghunPark,Member,IEEE,NirShlezinger,Member,IEEE,YoninaC.EldarFellow,IEEEandNamyoonLee,SeniorMember,IEEEAbstractSimultaneouslocalizationandmapping(SLAM)isamethodthatconstructsamapofanunknownenvironme...

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1 Split-KalmanNet A Robust Model-Based Deep Learning Approach for SLAM

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